Optimal semiclassical spectral asymptotics for differential operators with non-smooth coefficients

Abstract

We consider differential operators defined as Friedrichs extensions of quadratic forms with non-smooth coefficients. We prove a two-term optimal asymptotic for the Riesz means of these operators and thereby also reprove an optimal Weyl law under certain regularity conditions. The methods used are then extended to consider more general admissible operators perturbed by a rough differential operator and to obtain optimal spectral asymptotics again under certain regularity conditions. For the Weyl law, we assume that the coefficients are differentiable with Hölder continuous derivatives, while for the Riesz means we assume that the coefficients are twice differentiable with Hölder continuous derivatives.

1 Introduction

In this paper, we will study (uniformly) elliptic self-adjoint differential operators $A(\hbar )$ acting in $L^2({\mathbb {R}}^d)$ defined as Friedrichs extensions of sesquilinear forms given by

$$\begin{aligned} {\mathcal {A}}_\hbar [\varphi ,\psi ] = \sum _{\left| \alpha \right| ,\left| \beta \right| \le m} \int _{{\mathbb {R}}^d} a_{\alpha \beta }(x) (\hbar D_x)^\beta \varphi (x) \overline{(\hbar D_x)^\alpha \psi (x)} \, dx, \qquad \varphi ,\psi \in {\mathcal {D}}( {\mathcal {A}}_\hbar ), \end{aligned}$$

(1.1)

where $\hbar >0$ is the semiclassical parameter, ${\mathcal {D}}( {\mathcal {A}}_\hbar )$ is the associated form domain. For $\alpha \in {\mathbb {N}}_0^d$, we have used the standard notation

$$\begin{aligned} (\hbar D_x)^\alpha = \prod _{j=1}^d (-i\hbar \partial _{x_j})^{\alpha _j}. \end{aligned}$$

Our exact assumptions on the functions $a_{\alpha \beta }$ will be stated later in Assumption 1.2. For $\gamma$ in [0, 1] we will analyse the asymptotics, as $\hbar$ tends to zero, of the traces

$$\begin{aligned} \text {Tr} [(A(\hbar ))_{-}^\gamma ], \end{aligned}$$

(1.2)

where we have used the notation $(x)_{-} = \max (0,-x)$. In the case $\gamma =0$ we use the convention that $(x)_{-}^0 = {\varvec{1}}_{(-\infty ,0]}(x)$, where ${\varvec{1}}_{(-\infty ,0]}$ is the characteristic function for the set $(-\infty ,0]$. There are no technical obstructions in considering higher values of $\gamma$, but more regularity of the coefficients will be needed to obtain sharp remainder estimates.

In the literature, these kinds of traces are called Riesz means when $\gamma >0$, and counting functions when $\gamma =0$. Both objects have been studied extensively in the literature in different variations. For background on the Weyl law (asymptotic expansions of the counting function) and the connection to physics we refer the reader to the surveys [1,2,3].

In the case of smooth coefficients and with some regularity conditions it was first proven in [4] by Helffer and Robert that an estimate of the following type

$$\begin{aligned} \big | \text {Tr} [{\varvec{1}}_{(-\infty ,\lambda _0]}(A(\hbar ))] - \frac{1}{(2\pi \hbar )^d} \int _{{\mathbb {R}}^{2d}} {\varvec{1}}_{(-\infty ,\lambda _0]}(a_0(x,p)) \, dxdp \big | \le C\hbar ^{1-d}, \end{aligned}$$

(1.3)

holds, where

$$\begin{aligned} a_0(x,p) = \sum _{\left| \alpha \right| ,\left| \beta \right| \le m} a_{\alpha \beta }(x)p^{\alpha +\beta }, \end{aligned}$$

(1.4)

and $\lambda _0$ is a number such that there exists a number $\lambda >\lambda _0$ for which $a_0^{-1}((-\infty ,\lambda ])$ is compact. Moreover, $\lambda _0$ is assumed to be non-critical for $a_0(x,p)$. A number $\lambda _0$ is a non-critical value when

$$\begin{aligned} |\nabla a_0(x,p)| \ge c>0 \quad \text {for all } (x,p) \in a_0^{-1}(\{\lambda _0\}). \end{aligned}$$

From here on, we will always use the notation that $a_0(x,p)$ is given by (1.4), when we are working with a sesquilinear form ${\mathcal {A}}_\hbar$ given by (1.1). In fact, Helffer and Robert proved (1.3) for a larger class of pseudo-differential operators in [4]. The high energy analog was proven by Hörmander in [5], where the operator was defined to act in $L^2(M)$, and M is a smooth compact manifold without boundary. For the Riesz means an asymptotic expansion was obtained by Helffer and Robert in [6] under the same conditions as above and again, for this larger class of pseudo-differential operators. For the high energy case, Riesz means were studied by Hörmander in [7].

When considering the assumptions above one question immediately arises:

• 1. labelMainspsquestionsps1 What happens if the coefficients are not smooth? Can such results still be proven with optimal errors?

A considerable body of literature have been devoted to these questions. Most prominent are the works of Ivrii [2, 8,9,10,11,12,13,14] and Zielinski [15,16,17,18,19,20,21,22,23]. A more detailed review of the literature will be given in section 1.3 after presenting the result obtained in this work.

1.1 Assumptions and main results I

Before we state our assumptions and results on the Riesz means we need the following definition to clarify terminology.

Definition 1.1

For k in ${\mathbb {N}}$ and $\mu$ in (0, 1] we denote by $C^{k,\mu }({\mathbb {R}}^d)$ the subspace of $C^{k}({\mathbb {R}}^d)$ defined by

$$\begin{aligned} \begin{aligned} C^{k,\mu }({\mathbb {R}}^d) = \big \{ f\in C^{k}({\mathbb {R}}^d) \, \big | \, \exists C>0:&|\partial _x^{\alpha } f(x) - \partial _x^{\alpha } f(y) | \le C |x-y|^{\mu } \\&\forall \alpha \in {\mathbb {N}}^d \text { with } \left| \alpha \right| =k \text { and } \forall x,y\in {\mathbb {R}}^d \big \}. \end{aligned} \end{aligned}$$

In the case $\mu =0$ we use the convention

$$\begin{aligned} C^{k,0}({\mathbb {R}}^d) = C^{k}({\mathbb {R}}^d). \end{aligned}$$

We will make the following assumption on the sesquilinear form, for both the Weyl law and the Riesz means.

Assumption 1.2

(Tempered variation model) Let $(k,\mu )$ be numbers in ${\mathbb {N}}\times [0,1]$ and ${\mathcal {A}}_\hbar$ be a sesquilinear form given by

$$\begin{aligned} {\mathcal {A}}_\hbar [\varphi ,\psi ] = \sum _{\left| \alpha \right| ,\left| \beta \right| \le m} \int _{{\mathbb {R}}^d} a_{\alpha \beta }(x) (\hbar D_x)^\beta \varphi (x) \overline{(\hbar D_x)^\alpha \psi (x)} \, dx, \qquad \varphi ,\psi \in {\mathcal {D}}( {\mathcal {A}}_\hbar ). \end{aligned}$$

Assume the coefficients $a_{\alpha \beta }(x)$ are in $C^{k,\mu }({\mathbb {R}}^d)$ and satisfy that $a_{\alpha \beta }(x) = \overline{a_{\beta \alpha }(x)}$ for all multi-indices $\alpha$ and $\beta$. Moreover, for all multi-indices $\alpha$ and $\beta$ assume that

• 1. labelass.symbolsps1 There is a $\zeta _0>0$ such that $\min _{x\in {\mathbb {R}}^d}(a_{\alpha \beta }(x))> - \zeta _0$.

• 2. There is a $\zeta _1>\zeta _0$ and $C_1,M>0$ such that

  $$\begin{aligned} a_{\alpha \beta }(x) + \zeta _1 \le C_1(a_{\alpha \beta }(y)+\zeta _1)(1+|x-y|)^M, \end{aligned}$$

  for all x, y in ${\mathbb {R}}^d$.

• 3. For all $\delta$ in ${\mathbb {N}}_0^d$ with $|\delta |\le k$ there is a $c_\delta >0$ such that

  $$\begin{aligned} \left| \partial _{x}^\delta a_{\alpha \beta }(x) \right| \le c_\delta (a_{\alpha \beta }(x) + \zeta _1). \end{aligned}$$

Suppose finally that there exists a constant $C_2$ such that

$$\begin{aligned} \sum _{\left| \alpha \right| =\left| \beta \right| = m} a_{\alpha \beta }(x) p^{\alpha +\beta } \ge C_2 \left| p \right| ^{2m}, \end{aligned}$$

(1.5)

for all (x, p) in ${\mathbb {R}}^d_x\times {\mathbb {R}}_p^d$.

The assumption for the coefficients to be in $C^{k,\mu }({\mathbb {R}}^d)$ is is needed to get optimal asymptotic results. The assumption $a_{\alpha \beta }(x) = \overline{a_{\beta \alpha }(x)}$ is made to ensure the form is symmetric. The assumptions 1–3 ensure that the coefficients are of tempered variation. Assumption 2 is the condition that for all $\alpha$ and $\beta$, $a_{\alpha \beta }(x) + \zeta _1$ is a tempered weight. Note that due to the semiclassical structure of the problem these assumptions are needed to be true for all $\alpha$ and $\beta$ and not just $|\alpha |=|\beta |=m$, as it is the case in the high energy asymptotics. The “ellipticity” assumption (1.5) helps ensure the existence of eigenvalues.

We can now state our two main results. Firstly we have that

Theorem 1.3

Let $A(\hbar )$ be the Friedrichs extension of a sesquilinear form ${\mathcal {A}}_\hbar$ which satisfies Assumption 1.2 with the numbers $(1,\mu )$ where $\mu >0$. Suppose there is a $\nu >0$ such that the set $a_0^{-1}((-\infty ,\nu ])$ is compact and there is $c>0$ such that

$$\begin{aligned} |\nabla _p a_0(x,p)| \ge c \quad \text {for all } (x,p)\in a_0^{-1}(\{0\}). \end{aligned}$$

(1.6)

Then we have

$$\begin{aligned} \big |\text {Tr} [{\varvec{1}}_{(-\infty ,0]}(A(\hbar ))] - \frac{1}{(2\pi \hbar )^d} \int _{{\mathbb {R}}^{2d}} {\varvec{1}}_{(-\infty ,0]}( a_{0}(x,p)) \,dx dp \big | \le C \hbar ^{1-d}, \end{aligned}$$

for all sufficiently small $\hbar$.

Remark 1.4

The assumption that there is a $\nu >0$ such that the set $a_0^{-1}((-\infty ,\nu ])$ is compact is needed to ensure that we only have pure point spectrum in $(-\infty ,0]$. Due to the ellipticity assumption this is in fact an assumption on the coefficients $a_{\alpha \beta }(x)$.

Furthermore, for the Reisz means with $\gamma$ in (0, 1] we have

Theorem 1.5

Assume $\gamma$ is in (0, 1] and let $A(\hbar )$ be the Friedrichs extension of a sesquilinear form ${\mathcal {A}}_\hbar$ which satisfies Assumption 1.2 with the numbers $(2,\mu )$. If $\gamma =1$ we suppose $\mu >0$ and if $\gamma <1$ we suppose $\mu =0$. Lastly, suppose there is a $\nu >0$ such that the set $a_0^{-1}((-\infty ,\nu ])$ is compact and that there is $c>0$ such that

$$\begin{aligned} |\nabla _p a_0(x,p)| \ge c \quad \text {for all } (x,p)\in a_0^{-1}(\{0\}). \end{aligned}$$

(1.7)

Then we have

$$\begin{aligned} \big |\text {Tr} [(A(\hbar ))^\gamma _{-}] - \frac{1}{(2\pi \hbar )^d} \int _{{\mathbb {R}}^{2d}} ( a_{0}(x,p))^{\gamma }_{-} +\hbar \gamma a_1(x,p) ( a_{0}(x,p))^{\gamma -1}_{-} \,dx dp \big | \le C \hbar ^{1+\gamma -d}, \end{aligned}$$

for all sufficiently small $\hbar$ where $a_1(x,p)$ is defined as

$$\begin{aligned} a_{1}(x,p) = i \sum _{|\alpha |, |\beta |\le m}\sum _{j=1}^d \frac{\beta _j-\alpha _j}{2} \partial _{x_j} a_{\alpha \beta } (x) p^{\alpha +\beta -\eta _j}, \end{aligned}$$

(1.8)

and $\eta _j$ is the multi-index with all entries equal zero except the j’th which is equal one.

Remark 1.6

In the case when $\gamma \le \frac{1}{3}$ we may assume that the coefficients are in $C^{1,\mu }({\mathbb {R}}^d)$, with $\mu \ge \frac{2\gamma }{1-\gamma }$ and still obtain sharp estimates. Further details are given in Remark 6.7.

There is no technical obstruction in considering larger values of $\gamma$. For a $\gamma >1$ we will need the coefficients to be in $C^{\lceil \gamma \rceil ,0}$ if $\gamma$ is not an integer in general. If $\gamma$ is an integer we will need the coefficients to be in $C^{\gamma ,\mu }({\mathbb {R}}^d)$ for some $\mu >0$. For both cases, the expansions will also consist of $\lceil \gamma \rceil +1$ terms, where these terms can be calculated explicitly.

Example 1.7

Let A(x) be a $d\times d$–dimensional symmetric matrix and suppose that A(x) is positive definite for all x in ${\mathbb {R}}^d$. We assume that each entry $a_{ij}(x)$ of A(x) is in $C^{1,\mu }({\mathbb {R}}^d)$, where $\mu >0$, and satisfies assumption 1–3 from Assumption 1.2. We will further assume that $\det (A(x)) \ge c(1+ |x|^2)^{d+1}$ for all x. We want to estimate the number of eigenvalues less than or equal to $E^2$ for the second order differential operator with quadratic form

$$\begin{aligned} \langle A(x)(-i\nabla _x) \varphi , (-i\nabla _x)\varphi \rangle _{L^2({\mathbb {R}}^d)}. \end{aligned}$$

(1.9)

In this example we will denote this number by ${\mathcal {N}}(E^2)$. This is equivalent to counting the number of eigenvalues less than or equal to zero for the operator with quadratic form

$$\begin{aligned} \langle A(x)(-iE^{-1}\nabla _x) \varphi , (-iE^{-1}\nabla _x)\varphi \rangle _{L^2({\mathbb {R}}^d)} -\langle \varphi , \varphi \rangle _{L^2({\mathbb {R}}^d)}. \end{aligned}$$

(1.10)

Treating $E^{-1}$ as the semiclassical parameter, it is a simple calculation to check that all assumptions for Theorem 1.3 are satisfied. This gives us that

$$\begin{aligned} \big |{\mathcal {N}}(E^2) - \frac{1}{(2\pi )^d} \int _{{\mathbb {R}}^{2d}} {\varvec{1}}_{(-\infty ,E^2]}( \langle A(x)p,p\rangle ) \,dx dp \big | \le C E^{1-d}, \end{aligned}$$

for E suffciently large. Note that the phase space integral has this more familiar form

$$\begin{aligned} \frac{1}{(2\pi )^d} \int _{{\mathbb {R}}^{2d}} {\varvec{1}}_{(-\infty ,E^2]}( \langle A(x)p,p\rangle ) \,dx dp = E^d \frac{\text {Vol} _d(B(0,1))}{(2\pi )^d} \int _{{\mathbb {R}}^d} \frac{1}{\sqrt{\det (A(x))}} \, dx, \end{aligned}$$

where $\text {Vol} _d(B(0,1))$ is the d-dimensional volume of the unit ball.

1.2 Assumptions and main results II

The main motivation and the novel results, in this work are, to the best of the author’s knowledge, the following results, where we consider the spectral asymptotic of admissible operators perturbed by differential operators defined as Friedrichs extensions of sesquilinear forms with non-smooth coefficients. We will need some regularity conditions on the admissible operator, which will be the same as the assumption made in [24, Chapter 3]. We will here recall them:

Assumption 1.8

For $\hbar$ in $(0,\hbar _0]$ let $B(\hbar )$ be an admissible operator with tempered weight m and symbol

$$\begin{aligned} b_\hbar (x,p) = \sum _{j\ge 0} \hbar ^j b_j(x,p). \end{aligned}$$

(1.11)

Assume that

• 1. labelass.opsps1 The operator $B(\hbar )$ is symmetric on ${\mathcal {S}}({\mathbb {R}}^d)$ for all $\hbar$ in $(0,\hbar _0]$.

• 2. There is a $\zeta _0>0$ such that

  $$\begin{aligned} \min _{(x,p)\in {\mathbb {R}}^d\times {\mathbb {R}}^d}(b_0(x,p))> - \zeta _0. \end{aligned}$$

• 3. There is a $\zeta _1>\zeta _0$ such that $b_0+\zeta _1$ is a tempered weight and $b_j$ is in $\Gamma _{0,1}^{b_0+\zeta _1} \left( {\mathbb {R}}^{2d}\right)$.

The definition of a tempered weight is recalled in Sect. 3, where the definition of the space $\Gamma _{0,1}^{b_0+\zeta _1} \left( {\mathbb {R}}^{2d}\right)$ is also given. We can now state the second set of main theorems.

Theorem 1.9

Let $B(\hbar )$ be an admissible operator satisfying Assumption 1.8 with $\hbar$ in $(0,\hbar _0]$ and ${\mathcal {A}}_\hbar$ be a sesquilinear form satisfying Assumption 1.2 with the numbers $(1,\mu )$ with $\mu >0$.

Define the operator ${\tilde{B}}(\hbar )$ as the Friedrichs extension of the form sum $B(\hbar ) + {\mathcal {A}}_\hbar$ and let

$$\begin{aligned} {\tilde{b}}_0(x,p) = b_0(x,p) + \sum _{\left| \alpha \right| ,\left| \beta \right| \le m} a_{\alpha \beta }(x)p^{\alpha +\beta }. \end{aligned}$$

(1.12)

We suppose there is a $\nu$ such that ${\tilde{b}}_0^{-1}((-\infty ,\nu ])$ is compact and there is $c>0$ such that

$$\begin{aligned} |\nabla _p {\tilde{b}}_0(x,p)| \ge c \quad \text {for all } (x,p)\in {\tilde{b}}_0^{-1}(\{0\}). \end{aligned}$$

(1.13)

Then we have

$$\begin{aligned} \big |\text {Tr} [{\varvec{1}}_{(-\infty ,0]}({\tilde{B}}(\hbar ))] - \frac{1}{(2\pi \hbar )^d} \int _{{\mathbb {R}}^{2d}} {\varvec{1}}_{(-\infty ,0]}( {\tilde{b}}_0(x,p)) \,dx dp \big | \le C \hbar ^{1-d}, \end{aligned}$$

for all sufficiently small $\hbar$.

We also obtain asymptotic expansions for Riesz means for this type of operator.

Theorem 1.10

Assume $\gamma$ is in (0, 1]. Let $B(\hbar )$ be an admissible operator satisfing Assumption 1.8 with $\hbar$ in $(0,\hbar _0]$ and ${\mathcal {A}}_\hbar$ be a sesquilinear form satisfying Assumption 1.2 with the numbers $(2,\mu )$. If $\gamma =1$ we suppose $\mu >0$ and if $\gamma <1$ we may suppose $\mu =0$.

Define the operator ${\tilde{B}}(\hbar )$ as the Friedrichs extension of the form sum $B(\hbar ) + {\mathcal {A}}_\hbar$ and let

$$\begin{aligned} {\tilde{b}}_0(x,p) = b_0(x,p) + \sum _{\left| \alpha \right| ,\left| \beta \right| \le m} a_{\alpha \beta }(x)p^{\alpha +\beta }. \end{aligned}$$

(1.14)

We suppose there is a $\nu$ such that ${\tilde{b}}_0^{-1}((-\infty ,\nu ])$ is compact and there is $c>0$ such that

$$\begin{aligned} |\nabla _p {\tilde{b}}_0(x,p)| \ge c \quad \text {for all } (x,p)\in {\tilde{b}}_0^{-1}(\{0\}), \end{aligned}$$

(1.15)

Then we have

$$\begin{aligned}{} & {} \big |\text {Tr} [({\tilde{B}}(\hbar ))^\gamma _{-}] - \frac{1}{(2\pi \hbar )^d} \int _{{\mathbb {R}}^{2d}} ( {\tilde{b}}_{0}(x,p))^{\gamma }_{-} +\hbar \gamma {\tilde{b}}_1(x,p) ( {\tilde{b}}_{0}(x,p))^{\gamma -1}_{-} \,dx dp \big | \\ {}{} & {} \le C \hbar ^{1+\gamma -d}, \end{aligned}$$

for all sufficiently small $\hbar$ where the symbol ${\tilde{b}}_1(x,p)$ is defined as

$$\begin{aligned} {\tilde{b}}_{1}(x,p) = b_1(x,p)+ i \sum _{|\alpha |, |\beta |\le m}\sum _{j=1}^d \frac{\beta _j-\alpha _j}{2} \partial _{x_j} a_{\alpha \beta } (x) p^{\alpha +\beta -\eta _j}. \end{aligned}$$

(1.16)

1.3 Previous work

The first results with an optimal Weyl law without full regularity was proven in the papers [15,16,17,18] by Zielinski. In these papers Zielinski obtained an optimal Weyl law under the assumption that the coefficients are differentiable with Lipschitz continuous first derivative. Zielinski did not consider the semiclassical setting in those papers. These results were generalised by Ivrii in the semiclassical setting in [13]. Here the coefficients are assumed to be differentiable and with a Hölder continuous first derivative. This was further generalised by Bronstein and Ivrii in [8], where they reduced the assumptions further by assuming the first derivative to have modulus of continuity ${\mathcal {O}}(|\log (x-y)|^{-1})$. All these papers considered differential operators acting in $L^2(M)$, where M is a compact manifold either with or without boundary. In [19], Zielinski considers the semiclassical setting with differential operators acting in $L^2({\mathbb {R}}^d)$ and proves an optimal Weyl Law under the assumption that all coefficients are one time differentiable with a Hölder continuous derivative. Moreover, he assumes that the coefficients and the derivatives are bounded. However, he remarkes that it is possible to consider unbounded coefficients in a framework of tempered variation models. As we have seen above this is indeed the case and thereby this minor technical generalisation is proven.

Another question one could ask is: Can such results be obtained without a non-critical condition? This question has also been studied in the literature. The result is, that it is possible to prove, for Schrödinger operators, optimal Weyl laws without a non-critical condition by using a multiscale argument (see [8,9,10,11]). This approach is also described in [25]. This multiscale argument can be seen as a discrete approach and a continuous version has been proved and used in [26]. The essence of this approach is to localise and introduce a locally non-critical condition by unitary conjugation. Then, by an optimal Weyl law with a non-critical condition, one locally obtains the right asymptotics. The last step is to average out the localisations. Ivrii has also considered multiscale analysis for higher order differential operators but, to treat these cases, extra assumptions on the Hessian of the principal symbol is needed (see [10, 12]). There is also another approach by Zielinski (see [20,21,22]), where he proves optimal Weyl laws without a non-critical condition, but with an extra assumption on a specific phase space volume.

In the case of Riesz means, sharp remainder estimates for the asymptotic expansions have been obtained by Bronstein and Ivrii in [8], for the case of differential operators acting in $L^2(M)$, where M is a compact closed manifold. However, it is mentioned in a remark in [8] that the results of the paper need to be combined with the methods in section 4.3 in [9] to obtain the reminder estimate for the Riesz means. Results on Riesz means can also be found in [10, Chapter 4] by Ivrii. Again, this is for differential operators acting in $L^2(M)$, where M is a compact closed manifold. In chapter 11 of [10], also unbounded domains are considered, but in the large eigenvalue/high energy case. In [10, Chapter 11] all results are stated for “smooth” coefficients. However, it is remarked that the case of non-smooth coefficients is left to the reader. The results given here on the Reisz means for differential operators associated to quadratic forms are a generalisation of previous known results.

To summarise the relation between the results obtained in this manuscript and previous known results: the results obtained here in Theorem 1.3 are a technical generalisation of previous known results, as we allow for unbounded coefficients and non-compact domains. The results obtained in Theorem 1.5 have been mentioned in a remark by Bronstein and Ivrii in [8] for operators on compact bounded manifolds. However, they have not been written and proven previously in the case of unbounded coefficients and on non-compact domains before, to the best of the author’s knowledge. The second set of main results (Theorems 1.9 and 1.10) has not been discussed/treated in the existing literature–again to the best of the author’s knowledge.

2 Preliminaries and notation

2.1 Notation and definitions

Here, we will mainly set up the non-standard notation used and some definitions. In the following, we will use the notation

$$\begin{aligned} \lambda (x) = (1+\left| x \right| ^2)^{\frac{1}{2}}, \end{aligned}$$

for x in ${\mathbb {R}}^d$ and not the usual bracket notation. Moreover, for more vectors $x_1,x_2,x_3$ from ${\mathbb {R}}^d$ we will use the convention

$$\begin{aligned} \lambda (x_1,x_2,x_3) = (1+\left| x_1 \right| ^2+\left| x_2 \right| ^2+\left| x_3 \right| ^2)^{\frac{1}{2}}, \end{aligned}$$

and similar in the case of 2 or even more vectors. For the natural numbers, we use the following conventions

$$\begin{aligned} {\mathbb {N}}= \{1,2,3,\dots \} \quad \text {and}\quad {\mathbb {N}}_0 = \{0\} \cup {\mathbb {N}}. \end{aligned}$$

When working with the Fourier transform, we will use the following semiclassical version for $\hbar >0$

$$\begin{aligned} {\mathcal {F}}_\hbar [ \varphi ](p) {{:}{=}} \int _{{\mathbb {R}}^d} e^{-i\hbar ^{-1} \langle x,p \rangle }\varphi (x) \, dx, \end{aligned}$$

and with inverse given by

$$\begin{aligned} {\mathcal {F}}_\hbar ^{-1}[\psi ] (x) {{:}{=}} \frac{1}{(2\pi \hbar )^d} \int _{{\mathbb {R}}^d} e^{i\hbar ^{-1} \langle x,p \rangle }\psi (p) \, dp, \end{aligned}$$

where $\varphi$ and $\psi$ are elements of ${\mathcal {S}}({\mathbb {R}}^d)$.

We will by ${\mathcal {L}}({\mathcal {B}}_1,{\mathcal {B}}_2)$ denote the linear bounded operators from the space ${\mathcal {B}}_1$ into ${\mathcal {B}}_2$ and ${\mathcal {L}}({\mathcal {B}}_1)$ denotes the linear bounded operators from the space ${\mathcal {B}}_1$ into itself. For an operator A acting in a Hilbert space we will denote the spectrum of A by

$$\begin{aligned} \text {spec} (A). \end{aligned}$$

As we will later need stationary phase asymptotics, we will for sake of completeness recall it here.

Proposition 2.1

Let B be a invertible, symmetric real $n\times n$ matrix and $(u,v)\rightarrow a(u,v;\hbar )$ be a function in $C^\infty ({\mathbb {R}}_u^{d} \times {\mathbb {R}}_v^{n})$ for all $\hbar$ in $(0,\hbar _0]$. We suppose $v\rightarrow a(u,v;\hbar )$ has compact support for all u in ${\mathbb {R}}_u^{d}$ and $\hbar$ in $(0,\hbar _0]$. Moreover we let

$$\begin{aligned} I(u;a,B,\hbar ) = \int _{{\mathbb {R}}^{n}} e^{\frac{i}{2\hbar } \langle B v,v\rangle } a(u,v;\hbar ) \, dv. \end{aligned}$$

Then for each N in ${\mathbb {N}}$ we have

$$\begin{aligned} \begin{aligned}&I(u;a,B,\hbar )\\&= (2\pi \hbar )^{\frac{n}{2}} \frac{e^{i\frac{\pi }{4} \text {sgn} (B)}}{\left| \det (B) \right| ^{\frac{1}{2}}} \sum _{j=0}^N \frac{\hbar ^j}{j!} \Big (\frac{\langle B^{-1} D_v,D_v\rangle }{2i}\Big )^j a(u,v;\hbar ) \Big |_{v=0} + \hbar ^{N+1} R_{N+1}(u;\hbar ), \end{aligned} \end{aligned}$$

where $\text {sgn} (B)$ is the difference between the number of positive and negative eigenvalues of B. Moreover there exists a constant $c_n$ only depending on the dimension such that the error term $R_{N+1}$ satisfies the bound

$$\begin{aligned} \left| R_{N+1}(u;\hbar ) \right| \le c_n \bigg \Vert \frac{\langle B^{-1} D_v,D_v\rangle ^{N+1}}{(N+1)!} a(u,v;\hbar ) \bigg \Vert _{H^{[\frac{n}{2}]+1}({\mathbb {R}}_v^n)}, \end{aligned}$$

where $[\tfrac{n}{2}]$ is the integer part of $\tfrac{n}{2}$ and $\Vert \cdot \Vert _{H^{[\tfrac{n}{2}]+1}({\mathbb {R}}_v^n)}$ is the Sobolev norm.

A proof of the proposition can be found in e.g. [23, 24]. For the sesquilinear forms we consider we will use the following terminology.

Definition 2.2

For a sesquilinear form ${\mathcal {A}}_\hbar$ given by

$$\begin{aligned} {\mathcal {A}}_\hbar [\varphi ,\psi ] = \sum _{\left| \alpha \right| ,\left| \beta \right| \le m} \int _{{\mathbb {R}}^d} a_{\alpha \beta }(x) (\hbar D_x)^\beta \varphi (x) \overline{(\hbar D_x)^\alpha \psi (x)} \, dx, \qquad \varphi ,\psi \in {\mathcal {D}}( {\mathcal {A}}_\hbar ), \end{aligned}$$

we call a number E in ${\mathbb {R}}$ non-critical if there exists $c>0$ such that

$$\begin{aligned} |\nabla _p a_0(x,p)| \ge c \quad \text {for all } (x,p)\in a_0^{-1}(\{E\}), \end{aligned}$$

where

$$\begin{aligned} a_0(x,p) = \sum _{\left| \alpha \right| ,\left| \beta \right| \le m} a_{\alpha \beta }(x)p^{\alpha +\beta }. \end{aligned}$$

Definition 2.3

A sesquilinear form ${\mathcal {A}}_\hbar$ given by

$$\begin{aligned} {\mathcal {A}}_\hbar [\varphi ,\psi ] = \sum _{\left| \alpha \right| ,\left| \beta \right| \le m} \int _{{\mathbb {R}}^d} a_{\alpha \beta }(x) (\hbar D_x)^\beta \varphi (x) \overline{(\hbar D_x)^\alpha \psi (x)} \, dx, \qquad \varphi ,\psi \in {\mathcal {D}}( {\mathcal {A}}_\hbar ), \end{aligned}$$

is called elliptic if there exists a strictly positive constant C such that

$$\begin{aligned} \sum _{\left| \alpha \right| =\left| \beta \right| = m} a_{\alpha \beta }(x) p^{\alpha +\beta } \ge C \left| p \right| ^{2m}, \end{aligned}$$

(2.1)

for all (x, p) in ${\mathbb {R}}^d_x\times {\mathbb {R}}_p^d$.

2.2 Approximation of quadratic forms

Here, we will construct our approximating (framing) quadratic forms, and prove that some properties, of the original quadratic form, can be inherited by the approximations. The construction is similar to the constructions used in [8, 9, 12, 14, 19] for their framing operators. The first example, of approximation of non-smooth coefficients in this way, appeared in [27], to the author’s knowledge. The crucial part in this construction is Proposition 2.4, for which a proof can be found in [8, 10].

Proposition 2.4

Let f be in $C^{k,\mu }({\mathbb {R}}^d)$ for a $\mu$ in [0, 1]. Then for every $\varepsilon >0$ there exists a function $f_\varepsilon$ in $C^\infty ({\mathbb {R}}^d)$ such that

$$\begin{aligned} \begin{aligned} \left| \partial _x^\alpha f_\varepsilon (x) -\partial _x^\alpha f(x) \right|&\le {} C_\alpha \varepsilon ^{k+\mu -\left| \alpha \right| } \qquad \left| \alpha \right| \le k, \\ \left| \partial _x^\alpha f_\varepsilon (x) \right|&\le {} C_\alpha \varepsilon ^{k+\mu -\left| \alpha \right| } \qquad \left| \alpha \right| \ge k+1, \end{aligned} \end{aligned}$$

(2.2)

where the constants is independent of $\varepsilon$.

The function $f_\varepsilon$ is a smoothing (mollification) of f. Usually this is done by convolution with a compactly supported smooth function. However, here one uses a Schwartz function in the convolution in order to ensure the stated error terms. The convolution with a compactly supported smooth function will in most cases “only” give an error of order $\varepsilon$. The ideas used to construct the approximating or framing quadratic forms are of the same type as the ideas used to construct the framing operators in [19].

Proposition 2.5

Let ${\mathcal {A}}_\hbar$ be an elliptic sesquilinear form given by

$$\begin{aligned} {\mathcal {A}}_\hbar [\varphi ,\psi ] = \sum _{\left| \alpha \right| ,\left| \beta \right| \le m} \int _{{\mathbb {R}}^d} a_{\alpha \beta }(x) (\hbar D_x)^\beta \varphi (x) \overline{(\hbar D_x)^\alpha \psi (x)} \, dx, \qquad \varphi ,\psi \in {\mathcal {D}}( {\mathcal {A}}_\hbar ). \end{aligned}$$

Assume the coefficients $a_{\alpha \beta }(x)$ are in $C^{k,\mu }({\mathbb {R}}^d)$, for a pair $(k,\mu )$ in ${\mathbb {N}}\times [0,1]$, bounded from below and satisfies that $a_{\alpha \beta }(x) = \overline{a_{\beta \alpha }(x)}$ for all multi-indices $\alpha$ and $\beta$. Then for all $\varepsilon >0$ there exists a pair of sesquilinear forms ${\mathcal {A}}_{\hbar ,\varepsilon }^{+}$ and ${\mathcal {A}}_{\hbar ,\varepsilon }^{-}$ such that

$$\begin{aligned} {\mathcal {A}}_{\hbar ,\varepsilon }^{-}[\varphi ,\varphi ]\le {\mathcal {A}}_{\hbar }[\varphi ,\varphi ]\le {\mathcal {A}}_{\hbar ,\varepsilon }^{+}[\varphi ,\varphi ] \end{aligned}$$

(2.3)

for all $\varphi$ in ${\mathcal {D}}( {\mathcal {A}}_\hbar )$. Moreover, for all sufficiently small $\varepsilon$ the sesquilinear forms ${\mathcal {A}}_{\hbar ,\varepsilon }^{\pm }$ will be elliptic. If E is a non-critical value of ${\mathcal {A}}_{\hbar }$ then E will also be non-critical for ${\mathcal {A}}_{\hbar ,\varepsilon }^{\pm }$ for all $\varepsilon$ sufficiently small. The sesquilinear forms ${\mathcal {A}}_{\hbar ,\varepsilon }^{\pm }$ is explicit given by

$$\begin{aligned} \begin{aligned} {\mathcal {A}}_{\hbar ,\varepsilon }^{\pm }[\varphi ,\psi ] = {}&\sum _{\left| \alpha \right| ,\left| \beta \right| \le m} \int _{{\mathbb {R}}^d} a_{\alpha \beta }^\varepsilon (x) (\hbar D_x)^\beta \varphi (x) \overline{(\hbar D_x)^\alpha \psi (x)} \, dx \\&\pm C_1 \varepsilon ^{k+\mu } \sum _{\left| \alpha \right| \le m} \int _{{\mathbb {R}}^d} (\hbar D_x)^\alpha \varphi (x) \overline{(\hbar D_x)^\alpha \psi (x)} \, dx, \qquad \varphi ,\psi \in {\mathcal {D}}( {\mathcal {A}}_\hbar ), \end{aligned} \end{aligned}$$

(2.4)

where $a_{\alpha \beta }^\varepsilon (x)$ are the smoothed functions of $a_{\alpha \beta }(x)$ according to Proposition 2.4 and $C_1$ is some positive constant.

Proof

We start by considering the form ${\mathcal {A}}_{\hbar ,\varepsilon }$ given by

$$\begin{aligned} {\mathcal {A}}_{\hbar ,\varepsilon }[\varphi ,\psi ] = \sum _{\left| \alpha \right| ,\left| \beta \right| \le m} \int _{{\mathbb {R}}^d} a_{\alpha \beta }^\varepsilon (x) (\hbar D_x)^\beta \varphi (x) \overline{(\hbar D_x)^\alpha \psi (x)} \, dx, \end{aligned}$$

where we have replaced the coefficients of ${\mathcal {A}}_{\hbar }$ with smooth functions made according to Proposition 2.4. For $\varphi$ in ${\mathcal {D}}( {\mathcal {A}}_{\hbar }) \cap {\mathcal {D}}( {\mathcal {A}}_{\hbar ,\varepsilon })$ we have by Cauchy-Schwarz inequality that

$$\begin{aligned}{} & {} | {\mathcal {A}}_{\hbar }[\varphi ,\varphi ] - {\mathcal {A}}_{\hbar ,\varepsilon }[\varphi ,\varphi ] | \nonumber \\{} & {} \le {} \sum _{\left| \alpha \right| ,\left| \beta \right| \le m}| \langle (a_{\alpha \beta }-a_{\alpha \beta }^\varepsilon )(\hbar D)^\beta \varphi , (\hbar D)^\alpha \varphi \rangle | \nonumber \\{} & {} \le {} \sum _{\left| \alpha \right| ,\left| \beta \right| \le m} \frac{1}{2\varepsilon ^{k+\mu }} \Vert (a_{\alpha \beta }-a_{\alpha \beta }^\varepsilon )(\hbar D)^\beta \varphi \Vert _{L^2({\mathbb {R}}^d)}^2 + \frac{\varepsilon ^{k+\mu }}{2} \Vert (\hbar D)^\alpha \varphi \Vert _{L^2({\mathbb {R}}^d)}^2 \nonumber \\{} & {} \le {} c \varepsilon ^{k+\mu } \sum _{\left| \alpha \right| \le m} \langle (\hbar D)^{\alpha } \varphi , (\hbar D)^{\alpha } \varphi \rangle , \end{aligned}$$

(2.5)

where we in the last inequality have used Proposition 2.4. From this inequality, we get that ${\mathcal {D}}( {\mathcal {A}}_{\hbar })= {\mathcal {D}}( {\mathcal {A}}_{\hbar ,\varepsilon })$. We recognise the last bound in (2.5) as the quadratic form associated to $(I-\hbar ^2\Delta )^m$. Hence for sufficiently large value of the constant $C_1$ we can choose the approximating forms ${\mathcal {A}}_{\hbar ,\varepsilon }^{\pm }$ to be given by (2.4) such that

$$\begin{aligned} {\mathcal {A}}_{\hbar ,\varepsilon }^{-}[\varphi ,\varphi ]\le {\mathcal {A}}_{\hbar }[\varphi ,\varphi ]\le {\mathcal {A}}_{\hbar ,\varepsilon }^{+}[\varphi ,\varphi ], \end{aligned}$$

(2.6)

for all $\varphi$ in ${\mathcal {D}}( {\mathcal {A}}_\hbar )$.

In order to show, that we can choose the forms ${\mathcal {A}}_{\hbar ,\varepsilon }^{\pm }$ elliptic, we note that

$$\begin{aligned} \begin{aligned} \sum _{\left| \alpha \right| =\left| \beta \right| = m}&a_{\alpha \beta }^\varepsilon (x) p^{\alpha +\beta } \pm C_1\varepsilon ^{k+\mu }\left| p \right| ^{2m} \\&= \sum _{\left| \alpha \right| =\left| \beta \right| = m} (a_{\alpha \beta }^\varepsilon (x) -a_{\alpha \beta } (x) ) p^{\alpha +\beta } +\sum _{\left| \alpha \right| =\left| \beta \right| = m} a_{\alpha \beta } (x) p^{\alpha +\beta } \pm C_1 \varepsilon ^{k+\mu }\left| p \right| ^{2m} \\&\ge \,(C-(C_1-c)\varepsilon ^{k+\mu }) \left| p \right| ^{2m} \ge {\tilde{C}} \left| p \right| ^{2m}, \end{aligned} \end{aligned}$$

(2.7)

for sufficiently small $\varepsilon$ and all (x, p) in ${\mathbb {R}}_x^d\times {\mathbb {R}}_p^d$. This gives us that both forms ${\mathcal {A}}_{\hbar ,\varepsilon }^{\pm }$ are elliptic.

For the last part we assume E is a non-critical value for the form ${\mathcal {A}}_{\hbar }$. That is there exist a $c>0$ such that

$$\begin{aligned} |\nabla _p a_0(x,p)| \ge c \quad \text {for all } (x,p)\in a_0^{-1}(\{E\}), \end{aligned}$$

where

$$\begin{aligned} a_0(x,p) = \sum _{\left| \alpha \right| ,\left| \beta \right| \le m} a_{\alpha \beta }(x)p^{\alpha +\beta }. \end{aligned}$$

In order to prove, that E is a non-critical value for the framing forms, we need to find an expression for $a_{\varepsilon ,0}^{-1}(\{E\})$ for the framing operators. Here, we have omitted the $+$ and − in the notation. By the ellipticity, in the following calculations we can assume, with out loss of generality, that p belongs to a bounded set. We have

$$\begin{aligned} \begin{aligned}&a_{\varepsilon ,0}(x,p) \\&={} \sum _{\left| \alpha \right| ,\left| \beta \right| \le m} (a_{\alpha \beta }^\varepsilon (x) - a_{\alpha \beta }(x))p^{\alpha +\beta } \pm C_1 \varepsilon ^{k+\mu }(1+p^2)^m + \sum _{\left| \alpha \right| ,\left| \beta \right| \le m} a_{\alpha \beta }(x)p^{\alpha +\beta }. \end{aligned} \end{aligned}$$

(2.8)

Since we can assume p to be in a compact set we have that

$$\begin{aligned} \Big | \sum _{\left| \alpha \right| ,\left| \beta \right| \le m} (a_{\alpha \beta }^\varepsilon (x) - a_{\alpha \beta }(x))p^{\alpha +\beta } \pm C_1 \varepsilon ^{k+\mu }(1+p^2)^m \Big | \le C \varepsilon ^{k+\mu }. \end{aligned}$$

This combined with (2.8) implies the inclusion

$$\begin{aligned} \big \{ (x,p) \in {\mathbb {R}}^{2d} \, |\, a_{\varepsilon ,0}(x,p) =E \big \} \subseteq \big \{ (x,p) \in {\mathbb {R}}^{2d} \, |\, | a_{0}(x,p) -E | \le C \varepsilon ^{k+\mu } \big \}. \end{aligned}$$

Hence by continuity of $a_0(x,p)$ we get for all sufficiently small $\varepsilon$ the inclusion

$$\begin{aligned} \big \{ (x,p) \in {\mathbb {R}}^{2d} \, |\, a_{\varepsilon ,0}(x,p) =E \big \} \subseteq \big \{ (x,p) \in {\mathbb {R}}^{2d} \, |\, \left| \nabla _p a_{0}(x,p) \right| \ge \tfrac{c}{2} \big \}. \end{aligned}$$

(2.9)

For a point $(x,p)\in \big \{ (x,p) \in {\mathbb {R}}^{2d} \, |\, a_{\varepsilon ,0}(x,p) =E \big \}$ we have

$$\begin{aligned} \begin{aligned}&\nabla _p a_{\varepsilon ,0}(x,p) \\&={} \sum _{\left| \alpha \right| ,\left| \beta \right| \le m} (a_{\alpha \beta }^\varepsilon (x) - a_{\alpha \beta }(x)) \nabla _p p^{\alpha +\beta } \pm C_1 \varepsilon ^{k+\mu } \nabla _p (1+p^2)^m + \nabla _p a_{0}(x,p). \end{aligned} \end{aligned}$$

(2.10)

Since we can assume p to be contained in a compact set, we have that

$$\begin{aligned} \Big | \sum _{\left| \alpha \right| ,\left| \beta \right| \le m} (a_{\alpha \beta }^\varepsilon (x) - a_{\alpha \beta }(x)) \nabla _p p^{\alpha +\beta } \pm C_1 \varepsilon ^{k+\mu } \nabla _p (1+p^2)^m \Big | \le C \varepsilon ^{k+\mu }. \end{aligned}$$

Combining this with (2.9) and (2.10) we get that

$$\begin{aligned} |\nabla _p a_{\varepsilon ,0}(x,p)| \ge |\nabla _p a_{0}(x,p)| - C \varepsilon ^{k+\mu } \ge \frac{c}{2} - C \varepsilon ^{k+\mu } \ge \frac{c}{4}, \end{aligned}$$

(2.11)

where the last inequality is for $\varepsilon$ sufficiently small. This inequality proves E is also a non-critical value of the framing forms. $\square$

The framing forms constructed, in the previous proposition, are forms with smooth coefficients. But, when we take derivatives of these coefficients, we start to get negative powers of $\varepsilon$ from some point. Hence, we cannot associate a “classic” pseudo-differential operator to the form. We will instead in the following sections consider a “rough” theory for pseudo-differential operators. Here we will see that it is in fact possible to verify most of the results from classic theory of pseudo-differential operators in this rough theory. After this has been developed, we will return to these framing forms.

As a last remark of this section, note that there is no unique way to construct these framing forms.

3 Definitions and properties of rough pseudo-differential operators

In this section we will, inspired by the approximation results in the previous section, define a class of pseudo-differential operators with rough symbols. We will further state and prove some of the properties for these operators. The definitions are very similar to the definitions in the monograph [24] and we will see that the properties of these operators can be deduced from the results in the monograph [24].

3.1 Definition of rough operators and rough symbols

We start, for the sake of completeness, by recalling the definition of a tempered weight function.

Definition 3.1

A tempered weight function on ${\mathbb {R}}^D$ is a continuous function

$$\begin{aligned} m:{\mathbb {R}}^D \rightarrow [0,\infty [, \end{aligned}$$

for which there exist positive constants $C_0$, $N_0$ such that for all points $x_1$ in ${\mathbb {R}}^D$ the estimate

$$\begin{aligned} m(x) \le C_0 m(x_1) \lambda (x_1-x)^{N_0}, \end{aligned}$$

holds for all points x in ${\mathbb {R}}^D$.

For our purpose here, we will consider the cases where $D=2d$ or $D=3d$. These types of functions are in the literature sometimes called order functions. This is the case in the monographs [23, 28]. But we have chosen the name tempered weights to align with the terminology in the monographs [24, 29]. We can now define the symbols, we will be working with.

Definition 3.2

(Rough symbol) Let $\Omega \subseteq {\mathbb {R}}_x^d \times {\mathbb {R}}_p^d \times {\mathbb {R}}_y^d$ be open, $\rho$ be in [0, 1], $\varepsilon >0$, $\tau$ be in ${\mathbb {Z}}$ and m a tempered weight function on ${\mathbb {R}}_x^d \times {\mathbb {R}}_p^d \times {\mathbb {R}}_y^d$. We call a function $a_\varepsilon$ a rough symbol of regularity $\tau$ with weights $(m,\rho , \varepsilon )$ if $a_\varepsilon$ is in $C^{\infty }(\Omega )$ and satisfies that

$$\begin{aligned} \begin{aligned} |\partial _x^\alpha \partial _p^\beta \partial _y^\gamma a_\varepsilon (x,p,y)| \le C_{\alpha \beta \gamma } \varepsilon ^{\min (0,\tau -\left| \alpha \right| -\left| \gamma \right| )} m (x,p,y) \lambda (x,p,y)^{-\rho (\left| \alpha \right| +\left| \beta \right| +\left| \gamma \right| )}, \end{aligned} \end{aligned}$$

(3.1)

for all (x, p, y) in $\Omega$ and $\alpha$, $\beta$, $\gamma$ in ${\mathbb {N}}^d$, where the constants $C_{\alpha \beta \gamma }$’s do not depend on $\varepsilon$. The space of these functions is denoted $\Gamma _{\rho ,\varepsilon }^{m,\tau } \left( \Omega \right)$.

Remark 3.3

The space $\Gamma _{\rho ,\varepsilon }^{m,\tau } \left( \Omega \right)$ can be turned into a Fréchet space with semi norms associated to the estimates in (3.1). It is important to note that the semi norms on $\Gamma _{\rho ,\varepsilon }^{m,\tau } \left( \Omega \right)$ should be chosen weighted such that the norms associated to a set of numbers $\alpha ,\beta ,\gamma$ will be bounded by the constant $C_{\alpha \beta \gamma }$ and hence independent of $\varepsilon$.

If $\varepsilon$ is equal to 1, then these symbols are the same as the symbols defined in the monograph [24] (Definition II-10). We will always assume $\varepsilon \le 1$ as we are interested in the cases of very small $\varepsilon$.

We will later call a function $a_\varepsilon (x,p)$ or $b_\varepsilon (p,y)$ a rough symbol if it satisfies the above definition in the two variables x and p or p and y. This more general definition is made in order to define the different forms of quantisation and the interpolation between them. If we say a symbol of regularity $\tau$ with tempered weight m we implicit assume that $\rho =0$. This type of rough symbols is contained in the class of rough symbols considered in [10, Section 2.3 and 4.6].

Remark 3.4

We will later assume, that a rough symbol is a tempered weight. When this is done, we will implicitly assume, that the constants from the definition of a tempered weight is independent of $\varepsilon$. This is an important assumption since we need the estimates we make to be uniform for $\hbar$ in $(0,\hbar _0]$ with $\hbar _0>0$ sufficiently small. Then for a choice of $\delta$ in (0, 1), we also need the estimates to be uniform for $\varepsilon$ in $[\hbar ^{1-\delta },1]$.

Essentially the constants will be uniform for both $\hbar$ in $(0,\hbar _0]$ and $\varepsilon$ in (0, 1], but if $\varepsilon \le \hbar$ then the estimates will diverge in the semiclassical parameter. Hence we will assume the lower bound on $\varepsilon$. The assumption that $\varepsilon \ge \hbar ^{1-\delta }$ is called a microlocal uncertainty principal in [9, Chapter 6 Section 4] and [10, Vol I Section 1.1]. It is also in both cases mentioned in the introduction. In [9, 10] there are two parameter instead of just one. This other parameter can be used to scale in the p-variable.

As we are interested in asymptotic expansions in the semiclassical parameter, we will define $\hbar$-$\varepsilon$-admissible symbols. These are symbols depending on the semiclassical parameter $\hbar$, for which we can make an expansion in $\hbar$.

Definition 3.5

We use the notation from Definition 3.2. We call a symbol $a_\varepsilon (\hbar )$ $\hbar$-$\varepsilon$-admissible of regularity $\tau$ with weights $(m,\rho , \varepsilon )$ in $\Omega$, if for fixed $\varepsilon$ and a $\hbar _0>0$ the map that takes $\hbar$ into $a_\varepsilon (\hbar )$ is smooth from $(0,\hbar _0]$ into $\Gamma _{\rho ,\varepsilon }^{m,\tau } \left( \Omega \right)$ such that there exists a $N_0$ in ${\mathbb {N}}$ such for all $N\ge N_0$ we have

$$\begin{aligned} a_\varepsilon (x,p,y;\hbar ) = a_{\varepsilon ,0}(x,p,y) + \hbar a_{\varepsilon ,1}(x,p,y) + \cdots + \hbar ^N a_{\varepsilon ,N}(x,p,y) + \hbar ^{N+1} r_{\varepsilon ,N}(x,p,y;\hbar ), \end{aligned}$$

where $a_{\varepsilon ,j}$ is in $\Gamma _{\rho ,\varepsilon ,-2j}^{m,\tau _j} \left( \Omega \right)$ with the notation $\tau _j=\tau -j$ and $r_{\varepsilon ,N}$ is a symbol satisfying the bounds

$$\begin{aligned} \begin{aligned} \hbar ^{N+1} |\partial _x^\alpha&\partial _p^\beta \partial _y^\gamma r_{\varepsilon ,N}(x,p,y;\hbar )| \\&\le {} C_{\alpha \beta \gamma } \hbar ^{\kappa _1(N)} \varepsilon ^{-\left| \alpha \right| -\left| \gamma \right| }m(x,y,p) \lambda (x,y,p)^{-\rho (\kappa _2(N)+ \left| \alpha \right| +\left| \beta \right| +\left| \gamma \right| )}, \end{aligned} \end{aligned}$$

where $\kappa _1$ is a positive strictly increasing function and $\kappa _2$ is non-decreasing function. For k in ${\mathbb {Z}}$ $\Gamma _{\rho ,\varepsilon ,k}^{m,\tau } \left( \Omega \right)$ is the space of rough symbols of regularity $\tau$ with weights $(m(1+|(x,y,p)|)^{k\rho },\rho , \varepsilon )$.

Remark 3.6

We will also use the terminology $\hbar$-$\varepsilon$-admissible for symbols in two variables, where the definition is the same just in two variables. This definition is slightly different from the “usual” definition of an $\hbar$-admissible symbol [24, Definition II-11]. One difference is in the error term.

The functions $\kappa _1$ and $\kappa _2$ will in most cases be dependent on the regularity $\tau$, the dimension d and the tempered weight function through the constants in the definition of a tempered weight. It should be noted that the function $\kappa _2$ might be constant negative.

We will now define the pseudo-differential operators associated with the rough symbols. We will call them rough pseudo-differential operators.

Definition 3.7

Let m be a tempered weight function on ${\mathbb {R}}^d_x\times {\mathbb {R}}^d_p\times {\mathbb {R}}_y^d$, $\rho$ in [0, 1], $\varepsilon >0$ and $\tau$ in ${\mathbb {Z}}$. For a rough symbol $a_\varepsilon$ in $\Gamma _{\rho ,\varepsilon }^{m,\tau }({\mathbb {R}}^d_x\times {\mathbb {R}}^d_p\times {\mathbb {R}}_y^d)$ we associate the operator $\text {Op} _\hbar (a_\varepsilon )$ defined by

$$\begin{aligned} \text {Op} _\hbar (a_\varepsilon )\psi (x) = \frac{1}{(2\pi \hbar )^{d}} \int _{{\mathbb {R}}^{2d}} e^{i \hbar ^{-1} \langle {x-y},{p}\rangle } a_\varepsilon ( x,p,y) \psi (y) \, d y \, d p, \end{aligned}$$

for $\psi$ in ${\mathcal {S}}({\mathbb {R}}^d)$.

Remark 3.8

We use the notation from Definition 3.7. We remark that the integral in the definition of $\text {Op} _\hbar (a_\varepsilon )\psi (x)$ shall be considered as an oscillating integral. By applying the techniques for oscillating integrals it can be proven that $\text {Op} _\hbar (a_\varepsilon )$ is a continuous linear operator from ${\mathcal {S}}({\mathbb {R}}^d)$ into itself. The proof of this is analogous to the proof in [24] in the non-rough case. By duality it can also be defined as an operator from ${\mathcal {S}}'({\mathbb {R}}^d)$ into ${\mathcal {S}}'({\mathbb {R}}^d)$.

Definition 3.9

We call an operator $A_\varepsilon (\hbar )$ from ${\mathcal {L}}({\mathcal {S}}({\mathbb {R}}^d),L^2({\mathbb {R}}^d))$ $\hbar$-$\varepsilon$-admissible of regularity $\tau \ge 0$ with tempered weight m if for fixed $\varepsilon$ and a $\hbar _0>0$ the map

$$\begin{aligned} A_\varepsilon : (0,\hbar _0] \rightarrow {\mathcal {L}}({\mathcal {S}}({\mathbb {R}}^d),L^2({\mathbb {R}}^d)) \end{aligned}$$

is smooth and there exists a sequence $a_{\varepsilon ,j}$ in $\Gamma _{0,\varepsilon }^{m,{\tau _j}}({\mathbb {R}}^d_x\times {\mathbb {R}}^d_p\times {\mathbb {R}}_y^d)$, where $\tau _0=\tau$ and $\tau _{j+1}= \tau _j-1$ and a sequence $R_N$ in ${\mathcal {L}}(L^2({\mathbb {R}}^d))$ such that for $N\ge N_0$, $N_0$ sufficient large,

$$\begin{aligned} A_\varepsilon (\hbar )=\sum _{j=0}^N \hbar ^j \text {Op} _\hbar ( a_{\varepsilon ,j}) + \hbar ^{N+1} R_N(\varepsilon ,\hbar ), \end{aligned}$$

(3.2)

and

$$\begin{aligned} \hbar ^{N+1} \Vert R_N(\varepsilon ,\hbar ) \Vert _{{\mathcal {L}}(L^2({\mathbb {R}}^d))} \le \hbar ^{\kappa (N)} C_N, \end{aligned}$$

for a strictly positive increasing function $\kappa$.

Remark 3.10

By the results in Theorem 3.25 we have that if the tempered weight function m is in $L^\infty ({\mathbb {R}}^d)$. Then for a $\hbar$-$\varepsilon$-admissible symbol $a_\varepsilon (\hbar )$ of regularity $\tau \ge 0$ with tempered weight m the operator $A_\varepsilon (\hbar )=\text {Op} _\hbar (a_\varepsilon (\hbar ))$ is a $\hbar$-$\varepsilon$-admissible operator of regularity $\tau$.

Remark 3.11

When we have an operator $A_\varepsilon (\hbar )$ with an expansion

$$\begin{aligned} A_\varepsilon (\hbar )=\sum _{j\ge 0} \hbar ^j \text {Op} _\hbar ( a_{\varepsilon ,j}), \end{aligned}$$

where the sum is understood as a formal sum. That is for all N sufficiently large there exists $R_N$ in ${\mathcal {L}}(L^2({\mathbb {R}}^d))$, such that the operator is of the same form as in (3.2). Then we call the symbol $a_{\varepsilon ,0}$ the principal symbol and the symbol $a_{\varepsilon ,1}$ the subprincipal symbol.

Definition 3.12

Let $A_\varepsilon (\hbar )$ be a $\hbar$-$\varepsilon$-admissible of regularity $\tau$ with tempered weight m. For any t in [0, 1] we call all $\hbar$-$\varepsilon$-admissible symbols $b_\varepsilon (\hbar )$ in $\Gamma _{0,\varepsilon }^{m,\tau }({\mathbb {R}}^d_x\times {\mathbb {R}}^d_p)$ such that,

$$\begin{aligned} A_\varepsilon (\hbar ) \psi (x) = \frac{1}{(2\pi \hbar )^{d}} \int _{{\mathbb {R}}^{2d}} e^{i \hbar ^{-1} \langle {x-y},{p}\rangle } b_\varepsilon ( (1-t)x +ty,p;\hbar ) \psi (y) \, d y \, d p, \end{aligned}$$

for all $\psi \in {\mathcal {S}}({\mathbb {R}}^d)$ and all $\hbar \in ]0,\hbar _0]$, where $\hbar _0$ is a strictly positive number, rough t-$\varepsilon$-symbols of regularity $\tau$ associated to $A_\varepsilon (\hbar )$.

Notation 3.13

In general for a symbol $b_\varepsilon (\hbar )$ in $\Gamma _{\rho ,\varepsilon }^{m,\tau }({\mathbb {R}}^d_x\times {\mathbb {R}}^d_p)$ and $\psi$ in ${\mathcal {S}}({\mathbb {R}}^d)$ we will use the notation

$$\begin{aligned} \text {Op} _{\hbar , t }(b_\varepsilon )\psi (x) = \frac{1}{(2\pi \hbar )^d} \int _{{\mathbb {R}}^{2d}} e^{i\hbar ^{-1} \langle x-y,p\rangle } b_\varepsilon ( (1-t)x +ty,p;\hbar ) \psi (y) \, d y \, d p. \end{aligned}$$

We have the special case of Weyl quantisation when $t=\frac{1}{2}$, which is the one we will work the most with. In this case we write

$$\begin{aligned} \text {Op} _{\hbar ,\frac{1}{2}}(b_\varepsilon ) = \text {Op} _\hbar ^{\text {w} }(b_\varepsilon ). \end{aligned}$$

For some applications, we will need stronger assumptions than $\hbar$-$\varepsilon$-admissibility of our operators. The operators satisfying these stronger assumptions will be called strongly $\hbar$-$\varepsilon$-admissible operators with some regularity. As an example we could consider a symbol $a_\varepsilon (x,p)$ in $\Gamma _{\rho ,\varepsilon }^{m,\tau } ({\mathbb {R}}_x^{d}\times {\mathbb {R}}^d_p)$. For this symbol define ${\tilde{a}}_\varepsilon (x,p,y) =a_\varepsilon (tx+(1-t)y,p)$ and ask if this symbol is in $\Gamma _{\rho ,\varepsilon }^{{\tilde{m}},\tau } ({\mathbb {R}}_x^{d}\times {\mathbb {R}}^d_p\times {\mathbb {R}}^d_y)$, where ${\tilde{m}}(x,p,y)=m(tx+(1-t)y,p)$. The answer will not in general be positive. Hence, in general, we can not ensure decay in the variables (x, p, y) when viewing a function of (x, p) as a function of (x, p, y). With this in mind we define a new class of symbols and strongly $\hbar$-$\varepsilon$-admissible operators.

Definition 3.14

A symbol $a_\varepsilon$ belongs to the class ${\tilde{\Gamma }}_{\rho ,\varepsilon }^{m,\tau } ({\mathbb {R}}_x^{d}\times {\mathbb {R}}^d_p\times {\mathbb {R}}^d_y)$ if $a_\varepsilon$ is in $\Gamma _{0,\varepsilon }^{m,\tau } ({\mathbb {R}}_x^{d}\times {\mathbb {R}}^d_p\times {\mathbb {R}}^d_y)$ and there exists a positive $\nu$ such that

$$\begin{aligned} a_\varepsilon \in \Gamma _{\rho ,\varepsilon }^{m,\tau } (\Omega _\nu ), \end{aligned}$$

where $\Omega _\nu =\{(x,p,y)\in {\mathbb {R}}^{3d} \,|\, \left| x-y \right| <\nu \}$.

Definition 3.15

We call the family of operators $A_\varepsilon (\hbar )=\text {Op} _\hbar (a_\varepsilon (\hbar ))$ strongly $\hbar$-$\varepsilon$-admissible of regularity $\tau$, if $a_\varepsilon (\hbar )$ is an $\hbar$-$\varepsilon$-admissible symbol of regularity $\tau$ with respect to the weights $(m,0,\varepsilon )$ on ${\mathbb {R}}_x^{d}\times {\mathbb {R}}^d_p\times {\mathbb {R}}^d_y$ and the weights $(m,\rho ,\varepsilon )$ on $\Omega _\nu =\{(x,p,y)\in {\mathbb {R}}^{3d} \,|\, \left| x-y \right| <\nu \}$ for a positive $\nu$.

Remark 3.16

We note that a strongly $\hbar$-$\varepsilon$-admissible operator is also $\hbar$-$\varepsilon$-admissible. But as a consequence of the definition, the error term of a strongly $\hbar$-$\varepsilon$-admissible operator will be a pseudo-differential operator and not just a bounded operator as for the $\hbar$-$\varepsilon$-admissible operators.

Before, we start proving/stating results about these operators, we make the following observation.

Observation 3.17

Let m be a tempered weight function on ${\mathbb {R}}^d_x\times {\mathbb {R}}^d_p\times {\mathbb {R}}_y^d$, $\rho$ in [0, 1], $0<\varepsilon \le 1$ and $\tau$ in ${\mathbb {N}}_0$. Consider a rough symbol $a_\varepsilon$ in $\Gamma _{\rho ,\varepsilon }^{m,\tau }({\mathbb {R}}^d_x\times {\mathbb {R}}^d_p\times {\mathbb {R}}_y^d)$. We suppose that there is a $\delta$ in (0, 1) such that $\varepsilon \ge \hbar ^{1-\delta }$ and consider the operator $\text {Op} _\hbar (a_\varepsilon )$ associated with $a_\varepsilon$. We define the unitary dilation operator ${\mathcal {U}}_{\varepsilon }$ by

$$\begin{aligned} {\mathcal {U}}_{\varepsilon } f(x) = \varepsilon ^{d/2} f(\varepsilon x), \end{aligned}$$

for f in $L^2({\mathbb {R}}^d)$. Moreover we set

$$\begin{aligned} a_{\varepsilon }^{\#}(x,p,y,\hbar ) = a_\varepsilon \left( \varepsilon x, \tfrac{\hbar ^{1-\delta }}{\varepsilon } p,\varepsilon y\right) . \end{aligned}$$

We observe that with this operator and the scaled symbol $a_{\varepsilon }^{\#}$ we obtain the following equality

$$\begin{aligned} {\mathcal {U}}_{\varepsilon } \text {Op} _\hbar (a_\varepsilon (x,p,y)) {\mathcal {U}}_{\varepsilon }^{*} = \text {Op} _{\hbar ^{\delta }}(a_{\varepsilon }^{\#}(x,p,y,\hbar )). \end{aligned}$$

This equality follow by writing the kernels of the operators and perform some simple changes of variables. Since we have that $a_\varepsilon$ is in $\Gamma _{\rho ,\varepsilon }^{m,\tau }({\mathbb {R}}^d_x\times {\mathbb {R}}^d_p\times {\mathbb {R}}_y^d)$ we get for all $\alpha , \beta , \gamma$ in ${\mathbb {N}}^d_0$ that

$$\begin{aligned} \begin{aligned} |\partial _x^\alpha \partial _p^\beta \partial _y^\gamma a_{\varepsilon }^{\#}(x,p,y,\hbar ) |&\le {} C_{\alpha \beta \gamma } \varepsilon ^{\min (0,\tau -\left| \alpha \right| -\left| \gamma \right| )+\left| \alpha \right| +\left| \gamma \right| } \left( \tfrac{\hbar ^{1-\delta }}{\varepsilon }\right) ^{\left| \beta \right| } m (\varepsilon x, \tfrac{\hbar ^{1-\delta }}{\varepsilon } p,\varepsilon y) \\&\quad \times \lambda \left( \varepsilon x, \tfrac{\hbar ^{1-\delta }}{\varepsilon } p,\varepsilon y\right) ^{-\rho (\left| \alpha \right| +\left| \beta \right| +\left| \gamma \right| )} \\&\le {} C_{\alpha \beta \gamma }C_0 \varepsilon ^{\min (0,\tau -\left| \alpha \right| -\left| \gamma \right| )+\left| \alpha \right| +\left| \gamma \right| } \left( \tfrac{\hbar ^{1-\delta }}{\varepsilon }\right) ^{\left| \beta \right| } m ( x, p, y) \\&\quad \times \lambda \left( (1-\varepsilon ) x,(1- \tfrac{\hbar ^{1-\delta }}{\varepsilon }) p,(1-\varepsilon ) y\right) ^{N_0} \\&\le {} {\tilde{C}}_{\alpha \beta \gamma } {\tilde{m}} (x,p,y), \end{aligned} \end{aligned}$$

where we have used that $\hbar ^{1-\delta } \le \varepsilon \le 1$ and used the notation ${\tilde{m}} = m\lambda ^{N_0}$. This shows that $a_{\varepsilon }^{\#}$ is in $\Gamma _{0,1}^{{\tilde{m}},0}({\mathbb {R}}^d_x\times {\mathbb {R}}^d_p\times {\mathbb {R}}_y^d)$ by definition of the set. As mentioned earlier this is the type of symbols usually considered in the literature.

In what follows, we will use this observation to establish the symbolic calculus for the rough pseudo-differential operators. Here, we will mainly focus on the results we will need later in the proofs of our main theorems.

We will now prove a connection between operators with symbols in the class ${\tilde{\Gamma }}_{\rho ,\varepsilon }^{m,\tau } ({\mathbb {R}}_x^{d}\times {\mathbb {R}}^d_p\times {\mathbb {R}}^d_y)$ and t-quantised operators.

Theorem 3.18

Let $a_\varepsilon$ be a symbol in ${\tilde{\Gamma }}_{\rho ,\varepsilon }^{m,\tau } ({\mathbb {R}}_x^{d}\times {\mathbb {R}}^d_p\times {\mathbb {R}}^d_y)$ of regularity $\tau \ge 0$ with weights $(m,\rho ,\varepsilon )$ and

$$\begin{aligned} A_\varepsilon (\hbar ) \psi (x) = \frac{1}{(2\pi \hbar )^d} \int _{{\mathbb {R}}^{2d}} e^{i\hbar ^{-1}\langle x-y, p \rangle } a_\varepsilon (x,p,y) \psi (y) \, dy \,dp. \end{aligned}$$

We suppose there is a $\delta$ in (0, 1) such that $\varepsilon \ge \hbar ^{1-\delta }$. Then for every t in [0, 1] the t-quantisation of $A_{\varepsilon }(\hbar )$ is given by the unique symbol $b_{t,\varepsilon }$ of regularity $\tau$ with weights $({\tilde{m}},\rho ,\varepsilon )$, where ${\tilde{m}}(x,p)=m(x,x,p)$. The symbol $b_{t,\varepsilon }$ is defined by the oscillating integral

$$\begin{aligned} b_{t,\varepsilon }(x,p,\hbar ) = \frac{1}{(2\pi \hbar )^d} \int _{{\mathbb {R}}^{2d}} e^{i\hbar ^{-1}\langle u,q \rangle } a_\varepsilon (x+tu,p+q,x-(1-t)u) \, dq \,du. \end{aligned}$$

The symbol $b_{t,\varepsilon }$ has the following asymptotic expansion

$$\begin{aligned} b_{t,\varepsilon }(x,p;\hbar ) = \sum _{j=0}^N \hbar ^j a_{\varepsilon ,j}(x,p) + \hbar ^{N+1} r_{\varepsilon ,N+1}(x,p;\hbar ), \end{aligned}$$

where

$$\begin{aligned} a_{\varepsilon ,j} (x,p) = \frac{(-i)^j}{j!} \langle D_u, D_p\rangle ^j a_\varepsilon (x+tu, p,x-(1-t)u) \Big |_{u=0}, \end{aligned}$$

and the error term satisfies that

$$\begin{aligned} \hbar ^{N+1} |\partial _x^\alpha \partial _p^\beta r_{\varepsilon ,N+1}(x,p,\hbar ) | \le C_{d,N,\alpha ,\beta } \hbar ^{N +1} \varepsilon ^{-(\tau -N-2-d-\left| \alpha \right| )_{-}} m(x,p,x) \lambda (x,p)^{\rho N_0 }, \end{aligned}$$

for all $\alpha$ and $\beta$ in ${\mathbb {N}}^d$. In particular we have that

$$\begin{aligned} \begin{aligned} a_{\varepsilon ,0}(x,p)&= a_\varepsilon (x,p,x) \\ a_{\varepsilon ,1} (x,p)&= (1-t) (\nabla _y D_p a_\varepsilon )(x,p,x) - t (\nabla _x D_p a_\varepsilon ) (x,p,x). \end{aligned} \end{aligned}$$

Remark 3.19

It can be noted that in order for the error term not to diverge, when the semiclassical parameter tends to zero, one needs to take N such that

$$\begin{aligned} \tau -1-d + \delta (N+2+d) \ge 0. \end{aligned}$$

If the symbol is a polynomial in one of the variables or both, then the asymptotic expansion will be exact and a finite sum. This is in particular the case when “ordinary” differential operators are considered.

Proof

Let ${\mathcal {U}}_{\varepsilon }$ be the unitary dilation operator as defined in Observation 3.17 and define the operator $A_\varepsilon ^{\#}(\hbar )$ by conjugation with ${\mathcal {U}}_{\varepsilon }$. This new operator is defined by

$$\begin{aligned} A_\varepsilon ^{\#}(\hbar ) \psi (x) = \frac{1}{(2\pi \hbar ^{\delta })^d} \int _{{\mathbb {R}}^{2d}} e^{i\hbar ^{-\delta }\langle x-y, p \rangle } a_\varepsilon ^{\#}(x,p,y) \psi (y) \, dy \,dp, \end{aligned}$$

where $\psi$ is a Schwartz function. From Observation 3.17 we have that $a_\varepsilon ^{\#}$ is in ${\tilde{\Gamma }}_{0,1}^{m,0}({\mathbb {R}}^d_x\times {\mathbb {R}}^d_p\times {\mathbb {R}}_y^d)$. So by [24, Theorem II-27] the result is true for the operator $A_\varepsilon ^{\#}(\hbar )$. Due to the identity

$$\begin{aligned} A_\varepsilon (\hbar ) = {\mathcal {U}}_{\varepsilon }^{*} A_\varepsilon ^{\#}(\hbar ) {\mathcal {U}}_{\varepsilon }, \end{aligned}$$

what remains is to conjugate the terms in the representation of $A_\varepsilon ^{\#}(\hbar )$ by ${\mathcal {U}}_{\varepsilon }^{*}$. First we observe that

$$\begin{aligned} \begin{aligned}&b_{t,\varepsilon }^{\#}(\varepsilon ^{-1} x, \tfrac{\varepsilon }{\hbar ^{1-\delta }} p;\hbar ^{\delta }) \\&={} \frac{1}{(2\pi \hbar ^{\delta })^d} \int _{{\mathbb {R}}^{2d}} e^{i\hbar ^{-\delta }\langle u,q \rangle } a_\varepsilon ^{\#} (\varepsilon ^{-1} x+tu,\tfrac{\varepsilon }{\hbar ^{1-\delta }}p+q,\varepsilon ^{-1}x-(1-t)u,\hbar ) \, dq \,du \\&={} \frac{1}{(2\pi \hbar ^{\delta })^d} \int _{{\mathbb {R}}^{2d}} e^{i\hbar ^{-\delta }\langle u,q \rangle } a_\varepsilon (x+t \varepsilon u,p+ \tfrac{\hbar ^{1-\delta }}{\varepsilon }q,x-(1-t) \varepsilon u) \, dq \,du \\&={} \frac{1}{(2\pi \hbar )^d} \int _{{\mathbb {R}}^{2d}} e^{i\hbar ^{-1}\langle u,q \rangle } a_\varepsilon (x+t u,p+ q,x-(1-t) u) \, dq \,du \\&={} b_{t,\varepsilon }(x,p,\hbar ). \end{aligned} \end{aligned}$$

(3.3)

Next by using the identity obtained in (3.3) we get that

$$\begin{aligned} A_\varepsilon (\hbar ){} & {} = {\mathcal {U}}_{\varepsilon }^{*} A_\varepsilon ^{\#}(\hbar ) {\mathcal {U}}_{\varepsilon } = {\mathcal {U}}_{\varepsilon }^{*} \text {Op} _{\hbar ^{\delta },t}(b_{t,\varepsilon }^{\#}) {\mathcal {U}}_{\varepsilon } \nonumber \\{} & {} = \text {Op} _{\hbar , t }(b_{t,\varepsilon }^{\#}(\varepsilon ^{-1} x, \tfrac{\varepsilon }{\hbar ^{1-\delta }} p;\hbar ^{\delta }) ) = \text {Op} _{\hbar , t }(b_{t,\varepsilon }). \end{aligned}$$

(3.4)

From the asymptotic expansion of $b_{t,\varepsilon }^{\#}(x,p;\hbar ^\delta )$ obtained in [24, Theorem II-27] we have that

$$\begin{aligned} b_{t,\varepsilon }^{\#}(x,p;\hbar ^\delta ) = \sum _{j=0}^N (\hbar ^\delta )^j a_{\varepsilon ,j}^{\#}(x,p) + (\hbar ^\delta )^{N+1} r_{\varepsilon ,N+1}^{\#}(x,p;\hbar ^{\delta }). \end{aligned}$$

(3.5)

In order to arrive at the stated expansion of $b_t(x,p,\hbar )$ we have to find an expression of $a_{\varepsilon ,j}^{\#}(\varepsilon ^{-1}x,\tfrac{\varepsilon }{\hbar ^{1-\delta }} p,\hbar )$ in terms of $a_\varepsilon$. By definition of $a_{\varepsilon ,j}^{\#}$ we have that

$$\begin{aligned} a_{\varepsilon ,j}^{\#} (\varepsilon ^{-1}x, \tfrac{\varepsilon }{\hbar ^{1-\delta }}p,\hbar ) = \frac{(-i)^j}{j!} \hbar ^{(1-\delta ) j} \langle D_u , D_p\rangle ^j a_\varepsilon ( x+t u, p, x-(1-t)u) \Big |_{u=0}. \end{aligned}$$

(3.6)

What remains is to prove that the error term satisfies the desired estimate. The error term is given by

$$\begin{aligned} r_{\varepsilon ,N+1}(x,p;\hbar ) = r_{\varepsilon ,N+1}^{\#}(\varepsilon ^{-1}x,\tfrac{\varepsilon }{\hbar ^{1-\delta }}p;\hbar ^{\delta }). \end{aligned}$$

(3.7)

In order to see that this function satisfies the stated estimate one needs to go back to the proof of [24, Theorem II-27] and consider the exact definition of the function. But from here it is a straight forward argument to see that the desired bound is indeed true. Combining (3.3), (3.4) (3.5), (3.6) and (3.7) we obtain the desired result.

Alternatively one can also follow the proof in [24, Theorem II-27] and do the full stationary phase argument. This gives a slightly longer proof but one obtains the same results. $\square$

From this Theorem we immediately obtain the following Corollary.

Corollary 3.20

Let $t_1$ be in [0, 1] and $b_{t_1}$ be a $t_1$-$\varepsilon$-symbol of regularity $\tau \ge 0$ with weights $(m,\rho ,\varepsilon )$ and suppose $\varepsilon \ge \hbar ^{1-\delta }$ for a $\delta$ in (0, 1). Let $A_\varepsilon (\hbar )$ be the associated operator acting on a Schwarzt function by the formula

$$\begin{aligned} A_\varepsilon (\hbar ) \psi (x) = \frac{1}{(2\pi \hbar )^d} \int _{{\mathbb {R}}^{2d}} e^{i\hbar ^{-1}\langle x-y, p\rangle } b_{t_1,\varepsilon }((1-t_1)x+t_1y,p) \psi (y) \,dy \,dp. \end{aligned}$$

Then for every $t_2$ in [0, 1] we can associate an admissible $t_2$-$\varepsilon$-symbol given by the expansion

$$\begin{aligned} b_{t_2,\varepsilon }(\hbar ) = \sum _{j=0}^N \hbar ^j b_{t_2,\varepsilon ,j}+ \hbar ^{N+1} r_{\varepsilon ,N+1}(x,p;\hbar ), \end{aligned}$$

where

$$\begin{aligned} b_{t_2,\varepsilon ,j}(x,p) = \frac{( t_1-t_2)^j}{j!} (\nabla _x D_p)^j b_{t_1,\varepsilon }(x,p), \end{aligned}$$

and the error term satisfies that

$$\begin{aligned} \hbar ^{N+1} |\partial _x^\alpha \partial _p^\beta r_{\varepsilon ,N+1}(x,p,\hbar ) | \le C_{d,N,\alpha ,\beta } \hbar ^{N +1} \varepsilon ^{-(\tau -N-2-d-\left| \alpha \right| )_{-}} m(x,p) \lambda (x,p)^{\rho N_0 }, \end{aligned}$$

for all $\alpha$ and $\beta$ in ${\mathbb {N}}^d$, the number $N_0$ is the number connected to the tempered weight m.

This corollary can also be proven directly by considering the kernel as an oscillating integral and the integrant as a function in the variable $t_1$. To obtain the corollary do a Taylor expansion in $t_1$ at the point $t_2$, then perform integration by parts a number of times and then one would recover the result.

3.2 Composition of rough pseudo-differential operators

With the rough pseudo-differential operators defined and the ability to interpolate between the different quantisations, our next aim is to prove results concerning the composition of rough pseudo-differential operators. This is done in the following theorem. We will here omit the proof as it is analogous to the proof of Theorem 3.18. The idea is to conjugate the operators with ${\mathcal {U}}_\varepsilon$ use the results from e.g. [24] and then conjugate with ${\mathcal {U}}_\varepsilon ^{*}$.

Theorem 3.21

Let $A_\varepsilon (\hbar )$ and $B_\varepsilon (\hbar )$ be two t-quantised operators given by

$$\begin{aligned} A_\varepsilon (\hbar ) \psi (x) = \frac{1}{(2\pi \hbar )^d} \int _{{\mathbb {R}}^{2d}}e^{i\hbar ^{-1}\langle x-z, p\rangle } a_\varepsilon ((1-t)x+tz,p) \psi (z) \,dz \,dp \end{aligned}$$

and

$$\begin{aligned} B_\varepsilon (\hbar ) \psi (z) = \frac{1}{(2\pi \hbar )^d} \int _{{\mathbb {R}}^{2d}}e^{i\hbar ^{-1}\langle z-y, q\rangle } b_\varepsilon ((1-t)z+ty,q) \psi (y) \,dy \,dq, \end{aligned}$$

where $a_\varepsilon$ and $b_\varepsilon$ are two rough symbols of regularity $\tau _1,\tau _2 \ge 0$ with weights $(m_1, \rho , \varepsilon )$ and $(m_2, \rho , \varepsilon )$ respectively. We suppose there exists a number $\delta \in (0,1)$ such that $\varepsilon \ge \hbar ^{1-\delta }$. Then the operator $C_\varepsilon (\hbar ) = A_\varepsilon (\hbar ) \circ B_\varepsilon (\hbar )$ is strongly $\hbar$-$\varepsilon$-admissible and $C_ \varepsilon (\hbar ) = \text {Op} _{\hbar , t }(c_\varepsilon )$, where $c_\varepsilon$ is a rough admissible symbol of regularity $\tau = \min (\tau _1,\tau _2)$ with weights $(m_1m_2, \rho , \varepsilon )$. The symbol $c_\varepsilon$ satisfies the following: For every $N \ge N_\delta$ we have

$$\begin{aligned} c_\varepsilon (\hbar ) = \sum _{j=0}^N \hbar ^j c_{\varepsilon ,j} + \hbar ^{N+1} r_{\varepsilon ,N+1}(a_\varepsilon ,b_\varepsilon ;\hbar ) \end{aligned}$$

with

$$\begin{aligned} c_{\varepsilon ,j}(x,p) = \frac{(i\sigma (D_u,D_\mu ;D_v,D_\nu ))^j}{j!} [{\tilde{a}}_\varepsilon (x,p;u,v,\mu ,\nu ){\tilde{b}}_\varepsilon (x,p;u,v,\mu ,\nu )]\Big |_{\begin{array}{c} u=v=0\\ \mu =\nu =0 \end{array}}, \end{aligned}$$

where

$$\begin{aligned} \begin{aligned} \sigma (u,\mu ;v,\nu )&= \langle v,\mu \rangle - \langle u, \nu \rangle \\ {\tilde{a}}_\varepsilon (x,p;u,v,\mu ,\nu )&= a_\varepsilon (x+tv +t(1-t)u,\nu +(1-t)\mu +p) \\ {\tilde{b}}_\varepsilon (x,p;u,v,\mu ,\nu )&= b_\varepsilon (x+(1-t)v - t(1-t)u,\nu -t\mu +p). \end{aligned} \end{aligned}$$

Moreover the error term $r_{\varepsilon ,N+1}(a_\varepsilon ,b_\varepsilon ;\hbar )$ satisfies that for every multi-indices $\alpha ,\beta$ in ${\mathbb {N}}^d$ there exist a constant $C(N,\alpha ,\beta )$ independent of $a_\varepsilon$ and $b_\varepsilon$ and a natural number M such that:

$$\begin{aligned} \begin{aligned} \hbar ^{N+1}|\partial ^\alpha _x\partial ^\beta _p r_{\varepsilon ,N+1}(a_\varepsilon ,b_\varepsilon ;x,p,\hbar )|&\le {} C \varepsilon ^{-\left| \alpha \right| }\hbar ^{\delta (\tau -N-2d-2)_{-} + \tau -2d-1} {\mathcal {G}}_{M,\tau }^{\alpha ,\beta }(a_\varepsilon ,m_1,b_\varepsilon ,m_2) \\&\times m_1(x,\xi ) m_2(x,\xi ) \lambda (x,\xi )^{-\rho ({\tilde{N}}(M) + \left| \alpha \right| + \left| \beta \right| )}, \end{aligned} \end{aligned}$$

where

$$\begin{aligned} \begin{aligned} {\mathcal {G}}_{M,\tau }^{\alpha ,\beta }(a_\varepsilon ,m_1,b_\varepsilon ,m_2)&= \sup _{{\begin{array}{c} \left| \gamma _1 + \gamma _2 \right| +\left| \eta _1 + \eta _2 \right| \le M \\ (x,\xi ) \in {\mathbb {R}}^{2d} \end{array}}} \varepsilon ^{(\tau -M)_{-}+\left| \alpha \right| } \lambda (x,\xi )^{\rho (\left| \gamma _1 + \gamma _2 \right| +\left| \eta _1 + \eta _2 \right| )} \\&\qquad \times \frac{ \left| \partial _x^\alpha \partial _\xi ^\beta ( \partial _{x}^{\gamma _1} \partial _{\xi }^{\eta _1}a_\varepsilon (x,\xi ) \partial _{x}^{\gamma _2} \partial _{\xi }^{\eta _2}b_\varepsilon (x,\xi )) \right| }{m_1(x,\xi ) m_2(x,\xi )}. \end{aligned} \end{aligned}$$

The function ${\tilde{N}}(M)$ is also depending on the weights $m_1$, $m_2$ and the dimension d.

Remark 3.22

The number $N_\delta$ is explicit and it is the smallest number such that

$$\begin{aligned} \delta (N_\delta +2d + 2-\tau ) +\tau > 2d+1. \end{aligned}$$

This restriction is made in order to ensure that the error term is estimated by the semiclassical parameter raised to a positive power. The main difference between this result and the classical analog is that for this new class there is a minimum of terms in the expansion of the symbol for the composition in order to obtain an error that does not diverge as $\hbar \rightarrow 0$.

The form of the c’s we obtain in the theorem is sometimes expressed as

$$\begin{aligned} \begin{aligned} c_\varepsilon (x,\xi ;\hbar ) =&e^{i\hbar \sigma (D_u,D_\mu ;D_v,D_\nu )} \big [ a_\varepsilon (x+tv+t(1-t)u,\nu +(1- t)\mu + \xi ) \\ \hbox {}&\times b_\varepsilon (x+(1-t)v - t(1-t)u,\nu - t\mu + \xi ) \big ] \Big |_{\begin{array}{c} u=v=0 \\ \mu =\nu =0 \end{array}}. \end{aligned} \end{aligned}$$

Remark 3.23

(Particular cases of Theorem 3.21) We will see the 3 most important cases for this presentation of the composition for t-quantised operators. We suppose the assumptions of Theorem 3.21 are satisfied.

$\varvec{t=0}$: In this case the amplitude will be independent of u hence we have

$$\begin{aligned} c_\varepsilon (x,p;\hbar ) = e^{i\hbar \langle D_y,D_q\rangle } [ a_\varepsilon (x,q) b_\varepsilon (y,p) ] \Big |_{\begin{array}{c} y=x \\ p=q \end{array}}. \end{aligned}$$

This gives the formula

$$\begin{aligned} c_{\varepsilon ,j}(x,p) = \sum _{\left| \alpha \right| =j} \frac{1}{\alpha !} \partial _p^\alpha a_\varepsilon (x,p) D_x^\alpha b_\varepsilon (x,p). \end{aligned}$$

$\varvec{t=1}$: This case is similar to the one above, except a change of signs. The composition formula is given by

$$\begin{aligned} c_\varepsilon (x,p;\hbar ) = e^{-i\hbar \langle D_y,D_q\rangle } [ a_\varepsilon (y,p) b_\varepsilon (x,q) ] \Big |_{\begin{array}{c} y=x \\ p=q \end{array}}. \end{aligned}$$

This gives the formula

$$\begin{aligned} c_{\varepsilon ,j}(x,p) = (-1)^j \sum _{\left| \alpha \right| =j} \frac{1}{\alpha !} D_x^\alpha a_\varepsilon (x,p) \partial _p^\alpha b_\varepsilon (x,p). \end{aligned}$$

$\varvec{t=}\frac{{\varvec{1}}}{{\varvec{2}}}$ (Weyl-quatisation): After a bit of additional work we can arrive at the usual formula in this case as well. That is we obtain the expression

$$\begin{aligned} c_\varepsilon (x,p;\hbar ) = e^{i\frac{\hbar }{2} \sigma (D_x,D_p;D_y,D_q)} [ a_\varepsilon (x,p) b_\varepsilon (y,q) ] \Big |_{\begin{array}{c} y=x \\ p=q \end{array}} \end{aligned}$$

with

$$\begin{aligned} c_{\varepsilon ,j}(x,p) = \Big (\frac{i}{2} \Big )^j \frac{1}{j!} [\sigma (D_x,D_p;D_y,D_q)]^j a_\varepsilon (x,p) b_\varepsilon (y,q) \Big |_{\begin{array}{c} y=x \\ p=q \end{array}}. \end{aligned}$$

The last equation can be rewritten by some algebra to the classic formula

$$\begin{aligned} c_{\varepsilon ,j}(x,p) = \sum _{\left| \alpha \right| +\left| \beta \right| =j} \frac{1}{\alpha !\beta !}\Big (\frac{1}{2} \Big )^{\left| \alpha \right| }\Big (-\frac{1}{2} \Big )^{\left| \beta \right| } (\partial _p^\alpha D_x^\beta a_\varepsilon ) (\partial _p^\beta D_x^\alpha b_\varepsilon )(x,p). \end{aligned}$$

In all three cases we can note that the symbols for the compositions of operators are the same as in the non-rough case.

We now have composition of operators given by a single symbol. The next result generalises the previous to composition of strongly $\hbar$-$\varepsilon$-admissible operators. Moreover it verifies that the strongly $\hbar$-$\varepsilon$-admissible operators form an algebra. More precisely we have.

Theorem 3.24

Let $A_\varepsilon (\hbar )$ and $B_\varepsilon (\hbar )$ be two strongly $\hbar$-$\varepsilon$-admissible operators of regularity $\tau _a\ge 0$ and $\tau _b\ge 0$. with weights $(m_1, \rho , \varepsilon )$ and $(m_2, \rho , \varepsilon )$ respectively and of the form

$$\begin{aligned} A_\varepsilon (\hbar ) = \text {Op} _\hbar ^{\text {w} }(a_\varepsilon ) \quad \text {and}\quad B_\varepsilon (\hbar ) = \text {Op} _\hbar ^{\text {w} }(b_\varepsilon ). \end{aligned}$$

We suppose $\varepsilon \ge \hbar ^{1-\delta }$ for a $\delta$ in (0, 1) and let $\tau =\min (\tau _a,\tau _b)$. Then is $C_\varepsilon (\hbar )=A_\varepsilon (\hbar )\circ B_\varepsilon (\hbar )$ a strongly $\hbar$-$\varepsilon$-admissible operators of regularity $\tau \ge 0$ with weights $(m_1 m_2, \rho , \varepsilon )$. The symbol $c_\varepsilon (x,p;\hbar )$ of $C_\varepsilon (\hbar )$ has for $N\ge N_\delta$ the expansion

$$\begin{aligned} c_\varepsilon (x,p;\hbar ) = \sum _{j=0}^N \hbar ^j c_{\varepsilon ,j}(x,p) + \hbar ^{N+1} {\mathcal {R}}_\varepsilon (a_\varepsilon (\hbar ), b_\varepsilon (\hbar );\hbar ), \end{aligned}$$

where

$$\begin{aligned} c_{\varepsilon ,j}(x,p) = \sum _{\left| \alpha \right| +\left| \beta \right| +k+l=j} \frac{1}{\alpha !\beta !}\Big (\frac{1}{2} \Big )^{\left| \alpha \right| }\Big (-\frac{1}{2} \Big )^{\left| \beta \right| } (\partial _p^\alpha D_x^\beta a_{\varepsilon ,k}) (\partial _p^\beta D_x^\alpha b_{\varepsilon ,l})(x,p). \end{aligned}$$

The symbols $a_{\varepsilon ,k}$ and $b_{\varepsilon ,l}$ are from the expansion of $a_\varepsilon$ and $b_\varepsilon$ respectively. Let

$$\begin{aligned} a_\varepsilon (x,p) = \sum _{k=0}^N \hbar ^j a_{\varepsilon ,j}(x,p) + \hbar ^{N+1} r_{\varepsilon ,N+1}(a_\varepsilon ,x,p;\hbar ) \end{aligned}$$

and equvalint for $b_\varepsilon (x,p)$. Then for every multi-indices $\alpha$, $\beta$ there exists a constant $C(\alpha ,\beta ,N)$ independent of $a_\varepsilon$ and $b_\varepsilon$ and an integer M such that

$$\begin{aligned} \begin{aligned} \hbar&^{N+1} |\partial _x^\alpha \partial _p^\beta {\mathcal {R}}_\varepsilon (a_\varepsilon (\hbar ), b_\varepsilon (\hbar );x,p;\hbar ) |\\&\le C(\alpha ,\beta ,N) \hbar ^{\delta (\tau -N-2d-2)_{-}+ \tau - 2d - 1} \varepsilon ^{-\left| \alpha \right| } m_1(x,p) m_2(x,p) \lambda (x,p)^{-\rho ({\tilde{N}}(M)+\left| \alpha \right| +\left| \beta \right| )}\\&\times \left[ \sum _{j=0}^N \{ {\mathcal {G}}^{\alpha ,\beta }_{M,\tau }(a_{\varepsilon ,j},m_1, r_{\varepsilon ,N+1}(b_\varepsilon (\hbar )),m_2) + {\mathcal {G}}^{\alpha ,\beta }_{M,\tau }(r_{\varepsilon ,N+1}(a_{\varepsilon }(\hbar )),m_1,b_{\varepsilon ,j},m_2)\}\right. \\&\left. + \; \sum _{{N\le j+k \le 2N}} \; {\mathcal {G}}^{\alpha ,\beta }_{M,\tau }(a_{\varepsilon ,j},m_1,b_{\varepsilon ,k},m_2) +{\mathcal {G}}^{\alpha ,\beta }_{M,\tau }(r_{\varepsilon ,N+1}(a_{\varepsilon }(\hbar )),m_1, r_{\varepsilon ,N+1}(b_\varepsilon (\hbar )),m_2) \right] , \end{aligned} \end{aligned}$$

where

$$\begin{aligned} \begin{aligned}&{\mathcal {G}}_{M,\tau }^{\alpha ,\beta }(a_\varepsilon ,m_1,b_\varepsilon ,m_2) \\&= \sup _{{\begin{array}{c} \left| \gamma _1 + \gamma _2 \right| +\left| \eta _1 + \eta _2 \right| \le M \\ (x,\xi ) \in {\mathbb {R}}^{2d} \end{array}}} \varepsilon ^{(\tau -M)_{-}+\left| \alpha \right| } \frac{ \left| \partial _x^\alpha \partial _\xi ^\beta ( \partial _{x}^{\gamma _1} \partial _{\xi }^{\eta _1}a_\varepsilon (x,\xi ) \partial _{x}^{\gamma _2} \partial _{\xi }^{\eta _2}b_\varepsilon (x,\xi )) \right| }{m_1(x,\xi ) m_2(x,\xi )}\\&\quad \times \lambda (x,\xi )^{\rho (\left| \gamma _1 + \gamma _2 \right| +\left| \eta _1 + \eta _2 \right| )}. \end{aligned} \end{aligned}$$

The function ${\tilde{N}}(M)$ is also depending on the weights $m_1$, $m_2$ and the dimension d.

The proof of this theorem is an application of Theorem 3.21 a number of times and recalling that the error operator of a strongly $\hbar$-$\varepsilon$-admissible operator of some regularity is a quantised pseudo-differential operator.

3.3 Rough pseudo-differential operators as operators on $L^2({\mathbb {R}}^d)$

So far we have only considered operators acting on ${\mathcal {S}}({\mathbb {R}}^d)$ or ${\mathcal {S}}'({\mathbb {R}}^d)$. Hence they can be viewed as unbounded operators acting in $L^2({\mathbb {R}}^d)$ with domain ${\mathcal {S}}({\mathbb {R}}^d)$. The question is then, when is this a bounded operator? The first theorem of this section gives a criteria for when the operator can be extended to a bounded operator. This theorem is a Calderon-Vaillancourt type theorem and the proof uses the Calderon-Vaillancourt Theorem for the non-rough pseudo-differential operators. We will not recall this theorem but refer to [23, 24, 28].

Theorem 3.25

Let $a_\varepsilon$ be in $\Gamma _{0,\varepsilon }^{m,\tau }({\mathbb {R}}^d_x\times {\mathbb {R}}_p^d)$, where m is a bounded tempered weight function, $\tau \ge 0$ and suppose there exists a $\delta$ in (0, 1) such that $\varepsilon \ge \hbar ^{1-\delta }$. Then there exists a constant $C_d$ and an integer $k_d$ only depending on the dimension such that

$$\begin{aligned} \Vert \text {Op} _\hbar ^{\text {w} }(a_\varepsilon )\psi \Vert _{L^2({\mathbb {R}}^d)} \le C_d \sup _{\begin{array}{c} \left| \alpha \right| ,\left| \beta \right| \le k_d \\ (x,p)\in {\mathbb {R}}^{2d} \end{array}} \varepsilon ^{\left| \alpha \right| } \left| \partial _x^\alpha \partial _p^\beta a_\varepsilon (x,p) \right| \Vert \psi \Vert _{L^2({\mathbb {R}}^d)}, \end{aligned}$$

for all $\psi$ in ${\mathcal {S}}({\mathbb {R}}^d)$. Especially $\text {Op} _\hbar ^{\text {w} }(a_\varepsilon )$ can be extended to a bounded operator on $L^2({\mathbb {R}}^d)$.

Proof

Let ${\mathcal {U}}_\varepsilon$ be the unitary dilation operator as defined in Observation 3.17. We have that

$$\begin{aligned} \begin{aligned} \Vert \text {Op} _\hbar ^{\text {w} }(a_\varepsilon )\psi \Vert _{L^2({\mathbb {R}}^d)}&= \Vert {\mathcal {U}}_\varepsilon ^{*}{\mathcal {U}}_\varepsilon \text {Op} _\hbar ^{\text {w} }(a_\varepsilon ) {\mathcal {U}}_\varepsilon ^{*}{\mathcal {U}}_\varepsilon \psi \Vert _{L^2({\mathbb {R}}^d)} = \Vert \text {Op} _\hbar ^{\text {w} }(a_\varepsilon ^{\#}) {\mathcal {U}}_\varepsilon \psi \Vert _{L^2({\mathbb {R}}^d)} . \end{aligned} \end{aligned}$$

(3.8)

By our assumptions and Observation 3.17, we note that the symbol $a_\varepsilon ^{\#}$ satisfies the assumptions of the classical Calderon-Vaillancourt theorem. This gives us a constant $C_d$ and an integer $k_d$ only depending on the dimension such that

$$\begin{aligned} \begin{aligned} \Vert \text {Op} _\hbar ^{\text {w} }(a_\varepsilon ^{\#}) {\mathcal {U}}_\varepsilon \psi \Vert _{L^2({\mathbb {R}}^d)} \le C_d \sup _{\begin{array}{c} \left| \alpha \right| ,\left| \beta \right| \le k_d \\ (x,p)\in {\mathbb {R}}^{2d} \end{array}} \left| \partial _x^\alpha \partial _p^\beta a_\varepsilon ^{\#}(x,p,\hbar ) \right| \Vert \psi \Vert _{L^2({\mathbb {R}}^d)}, \end{aligned} \end{aligned}$$

(3.9)

where we have used the unitarity of ${\mathcal {U}}_\varepsilon$. By the definition of $a_\varepsilon ^{\#}$ we have that

$$\begin{aligned} \sup _{\begin{array}{c} \left| \alpha \right| ,\left| \beta \right| \le k_d \\ (x,p)\in {\mathbb {R}}^{2d} \end{array}} \left| \partial _x^\alpha \partial _p^\beta a_\varepsilon ^{\#}(x,p,\hbar ) \right|{} & {} ={} \sup _{\begin{array}{c} \left| \alpha \right| ,\left| \beta \right| \le k_d \\ (x,p)\in {\mathbb {R}}^{2d} \end{array}} \left| \partial _x^\alpha \partial _p^\beta a_\varepsilon (\varepsilon x, \tfrac{\hbar ^{1-\delta }}{\varepsilon } p) \right| \nonumber \\{} & {} \le {} \sup _{\begin{array}{c} \left| \alpha \right| ,\left| \beta \right| \le k_d \\ (x,p)\in {\mathbb {R}}^{2d} \end{array}} \varepsilon ^{\left| \alpha \right| } \left| \partial _x^\alpha \partial _p^\beta a_\varepsilon ( x, p) \right| . \end{aligned}$$

(3.10)

Combining (3.8), (3.9) and (3.10) we arrive at the desired result. This completes the proof. $\square$

We can now give a criteria for the rough pseudo-differential operators to be trace class. The criteria will be sufficient but not necessary. Hence it does not provide a full characteristic for the set of rough pseudo-differential operators, which are trace class.

Theorem 3.26

There exists a constant C(d) only depending on the dimension such that

$$\begin{aligned} \Vert \text {Op} _\hbar ^{\text {w} }(a_\varepsilon ) \Vert _{\text {Tr} } \le \frac{ C(d)}{\hbar ^d} \sum _{\left| \alpha \right| +\left| \beta \right| \le 2d+2} \varepsilon ^{\left| \alpha \right| } \hbar ^{\delta \left| \beta \right| } \int _{{\mathbb {R}}^{2d}} |\partial _x^\alpha \partial _p^\beta a_\varepsilon (x,p)| \,dxdp. \end{aligned}$$

for every $a_\varepsilon$ in $\Gamma _{0,\varepsilon }^{m,\tau }({\mathbb {R}}^d_x\times {\mathbb {R}}_p^d)$ with $\tau \ge 0$.

Proof

Let ${\mathcal {U}}_\varepsilon$ be the unitary dilation operator as defined in Observation 3.17. We have by the unitary invariance of the trace norm that

$$\begin{aligned} \Vert \text {Op} _\hbar ^{\text {w} }(a_\varepsilon ) \Vert _{\text {Tr} } = \Vert {\mathcal {U}}_\varepsilon \text {Op} _\hbar ^{\text {w} }(a_\varepsilon ) {\mathcal {U}}_\varepsilon ^{*} \Vert _{\text {Tr} } = \Vert \text {Op} _\hbar ^{\text {w} }(a_\varepsilon ^{\#}) \Vert _{\text {Tr} }. \end{aligned}$$

From our assumptions and Observation 3.17, we get that $a_\varepsilon ^{\#}$ satisfies the assumption for [24, Theorem II-49]. From this theorem we get the existence of a constant C(d) only depending on the dimension such that

$$\begin{aligned} \Vert \text {Op} _\hbar ^{\text {w} }(a_\varepsilon ^{\#}) \Vert _{\text {Tr} } \le \frac{ C(d)}{\hbar ^{\delta d}} \sum _{\left| \alpha \right| +\left| \beta \right| \le 2d+2} \hbar ^{\delta \left| \beta \right| } \int _{{\mathbb {R}}^{2d}} |\partial _x^\alpha \partial _p^\beta a_\varepsilon ^{\#}(x,p,\hbar )] \,dxdp. \end{aligned}$$

(3.11)

By the definition of $\partial _x^\alpha \partial _p^\beta a_\varepsilon (\varepsilon x, \tfrac{\hbar ^{1-\delta }}{\varepsilon } p)$, we have that

$$\begin{aligned} \begin{aligned} \int _{{\mathbb {R}}^{2d}} |\partial _x^\alpha \partial _p^\beta a_\varepsilon ^{\#}(x,p,\hbar )| \,dxdp&= \int _{{\mathbb {R}}^{2d}}|\partial _x^\alpha \partial _p^\beta a_\varepsilon (\varepsilon x, \tfrac{\hbar ^{1-\delta }}{\varepsilon } p)| \,dxdp \\&= \varepsilon ^{\left| \alpha \right| } \left( \tfrac{\hbar ^{1-\delta }}{\varepsilon }\right) ^{\left| \beta \right| } \int _{{\mathbb {R}}^{2d}} |[\partial _x^\alpha \partial _p^\beta a_\varepsilon ](\varepsilon x, \tfrac{\hbar ^{1-\delta }}{\varepsilon } p)| \,dxdp \\&= \hbar ^{(\delta -1)d} \varepsilon ^{\left| \alpha \right| } \left( \tfrac{\hbar ^{1-\delta }}{\varepsilon }\right) ^{\left| \beta \right| } \int _{{\mathbb {R}}^{2d}} |[\partial _x^\alpha \partial _p^\beta a_\varepsilon ](x, p)| \,dxdp. \end{aligned} \end{aligned}$$

(3.12)

Combining (3.11) and (3.12), we get that

$$\begin{aligned} \Vert \text {Op} _\hbar ^{\text {w} }(a_\varepsilon ) \Vert _{\text {Tr} } \le \frac{ C(d)}{\hbar ^d} \sum _{\left| \alpha \right| +\left| \beta \right| \le 2d+2} \varepsilon ^{\left| \alpha \right| } \hbar ^{\delta \left| \beta \right| } \int _{{\mathbb {R}}^{2d}} |\partial _x^\alpha \partial _p^\beta a_\varepsilon (x,p)| \,dxdp. \end{aligned}$$

This is the desired estimate and this concludes the proof. $\square$

The previous theorem gives us a sufficient condition for the rough pseudo-differential operators to be trace class. The next theorem gives a formula for the trace of a rough pseudo-differential operator.

Theorem 3.27

Let $a_\varepsilon$ be in $\Gamma _{0,\varepsilon }^{m,\tau }({\mathbb {R}}^d_x\times {\mathbb {R}}_p^d)$ with $\tau \ge 0$ and suppose $\partial _x^\alpha \partial _p^\beta a_\varepsilon (x,p)$ is an element of $L^1({\mathbb {R}}^d_x\times {\mathbb {R}}_p^d)$ for all $\left| \alpha \right| +\left| \beta \right| \le 2d+2$. Then $\text {Op} _\hbar ^{\text {w} }(a_\varepsilon )$ is trace class and

$$\begin{aligned} \text {Tr} (\text {Op} _\hbar ^{\text {w} }(a_\varepsilon ))=\frac{1}{(2\pi \hbar )^d} \int _{{\mathbb {R}}^{2d}} a_\varepsilon (x,p) \,dxdp. \end{aligned}$$

Proof

That our operator is trace class under the assumptions follows from Theorem 3.26. To obtain the formula for the trace let ${\mathcal {U}}_\varepsilon$ be the unitary dilation operator as defined in Observation 3.17. Then by [24, Theorem II-53] and Observation 3.17 we get that

$$\begin{aligned} \begin{aligned} \text {Tr} (\text {Op} _\hbar ^{\text {w} }(a_\varepsilon ))&= \text {Tr} ({\mathcal {U}}_\varepsilon \text {Op} _\hbar ^{\text {w} }(a_\varepsilon ){\mathcal {U}}_\varepsilon ^{*}) = \text {Tr} (\text {Op} _\hbar ^{\text {w} }(a_\varepsilon ^{\#})) \\&= \frac{1}{(2\pi \hbar ^\delta )^d} \int _{{\mathbb {R}}^{2d}} a_\varepsilon (\varepsilon x, \tfrac{\hbar ^{1-\delta }}{\varepsilon } p) \,dxdp = \frac{1}{(2\pi \hbar )^d} \int _{{\mathbb {R}}^{2d}} a_\varepsilon (x,p) \,dxdp. \end{aligned} \end{aligned}$$

This identity concludes the proof. $\square$

The last result in this section is a sharp Gårdinger inequality, which we will need later.

Theorem 3.28

Let $a_\varepsilon$ be a bounded rough symbol of regularity $\tau \ge 0$ which satisfies

$$\begin{aligned} a_\varepsilon (x,p) \ge 0 \quad \text {for all } (x,p) \in {\mathbb {R}}_x^d\times {\mathbb {R}}^d_p, \end{aligned}$$

and suppose there exist $\delta \in (0,1)$ such that $\varepsilon >\hbar ^{1-\delta }$. Then there exists a $C_0>0$ and $\hbar _0>0$ such that

$$\begin{aligned} \langle \text {Op} _\hbar ^{\text {w} }(a_\varepsilon ) g,g \rangle \ge -\hbar ^{\delta } C \Vert g \Vert _{L^2({\mathbb {R}}^d)}, \end{aligned}$$

for all g in $L^2({\mathbb {R}}^d)$ and $\hbar$ in $(0,\hbar _0]$.

Proof

Again let ${\mathcal {U}}_\varepsilon$ be the unitary dilation operator as defined in Observation 3.17. After conjugation with this operator, we will be able to use the “usual” semiclassical sharp Gårdinger inequality (see e.g. [23, Theorem 4.32]) with the semiclassical parameter $\hbar ^\delta$. We have that

$$\begin{aligned} \langle \text {Op} _\hbar ^{\text {w} }(a_\varepsilon ) g,g \rangle{} & {} = \langle {\mathcal {U}}_\varepsilon \text {Op} _\hbar ^{\text {w} }(a_\varepsilon ){\mathcal {U}}_\varepsilon ^{*}{\mathcal {U}}_\varepsilon g,{\mathcal {U}}_\varepsilon g \rangle = \langle \text {Op} _\hbar ^{\text {w} }(a_\varepsilon ^{\#}){\mathcal {U}}_\varepsilon g,{\mathcal {U}}_\varepsilon g \rangle \nonumber \\{} & {} \ge - C_0 \hbar ^{\delta } \Vert {\mathcal {U}}_\varepsilon g \Vert _{L^2({\mathbb {R}}^d)} = - C_0 \hbar ^{\delta } \Vert g \Vert _{L^2({\mathbb {R}}^d)}. \end{aligned}$$

(3.13)

The existence of the numbers $C_0$ and $\hbar _0$ is ensured by the “usual” semiclassical sharp Gårdinger inequality. This is the desired estimate and this ends the proof. $\square$

4 Self-adjointness and functional calculus for rough pseudo-differential operators

In this section we will establish a functional calculus for rough pseudo-differential operators. For this to be well defined we will also give a set of assumptions that ensure essential self-adjointness of the operators. We could as in the previous section use unitary conjugations to obtain non-rough operators and use the results for this type of operators. We have chosen not to do this, since in the construction we will need to control polynomials in the derivatives of our symbols. So, to make it transparent that the regularity of these polynomials is as desired we have chosen to do the full construction. The construction is based on the results of Helffer–Robert in [4] and this approach is also described in [24]. The method they used is based on the Mellin transform, where we will use the Helffer-Sjöstrand formula instead. The construction of a functional calculus using this formula can be found in the monographs [23, 28]. There is also a construction of the functional calculus using Fourier theory in [10], but we have not tried to adapt this to the case studied here.

4.1 Essential self-adjointness of rough pseudo-differential operators

First we will give criteria for the operator to be lower semi-bounded and essential self-adjoint.

Assumption 4.1

Let $A_\varepsilon (\hbar )$ be a $\hbar$-$\varepsilon$-admissible operator of regularity $\tau$ and suppose that

• 1. labelB.H.1 $A_\varepsilon (\hbar )$ is symmetric on ${\mathcal {S}}({\mathbb {R}}^n)$ for all $\hbar$ in $]0,\hbar _0]$.

• 2. The principal symbol $a_{\varepsilon ,0}$ satisfies that

  $$\begin{aligned} \min _{(x,p) \in {\mathbb {R}}^{2n}} a_{\varepsilon ,0}(x,p) = \zeta _0 > -\infty . \end{aligned}$$

• 3. Let $\zeta _1 < \zeta _0$ and $\zeta _1 \le 0$. Then $a_{\varepsilon ,0} - \zeta _1$ is a tempered weight function with constants independent of $\varepsilon$ and

  $$\begin{aligned} a_{\varepsilon ,j} \in \Gamma _{0,\varepsilon }^{a_{\varepsilon ,0} - \zeta _1,\tau -j} \left( {\mathbb {R}}_x^{d}\times {\mathbb {R}}^d_p\right) , \end{aligned}$$

  for all j in ${\mathbb {N}}$.

Theorem 4.2

Let $A_\varepsilon (\hbar )$, for $\hbar$ in $(0,\hbar _0]$, be a $\hbar$-$\varepsilon$-admissible operator of regularity $\tau \ge 1$ with tempered weight m and symbol

$$\begin{aligned} a_\varepsilon (\hbar ) = \sum _{j\ge 0} \hbar ^j a_{\varepsilon ,j}. \end{aligned}$$

Suppose that $A_\varepsilon (h)$ satisfies Assumption 4.1. Then there exists $\hbar _1$ in $(0,\hbar _0]$ such that for all $\hbar$ in $(0,\hbar _1]$ $A_\varepsilon (\hbar )$ is essential self-adjoint and lower semi-bounded.

Proof

We let $t<\zeta _0$, where $\zeta _0$ is the number from Assumption 4.1. For this t we define the symbol

$$\begin{aligned} b_{\varepsilon ,t}(x,p) = \frac{1}{a_{\varepsilon ,0}(x,p) -t}. \end{aligned}$$

By assumption we have that $b_{\varepsilon ,t} \in \Gamma _{0,\varepsilon }^{(a_{\varepsilon ,0} - \gamma _1)^{-1},\tau }({\mathbb {R}}_x^d\times {\mathbb {R}}^d_p)$. For N sufficiently large we get by assumption that

$$\begin{aligned} \begin{aligned} (A_\varepsilon (\hbar ) - t) \text {Op} _\hbar ^{\text {w} }(b_{\varepsilon ,t})&={} \Big [ \text {Op} _\hbar ^{\text {w} }( a_{\varepsilon ,j} -t) + \sum _{j=1}^N \hbar ^j \text {Op} _\hbar ^{\text {w} }( a_{\varepsilon ,j}) + \hbar ^{N+1} R_N(\hbar ) \Big ]\text {Op} _\hbar ^{\text {w} }(b_{\varepsilon ,t}) \\&={} I + \hbar S_N(\varepsilon ,\hbar ). \end{aligned} \end{aligned}$$

The formula for composition of operators and the Calderon-Vaillancourt theorem give us that the operator $S_N$ satisfies the estimate

$$\begin{aligned} \sup _{\hbar \in (0,\hbar _0]} \Vert S_N(\varepsilon ,\hbar ) \Vert _{{\mathcal {L}}(L^2({\mathbb {R}}^d))}<\infty . \end{aligned}$$

(4.1)

We note that if $\hbar$ is chosen such that $\hbar \Vert S_N(\varepsilon ,\hbar ) \Vert _{{\mathcal {L}}(L^2({\mathbb {R}}^d))}<1$ then the operator $I + \hbar S_N(\varepsilon ,\hbar )$ will be invertible.

The Calderon-Vaillancourt theorem gives us that $\text {Op} _\hbar ^{\text {w} }(b_{\varepsilon ,t})$ is a bounded operator. This implies that the expression $\text {Op} _\hbar ^{\text {w} }(b_{\varepsilon ,t}) ( I + \hbar S_N(\varepsilon ,\hbar ))^{-1}$ is a well defined bounded operator. Hence we have that the operator $(A_\varepsilon (\hbar ) - t)$ maps its domain surjective onto all of $L^2({\mathbb {R}}^d)$. By [30, Proposition 3.11] this implies that $A_\varepsilon (\hbar )$ is essential self-adjoint.

Since we have for all $t<\zeta _0$ that $(A_\varepsilon (\hbar ) - t)$ maps its domain surjective onto all of $L^2({\mathbb {R}}^d)$ they are all in the resolvent set and hence the operator has to be lower semi-bounded. $\square$

4.2 The resolvent of a rough pseudo-differential operator

A main part in the construction of the functional calculus is to prove that the resolvent of a $\hbar$-$\varepsilon$-admissible operator of regularity $\tau$, which satisfies Assumption 4.1, is an operator of the same type. This is the content of the following Theorem.

Theorem 4.3

Let $A(\hbar )$, for $\hbar$ in $(0,\hbar _0]$, be a $\hbar$-$\varepsilon$-admissible operator of regularity $\tau \ge 1$ with tempered weight m and symbol

$$\begin{aligned} a_\varepsilon (\hbar ) = \sum _{j\ge 0} \hbar ^j a_{\varepsilon ,j}. \end{aligned}$$

Suppose that $A_\varepsilon (h)$ satisfies Assumption 4.1 with the numbers $\zeta _0$ and $\zeta _1$. For z in ${\mathbb {C}}{\setminus }[\zeta _1,\infty )$ we define the sequence of symbols

$$\begin{aligned} \begin{aligned} b_{\varepsilon ,z,0}&= (a_{\varepsilon ,0} - z)^{-1} \\ b_{\varepsilon ,z,j+1}&= - b_{\varepsilon ,z,0} \cdot \sum _{\begin{array}{c} l + \left| \alpha \right| + \left| \beta \right| + k = j+1\\ 0\le l \le j \end{array}} \frac{1}{\alpha ! \beta !} \frac{1}{2^{\left| \alpha \right| }} \frac{1}{(-2)^{\left| \beta \right| }} (\partial _p^{\alpha } D_x^{\beta } a_{\varepsilon ,k}) (\partial _p^{\beta } D_x^{\alpha } b_{\varepsilon ,z,l}), \end{aligned} \end{aligned}$$

(4.2)

for $j\ge 1$. Moreover we define

$$\begin{aligned} B_{\varepsilon ,z,M}(\hbar ) = \sum _{j=0}^M \hbar ^j b_{\varepsilon ,z,j}. \end{aligned}$$

Then for N in ${\mathbb {N}}$ we have that

$$\begin{aligned} (A_\varepsilon (h) - z ) \text {Op} _\hbar ^{\text {w} }B_{\varepsilon ,z,N} = I + h^{N+1} \Delta _{\varepsilon ,z,N+1}(h), \end{aligned}$$

(4.3)

with

$$\begin{aligned} \hbar ^{N+1}\Vert \Delta _{\varepsilon ,z,N+1}(h) \Vert _{{\mathcal {L}}(L^2({\mathbb {R}}^d))} \le C \hbar ^{\kappa (N)} \left( \frac{\left| z \right| }{\text {dist} (z,[\zeta _1,\infty ))} \right) ^{q(N)}, \end{aligned}$$

(4.4)

where $\kappa$ is a positive strictly increasing function and q(N) is a positive integer depending on N. In particular we have for all z in ${\mathbb {C}}{\setminus }[\zeta _1,\infty )$ and all $\hbar$ in $(0,\hbar _1]$ ($\hbar _1$ sufficient small and independent of z), that $(A_\varepsilon (h) - z)^{-1}$ is a $\hbar$-$\varepsilon$-admissible operator with respect to the tempered weight $(a_{\varepsilon ,0}-\zeta _1)^{-1}$ and of regularity $\tau$ with symbol:

$$\begin{aligned} B_{\varepsilon ,z}(\hbar ) = \sum _{j\ge 0} \hbar ^j b_{\varepsilon ,z,j}. \end{aligned}$$

(4.5)

Before we can prove the theorem we will need some lemma’s with the same setting. It is in these lemma’s we will see, that the symbol for the resolvent has the same regularity. From these lemmas we can also find the explicit formulas for every symbol in the expansion.

Lemma 4.4

Let the setting be as in Theorem 4.3. For every j in ${\mathbb {N}}$ we have

$$\begin{aligned} b_{\varepsilon ,z,j} = \sum _{k=1}^{2j-1} d_{\varepsilon , j ,k} b_{\varepsilon ,z,0}^{k+1}, \end{aligned}$$

(4.6)

where $d_{\varepsilon , j,k}$ are universal polynomials in $\partial _p^\alpha \partial _x^\beta a_{\varepsilon ,l}$ for $\left| \alpha \right| +\left| \beta \right| +l\le j$ and $d_{\varepsilon , j,k} \in \Gamma _{0,\varepsilon }^{(a_0-\zeta _1)^k,\tau -j}$ for all k, $1\le k \le 2j-1$. In particular we have that

$$\begin{aligned} b_{\varepsilon ,z,1}=-a_{\varepsilon ,1} b_{\varepsilon ,z,0}^2. \end{aligned}$$

In order to prove this Lemma we will need the following Lemma:

Lemma 4.5

Let the setting be as in Lemma 4.4. For any j and k in ${\mathbb {N}}$ we let $d_{\varepsilon , j,k} b_{\varepsilon ,z,0}^{k+1}$ be one of the elements in the expansion of $b_{\varepsilon ,z,j}$. Then for all multi-indices $\alpha$ and $\beta$ it holds that

$$\begin{aligned} \partial _p^\beta \partial _x^\alpha d_{\varepsilon , j,k} b_{\varepsilon ,z,0}^{k+1} = \sum _{n=0}^{\left| \alpha \right| +\left| \beta \right| } {\tilde{d}}_{\varepsilon , j,k, n, \alpha , \beta } b_{\varepsilon ,z,0}^{k+1+n}, \end{aligned}$$

where ${\tilde{d}}_{\varepsilon , j,k, n, \alpha , \beta }$ are polynomials in $\partial _x^{\alpha '} \partial _p^{\beta '} a_{\varepsilon ,k}$ with $\left| \alpha ' \right| +\left| \beta ' \right| +k \le j +\left| \alpha \right| +\left| \beta \right|$ of degree at most $k+n$. They are of regularity at least $\tau -j-\left| \alpha \right|$, and they are determined only by $\alpha$, $\beta$, $a_{\varepsilon ,0}$ and $d_{\varepsilon , j,k}$.

Proof

The proof is an application of Theorem A.1 (Faà di Bruno formula) and the Corollary A.2 to the formula. For our $\alpha ,\beta$ we have by the Leibniz’s formula that:

$$\begin{aligned} \begin{aligned} \partial _p^\beta \partial _x^\alpha d_{\varepsilon , j,k} b_{\varepsilon ,z,0}^{k+1}&={} \partial _p^\beta \big \{ (\partial _x^{\alpha } d_{\varepsilon , j,k}) b_{\varepsilon ,z,0}^{k+1} + \sum _{\begin{array}{c} \left| \gamma \right| =1\\ \gamma \le \alpha \end{array}}^{\left| \alpha \right| } \left( {\begin{array}{c}\alpha \\ \gamma \end{array}}\right) \partial _x^{\alpha -\gamma } d_{\varepsilon , j,k} \partial _x^\gamma b_{\varepsilon ,z,0}^{k+1} \big \} \\&={} (\partial _p^\beta \partial _x^{\alpha } d_{\varepsilon , j,k})b_{\varepsilon ,z,0}^{k+1} + \sum _{\begin{array}{c} \left| \eta \right| =1\\ \eta \le \beta \end{array}}^{\left| \beta \right| }\left( {\begin{array}{c}\beta \\ \eta \end{array}}\right) \partial _p^{\beta -\eta } \partial _x^{\alpha } d_{\varepsilon , j,k} \partial _p^\eta b_{\varepsilon ,z,0}^{k+1} \\&+ \sum _{\begin{array}{c} \left| \gamma \right| =1\\ \gamma \le \alpha \end{array}}^{\left| \alpha \right| } \left( {\begin{array}{c}\alpha \\ \gamma \end{array}}\right) (\partial _p^\beta \partial _x^{\alpha -\gamma } d_{\varepsilon , j,k}) \partial _x^\gamma b_{\varepsilon ,z,0}^{k+1} \\&+ \sum _{\begin{array}{c} \left| \eta \right| =1\\ \eta \le \beta \end{array}}^{\left| \beta \right| } \sum _{\begin{array}{c} \left| \gamma \right| =1\\ \gamma \le \alpha \end{array}}^{\left| \alpha \right| } \left( {\begin{array}{c}\beta \\ \eta \end{array}}\right) \left( {\begin{array}{c}\alpha \\ \gamma \end{array}}\right) (\partial _p^{\beta -\eta }\partial _x^{\alpha -\gamma } d_{\varepsilon , j,k}) \partial _p^\eta \partial _x^\gamma b_{\varepsilon ,z,0}^{k+1}. \end{aligned} \end{aligned}$$

We will here consider each of the three sums separately for the first we get by the Faà di Bruno formula (Theorem A.1)

$$\begin{aligned} \begin{aligned} \sum _{\begin{array}{c} \left| \eta \right| =1\\ \eta \le \beta \end{array}}^{\left| \beta \right| } \left( {\begin{array}{c}\beta \\ \eta \end{array}}\right) \partial _p^{\beta -\eta } \partial _x^{\alpha } d_{\varepsilon , j,k} \partial _p^\eta b_{\varepsilon ,z,0}^{k+1}&={} \sum _{\begin{array}{c} \left| \eta \right| =1\\ \eta \le \beta \end{array}}^{\left| \beta \right| }\left( {\begin{array}{c}\beta \\ \eta \end{array}}\right) \partial _p^{\beta -\eta } \partial _x^{\alpha } d_{\varepsilon , j,k} \sum _{{n}=1}^{\left| \eta \right| } (-1)^{n} \frac{(k+n)!}{k!} b_{\varepsilon ,z,0}^{k+1+ n} \\&\quad \times \sum _{{\begin{array}{c} \eta _1+\cdots +\eta _n=\eta \\ \left| \eta _i \right|>0 \end{array}}} C_{\eta _1,\dots ,\eta _n} \partial _p^{\eta _1}a_{\varepsilon ,0} \cdots \partial _p^{\eta _n}a_{\varepsilon ,0} \\&= \sum _{n_\beta =1}^{\left| \beta \right| } \Big \{ \sum _{\begin{array}{c} \left| \eta \right| \ge n_\beta \\ \eta \le \beta \end{array}}^{\left| \beta \right| } C_{k,n_\beta ,\beta ,\eta } \partial _p^{\beta -\eta } \partial _x^{\alpha } d_{\varepsilon , j,k} \\&\quad \times \sum _{\begin{array}{c} \eta _1+\cdots +\eta _{n_\beta }=\eta \\ \left| \eta _i \right| >0 \end{array}} C_{\eta _1,\dots ,\eta _n} \partial _p^{\eta _1}a_{\varepsilon ,0} \cdots \partial _p^{\eta _{n_\beta }}a_{\varepsilon ,0} \Big \}b_{\varepsilon ,z,0}^{k+1+ n_\beta } \\&= \sum _{n_\beta =1}^{\left| \beta \right| } {\tilde{d}}_{\varepsilon ,j,k,\alpha ,\beta ,n_\beta } b_{\varepsilon ,z,0}^{k+1+ n_\beta }, \end{aligned} \end{aligned}$$

where C’s with some indices are constants depending on those indices. This calculation gives that we have a polynomial structure, where the coefficients are the polynomials ${\tilde{d}}_{\varepsilon ,j,k,\alpha ,\beta ,n_\beta }$, which themselves are polynomials in $\partial _x^{\alpha '} \partial _p^{\beta '} a_{\varepsilon ,k}$ with $\left| \alpha ' \right| +\left| \beta ' \right| +k \le j +\left| \alpha \right| +\left| \beta \right|$ and of regularity $\tau -j-\left| \alpha \right|$. For the second sum we again use Faà di Bruno formula (Theorem A.1) and get

$$\begin{aligned} \sum _{\begin{array}{c} \left| \gamma \right| =1\\ \gamma \le \alpha \end{array}}^{\left| \alpha \right| } \left( {\begin{array}{c}\alpha \\ \gamma \end{array}}\right) (\partial _p^\beta \partial _x^{\alpha -\gamma } d_{\varepsilon , j,k}) \partial _x^\gamma b_{\varepsilon ,z,0}^{k+1}{} & {} ={} \sum _{\begin{array}{c} \left| \gamma \right| =1\\ \gamma \le \alpha \end{array}}^{\left| \alpha \right| }\left( {\begin{array}{c}\alpha \\ \gamma \end{array}}\right) \partial _p^{\beta } \partial _x^{\alpha -\gamma } d_{\varepsilon , j,k} \sum _{{n}=1}^{\left| \gamma \right| } (-1)^{n} \frac{(k+n)!}{k!} b_{\varepsilon ,z,0}^{k+1+ n} \\{} & {} \quad \times \sum _{{\begin{array}{c} \gamma _1+\cdots +\gamma _n=\gamma \\ \left| \gamma _i \right|>0 \end{array}}} C_{\gamma _1,\dots ,\gamma _n} \partial _x^{\gamma _1}a_{\varepsilon ,0} \cdots \partial _x^{\gamma _n}a_{\varepsilon ,0} \\{} & {} = \sum _{n_\alpha =1}^{\left| \alpha \right| } b_{\varepsilon ,z,0}^{k+1+ n_\alpha } \Big \{ \sum _{\begin{array}{c} \left| \gamma \right| \ge n_\alpha \\ \gamma \le \alpha \end{array}}^{\left| \alpha \right| } C_{k,n_\alpha ,\alpha ,\gamma } \partial _p^{\beta } \partial _x^{\alpha -\gamma } d_{\varepsilon , j,k} \\{} & {} \quad \times \sum _{{\begin{array}{c} \gamma _1+\cdots +\gamma _{n_\alpha }=\gamma \\ \left| \gamma _i \right| >0 \end{array}}} C_{\gamma _1,\dots ,\gamma _n} \partial _x^{\gamma _1}a_{\varepsilon ,0} \cdots \partial _x^{\gamma _{n_\alpha }}a_{\varepsilon ,0} \Big \} \\{} & {} = \sum _{n_\alpha =1}^{\left| \alpha \right| } {\tilde{d}}_{\varepsilon ,j,k,\alpha ,\beta ,n_\alpha } b_{\varepsilon ,z,0}^{k+1+ n_\alpha }, \end{aligned}$$

where again C’s with some indices are constants depending on those indices. We have again the polynomial structure, where the coefficients ${\tilde{d}}_{\varepsilon ,j,k,\alpha ,\beta ,n_\alpha }$ are polynomials in $\partial _x^{\alpha '} \partial _p^{\beta '} a_{\varepsilon ,k}$ with $\left| \alpha ' \right| +\left| \beta ' \right| +k \le j +\left| \alpha \right| +\left| \beta \right|$, and they are of at least regularity $\tau -j-\left| \alpha \right|$.

For the last sum we need a slightly modified version of the Faà di Bruno formula, which is Corollary A.2. If we use this, we get that

$$\begin{aligned} \begin{aligned}&\sum _{\begin{array}{c} \left| \eta \right| =1\\ \eta \le \beta \end{array}}^{\left| \beta \right| } \sum _{\begin{array}{c} \left| \gamma \right| =1\\ \gamma \le \alpha \end{array}}^{\left| \alpha \right| } \left( {\begin{array}{c}\beta \\ \eta \end{array}}\right) \left( {\begin{array}{c}\alpha \\ \gamma \end{array}}\right) (\partial _p^{\beta -\eta }\partial _x^{\alpha -\gamma } d_{\varepsilon , j,k}) \partial _p^\eta \partial _x^\gamma b_{\varepsilon ,z,0}^{k+1} \\&={} \sum _{\begin{array}{c} \left| \eta \right| =1\\ \eta \le \beta \end{array}}^{\left| \beta \right| } \sum _{\begin{array}{c} \left| \gamma \right| =1\\ \gamma \le \alpha \end{array}}^{\left| \alpha \right| } \left( {\begin{array}{c}\beta \\ \eta \end{array}}\right) \left( {\begin{array}{c}\alpha \\ \gamma \end{array}}\right) (\partial _p^{\beta -\eta }\partial _x^{\alpha -\gamma } d_{\varepsilon , j,k}) \sum _{n=1}^{\left| \eta \right| +\left| \gamma \right| } C_n b_{\varepsilon ,z,0}^{k+1+n} \\&\quad \times \sum _{{\mathcal {I}}_n(\gamma ,\eta )}C_{\gamma _1\cdots \gamma _k}^{\eta _1\cdots \eta _k} \partial _p^{\eta _1} \partial _x^{\gamma _1} a_{\varepsilon ,0} \cdots \partial _p^{\eta _n} \partial _x^{\gamma _n} a_{\varepsilon ,0} \\&= \sum _{n=1}^{\left| \alpha \right| +\left| \beta \right| } b_{\varepsilon ,z,0}^{k+1+n} \\&\quad \times \left\{ \sum _{\begin{array}{c} \left| \eta \right| + \left| \gamma \right| \ge n \\ \eta \le \beta ,\ \gamma \le \alpha \end{array}}^{\left| \beta \right| +\left| \alpha \right| } \sum _{{\mathcal {I}}_n(\gamma ,\eta )} C_{k,n,\alpha ,\beta ,\gamma ,\eta } (\partial _p^{\beta -\eta }\partial _x^{\alpha -\gamma } d_{\varepsilon , j,k}) \partial _p^{\eta _1} \partial _x^{\gamma _1} a_{\varepsilon ,0} \cdots \partial _p^{\eta _n} \partial _x^{\gamma _n} a_{\varepsilon ,0} \right\} \\&={} \sum _{n=1}^{\left| \alpha \right| +\left| \beta \right| } {\tilde{d}}_{\varepsilon ,j,k,n,\alpha ,\beta } b_{\varepsilon ,z,0}^{k+1+n}, \end{aligned} \end{aligned}$$

where ${\tilde{d}}_{\varepsilon ,j,k,n,\alpha ,\beta }$ are polynomials in $\partial _x^{\alpha '} \partial _p^{\beta '} a_{\varepsilon ,k}$ with $\left| \alpha ' \right| +\left| \beta ' \right| +k \le j +\left| \alpha \right| +\left| \beta \right|$ of degree at most $k+n$, and they are of regularity at least $\tau -j-\left| \alpha \right|$. If we combine all of the above calculations we get the desired result:

$$\begin{aligned} \partial _p^\beta \partial _x^\alpha d_{\varepsilon , j,k} b_{\varepsilon ,z,0}^{k+1} = \sum _{n=0}^{\left| \alpha \right| +\left| \beta \right| } {\tilde{d}}_{\varepsilon , j,k, n, \alpha , \beta } b_{\varepsilon ,z,0}^{k+1+n}, \end{aligned}$$

where ${\tilde{d}}_{\varepsilon , j,k, n, \alpha , \beta }$ are polynomials in $\partial _x^{\alpha '} \partial _p^{\beta '} a_{\varepsilon ,k}$ with $\left| \alpha ' \right| +\left| \beta ' \right| +k \le j +\left| \alpha \right| +\left| \beta \right|$ of degree at most $k+n$, and they are of regularity at least $\tau -j-\left| \alpha \right|$. The form of the polynomials is entirely determined by the multi-indices $\alpha ,\beta$, the symbol $a_{\varepsilon ,0}$ and the polynomial $d_{\varepsilon , j,k}$. $\square$

Proof of Lemma 4.4

The proof will be induction in the parameter j. We start by considering the case $j=1$, where we by definition of $b_{\varepsilon ,z,1}$ have

$$\begin{aligned} \begin{aligned} b_{\varepsilon ,z,1}&= - b_{\varepsilon ,z,0} \cdot \sum _{ \left| \alpha \right| + \left| \beta \right| + k = 1} \frac{1}{\alpha ! \beta !} \frac{1}{2^{\left| \alpha \right| }} \frac{1}{(-2)^{\left| \beta \right| }} (\partial _p^{\alpha } D_x^{\beta } a_{\varepsilon ,k}) (\partial _p^{\beta } D_x^{\alpha } b_{\varepsilon ,z,0}) \\&= - b_{\varepsilon ,z,0} \big ( a_{\varepsilon ,1}b_{\varepsilon ,z,0} -\frac{i}{2} \sum _{n=1}^d \partial _{p_n} a_{\varepsilon ,0} \partial _{x_n} a_{\varepsilon ,0}b_{\varepsilon ,z,0}^2 - \partial _{x_n} a_{\varepsilon ,0}\partial _{p_n} a_{\varepsilon ,0} b_{\varepsilon ,z,0}^2 \big ) \\&= - a_{\varepsilon ,1}b_{\varepsilon ,z,0}^2. \end{aligned} \end{aligned}$$

This calculation verifies the form of $b_{\varepsilon ,z,1}$ stated in the lemma. Moreover, it varifies that $b_{\varepsilon ,z,1}$ has the form given by (4.6) with $d_{\varepsilon ,1,1}=-a_{\varepsilon ,1}$, which is in the symbol class $\Gamma _{0,\varepsilon }^{(a_0-\zeta _1),\tau -1}$ by assumption.

Assume the lemma to be correct for $b_{\varepsilon ,z,j}$ and consider $b_{\varepsilon ,z,j+1}$. By the definition of $b_{\varepsilon ,z,j+1}$ and our assumption we have

$$\begin{aligned} \begin{aligned} b_{\varepsilon ,z,j+1}&={} - b_{\varepsilon ,z,0} \cdot \sum _{\begin{array}{c} l + \left| \alpha \right| + \left| \beta \right| + k = j+1\\ 0\le l \le j \end{array}} \frac{1}{\alpha ! \beta !} \frac{1}{2^{\left| \alpha \right| }} \frac{1}{(-2)^{\left| \beta \right| }} (\partial _p^{\alpha } D_x^{\beta } a_{\varepsilon ,k}) (\partial _p^{\beta } D_x^{\alpha } b_{\varepsilon ,z,l}) \\&={} - b_{\varepsilon ,z,0} \left\{ \sum _{\left| \alpha \right| + \left| \beta \right| + k = j+1} C_{\alpha ,\beta } (\partial _p^{\alpha } D_x^{\beta } a_{\varepsilon ,k}) (\partial _p^{\beta } D_x^{\alpha } b_{\varepsilon ,z,0}) \right. \\ \hbox {}&\left. \qquad + \sum _{l=1}^j \sum _{m=1}^{2l-1} \sum _{l+\left| \alpha \right| + \left| \beta \right| + k = j+1} C_{\alpha ,\beta } (\partial _p^{\alpha } D_x^{\beta } a_{\varepsilon ,k}) (\partial _p^{\beta } D_x^{\alpha }d_{\varepsilon , l,m} b_{\varepsilon ,z,0}^{m+1})\right\} , \end{aligned} \end{aligned}$$

where $C_{\alpha ,\beta }$ is some constant depending on $\alpha$ and $\beta$. We will consider each of the sums separately. To calculate the first sum, we get by applying Corollary A.2 that

$$\begin{aligned} \begin{aligned}&\sum _{\left| \alpha \right| + \left| \beta \right| + k = j+1} C_{\alpha ,\beta } (\partial _p^{\alpha } D_x^{\beta } a_{\varepsilon ,k}) (\partial _p^{\beta } D_x^{\alpha } b_{\varepsilon ,z,0}) - C_{\alpha ,\beta } a_{\varepsilon ,j+1} b_{\varepsilon ,z,0} \\&={} \sum _{\begin{array}{c} \left| \alpha \right| + \left| \beta \right| + k = j+1 \\ \left| \alpha \right| + \left| \beta \right| \ge 1 \end{array}} \left\{ C_{\alpha ,\beta } (\partial _p^{\alpha } D_x^{\beta } a_{\varepsilon ,k}) \sum _{n=1}^{\left| \alpha \right| +\left| \beta \right| } C_n b_{\varepsilon ,z,0}^{n+1} \right. \\&\left. \quad \times \sum _{{\mathcal {I}}_n(\alpha ,\beta )}C_{\alpha _1\cdots \alpha _n}^{\beta _1\cdots \beta _n} \partial _p^{\beta _1} D_x^{\alpha _1} a_{\varepsilon ,0} \cdots \partial _p^{\beta _n} D_x^{\alpha _n} a_{\varepsilon ,0}\right\} \\&= \sum _{n=1}^{j+1} \left\{ \sum _{\begin{array}{c} \left| \alpha \right| + \left| \beta \right| + k = j+1 \\ \left| \alpha \right| + \left| \beta \right| \ge n \end{array}} \sum _{{\mathcal {I}}_n(\alpha ,\beta )} C_{\alpha ,\beta ,n} (\partial _p^{\alpha } D_x^{\beta } a_{\varepsilon ,k}) \partial _p^{\beta _1} D_x^{\alpha _1} a_{\varepsilon ,0} \cdots \partial _p^{\beta _n} D_x^{\alpha _n} a_{\varepsilon ,0} \right\} b_{\varepsilon ,z,0}^{n+1} \\&= \sum _{n=1}^{j+1} {\tilde{d}}_{\varepsilon ,j,n,\alpha ,\beta } b_{\varepsilon ,z,0}^{n+1} - C_{\alpha ,\beta } a_{\varepsilon ,j+1} b_{\varepsilon ,z,0}, \end{aligned} \end{aligned}$$

where ${\tilde{d}}_{\varepsilon , j, n, \alpha , \beta }$ are polynomials in $\partial _x^{\alpha '} \partial _p^{\beta '} a_{\varepsilon ,k}$ with $\left| \alpha ' \right| +\left| \beta ' \right| +k \le j + 1$ of degree $n+1$ and of regularity at least $\tau -j-1$. C’s with some indices are constants depending on those indices. The index set ${\mathcal {I}}_n(\alpha ,\beta )$ is defined by

$$\begin{aligned} \begin{aligned} {\mathcal {I}}_n(\alpha ,\beta ) = \{(\alpha _1,\dots ,\alpha _n,&\beta _1,\dots ,\beta _n) \in {\mathbb {N}}^{2nd} \ \\&| \ \sum _{l=1}^n \alpha _l=\alpha , \, \sum _{l=1}^n \beta _l=\beta , \, \max (\left| \alpha _l \right| ,\left| \beta _l \right| ) \ge 1 \, \forall l \}. \end{aligned} \end{aligned}$$

The form of the polynomials is determined by the multi-indices $\alpha$ and $\beta$. Moreover we have that the polynomials ${\tilde{d}}_{\varepsilon , j, n, \alpha , \beta }$ are elements of $\Gamma _{0,\varepsilon }^{(a_0-\zeta _1)^n,\tau -j-1}$.

If we now consider the triple sum and apply Lemma 4.5 we get that

$$\begin{aligned} \begin{aligned}&\sum _{l=1}^j \sum _{m=1}^{2l-1} \sum _{l+\left| \alpha \right| + \left| \beta \right| + k = j+1} C_{\alpha ,\beta } (\partial _p^{\alpha } D_x^{\beta } a_{\varepsilon ,k}) (\partial _p^{\beta } D_x^{\alpha }d_{\varepsilon , l,m} b_{\varepsilon ,z,0}^{m+1}) \\&={} \sum _{l=1}^j \sum _{m=1}^{2l-1} \sum _{l+\left| \alpha \right| + \left| \beta \right| + k = j+1} C_{\alpha ,\beta } (\partial _p^{\alpha } D_x^{\beta } a_{\varepsilon ,k}) (-i)^{\left| \alpha \right| } \sum _{n=0}^{ \left| \alpha \right| +\left| \beta \right| } {\tilde{d}}_{\varepsilon , l,m, n, \alpha ,\beta } b_{\varepsilon ,z,0}^{m+1+n} \\&={} \sum _{m=1}^{2j-1} \sum _{l =\lceil \frac{m-1}{2} \rceil +1 }^j \sum _{l+\left| \alpha \right| + \left| \beta \right| + k = j+1} \sum _{n=0}^{ \left| \alpha \right| +\left| \beta \right| } C_{\alpha ,\beta } (\partial _p^{\alpha } D_x^{\beta } a_{\varepsilon ,k}) (-i)^{\left| \alpha \right| } {\tilde{d}}_{\varepsilon , l,m, n, \alpha ,\beta } b_{\varepsilon ,z,0}^{m+1+n} \\&={}\sum _{m=1}^{2j-1} \sum _{l =\lceil \frac{m-1}{2} \rceil +1 }^j \sum _{n=0}^{ j+1-l} \sum _{\left| \alpha \right| + \left| \beta \right| = n} C_{\alpha ,\beta } (\partial _p^{\alpha } D_x^{\beta } a_{\varepsilon ,j+1-l-n}) (-i)^{\left| \alpha \right| } {\tilde{d}}_{\varepsilon , l,m, n, \alpha ,\beta } b_{\varepsilon ,z,0}^{m+1+n}, \end{aligned} \end{aligned}$$

where the ${\tilde{d}}_{\varepsilon , l,m, n \alpha ,\beta }$’s are the polynomials from Lemma 4.5. Due to the intimal constraint $l+\left| \alpha \right| + \left| \beta \right| + k = j+1$, the way we have expressed the sums ensures that k is uniquely determined by j, l and n via the relation $k=j+1-l-n$. Hence we have written the number $j+1-l-n$ instead of k. From Lemma 4.5 we have that ${\tilde{d}}_{\varepsilon , l,m, n, \alpha ,\beta }$ are polynomials in $\partial _x^{\alpha '} \partial _p^{\beta '} a_{\varepsilon ,m}$ with $\left| \alpha ' \right| +\left| \beta ' \right| +m \le l+n\le j+1$ of degree $n+m$ and with regularity at least $\tau -l-\left| \alpha \right|$. Hence the factors $C_{\alpha ,\beta } (\partial _p^{\alpha } D_x^{\beta } a_{\varepsilon ,j+1-l-n}) (-i)^{\left| \alpha \right| } {\tilde{d}}_{\varepsilon , l,m, n, \alpha ,\beta }$ will be polynomials in $\partial _x^{\alpha '} \partial _p^{\beta '} a_{\varepsilon ,m}$ with $\left| \alpha ' \right| +\left| \beta ' \right| +m\le j+1$ of degree $n+m+1$. The regularity of the terms will be at least

$$\begin{aligned} \tau - l-\left| \alpha \right| - (j+1-l-n) - \left| \beta \right| = \tau -(j+1), \end{aligned}$$

where most terms will have more regularity. By rewriting and renaming some of the terms we get the following equality

$$\begin{aligned} \sum _{l=1}^j \sum _{m=1}^{2l-1} \sum _{l+\left| \alpha \right| + \left| \beta \right| + k = j+1} C_{\alpha ,\beta } (\partial _p^{\alpha } D_x^{\beta } a_{\varepsilon ,k}) (\partial _p^{\beta } D_x^{\alpha }d_{\varepsilon , l,m} b_{\varepsilon ,z,0}^{m+1}) = \sum _{n=1}^{2j} {\tilde{d}}_{\varepsilon ,j,n,\alpha ,\beta } b_{\varepsilon ,z,0}^{n+1}, \end{aligned}$$

where ${\tilde{d}}_{\varepsilon ,j,n,\alpha ,\beta }$ again are polynomials in $\partial _x^{\alpha '} \partial _p^{\beta '} a_{\varepsilon ,k}$ with $\left| \alpha ' \right| +\left| \beta ' \right| +k \le j+1$ of degree $n+1$ of regularity at least $\tau -(j+1)$. By combining these calculation we arrive at the expression

$$\begin{aligned} \begin{aligned} b_{\varepsilon ,z,j+1}&={} - b_{\varepsilon ,z,0} \cdot \sum _{\begin{array}{c} l + \left| \alpha \right| + \left| \beta \right| + k = j+1\\ 0\le l \le j \end{array}} \frac{1}{\alpha ! \beta !} \frac{1}{2^{\left| \alpha \right| }} \frac{1}{(-2)^{\left| \beta \right| }} (\partial _p^{\alpha } D_x^{\beta } a_{\varepsilon ,k}) (\partial _p^{\beta } D_x^{\alpha } b_{\varepsilon ,z,l}) \\&={} - b_{\varepsilon ,z,0} \Big \{ \sum _{n=0}^{j+1} {\tilde{d}}_{\varepsilon ,j,n,\alpha ,\beta } b_{\varepsilon ,z,0}^{n+1} + \sum _{n=1}^{2j} {\tilde{d}}_{\varepsilon ,j,n,\alpha ,\beta } b_{\varepsilon ,z,0}^{n+1}\Big \} \\&={} \sum _{k=1}^{2j+1} d_{\varepsilon ,j+1,k} b_{\varepsilon ,z,0}^{k+1}, \end{aligned} \end{aligned}$$

where the polynomials $d_{\varepsilon ,j+1,k}$ are universal polynomials in $\partial _x^{\alpha '} \partial _p^{\beta '} a_{\varepsilon ,k}$ with $\left| \alpha ' \right| +\left| \beta ' \right| +k \le j+1$ of degree k and with regularity at least $\tau -j-1$. Hence they are elements of $\Gamma _{0,\varepsilon }^{(a_0-\zeta _1)^k,\tau -(j+1)}$. This ends the proof. $\square$

Lemma 4.6

Let the setting be as in Theorem 4.3. For every j in ${\mathbb {N}}$ and $\alpha$, $\beta$ in ${\mathbb {N}}^d_0$ there exists a number $C_{j,\alpha ,\beta }>0$ such that

$$\begin{aligned} \left| \partial _p^\beta \partial _x^\alpha b_{\varepsilon ,z,j} \right| \le C_{j,\alpha ,\beta } \varepsilon ^{-(\tau -j-\left| \alpha \right| )_{-}} (a_{\varepsilon ,0}-\zeta _1)^{-1} \left( \frac{\left| z-\zeta _1 \right| }{\text {dist} (z,[\zeta _1,\infty ))}\right) ^{2j+\left| \alpha \right| +\left| \beta \right| }, \end{aligned}$$

for all $z \in {\mathbb {C}}\setminus [\zeta _1,\infty )$ and all $(x,p) \in {\mathbb {R}}^d_x\times {\mathbb {R}}^d_p$.

Proof

We start by considering the fraction $\frac{\left| a_{\varepsilon ,0}-\zeta _1 \right| }{\left| a_{\varepsilon ,0}-z \right| }$. We will consider two cases depending on the real part of z. If $\text {Re} (z)<\zeta _1$, then

$$\begin{aligned} \frac{\left| a_{\varepsilon ,0}-\zeta _1 \right| }{\left| a_{\varepsilon ,0}-z \right| } \le 1 \le \frac{\left| z-\zeta _1 \right| }{\text {dist} (z,[\zeta _1,\infty ))}. \end{aligned}$$

If instead $\text {Re} (z)\ge \zeta _1$ and $\left| \text {Im} {z} \right| >0$ we have by the law of sines that

$$\begin{aligned} \frac{\left| a_{\varepsilon ,0} - z \right| }{\sin (\phi _1)} = \frac{\left| a_{\varepsilon ,0}-\zeta _1 \right| }{\sin (\phi _2)} \ge \left| a_{\varepsilon ,0}-\zeta _1 \right| , \end{aligned}$$

where $\phi _1$ and $\phi _2$ are angles in the triangle with vertices $\zeta _1$, $a_{\varepsilon ,0}$ and z. We have in the above estimate used that $0< \sin (\phi _2)\le 1$. If we apply this inequality and the law of sines again, we arrive at the following expression

$$\begin{aligned} \frac{\left| a_{\varepsilon ,0}-\zeta _1 \right| }{\left| a_{\varepsilon ,0}-z \right| } \le \frac{1}{\sin (\phi _1)} = \frac{\left| z-\zeta _1 \right| }{\left| \text {Im} (z) \right| }. \end{aligned}$$

Combining these two cases we get the estimate

$$\begin{aligned} \frac{\left| a_{\varepsilon ,0}-\zeta _1 \right| }{\left| a_{\varepsilon ,0}-z \right| } \le \frac{\left| z-\zeta _1 \right| }{\text {dist} (z,[\zeta _1,\infty ))} \quad \quad \text {For all }z \in {\mathbb {C}}\setminus [\zeta _1,\infty ). \end{aligned}$$

(4.7)

If we now consider a given $b_{\varepsilon ,z,j}$ and $\alpha ,\beta$ in ${\mathbb {N}}^d_0$. Lemma 4.4 and Lemma 4.5 gives us that

$$\begin{aligned} \partial _p^\beta \partial _x^\alpha b_{\varepsilon ,z,j} = \sum _{k=1}^{2j-1} \partial _p^\beta \partial _x^\alpha (d_{\varepsilon ,j,k} b_{\varepsilon ,z,0}^{k+1}) = \sum _{k=1}^{2j-1} \sum _{n=0}^{\left| \alpha \right| +\left| \beta \right| } {\tilde{d}}_{\varepsilon , j,k, n, \alpha , \beta } b_{\varepsilon ,z,0}^{k+1+n}, \end{aligned}$$

with ${\tilde{d}}_{\varepsilon ,j,k,\alpha ,\beta }$ in $\Gamma _{0,\varepsilon }^{(a_0-\zeta _1)^{k+n},\tau -j-\left| \alpha \right| }$. By taking absolute value and applying (4.7) we get that

$$\begin{aligned} \begin{aligned} \left| \partial _p^\beta \partial _x^\alpha b_{\varepsilon ,z,j} \right|&\le {} \sum _{k=1}^{2j-1} \sum _{n=0}^{\left| \alpha \right| +\left| \beta \right| } \left| {\tilde{d}}_{\varepsilon , j,k, n, \alpha , \beta } b_{\varepsilon ,z,0}^{k+1+n} \right| \\&\le {} \left| b_{\varepsilon ,z,0} \right| \sum _{k=1}^{2j-1} \sum _{n=0}^{\left| \alpha \right| +\left| \beta \right| } \varepsilon ^{-(\tau -j-\left| \alpha \right| )_{-}} c_{j,k,\alpha ,\beta } \left( \frac{\left| a_{\varepsilon ,0}-\zeta _1 \right| }{\left| a_{\varepsilon ,0}-z \right| }\right) ^{k+n} \\&\le {} C_{j,\alpha ,\beta } \varepsilon ^{-(\tau -j-\left| \alpha \right| )_{-}} (a_{\varepsilon ,0}-\zeta _1)^{-1} \left( \frac{\left| z-\zeta _1 \right| }{\text {dist} (z,[\zeta _1,\infty ))}\right) ^{2j+\left| \alpha \right| +\left| \beta \right| }, \end{aligned} \end{aligned}$$

where we have use that

$$\begin{aligned} \left| b_{\varepsilon ,z,0} \right| = \frac{(a_{\varepsilon ,0}-\zeta _1)}{|a_{\varepsilon ,0}-z|(a_{\varepsilon ,0}-\zeta _1)} \le \frac{1}{(a_{\varepsilon ,0}-\zeta _1)} \left( \frac{\left| z-\zeta _1 \right| }{\text {dist} (z,[\zeta _1,\infty ))}\right) . \end{aligned}$$

We have now obtained the desired estimate and this ends the proof. $\square$

Proof of Theorem 4.3

By Lemma 4.4 the symbols $b_{\varepsilon ,z,j}$ are in the class $\Gamma _{0,\varepsilon }^{(a_0-\zeta _1)^{-1},\tau -j}$ for every j in ${\mathbb {N}}$, where $b_{\varepsilon ,z,j}$ is defined (4.2). Hence we have that

$$\begin{aligned} B_{\varepsilon ,z,N}(\hbar ) = \sum _{j=0}^N \hbar ^j b_{\varepsilon ,z,j}. \end{aligned}$$

is a well defined symbol for every N in ${\mathbb {N}}$. Moreover, as $(a_0-\zeta _1)^{-1}$ is a bounded function, we have by Theorem 3.25, that $\text {Op} _\hbar ^{\text {w} }(B_{\varepsilon ,z,N}(\hbar ))$ is a bounded operator. Now for N sufficiently large, we have by assumption that

$$\begin{aligned} A_\varepsilon (\hbar ) -z = \text {Op} _\hbar ^{\text {w} }( a_{\varepsilon ,0}-z)+ \sum _{k=1}^N \hbar ^k \text {Op} _\hbar ^{\text {w} }( a_{\varepsilon ,k}) + \hbar ^{N+1} R_N(\varepsilon ,\hbar ), \end{aligned}$$

where the error term satisfies

$$\begin{aligned} \hbar ^{N+1} \Vert R_N(\varepsilon ,\hbar ) \Vert _{{\mathcal {L}}(L^2({\mathbb {R}}^d))} \le \hbar ^{\kappa (N)} C_N, \end{aligned}$$

for a positive strictly increasing function $\kappa$. If we consider the composition of $A_\varepsilon (\hbar )-z$ and $\text {Op} _\hbar ^{\text {w} }(B_{\varepsilon ,z,N}(\hbar ))$ we get

$$\begin{aligned}{} & {} (A_\varepsilon (\hbar )-z) \text {Op} _\hbar ^{\text {w} }(B_{\varepsilon ,z,N}(\hbar )) \nonumber \\{} & {} = \text {Op} _\hbar ^{\text {w} }( a_{\varepsilon ,0}-z) \sum _{j=0}^N \hbar ^{j} \text {Op} _\hbar ^{\text {w} }(b_{\varepsilon ,z,j}) + \sum _{k=1}^N \hbar ^{k} \text {Op} _\hbar ^{\text {w} }( a_{\varepsilon ,k}) \sum _{j=0}^N \hbar ^{j} \text {Op} _\hbar ^{\text {w} }(b_{\varepsilon ,z,j}) \nonumber \\{} & {} \quad + \sum _{j=0}^N \hbar ^{N+1+j} R_N(\varepsilon ,\hbar ) \text {Op} _\hbar ^{\text {w} }(b_{\varepsilon ,z,j}). \end{aligned}$$

(4.8)

If we consider the first part then this corresponds to a composition of two strongly $\hbar$-$\varepsilon$-admissible operators. As we want to apply Theorem 3.24 we need to ensure N satisfies the inequality

$$\begin{aligned} \delta (N +2d + 2-\tau ) +\tau > 2d+1. \end{aligned}$$

As this is the condition, that ensures a positive power of the semiclassical parameter $\hbar$ in front of the error term. If N is sufficiently large, then by Theorem 3.24 we have

$$\begin{aligned} \begin{aligned}&\text {Op} _\hbar ^{\text {w} }( a_{\varepsilon ,0}-z) \sum _{j=0}^N \hbar ^{j} \text {Op} _\hbar ^{\text {w} }(b_{\varepsilon ,z,j}) + \sum _{k=1}^N \hbar ^{k} \text {Op} _\hbar ^{\text {w} }( a_{\varepsilon ,k}) \sum _{j=0}^N \hbar ^{j} \text {Op} _\hbar ^{\text {w} }(b_{\varepsilon ,z,j}) \\&= \sum _{l=0}^N \hbar ^l \text {Op} _\hbar ^{\text {w} }(c_{\varepsilon ,l}) + \hbar ^{N+1} \text {Op} _\hbar ^{\text {w} }({\mathcal {R}}_\varepsilon (a_\varepsilon (\hbar ), B_{\varepsilon ,z,N}(\hbar );\hbar )), \end{aligned} \end{aligned}$$

where ${\mathcal {R}}_\varepsilon (a_\varepsilon (\hbar ),B_{\varepsilon ,z,N}(\hbar );\hbar )$ is the “error symbol” from Theorem 3.24 and

$$\begin{aligned} c_{\varepsilon ,l}(x,p) = \sum _{\left| \alpha \right| +\left| \beta \right| +k+j=l} \frac{1}{\alpha !\beta !}\Big (\frac{1}{2} \Big )^{\left| \alpha \right| }\Big (-\frac{1}{2} \Big )^{\left| \beta \right| } (\partial _p^\alpha D_x^\beta (a_{\varepsilon ,k} -z\delta _k)) (\partial _p^\beta D_x^\alpha b_{\varepsilon ,z,j})(x,p), \end{aligned}$$

where $\delta _k=1$ if $k=0$ and otherwise $\delta _k=0$. The error symbol satisfies by Theorem 3.24, that for every multi-indices $\alpha$, $\beta$ in ${\mathbb {N}}^d_0$, there exists a constant $C(\alpha ,\beta ,N)$ independent of $a_\varepsilon$ and $B_{\varepsilon ,z,N}$ and an integer M such that

$$\begin{aligned} \begin{aligned}&\hbar ^{N+1} |\partial _x^\alpha \partial _p^\beta {\mathcal {R}}_\varepsilon (a_\varepsilon (\hbar ), b_\varepsilon (\hbar );x,p;\hbar ) | \\&\le {} C(\alpha ,\beta ,N) \hbar ^{\delta (\tau -N-2d-2)_{-} + \tau -2d-1} \varepsilon ^{-\left| \alpha \right| } \\&\quad \times \sum _{ j+k \le 2N} {\mathcal {G}}^{\alpha ,\beta }_{M,\tau -j-k}(a_{\varepsilon ,k},(a_{\varepsilon ,0}-\zeta _1),b_{\varepsilon ,z,j},(a_{\varepsilon ,0}-\zeta _1)^{-1}), \end{aligned} \end{aligned}$$

where the functions ${\mathcal {G}}^{\alpha ,\beta }_{M,\tau -j-k}$ are as defined in Theorem 3.24. By Lemma 4.6 we have for all $j+k\le 2N$ that

$$\begin{aligned} \begin{aligned}&{\mathcal {G}}^{\alpha ,\beta }_{M,\tau -j-k}(a_{\varepsilon ,k},(a_{\varepsilon ,0}-\zeta _1),b_{\varepsilon ,z,j},(a_{\varepsilon ,0}-\zeta _1)^{-1}) \\&={} \sup _{\begin{array}{c} \left| \gamma _1 + \gamma _2 \right| +\left| \eta _1 + \eta _2 \right| \le M \\ (x,\xi ) \in {\mathbb {R}}^{2d} \end{array}} \varepsilon ^{(\tau -j - k -M)_{-}+\left| \alpha \right| } \left| \partial _x^\alpha \partial _\xi ^\beta ( \partial _{x}^{\gamma _1} \partial _{\xi }^{\eta _1}a_{\varepsilon ,k}(x,\xi ) \partial _{x}^{\gamma _2} \partial _{\xi }^{\eta _2}b_{\varepsilon ,z,j}(x,\xi )) \right| \\&\le {} C_{\alpha ,\beta ,M}\; \sup _{{\begin{array}{c} \left| \gamma _1 + \gamma _2 \right| +\left| \eta _1 + \eta _2 \right| \le M \\ (x,\xi ) \in {\mathbb {R}}^{2d} \end{array}}} \; \varepsilon ^{(\tau -j - k -M)_{-}-(\tau -k-\gamma _1)_{-}-(\tau -j-\gamma _2)_{-}}\\&\quad \left( \frac{\left| z-\zeta _1 \right| }{\text {dist} (z,[\zeta _1,\infty ))}\right) ^{2j+M+\left| \alpha \right| +\left| \beta \right| } \\&\le {} C_{\alpha ,\beta ,M} \left( \frac{\left| z-\zeta _1 \right| }{\text {dist} (z,[\zeta _1,\infty ))}\right) ^{2j+M+\left| \alpha \right| +\left| \beta \right| }. \end{aligned} \end{aligned}$$

Now by Theorem 3.25 there exists a number $M_d$ such that

$$\begin{aligned} \hbar ^{N+1}\Vert \text {Op} _\hbar ^{\text {w} }({\mathcal {R}}_\varepsilon (a_\varepsilon (\hbar ), B_\varepsilon ,z,N(\hbar );\hbar )) \Vert _{{\mathcal {L}}(L^2({\mathbb {R}}^d))} \le C \hbar ^{\kappa (N)} \left( \frac{\left| z-\zeta _1 \right| }{\text {dist} (z,[\zeta _1,\infty ))}\right) ^{M_d}. \end{aligned}$$

If we now consider the symbols $c_{\varepsilon ,l}(x,p)$ for $0\le l \le N$, then for $l=0$ we have

$$\begin{aligned} c_{\varepsilon ,0}(x,p) = (a_{\varepsilon ,0}(x,p)-z) b_{\varepsilon ,z,0}(x,p) =1, \end{aligned}$$

by definition of $b_{\varepsilon ,z,0}(x,p)$. Now for $1\le l \le N$ we have

$$\begin{aligned} \begin{aligned} c_{\varepsilon ,l}&={} \sum _{\left| \alpha \right| +\left| \beta \right| +k+j=l} \frac{1}{\alpha !\beta !}\Big (\frac{1}{2} \Big )^{\left| \alpha \right| }\Big (-\frac{1}{2} \Big )^{\left| \beta \right| } (\partial _p^\alpha D_x^\beta (a_{\varepsilon ,k} -z\delta _k)) (\partial _p^\beta D_x^\alpha b_{\varepsilon ,z,j}) \\&={} \sum _{\begin{array}{c} \left| \alpha \right| +\left| \beta \right| +k+j=l\\ 0\le j\le l-1 \end{array}} \frac{1}{\alpha !\beta !}\Big (\frac{1}{2} \Big )^{\left| \alpha \right| }\Big (-\frac{1}{2} \Big )^{\left| \beta \right| } (\partial _p^\alpha D_x^\beta a_{\varepsilon ,k}) (\partial _p^\beta D_x^\alpha b_{\varepsilon ,z,j}) +(a_{\varepsilon ,0}-z) b_{\varepsilon ,z,l} \\&={} \sum _{\begin{array}{c} \left| \alpha \right| +\left| \beta \right| +k+j=l\\ 0\le j\le l-1 \end{array}} \frac{1}{\alpha !\beta !}\Big (\frac{1}{2} \Big )^{\left| \alpha \right| }\Big (-\frac{1}{2} \Big )^{\left| \beta \right| } (\partial _p^\alpha D_x^\beta a_{\varepsilon ,k}) (\partial _p^\beta D_x^\alpha b_{\varepsilon ,z,j}) \\&\quad -\sum _{\begin{array}{c} \left| \alpha \right| + \left| \beta \right| + k +j = l\\ 0\le j \le l-1 \end{array}} \frac{1}{\alpha ! \beta !} \frac{1}{2^{\left| \alpha \right| }} \frac{1}{(-2)^{\left| \beta \right| }} (\partial _p^{\alpha } D_x^{\beta } a_{\varepsilon ,k}) (\partial _p^{\beta } D_x^{\alpha } b_{\varepsilon ,z,j}) \\&= 0, \end{aligned} \end{aligned}$$

by definition of $b_{\varepsilon ,z,l}$. We have in the above calculation used that it is only when $j=l$, that $k=|\alpha |+|\beta |=0$. These two equalities imply that

$$\begin{aligned} \sum _{k,j=0}^N \hbar ^{k+j} \text {Op} _\hbar ^{\text {w} }( a_{\varepsilon ,k}-z\delta _k) \text {Op} _\hbar ^{\text {w} }(b_{\varepsilon ,z,j}) = I + \hbar ^{N+1}\text {Op} _\hbar ^{\text {w} }({\mathcal {R}}_\varepsilon (a_\varepsilon (\hbar ), B_\varepsilon ,z,N(\hbar );\hbar )). \end{aligned}$$

(4.9)

This was the first part of equation (4.8). Let us consider the second part of (4.8):

$$\begin{aligned} \sum _{j=0}^N \hbar ^{N+1+j} R_N(\varepsilon ,\hbar ) \text {Op} _\hbar ^{\text {w} }(b_{\varepsilon ,z,j}). \end{aligned}$$

By Theorem 3.25 and Lemma 4.6 there exist constants $M_d$ and C such that

$$\begin{aligned} \hbar ^j \Vert \text {Op} _\hbar ^{\text {w} }(b_{\varepsilon ,z,j}) \Vert _{{\mathcal {L}}(L^2({\mathbb {R}}^d))} \le C \left( \frac{\left| z-\zeta _1 \right| }{\text {dist} (z,[\zeta _1,\infty ))}\right) ^{2j+M_d} \end{aligned}$$

for all j in $\{0,\dots ,N\}$. Hence by assumption we have that

$$\begin{aligned} \sum _{j=0}^N \hbar ^{N+1+j} \Vert R_N(\varepsilon ,\hbar ) \text {Op} _\hbar ^{\text {w} }(b_{\varepsilon ,z,j}) \Vert _{{\mathcal {L}}(L^2({\mathbb {R}}^d))} \le C \hbar ^{\kappa (N)} \left( \frac{\left| z-\zeta _1 \right| }{\text {dist} (z,[\zeta _1,\infty ))}\right) ^{q(N)}. \end{aligned}$$

Now by combining this with (4.8) and (4.9) we get that

$$\begin{aligned} (A_\varepsilon (h) - z ) \text {Op} _\hbar ^{\text {w} }B_{\varepsilon ,z,N} = I + h^{N+1} \Delta _{z,N+1}(h) \end{aligned}$$

with

$$\begin{aligned} \hbar ^{N+1}\Vert \Delta _{z,N+1}(h) \Vert _{{\mathcal {L}}(L^2({\mathbb {R}}^d))} \le C \hbar ^{\kappa (N)} \left( \frac{\left| z \right| }{\text {dist} (z,[\zeta _1,\infty ))} \right) ^{q(N)}, \end{aligned}$$

where $\kappa$ is a positive strictly increasing function and q(N) is a positive integer depending on N. This is the desired form and this ends the proof. $\square$

4.3 Functional calculus for rough pseudo-differential operators

We are now almost ready to construct/prove a functional calculus for operators satisfying Assumption 4.1. First, we need to settle some terminology and recall a theorem.

Definition 4.7

For a smooth function $f:{\mathbb {R}}\rightarrow {\mathbb {R}}$ we define the almost analytical extension ${\tilde{f}}:{\mathbb {C}}\rightarrow {\mathbb {C}}$ of f by

$$\begin{aligned} {\tilde{f}}(x+iy) = \left( \sum _{r=0}^n f^{(r)}(x)\frac{(iy)^r}{r!} \right) \sigma (x,y), \end{aligned}$$

where $n \ge 1$ and

$$\begin{aligned} \sigma (x,y) = \omega \left( \tfrac{y}{\lambda (x)}\right) , \end{aligned}$$

for some smooth function $\omega$, defined on ${\mathbb {R}}$ such that $\omega (t)=1$ for $\left| t \right| \le 1$ and $\omega (t)=0$ for $\left| t \right| \ge 2$. Moreover we will use the notation

$$\begin{aligned} \begin{aligned} {\bar{\partial }} {\tilde{f}}(x+iy)&\,{{:}{=}}\, \frac{1}{2} \left( \frac{\partial {\tilde{f}}}{\partial x} + i \frac{\partial {\tilde{f}}}{\partial y} \right) \\&= \frac{1}{2} \left( \sum _{r=0}^n f^{(r)}(x)\frac{(iy)^r}{r!} \right) ( \sigma _x(x,y) + i \sigma _y(x,y)) \\ {}&\quad + \frac{1}{2} f^{(n+1)}(x)\frac{(iy)^n}{n!} \sigma (x,y), \end{aligned} \end{aligned}$$

where $\sigma _x$ and $\sigma _y$ are the partial derivatives of $\sigma$ with respect to x and y respectively.

Remark 4.8

The above choice is one way to define an almost analytic extension and it is not unique. Once an n has been fixed the extension has the property that

$$\begin{aligned} |{\bar{\partial }} {\tilde{f}}(x+iy)| ={\mathcal {O}}( \left| y \right| ^n) \end{aligned}$$

as $y\rightarrow 0$. Hence when making calculation a choice has to be made concerning how fast $|{\bar{\partial }} {\tilde{f}}|$ vanishes when approaching the real axis. If f is a $C_0^\infty ({\mathbb {R}})$ function one can find an almost analytic extension ${\tilde{f}}$ in $C_0^\infty ({\mathbb {C}})$ such $f(x)={\tilde{f}}(x)$ for x in ${\mathbb {R}}$ and

$$\begin{aligned} |{\bar{\partial }} {\tilde{f}}(x+iy)| \le C_N \left| y \right| ^N, \quad \text {for all } N\ge 0. \end{aligned}$$

without chancing the extension. This type of extension could be based on a Fourier transform hence it may not work for a general smooth function. For details see [28, Chapter 8] or [23, Chapter 3].

The type of functions for which we can construct a functional calculus is introduced in the next definition:

Definition 4.9

For $\rho$ in ${\mathbb {R}}$ we define the set $S^\rho$ to be the set of smooth functions $f:{\mathbb {R}}\rightarrow {\mathbb {R}}$ such that

$$\begin{aligned} |f^{(r)}(x)| \,{{:}{=}}\, \Big |\frac{d^r f}{dx^r}(x) \Big | \le c_r \lambda (x)^{\rho -r} \end{aligned}$$

for some $c_r< \infty$, all x in ${\mathbb {R}}$ and all integers $r\ge 0$. Moreover we define ${\mathcal {W}}$ by

$$\begin{aligned} {\mathcal {W}} \,{{:}{=}}\, \bigcup _{\rho <0} S^\rho . \end{aligned}$$

We can now recall the form of the spectral theorem which we will use:

Theorem 4.10

(The Helffer-Sjöstrand formula) Let H be a self-adjoint operator acting on a Hilbert space ${\mathscr {H}}$ and f a function from ${\mathcal {W}}$. Moreover let ${\tilde{f}}$ be an almost analytic extension of f with n terms. Then the bounded operator f(H) is given by the equation

$$\begin{aligned} f(H) =- \frac{1}{\pi } \int _{\mathbb {C}}{\bar{\partial }}{\tilde{f}}(z) (z-H)^{-1} \, L(dz), \end{aligned}$$

where $L(dz)=dxdy$ is the Lebesgue measure on ${\mathbb {C}}$. The formula holds for all numbers $n\ge 1$.

A proof of the above theorem can be found in e.g. [28] or [31]. We are now ready to state and prove the functional calculus for a certain class of rough pseudo-differential operators.

Theorem 4.11

Let $A_\varepsilon (\hbar )$, for $\hbar$ in $(0,\hbar _0]$, be a $\hbar$-$\varepsilon$-admissible operator of regularity $\tau \ge 1$ and with symbol

$$\begin{aligned} a_\varepsilon (\hbar ) = \sum _{j\ge 0} \hbar ^j a_{\varepsilon ,j}. \end{aligned}$$

Suppose that $A_\varepsilon (h)$ satisfies Assumption 4.1. Then for any function f from ${\mathcal {W}}$, $f(A_\varepsilon (h))$ is a $\hbar$-$\varepsilon$-admissible operator of regularity $\tau$ with respect to a constant tempered weight function. $f(A_\varepsilon (h))$ has the symbol

$$\begin{aligned} a_{\varepsilon }^f(\hbar ) = \sum _{j\ge 0} \hbar ^j a_{\varepsilon ,j}^f, \end{aligned}$$

where

$$\begin{aligned} a_{\varepsilon ,0}^f{} & {} = f(a_{\varepsilon ,0}), \nonumber \\ a_{\varepsilon ,j}^f{} & {} = \sum _{k=1}^{2j-1} \frac{(-1)^k}{k!} d_{\varepsilon ,j,k} f^{(k)}(a_{\varepsilon ,0}) \quad \quad \text {for }j\ge 1, \end{aligned}$$

(4.10)

the symbols $d_{\varepsilon ,j,k}$ are the polynomials from Lemma 4.4. Especially we have

$$\begin{aligned} a_{\varepsilon ,1}^f = a_{\varepsilon ,1}f^{(1)}(a_{\varepsilon ,0}). \end{aligned}$$

The proof is an application of Theorem 4.10, and the fact that the resolvent is a $\hbar$-$\varepsilon$-admissible operator as well.

Proof

By Theorem 4.2 the operator $A_\varepsilon (\hbar )$ is essentially self-adjoint for sufficiently small $\hbar$. Hence Theorem 4.10 gives us

$$\begin{aligned} f(A_\varepsilon (\hbar )) =- \frac{1}{\pi } \int _{\mathbb {C}}\bar{\partial }{\tilde{f}}(z) (z-A_\varepsilon (\hbar ))^{-1} \, L(dz), \end{aligned}$$

where ${\tilde{f}}$ is an almost analytic extension of f. For the almost analytic extension of f we will need a sufficiently large number of terms, which we assume to have chosen from the start. Theorem 4.3 gives that the resolvent is a $\hbar$-$\varepsilon$-admissible operator and the explicit form of it as well. Hence

$$\begin{aligned} \begin{aligned} f(A_\varepsilon (\hbar ))&={} \frac{1}{\pi } \int _{\mathbb {C}}\bar{\partial }{\tilde{f}}(z) \sum _{j=0}^M \hbar ^j \text {Op} _\hbar ^{\text {w} }(b_{\varepsilon ,z,j}) \, L(dz) \\&\quad - \frac{1}{\pi } \int _{\mathbb {C}}{\bar{\partial }}{\tilde{f}}(z) h^{N+1} (z-A_\varepsilon (\hbar ))^{-1}\Delta _{\varepsilon ,z,N+1}(\hbar ) \, L(dz), \end{aligned} \end{aligned}$$

where the symbols $b_{\varepsilon ,z,j}$ and the operator $\Delta _{\varepsilon ,z,N+1}(\hbar )$ are as defined in Theorem 4.3. If we start by considering the error term we have by Theorem 4.3 the estimate

$$\begin{aligned} \begin{aligned} \Vert (z-A_\varepsilon (\hbar ))^{-1}\Delta _{\varepsilon ,z,N+1}(\hbar ) \Vert _{{\mathcal {L}}(L^2({\mathbb {R}}^d))}&\le {} C \hbar ^{\kappa (N)} \frac{1}{\left| \text {Im} (z) \right| } \left( \frac{\left| z \right| }{\text {dist} (z,[\zeta _1,\infty ))} \right) ^{q(N)} \\&\le {} C \hbar ^{\kappa (N)} \frac{\left| z \right| ^{q(N)}}{\left| \text {Im} (z) \right| ^{q(N)+1} }, \end{aligned} \end{aligned}$$

for N sufficiently large. Where q(N) is an integer dependent on the number N. We have that

$$\begin{aligned} |{\bar{\partial }}{\tilde{f}}(z)| \le c_1 \sum _{r=0}^n |{\tilde{f}}^{(r)}(\text {Re} (z))| \lambda (\text {Re} (z))^{r-1} {\varvec{1}}_U(z) + c_2 |{\tilde{f}}^{(n+1)}(\text {Re} (z))| |\text {Im} (z)|^n {\varvec{1}}_V(z), \end{aligned}$$

where

$$\begin{aligned} U&=\{z \in {\mathbb {C}}\, |\, \lambda (\text {Re} (z))< \left| \text {Im} (z) \right| <2\lambda (\text {Re} (z))\}, \end{aligned}$$

and

$$\begin{aligned} V&=\{z \in {\mathbb {C}}\, | \, 0< \left| \text {Im} (z) \right| <2\lambda (\text {Re} (z))\}. \end{aligned}$$

This estimate follows directly from the definition of ${\tilde{f}}$. By combining these estimates and the definition of the class of functions ${\mathcal {W}}$ we have

$$\begin{aligned} \Big \Vert \frac{1}{\pi } \int _{\mathbb {C}}{\bar{\partial }}{\tilde{f}}(z) h^{N+1} (z-A_\varepsilon (\hbar ))^{-1}\Delta _{\varepsilon ,z,N+1}(\hbar ) \, L(dz)\Big \Vert _{{\mathcal {L}}(L^2({\mathbb {R}}^d))} \le C \hbar ^{\kappa (N)}. \end{aligned}$$

What remains to prove the following equality

$$\begin{aligned} \sum _{j=0}^M \hbar ^j \frac{1}{\pi } \int _{\mathbb {C}}\bar{\partial }{\tilde{f}}(z) \text {Op} _\hbar ^{\text {w} }(b_{\varepsilon ,z,j}) \, L(dz) = \sum _{j=0}^M \hbar ^j \text {Op} _\hbar ^{\text {w} }(a_{\varepsilon ,j}^f), \end{aligned}$$

(4.11)

where the symbols $a_{\varepsilon ,j}^f$ are as defined in the statement. We will only consider one of the terms, as the rest is treated analogously. Hence we need to establish the equality

$$\begin{aligned} \frac{1}{\pi } \int _{\mathbb {C}}{\bar{\partial }}{\tilde{f}}(z) \text {Op} _\hbar ^{\text {w} }(b_{\varepsilon ,z,j}) \, L(dz) = \text {Op} _\hbar ^{\text {w} }(a_{\varepsilon ,j}^f). \end{aligned}$$

As both operators are bounded, we need only establish the equality weekly for a dense subset of $L^2({\mathbb {R}}^d)$. Hence let $\varphi$ and $\psi$ be two functions from $C_0^\infty ({\mathbb {R}}^d)$ and a j be given. We have that

$$\begin{aligned} \langle \frac{1}{\pi } \int _{\mathbb {C}}{\bar{\partial }}{\tilde{f}}(z) \text {Op} _\hbar ^{\text {w} }(b_{\varepsilon ,z,j}) \, L(dz) \varphi , \psi \rangle = \frac{1}{\pi } \int _{\mathbb {C}}{\bar{\partial }}{\tilde{f}}(z) \langle \text {Op} _\hbar ^{\text {w} }(b_{\varepsilon ,z,j}) \varphi , \psi \rangle \, L(dz). \end{aligned}$$

(4.12)

Considering just the inner product $\langle \text {Op} _\hbar ^{\text {w} }(b_{\varepsilon ,z,j}) \varphi , \psi \rangle$, we have that

$$\begin{aligned} \begin{aligned} \langle \text {Op} _\hbar ^{\text {w} }&(b_{\varepsilon ,z,j}) \varphi , \psi \rangle \\&={} \frac{1}{(2\pi \hbar )^d}\int _{{\mathbb {R}}^{3d}} e^{i\hbar ^{-1}\langle x-y,p\rangle } b_{\varepsilon ,z,j}(\tfrac{x+y}{2},p) \varphi (y) {\overline{\psi }}(x) \, dydpdx \\&={} \lim _{\sigma \rightarrow \infty } \frac{1}{(2\pi \hbar )^d} \int _{{\mathbb {R}}^{3d}} e^{i\hbar ^{-1}\langle x-y,p\rangle } g_\sigma (x,y,p) b_{\varepsilon ,z,j}(\tfrac{x+y}{2},p) \varphi (y) {\overline{\psi }}(x) \, dydpdx, \end{aligned} \end{aligned}$$

(4.13)

where the function g is a positive Schwartz function bounded by 1 and identical 1 in a neighbourhood of 0. We have set $g_{\sigma }(x,y,p)=g(\tfrac{x}{\sigma },\tfrac{y}{\sigma },\tfrac{p}{\sigma })$. The next step in the proof is to apply the dominated convergence theorem to move the limit outside the integral over z.

We let $\chi$ be in $C_0^\infty ({\mathbb {R}}^d)$ such that $\chi (p)=1$ for $\left| p \right| \le 1$ and $\chi (p)=0$ for $\left| p \right| \ge 2$. With this function we have

$$\begin{aligned} \begin{aligned}&\frac{1}{(2\pi \hbar )^d}\int _{{\mathbb {R}}^{3d}} e^{i\hbar ^{-1}\langle x-y,p\rangle } g_\sigma (x,y,p) b_{\varepsilon ,z,j}(\tfrac{x+y}{2},p) \varphi (y) {\overline{\psi }}(x) \, dydpdx \\&= \frac{1}{(2\pi \hbar )^d}\big [ \int _{{\mathbb {R}}^{3d}}e^{i\hbar ^{-1}\langle x-y,p\rangle } g_\sigma (x,y,p) \chi (p) b_{\varepsilon ,z,j}(\tfrac{x+y}{2},p) \varphi (y) {\overline{\psi }}(x) \, dydpdx \\&\quad + \int _{{\mathbb {R}}^{3d}} e^{i\hbar ^{-1}\langle x-y,p\rangle } g_\sigma (x,y,p)(1-\chi (p)) b_{\varepsilon ,z,j}(\tfrac{x+y}{2},p) \varphi (y) {\overline{\psi }}(x) \, dydpdx\big ]. \end{aligned} \end{aligned}$$

(4.14)

By Lemma 4.6 we have

$$\begin{aligned}{} & {} \Big |\int _{{\mathbb {R}}^{3d}}e^{i\hbar ^{-1}\langle x-y,p\rangle } g_\sigma (x,y,p) \chi (p) b_{\varepsilon ,z,j}(\tfrac{x+y}{2},p) \varphi (y) {\overline{\psi }}(x) \, dydpdx \Big | \nonumber \\{} & {} \le C_j \varepsilon ^{-(\tau -j)_{-}} \left( \frac{\left| z-\zeta _1 \right| }{\left| \text {Im} (z) \right| }\right) ^{2j}, \end{aligned}$$

(4.15)

where the $\zeta _1$ is the number from Assumption 4.1. The factor $\varepsilon ^{-(\tau -j)_{-}}$ is not an issue as the operator we consider has $\hbar ^j$ in front. We have just omitted to write this factor. This bound is clearly independent of $\sigma$. We now need to bound the term with $1-\chi (p)$. Here we use that on the support of $1-\chi (p)$ we have $\left| p \right| >1$. Hence the operator

$$\begin{aligned} M= \frac{(-i\hbar )^{2d}}{ \left| p \right| ^{2d}}\left( \sum _{k=1}^d \partial _{y_k}^2\right) ^d = \frac{(-i\hbar )^{2d}}{ \left| p \right| ^{2d}} \sum _{\left| \alpha \right| =d} \partial _{y}^{2\alpha }, \end{aligned}$$

is well defined, when acting on functions supported in $\text {supp} (1-\chi )$. We then obtain that

$$\begin{aligned} \begin{aligned}&\int _{{\mathbb {R}}^{3d}} e^{i\hbar ^{-1}\langle x-y,p\rangle } g_\sigma (x,y,p) (1-\chi (p)) b_{\varepsilon ,z,j}(\tfrac{x+y}{2},p) \varphi (y) {\overline{\psi }}(x) \, dydpdx \\&= \int _{{\mathbb {R}}^{3d}} (M e^{i\hbar ^{-1}\langle x-y,p\rangle }) g_\sigma (x,y,p) (1-\chi (p)) b_{\varepsilon ,z,j}(\tfrac{x+y}{2},p) \varphi (y) {\overline{\psi }}(x) \, dydpdx \\&= \int _{{\mathbb {R}}^{3d}} e^{i\hbar ^{-1}\langle x-y,p\rangle }(1-\chi (p)) M^t(g_\sigma (x,y,p) b_{\varepsilon ,z,j}(\tfrac{x+y}{2},p) \varphi (y)){\overline{\psi }}(x) \, dydpdx, \end{aligned} \end{aligned}$$

where we have used that $M e^{i\hbar ^{-1}\langle x-y,p\rangle } = e^{i\hbar ^{-1}\langle x-y,p\rangle }$. Just considering the action of $M^t$ inside the integral, we have by Leibniz’s formula that

$$\begin{aligned} \begin{aligned}&|M^tg_\sigma (x,y,p) b_{\varepsilon ,z,j}(\tfrac{x+y}{2},p) \varphi (y) | \\&= \frac{\hbar ^{2d}}{ \left| p \right| ^{2d}} \Big | \sum _{\left| \alpha \right| =d} \sum _{\beta \le 2\alpha } \partial _{y}^{2\alpha -\beta }(g_\sigma (x,y,p) \varphi (y)) \partial _{y}^{\beta } b_{\varepsilon ,z,j}(\tfrac{x+y}{2},p) \Big | \\&\le C_j \frac{\hbar ^{2d}}{ \left| p \right| ^{2d}} {\varvec{1}}_{\text {supp} (\varphi )}(y) \sum _{\left| \alpha \right| =d} \sum _{\beta \le 2\alpha } \varepsilon ^{-(\tau -j-\left| \beta \right| )_{-}} \left( \frac{\left| z-\zeta _1 \right| }{\left| \text {Im} (z) \right| }\right) ^{2j + \left| \beta \right| } \\&\le C \frac{\hbar ^{2d}}{ \left| p \right| ^{2d}} \varepsilon ^{-(\tau -j-2d)_{-}} {\varvec{1}}_{\text {supp} (\varphi )}(y) \left( 1+\frac{\left| z-\zeta _1 \right| }{\left| \text {Im} (z) \right| }\right) ^{2j + 2d}, \end{aligned} \end{aligned}$$

where we again have used Lemma 4.6. This implies the estimate

$$\begin{aligned} \begin{aligned}&\Big |\int _{{\mathbb {R}}^{3d}} e^{i\hbar ^{-1}\langle x-y,p\rangle } g_\sigma (x,y,p) (1-\chi (p)) b_{\varepsilon ,z,j}(\tfrac{x+y}{2},p) \varphi (y) {\overline{\psi }}(x) \, dydpdx \Big | \\&\le C_j \varepsilon ^{-(\tau -j)_{-}} \left( 1+\frac{\left| z-\zeta _1 \right| }{\left| \text {Im} (z) \right| }\right) ^{2j + 2d}. \end{aligned} \end{aligned}$$

If we combine this estimate with (4.14) and (4.15) we have

$$\begin{aligned} \begin{aligned}&\Big | \frac{1}{(2\pi \hbar )^d} \int _{{\mathbb {R}}^{3d}} e^{i\hbar ^{-1}\langle x-y,p\rangle } g_\sigma (x,y,p) b_{\varepsilon ,z,j}(\tfrac{x+y}{2},p) \varphi (y) {\overline{\psi }}(x) \, dydpdx \Big | \\&\le C \varepsilon ^{-(\tau -j)_{-}} \left( 1+\frac{\left| z-\zeta _1 \right| }{\left| \text {Im} (z) \right| }\right) ^{2j + 2d}. \end{aligned} \end{aligned}$$

As above we have

$$\begin{aligned} \Big | \int _{\mathbb {C}}{\bar{\partial }}{\tilde{f}}(z) \left( 1+\frac{\left| z-\zeta _1 \right| }{\left| \text {Im} (z) \right| }\right) ^{2j + 2d} \, L(dz)\Big | < \infty . \end{aligned}$$

Hence we can apply dominated convergence. By an analogous argument, we can also apply Fubini’s Theorem. This gives us that

$$\begin{aligned}{} & {} \frac{1}{\pi } \int _{\mathbb {C}}{\bar{\partial }}{\tilde{f}}(z) \langle \text {Op} _\hbar ^{\text {w} }(b_{\varepsilon ,z,j}) \varphi , \psi \rangle \, L(dz) \nonumber \\{} & {} ={} \lim _{\sigma \rightarrow \infty } \frac{1}{(2\pi \hbar )^d} \int _{{\mathbb {R}}^{3d}} e^{i\hbar ^{-1}\langle x-y,p\rangle } g_\sigma (x,y,p) \nonumber \\{} & {} \quad \times \frac{1}{\pi } \int _{\mathbb {C}}\bar{\partial }{\tilde{f}}(z) b_{\varepsilon ,z,j}(\tfrac{x+y}{2},p) \, L(dz) \varphi (y) {\overline{\psi }}(x) \, dydpdx. \end{aligned}$$

(4.16)

If we only consider the integral over z, then we have by a Cauchy formula and Lemma 4.4 that

$$\begin{aligned} \begin{aligned}&\frac{1}{\pi } \int _{\mathbb {C}}{\bar{\partial }}{\tilde{f}}(z) b_{\varepsilon ,z,j}(\tfrac{x+y}{2},p) \, L(dz) \\&= \frac{1}{\pi } \int _{\mathbb {C}}{\bar{\partial }}{\tilde{f}}(z) \sum _{k=1}^{2j-1} d_{\varepsilon , j,k}(\tfrac{x+y}{2},p) b_{\varepsilon ,z,0}^{k+1}(\tfrac{x+y}{2},p) \, L(dz) \\&= \sum _{k=1}^{2j-1} d_{\varepsilon , j,k}(\tfrac{x+y}{2},p) \frac{1}{\pi } \int _{\mathbb {C}}{\bar{\partial }}{\tilde{f}}(z) \Big (\frac{1}{a_{\varepsilon ,0}(\tfrac{x+y}{2},p)-z}\Big )^{k+1}\, L(dz) \\&= \sum _{k=1}^{2j-1} \frac{(-1)^k}{k!} d_{\varepsilon , j,k}(\tfrac{x+y}{2},p) f^{(k)}( a_{\varepsilon ,0}(\tfrac{x+y}{2},p)) = a_{\varepsilon ,j}^f(\tfrac{x+y}{2},p), \end{aligned} \end{aligned}$$

which is the desired form of $a_{\varepsilon ,j}^f$ given in (4.10). Now combining this with (4.12), (4.13) and (4.16) we arrive at

$$\begin{aligned} \langle \frac{1}{\pi } \int _{\mathbb {C}}{\bar{\partial }}{\tilde{f}}(z) \text {Op} _\hbar ^{\text {w} }(b_{\varepsilon ,z,j}) \, L(dz) \varphi , \psi \rangle = \langle \text {Op} _\hbar ^{\text {w} }(a_{\varepsilon ,j}^f) \varphi , \psi \rangle . \end{aligned}$$

The remaning j’ can be treated analogously and hence we obtain the equality (4.11). This ends the proof. $\square$

4.4 Applications of functional calculus

With the functional calculus, we can now prove some useful theorems and lemmas. One of them is an asymptotic expansion of certain traces.

Theorem 4.12

Let $A_\varepsilon (\hbar )$, for $\hbar$ in $(0,\hbar _0]$, be a $\hbar$-$\varepsilon$-admissible operator of regularity $\tau \ge 1$ and with symbol

$$\begin{aligned} a_\varepsilon (\hbar ) = \sum _{j\ge 0} \hbar ^j a_{\varepsilon ,j}. \end{aligned}$$

Suppose that $A_\varepsilon (\hbar )$ satisfies Assumption 4.1. Let $E_1<E_2$ be two real numbers and suppose there exists an $\eta >0$ such that $a_{\varepsilon ,0}^{-1}([E_1 - \eta , E_2+\eta ])$ is compact. Then we have

$$\begin{aligned} \text {spec} (A_\varepsilon (\hbar ))\cap [E_1.E_2] \subseteq \text {spec} _{pp}(A_\varepsilon (\hbar )), \end{aligned}$$

(4.17)

for $\hbar$ sufficiently small, where $\text {spec} _{pp}(A_\varepsilon (\hbar ))$ is the pure point spectrum of $A_\varepsilon (\hbar )$.

Proof

Let f and g be in $C_0^\infty ((E_1-\eta ,E_2+\eta ))$ such $g(t)=1$ for $t\in [E_1,E_2]$ and $f(t)=1$ for t in $\text {supp} (g)$. By Theorem 4.11 we have

$$\begin{aligned} f(A_\varepsilon (\hbar )) = A_{\varepsilon ,f,N}(\hbar ) + \hbar ^{N+1} R_{N+1,f}(\varepsilon ,\hbar ), \end{aligned}$$

(4.18)

where the terms $A_{\varepsilon ,f,N}(\hbar )$ consist of the first N terms in the expansion in $\hbar$ of $f(A_\varepsilon (\hbar ))$. We get by (4.18) and the definition of g and f that

$$\begin{aligned} g(A_\varepsilon (\hbar )) (I - \hbar ^{N+1} R_{N+1,f}(\varepsilon ,\hbar ))= g(A_\varepsilon (\hbar ))A_{\varepsilon ,f,N}(\hbar ). \end{aligned}$$

Hence for $\hbar$ sufficiently small we have

$$\begin{aligned} g(A_\varepsilon (\hbar )) = g(A_\varepsilon (\hbar ))A_{\varepsilon ,f,N}(\hbar ) (I - \hbar ^{N+1} R_{N+1,f}(\varepsilon ,\hbar ))^{-1}, \end{aligned}$$

thereby we have the inequality

$$\begin{aligned} \Vert g(A_\varepsilon (\hbar )) \Vert _{\text {Tr} } \le c \Vert g \Vert _{\infty } \Vert A_{\varepsilon ,f,N}(\hbar ) \Vert _{\text {Tr} } \le C \hbar ^{-d}, \end{aligned}$$

(4.19)

where we have used Theorem 3.26. Since ${\varvec{1}}_{[E_1,E_2]}(t)\le g(t)$ we have that ${\varvec{1}}_{[E_1,E_2]}(A_\varepsilon (\hbar ))$ is a trace class operator by (4.19). This implies the inclusion

$$\begin{aligned} \text {spec} (A_\varepsilon (\hbar ))\cap [E_1,E_2] \subseteq \text {spec} _{pp}(A_\varepsilon (\hbar )), \end{aligned}$$

for $\hbar$ sufficiently small. This ends the proof. $\square$

Theorem 4.13

Let $A_\varepsilon (\hbar )$, for $\hbar$ in $(0,\hbar _0]$, be a $\hbar$-$\varepsilon$-admissible operator of regularity $\tau \ge 1$ and with symbol

$$\begin{aligned} a_\varepsilon (\hbar ) = \sum _{j\ge 0} \hbar ^j a_{\varepsilon ,j}. \end{aligned}$$

Suppose that $A_\varepsilon (\hbar )$ satisfies Assumption 4.1. Let $E_1<E_2$ be two real numbers and suppose there exists an $\eta >0$ such that $a_{\varepsilon ,0}^{-1}([E_1 - \eta , E_2+\eta ])$ is compact. Then for every f in $C_0^\infty ((E_1,E_2))$ and any $N_0$ in ${\mathbb {N}}$ there exists an N in ${\mathbb {N}}$ such that

$$\begin{aligned} |\text {Tr} [f(A_\varepsilon (\hbar ))] -\frac{1}{(2\pi \hbar )^d} \sum _{j=0}^N \hbar ^j T_j(f,A_\varepsilon (\hbar )) | \le C \hbar ^{N_0+1-d}, \end{aligned}$$

for all sufficiently small $\hbar$. The $T_j$’s are given by

$$\begin{aligned} T_j(f,A_\varepsilon (\hbar ) ) = \int _{{\mathbb {R}}^{2d}} \sum _{k=1}^{2j-1} \frac{(-1)^k}{k!} d_{\varepsilon ,j,k} f^{(k)}(a_{\varepsilon ,0}) \, dxdp, \end{aligned}$$

where $d_{\varepsilon ,j,k}$ are the polynomials from Lemma 4.4. In particular we have

$$\begin{aligned} T_0(f,A_\varepsilon (\hbar ) )&={} \int _{{\mathbb {R}}^{2d}} f(a_{\varepsilon ,0}) \, dxdp \end{aligned}$$

and

$$\begin{aligned} T_1(f,A_\varepsilon (\hbar ) ) =&\int _{{\mathbb {R}}^{2d}} a_{\varepsilon ,1} f^{(1)}(a_{\varepsilon ,0}) \, dxdp. \end{aligned}$$

The proof is an application of Theorem 4.11 which gives the form of the operator $f(A_\varepsilon (\hbar ))$ combined with the trace formula from Theorem 3.27 and we use some of the same ideas as in the proof of Theorem 4.12.

Proof

Let f in $C_0^\infty ((E_1,E_2))$ be given and fix a g in $C_0^\infty ((E_1-\eta ,E_2+\eta ))$ such that $g(t)=1$ for $t\in [E_1,E_2]$. By Theorem 4.11 we have

$$\begin{aligned} f(A_\varepsilon (\hbar )) = A_{\varepsilon ,f,N}(\hbar ) + \hbar ^{N+1} R_{N+1,f}(\varepsilon ,\hbar ), \end{aligned}$$

and

$$\begin{aligned} g(A_\varepsilon (\hbar )) = A_{\varepsilon ,g,N}(\hbar ) + \hbar ^{N+1} R_{N+1,g}(\varepsilon ,\hbar ), \end{aligned}$$

where the terms $A_{\varepsilon ,f,N}(\hbar )$ and $A_{\varepsilon ,g,N}(\hbar )$ consist of the first N terms in the expansion in $\hbar$ of $f(A_\varepsilon (\hbar ))$ and $g(A_\varepsilon (\hbar ))$ respectively. Since $f(A_\varepsilon (\hbar )) g(A_\varepsilon (\hbar )) = f(A_\varepsilon (\hbar ))$ we have

$$\begin{aligned} \begin{aligned}&f(A_\varepsilon (\hbar )) \\&= (A_{\varepsilon ,f,N}(\hbar ) + \hbar ^{N+1} R_{N+1,f}(\varepsilon ,\hbar ))( A_{\varepsilon ,g,N}(\hbar ) + \hbar ^{N+1} R_{N+1,g}(\varepsilon ,\hbar )) \\&= A_{\varepsilon ,f,N}(\hbar ) A_{\varepsilon ,g,N}(\hbar ) +\hbar ^{N+1}[ A_{\varepsilon ,f,N}(\hbar ) R_{N+1,g}(\varepsilon ,\hbar ) + R_{N+1,f}(\varepsilon ,\hbar )A_{\varepsilon ,g,N}(\hbar ) ]. \end{aligned} \end{aligned}$$

By Theorem 3.26 we have that

$$\begin{aligned} \Vert f(A_\varepsilon (\hbar )) - A_{\varepsilon ,f,N}(\hbar ) A_{\varepsilon ,g,N}(\hbar ) \Vert _{\text {Tr} } \le C \hbar ^{\kappa (N)-d} \end{aligned}$$

(4.20)

as $\hbar \rightarrow 0$. Hence taking N sufficiently large we can consider the composition of the operators $A_{\varepsilon ,f,N}(\hbar ) A_{\varepsilon ,g,N}(\hbar )$ instead of $f(A_\varepsilon (\hbar ))$. By the choice of g and Theorem 3.24 (composition of operators) we have

$$\begin{aligned} A_{\varepsilon ,f,N}(\hbar ) A_{\varepsilon ,g,N}(\hbar ){} & {} = \sum _{j=0}^N \hbar ^j \text {Op} _\hbar ^{\text {w} }(a_{\varepsilon ,j}^f) + \hbar ^{N+1} R_{N,f,g}(\varepsilon , \hbar ,a_\varepsilon ) \nonumber \\{} & {} =A_{\varepsilon ,f,N}(\hbar ) + \hbar ^{N+1} R_{N,f,g}(\varepsilon , \hbar ,a_\varepsilon ). \end{aligned}$$

(4.21)

Hence we have by Theorem 3.26 that

$$\begin{aligned} \Vert A_{\varepsilon ,f,N}(\hbar ) - A_{\varepsilon ,f,N}(\hbar ) A_{\varepsilon ,g,N}(\hbar ) \Vert _{\text {Tr} } \le C \hbar ^{\kappa (N)-d}, \end{aligned}$$

(4.22)

where we have used that the error term in (4.21) is a $\hbar$-pseudo-differential operator, which follows from Theorem 3.24. Theorem 3.27 now gives

$$\begin{aligned} \text {Tr} [A_{\varepsilon ,f,N}(\hbar ) ]{} & {} = \sum _{j=0}^N \hbar ^j \text {Tr} [\text {Op} _\hbar ^{\text {w} }(a_{\varepsilon ,j}^f)] \nonumber \\{} & {} =\frac{1}{(2\pi \hbar )^d} \sum _{j=0}^N \hbar ^j \int _{{\mathbb {R}}^{2d}} \sum _{k=1}^{2j-1} \frac{(-1)^k}{k!} d_{\varepsilon ,j,k} f^{(k)}(a_{\varepsilon ,0}) \, dxdp. \end{aligned}$$

(4.23)

The estimates obtained in (4.20) and (4.22) implies that

$$\begin{aligned} \left| \text {Tr} [f(A_\varepsilon (\hbar ))]- \text {Tr} [A_{\varepsilon ,f,N}(\hbar ) ] \right| \le C \hbar ^{\kappa (N)-2d-1}. \end{aligned}$$

(4.24)

Hence by choosing N sufficiently large and combining (4.23) and (4.24), we get the desired estimate. $\square$

The next Lemmas will be usefull in the proof of the Weyl law. Both of these Lemmas are proven by applying the functional calculus and the results on compositions of operators.

Lemma 4.14

Let $A_\varepsilon (\hbar )$, for $\hbar$ in $(0,\hbar _0]$, be a $\hbar$-$\varepsilon$-admissible operator of regularity $\tau \ge 1$ and with symbol

$$\begin{aligned} a_\varepsilon (\hbar ) = \sum _{j\ge 0} \hbar ^j a_{\varepsilon ,j}. \end{aligned}$$

Suppose that $A_\varepsilon (h)$ satisfies Assumption 4.1. Let $E_1<E_2$ be two real numbers and suppose there exists an $\eta >0$ such that $a_{\varepsilon ,0}^{-1}([E_1 - \eta , E_2+\eta ])$ is compact. Let f be in $C_0^\infty ((E_1,E_2))$ and suppose $\theta$ is in $C_0^\infty ({\mathbb {R}}^d_x\times {\mathbb {R}}^d_p)$, $\text {supp} (\theta )\subset a_{\varepsilon ,0}^{-1}([E_1 - \eta , E_2+\eta ])$ and $\theta (x,p)=1$ for all $(x,p)\in \text {supp} (f(a_{0,\varepsilon }))$. Then we have the bound

$$\begin{aligned} \Vert (1-\text {Op} _\hbar ^{\text {w} }(\theta )) f(A_\varepsilon (\hbar )) \Vert _{\text {Tr} } \le C_N \hbar ^N, \end{aligned}$$

for every N in ${\mathbb {N}}$.

Proof

We choose g in $C_0^\infty ((E_1,E_2))$ such that $g(t)f(t)=f(t)$. We then have that

$$\begin{aligned} \Vert (1-\text {Op} _\hbar ^{\text {w} }(\theta )) f(A_\varepsilon (\hbar )) \Vert _{\text {Tr} } \le \Vert (1-\text {Op} _\hbar ^{\text {w} }(\theta )) f(A_\varepsilon (\hbar )) \Vert _{{\mathcal {L}}(L^2({\mathbb {R}}^d))} \Vert g(A_\varepsilon (\hbar )) \Vert _{\text {Tr} }. \end{aligned}$$

By arguing as in the proof of Theorem 4.13 and applying Theorem 3.26 we get

$$\begin{aligned} \Vert g(A_\varepsilon (\hbar )) \Vert _{\text {Tr} } \le C_d \hbar ^{-d}. \end{aligned}$$

Form Theorem 4.11 we have that, $f(A_\varepsilon (\hbar ))$ is $\hbar$-$\varepsilon$-admissible operator with symbols

$$\begin{aligned} a_{\varepsilon }^f(\hbar ) = \sum _{j\ge 0} \hbar ^j a_{\varepsilon ,j}^f, \end{aligned}$$

where

$$\begin{aligned} a_{\varepsilon ,j}^f = \sum _{k=1}^{2j-1} \frac{(-1)^k}{k!} d_{\varepsilon ,j,k} f^{(k)}(a_{\varepsilon ,0}), \end{aligned}$$

the symbols $d_{\varepsilon ,j,k}$ are the polynomials from Lemma 4.4. The supports, of these functions, are disjoint from the support of the symbol for the operator $(1-\text {Op} _\hbar ^{\text {w} }(\theta ))$. Hence by Theorem 3.24 we get the desired estimate. $\square$

Lemma 4.15

Let $A_\varepsilon (\hbar )$, for $\hbar$ in $(0,\hbar _0]$, be a $\hbar$-$\varepsilon$-admissible operator of regularity $\tau \ge 1$ and with symbol

$$\begin{aligned} a_\varepsilon (\hbar ) = \sum _{j\ge 0} \hbar ^j a_{\varepsilon ,j}. \end{aligned}$$

Suppose that $A_\varepsilon (h)$ satisfies Assumption 4.1. Let $E_1<E_2$ be two real numbers and suppose there exists an $\eta >0$ such that $a_{\varepsilon ,0}^{-1}([E_1 - \eta , E_2+\eta ])$ is compact. Suppose $\theta$ is in $C_0^\infty ({\mathbb {R}}^d_x\times {\mathbb {R}}^d_p)$ such that $\text {supp} (\theta )\subset a_{\varepsilon ,0}^{-1}((E_1, E_2 ))$.

Then for every f in $C_0^\infty ([E_1-\eta ,E_2+\eta ])$ such $f(t)=1$ for all t in $[E_1-\frac{\eta }{2},E_2+\frac{\eta }{2}]$ the bound

$$\begin{aligned} \Vert \text {Op} _\hbar ^{\text {w} }(\theta )(1- f(A_\varepsilon (\hbar ))) \Vert _{{\mathcal {L}}(L^2({\mathbb {R}}^d))} \le C_N \hbar ^N, \end{aligned}$$

is true for every N in ${\mathbb {N}}$

Proof

Theorem 4.11 gives us that $f(A_\varepsilon (\hbar ))$ is $\hbar$-$\varepsilon$-admissible operator with symbols

$$\begin{aligned} a_{\varepsilon }^f(\hbar ) = \sum _{j\ge 0} \hbar ^j a_{\varepsilon ,j}^f, \end{aligned}$$

where

$$\begin{aligned} a_{\varepsilon ,j}^f = \sum _{k=1}^{2j-1} \frac{(-1)^k}{k!} d_{\varepsilon ,j,k} f^{(k)}(a_{\varepsilon ,0}), \end{aligned}$$

the symbols $d_{\varepsilon ,j,k}$ are the polynomials from Lemma 4.4. Hence we have that the principal symbol of $(1- f(A_\varepsilon (\hbar )))$ is $1-f(a_{\varepsilon ,0})$. By assumption we then have that the support of $\theta$ and the support of every symbol in $f(A_\varepsilon (\hbar ))$ are disjoint. Hence Theorem 3.24 implies the desired estimate. $\square$

5 Microlocal approximation and properties of propagators

In this section we will study the solution to the operator valued Cauchy problem:

$$\begin{aligned} {\left\{ \begin{array}{ll} \hbar \partial _t U(t,\hbar ) - i U(t,\hbar ) A_\varepsilon (\hbar ) = 0 &{} t \ne 0 \\ U(0,\hbar ) = \theta (x,\hbar D) &{} t=0, \end{array}\right. } \end{aligned}$$

where $A_\varepsilon$ is self-adjoint and the symbol $\theta$ is in $C_0^\infty ({\mathbb {R}}^d_x\times {\mathbb {R}}^d_p)$. In particular we will only consider the case where $A_\varepsilon (\hbar )$ is a $\hbar$-$\varepsilon$-admissible operator of regularity $\tau \ge 1$ which satisfies Assumption 4.1. Hence for sufficiently small $\hbar$ the operator $A_\varepsilon (\hbar )$ is essentially self-adjoint by Theorem 4.2. It is well-known, that the solution to the operator valued Cauchy problem is the microlocalised propagator $\theta (x,\hbar D)e^{i\hbar ^{-1}tA_\varepsilon (\hbar )}$.

We are interested in the propagators, as they appear in a smoothing of functions applied to the operators we consider. In this smoothing procedure we need to understand the behaviour of the propagator for t in a small interval around zero. Usually this is done by constructing a specific Fourier integral operator (FIO) as an approximation to the propagator. In the “usual” construction the phase function is the solution to the Hamilton-Jacobi equations associated with the principal symbol. For our setup this FIO approximation is not desirable, as we cannot control the number of derivatives in the space variables and hence, we cannot be certain about how the operator behaves. Instead we will use a different microlocal approximation for times in $[-\hbar ^{1-\frac{\delta }{2}},\hbar ^{1-\frac{\delta }{2}}]$.

The construction of the approximation is recursive and inspired by the construction in the works of Zielinski. If the construction is compared to the approximation in the works of Ivrii, one can note that Ivrii’s construction is successive.

Our objective is to construct the approximation $U_N(t, \hbar )$ such that

$$\begin{aligned} \Vert \hbar \partial _t U_N(t, \hbar ) - i U_N(t, \hbar ) A_\varepsilon \Vert _{{\mathcal {L}}(L^2({\mathbb {R}}^d))}, \end{aligned}$$

is small and the trace of the operator has the “right” asymptotic behaviour. The kernel of the approximation will have the following form

$$\begin{aligned} (x,y) \rightarrow \frac{1}{(2\pi \hbar )^d} \int _{{\mathbb {R}}^d} e^{i \hbar ^{-1} \langle x-y , p \rangle } e^{ i t \hbar ^{-1} a_\varepsilon (x,p)} \sum _{j=0}^N (it\hbar ^{-1})^j u_j(x,p,\hbar ,\varepsilon ) \, dp, \end{aligned}$$

where N is chosen such that, the error is of a desired order, and the $u_j$’s are compactly supported rough functions in x. A priori these operators look singular in the sense that for each term in the sum we have an increasing power of the factor $\hbar ^{-1}$. What we will see in the following Theorem, is that the since each power of $\hbar ^{-1}$ comes with a power of t, then for t in $[-\hbar ^{1-\frac{\delta }{2}},\hbar ^{1-\frac{\delta }{2}}]$ it has the desired properties.

Theorem 5.1

Let $A_\varepsilon (\hbar )$ be a $\hbar$-$\varepsilon$-admissible operator of regularity $\tau \ge 1$ with tempered weight m which is self-adjoint for all $\hbar$ in $(0,\hbar _0]$, for $\hbar _0>0$ and with $\varepsilon \ge \hbar ^{1-\delta }$ for a $\delta \in (0,1)$. Let $\theta (x,p)$ be a function in $C_0^\infty ({\mathbb {R}}^d_x\times {\mathbb {R}}^d_p)$. Then for all $N_0\in {\mathbb {N}}$ there exist an operator $U_N(t,\varepsilon ,\hbar )$ with integral kernel

$$\begin{aligned} \begin{aligned} K_{U_N}(x,y,t&,\varepsilon ,\hbar ) = \frac{1}{(2\pi \hbar )^d} \int _{{\mathbb {R}}^d} e^{i \hbar ^{-1} \langle x-y,p\rangle } e^{ i t \hbar ^{-1} a_{\varepsilon ,0}(x,p)} \sum _{j=0}^N (it\hbar ^{-1})^j u_j(x,p,\hbar ,\varepsilon ) \, dp, \end{aligned} \end{aligned}$$

such that $U_N$ satisfies the following bound:

$$\begin{aligned} \sup _{t\in [-\hbar ^{1-\frac{\delta }{2}},\hbar ^{1-\frac{\delta }{2}}]}\Vert \hbar \partial _t U_N(t, \hbar ) - i U_N(t, \hbar ) A_\varepsilon \Vert _{{\mathcal {L}}(L^2({\mathbb {R}}^d))} \le C \hbar ^{N_0}. \end{aligned}$$

For the terms in the sum we have that $u_j(x,p,\hbar ,\varepsilon ) \in C_0^\infty ({\mathbb {R}}_x^d\times {\mathbb {R}}_p^d)$ for all j. In particular we have that $u_0(x,p,\hbar ,\varepsilon )=\theta (x,p)$. The terms will satisfy the bounds

$$\begin{aligned} \left| \partial _x^\beta \partial _p^\alpha u_j(x,p,\hbar ,\varepsilon ) \right| \le {\left\{ \begin{array}{ll} C_{\alpha \beta } &{} j=0 \\ C_{\alpha \beta } \hbar \varepsilon ^{-\left| \beta \right| } &{}j=1 \\ C_{\alpha \beta } \hbar ^{1+\delta (j-2)} \varepsilon ^{-\left| \beta \right| } &{}j\ge 2, \end{array}\right. } \end{aligned}$$

for all $\alpha$ and $\beta$ in ${\mathbb {N}}^d_0$ in the case $\tau =1$. For $\tau \ge 2$ the $u_j$’s satisfy the bounds

$$\begin{aligned} \left| \partial _x^\beta \partial _p^\alpha u_j(x,p,\hbar ,\varepsilon ) \right| \le {\left\{ \begin{array}{ll} C_{\alpha \beta } &{} j=0 \\ C_{\alpha \beta } \hbar \varepsilon ^{-\left| \beta \right| } &{}j=1,2 \\ C_{\alpha \beta } \hbar ^{2+\delta (j-3)} \varepsilon ^{-\left| \beta \right| } &{}j\ge 3, \end{array}\right. } \end{aligned}$$

for all $\alpha$ and $\beta$ in ${\mathbb {N}}^d_0$.

Remark 5.2

If the operator satisfies Assumption 4.1, then by Theorem 4.2 the operator will be essentially self-adjoint for all sufficiently small $\hbar$. Hence Assumption 4.1 would be sufficient but not necessary for the above theorem to be true. The number N is explicitly dependent on $N_0$, d and $\delta$ which follows directly from the proof.

Proof

We start by fixing N such that

$$\begin{aligned} \min \big ( 1+\delta \big (\tfrac{N}{2} -1\big ) - d, 2+\delta \big (\tfrac{N}{2} -3\big ) - d \big ) \ge N_0. \end{aligned}$$

By assumption we have for sufficiently large M in ${\mathbb {N}}$ the following form of $A_\varepsilon (\hbar )$

$$\begin{aligned} A_\varepsilon (\hbar ) = \sum _{j=0}^M \hbar ^j \text {Op} _\hbar ^{\text {w} }(a_{\varepsilon ,j}) + \hbar ^{M+1} R_M(\varepsilon ,\hbar ). \end{aligned}$$

(5.1)

We can choose and fix M such the following estimate is true

$$\begin{aligned} \hbar ^{M+1}\Vert R_M(\varepsilon ,\hbar ) \Vert _{{\mathcal {L}}(L^2({\mathbb {R}}^d))} \le C_M \hbar ^{N_0}. \end{aligned}$$

With this M we consider the sum in the expression of $A_\varepsilon (\hbar )$. By Corollary 3.20 there exists a sequence $\{{\tilde{a}}_{\varepsilon ,j}\}_{j\in {\mathbb {N}}}$ of symbols where ${\tilde{a}}_{\varepsilon ,j}$ is of regularity $\tau -j$ and a ${\tilde{M}}$ such

$$\begin{aligned} \sum _{j=0}^M \hbar ^j \text {Op} _\hbar ^{\text {w} }(a_{\varepsilon ,j}) = \sum _{j=0}^{{\tilde{M}}} \hbar ^j \text {Op} _{\hbar ,1}({\tilde{a}}_{\varepsilon ,j}) + \hbar ^{{\tilde{M}}+1} {\tilde{R}}_{{\tilde{M}}}(\varepsilon ,\hbar ), \end{aligned}$$

(5.2)

where $a_{\varepsilon ,0}={\tilde{a}}_{\varepsilon ,0}$ and

$$\begin{aligned} \hbar ^{{\tilde{M}}+1}\Vert {\tilde{R}}_{{\tilde{M}}}(\varepsilon ,\hbar ) \Vert _{{\mathcal {L}}(L^2({\mathbb {R}}^d))} \le C_{{\tilde{M}}} \hbar ^{N_0}. \end{aligned}$$

We will for the reminder of the proof use the notation

$$\begin{aligned} {\tilde{a}}_{\varepsilon }(x,p) = \sum _{j=0}^{{\tilde{M}}} \hbar ^j {\tilde{a}}_{\varepsilon ,j}(x,p). \end{aligned}$$

(5.3)

The function ${\tilde{a}}_{\varepsilon }(x,p)$ is a rough function of regularity $\tau$. These choices and definitions will become important again at the end of the proof.

For our fixed N we define the operator $\hbar \partial _t - i {\mathcal {P}}_N: C^\infty ({\mathbb {R}}_t \times {\mathbb {R}}^d_x \times {\mathbb {R}}_p^d) \rightarrow C^\infty ({\mathbb {R}}_t \times {\mathbb {R}}^d_x \times {\mathbb {R}}_p^d)$, where

$$\begin{aligned} {\mathcal {P}}_N b(t,x,p) = \sum _{\left| \alpha \right| \le N} \frac{(-i\hbar )^{\left| \alpha \right| }}{\alpha !} \partial _p^\alpha \{b(t,x,p) \partial _x^\alpha {\tilde{a}}_\varepsilon (x,p)\} \end{aligned}$$

for a $b\in C^\infty ({\mathbb {R}}_t\times {\mathbb {R}}^d_x\times {\mathbb {R}}^d_p)$. First step is to observe how the operator $\hbar \partial _t - i {\mathcal {P}}_N$ acts on $e^{ i t \hbar ^{-1} a_{\varepsilon ,0}(x,p)}\psi (x,p)$ for $\psi \in C_0^\infty ({\mathbb {R}}^d_x\times {\mathbb {R}}^d_p)$. We will in the following calculation OMIT THE DEPendence on the variables x and p. By Leibniz’s formula and the chain rule we get

$$\begin{aligned}{} & {} (\hbar \partial _t - i {\mathcal {P}}_N) e^{it\hbar ^{-1} a_{\varepsilon ,0}} \psi = \hbar \partial _t e^{it\hbar ^{-1} a_{\varepsilon ,0}} \psi - i \sum _{\left| \alpha \right| \le N} \frac{(-i\hbar )^{\left| \alpha \right| }}{\alpha !} \partial _p^\alpha \{ e^{it\hbar ^{-1} a_{\varepsilon ,0}} \psi \partial _x^\alpha {\tilde{a}}_\varepsilon \} \nonumber \\{} & {} ={}e^{it\hbar ^{-1} a_{\varepsilon ,0}}\Bigg [i \psi (a_{\varepsilon ,0}-{\tilde{a}}_\varepsilon ) - i \sum _{\left| \alpha \right| =1}^{N} \frac{(-i\hbar )^{\left| \alpha \right| }}{\alpha !} \partial _p^\alpha \{\psi \partial _x^\alpha {\tilde{a}}_\varepsilon \} - i \sum _{k=1}^{N } (it\hbar ^{-1})^k \nonumber \\{} & {} \quad \times \sum _{\left| \alpha \right| =k }^{N} \frac{(-i\hbar )^{\left| \alpha \right| }}{\alpha !} \sum _{\begin{array}{c} \beta _1 + \cdots + \beta _k \le \alpha \\ \left| \beta _j \right| >0 \end{array}} c_{\alpha ,\beta _1\cdots \beta _k} \prod _{j=1}^k \partial _p^{\beta _j}a_{\varepsilon ,0} \partial _p^{\alpha -(\beta _1+\cdots +\beta _k )} \{\psi \partial _x^\alpha {\tilde{a}}_\varepsilon \}\Bigg ] \nonumber \\{} & {} ={} i e^{it\hbar ^{-1} a_{\varepsilon ,0}} \sum _{k=0}^N (it\hbar ^{-1})^k q_k(\psi , x,p,\hbar ,\varepsilon ). \end{aligned}$$

(5.4)

From this we note that after acting with $\hbar \partial _t - i {\mathcal {P}}_N$ on $e^{ i t \hbar ^{-1} a_{\varepsilon ,0}(x,p)}\psi (x,p)$ we get $i e^{it\hbar ^{-1} a_{\varepsilon ,0}}$ times a polynomial in $it\hbar ^{-1}$ with coefficients in $C_0^\infty ({\mathbb {R}}^d_x\times {\mathbb {R}}^d_p)$ depending on $\psi$ ${\tilde{a}}_\varepsilon$, $\hbar$ and $\varepsilon$. We will in the following need estimates for the terms $|q_k(\psi , x,p,\hbar ,\varepsilon )|$ for each k. For $k=0$ we get that

$$\begin{aligned}{} & {} |q_0(\psi , x,p,\hbar ,\varepsilon )| = |\psi ( a_{\varepsilon ,0}-{\tilde{a}}_\varepsilon ) + \sum _{\left| \alpha \right| =1}^{N} \frac{(-i\hbar )^{\left| \alpha \right| }}{\alpha !} \partial _p^\alpha \{\psi \partial _x^\alpha {\tilde{a}}_\varepsilon \}| \nonumber \\{} & {} \le c_1\hbar + \sum _{\left| \alpha \right| =1}^{N} \frac{\hbar ^{\left| \alpha \right| }}{\alpha !}\left| \partial _p^\alpha \{\psi \partial _x^\alpha {\tilde{a}}_\varepsilon \} \right| \le c_1\hbar + \sum _{\left| \alpha \right| =1}^{\tau } \frac{\hbar ^{\left| \alpha \right| }}{\alpha !}c_\alpha +\sum _{\left| \alpha \right| =\tau +1}^{N} \frac{\hbar ^{\left| \alpha \right| }}{\alpha !}c_\alpha \varepsilon ^{\tau -\left| \alpha \right| } \nonumber \\{} & {} \le c_1\hbar + \sum _{\left| \alpha \right| =1}^{\tau } \frac{\hbar ^{\left| \alpha \right| }}{\alpha !}c_\alpha +\sum _{\left| \alpha \right| =\tau +1}^{N} \frac{\hbar ^{\left| \alpha \right| }}{\alpha !}c_\alpha \hbar ^{(1-\delta )(\tau -\left| \alpha \right| )} \le C \hbar , \end{aligned}$$

(5.5)

where C depends on the p-derivatives of $\psi$ and $\partial _x^\alpha a_\varepsilon$ on the support of $\psi$ for $\left| \alpha \right| \le N$. For $1\le k \le \tau$ we have

$$\begin{aligned}{} & {} |q_k(\psi , x,p,\hbar ,\varepsilon )| \nonumber \\{} & {} ={} |\sum _{\left| \alpha \right| =k }^{N} \frac{(-i\hbar )^{\left| \alpha \right| }}{\alpha !} \sum _{\begin{array}{c} \beta _1 + \cdots + \beta _k \le \alpha \\ \left| \beta _j \right| >0 \end{array}} c_{\alpha ,\beta _1\cdots \beta _k} \prod _{j=1}^k \partial _p^{\beta _j}a_{\varepsilon ,0} \partial _p^{\alpha -(\beta _1+\cdots +\beta _k )} \{\psi \partial _x^\alpha {\tilde{a}}_\varepsilon \}| \nonumber \\{} & {} \le {} \sum _{\left| \alpha \right| =k }^{\tau } c_\alpha \frac{\hbar ^{\left| \alpha \right| }}{\alpha !} + \sum _{\left| \alpha \right| =\tau +1 }^{N} c_\alpha \frac{\hbar ^{\left| \alpha \right| }}{\alpha !} \varepsilon ^{\tau -\left| \alpha \right| } \nonumber \\{} & {} \le {} \sum _{\left| \alpha \right| =k }^{\tau } c_\alpha \frac{\hbar ^{\left| \alpha \right| }}{\alpha !} + \sum _{\left| \alpha \right| =\tau +1 }^{N} c_\alpha \frac{\hbar ^{\left| \alpha \right| }}{\alpha !} \hbar ^{(1-\delta )(\tau -\left| \alpha \right| )} \le C \hbar ^k, \end{aligned}$$

(5.6)

where C depends on the p-derivatives of $\psi$ and $\partial _x^\alpha a_\varepsilon$ on the support of $\psi$. For $\tau < k \le N$ we have

$$\begin{aligned}{} & {} |q_k(\psi , x,p,\hbar ,\varepsilon )| \nonumber \\{} & {} ={} |\sum _{\left| \alpha \right| =k }^{N} \frac{(-i\hbar )^{\left| \alpha \right| }}{\alpha !} \sum _{\begin{array}{c} \beta _1 + \cdots + \beta _k \le \alpha \\ \left| \beta _j \right| >0 \end{array}} c_{\alpha ,\beta _1\cdots \beta _k} \prod _{j=1}^k \partial _p^{\beta _j}a_{\varepsilon ,0} \partial _p^{\alpha -(\beta _1+\cdots +\beta _k )} \{\psi \partial _x^\alpha {\tilde{a}}_\varepsilon \}| \nonumber \\{} & {} \le {} \sum _{\left| \alpha \right| =k }^{N} c_\alpha \frac{\hbar ^{\left| \alpha \right| }}{\alpha !} \varepsilon ^{\tau -\left| \alpha \right| } \le \ \sum _{\left| \alpha \right| =k }^{N} c_\alpha \frac{\hbar ^{\left| \alpha \right| }}{\alpha !} \hbar ^{(1-\delta )(\tau -\left| \alpha \right| )} \le C \hbar ^{\tau + (k-\tau )\delta }, \end{aligned}$$

(5.7)

where C depends on the p-derivatives of $\psi$ and $\partial _x^\alpha a_\varepsilon$ on the support of $\psi$. It is important that the coefficients only depend on derivatives in p for the function we apply the operator to. One should also note that if $\psi$ had $\hbar$ to some power multiplied to it. Then it should be multiplied to the new power obtained. In the reminder of the proof we will continue to denote the coefficients obtained by acting with $\hbar \partial _t - i {\mathcal {P}}_N$ by $q_k$ and the exact form can be found in (5.4).

We are now ready to start constructing the kernel. We set $u_0(x,p,\hbar ,\varepsilon )=\theta (x,p)$ which gives the first term. In order to find $u_1$ we act with the operator $\hbar \partial _t - i {\mathcal {P}}_N$ on the function $e^{ i t \hbar ^{-1} a_{\varepsilon ,0}} u_0(x,p,\hbar ,\varepsilon )$, where we in the reminder of the construction of the approximation will omit writing the dependence of the variables (x, p) in the exponential. By (5.4) we get

$$\begin{aligned} (\hbar \partial _t - i {\mathcal {P}}_N)e^{ i t \hbar ^{-1} a_{\varepsilon ,0}} u_0(x,p,\hbar ,\varepsilon ) = i e^{it\hbar ^{-1} a_{\varepsilon ,0}} \sum _{k=0}^N (it\hbar ^{-1})^k q_k(u_0, x,p,\hbar ,\varepsilon ). \end{aligned}$$

This would not lead to the desired estimate. So we now take

$$\begin{aligned} u_1(x,p,\hbar ,\varepsilon ) = - q_0(u_0, x,p,\hbar ,\varepsilon ). \end{aligned}$$

We can note by the previous estimates (5.5) we have that

$$\begin{aligned} \left| u_1(x,p,\hbar ,\varepsilon ) \right| = \left| q_0(u_0, x,p,\hbar ,\varepsilon ) \right| \le \hbar C. \end{aligned}$$

(5.8)

Acting with the operator $\hbar \partial _t - i {\mathcal {P}}_N$ on $e^{ i t \hbar ^{-1} a_{\varepsilon ,0}}(u_0(x,p,\hbar ,\varepsilon ) + i t \hbar ^{-1} u_1(x,p,\hbar ,\varepsilon ))$ we obtain according to (5.4) that

$$\begin{aligned} \begin{aligned}&(\hbar \partial _t - i {\mathcal {P}}_N) (e^{ i t \hbar ^{-1} a_{\varepsilon ,0}}(u_0(x,p,\hbar ,\varepsilon ) + i t \hbar ^{-1} u_1(x,p,\hbar ,\varepsilon ))) \\&= i e^{it\hbar ^{-1} a_{\varepsilon ,0}} \sum _{k=0}^N (it\hbar ^{-1})^k q_k(u_0, x,p,\hbar ,\varepsilon ) + i e^{ i t \hbar ^{-1} a_{\varepsilon ,0}} u_1(x,p,\hbar ,\varepsilon ) \\&\quad + i t \hbar ^{-1} i e^{it\hbar ^{-1} a_{\varepsilon ,0}} \sum _{k=0}^N (it\hbar ^{-1})^k q_k(u_1, x,p,\hbar ,\varepsilon ) \\&= i e^{it\hbar ^{-1} a_{\varepsilon ,0}} \left( \sum _{k=1}^N (it\hbar ^{-1})^k q_k(u_0, x,p,\hbar ,\varepsilon ) + \sum _{k=0}^N (it\hbar ^{-1})^{k+1} q_k(u_1, x,p,\hbar ,\varepsilon )\right) . \end{aligned} \end{aligned}$$

Now taking $u_2(x,p,\hbar ,\varepsilon )= - \frac{1}{2} (q_1(u_0,x,p,\hbar ,\varepsilon ) + q_0(u_1,x,p,\hbar ,\varepsilon ))$ and acting with the operator $\hbar \partial _t - i {\mathcal {P}}_N$, according to (5.4), we get that

$$\begin{aligned} \begin{aligned} (\hbar \partial _t&- i {\mathcal {P}}_N) e^{ i t \hbar ^{-1} a_{\varepsilon ,0}}\sum _{j=0}^2 (it\hbar ^{-1})^j u_j(x,p,\hbar ,\varepsilon ) \\&= i e^{it\hbar ^{-1} a_{\varepsilon ,0}} \left[ \sum _{j=0}^2 \sum _{k=0}^N (it\hbar ^{-1})^{k+j} q_k(u_j, x,p,\hbar ,\varepsilon ) + \sum _{j=1}^2 j (it\hbar ^{-1})^{j-1} u_j(x,p,\hbar ,\varepsilon ) \right] \\&= i e^{it\hbar ^{-1} a_{\varepsilon ,0}} \sum _{j=0}^2 \sum _{k=2-j}^N (it\hbar ^{-1})^{k+j} q_k(u_j, x,p,\hbar ,\varepsilon ). \end{aligned} \end{aligned}$$

We note that the “lowest” power of $it\hbar ^{-1}$ is 2, hence we must use these terms to construct $u_3$. Moreover, we note that by (5.5) and (5.6) we have

$$\begin{aligned} \left| u_2(x,p,\hbar ,\varepsilon ) \right| =\tfrac{1}{2} \left| q_1(u_0,x,p,\hbar ,\varepsilon ) + q_0(u_1,x,p,\hbar ,\varepsilon ) \right| \le \tfrac{1}{2} C (\hbar +\hbar ^2) \le C\hbar , \end{aligned}$$

(5.9)

and $u_2$ is a smooth compactly supported function in the variables x and p. Generally for $2\le j\le N$ we have

$$\begin{aligned} u_j(x,p,\hbar ,\varepsilon ) = - \frac{1}{j} \sum _{k=0}^{j-1} q_{j-1-k}(u_k,x,p,\hbar ,\varepsilon ). \end{aligned}$$

We now need estimates for these terms. In the case $\tau =1$ the next step will be empty, but for $\tau \ge 2$ it is needed. For $\tau \ge 2$ we have

$$\begin{aligned} |u_3(x,p,\hbar ,\varepsilon )| \le \frac{1}{3} \sum _{k=0}^{2} |q_{2-k}(u_k,x,p,\hbar ,\varepsilon )| \le C\hbar ^2, \end{aligned}$$

where we have used (5.5), (5.6), (5.8) and (5.9). For the rest of the $u_j$’s we split in the two cases $\tau =1$ or $\tau \ge 2$. First the cases $\tau =1$ for $2\le j\le N$ the estimate is

$$\begin{aligned} \left| u_j(x,p,\hbar ,\varepsilon ) \right| \le C \hbar ^{1+\delta (j-2)} \end{aligned}$$

Note that $u_2$ satisfies the above equation hence if we assume it okay for $j-1$ between 2 and $N-1$ we want to show the above estimate for j. We note that for $j\ge 5$

$$\begin{aligned} \begin{aligned}&|{u_{j}(x,p,\hbar ,\varepsilon )} | \\&\le {} \frac{1}{j} \sum _{k=0}^{j-1} \left| q_{j-1-k}(u_k,x,p,\hbar ,\varepsilon ) \right| \\&\le {} C( \left| q_{j-1}(u_0,x,p,\hbar ,\varepsilon ) \right| + \left| q_{j-2}(u_1,x,p,\hbar ,\varepsilon ) \right| \\&\quad +\sum _{k=2}^{j-3} \left| q_{j-1-k}(u_k,x,p,\hbar ,\varepsilon ) \right| + \left| q_{1}(u_{j-2},x,p,\hbar ,\varepsilon ) \right| + \left| q_{0}(u_{j-1},x,p,\hbar ,\varepsilon ) \right| ) \\&\le {} C(\hbar ^{1+\delta (j-2)} + \hbar ^{2+\delta (j-3)} + \sum _{k=2}^{j-3} \hbar ^{1+\delta (j-1-k-1) + 1+\delta (k-2)} + \hbar ^{2+\delta (j-4)}+ \hbar ^{2+\delta (j-3)}) \\&\le {} C(\hbar ^{1+\delta (j-2)} + \hbar ^{2+\delta (j-3)} + \hbar ^{2+\delta (j-4)} + \hbar ^{2+\delta (j-3)}) \le C \hbar ^{1+\delta (j-2)}, \end{aligned} \end{aligned}$$

where we have used (5.5), (5.7) and the induction assumption. The cases $j=3$ and $j=4$ are estimated analogously.

Now the case $\tau \ge 2$ which we will treat as $\tau =2$, here the estimate is

$$\begin{aligned} \left| u_j(x,p,\hbar ,\varepsilon ) \right| \le C \hbar ^{2+\delta (j-3)} \end{aligned}$$

for $3\le j \le N$. To prove this bound is the same as in the case of $\tau =1$. In order to prove the bound with the derivatives as stated in the theorem the above arguments are repeated with a number of derivatives on the $u_j$’s otherwise it is analogous.

What remains is to prove this construction satisfies the bound

$$\begin{aligned} \sup _{t\in [-\hbar ^{1-\frac{\delta }{2}},\hbar ^{1-\frac{\delta }{2}}]} \Vert \hbar \partial _t U_N(t, \hbar ) - i U_N(t, \hbar ) A_\varepsilon \Vert _{{\mathcal {L}}(L^2({\mathbb {R}}^d))} \le C \hbar ^{N_0}. \end{aligned}$$

Here we only consider the case $\tau =1$ as the cases $\tau \ge 2$ will have better estimates. From the above estimates on $u_k(x,p,\hbar ,\varepsilon )$ we have for k in $\{0,\dots ,N\}$ and $\left| t \right| \le \hbar ^{1-\frac{\delta }{2}}$ that

$$\begin{aligned} \left| (it\hbar ^{-1})^k u_k(x,p,\hbar ,\varepsilon ) \right| \le {\left\{ \begin{array}{ll} C &{}k=0 \\ C \hbar ^{1-\frac{\delta }{2}} &{} k=1 \\ C \hbar ^{1 + \delta (\frac{k}{2} -2)} &{} k\ge 2. \end{array}\right. } \end{aligned}$$

(5.10)

The first step is to apply the operator $\hbar \partial _t - i {\mathcal {P}}_N$ on then “full” kernel and see what error this produces. By construction we have

$$\begin{aligned} (\hbar \partial _t - i {\mathcal {P}}_N)e^{ i t \hbar ^{-1} a_{\varepsilon ,0}}\sum _{k=0}^N (it\hbar ^{-1})^k u_k(x,p,\hbar ,\varepsilon ) = \sum _{j=0}^N \sum _{k=N-j}^N (it\hbar ^{-1})^{k+j} q_k(u_j, x,p,\hbar ,\varepsilon ). \end{aligned}$$

If we start by considering j equal 0 and 1 we note that:

$$\begin{aligned}{} & {} \Big |\sum _{j=0}^1 \sum _{k=N-j}^N (it\hbar ^{-1})^{k+j} q_k(u_j, x,p,\hbar ,\varepsilon ) \Big | \nonumber \\{} & {} \le C(\hbar ^{-\frac{\delta }{2}N} \hbar ^{1+\delta (N-1)} + \hbar ^{-\frac{\delta }{2}N} \hbar ^{2+\delta (N-2)} + \hbar ^{-\frac{\delta }{2}(N+1)} \hbar ^{2+\delta (N-1)}) \nonumber \\{} & {} \le C( \hbar ^{1+ \delta (\frac{N}{2} -1)} + \hbar ^{2+ \delta (\frac{N}{2} -2)} + \hbar ^{2+ \frac{\delta }{2}(N -3)}) \nonumber \\{} & {} \le {\tilde{C}} \hbar ^{1+ \delta (\frac{N}{2} -1)} \le {\tilde{C}} \hbar ^{N_0+d}, \end{aligned}$$

(5.11)

where we have used (5.10) and our choice of N. For the rest of the terms we have that

$$\begin{aligned}{} & {} \Big | \sum _{j=2}^N \sum _{k=N-j}^N (it\hbar ^{-1})^{k+j} q_k(u_j, x,p,\hbar ,\varepsilon ) \Big | \nonumber \\{} & {} \le C \Big [ \sum _{j=2}^N \sum _{k=\max (N-j,1)}^N \hbar ^{-\frac{\delta }{2}(k+j)} \hbar ^{2+\delta (k-1) + \delta (j-2)} + \hbar ^{-\frac{\delta }{2}N} \hbar ^{2+ \delta (N-2)}\Big ] \nonumber \\{} & {} \le C\Big [ \sum _{j=2}^N \sum _{k=\max (N-j,1)}^N \hbar ^{2 + \delta (\frac{j+k}{2} -3)} + \hbar ^{2+ \delta (\frac{N}{2}-2)}\Big ] \le {\tilde{C}} \hbar ^{N_0+d}, \end{aligned}$$

(5.12)

where we have used (5.10), our choice of N and that in the double sum $k+j\ge N$. When (5.11) and (5.12) are combined we have that

$$\begin{aligned} \big |(\hbar \partial _t - i {\mathcal {P}}_N)e^{ i t \hbar ^{-1} a_{\varepsilon ,0}}\sum _{k=0}^N (it\hbar ^{-1})^k u_k(x,p,\hbar ,\varepsilon ) \big | \le C \hbar ^{N_0+d}, \end{aligned}$$

(5.13)

where $\left| t \right| \le \hbar ^{1-\frac{\delta }{2}}$. We now let $U_N(t,\hbar )$ be the operator with the integral kernel:

$$\begin{aligned} K_{U_N}(x,y,t,\varepsilon ,\hbar ) = \frac{1}{(2\pi \hbar )^d} \int _{{\mathbb {R}}^d} e^{i \hbar ^{-1} \langle x-y , p \rangle } e^{ i t \hbar ^{-1} a_{\varepsilon ,0}(x,p)} \sum _{j=0}^N (it\hbar ^{-1})^j u_j(x,p,\hbar ,\varepsilon ) \, dp, \end{aligned}$$

which is well defined due to our previous estimates (5.10). In particular we have that it is a bounded operator by the Schur test. We now need to find an expression for

$$\begin{aligned} \hbar \partial _t U_N(t, \hbar ) - i U_N(t, \hbar ) A_\varepsilon (\hbar ). \end{aligned}$$

At the beginning of the proof, we wrote the operator $A_\varepsilon (\hbar )$ in two different ways (5.1) and (5.2). If we combine these we have

$$\begin{aligned} \begin{aligned} A_\varepsilon (\hbar )&= \sum _{j=0}^{{\tilde{M}}} \hbar ^j \text {Op} _{\hbar ,1}({\tilde{a}}_{\varepsilon ,j}) + \hbar ^{{\tilde{M}}+1} {\tilde{R}}_{{\tilde{M}}}(\varepsilon ,\hbar ) + \hbar ^{M+1} R_M(\varepsilon ,\hbar ) \\&= \text {Op} _{\hbar ,1}({\tilde{a}}_{\varepsilon }) + \hbar ^{{\tilde{M}}+1} {\tilde{R}}_{{\tilde{M}}}(\varepsilon ,\hbar ) + \hbar ^{M+1} R_M(\varepsilon ,\hbar ), \end{aligned} \end{aligned}$$

where the two reminder terms satisfy

$$\begin{aligned} \Vert \hbar ^{{\tilde{M}}+1}{\tilde{R}}_{{\tilde{M}}}(\varepsilon ,\hbar ) + \hbar ^{M+1} R_M(\varepsilon ,\hbar ) \Vert _{{\mathcal {L}}(L^2({\mathbb {R}}^d))} \le C \hbar ^{N_0}. \end{aligned}$$

If we use this form of $A_\varepsilon (\hbar )$ we have

$$\begin{aligned} \begin{aligned}&\hbar \partial _t U_N(t, \hbar ) - i U_N(t, \hbar ) A_\varepsilon (\hbar ) \\&= \hbar \partial _t U_N(t, \hbar ) - i U_N(t,\hbar ) \text {Op} _{\hbar ,1}({\tilde{a}}_{\varepsilon }) - i\hbar ^{{\tilde{M}}+1} U_N(t, \hbar ) {\tilde{R}}_{{\tilde{M}}}(\varepsilon ,\hbar ) \\&\quad - i \hbar ^{M+1} U_N(t, \hbar )R_M(\varepsilon ,\hbar ). \end{aligned} \end{aligned}$$

When considering the operator norm of the two last terms we have

$$\begin{aligned} \Vert \hbar ^{{\tilde{M}}+1} U_N(t, \hbar ){\tilde{R}}_{{\tilde{M}}}(\varepsilon ,\hbar ) + \hbar ^{M+1} U_N(t, \hbar ) R_M(\varepsilon ,\hbar ) \Vert _{{\mathcal {L}}(L^2({\mathbb {R}}^d))} \le C \hbar ^{N_0} \end{aligned}$$

(5.14)

as $U_N(t, \hbar )$ is a bounded operator. What remains is the expression

$$\begin{aligned} \hbar \partial _t U_N(t, \hbar ) - i U_N(t, \hbar ) \text {Op} _{\hbar ,1}({\tilde{a}}_{\varepsilon }). \end{aligned}$$

The rules for composition of kernels gives, by a straight forward calculation, that the kernel of the above expression is

$$\begin{aligned} \begin{aligned}&K(x,y;\varepsilon ,\hbar ) \\&{:}{=}{}\frac{1}{(2\pi \hbar )^d} \int _{{\mathbb {R}}^d} e^{i \hbar ^{-1} \langle x-y , p \rangle } (\hbar \partial _t - i{\tilde{a}}_\varepsilon (y,p)) e^{ i t \hbar ^{-1} a_{\varepsilon ,0}(x,p)} \sum _{j=0}^N (it\hbar ^{-1})^j u_j(x,p,\hbar ,\varepsilon ) \, dp. \end{aligned} \end{aligned}$$

By performing a Taylor expansion of ${\tilde{a}}_\varepsilon$ in the variable y centred at x we obtain that

$$\begin{aligned} \begin{aligned} {\tilde{a}}_\varepsilon (y,p)&={} \sum _{\left| \alpha \right| \le N} \frac{(y-x)^\alpha }{\alpha !} \partial _x^\alpha {\tilde{a}}_\varepsilon (x,p) \\&+ \sum _{\left| \alpha \right| =N+1} (N+1) \frac{(y-x)^\alpha }{\alpha !} \int _0^1 (1-s)^N \partial _x^\alpha {\tilde{a}}_\varepsilon (x + s(y-x),p) \, ds. \end{aligned} \end{aligned}$$

We replace ${\tilde{a}}_\varepsilon (y,p)$ by the above Taylor expansion in the kernel. First, we consider the part of the kernel with the first sum. Here, we have that

$$\begin{aligned} \begin{aligned}&\int _{{\mathbb {R}}^d} e^{i \hbar ^{-1} \langle x-y,p \rangle } \big [ \hbar \partial _t -i\sum _{\left| \alpha \right| \le N} \frac{(y-x)^\alpha }{\alpha !} \partial _x^\alpha {\tilde{a}}_\varepsilon (x,p) \big ] e^{ i t \hbar ^{-1}a_{\varepsilon ,0}(x,p)} \\&\quad \times \sum _{j=0}^N (it\hbar ^{-1})^j u_j(x,p,\hbar ,\varepsilon ) \big ] \, dp \\&={} \int _{{\mathbb {R}}^d} e^{i \hbar ^{-1} \langle x-y,p \rangle }[ \hbar \partial _t e^{ i t \hbar ^{-1} a_{\varepsilon ,0}(x,p)}\sum _{j=0}^N (it\hbar ^{-1})^j u_j(x,p,\hbar ,\varepsilon ) \\&\quad -i\sum _{\left| \alpha \right| \le N} \frac{(-i\hbar )^{\left| \alpha \right| }}{\alpha !} \partial _p^\alpha [ \partial _x^\alpha {\tilde{a}}_\varepsilon (x,p) e^{ i t \hbar ^{-1}a_{\varepsilon ,0}(x,p)}\sum _{j=0}^N (it\hbar ^{-1})^j u_j(x,p,\hbar ,\varepsilon )]] \, dp \\&={} \int _{{\mathbb {R}}^d} e^{i \hbar ^{-1} \langle x-y,p \rangle } ( \hbar \partial _t- i {\mathcal {P}}_N) [ e^{ i t \hbar ^{-1} a_\varepsilon (x,p)} \sum _{j=0}^N (it\hbar ^{-1})^j u_j(x,p,\hbar ,\varepsilon )] \, dp, \end{aligned} \end{aligned}$$

where we have used the identity $(y-x)^\alpha e^{i \hbar ^{-1} \langle x-y,p \rangle }= (-i\hbar )^\alpha \partial _p^\alpha e^{i \hbar ^{-1} \langle x-y,p \rangle }$, integration by parts and omitted the pre-factor $(2\pi \hbar )^{-d}$. When considering the part of the kernel with the error term, we have that

$$\begin{aligned}&\frac{-i}{(2\pi \hbar )^d} \int _{{\mathbb {R}}^d} e^{i \hbar ^{-1} \langle x-y,p \rangle } e^{ i t \hbar ^{-1} a_\varepsilon (x,p)} \sum _{j=0}^N (it\hbar ^{-1})^j u_j(x,p,\hbar ,\varepsilon ) \\&\quad \times \sum _{\left| \alpha \right| =N+1} (N+1) \frac{(y-x)^\alpha }{\alpha !} \int _0^1 (1-s)^N \partial _x^\alpha {\tilde{a}}_\varepsilon (x + s(y-x),p) \, ds dp \\&=\frac{-i}{(2\pi \hbar )^d} \int _{{\mathbb {R}}^d} e^{i \hbar ^{-1} \langle x-y,p \rangle } \sum _{\left| \alpha \right| =N+1} (N+1) \frac{(-i\hbar )^\alpha }{\alpha !} \partial _p^\alpha [ e^{ - i t \hbar ^{-1} a_\varepsilon (x,p)} \sum _{j=0}^N (-it\hbar ^{-1})^j \\&\quad \times u_j(x,p,\hbar ,\varepsilon ) \int _0^1 (1-s)^N \partial _x^\alpha {\tilde{a}}_\varepsilon (x + s(y-x),p) \, ds] dp, \end{aligned}$$

where we again have used the identity $(y-x)^\alpha e^{i \hbar ^{-1} \langle x-y,p \rangle }= (-i\hbar )^\alpha \partial _p^\alpha e^{i \hbar ^{-1} \langle x-y,p \rangle }$ and integration by parts. Combining the two expressions we get that

$$\begin{aligned} \begin{aligned}&K(x,y;\varepsilon ,\hbar ) \\&= \frac{1}{(2\pi \hbar )^d} \int _{{\mathbb {R}}^d} e^{i \hbar ^{-1} \langle x-y,p \rangle } ( \hbar \partial _t- i {\mathcal {P}}_N) [ e^{ i t \hbar ^{-1} a_\varepsilon (x,p)} \sum _{j=0}^N (it\hbar ^{-1})^j u_j(x,p,\hbar ,\varepsilon )] \, dp \\&\quad + \frac{-i}{(2\pi \hbar )^d} \int _{{\mathbb {R}}^d} e^{i \hbar ^{-1} \langle x-y,p \rangle } \sum _{\left| \alpha \right| =N+1} (N+1) \frac{(-i\hbar )^\alpha }{\alpha !} \partial _p^\alpha [ e^{ i t \hbar ^{-1} a_\varepsilon (x,p)} \sum _{j=0}^N (it\hbar ^{-1})^j \\&\quad \times u_j(x,p,\hbar ,\varepsilon ) \int _0^1 (1-s)^N \partial _x^\alpha {\tilde{a}}_\varepsilon (x + s(y-x),p) \, ds] dp. \end{aligned} \end{aligned}$$

(5.15)

In order to estimate the operator norm, we will divide the kernel into two parts. We do this by considering a part localised in y and the reminder. To localise in y we let $\psi$ be a smooth function on ${\mathbb {R}}^d$ such $\psi (y) = 1$ on the set $\{y\in {\mathbb {R}}^d \,|\, \text {dist} [y,\text {supp} _x(\theta )]\le 1\}$, $0\le \psi (y)\le 1$ for all $y\in {\mathbb {R}}^d$ and supported in the set $\{y\in {\mathbb {R}}^d \,|\, \text {dist} [y,\text {supp} _x(\theta )]\le 2\}$. With this function our kernel can be written as

$$\begin{aligned} K(x,y;\varepsilon ,\hbar ) = K(x,y;\varepsilon ,\hbar ) \psi (y) + K(x,y;\varepsilon ,\hbar ) (1-\psi (y)). \end{aligned}$$

(5.16)

If we consider the part multiplied by $\psi (y)$ then this part has the form as in (5.15) but each term is multiplied by $\psi (y)$. By the estimate in (5.13) we have for the first part of $K(x,y;\varepsilon ,\hbar ) \psi (y)$ the following estimate

$$\begin{aligned}{} & {} \Big | \int _{{\mathbb {R}}^d} e^{i \hbar ^{-1} \langle x-y,p \rangle } ( \hbar \partial _t- i {\mathcal {P}}_N) [ e^{ i t \hbar ^{-1} a_{\varepsilon ,0}(x,p)} \sum _{j=0}^N (it\hbar ^{-1})^j u_j(x,p,\hbar ,\varepsilon )]\psi (y) \, dp \Big | \nonumber \\{} & {} \le \psi (x)\psi (y) C \hbar ^{N_0+d}. \end{aligned}$$

(5.17)

For the second part of $K(x,y;\varepsilon ,\hbar ) \psi (y)$, we have by Leibniz’s formula and Faà di Bruno formula (Theorem A.1) for each term in the sum over $\alpha$ that

$$\begin{aligned}&\frac{(-i\hbar )^{\left| \alpha \right| }}{\alpha !} \partial _p^\alpha \big [e^{ i t \hbar ^{-1} a_{\varepsilon ,0}(x,p)} \sum _{j=0}^N (it\hbar ^{-1})^j u_j(x,p,\hbar ,\varepsilon ) \\&\quad \times \int _0^1 (1-s)^N \partial _x^\alpha {\tilde{a}}_\varepsilon (x + s(y-x),p) \, ds\big ]\psi (y) \\&={}\sum _{j=0}^N \frac{(-i\hbar )^{\left| \alpha \right| }}{\alpha !} e^{ i t \hbar ^{-1} a_{\varepsilon ,0}(x,p)} \sum _{k=0}^{N+1} (it\hbar ^{-1})^{k+j} \quad \sum _{{\begin{array}{c} \beta _1 + \cdots + \beta _k \le \alpha \\ \left| \beta _j \right| >0 \end{array}}} \quad c_{\alpha ,\beta _1\cdots \beta _k} \prod _{n=1}^k \partial _p^{\beta _n}a_{\varepsilon ,0}(x,p) \\&\times \partial _p^{\alpha -(\beta _1+\cdots +\beta _k )} [u_j(x,p,\hbar ,\varepsilon ) \int _0^1 (1-s)^N \partial _x^\alpha {\tilde{a}}_\varepsilon (x + s(y-x),p) \, ds]\psi (y). \end{aligned}$$

We note that for j equal 0 we have an estimate of the following form:

$$\begin{aligned}&\Big |\frac{(-i\hbar )^{\left| \alpha \right| }}{\alpha !} e^{ i t \hbar ^{-1} a_{\varepsilon ,0}(x,p)} \sum _{k=0}^{N+1} (it\hbar ^{-1})^k \sum _{\begin{array}{c} \beta _1 + \cdots + \beta _k \le \alpha \nonumber \\ \left| \beta _j \right| >0 \end{array}} c_{\alpha ,\beta _1\cdots \beta _k} \prod _{n=1}^k \partial _p^{\beta _n}a_{\varepsilon ,0}(x,p)\nonumber \\&\quad \times \partial _p^{\alpha -(\beta _1+\cdots +\beta _k )} [u_0(x,p,\hbar ,\varepsilon ) \int _0^1 (1-s)^N \partial _x^\alpha {\tilde{a}}_\varepsilon (x + s(y-x),p) \, ds]\psi (y)\Big |\nonumber \\&\le {} C \sum _{k=0}^{N+1} \hbar ^{N+1} \hbar ^{-\frac{\delta }{2}k} \varepsilon ^{-N} \psi (x)\psi (y) \le C \hbar ^{N+1}\hbar ^{-\frac{\delta }{2}(N+1)} \hbar ^{-N +\delta N}\psi (x)\psi (y)\nonumber \\&\le {} C \hbar ^{N_0+d}\psi (x)\psi (y). \end{aligned}$$

(5.18)

We note that for j equal 1 we have an error of the following form:

$$\begin{aligned}&\Big |\frac{(-i\hbar )^{\left| \alpha \right| }}{\alpha !} e^{ i t \hbar ^{-1} a_{\varepsilon ,0}(x,p)} \sum _{k=0}^{N+1} (it\hbar ^{-1})^{k+1} \sum _{\begin{array}{c} \beta _1 + \cdots + \beta _k \le \alpha \nonumber \\ \left| \beta _j \right| >0 \end{array}} c_{\alpha ,\beta _1\cdots \beta _k} \prod _{n=1}^k \partial _p^{\beta _n}a_{\varepsilon ,0}(x,p) \nonumber \\&\times \partial _p^{\alpha -(\beta _1+\cdots +\beta _k )} [u_1(x,p,\hbar ,\varepsilon ) \int _0^1 (1-s)^N \partial _x^\alpha {\tilde{a}}_\varepsilon (x + s(y-x),p) \, ds]\psi (y)\Big | \nonumber \\&\le C \sum _{k=0}^{N+1} \hbar ^{N+1} \hbar ^{-\frac{\delta }{2}(k+1)} \hbar \varepsilon ^{-N}\psi (x)\psi (y) \le C \hbar ^{N+2}\hbar ^{-\frac{\delta }{2}(N+2)} \hbar ^{-N +\delta N}\psi (x)\psi (y) \nonumber \\&\le C \hbar ^{N_0+d}\psi (x)\psi (y), \end{aligned}$$

(5.19)

where we have used the estimate $|u_1(x,p,\hbar ,\varepsilon )|\le c\hbar$. We note that for j greater than or equal to 2, we have an error of the following form:

$$\begin{aligned}&\Big |\frac{(-i\hbar )^{\left| \alpha \right| }}{\alpha !} e^{ i t \hbar ^{-1} a_{\varepsilon ,0}(x,p)} \sum _{k=0}^{N+1} (it\hbar ^{-1})^{k+j} \sum _{\begin{array}{c} \beta _1 + \cdots + \beta _k \le \alpha \nonumber \\ \left| \beta _j \right| >0 \end{array}} c_{\alpha ,\beta _1\cdots \beta _k} \prod _{n=1}^k \partial _p^{\beta _n}a_{\varepsilon ,0}(x,p) \nonumber \\&\quad \times \partial _p^{\alpha -(\beta _1+\cdots +\beta _k )} [u_j(x,p,\hbar ,\varepsilon ) \int _0^1 (1-s)^N \partial _x^\alpha {\tilde{a}}_\varepsilon (x + s(y-x),p) \, ds]\psi (y) \Big | \nonumber \\&\le C \sum _{k=0}^{N+1} \hbar ^{N+1} \hbar ^{-\frac{\delta }{2}(k+j)} \hbar ^{1+\delta (j-2)} \varepsilon ^{-N} \psi (x)\psi (y) \nonumber \\&\le C \hbar ^{N+1} \hbar ^{-\frac{\delta }{2}(N+1+j)}\hbar ^{1+\delta (j-2)} \hbar ^{-N+\delta N}\psi (x)\psi (y) \le C \hbar ^{2 + \frac{\delta }{2}(N + j -1) -2\delta }\psi (x)\psi (y) \nonumber \\&\le C \hbar ^{N_0+d }\psi (x)\psi (y), \end{aligned}$$

(5.20)

where we have used the estimate $|u_j(x,p,\hbar ,\varepsilon )|\le c \hbar ^{1+\delta (j-2)}$. Now by combining (5.18), (5.19) and (5.20) we arrive at

$$\begin{aligned} \begin{aligned}&\Bigl | \int _{{\mathbb {R}}^d} e^{i \hbar ^{-1} \langle x-y,p \rangle } \sum _{\left| \alpha \right| =N+1} (N+1) \frac{(-i\hbar )^\alpha }{\alpha !} \partial _p^\alpha [ e^{ i t \hbar ^{-1} a_\varepsilon (x,p)} \sum _{j=0}^N (it\hbar ^{-1})^j \\&\quad \times u_j(x,p,\hbar ,\varepsilon ) \int _0^1 (1-s)^N \partial _x^\alpha {\tilde{a}}_\varepsilon (x + s(y-x),p) \, ds]\psi (y) \,dp\Bigr | \\&\le {} C \hbar ^{N_0+d}\psi (x)\psi (y). \end{aligned} \end{aligned}$$

Combining this estimate with (5.17) we have that

$$\begin{aligned} | K(x,y;\varepsilon ,\hbar ) \psi (y)| \le C \hbar ^{N_0}\psi (x)\psi (y). \end{aligned}$$

(5.21)

Now we turn to the term $K(x,y;\varepsilon ,\hbar )(1- \psi (y))$. On the support of this kernel we have

$$\begin{aligned} 1\le \left| x-y \right| \end{aligned}$$

due to the definition of $\psi$. This imply we can divide by the difference between x and y where the kernel is supported. The idea is now to multiply the kernel with $\frac{\left| x-y \right| }{\left| x-y \right| }$ to an appropriate power $\eta$. We take $\eta$ such

$$\begin{aligned} \frac{m(x+s(y-x),p)}{\left| x-y \right| ^{2\eta }} \le \frac{C}{\left| x-y \right| ^{d+1}} \quad \text {for } (x,p) \in \text {supp} (\theta ), \end{aligned}$$

where m is the tempered weight function associated to our operator. The existence of such a $\eta$ is ensured by the definition of the tempered weight. By (5.15) the kernel $K(x,y;\varepsilon ,\hbar )(1- \psi (y))$ is of the form

$$\begin{aligned} \begin{aligned} \frac{1}{(2\pi \hbar )^d} \int _{{\mathbb {R}}^d} e^{i \hbar ^{-1} \langle x-y,p \rangle } \varphi (x,y,p;\hbar ,\varepsilon ) (1- \psi (y))dp, \end{aligned} \end{aligned}$$

where the exact form of $\varphi$ is not important at the moment. Now for our choice of $\eta$ we have

$$\begin{aligned} \begin{aligned}&\int _{{\mathbb {R}}^d} e^{i \hbar ^{-1} \langle x-y,p \rangle } \varphi (x,y,p;\hbar ,\varepsilon ) (1- \psi (y))dp \\&= \int _{{\mathbb {R}}^d} e^{i \hbar ^{-1} \langle x-y,p \rangle } \frac{\left| x-y \right| ^{2\eta }}{\left| x-y \right| ^{2\eta }} \varphi (x,y,p;\hbar ,\varepsilon ) (1- \psi (y))dp \\&= \int _{{\mathbb {R}}^d} (-i\hbar )^{2\eta } \sum _{\left| \gamma \right| =\eta } c_\gamma \partial _p^{2\gamma } (e^{i \hbar ^{-1} \langle x-y,p \rangle }) \frac{1}{\left| x-y \right| ^{2\eta }} \varphi (x,y,p;\hbar ,\varepsilon ) (1- \psi (y))dp \\&= \int _{{\mathbb {R}}^d} e^{i \hbar ^{-1} \langle x-y,p \rangle } \frac{1- \psi (y)}{\left| x-y \right| ^{2\eta }}\sum _{\left| \gamma \right| =\eta } c_\gamma (i\hbar )^{2\eta } \partial _p^{2\gamma }\varphi (x,y,p;\hbar ,\varepsilon ) dp. \end{aligned} \end{aligned}$$

By analogous estimates to the estimate used above we have

$$\begin{aligned} \begin{aligned}&\Big |\frac{1- \psi (y)}{\left| x-y \right| ^{2\eta }}\sum _{\left| \gamma \right| =\eta } (i\hbar )^{2\eta } \partial _p^{2\gamma }\varphi (x,y,p;\hbar ,\varepsilon )\Big | \\&\le C \hbar ^{2\eta (1-\frac{\delta }{2})+1 +\frac{ \delta }{2}(N -1)} \frac{1- \psi (y)}{\left| x-y \right| ^{d+1}} {\varvec{1}}_{\text {supp} (\theta )}(x,p), \end{aligned} \end{aligned}$$

where the term $\hbar ^{-\eta \delta }$ is due to the exponentials $e^{ i t \hbar ^{-1} a_\varepsilon (x,p)}$ in $\varphi$. These give $it\hbar ^{-1}$, when we take a derivative with respect to $p_j$ for all j in $\{1,\dots ,d\}$. Since we have that $|t|\le \hbar ^{1-\frac{\delta }{2}}$ we get the estimate $|t\hbar ^{-1}| \le \hbar ^{-\frac{\delta }{2}}$. The rest of the powers in $\hbar$ can be found similar to above. Hence we have

$$\begin{aligned} \begin{aligned} |K(x,y;\varepsilon ,\hbar )(1- \psi (y))|&\le {} C \hbar ^{2\eta (1-\frac{\delta }{2})+1 +\frac{ \delta }{2}(N -1) -d} {\varvec{1}}_{\text {supp} _x(\theta )}(x) \frac{1- \psi (y)}{\left| x-y \right| ^{d+1}} \\ \le {}&C \hbar ^{N_0} \psi (x) \frac{1- \psi (y)}{\left| x-y \right| ^{d+1}}. \end{aligned} \end{aligned}$$

By combining this with (5.16) and (5.21) we have

$$\begin{aligned} |K(x,y;\varepsilon ,\hbar )| \le C \hbar ^{N_0}\Big [\psi (x)\psi (y) + {\varvec{1}}_{\text {supp} _x(\theta )}(x) \frac{1- \psi (y)}{\left| x-y \right| ^{d+1}}\Big ]. \end{aligned}$$

(5.22)

We have by definition of $\psi$ the estimates

$$\begin{aligned} \begin{aligned}&\sup _{x\in {\mathbb {R}}^d} \int _{{\mathbb {R}}^d} |\psi (x)\psi (y)+ {\varvec{1}}_{\text {supp} _x(\theta )}(x) \frac{1- \psi (y)}{\left| x-y \right| ^{d+1}}| \, dy \le c + \int _{\left| y \right| \ge 1} \frac{1}{\left| y \right| ^{d+1}} \, dy \le C_1 \\&\sup _{y\in {\mathbb {R}}^d} \int _{{\mathbb {R}}^d} |\psi (x)\psi (y)+ {\varvec{1}}_{\text {supp} _x(\theta )}(x) \frac{1- \psi (y)}{\left| x-y \right| ^{d+1}}| \, dx \le C_2 \end{aligned} \end{aligned}$$

These estimates combined with the Schur test, (5.14) and (5.22) give us that

$$\begin{aligned} \Vert \hbar \partial _t U_N(t, \hbar ) - i U_N(t, \hbar ) A_\varepsilon \Vert _{{\mathcal {L}}(L^2({\mathbb {R}}^d))} \le C \hbar ^{N_0} \end{aligned}$$

for $\left| t \right| \le \hbar ^{1-\frac{\delta }{2}}$. As this is the desired estimate, this concludes the proof. $\square$

In the previous proof we constructed a microlocal approximation for the propagator for short times dependent on $\hbar$. It would be preferable not to have this dependence of $\hbar$ in the time. In the following Lemma we prove, that under a non-critical condition on the principal symbol, a localised trace of the approximation becomes negligible.

Lemma 5.3

Let $A_\varepsilon (\hbar )$ be a $\hbar$-$\varepsilon$-admissible operator of regularity $\tau \ge 1$ with tempered weight m which is self-adjoint for all $\hbar$ in $(0,\hbar _0]$ and with $\varepsilon \ge \hbar ^{1-\delta }$ for a $\delta \in (0,1)$. Let $\theta (x,p)$ be a function in $C_0^\infty ({\mathbb {R}}^d_x\times {\mathbb {R}}^d_p)$. Suppose

$$\begin{aligned} \left| \nabla _p a_{\varepsilon ,0}(x,p) \right| \ge c>0 \quad \text {for all } (x,p)\in \text {supp} (\theta ), \end{aligned}$$

where $a_{\varepsilon ,0}$ is the principal symbol of $A_\varepsilon (\hbar )$. Moreover let the operator $U_N(t,\hbar )$ be the one constructed in Theorem 5.1 with the function $\theta$. Then for $\left| t \right| \in [\frac{1}{2}\hbar ^{1-\frac{\delta }{2}},1]$ and every $N_0$ in ${\mathbb {N}}$ it holds that

$$\begin{aligned} \left| \text {Tr} [U_N(t,\hbar ) \text {Op} _{\hbar ,1}(\theta )] \right| \le C \hbar ^{N_0} \end{aligned}$$

for a constant $C>0$, which depends on the constant from the non-critical condition.

Proof

Recall that the kernel of $U_N(t,\hbar )$ is given by

$$\begin{aligned} K_{U_N}(x,y,t,\varepsilon ,\hbar ) = \frac{1}{(2\pi \hbar )^d} \int _{{\mathbb {R}}^d} e^{i \hbar ^{-1} \langle x-y,p\rangle } e^{ i t \hbar ^{-1} a_\varepsilon (x,p)} u_N(x,p,t,\hbar ,\varepsilon ) \, dp, \end{aligned}$$

where

$$\begin{aligned} u_N(x,p,t,\hbar ,\varepsilon ) = \sum _{j=0}^N (it\hbar ^{-1})^j u_j(x,p,\hbar ,\varepsilon ). \end{aligned}$$

From Theorem 5.1 we have the estimate

$$\begin{aligned} \sup _{x,p} \sup _{\left| t \right| \le 1}| \partial _x^\alpha \partial _p^\beta u_N(x,p,t,\hbar ,\varepsilon ) | \le C_{\alpha \beta } \hbar ^{(\delta -1)N} \varepsilon ^{-\left| \alpha \right| }. \end{aligned}$$

(5.23)

This initial estimate is a priori not desirable as it implies the trace is of order $\hbar ^{(\delta -1)N-d}$. Due to the form of the kernels for the operators $U_N(t,\hbar )$ and $\text {Op} _{\hbar ,1}(\theta )$ it immediately follows that the kernel for the composition is given by

$$\begin{aligned} (x,y) \mapsto \frac{1}{(2\pi \hbar )^d} \int _{{\mathbb {R}}^d} e^{i \hbar ^{-1} \langle x-y,p\rangle } e^{ i t \hbar ^{-1} a_\varepsilon (x,p)} u_N(x,p,t,\hbar ,\varepsilon ) \theta (y,p) \, dp. \end{aligned}$$

With this expression for the kernel we get that the trace is given by

$$\begin{aligned} \text {Tr} [U_N(t,\hbar ) \text {Op} _{\hbar ,1}(\theta )] = \frac{1}{(2\pi \hbar )^d} \int _{{\mathbb {R}}^{2d}} e^{ i t \hbar ^{-1} a_\varepsilon (x,p)} u_N(x,p,t,\hbar ,\varepsilon ) \theta (x,p) \, dxdp. \end{aligned}$$

Since we suppose $\left| \nabla _p a_{\varepsilon ,0} \right| \ge c>0$ on the support of $\theta (x,p)$ we have that

$$\begin{aligned} \begin{aligned}&\text {Tr} [U_N(t,\hbar ) \text {Op} _{\hbar ,1}(\theta )] \\&={} \frac{1}{(2\pi \hbar )^d} \int _{{\mathbb {R}}^{2d}} \frac{\sum _{j=1}^d (\partial _{p_j}a_{\varepsilon ,0}(x,p))^2}{\left| \nabla _p a_{\varepsilon ,0}(x,p) \right| ^2} e^{ i t \hbar ^{-1} a_\varepsilon (x,p)} u_N(x,p,t,\hbar ,\varepsilon ) \theta (x,p) \, dxdp \\&={}\frac{-i\hbar t^{-1}}{(2\pi \hbar )^d} \sum _{j=1}^d \int _{{\mathbb {R}}^{2d}} \partial _{p_j}e^{ i t \hbar ^{-1} a_\varepsilon (x,p)} \frac{\partial _{p_j}a_{\varepsilon ,0}(x,p)}{\left| \nabla _pa_{\varepsilon ,0}(x,p) \right| ^2} u_N(x,p,t,\hbar ,\varepsilon ) \theta (x,p) \, dxdp \\&={} \frac{i\hbar t^{-1}}{(2\pi \hbar )^d} \sum _{j=1}^d \int _{{\mathbb {R}}^{2d}} e^{ i t \hbar ^{-1} a_{\varepsilon ,0}(x,p)} \partial _{p_j} \left[ \frac{\partial _{p_j}a_{\varepsilon ,0}(x,p)}{\left| \nabla _pa_{\varepsilon ,0}(x,p) \right| ^2} u_N(x,p,t,\hbar ,\varepsilon ) \theta (x,p)\right] \, dxdp. \end{aligned} \end{aligned}$$

(5.24)

Combining (5.23), (5.24) and our assumptions on t we see that we have gained $\hbar ^{\frac{\delta }{2}}$ compared to our naive first estimate. To obtain the desired estimate we iterate the argument in (5.24) until an error of the desired order has been obtained. This concludes the proof. $\square$

The previous Lemma showed that under a non-critical assumption on the principal symbol a localised trace of our approximation becomes negligible. But we would also need a result similar to this for the true propagator. Actually this can be proven in a setting for which we will need it, which is the content of the next Thoerem. An observation of this type was first made by Ivrii (see [13]). Here we will follow the proof of such a statement as made by Dimassi and Sjöstrand in [28]. The statement is:

Theorem 5.4

Let $A_\varepsilon (\hbar )$ be a $\hbar$-$\varepsilon$-admissible operator of regularity $\tau \ge 1$ which satisfies Assumption 4.1, has a bounded principal symbol and suppose there exists a $\delta$ in (0, 1) such that $\varepsilon \ge \hbar ^{1-\delta }$. Furthermore, suppose there exists a number $\eta >0$ such $a_{\varepsilon ,0}^{-1}([-2\eta ,2\eta ])$ is compact and a constant $c>0$ such that

$$\begin{aligned} \left| \nabla _p a_{\varepsilon ,0}(x,p) \right| \ge c \quad \text {for all } (x,p) \in a_{\varepsilon ,0}^{-1}([-2\eta ,2\eta ]), \end{aligned}$$

where $a_{\varepsilon ,0}$ is the principal symbol of $A_\varepsilon (\hbar )$. Let f be in $C_0^\infty ((-\eta ,\eta ))$ and $\theta$ be in $C_0^\infty ({\mathbb {R}}^d_x\times {\mathbb {R}}^d_p)$ such that $\text {supp} (\theta )\subset a_{\varepsilon ,0}^{-1}((-\eta ,\eta ))$.

Then there exists a constant $T_0>0$ such that if $\chi$ is in $C_0^\infty ((\frac{1}{2} \hbar ^{1-\gamma },T_0))$ for a $\gamma$ in $(0,\delta ]$, then for every N in ${\mathbb {N}}$, we have

$$\begin{aligned} \left| \text {Tr} [\text {Op} _\hbar ^{\text {w} }(\theta )f(A_\varepsilon (\hbar )) {\mathcal {F}}_\hbar ^{-1}[\chi ](s-A_\varepsilon (\hbar ))\text {Op} _\hbar ^{\text {w} }(\theta )] \right| \le C_N \hbar ^N, \end{aligned}$$

uniformly for s in $(-\eta ,\eta )$.

Remark 5.5

Theorems of this type for non-regular operators can be found in the works of Ivrii see [10] and Zielinski see [21]. In both cases the proof of such theorems is different from the one we present here. The techniques used by both are based on the propagation of singularities. The propagation of singularities is not directly present in the proof presented here, but hidden in the techniques used.

In both [28] and [10] they assume the symbol to be microhyperbolic in some direction. It might also be possible to extend the Theorem here to a general microhyperbolic assumption instead of the non-critical assumption. The challenge in this will be, that for the proof to work under a general microhyperbolic assumption the symbol should be changed so that microhyperbolic assumption similar to the non-critical assumption is achieved. This change might be problematic to do since it could mix the rough and non-rough variables.

The localising operators $\text {Op} _\hbar ^{\text {w} }(\theta )$ could be omitted if, before the first step of the proof, Lemma 4.14 is used to introduce the localisation operators. We have chosen to state the theorem with them since when we will apply the theorem, we have the localisations.

Proof

We start by remarking that it suffices to show the estimate with a function $\chi _\xi (t) = \chi (\frac{t}{\xi })$, where $\chi$ is in $C_0^\infty ((\frac{1}{2},1))$ uniformly for $\xi$ in $[\hbar ^{1-\gamma },T_0]$. Indeed assume such an estimate has been proven. We can split the interval $(\frac{1}{2} \hbar ^{1-\gamma },T_0)$ in $\frac{2T_0}{\hbar ^{1-\gamma }}$ intervals of size $\frac{1}{2}\hbar ^{1-\gamma }$ and make a partition of unity, where each member is supported in one of these intervals. By linearity of the inverse Fourier transform and trace we would have that

$$\begin{aligned} \begin{aligned} \big |\text {Tr} [\text {Op} _\hbar ^{\text {w} }&(\theta )f(A_\varepsilon (\hbar )) {\mathcal {F}}_\hbar ^{-1}[\chi ](s-A_\varepsilon (\hbar ))\text {Op} _\hbar ^{\text {w} }(\theta )]\big | \\&\le {} \sum _{j=1}^{M(\hbar )} \left| \text {Tr} [\text {Op} _\hbar ^{\text {w} }(\theta )f(A_\varepsilon (\hbar )) {\mathcal {F}}_\hbar ^{-1}[\chi _{\xi _j}](s-A_\varepsilon (\hbar ))\text {Op} _\hbar ^{\text {w} }(\theta )] \right| \le {\tilde{C}}_N \hbar ^{N-1+\delta }. \end{aligned} \end{aligned}$$

Hence we will consider the trace

$$\begin{aligned} \text {Tr} [\text {Op} _\hbar ^{\text {w} }(\theta )f(A_\varepsilon (\hbar )) {\mathcal {F}}_\hbar ^{-1}[\chi _\xi ](s-A_\varepsilon (\hbar ))\text {Op} _\hbar ^{\text {w} }(\theta )], \end{aligned}$$

with $\chi _\xi (t) = \chi (\frac{t}{\xi })$, where $\chi$ is in $C_0^\infty ((\frac{1}{2},1))$ and $\xi$ in $[\hbar ^{1-\gamma },T_0]$. For the rest of the proof we let a N in ${\mathbb {N}}$ be given as the error we desire.

Without loss of generality we can assume $\theta =\sum _k \theta _k$, where the $\theta _k$’s satisfies that if $\text {supp} (\theta _k)\cap \text {supp} (\theta _l)\ne \emptyset$ then there exists j in $\{1,\dots ,d\}$ such $|\partial _{p_j} a_{\varepsilon ,0}(x,p)|>{\tilde{c}}$ on the set $\text {supp} (\theta _k)\cup \text {supp} (\theta _l)$. With this splitting of $\theta$ we have

$$\begin{aligned} \begin{aligned} \text {Tr} [\text {Op} _\hbar ^{\text {w} }(\theta )&f(A_\varepsilon (\hbar )) {\mathcal {F}}_\hbar ^{-1}[\chi _\xi ](s-A_\varepsilon (\hbar ))\text {Op} _\hbar ^{\text {w} }(\theta )] \\&= \sum _{k} \sum _{l} \text {Tr} [\text {Op} _\hbar ^{\text {w} }(\theta _k)f(A_\varepsilon (\hbar )) {\mathcal {F}}_\hbar ^{-1}[\chi _\xi ](s-A_\varepsilon (\hbar ))\text {Op} _\hbar ^{\text {w} }(\theta _l)]. \end{aligned} \end{aligned}$$

By the cyclicity of the trace and the formulas for compositions of pseudo-differential operators we observe if $\text {supp} (\theta _k)\cap \text {supp} (\theta _l)=\emptyset$, then the term is negligible. Hence it remains to consider terms with $\text {supp} (\theta _k)\cap \text {supp} (\theta _l)\ne \emptyset$. All terms of this form are estimated with analogous techniques but some different indicies. Hence we will suppose, that we have a pair $\theta _k$ and $\theta _l$ of functions such that $\text {supp} (\theta _k)\cap \text {supp} (\theta _l)\ne \emptyset$ and $|\partial _{p_1} a_{\varepsilon ,0}(x,p)|>{\tilde{c}}$ on the set $\text {supp} (\theta _k)\cup \text {supp} (\theta _l)$. This implies we either have $\partial _{p_1} a_{\varepsilon ,0}(x,p)>{\tilde{c}}$ or $-\partial _{p_1} a_{\varepsilon ,0}(x,p)>{\tilde{c}}$. We suppose we are in the first case. The other case is treated in the same manner but with a change of some signs.

To sum up we have reduced to the case where we consider

$$\begin{aligned} \text {Tr} [\text {Op} _\hbar ^{\text {w} }(\theta _k)f(A_\varepsilon (\hbar )) {\mathcal {F}}_\hbar ^{-1}[\chi _\xi ](s-A_\varepsilon (\hbar ))\text {Op} _\hbar ^{\text {w} }(\theta _l)], \end{aligned}$$

(5.25)

with $\partial _{p_1} a_{\varepsilon ,0}(x,p)>{\tilde{c}}$ on the the set $\text {supp} (\theta _k)\cup \text {supp} (\theta _l)$. In order to estimate (5.25) we will need an auxiliary function. Let $\psi$ be in $C^\infty ({\mathbb {R}})$ such $\psi (t)=1$ for $t\le 1$ and $\psi (t)=0$ for $t\ge 2$. Moreover, let M be a sufficiently large constant which will be fixed later and put

$$\begin{aligned} \psi _{\mu _1}(z) = \psi \Big (\tfrac{\text {Im} (z)}{\mu _1}\Big ), \end{aligned}$$

where $\mu _1=\frac{M\hbar }{\xi }\log (\frac{1}{\hbar })$. With this function we have

$$\begin{aligned} |{\bar{\partial }}({\tilde{f}}\psi _{\mu _1})| \le {\left\{ \begin{array}{ll} C_N \left| \text {Im} (z) \right| ^N, &{} \text {if } \text {Im} (z)<0 \\ C_N \psi _{\mu _1}(z) \left| \text {Im} (z) \right| ^N + \mu _1^{-1} {\varvec{1}}_{[1,2]}(\tfrac{\text {Im} (z)}{\mu _1}) , &{} \text {if } \text {Im} (z)\ge 0, \end{array}\right. } \end{aligned}$$

(5.26)

for any N in ${\mathbb {N}}$, where ${\tilde{f}}$ is an almost analytic extension of f. We can use Theorem 4.10 for the operator $A_\varepsilon (\hbar )$ on the function $({\tilde{f}}\psi _{\mu _1})(z) {\mathcal {F}}_\hbar ^{-1}[\chi _\xi ](s-z)$. This gives

$$\begin{aligned} ({\tilde{f}}\psi _{\mu _1})(A_\varepsilon (\hbar ))&{\mathcal {F}}_\hbar ^{-1}[\chi _\xi ](s-A_\varepsilon (\hbar )) \\&= -\frac{1}{\pi } \int _{\mathbb {C}}{\bar{\partial }}({\tilde{f}}\psi _{\mu _1})(z) {\mathcal {F}}_\hbar ^{-1}[\chi _{\xi }](s-z) (z-A_\varepsilon (\hbar ))^{-1} \, L(dz), \end{aligned}$$

where we have used that ${\mathcal {F}}_\hbar ^{-1}[\chi _\xi ](s-z)$ is an analytic function in z. Hence the trace we consider is

$$\begin{aligned}&\text {Tr} [\text {Op} _\hbar ^{\text {w} }(\theta _k)f(A_\varepsilon (\hbar )) {\mathcal {F}}_\hbar ^{-1}[\chi _\xi ](s-A_\varepsilon (\hbar ))\text {Op} _\hbar ^{\text {w} }(\theta _l)] \nonumber \\&= -\frac{1}{\pi } \int _{\mathbb {C}}{\bar{\partial }}({\tilde{f}}\psi _{\mu _1})(z) {\mathcal {F}}_\hbar ^{-1}[\chi _{\xi }](s-z) \text {Tr} [\text {Op} _\hbar ^{\text {w} }(\theta _k)(z-A_\varepsilon (\hbar ))^{-1} \text {Op} _\hbar ^{\text {w} }(\theta _l)]\, L(dz) \nonumber \\&= -\frac{1}{\pi } \int _{\text {Im} (z)<0} \cdots \, L(dz) -\frac{1}{\pi } \int _{\text {Im} (z)\ge 0} \cdots \, L(dz) \end{aligned}$$

(5.27)

We shortly investigate each of the integrals. Firstly we note the bound

$$\begin{aligned} \left| \text {Tr} [\text {Op} _\hbar ^{\text {w} }(\theta _k)(z-A_\varepsilon (\hbar ))^{-1} \text {Op} _\hbar ^{\text {w} }(\theta _l)] \right| \le \frac{C}{\hbar ^d \left| \text {Im} (z) \right| }. \end{aligned}$$

If we consider the integral over the negative imaginary part, we have that

$$\begin{aligned} \begin{aligned} \Big |\frac{1}{\pi } \int _{\text {Im} (z)<0}&{\bar{\partial }}({\tilde{f}}\psi _{\mu _1})(z) {\mathcal {F}}_\hbar ^{-1}[\chi _{\xi }](s-z) \text {Tr} [\text {Op} _\hbar ^{\text {w} }(\theta _k)(z-A_\varepsilon (\hbar ))^{-1} \text {Op} _\hbar ^{\text {w} }(\theta _l)]\, L(dz)\Big | \\&\le \frac{C_{2N}\xi }{\pi \hbar ^{d+1}} \int _{\text {Im} (z)<0}\frac{ \text {Im} (z)^{2N}}{|\text {Im} (z)|} e^{\frac{\xi \text {Im} (z)}{\hbar }}\, d\text {Im} (z) \\&\le \frac{C_{2N}}{\pi \hbar ^d} (\frac{\hbar }{\xi })^{2N-2} \le {\tilde{C}} \hbar ^{(2N-2)\gamma - d}, \end{aligned} \end{aligned}$$

for any N in ${\mathbb {N}}$. We have in the above calculation used integration by parts and the estimate

$$\begin{aligned} \left| {\mathcal {F}}_\hbar ^{-1}[\chi _{\xi }](s-z) \right| \le C \frac{\xi }{\hbar } e^{\frac{\xi \text {Im} (z)}{\hbar }}. \end{aligned}$$

The above estimate implies that the contribution to the trace from the negative integral is negligible. If we split the integral over positive imaginary part up according to $\mu _1$ we have by (5.26) the estimate

$$\begin{aligned} \begin{aligned} \Big |\frac{1}{\pi }&\int _{0\le \text {Im} (z)\le \mu _1} {\bar{\partial }}({\tilde{f}}\psi _{\mu _1})(z) {\mathcal {F}}_\hbar ^{-1}[\chi _{\xi }](s-z) \text {Tr} [\text {Op} _\hbar ^{\text {w} }(\theta _k)(z-A_\varepsilon (\hbar ))^{-1} \text {Op} _\hbar ^{\text {w} }(\theta _l)]\, L(dz)\Big | \\&\le {} \frac{C_{2{\tilde{N}}}\xi }{\pi \hbar ^{d+1}} \int _{0\le \text {Im} (z)\le \mu _1} \psi _{\mu _1}(z) \left| \text {Im} (z) \right| ^{{\tilde{N}}} e^{\frac{\xi \text {Im} (z)}{\hbar }}\, d\text {Im} (z) \\&\le {} {\tilde{C}} \frac{\xi }{\hbar ^{d+1}} \mu _1^{{\tilde{N}}+1} e^{\frac{\xi \mu _1}{\hbar }} \le {\tilde{C}} \frac{\xi }{\hbar ^{d+1}} \Big (\frac{M\hbar }{\xi }\log \left( \frac{1}{\hbar }\right) \Big )^{{\tilde{N}}+1} e^{M\log (\hbar ^{-1})} \le {\tilde{C}} M \hbar ^{({\tilde{N}}+1)\gamma -M- d-1} , \end{aligned} \end{aligned}$$

(5.28)

for any ${\tilde{N}}\in {\mathbb {N}}$, where we have used that $\mu _1=\frac{M\hbar }{\xi }\log (\frac{1}{\hbar })$. Hence this term also becomes negligible, when ${\tilde{N}}$ is chosen sufficiently large depending on M. What remains from (5.27) is the expression

$$\begin{aligned} -\frac{1}{\pi } \int _{\text {Im} (z)>\mu _1} {\bar{\partial }}({\tilde{f}}\psi _{\mu _1})(z) {\mathcal {F}}_\hbar ^{-1}[\chi _{\xi }](s-z) \text {Tr} [\text {Op} _\hbar ^{\text {w} }(\theta _k)(z-A_\varepsilon (\hbar ))^{-1} \text {Op} _\hbar ^{\text {w} }(\theta _l)]\, L(dz). \end{aligned}$$

(5.29)

The next step is to change the principal symbol of our operator, so that it becomes global microhyperbolic in the direction $({\varvec{0}};(1,0,\dots ,0))$, where ${\varvec{0}}$ is the d-dimensional vector with only zeros. We let $\varphi _2$ be a function in $C^\infty _0({\mathbb {R}}_x^d\times {\mathbb {R}}_p^d)$ such that $\varphi _2(x,p)=1$ on a small neighbourhood of $\text {supp} (\theta _k)\cup \text {supp} (\theta _l)$ and has its support contained in the set

$$\begin{aligned} \big \{(x,p)\in {\mathbb {R}}^{2d}\,|\, |\partial _{p_1} a_{\varepsilon ,0}(x,p)|>\tfrac{{\tilde{c}}}{2} \big \}. \end{aligned}$$

Moreover, we let $\varphi _1$ be a function in $C^\infty _0({\mathbb {R}}_x^d\times {\mathbb {R}}_p^d)$ such $\varphi _1(x,p)=1$ on $\text {supp} (\varphi _2)$ and such that

$$\begin{aligned} \text {supp} (\varphi _1) \subseteq \big \{(x,p)\in {\mathbb {R}}^{2d}\,|\, |\partial _{p_1} a_{\varepsilon ,0}(x,p)|>\tfrac{{\tilde{c}}}{4} \big \}. \end{aligned}$$

(5.30)

With these functions we define the symbol

$$\begin{aligned} {\tilde{a}}_{\varepsilon ,0} (x,p) = a_{\varepsilon ,0} (x,p) \varphi _1(x,p) + C(1-\varphi _2(x,p)), \end{aligned}$$

where the constant C is chosen such that ${\tilde{a}}_{\varepsilon ,0} (x,p)\ge 1 + \eta$ outside the support of $\varphi _2(x,p)$. We have that

$$\begin{aligned} \partial _{p_1}{\tilde{a}}_{\varepsilon ,0} (x,p) = (\partial _{p_1} a_{\varepsilon ,0}) (x,p) \varphi _1(x,p) + a_{\varepsilon ,0} (x,p) (\partial _{p_1}\varphi _1)(x,p) - C\partial _{p_1}\varphi _2(x,p). \end{aligned}$$

Hence there exist constants $c_0$ and $c_1$ so that

$$\begin{aligned} \partial _{p_1}{\tilde{a}}_{\varepsilon ,0} (x,p) \ge c_0 - c_1({\tilde{a}}_{\varepsilon ,0} (x,p))^2, \end{aligned}$$

(5.31)

for all (x, p) in ${\mathbb {R}}^{2d}$. To see this, we observe that on $\text {supp} (\theta _k)\cup \text {supp} (\theta _l)$, we have the inequality

$$\begin{aligned} \partial _{p_1}{\tilde{a}}_{\varepsilon ,0} (x,p)\ge {\tilde{c}}. \end{aligned}$$

By continuity there exists an open neighbourhood $\Omega$ of $\text {supp} (\theta _k)\cup \text {supp} (\theta _l)$ such that $\partial _{p_1}{\tilde{a}}_{\varepsilon ,0} (x,p)\ge \frac{{\tilde{c}}}{3}$ and $(1-\varphi _2)\ne 0$ on $\Omega ^c$. Hence outside $\Omega$ we get the bound

$$\begin{aligned} \partial _{p_1}{\tilde{a}}_{\varepsilon ,0} (x,p) \ge c_0 - c_1({\tilde{a}}_{\varepsilon ,0} (x,p))^2, \end{aligned}$$

by choosing $c_1$ sufficiently large. This estimate ensures that our new symbol is global microhyperbolic in the direction $({\varvec{0}};(1,0,\dots ,0))$.

Our assumptions on the operator ${A}_\varepsilon (\hbar )$ imply the form

$$\begin{aligned} A_\varepsilon (\hbar ) = \sum _{j=0}^{N_0} \hbar ^j \text {Op} _\hbar ^{\text {w} }(a_{\varepsilon ,j}) + \hbar ^{N_0+1} R_{N_0}(\hbar ,\varepsilon ), \end{aligned}$$

where $N_0$ is chosen such that

$$\begin{aligned} \hbar ^{N_0+1}\Vert R_{N_0}(\hbar ,\varepsilon ) \Vert _{{\mathcal {L}}(L^2({\mathbb {R}}^d))} \le C \hbar ^{{\tilde{N}}+d}, \end{aligned}$$

(5.32)

where ${\tilde{N}}$ will be determined later, such that the total error is of the order $\hbar ^N$. By ${\tilde{A}}_\varepsilon (\hbar )$ we denote the operator obtained by taking the $N_0$ first terms of $A_\varepsilon (\hbar )$ and exchanging the principal symbol $a_{\varepsilon ,0}$ of $A_\varepsilon (\hbar )$ by ${\tilde{a}}_{\varepsilon ,0}$. Note that the operator ${\tilde{A}}_\varepsilon (\hbar )$ still satisfies Assumption 4.1, since the original symbols were assumed to be bounded. We have that

$$\begin{aligned} A_\varepsilon (\hbar )-{\tilde{A}}_\varepsilon (\hbar ) = \text {Op} _\hbar ^{\text {w} }(a_{\varepsilon ,0} -{\tilde{a}}_{\varepsilon ,0}) + \hbar ^{N_0+1} R_{N_0}(\hbar ,\varepsilon ), \end{aligned}$$

(5.33)

and by construction $a_{\varepsilon ,0} -{\tilde{a}}_{\varepsilon ,0}$ is supported away from $\text {supp} (\theta _k)\cup \text {supp} (\theta _l)$. Let ${\tilde{\theta }}\in C_0^\infty ({\mathbb {R}}_x^d\times {\mathbb {R}}_p^d)$ such that ${\tilde{\theta }}(x,p)=1$ for all $(x,p)\in \text {supp} (\theta _k)\cup \text {supp} (\theta _l)$ and $\text {supp} ({\tilde{\theta }})\cap \text {supp} (a_{\varepsilon ,0} -{\tilde{a}}_{\varepsilon ,0}) = \emptyset$. We have the following identity

$$\begin{aligned} \begin{aligned}&(z-A_\varepsilon (\hbar )) \text {Op} _\hbar ^{\text {w} }({\tilde{\theta }}) (z-{\tilde{A}}_\varepsilon (\hbar ))^{-1} \text {Op} _\hbar ^{\text {w} }(\theta _l) \\&= \text {Op} _\hbar ^{\text {w} }({\tilde{\theta }})\text {Op} _\hbar ^{\text {w} }(\theta _l) + \text {Op} _\hbar ^{\text {w} }({\tilde{\theta }}) ({\tilde{A}}_\varepsilon (\hbar )- A_\varepsilon (\hbar )) (z- {\tilde{A}}_\varepsilon (\hbar ))^{-1} \text {Op} _\hbar ^{\text {w} }(\theta _l) \\&\quad - [A_\varepsilon (\hbar ),\text {Op} _\hbar ^{\text {w} }({\tilde{\theta }})](z-{\tilde{A}}_\varepsilon (\hbar ))^{-1}\text {Op} _\hbar ^{\text {w} }({\tilde{\theta }}). \end{aligned} \end{aligned}$$

Using this identity and that $\text {Op} _\hbar ^{\text {w} }({\tilde{\theta }})\text {Op} _\hbar ^{\text {w} }(\theta _l)=\text {Op} _\hbar ^{\text {w} }(\theta _l) + \hbar ^{N_0} R_{N_0}(\hbar )$, where $\sup _{\hbar \in (0,1]} \Vert R_{N_0}(\hbar ) \Vert _{\text {Tr} } <\infty$ for all $N_0\in {\mathbb {N}}$, we obtain that

$$\begin{aligned} \begin{aligned}&\big \Vert \text {Op} _\hbar ^{\text {w} }(\theta _k) (z-A_\varepsilon (\hbar ))^{-1}\text {Op} _\hbar ^{\text {w} }(\theta _l) - \text {Op} _\hbar ^{\text {w} }(\theta _k) (z-{\tilde{A}}_\varepsilon (\hbar ))^{-1} \text {Op} _\hbar ^{\text {w} }(\theta _l)\big \Vert _{\text {Tr} } \\&\le {} \big \Vert \text {Op} _\hbar ^{\text {w} }(\theta _k) (z-A_\varepsilon (\hbar ))^{-1} \text {Op} _\hbar ^{\text {w} }({\tilde{\theta }}) ({\tilde{A}}_\varepsilon (\hbar )- A_\varepsilon (\hbar )) (z- {\tilde{A}}_\varepsilon (\hbar ))^{-1} \text {Op} _\hbar ^{\text {w} }(\theta _l) \big \Vert _{\text {Tr} } \\&\quad + \big \Vert \text {Op} _\hbar ^{\text {w} }(\theta _k) (z-A_\varepsilon (\hbar ))^{-1} [A_\varepsilon (\hbar ),\text {Op} _\hbar ^{\text {w} }({\tilde{\theta }})](z-{\tilde{A}}_\varepsilon (\hbar ))^{-1}\text {Op} _\hbar ^{\text {w} }(\theta _l)\big \Vert _{\text {Tr} } + \frac{\hbar ^{{\tilde{N}}}C_{{\tilde{N}}}}{|\text {Im} (z)|}, \end{aligned} \end{aligned}$$

(5.34)

where we will need to choose ${\tilde{N}}$ sufficiently large. Using the support properties of ${\tilde{\theta }}$, (5.32) and (5.33) we get the estimate

$$\begin{aligned}&\big \Vert \text {Op} _\hbar ^{\text {w} }(\theta _k) (z-A_\varepsilon (\hbar ))^{-1} \text {Op} _\hbar ^{\text {w} }({\tilde{\theta }}) ({\tilde{A}}_\varepsilon (\hbar )- A_\varepsilon (\hbar )) (z- {\tilde{A}}_\varepsilon (\hbar ))^{-1} \text {Op} _\hbar ^{\text {w} }(\theta _l) \big \Vert _{\text {Tr} } \nonumber \\&\le \frac{\hbar ^{{\tilde{N}}} C_{{\tilde{N}}}}{|\text {Im} (z)|^2}, \end{aligned}$$

(5.35)

where we again will choose ${\tilde{N}}$ sufficiently large later. To estimate the last term on the righthand side of (5.34), we introduce the function $G\in C_0^\infty ({\mathbb {R}}^d_x\times {\mathbb {R}}^d_p)$ such that $G\theta _k = \theta _k$ and $G{\tilde{\theta }} = G$. For all $\alpha >0$ we have, that the function $e^{\alpha G \log (\hbar ^{-1})}$ is a rough symbol, that is uniformly bounded. The operator $\text {Op} _\hbar ^{\text {w} }(e^{\alpha G \log (\hbar ^{-1})})$ is elliptic, and hence it will have an inverse, which we will denote by $\text {Op} _\hbar ^{\text {w} }(e^{\alpha G \log (\hbar ^{-1})})^{-1}$. For these operators, we have that

$$\begin{aligned} \text {Op} _\hbar ^{\text {w} }(e^{\alpha G \log (\hbar ^{-1})}) (z-A_\varepsilon (\hbar )) \text {Op} _\hbar ^{\text {w} }(e^{\alpha G \log (\hbar ^{-1})})^{-1} = z-A_\varepsilon (\hbar ) + \alpha \hbar \log (\hbar ^{-1}) R_\hbar (\varepsilon ), \end{aligned}$$

(5.36)

where $\sup _{\hbar \in (0,\hbar (\alpha )]}\Vert R_\hbar (\varepsilon ) \Vert _{{\mathcal {L}}(L^2({\mathbb {R}}^d)}<\infty$. Here $\hbar (\alpha )>0$ is some continuous function of $\alpha$. We get from (5.35) that the resolvent

$$\begin{aligned} (z-\text {Op} _\hbar ^{\text {w} }(e^{\alpha G \log (\hbar ^{-1})}) A_\varepsilon (\hbar ) \text {Op} _\hbar ^{\text {w} }(e^{\alpha G \log (\hbar ^{-1})})^{-1})^{-1} \end{aligned}$$

exists. Using the rules for compositions of operators we get that

$$\begin{aligned}{} & {} \big \Vert e^{\alpha \log (\hbar ^{-1})} \text {Op} _\hbar ^{\text {w} }(\theta _k) (z-A_\varepsilon (\hbar ))^{-1} [A_\varepsilon (\hbar ),\text {Op} _\hbar ^{\text {w} }({\tilde{\theta }})] \big \Vert _{{\mathcal {L}}(L^2({\mathbb {R}}^d))} \nonumber \\{} & {} \le {}\big \Vert \text {Op} _\hbar ^{\text {w} }(\theta _k) \text {Op} _\hbar ^{\text {w} }(e^{\alpha G \log (\hbar ^{-1})}) (z-A_\varepsilon (\hbar ))^{-1} \text {Op} _\hbar ^{\text {w} }(e^{\alpha G \log (\hbar ^{-1})})^{-1} \nonumber \\{} & {} \quad \times [A_\varepsilon (\hbar ),\text {Op} _\hbar ^{\text {w} }({\tilde{\theta }})] \big \Vert _{{\mathcal {L}}(L^2({\mathbb {R}}^d))} + \frac{C_{{\tilde{N}}} \hbar ^{{\tilde{N}}} }{|\text {Im} (z)|} \nonumber \\{} & {} ={} \big \Vert \text {Op} _\hbar ^{\text {w} }(\theta _k) \big [ z- \text {Op} _\hbar ^{\text {w} }(e^{\alpha G \log (\hbar ^{-1})})A_\varepsilon (\hbar ) \text {Op} _\hbar ^{\text {w} }(e^{\alpha G \log (\hbar ^{-1})})^{-1}\big ]^{-1} \nonumber \\{} & {} \quad \times [A_\varepsilon (\hbar ),\text {Op} _\hbar ^{\text {w} }({\tilde{\theta }})] \big \Vert _{{\mathcal {L}}(L^2({\mathbb {R}}^d))} + \frac{C_{{\tilde{N}}} \hbar ^{{\tilde{N}}} }{|\text {Im} (z)|} \nonumber \\{} & {} \le {} \frac{ C+ C_{{\tilde{N}}} \hbar ^{{\tilde{N}}} }{|\text {Im} (z)|}. \end{aligned}$$

(5.37)

We will be choosing the parameter $\alpha$ to be

$$\begin{aligned} \alpha = \min \Big ( C_1, \frac{\text {Im} (z)}{C_2 \hbar \log (\hbar ^{-1})} \Big ), \end{aligned}$$

where $C_1$ and $C_2$ are two positive constants. $C_1$ is to be chosen later. Whereas $C_2$ is chosen sufficiently large to ensure that inevitability of one operator implies inevitability of the other in (5.36). Note that $\alpha$ will be of order one for all $\text {Im} (z)\in [\mu _1,2\mu _1]$. With this choice for $\alpha$ we get from (5.37) that

$$\begin{aligned}{} & {} \big \Vert \text {Op} _\hbar ^{\text {w} }(\theta _k) (z-A_\varepsilon (\hbar ))^{-1} [A_\varepsilon (\hbar ),\text {Op} _\hbar ^{\text {w} }({\tilde{\theta }})](z-{\tilde{A}}_\varepsilon (\hbar ))^{-1}\text {Op} _\hbar ^{\text {w} }(\tilde{\theta _l})\big \Vert _{\text {Tr} } \nonumber \\{} & {} \le {} \frac{ C }{\hbar ^{d}|\text {Im} (z)|^2} \max \big ( \hbar ^{C_1}, e^{- \frac{\text {Im} (z)}{C_2 \hbar }} \big ). \end{aligned}$$

(5.38)

Now by combining (5.34), (5.35) and (5.38) we obtain that

$$\begin{aligned}{} & {} \big \Vert \text {Op} _\hbar ^{\text {w} }(\theta _k) (z-A_\varepsilon (\hbar ))^{-1}\text {Op} _\hbar ^{\text {w} }(\theta _l) - \text {Op} _\hbar ^{\text {w} }(\theta _k) (z-{\tilde{A}}_\varepsilon (\hbar ))^{-1} \text {Op} _\hbar ^{\text {w} }(\theta _l)\big \Vert _{\text {Tr} } \nonumber \\{} & {} \le {} \frac{ C }{\hbar ^{d}|\text {Im} (z)|^2} \max \big ( \hbar ^{C_1}, e^{- \frac{\text {Im} (z)}{C_2 \hbar }} \big )+ \frac{\hbar ^{{\tilde{N}}} C_{{\tilde{N}}}}{|\text {Im} (z)|^2}+ \frac{\hbar ^{{\tilde{N}}}C_{{\tilde{N}}}}{|\text {Im} (z)|}. \end{aligned}$$

(5.39)

From using (5.26), (5.38) and the estimate $|{\mathcal {F}}_\hbar ^{-1}[\chi _{\xi }](s-z) | \le C\xi \hbar ^{-1} e^{\frac{\xi \text {Im} (z)}{\hbar }}$ we get for $\text {Im} (z)>0$ the estimate

$$\begin{aligned} \begin{aligned}&\big \Vert {\bar{\partial }}({\tilde{f}}\psi _{\mu _1})(z) {\mathcal {F}}_\hbar ^{-1}[\chi _{\xi }](s-z) \text {Op} _\hbar ^{\text {w} }(\theta _k)\big ((z-A_\varepsilon (\hbar ))^{-1} - (z-{\tilde{A}}_\varepsilon (\hbar ))^{-1}\big ) \text {Op} _\hbar ^{\text {w} }(\theta _l) \big \Vert _{\text {Tr} } \\&\le {} C_{{\tilde{N}}} \psi _{\mu _1}(z) \xi \left| \text {Im} (z) \right| ^{{\tilde{N}}} e^{\frac{\xi \text {Im} (z)}{\hbar }} \left[ \frac{ C\Big ( \hbar ^{C_1} + e^{- \frac{\text {Im} (z)}{C_2 \hbar }} \Big ) }{\hbar ^{d+1}|\text {Im} (z)|^2} + \frac{\hbar ^{{\tilde{N}}-1} C_{{\tilde{N}}}}{|\text {Im} (z)|^2}+ \frac{\hbar ^{{\tilde{N}}-1}C_{{\tilde{N}}}}{|\text {Im} (z)|} \right] \\&\quad + \mu _1^{-1} \xi e^{\frac{\xi \text {Im} (z)}{\hbar }} {\varvec{1}}_{[1,2]}\left( \tfrac{\text {Im} (z)}{\mu _1}\right) \left[ \frac{ C \max \big ( \hbar ^{C_1} , e^{- \frac{\text {Im} (z)}{C_2 \hbar }} \big ) }{\hbar ^{d+1}|\text {Im} (z)|^2}+ \frac{\hbar ^{{\tilde{N}}-1} C_{{\tilde{N}}}}{|\text {Im} (z)|^2}+ \frac{\hbar ^{{\tilde{N}}-1}C_{{\tilde{N}}}}{|\text {Im} (z)|} \right] , \end{aligned} \end{aligned}$$

(5.40)

where we again will need to chose ${\tilde{N}}$ sufficiently large to obtain the desired error. By using analogously estimates to the ones used in (5.28) we obtain for any $N\in {\mathbb {N}}$ that

$$\begin{aligned}{} & {} \int _{\begin{array}{c} -\eta<\text {Re} (z) <\eta \\ \text {Im} (z)>\mu _1 \end{array}} C_{{\tilde{N}}} \psi _{\mu _1}(z) \left| \text {Im} (z) \right| ^{{\tilde{N}}} \xi e^{\frac{\xi \text {Im} (z)}{\hbar }} \nonumber \\{} & {} \times \left[ \frac{ C \big ( \hbar ^{C_1} + e^{- \frac{\text {Im} (z)}{C_2 \hbar }} \big ) }{\hbar ^{d+1}|\text {Im} (z)|^2}+ \frac{\hbar ^{{\tilde{N}}-1} C_{{\tilde{N}}}}{|\text {Im} (z)|^2}+ \frac{\hbar ^{{\tilde{N}}-1}C_{{\tilde{N}}}}{|\text {Im} (z)|} \right] \, L(dz) \nonumber \\{} & {} \le {} {\tilde{C}} M \hbar ^{({\tilde{N}}+1)\gamma -2M- d-1}, \end{aligned}$$

(5.41)

where we have had to choose ${\tilde{N}}$ large enough depending on M to obtain the error $\hbar ^N$. We have also used that the real part of z is contained in a bounded set due to the support properties of ${\tilde{f}}$. In the estimation of the integrals over the remaining terms in (5.40) we will consider two cases. Firstly, we have that

$$\begin{aligned}{} & {} \Big | \int _{\begin{array}{c} -\eta<\text {Re} (z)<\eta \\ \text {Im} (z)>\mu _1 \end{array}} \xi \mu _1^{-1} e^{\frac{\xi \text {Im} (z)}{\hbar }} {\varvec{1}}_{[1,2]}(\tfrac{\text {Im} (z)}{\mu _1}) \left[ \frac{\hbar ^{{\tilde{N}}-1} C_{{\tilde{N}}}}{|\text {Im} (z)|^2}+ \frac{\hbar ^{{\tilde{N}}-1}C_{{\tilde{N}}}}{|\text {Im} (z)|} \right] \, L(dz)\Big | \nonumber \\{} & {} \le \hbar ^{{\tilde{N}}-1} \xi C_{{\tilde{N}}} e^{\frac{\xi \mu _1}{\hbar }} \int _{\begin{array}{c} -\eta<\text {Re} (z)<\eta \\ \mu _1<\text {Im} (z)<2\mu _1 \end{array}} \frac{1}{\mu _1|\text {Im} (z)|^2}+ \frac{1}{\mu _1|\text {Im} (z)|} \, L(dz) \nonumber \\{} & {} \le \frac{ {\tilde{C}} \hbar ^{{\tilde{N}} -2M -3}}{ |\log (\hbar ^{-1})|^{2}}, \end{aligned}$$

(5.42)

where we in the last inequality have used that $\mu _1=\frac{M\hbar }{\xi }\log (\frac{1}{\hbar })$. Now by choosing ${\tilde{N}}$ depending on M sufficiently large we obtain that the integral is neglectable. In the remaining factor, we have the term $\max ( \hbar ^{C_1}, e^{- \frac{\text {Im} (z)}{C_2 \hbar }})$. If we are in the case, where $\hbar ^{C_1}$ is larger and $C_1$ is sufficiently large we can argue as in (5.42) and obtain that the integral is neglectable. Hence, we will consider the case, where $e^{- \frac{\text {Im} (z)}{C_2 \hbar }}$ is the larger factor. Here we have that

$$\begin{aligned}{} & {} \Big | \int _{\begin{array}{c} -\eta<\text {Re} (z)<\eta \\ \text {Im} (z)>\mu _1 \end{array}} \xi \mu _1^{-1} e^{\frac{\xi \text {Im} (z)}{\hbar }} {\varvec{1}}_{[1,2]}(\tfrac{\text {Im} (z)}{\mu _1}) \Big [ \frac{ C e^{- \frac{\text {Im} (z)}{C_2 \hbar }} }{\hbar ^{d+1}|\text {Im} (z)|^2} \Big ]\, L(dz)\Big | \nonumber \\{} & {} \le C \hbar ^{-d-1} \xi e^{-\mu _1\hbar ^{-1}(C_2^{-1} - T_0) } \int _{\begin{array}{c} -\eta<\text {Re} (z)<\eta \\ \mu _1<\text {Im} (z)<2\mu _1 \end{array}} \frac{1}{\mu _1|\text {Im} (z)|^2} \, L(dz) \nonumber \\{} & {} \le \frac{ {\tilde{C}} \hbar ^{M(\frac{1}{T_0 C_2}-1) -d-3}}{ |\log (\hbar ^{-1})|^2}, \end{aligned}$$

(5.43)

where we again have used how we chose $\mu _1$ and that $\xi \le T_0$. Moreover, for this to converge to zero as $\hbar$ goes to zero we need to assume that $T_0<\frac{1}{C_2}$. This is the point in the proof, where we use that $T_0$ has to be sufficiently small. Given $N\in {\mathbb {N}}$ we can, from combining the estimates in (5.40), (5.41), (5.42) and (5.43) by choosing M sufficiently large and then ${\tilde{N}}$ and $C_1$ sufficiently large, obtain that

$$\begin{aligned} \begin{aligned}&\Big | \frac{1}{\pi } \int _{\text {Im} (z)>\mu _1} {\bar{\partial }}({\tilde{f}}\psi _{\mu _1})(z) {\mathcal {F}}_\hbar ^{-1}[\chi _{\xi }](s-z) \text {Tr} [\text {Op} _\hbar ^{\text {w} }(\theta _k)(z-A_\varepsilon (\hbar ))^{-1} \text {Op} _\hbar ^{\text {w} }(\theta _l)]\, L(dz) \\&-\frac{1}{\pi } \int _{\text {Im} (z)>\mu _1} {\bar{\partial }}({\tilde{f}}\psi _{\mu _1})(z) {\mathcal {F}}_\hbar ^{-1}[\chi _{\xi }](s-z) \text {Tr} [\text {Op} _\hbar ^{\text {w} }(\theta _k)(z-{\tilde{A}}_\varepsilon (\hbar ))^{-1} \text {Op} _\hbar ^{\text {w} }(\theta _l)]\, L(dz)\Big | \\&\le C\hbar ^N. \end{aligned} \end{aligned}$$

(5.44)

The estimate in (5.44) gives us, that we without loss of generality may assume, that the operator is global micro-hyperbolic in the direction $({\varvec{0}};(1,0,\dots ,0))$ ((5.31) without the tildes). Hence for the reminder of the proof we will omit the tilde on the operator and its principal symbol, but instead assume the principal symbol to be global micro-hyperbolic in the direction $({\varvec{0}};(1,0,\dots ,0))$.

We have now changed the operator $A_\varepsilon (\hbar )$ such that the principal symbol is global micro-hyperbolic. But to get the desired estimate we need to change all our operators simultaneously. This is done by introducing an auxiliary variable in the symbols and make an almost analytic extension in this variable. Recall that the operator $A_{\varepsilon }(\hbar )$ is a sum of Weyl quantised pseudo-differential operators. In the following we let q(x, p) be one of our symbols and we let $q_t(x,p) = q(x,(p_1+t,p_2,\dots ,p_d))$. We now take t be complex and make an almost analytic extension ${\tilde{q}}_t$ of $q_t$ in t according to Definition 4.7 for $|\text {Im} (t)| <1$. The form of ${\tilde{q}}_t$ is

$$\begin{aligned} {\tilde{q}}_t(x,p) = \sum _{r=0}^{n} (\partial _{p_1}^r q)(x,(p_1+\text {Re} (t),p_2,\dots ,p_d)) \frac{(i\text {Im} (t))^r}{r!}, \end{aligned}$$

Recalling the identity

$$\begin{aligned} \text {Op} _\hbar ^{\text {w} }(q_{\text {Re} (t)}) = e^{-i\text {Re} (t)\hbar ^{-1}x_1} \text {Op} _\hbar ^{\text {w} }(q) e^{i\text {Re} (t)\hbar ^{-1}x_1}, \end{aligned}$$

we have that

$$\begin{aligned} \text {Op} _\hbar ^{\text {w} }({\tilde{q}}_t) = \sum _{r=0}^{n} \frac{(i\text {Im} (t))^r}{r!} e^{-i\text {Re} (t)\hbar ^{-1}x_1} \text {Op} _\hbar ^{\text {w} }(\partial _{p_1}^r q) e^{i\text {Re} (t)\hbar ^{-1}x_1}. \end{aligned}$$

(5.45)

If we take derivatives with respect to $\text {Re} (t)$ and $\text {Im} (t)$ in operator sense we see that

$$\begin{aligned} \frac{\partial }{\partial \text {Re} (t)} \text {Op} _\hbar ^{\text {w} }({\tilde{q}}_t)&= -\frac{i}{\hbar } \sum _{r=0}^{n} \frac{(i\text {Im} (t))^r}{r!} e^{-i\text {Re} (t)\hbar ^{-1}x_1} [x_1,\text {Op} _\hbar ^{\text {w} }(\partial _{p_1}^r q)] e^{i\text {Re} (t)\hbar ^{-1}x_1} \\&= \sum _{r=0}^{n} \frac{(i\text {Im} (t))^r}{r!} e^{-i\text {Re} (t)\hbar ^{-1}x_1} \text {Op} _\hbar ^{\text {w} }(\partial _{p_1}^{r+1} q) e^{i\text {Re} (t)\hbar ^{-1}x_1}, \end{aligned}$$

and

$$\begin{aligned} \frac{\partial }{\partial \text {Im} (t)} \text {Op} _\hbar ^{\text {w} }({\tilde{q}}_t)&= i \sum _{r=1}^{n} \frac{(i\text {Im} (t))^{r-1}}{(r-1)!} e^{-i\text {Re} (t)\hbar ^{-1}x_1} \text {Op} _\hbar ^{\text {w} }(\partial _{p_1}^r q) e^{i\text {Re} (t)\hbar ^{-1}x_1} \\&= i\sum _{r=0}^{n-1} \frac{(i\text {Im} (t))^{r}}{r!} e^{-i\text {Re} (t)\hbar ^{-1}x_1} \text {Op} _\hbar ^{\text {w} }(\partial _{p_1}^{r+1} q) e^{i\text {Re} (t)\hbar ^{-1}x_1}. \end{aligned}$$

In the above calculation the unbounded operator $x_1$ appears, but for all the symbols we consider the commutator $[x_1,\text {Op} _\hbar ^{\text {w} }(\partial _{p_1}^r q)]$ will be a bounded operator. This calculation gives that

$$\begin{aligned} \Big (\frac{\partial }{\partial \text {Re} (t)} + i \frac{\partial }{\partial \text {Im} (t)}\Big ) \text {Op} _\hbar ^{\text {w} }({\tilde{q}}_t) = \frac{(i\text {Im} (t))^{n}}{n!} e^{-i\text {Re} (t)\hbar ^{-1}x_1} \text {Op} _\hbar ^{\text {w} }(\partial _{p_1}^{n+1} q) e^{i\text {Re} (t)\hbar ^{-1}x_1} \end{aligned}$$

This implies the following estiamtes

$$\begin{aligned} \begin{aligned}&\big \Vert \frac{\partial }{\partial {\bar{t}}} \text {Op} _\hbar ^{\text {w} }({\tilde{\theta }}_{j,t})\big \Vert _{\text {Tr} } \le C_n\hbar ^{-d} \left| \text {Im} (t) \right| ^n \quad \quad \hbox { for}\ j=k,l \\&\big \Vert \frac{\partial }{\partial {\bar{t}}}{\tilde{A}}_\varepsilon (\hbar ;t)\big \Vert _{{\mathcal {L}}(L^2({\mathbb {R}}^d))} \le C_n \left| \text {Im} (t) \right| ^n, \end{aligned} \end{aligned}$$

(5.46)

for any n in ${\mathbb {N}}$ by choosing an almost analytic expansion of this order. The operator ${\tilde{A}}_\varepsilon (\hbar ;t)$ is the operator, where we have made the above construction for each symbol in the expansion of the operator. Moreover, we have by the construction of ${\tilde{A}}_\varepsilon (\hbar ;t)$ that

$$\begin{aligned} {\tilde{A}}_\varepsilon (\hbar ;t) =e^{-i\text {Re} (t)\hbar ^{-1}x_1} A_\varepsilon (\hbar ) e^{i\text {Re} (t)\hbar ^{-1}x_1} + i\text {Im} (t) B_\varepsilon (\hbar ;t), \end{aligned}$$

where $B_\varepsilon (\hbar ;t)$ is a bounded operator and its form is obtained from (5.45) with q replaced by the symbol of $A_\varepsilon (\hbar )$. This gives

$$\begin{aligned} z-{\tilde{A}}_\varepsilon (\hbar ;t) =(z-U^{*} A_\varepsilon (\hbar ) U)[I +(z-U^{*} A_\varepsilon (\hbar ) U)^{-1} i\text {Im} (t) B_\varepsilon (\hbar ;t)], \end{aligned}$$

where $U= e^{i\text {Re} (t)\hbar ^{-1}x_1}$. Hence if $\left| \text {Im} (t) \right| \le \tfrac{ \left| \text {Im} (z) \right| }{ C_1}$ the operator $z-{\tilde{A}}_\varepsilon (\hbar ;t)$ has an inverse, where $C_1\ge \Vert B_\varepsilon (\hbar ;t) \Vert _{{\mathcal {L}}(L^2({\mathbb {R}}^d))} +1$. This implies that the function $\eta (t,z)$, defined by

$$\begin{aligned} \eta (t,z) = \text {Tr} [\text {Op} _\hbar ^{\text {w} }({\tilde{\theta }}_{k,t})(z- {\tilde{A}}_\varepsilon (\hbar ;t))^{-1} \text {Op} _\hbar ^{\text {w} }({\tilde{\theta }}_{l,t})], \end{aligned}$$

is well defined for $\left| \text {Im} (t) \right| \le \tfrac{ \left| \text {Im} (z) \right| }{ C_1}$. The function has by construction the properties

$$\begin{aligned} \begin{aligned} \left| \eta (t,z) \right|&\le \frac{c}{\hbar ^d\left| \text {Im} (z) \right| } \\ |\Big (\frac{\partial }{\partial \text {Re} (t)} + i \frac{\partial }{\partial \text {Im} (t)}\Big ) \eta (t,z) |&\le \frac{c_n\left| \text {Im} (t) \right| ^n}{\hbar ^d\left| \text {Im} (z) \right| ^2}, \end{aligned} \end{aligned}$$

for n in ${\mathbb {N}}$. But by cyclicity of the trace the function $\eta (t,z)$ is independent of $\text {Re} (t)$. Hence we have

$$\begin{aligned} \left| \eta (\pm i \text {Im} (t),z) - \eta (0,z) \right| \le \frac{c_N\left| \text {Im} (t) \right| ^n}{\hbar ^d\left| \text {Im} (z) \right| ^2}, \end{aligned}$$

by the fundamental theorem of calculus. The construction of $\eta$ gives us that

$$\begin{aligned} \eta (0,z) = \text {Tr} [\text {Op} _\hbar ^{\text {w} }(\theta _{k})(z- A_\varepsilon (\hbar ))^{-1} \text {Op} _\hbar ^{\text {w} }(\theta _{l})]. \end{aligned}$$

Hence we can exchange the trace in (5.29) by $\eta (-i\frac{\mu _1}{C_1},z)$ with an error of the order $\hbar ^{\gamma n-2M-d}$. This is due to our choice of $\mu _1=\frac{M\hbar }{\xi } \log (\frac{1}{\hbar })$ in the start of the proof, and that the integral is only over a compact region, where $\left| \text {Im} (z) \right| >\mu _1$ due to the definition of $\psi _{\mu _1}$. It now remains to estimate the term

$$\begin{aligned} -\frac{1}{\pi } \int _{\text {Im} (z)>\mu _1} {\bar{\partial }}({\tilde{f}}\psi _{\mu _1})(z) {\mathcal {F}}_\hbar ^{-1}[\chi _{\xi }](s-z) \eta (-i\mu _2,z)\, L(dz), \end{aligned}$$

(5.47)

where

$$\begin{aligned} \eta (-i\mu _2,z) = \text {Tr} [ \text {Op} _\hbar ^{\text {w} }({\tilde{\theta }}_{k,-i\mu _2})(z- {\tilde{A}}_\varepsilon (\hbar ;-i\mu _2))^{-1} \text {Op} _\hbar ^{\text {w} }({\tilde{\theta }}_{l,-i\mu _2}) ], \end{aligned}$$

and $\mu _2=\frac{\mu _1}{C_1}$. From the construction of the almost analytic extension we have the following form of the principal symbol of $z- {\tilde{A}}_\varepsilon (\hbar ;-i\mu _2)$

$$\begin{aligned} z- {\tilde{a}}_{\varepsilon ,0}(x,p;-i\mu _2) = z - (a_{\varepsilon ,0}(x,p) - i\mu _2 (\partial _{p_1}a_{\varepsilon ,0})(x,p) + {\mathcal {O}}(\mu _2^2)). \end{aligned}$$

Let $z\in {\mathbb {C}}$ such that $-\frac{c_0\mu _2}{4}<\text {Im} (z)<0$ and $\left| \text {Re} (z) \right| <\eta$, where $c_0$ is the constant from the global micro-hyperbolicity (5.31). We have by the global micro-hyperbolicity and $\hbar$ sufficiently small that

$$\begin{aligned} \text {Im} (z- {\tilde{a}}_{\varepsilon ,0}(x,p;-i\mu _2)) \ge c_0\mu _2 +\text {Im} (z) - C \mu _2(\text {Re} (z)-a_{\varepsilon ,0}(x,p))^2. \end{aligned}$$

To see this recall how the principal symbol was changed, and that if $\text {Re} (z)-a_{\varepsilon ,0}(x,p)$ is zero or small then $(\partial _{p_1}a_{\varepsilon ,0})(x,p)>2c_0$. Hence, we have to assume $\hbar$ sufficiently small. This implies there exists a $C_2$ such we have the inequality

$$\begin{aligned} \begin{aligned} \text {Im} (z- {\tilde{a}}_{\varepsilon ,0}(x,p;-i\mu _2)&+ C_2 \mu _2 (\overline{z-{\tilde{a}}_{\varepsilon ,0}(x,p;-i\mu _2)})(z-{\tilde{a}}_{\varepsilon ,0}(x,p;-i\mu _2)) \\&\ge \frac{c_0}{2}\mu _2 +\text {Im} (z), \end{aligned} \end{aligned}$$

where we again assume $\hbar$ sufficiently small, and that all terms from the product in the above equation, which are not $(\text {Re} (z)-a_{\varepsilon ,0}(x,p))^2$, come with at least one extra $\mu _2$. Now by Theorem 3.28 we have for every g in $L^2({\mathbb {R}}^d)$ that

$$\begin{aligned} \begin{aligned} \text {Im}&(\langle \text {Op} _\hbar ^{\text {w} }(z- {\tilde{a}}_{\varepsilon ,0}(-i\mu _2)) g,g \rangle ) + C_2 \mu _2 \Vert \text {Op} _\hbar ^{\text {w} }(z-{\tilde{a}}_{\varepsilon ,0}(-i\mu _2))g \Vert _{L^2({\mathbb {R}}^d)}^2 \\&\ge \langle \text {Op} _\hbar ^{\text {w} }(\text {Im} (z- {\tilde{a}}_{\varepsilon ,0}(-i\mu _2)) + C_2 \mu _2 (\overline{z-{\tilde{a}}_{\varepsilon ,0}(-i\mu _2)})(z-{\tilde{a}}_{\varepsilon ,0}(-i\mu _2)) ) g,g \rangle \\&\quad - c \mu _2\hbar ^\delta \Vert g \Vert _{L^2({\mathbb {R}}^d)}^2 \\&\ge ( \frac{c_0\mu _2}{2} +\text {Im} (z) ) \Vert g \Vert _{L^2({\mathbb {R}}^d)}^2 - {\tilde{c}} ( \hbar ^\delta + \mu _2 \hbar ^\delta )\Vert g \Vert _{L^2({\mathbb {R}}^d)}^2 \ge \frac{c_0\mu _2}{6} \Vert g \Vert _{L^2({\mathbb {R}}^d)}^2, \end{aligned} \end{aligned}$$

for $\hbar$ sufficiently small. Moreover, by a Hölder inequality we have that

$$\begin{aligned} \begin{aligned} \frac{c_0\mu _2}{6}&\Vert g \Vert _{L^2({\mathbb {R}}^d)}^2 \\&\le {} \left| \langle \text {Op} _\hbar ^{\text {w} }(z- {\tilde{a}}_{\varepsilon ,0}(-i\mu _2)) g,g \rangle \right| + C_2 \mu _2 \Vert \text {Op} _\hbar ^{\text {w} }(z-{\tilde{a}}_{\varepsilon ,0}(-i\mu _2))g \Vert _{L^2({\mathbb {R}}^d)}^2 \\&\le {} \frac{c_0\mu _2}{12} \Vert g \Vert _{L^2({\mathbb {R}}^d)}^2 + ( \frac{6}{2c_0\mu _2} +C_2 \mu _2) \Vert \text {Op} _\hbar ^{\text {w} }(z-{\tilde{a}}_{\varepsilon ,0}(-i\mu _2))g \Vert _{L^2({\mathbb {R}}^d)}^2. \end{aligned} \end{aligned}$$

This shows that there exists a constant C such that

$$\begin{aligned} \frac{c_0\mu _2}{C} \Vert g \Vert _{L^2({\mathbb {R}}^d)} \le \Vert \text {Op} _\hbar ^{\text {w} }(z-{\tilde{a}}_{\varepsilon ,0}(-i\mu _2))g \Vert _{L^2({\mathbb {R}}^d)}, \end{aligned}$$

for all g in $L^2({\mathbb {R}}^d)$. Since $\text {Op} _\hbar ^{\text {w} }(z-{\tilde{a}}_{\varepsilon ,0}(-i\mu _2))$ is the principal part of ${\tilde{A}}_\varepsilon (\hbar ;-i\mu _2)$ and the rest comes with an extra $\hbar$ in front, since we have assumed regularity $\tau \ge 1$, the above estimate implies that

$$\begin{aligned} \frac{c_0\mu _2}{2C} \Vert g \Vert _{L^2({\mathbb {R}}^d)} \le \Vert (z-{\tilde{A}}_\varepsilon (\hbar ;-i\mu _2))g \Vert _{L^2({\mathbb {R}}^d)}, \end{aligned}$$

for $\hbar$ sufficiently small. We can do the above argument again for $\text {Im} (z)\ge 0$ and obtain the same result. The estimate implies that the set $\{ z\in {\mathbb {C}}\,|\, \text {Im} (z)>-\frac{c_0\mu _2}{4}\}$ is in the regularity set of ${\tilde{A}}_\varepsilon (\hbar ;-i\mu _2)$. Since $\{ z\in {\mathbb {C}}\,|\, \text {Im} (z)>-\frac{c_0\mu _2}{4}\}$ is connected we have that this is a subset of the resolvent set if just one point of the set is in the resolvent set. For a z in ${\mathbb {C}}$ with positive imaginary part and $\left| z \right| \ge 2\Vert {\tilde{A}}_\varepsilon (\hbar ;-i\mu _2) \Vert$ we have existence of $(z-{\tilde{A}}_\varepsilon (\hbar ;-i\mu _2))^{-1}$ as a Neumann series. Hence we can conclude that $(z-{\tilde{A}}_\varepsilon (\hbar ;-i\mu _2))^{-1}$ extends to a holomorphic function for z in ${\mathbb {C}}$ such $\text {Im} (z) \ge -\frac{c_0\mu _2}{4 C_1}$. This implies that

$$\begin{aligned} \begin{aligned} 0&=-\frac{1}{\pi }\int _{\mathbb {C}}({\tilde{f}}\psi _{\mu _1} \psi _{-\frac{c_0\mu _2}{4C_1}})(z) {\mathcal {F}}_\hbar ^{-1}[\chi _{\xi }](s-z) {\bar{\partial }} \eta (-i\mu _2,z) \, L(dz) \\&=\frac{1}{\pi }\int _{\mathbb {C}}{\bar{\partial }}({\tilde{f}}\psi _{\mu _1} \psi _{-\frac{c_0\mu _2}{4C_1}})(z) {\mathcal {F}}_\hbar ^{-1}[\chi _{\xi }](s-z) \eta (-i\mu _2,z) \, L(dz) \\&= \frac{1}{\pi }\int _{\text {Im} (z)\ge 0} {\bar{\partial }}({\tilde{f}}\psi _{\mu _1})(z) {\mathcal {F}}_\hbar ^{-1}[\chi _{\xi }](s-z) \eta (-i\mu _2,z) \, L(dz) \\&\quad + \frac{1}{\pi }\int _{\text {Im} (z)<0} {\bar{\partial }}({\tilde{f}}\psi _{\mu _1} \psi _{-\frac{c_0\mu _2}{4C_1}})(z) {\mathcal {F}}_\hbar ^{-1}[\chi _{\xi }](s-z) \eta (-i\mu _2,z) \, L(dz), \end{aligned} \end{aligned}$$

where we have used that $\psi _{-\frac{c_0\mu _2}{4C_1}}(z)=1$ for all z in ${\mathbb {C}}$ with $\text {Im} (z)\ge 0$. This equality gives us the following rewriting of (5.47)

$$\begin{aligned} \begin{aligned} -\frac{1}{\pi }&\int _{\text {Im} (z)>\mu _1} {\bar{\partial }}({\tilde{f}}\psi _{\mu _1})(z) {\mathcal {F}}_\hbar ^{-1}[\chi _{\xi }](s-z) \eta (-i\mu _2,z)\, L(dz) \\&= \frac{1}{\pi }\int _{\text {Im} (z)<0} {\bar{\partial }}({\tilde{f}}\psi _{\mu _1} \psi _{-\frac{c_0\mu _2}{4C_1}})(z) {\mathcal {F}}_\hbar ^{-1}[\chi _{\xi }](s-z) \eta (-i\mu _2,z) \, L(dz) + {\mathcal {O}}(\hbar ^{N_0}), \end{aligned} \end{aligned}$$

(5.48)

for any $N_0$ in ${\mathbb {N}}$. We have that

$$\begin{aligned} {\bar{\partial }}({\tilde{f}}\psi _{\mu _1} \psi _{-\frac{c_0\mu _2}{4C_1}})(z) = {\bar{\partial }}({\tilde{f}})(z) (\psi _{\mu _1} \psi _{-\frac{c_0\mu _2}{4C_1}})(z) + {\tilde{f}}\psi _{\mu _1}(z) {\bar{\partial }} \psi _{-\frac{c_0\mu _2}{4C_1}}(z), \end{aligned}$$

for $\text {Im} (z)<0$, where we have used that $\psi _{\mu _1}(z) = 1$ for $\text {Im} (z)\le 1$. The part of the integral on the right hand side of (5.48) with the derivative on ${\tilde{f}}$ will be small due to the same argumentation as previously in the proof. What remains is the part where the derivative is on $\psi _{-\frac{c_0\mu _2}{4C_1}}$. For this part we have that

$$\begin{aligned} \begin{aligned} \frac{1}{\pi } \Big |&\int _{\text {Im} (z)<0} {\tilde{f}}(z) {\bar{\partial }} \psi _{-\frac{c_0\mu _2}{4C_1}}(z) {\mathcal {F}}_\hbar ^{-1}[\chi _{\xi }](s-z) \eta (-i\mu _2,z) \, L(dz) \Big | \\&\le \frac{C}{\hbar ^d \left( \frac{M\hbar }{C_1\xi } \log (\frac{1}{\hbar })\right) ^2} \int _{\begin{array}{c} -\eta<\text {Re} (z)<\eta \\ -\frac{M\hbar c_0}{2C_1^2\xi }\log (\frac{1}{\hbar })<\text {Im} (z)< -\frac{M\hbar c_0}{4C_1^2\xi }\log (\frac{1}{\hbar }) \end{array}} \frac{\xi }{\hbar } e^{\frac{\xi \text {Im} (z)}{2\hbar }} \,L(dz) \\&= \frac{C}{\hbar ^d \left( \frac{M\hbar }{C_1\xi } \log (\frac{1}{\hbar })\right) ^2)} e^{-\frac{c_oM}{2C_1^2}\log (\frac{1}{\hbar })} = \frac{{\tilde{C}} \xi ^2}{\hbar ^{d+2} M^2 \log (\frac{1}{\hbar })^2} \hbar ^{\frac{c_o}{2C_1^2}M}. \end{aligned} \end{aligned}$$

Hence by choosing M sufficiently large we can make the above expression smaller than $\hbar ^N$ for any N in ${\mathbb {N}}$. This concludes the proof. $\square$

This Theorem actually imply a stronger version of it self, where the assumption of boundedness is not needed.

Corollary 5.6

Let $A_\varepsilon (\hbar )$ be a strongly $\hbar$-$\varepsilon$-admissible operator of regularity $\tau \ge 1$ which satisfies Assumption 4.1 and assume there exists a $\delta$ in (0, 1) such that $\varepsilon \ge \hbar ^{1-\delta }$. Suppose there exists a number $\eta >0$ such $a_{\varepsilon ,0}^{-1}([-2\eta ,2\eta ])$ is compact and a constant $c>0$ such that

$$\begin{aligned} \left| \nabla _p a_{\varepsilon ,0}(x,p) \right| \ge c \quad \text {for all } (x,p) \in a_{\varepsilon ,0}^{-1}([-2\eta ,2\eta ]), \end{aligned}$$

where $a_{\varepsilon ,0}$ is the principal symbol of $A_\varepsilon (\hbar )$. Let f be in $C_0^\infty ((-\eta ,\eta ))$ and $\theta$ be in $C_0^\infty ({\mathbb {R}}^d_x\times {\mathbb {R}}^d_p)$ such that $\text {supp} (\theta )\subset a_{\varepsilon ,0}^{-1}((-\eta ,\eta ))$. There exists a constant $T_0>0$ such that if $\chi$ is in $C_0^\infty ((\frac{1}{2} \hbar ^{1-\gamma },T_0))$ for a $\gamma$ in $(0,\delta ]$, then for every N in ${\mathbb {N}}$, we have

$$\begin{aligned} \left| \text {Tr} [\text {Op} _\hbar ^{\text {w} }(\theta )f(A_\varepsilon (\hbar )) {\mathcal {F}}_\hbar ^{-1}[\chi ](s-A_\varepsilon (\hbar ))\text {Op} _\hbar ^{\text {w} }(\theta )] \right| \le C_N \hbar ^N, \end{aligned}$$

uniformly for s in $(-\eta ,\eta )$.

Proof

We start by letting $\varphi \in C_0^\infty ({\mathbb {R}}^d_x\times {\mathbb {R}}^d_p)$ such that $\varphi (x,p)=1$ on the set $a_{\varepsilon ,0}^{-1}([-3/2\eta ,3/2\eta ])$ and $\text {supp} (\varphi )\subset a_{\varepsilon ,0}^{-1}([-2\eta ,2\eta ])$. Then we define the operator ${\tilde{A}}_\varepsilon (\hbar )$ as the operator with symbol

$$\begin{aligned} {\tilde{a}}_\varepsilon (\hbar ) = \varphi a_{\varepsilon ,0} + C(1-\varphi ) + \sum _{j\ge 1} \hbar ^j \varphi a_{\varepsilon ,j}, \end{aligned}$$

where C is chosen sufficiently large. This operator satisfies the assumptions in Theorem 5.4 and from this theorem we get for all $N\in {\mathbb {N}}$ that

$$\begin{aligned} \left| \text {Tr} [\text {Op} _\hbar ^{\text {w} }(\theta )f({\tilde{A}}_\varepsilon (\hbar )) {\mathcal {F}}_\hbar ^{-1}[\chi ](s-{\tilde{A}}_\varepsilon (\hbar ))\text {Op} _\hbar ^{\text {w} }(\theta )] \right| \le C_N \hbar ^N. \end{aligned}$$

(5.49)

Hence we are done, if we establish the following bound

$$\begin{aligned}&\big | \text {Tr} \big [\text {Op} _\hbar ^{\text {w} }(\theta )\big (f(A_\varepsilon (\hbar )) {\mathcal {F}}_\hbar ^{-1}[\chi ](s-A_\varepsilon (\hbar ))- f({\tilde{A}}_\varepsilon (\hbar )) {\mathcal {F}}_\hbar ^{-1}[\chi ](s-{\tilde{A}}_\varepsilon (\hbar ))\big )\text {Op} _\hbar ^{\text {w} }(\theta )\big ] \big |.\nonumber \\&\le C_N \hbar ^N \end{aligned}$$

(5.50)

As in the proof of Theorem 5.4 it will suffice to prove the following bound uniformly in $\xi$

$$\begin{aligned}&\big | \text {Tr} \big [\text {Op} _\hbar ^{\text {w} }(\theta )\big (f(A_\varepsilon (\hbar )) {\mathcal {F}}_\hbar ^{-1}[\chi _\xi ](s-A_\varepsilon (\hbar ))- f({\tilde{A}}_\varepsilon (\hbar )) {\mathcal {F}}_\hbar ^{-1}[\chi _\xi ](s-{\tilde{A}}_\varepsilon (\hbar ))\big )\text {Op} _\hbar ^{\text {w} }(\theta )\big ] \big |\nonumber \\&\le C_N \hbar ^N. \end{aligned}$$

(5.51)

With $\chi _\xi (t) = \chi (\frac{t}{\xi })$, where $\chi$ is in $C_0^\infty ((\frac{1}{2},1))$ and $\xi$ in $[\hbar ^{1-\gamma },T_0]$. From here the argument is almost analogous to the argument, where we obtained the global micro-hyperbolic condition, from the proof of Theorem 5.4. Again we let $\psi$ be in $C^\infty ({\mathbb {R}})$ such $\psi (t)=1$ for $t\le 1$ and $\psi (t)=0$ for $t\ge 2$. Moreover let M be a sufficiently large constant which will be fixed later and put

$$\begin{aligned} \psi _{\mu _1}(z) = \psi \Big (\tfrac{\text {Im} (z)}{\mu _1}\Big ), \end{aligned}$$

where $\mu _1=\frac{M\hbar }{\xi }\log (\frac{1}{\hbar })$. With this function we again have the estimates from (5.26) for the function ${\bar{\partial }}({\tilde{f}}\psi _{\mu _1})$. Again we can use the Helffer-Sjöstrand formula (Theorem 4.10) for both operators $A_\varepsilon (\hbar )$ and ${\tilde{A}}_\varepsilon (\hbar )$ on the function $({\tilde{f}}\psi _{\mu _1})(z){\mathcal {F}}_\hbar ^{-1}[\chi _\xi ](s-z)$. This gives us that

$$\begin{aligned}&\text {Tr} \big [\text {Op} _\hbar ^{\text {w} }(\theta )f(A_\varepsilon (\hbar )) {\mathcal {F}}_\hbar ^{-1}[\chi _\xi ](s-A_\varepsilon (\hbar ))\text {Op} _\hbar ^{\text {w} }(\theta )\big ] \nonumber \\&={}-\frac{1}{\pi } \int _{{\mathbb {C}}} {\bar{\partial }}({\tilde{f}}\psi _{\mu _1})(z) {\mathcal {F}}_\hbar ^{-1}[\chi _{\xi }](s-z) \text {Tr} [\text {Op} _\hbar ^{\text {w} }(\theta _k)(z-A_\varepsilon (\hbar ))^{-1} \text {Op} _\hbar ^{\text {w} }(\theta _l)]\, L(dz) \nonumber \\&={} -\frac{1}{\pi } \int _{\text {Im} (z)>\mu _1}\cdots \, L(dz) +{\mathcal {O}}(\hbar ^N), \end{aligned}$$

(5.52)

where the second equality follow from analogous arguments to those used in the proof of Theorem 5.4. The same identities are true for the operator ${\tilde{A}}_\varepsilon (\hbar )$. This gives us that

$$\begin{aligned}&\big | \text {Tr} \big [\text {Op} _\hbar ^{\text {w} }(\theta )\big (f(A_\varepsilon (\hbar )) {\mathcal {F}}_\hbar ^{-1}[\chi _\xi ](s-A_\varepsilon (\hbar ))- f({\tilde{A}}_\varepsilon (\hbar )) {\mathcal {F}}_\hbar ^{-1}[\chi _\xi ](s-{\tilde{A}}_\varepsilon (\hbar ))\big )\text {Op} _\hbar ^{\text {w} }(\theta )\big ] \big | \nonumber \\&\le \Big | \frac{1}{\pi } \int _{\text {Im} (z)>\mu _1} {\bar{\partial }}({\tilde{f}}\psi _{\mu _1})(z) {\mathcal {F}}_\hbar ^{-1}[\chi _{\xi }](s-z) \text {Tr} [\text {Op} _\hbar ^{\text {w} }(\theta )(z-A_\varepsilon (\hbar ))^{-1} \text {Op} _\hbar ^{\text {w} }(\theta )]\, L(dz) \nonumber \\&\quad -\frac{1}{\pi } \int _{\text {Im} (z)>\mu _1} {\bar{\partial }}({\tilde{f}}\psi _{\mu _1})(z) {\mathcal {F}}_\hbar ^{-1}[\chi _{\xi }](s-z) \text {Tr} [\text {Op} _\hbar ^{\text {w} }(\theta )(z-{\tilde{A}}_\varepsilon (\hbar ))^{-1} \text {Op} _\hbar ^{\text {w} }(\theta )]\, L(dz)\Big | \nonumber \\&\quad + C\hbar ^N. \end{aligned}$$

(5.53)

To estimate this difference we introduce the function ${\tilde{\theta }}\in C_0^\infty ({\mathbb {R}}^d_x\times {\mathbb {R}}^d_p)$ such that ${\tilde{\theta }}=1$ in a neighbourhood of $\text {supp} (\varphi )$. Then we use the resolvent formalism to obtain the estimate

$$\begin{aligned} \begin{aligned}&\big \Vert \text {Op} _\hbar ^{\text {w} }(\theta ) (z-{\tilde{A}}_\varepsilon (\hbar ))^{-1}\text {Op} _\hbar ^{\text {w} }(\theta ) - \text {Op} _\hbar ^{\text {w} }(\theta ) (z-A_\varepsilon (\hbar ))^{-1} \text {Op} _\hbar ^{\text {w} }(\theta )\big \Vert _{\text {Tr} } \\&\le {} \big \Vert \text {Op} _\hbar ^{\text {w} }(\theta ) (z-{\tilde{A}}_\varepsilon (\hbar ))^{-1} \text {Op} _\hbar ^{\text {w} }({\tilde{\theta }}) (A_\varepsilon (\hbar )- {\tilde{A}}_\varepsilon (\hbar )) (z- {\tilde{A}}_\varepsilon (\hbar ))^{-1} \text {Op} _\hbar ^{\text {w} }(\theta ) \big \Vert _{\text {Tr} } \\&+ \big \Vert \text {Op} _\hbar ^{\text {w} }(\theta ) (z-{\tilde{A}}_\varepsilon (\hbar ))^{-1} [{\tilde{A}}_\varepsilon (\hbar ),\text {Op} _\hbar ^{\text {w} }({\tilde{\theta }})](z-A_\varepsilon (\hbar ))^{-1}\text {Op} _\hbar ^{\text {w} }(\theta _l)\big \Vert _{\text {Tr} } + \frac{\hbar ^{{\tilde{N}}}C_{{\tilde{N}}}}{|\text {Im} (z)|}, \end{aligned} \end{aligned}$$

(5.54)

where we have used that $\text {supp} (\theta )\subset \text {supp} (\varphi )$. With this expression we can argue as in the proof of Theorem 5.4 and get that

$$\begin{aligned} \begin{aligned}&\big \Vert {\bar{\partial }}({\tilde{f}}\psi _{\mu _1})(z) {\mathcal {F}}_\hbar ^{-1}[\chi _{\xi }](s-z) \text {Op} _\hbar ^{\text {w} }(\theta _k)\big ((z-{\tilde{A}}_\varepsilon (\hbar ))^{-1} - (z-A_\varepsilon (\hbar ))^{-1}\big ) \text {Op} _\hbar ^{\text {w} }(\theta _l) \big \Vert _{\text {Tr} } \\&\le {} C_{{\tilde{N}}} \psi _{\mu _1}(z) \xi \left| \text {Im} (z) \right| ^{{\tilde{N}}} e^{\frac{\xi \text {Im} (z)}{\hbar }} \left[ \frac{ C \big ( \hbar ^{C_1} + e^{- \frac{\text {Im} (z)}{C_2 \hbar }} \big ) }{\hbar ^{d+1}|\text {Im} (z)|^2}+ \frac{\hbar ^{{\tilde{N}}-1} C_{{\tilde{N}}}}{|\text {Im} (z)|^2}+ \frac{\hbar ^{{\tilde{N}}-1}C_{{\tilde{N}}}}{|\text {Im} (z)|} \right] \\&\quad + \mu _1^{-1} \xi e^{\frac{\xi \text {Im} (z)}{\hbar }} {\varvec{1}}_{[1,2]}\left( \tfrac{\text {Im} (z)}{\mu _1}\right) \left[ \frac{ C \max \left( \hbar ^{C_1} , e^{- \frac{\text {Im} (z)}{C_2 \hbar }} \right) }{\hbar ^{d+1}|\text {Im} (z)|^2}+ \frac{\hbar ^{{\tilde{N}}-1} C_{{\tilde{N}}}}{|\text {Im} (z)|^2}+ \frac{\hbar ^{{\tilde{N}}-1}C_{{\tilde{N}}}}{|\text {Im} (z)|} \right] , \end{aligned} \end{aligned}$$

(5.55)

From this estimate we obtain the estimate

$$\begin{aligned} \begin{aligned}&\Big | \frac{1}{\pi } \int _{\text {Im} (z)>\mu _1} {\bar{\partial }}({\tilde{f}}\psi _{\mu _1})(z) {\mathcal {F}}_\hbar ^{-1}[\chi _{\xi }](s-z) \text {Tr} [\text {Op} _\hbar ^{\text {w} }(\theta _k)(z-A_\varepsilon (\hbar ))^{-1} \text {Op} _\hbar ^{\text {w} }(\theta _l)]\, L(dz) \\&-\frac{1}{\pi } \int _{\text {Im} (z)>\mu _1} {\bar{\partial }}({\tilde{f}}\psi _{\mu _1})(z) {\mathcal {F}}_\hbar ^{-1}[\chi _{\xi }](s-z) \text {Tr} [\text {Op} _\hbar ^{\text {w} }(\theta _k)(z-{\tilde{A}}_\varepsilon (\hbar ))^{-1} \text {Op} _\hbar ^{\text {w} }(\theta _l)]\, L(dz)\Big | \\&\le C\hbar ^N. \end{aligned} \end{aligned}$$

(5.56)

Finally by combining the estimates in (5.53) and (5.56) we obtain the estimate in (5.51). This concludes the proof. $\square$

6 Weyl law for rough pseudo-differential operators

In this section, we will prove a Weyl law for rough pseudo-differential operators, and we will do it with the approach used in [24]. Firstly, we will consider some asymptotic expansions of certain integrals.

Theorem 6.1

Let $A_\varepsilon (\hbar )$ be a $\hbar$-$\varepsilon$-admissible operator of regularity $\tau \ge 1$ which satisfies Assumption 4.1 and assume there exists a $\delta$ in (0, 1) such that $\varepsilon \ge \hbar ^{1-\delta }$. Suppose there exists $\eta >0$ such that $a_{0,\varepsilon }^{-1}([-2\eta ,2\eta ])$ is compact and every value in the interval $[-2\eta ,2\eta ]$ is non critical for $a_{0,\varepsilon }$, where $a_{\varepsilon ,0}$ is the principal symbol of $A_\varepsilon (\hbar )$. Let $\chi$ be in $C^\infty _0((-T_0,T_0))$ and $\chi =1$ in a neighbourhood of 0, where $T_0$ is the number from Corollary 5.6. Then for every f in $C_0^\infty ((-\eta ,\eta ))$ we have that

$$\begin{aligned} \int _{\mathbb {R}}\text {Tr} [f(A_\varepsilon (\hbar )) e^{it\hbar ^{-1}A_\varepsilon (\hbar )}]e^{-its\hbar ^{-1} } \chi (t) \,dt = (2\pi \hbar )^{1-d} \left[ \sum _{j=0}^{N_0} \hbar ^j \xi _j(s) + {\mathcal {O}}(\hbar ^{N})\right] . \end{aligned}$$

The error term is uniform with respect to $s \in (-\eta ,\eta )$. The number $N_0$ depends on the desired error. The functions $\xi _j(s)$ are smooth functions in s and are given by

$$\begin{aligned} \xi _0(s) = f(s) \int _{\{a_{\varepsilon ,0}=s\}} \frac{1}{\left| \nabla {a_{\varepsilon ,0}} \right| } \,dS_s, \\ \xi _j(s) = \sum _{k=1}^{2j-1}\frac{1}{k!} f(s) \partial ^k_s \int _{\{a_{\varepsilon ,0}=s\}} \frac{d_{\varepsilon ,j,k}}{\left| \nabla {a_{\varepsilon ,0}} \right| } \,dS_s, \end{aligned}$$

where the symbols $d_{\varepsilon ,j,k}$ are the polynomials from Lemma 4.4. In particular we have that

$$\begin{aligned} \begin{aligned} \xi _1 (s) = -f(s) \partial _s \int _{\{a_{\varepsilon ,0}=s\}} \frac{a_{\varepsilon ,1}}{\left| \nabla {a_{\varepsilon ,0}} \right| } \,dS_s. \end{aligned} \end{aligned}$$

Moreover, we have the a priori bounds

$$\begin{aligned} |\hbar ^j \xi _j(s) | \le {\left\{ \begin{array}{ll} C &{} \hbox { if}\ j=0 \\ C\hbar &{} \hbox { if}\ j=1 \\ C\hbar ^{1+\delta (j-2)} &{} \hbox { if}\ j\ge 2. \end{array}\right. } \end{aligned}$$

(6.1)

Remark 6.2

Suppose we are in the setting of Theorem 6.1. The statement of the theorem can be rephrased in terms of convolution of measures and a function. To see this let f be in $C_0^\infty ((-\eta ,\eta ))$, for this function we can define the function

$$\begin{aligned} \begin{aligned} M_f^0(\omega ;\hbar )&=\text {Tr} [f(A_\varepsilon (\hbar )) {\varvec{1}}_{(-\infty ,\omega ]}(A_\varepsilon (\hbar ))] \\&= \text {Tr} [f(A_\varepsilon (\hbar )) {\varvec{1}}_{[0,\infty )}(\omega -A_\varepsilon (\hbar ))] = \text {Tr} [f(A_\varepsilon (\hbar )) (\omega -A_\varepsilon (\hbar ))_{+}^0], \end{aligned} \end{aligned}$$

where $t_+=\max (0,t)$. We have, that $M_f^0(\omega ;\hbar )$ is a monotonic increasing function, hence it defines a measure in the natural way. If we consider the function

$$\begin{aligned} {\hat{\chi }}_\hbar (t) = \frac{1}{2\pi \hbar } \int _{\mathbb {R}}e^{it\hbar ^{-1}s} \chi (s) \, ds, \end{aligned}$$

then we have that

$$\begin{aligned}{} & {} {\hat{\chi }}_\hbar *dM_f^0(\cdot ;\hbar ) (s) \nonumber \\{} & {} ={} \sum _{e_j(\hbar )\in {\mathcal {P}}} {\hat{\chi }}_\hbar (s-e_j(\hbar ))f(e_j(\hbar )) =\text {Tr} [{\hat{\chi }}_\hbar (s-A_\varepsilon (\hbar ))f(A_\varepsilon (\hbar ))] \nonumber \\{} & {} ={}\frac{1}{2\pi \hbar }\int _{\mathbb {R}}\text {Tr} [f(A_\varepsilon (\hbar )) e^{it\hbar ^{-1}A_\varepsilon (\hbar )}]e^{-its\hbar ^{-1} } \chi (t) \,dt = \frac{1}{(2\pi \hbar )^{d}} \left[ \sum _{j=0}^{N_0} \hbar ^j \xi _j(s) + {\mathcal {O}}(\hbar ^{N})\right] .\nonumber \\ \end{aligned}$$

(6.2)

This formulation of the theorem will prove useful when we consider Riesz means.

The proof of the theorem is split in two parts. First is the existence of the expansion proven by a stationary phase theorem. Next is the form of the coefficients found by applying the functional calculus developed earlier.

Proof

In order to be in a situation, where we can apply the stationary phase theorem, we need to exchange the propagator with the approximation of it constructed in Sect. 5. As the construction required auxiliary localisation we need to introduce these. Let $\theta$ be in $C_0^\infty ({\mathbb {R}}_x^d\times {\mathbb {R}}^d_p)$ such that $\text {supp} (\theta )\subset a_{\varepsilon ,0}^{-1}((-\eta ,\eta ))$ and $\theta (x,p) = 1$ for all $(x,p)\in \text {supp} (f(a_{\varepsilon ,0}))$. Now by Lemma 4.14 we have that

$$\begin{aligned} \Vert (1-\text {Op} _\hbar ^{\text {w} }(\theta ))f(A_\varepsilon (\hbar )) e^{it\hbar ^{-1}A_\varepsilon (\hbar )} \Vert _{\text {Tr} } \le \Vert (1-\text {Op} _\hbar ^{\text {w} }(\theta ))f(A_\varepsilon (\hbar )) \Vert _{\text {Tr} } \le C_N\hbar ^N, \end{aligned}$$

(6.3)

for every N in ${\mathbb {N}}$. Hence we have that

$$\begin{aligned} \big |\text {Tr} [f(A_\varepsilon (\hbar ))e^{it\hbar ^{-1}A_\varepsilon (\hbar )}]-\text {Tr} [\text {Op} _\hbar ^{\text {w} }(\theta )f(A_\varepsilon (\hbar ))e^{it\hbar ^{-1}A_\varepsilon (\hbar )}] \big | \le C_N\hbar ^N, \end{aligned}$$

for any N in ${\mathbb {N}}$. This implies the identity

$$\begin{aligned}{} & {} \int _{\mathbb {R}}\text {Tr} [f(A_\varepsilon (\hbar )) e^{it\hbar ^{-1}A_\varepsilon (\hbar )}]e^{-its\hbar } \chi (t) \,dt \nonumber \\{} & {} = \int _{\mathbb {R}}\text {Tr} [ \text {Op} _\hbar ^{\text {w} }(\theta ) f(A_\varepsilon (\hbar )) e^{it\hbar ^{-1}A_\varepsilon (\hbar )}]e^{-its\hbar } \chi (t) \,dt +{\mathcal {O}}(\hbar ^N). \end{aligned}$$

(6.4)

In order to use the results of Sect. 5 we need also to localise in time. To do this we let $\chi _2$ be in $C_0^\infty ({\mathbb {R}})$ such that $\chi _2(t)=1$ for t in $[-\frac{1}{2} \hbar ^{1-\frac{\delta }{2}}, \frac{1}{2} \hbar ^{1-\frac{\delta }{2}}]$ and $\text {supp} (\chi _2) \subset [-\hbar ^{1-\frac{\delta }{2}},\hbar ^{1-\frac{\delta }{2}}]$. With this function we have that

$$\begin{aligned}{} & {} \int _{\mathbb {R}}\text {Tr} [ \text {Op} _\hbar ^{\text {w} }(\theta ) f(A_\varepsilon (\hbar )) e^{it\hbar ^{-1}A_\varepsilon (\hbar )}]e^{-its\hbar } \chi (t) \,dt \nonumber \\{} & {} = \int _{\mathbb {R}}\text {Tr} [ \text {Op} _\hbar ^{\text {w} }(\theta ) f(A_\varepsilon (\hbar )) e^{it\hbar ^{-1}A_\varepsilon (\hbar )}]e^{-its\hbar } \chi _2(t) \chi (t) \,dt \nonumber \\{} & {} \quad + \int _{\mathbb {R}}\text {Tr} [ \text {Op} _\hbar ^{\text {w} }(\theta ) f(A_\varepsilon (\hbar )) e^{it\hbar ^{-1}A_\varepsilon (\hbar )}]e^{-its\hbar }(1-\chi _2(t)) \chi (t)\,dt. \end{aligned}$$

(6.5)

We will use the notation ${\tilde{\chi }}(t)=(1-\chi _2(-t))\chi (-t)$ in the following. We start by considering the second term. Here we introduce an extra localisation analogous to how we introduced the first. Using the estimate in (6.3) and cyclicity of the trace again we have that

$$\begin{aligned} \text {Tr} [ \text {Op} _\hbar ^{\text {w} }(\theta ) f(A_\varepsilon (\hbar )) e^{it\hbar ^{-1}A_\varepsilon (\hbar )}] = \text {Tr} [ \text {Op} _\hbar ^{\text {w} }(\theta ) f(A_\varepsilon (\hbar )) e^{it\hbar ^{-1}A_\varepsilon (\hbar )}\text {Op} _\hbar ^{\text {w} }(\theta )] + C_N \hbar ^N. \end{aligned}$$

Now by Corollary 5.6 we have that

$$\begin{aligned}{} & {} \int _{\mathbb {R}}\text {Tr} [ \text {Op} _\hbar ^{\text {w} }(\theta ) f(A_\varepsilon (\hbar )) e^{it\hbar ^{-1}A_\varepsilon (\hbar )}\text {Op} _\hbar ^{\text {w} }(\theta )]e^{-its\hbar }{\tilde{\chi }}(-t)\,dt \nonumber \\{} & {} = 2\pi \hbar \text {Tr} [ \text {Op} _\hbar ^{\text {w} }(\theta ) f(A_\varepsilon (\hbar )) {\mathcal {F}}_\hbar ^{-1}[{\tilde{\chi }}](s-A_\varepsilon (\hbar )) \text {Op} _\hbar ^{\text {w} }(\theta )] \le {\tilde{C}}_N \hbar ^N, \end{aligned}$$

(6.6)

uniformly for s in $[-\eta ,\eta ]$ and any N in ${\mathbb {N}}$. What remains in (6.5) is the first term. For this term we change the quantisation of the localisation. By Corollary 3.20 we obtain for any N in ${\mathbb {N}}$ that

$$\begin{aligned} \text {Op} _\hbar ^{\text {w} }(\theta ) = \text {Op} _{\hbar ,0}(\theta _0^N) + \hbar ^{N+1} R_N(\hbar ), \end{aligned}$$

where $R_N$ is a bounded operator uniformly in $\hbar$, since $\theta$ is a non-rough symbol. Moreover, we have that

$$\begin{aligned} \theta _0^N(x,p) = \sum _{j=0}^N \frac{\hbar ^j}{j!} \left( -\frac{1}{2}\right) ^j (\nabla _x D_p)^j \theta (x,p). \end{aligned}$$

If we choose N sufficiently large (greater than or equal to 2) we can exchange $\text {Op} _\hbar ^{\text {w} }(\theta )$ by $\text {Op} _{\hbar ,0}(\theta _0^N)$ plus a negligible error. We will in the following omit the N on $\theta _0^N$. For the first term on the right hand side in (6.5) we have that $\left| t \right| \le \hbar ^{1-\frac{\delta }{2}}$. Now by Theorem 5.1 there exists $U_N(t,\varepsilon ,\hbar )$ with integral kernel

$$\begin{aligned} \begin{aligned} K_{U_N}(x,y,t,\varepsilon ,\hbar ) = \frac{1}{(2\pi \hbar )^d} \int _{{\mathbb {R}}^d} e^{i \hbar ^{-1} \langle x-y,p\rangle } e^{ i t \hbar ^{-1} a_{\varepsilon ,0}(x,p)} \sum _{j=0}^N (it\hbar ^{-1})^j u_j(x,p,\hbar ,\varepsilon ) \, dp, \end{aligned} \end{aligned}$$

such that

$$\begin{aligned} \sup _{\left| t \right| \le \hbar ^{1-\frac{\delta }{2}}}\Vert \hbar \partial _t U_N(t,\varepsilon , \hbar ) - i U_N(t,\varepsilon , \hbar ) A_\varepsilon (\hbar ) \Vert _{{\mathcal {L}}(L^2({\mathbb {R}}^d))} \le C \hbar ^{N_0}, \end{aligned}$$

(6.7)

and $U_N(0,\varepsilon ,\hbar ) = \text {Op} _{\hbar ,0}(\theta _0)$. We emphasise, that the number N in the operator $U_N$ is dependent on the error $N_0$. We observe that

$$\begin{aligned} \begin{aligned}&|\text {Tr} [ \text {Op} _{\hbar ,0}(\theta _0) e^{it\hbar ^{-1}A_\varepsilon (\hbar )} f(A_\varepsilon (\hbar ))] - \text {Tr} [ U_N(t,\varepsilon , \hbar ) f(A_\varepsilon (\hbar ))] | \\&={}| \text {Tr} \left[ \int _0^t \partial _s( U_N(t-s,\varepsilon , \hbar ) e^{is\hbar ^{-1}A_\varepsilon (\hbar )} f(A_\varepsilon (\hbar )) \,ds\right] | \\&={}| \text {Tr} \left[ \int _0^t (-(\partial _tU_N)(t-s,\varepsilon , \hbar ) + i\hbar ^{-1} U_N(t-s,\varepsilon , \hbar ) A_\varepsilon (\hbar ) ) e^{is\hbar ^{-1}A_\varepsilon (\hbar )} f(A_\varepsilon (\hbar )) \,ds\right] | \\&\le {}\frac{1}{ \hbar } \int _0^t \Vert \hbar \partial _sU_N(s,\varepsilon , \hbar ) - i U_N(s,\varepsilon , \hbar ) A_\varepsilon (\hbar ) \Vert _{{\mathcal {L}}(L^2({\mathbb {R}}^d))} \Vert e^{i(t-s)\hbar ^{-1}A_\varepsilon (\hbar )} f(A_\varepsilon (\hbar )) \Vert _{\text {Tr} } \,ds \\&\le {} C_N \hbar ^{N_0-d}, \end{aligned} \end{aligned}$$

where we have used (6.7). By combining this with (6.5) and (6.6) we have that

$$\begin{aligned} \int _{\mathbb {R}}{} & {} \text {Tr} [ \text {Op} _\hbar ^{\text {w} }(\theta ) f(A_\varepsilon (\hbar )) e^{it\hbar ^{-1}A_\varepsilon (\hbar )}]e^{-its\hbar ^{-1} } \chi _2(t) \chi (t) \,dt \nonumber \\{} & {} = \int _{\mathbb {R}}\text {Tr} [ U_N(t,\varepsilon , \hbar )f(A_\varepsilon (\hbar ))]e^{-its\hbar ^{-1} }\chi _2(t) \chi (t) \,dt + {\mathcal {O}}(\hbar ^N). \end{aligned}$$

(6.8)

Before we proceed we will change the quantisation of $f(A_\varepsilon (\hbar ))$. From Theorem 4.11 we have that

$$\begin{aligned} f(A_\varepsilon (\hbar )) = \sum _{j\ge 0} \hbar ^j \text {Op} _\hbar ^{\text {w} }(a_{\varepsilon ,j}^f), \end{aligned}$$

where

$$\begin{aligned} a_{\varepsilon ,j}^f = \sum _{k=1}^{2j-1} \frac{(-1)^k}{k!} d_{\varepsilon ,j,k} f^{(k)}(a_{\varepsilon ,0}), \end{aligned}$$

(6.9)

the symbols $d_{\varepsilon ,j,k}$ are the polynomials from Lemma 4.4. We choose a sufficiently large N and consider the first N terms of the operator $f(A_\varepsilon (\hbar ))$. For each of these terms we can use Corollary 3.20 and this yields

$$\begin{aligned} \text {Op} _\hbar ^{\text {w} }(a_{\varepsilon ,j}^f) =\text {Op} _{\hbar ,1}(a_{\varepsilon ,j}^{f,M}) + \hbar ^{M+1} R_M, \end{aligned}$$

where $\hbar ^{M+1} R_M$ is a bounded by $C_M \hbar ^N$ in the operator norm. The symbol $a_{\varepsilon ,j}^{f,M}$ is given by

$$\begin{aligned} a_{\varepsilon ,j}^{f,M} = \sum _{j=0}^M \frac{\hbar ^j}{j!} \left( \frac{1}{2}\right) ^j (\nabla _x D_p)^j a_{\varepsilon ,j}^{f}. \end{aligned}$$

By chossing N sufficiently large we can exchange $f(A_\varepsilon (\hbar ))$ by

$$\begin{aligned} \text {Op} _{\hbar ,1}({\tilde{a}}_{\varepsilon }^{f,M}) {:}{=}\sum _{j= 0}^N \hbar ^j \text {Op} _{\hbar ,1}(a_{\varepsilon ,j}^{f,M}), \end{aligned}$$

plus a negligible error, as $U_N(t,\varepsilon , \hbar )$ is trace class. Hence, we have the equality

$$\begin{aligned} \begin{aligned} \int _{\mathbb {R}}&\text {Tr} [ \text {Op} _\hbar ^{\text {w} }(\theta ) f(A_\varepsilon (\hbar )) e^{it\hbar ^{-1}A_\varepsilon (\hbar )}]e^{-its\hbar ^{-1} } \chi _2(t) \chi (t) \,dt \\&= \int _{\mathbb {R}}\text {Tr} [ U_N(t,\varepsilon , \hbar ) \text {Op} _{\hbar ,1}({\tilde{a}}_{\varepsilon }^{f,M}) ]e^{-its\hbar ^{-1} } \chi _2(t) \chi (t) \,dt + {\mathcal {O}}(\hbar ^N). \end{aligned} \end{aligned}$$

As we have the non-critical assumption, Lemma 5.3 gives us that, the trace in the above expression is negligible for $\frac{1}{2} \hbar ^{1-\frac{\delta }{2}} \le |t|\le T_0$. Hence we can omit the $\chi _2(t)$ in the expression and then we have that

$$\begin{aligned}{} & {} \int _{\mathbb {R}}\text {Tr} [ \text {Op} _\hbar ^{\text {w} }(\theta ) f(A_\varepsilon (\hbar )) e^{it\hbar ^{-1}A_\varepsilon (\hbar )}]e^{-its\hbar ^{-1} } \chi _2(t) \chi (t) \,dt \nonumber \\{} & {} = \int _{\mathbb {R}}\text {Tr} [ U_N(t,\varepsilon , \hbar ) \text {Op} _{\hbar ,1}({\tilde{a}}_{\varepsilon }^{f,M}) ]e^{-its\hbar ^{-1} } \chi (t) \,dt + {\mathcal {O}}(\hbar ^N). \end{aligned}$$

(6.10)

The two operators $U_N(t,\varepsilon , \hbar )$ and $\text {Op} _{\hbar ,1}({\tilde{a}}_{\varepsilon }^{f,M})$ are both given by kernels and the composition of the operators has the kernel

$$\begin{aligned} \begin{aligned}&K_{U_N(t,\varepsilon , \hbar )\text {Op} _{\hbar ,1}({\tilde{a}}_{\varepsilon }^{f,M})}(x,y) \\&= \frac{1}{(2\pi \hbar )^d} \int _{{\mathbb {R}}^d} e^{i \hbar ^{-1} \langle x-y,p\rangle } e^{ i t \hbar ^{-1} a_{\varepsilon ,0}(x,p)} \sum _{j=0}^N (it\hbar ^{-1})^j u_j(x,p,\hbar ,\varepsilon ) {\tilde{a}}_{\varepsilon }^{f,M} (y,p) \, dp. \end{aligned} \end{aligned}$$

We can now calculate the trace and we get that

$$\begin{aligned}{} & {} \int _{\mathbb {R}}\text {Tr} [ U_N(t,\varepsilon , \hbar ) \text {Op} _{\hbar ,1}({\tilde{a}}_{\varepsilon }^{f,M}) ]e^{-its\hbar ^{-1} } \chi (t) \,dt \nonumber \\{} & {} = \frac{1}{(2\pi \hbar )^{d}} \int _{{\mathbb {R}}^{2d+1}} \chi (t) e^{ i t \hbar ^{-1} (a_{\varepsilon ,0}(x,p)-s)} u(x,p,t,\hbar ,\varepsilon ) {\tilde{a}}_{\varepsilon }^{f,M} (x,p) \,dxdpdt, \end{aligned}$$

(6.11)

where

$$\begin{aligned} u(x,p,t,\hbar ,\varepsilon ) = \sum _{j=0}^N (it\hbar ^{-1})^j u_j(x,p,\hbar ,\varepsilon ). \end{aligned}$$

In order to evaluate the integral we will apply a stationary phase argument. We will use the theorem in t and one of the p coordinates, after using a partition of unity according to p. By assumption we have that $\left| \nabla _p a_{\varepsilon ,0} \right| >c$ on the support of $\theta$. Hence we can make a partition $\Omega _j$ such that $\partial _{p_j} a_{\varepsilon ,0}\ne 0$ on $\Omega _j$ and without loss of generality we can assume that $\Omega _j$ is connected. For this partition we choose a partition of the unit supported on each of the sets $\Omega _j$. When we have localised to each of these sets the calculation will be identical with some indices changed. Hence we assume that $\partial _{p_1} a_{\varepsilon ,0}\ne 0$ on the entire support of the integrant. We will now make a change of variables in the integral in the following way:

$$\begin{aligned} F:(x,p) \rightarrow (X,P)=(x_1,\dots ,x_d,a_{\varepsilon ,0}(x,p),p_2,\dots ,p_d). \end{aligned}$$

This transformation has the following Jacobian matrix

$$\begin{aligned} DF = \begin{pmatrix} I_d &{} 0_{d\times d} \\ \nabla _x a_{\varepsilon ,0}^t &{} \nabla _p a_{\varepsilon ,0}^t \\ 0_{d-1 \times d+1} &{} I_{d-1} \end{pmatrix}, \end{aligned}$$

where $I_d$ is the d-dimensional identity matrix, $\nabla _x a_\varepsilon ^t$ and $\nabla _p a_\varepsilon ^t$ are the transposed of the respective gradients. $0_{d\times d}$ is a $d\times d$ matrix with all entries zero and $0_{d-1 \times d+1}$ is a $d-1 \times d+1$ matrix with all entries zero. We note that

$$\begin{aligned} \det (DF)= \partial _{p_1}a_{\varepsilon ,0}, \end{aligned}$$

which is non zero by our assumptions. Hence the inverse map exists and we will denote it by $F^{-1}$. For the inverse we denote the part that gives p as a function of (X, P) by $F_2^{-1}$. By this change of variables we have that

$$\begin{aligned} \begin{aligned}&\int _{{\mathbb {R}}^{2d+1}}\chi (t) e^{ i t \hbar ^{-1} (a_{\varepsilon ,0}(x,p)-s)} u(x,p,t,\hbar ,\varepsilon ) {\tilde{a}}_{\varepsilon }^{f,M} (x,p) \,dxdpdt \\&= \int _{{\mathbb {R}}^{2d+1}} \chi (t) e^{ i t \hbar ^{-1} (P_1-s)} \frac{u(X,F_2^{-1}(X,P),t,\hbar ,\varepsilon ) {\tilde{a}}_{\varepsilon }^{f,M} (X,F_2^{-1}(X,P))}{\partial _{p_1}a_{\varepsilon ,0}(X,F_2^{-1}(X,P))} \,dXdPdt, \end{aligned} \end{aligned}$$

where we have omitted the prefactor $(2\pi \hbar )^{-d}$. If we perform the change of variables ${\tilde{P}}_1 = P_1-s$ we can apply the stationary phase method. Hence by stationary phase in the variables ${\tilde{P}}_1$ and t, (6.4), (6.5), (6.8), (6.10) and (6.11) we get that

$$\begin{aligned} \int _{\mathbb {R}}\text {Tr} [f(A_\varepsilon (\hbar )) e^{it\hbar ^{-1}A_\varepsilon (\hbar )}]e^{-its\hbar ^{-1} } \chi (t) \,dt = (2\pi \hbar )^{1-d} \left[ \sum _{j=0}^{N_0} \hbar ^j \xi _j(s) + {\mathcal {O}}(\hbar ^{N})\right] , \end{aligned}$$

(6.12)

uniformly for s in $(-\eta ,\eta )$. The a priori bounds given in the Theorem also follows directly from this application of stationary phase combined with the estimates on $u_j(x,p,\hbar ,\varepsilon )$ from Theorem 5.1. This ends the proof of the existence of the expansion.

From the above expression we have that $\xi _j(s)$ are smooth functions in s hence the above expression defines a distribution on $C^\infty _0((-\eta ,\eta ))$. So in order to find the expressions of the $\xi _j(s)$’s we consider the action of the distribution. We let $\varphi$ be in $C_0^\infty ((-\eta ,\eta ))$ and consider the expresion

$$\begin{aligned} \int _{{\mathbb {R}}^2} \text {Tr} [f(A_\varepsilon (\hbar )) e^{it\hbar ^{-1}A_\varepsilon (\hbar )}]e^{-its\hbar ^{-1}} \chi (t) \varphi (s) \,dt ds. \end{aligned}$$

(6.13)

Using that f is supported in the pure point spectrum of $A_\varepsilon (\hbar )$ we have that

$$\begin{aligned}{} & {} \int _{{\mathbb {R}}^2} \text {Tr} [f(A_\varepsilon (\hbar )) e^{it\hbar ^{-1}A_\varepsilon (\hbar )}]e^{-its\hbar ^{-1} } \chi (t) \varphi (s) \,dt ds \nonumber \\{} & {} = \text {Tr} [f(A_\varepsilon (\hbar )) \int _{\mathbb {R}}{\mathcal {F}}_1[\chi ](\tfrac{s}{\hbar }) \varphi (A_\varepsilon (\hbar )-s) \, ds ], \end{aligned}$$

(6.14)

where we have used Fubini’s theorem. That f is supported in the pure point spectrum follows from Theorem 4.12. If we consider the integral in the right hand side of (6.14) and let $\psi$ be in $C_0^\infty ((-2,2))$ such that $\psi (t)=1$ for $\left| t \right| \le 1$ we have

$$\begin{aligned} \begin{aligned} \int _{\mathbb {R}}{\mathcal {F}}_1[\chi ](\tfrac{s}{\hbar }) \varphi (A_\varepsilon (\hbar )-s) \, ds&={} \int _{{\mathbb {R}}^2} e^{-it s\hbar ^{-1}} \chi (t) \psi (s) \varphi (A_\varepsilon (\hbar )-s) \, ds dt \\&\quad + \int _{{\mathbb {R}}^2} e^{-it s\hbar ^{-1}} \chi (t)(1- \psi (s)) \varphi (A_\varepsilon (\hbar )-s) \, ds dt. \end{aligned} \end{aligned}$$

(6.15)

From the identity

$$\begin{aligned} \Big (\frac{i\hbar }{s}\Big )^n \partial _t^n e^{-it s\hbar ^{-1}} =e^{-it s\hbar ^{-1}}, \end{aligned}$$

integration by parts, the spectral theorem and that the function $(1- \psi (s))$ is support on $\left| s \right| \ge 1$, we have that

$$\begin{aligned} \Big \Vert \int _{{\mathbb {R}}^2} e^{-it s\hbar ^{-1}} \chi (t)(1- \psi (s)) \varphi (A_\varepsilon (\hbar )-s) \, ds dt \Big \Vert _{{\mathcal {L}}(L^2({\mathbb {R}}^d))}= C_N \hbar ^N, \end{aligned}$$

(6.16)

for any N in ${\mathbb {N}}$. Now for the first integral in the right hand side of (6.15) we have by Proposition 2.1 (Quadratic stationary phase) that

$$\begin{aligned} \int _{{\mathbb {R}}^2}{} & {} e^{-it s\hbar ^{-1}} \chi (t) \psi (s) \varphi (A_\varepsilon (\hbar )-s) \, ds dt \nonumber \\{} & {} = 2\pi \hbar \sum _{j=0}^N \hbar ^j \frac{(i)^j}{j!} \chi ^{(j)}(0) \varphi ^{(j)} (A_\varepsilon (\hbar )) + \hbar ^{N+1} R_{N+1}(\hbar ), \end{aligned}$$

(6.17)

where we have used that $\psi (0)=1$ and $\psi ^{(j)}(0)=0$ for all $j\in {\mathbb {N}}$. Moreover, we have from Theorem 2.1 the estimate

$$\begin{aligned} \left| R_{N+1}(\hbar ) \right| \le c \sum _{l+k=2} \int _{{\mathbb {R}}^2} | \chi ^{(N+1+l)}(t) \partial _s^{N+1+k} \psi (s) \varphi (A_\varepsilon (\hbar )-s)| \,dtds. \end{aligned}$$

As the integrants are supported on a compact set the integral will be convergent and since $\varphi$ is $C_0^\infty ({\mathbb {R}})$ we have by the spectral theorem that

$$\begin{aligned} \Vert R_{N+1}(\hbar ) \Vert _{{\mathcal {L}}(L^2({\mathbb {R}}^d))} \le C. \end{aligned}$$

(6.18)

If we now use that $\chi$ is 1 in a neighbourhood of 0 and combine (6.14)–(6.18) we have that

$$\begin{aligned}{} & {} \int _{{\mathbb {R}}^2} \text {Tr} [f(A_\varepsilon (\hbar )) e^{it\hbar ^{-1}A_\varepsilon (\hbar )}]e^{-its\hbar ^{-1} } \chi (t) \varphi (s) \,dt ds \nonumber \\{} & {} = 2\pi \hbar \text {Tr} [f(A_\varepsilon (\hbar )) \varphi (A_\varepsilon (\hbar )) ] + {\mathcal {O}}\big ( \hbar ^N\big ). \end{aligned}$$

(6.19)

Since both f and $\varphi$ are $C_0^\infty ((-\eta ,\eta ))$ functions we have by Theorem 4.13 the identity

$$\begin{aligned} \text {Tr} [f(A_\varepsilon (\hbar )) \varphi (A_\varepsilon (\hbar )) ] = \frac{1}{(2\pi \hbar )^d} \sum _{j=0}^N \hbar ^j T_j(f \varphi ,A_\varepsilon (\hbar )) + {\mathcal {O}}\big (\hbar ^{N+1-d}\big ). \end{aligned}$$

(6.20)

From Theorem 4.13 we have the exact form of the terms $T_j(f \varphi ,A_\varepsilon (\hbar ))$, which is given by

$$\begin{aligned} T_j(f \varphi ,A_\varepsilon (\hbar ) ) = {\left\{ \begin{array}{ll} \int _{{\mathbb {R}}^{2d}} (f\varphi )(a_{\varepsilon ,0}) \, dxdp &{} j=0 \\ \int _{{\mathbb {R}}^{2d}} a_{\varepsilon ,1} (f\varphi )^{(1)}(a_{\varepsilon ,0}) \, dxdp &{} j=1 \\ \int _{{\mathbb {R}}^{2d}} \sum _{k=1}^{2j-1} \frac{(-1)^k}{k!} d_{\varepsilon ,j,k} (f\varphi )^{(k)}(a_{\varepsilon ,0}) \, dxdp &{} j\ge 2, \end{array}\right. } \end{aligned}$$

where the symbols $d_{\varepsilon ,j,k}$ are the polynomials from Lemma 4.4. If we combine (6.12), (6.19) and (6.20) we get that

$$\begin{aligned} \int _{\mathbb {R}}\xi _j(s) \varphi (s) \, ds = T_j(f \varphi ,A_\varepsilon (\hbar ) ). \end{aligned}$$

If we consider $T_0(f \varphi ,A_\varepsilon (\hbar ) )$ we have that

$$\begin{aligned} \begin{aligned} T_0 (f \varphi ,A_\varepsilon (\hbar ) )&= \int _{{\mathbb {R}}^{2d}} (f\varphi )(a_{\varepsilon ,0}) \, dxdp \\&= \int _{\mathbb {R}}f(\omega ) \varphi (\omega ) \int _{\{a_{\varepsilon ,0}=\omega \}} \frac{1}{\left| \nabla {a_{\varepsilon ,0}} \right| } \,dS_\omega d\omega , \end{aligned} \end{aligned}$$

where $S_\omega$ is the euclidian surface measure on the surface in ${\mathbb {R}}_x^d\times {\mathbb {R}}_p^d$ given by the equation $a_{\varepsilon ,0}(x,p)=\omega$. If we now consider $T_j(f \varphi ,A_\varepsilon (\hbar ) )$ we have that

$$\begin{aligned} T_j(f \varphi ,A_\varepsilon (\hbar ) ){} & {} =\int _{{\mathbb {R}}^{2d}}\sum _{k=1}^{2j-1} \frac{(-1)^k}{k!} d_{\varepsilon ,j,k} (f\varphi )^{(k)}(a_{\varepsilon ,0}) \, dxdp \nonumber \\{} & {} = \sum _{k=1}^{2j-1}\frac{(-1)^k}{k!} \int _{{\mathbb {R}}} (f\varphi )^{(k)}(\omega ) \int _{\{a_{\varepsilon ,0}=\omega \}} \frac{d_{\varepsilon ,j,k}}{\left| \nabla {a_{\varepsilon ,0}} \right| } \,dS_\omega d\omega \nonumber \\{} & {} = \sum _{k=1}^{2j-1}\frac{1}{k!} \int _{{\mathbb {R}}} (f\varphi )(\omega ) \partial ^k_\omega \int _{\{a_{\varepsilon ,0}=\omega \}} \frac{d_{\varepsilon ,j,k}}{\left| \nabla {a_{\varepsilon ,0}} \right| } \,dS_\omega d\omega , \end{aligned}$$

(6.21)

where we in the last equality have made integration by parts. These equalities imply the stated form of the functions $\xi _j$. $\square$

We will now introduce some notation that will be useful for the rest of this section. We let $\chi$ be a function in $C_0^\infty ((-T_0,T_0))$, where $T_0$ is a sufficiently small number. The number will be the number $T_0$ from Corollary 5.6. We suppose $\chi$ is even and $\chi (t)=1$ for $|t|\le \frac{T_0}{2}$. We then set

$$\begin{aligned} {\hat{\chi }}_1(s) = \frac{1}{2\pi } \int _{{\mathbb {R}}} \chi (t) e^{-its} \, dt. \end{aligned}$$

We assume ${\hat{\chi }}_1\ge 0$ and that there is a $c>0$ such that ${\hat{\chi }}_1(t)\ge c$ in a small interval around 0. These assumptions can be guaranteed by replacing $\chi$ by $\chi *\chi$. With this we set

$$\begin{aligned} {\hat{\chi }}_\hbar (s) =\frac{1}{\hbar }{\hat{\chi }}_1(\tfrac{s}{\hbar }) = \frac{1}{2\pi \hbar } \int _{{\mathbb {R}}}\chi (t) e^{its\hbar ^{-1}} \, dt. \end{aligned}$$

Lemma 6.3

Assume we are in the setting of Theorem 6.1 and let g be in $L^1({\mathbb {R}})$ with $\text {supp} (g)\subset (-\eta ,\eta )$. Then for any $j\in {\mathbb {N}}_0$

$$\begin{aligned} \Big | \int _{{\mathbb {R}}} g(s) \xi _j(s) \,ds \Big | \le C\varepsilon ^{(\tau -j)_{-}} \Vert g \Vert _{L^1({\mathbb {R}})}. \end{aligned}$$

Proof

From Theorem 6.1 we have that $\xi _j(s)$ is bounded for all j. Hence by standard approximations it is sufficient to prove the statement for $g\in C_0^\infty ((-\eta ,\eta ))$, which we assume from here on.

That $\xi _j(s)$ is bounded for all j immediately give us the estimate for $j=0$. So we may assume $j\ge 1$. By definition of $\xi _j(s)$ we have that

$$\begin{aligned} \begin{aligned} \int _{{\mathbb {R}}} g(s) \xi _j(s) \,ds&= \sum _{k=1}^{2j-1}\frac{1}{k!} \int _{{\mathbb {R}}} g(s) f(s) \partial ^k_s \int _{\{a_{\varepsilon ,0}=s\}} \frac{d_{\varepsilon ,j,k}}{\left| \nabla {a_{\varepsilon ,0}} \right| } \,dS_s ds \\&= \sum _{k=1}^{2j-1}\frac{(-1)^k}{k!} \int _{{\mathbb {R}}^{2d}} (gf)^{(k)}( a_{\varepsilon ,0}(x,p)) d_{\varepsilon ,j,k}(x,p) \,dx dp \end{aligned} \end{aligned}$$

On the support of $fg(a_{\varepsilon ,0}(x,p))$ we have that $|\nabla _p a_{\varepsilon ,0}(x,p)|>c$. Hence we can make a partition $\{\Omega _j\}_{j=1}^d$ such that $\partial _{p_j} a_\varepsilon \ne 0$ on $\Omega _j$ and without loss of generality we can assume that $\Omega _j$ is connected. As in the proof of Theorem 6.1 we now choose a partition of unity supported on each $\Omega _j$ and split the integral accordingly. Hence without loss of generality we can assume $\partial _{p_1} a_\varepsilon \ne 0$ on the whole support. With the same notation as in the proof of Theorem 6.1 we perform the change of variables

$$\begin{aligned} F:(x,p) \rightarrow (X,P)=(x_1,\dots ,x_d,a_{\varepsilon ,0}(x,p),p_2,\dots ,p_d). \end{aligned}$$

This gives us that

$$\begin{aligned} \begin{aligned} \int _{{\mathbb {R}}} g(s) \xi _j(s) \,ds&= \sum _{k=1}^{2j-1}\frac{1}{k!} \int _{{\mathbb {R}}^{2d}} g( P_1) f( P_1) \partial _{P_1}^k \frac{d_{\varepsilon ,j,k}(X,F_2^{-1}(X,P))}{\partial _{p_1}a_{\varepsilon ,0}(X,F_2^{-1}(X,P))} \,dX dP, \end{aligned} \end{aligned}$$

where we after the change of variables have performed integration by parts. We have from the definition of the polynomials $d_{\varepsilon ,j,k}$ that they are of regularity $\tau -j$ and since we are integrating over a compact subset of ${\mathbb {R}}^{2d}$ we obtain the estimate

$$\begin{aligned} \Big | \int _{{\mathbb {R}}} g(s) \xi _j(s) \,ds \Big | \le C\varepsilon ^{(\tau -j)_{-}} \Vert g \Vert _{L^1({\mathbb {R}})}. \end{aligned}$$

This concludes the proof. $\square$

6.1 Weyl law for rough pseudo-differential operators

Before we state and prove the Weyl law for rough pseudo-differential operators we recall a Tauberian theorem from [24, Theorem V–13].

Theorem 6.4

Let $\tau _1<\tau _2$ and $\sigma _\hbar :{\mathbb {R}}\rightarrow {\mathbb {R}}$ be a family of increasing functions, where $\hbar$ is in (0, 1]. Moreover, let $d\ge 1$. Suppose that

• 1. $\sigma _\hbar (\tau ) =0$ for every $\tau \le \tau _1$.

• 2. $\sigma _\hbar (\tau )$ is constant for $\tau \ge \tau _2$.

• 3. $\sigma _\hbar (\tau ) = {\mathcal {O}}(\hbar ^{-d})$ as $\hbar \rightarrow 0$, and uniformly with respect to $\tau$ in ${\mathbb {R}}$.

• 4. $\partial _\tau \sigma _\hbar * {\hat{\chi }}_\hbar (\tau ) = {\mathcal {O}}(\hbar ^{-d})$ as $\hbar \rightarrow 0$ uniformly with respect to $\tau$ in ${\mathbb {R}}$.

where ${\hat{\chi }}_\hbar$ is defined as above. Then we have that

$$\begin{aligned} \sigma _\hbar (\tau )= \sigma _\hbar * {\hat{\chi }}_\hbar (\tau ) + {\mathcal {O}}(\hbar ^{1-d}), \end{aligned}$$

as $\hbar \rightarrow 0$ and uniformly with respect to $\tau$ in ${\mathbb {R}}$.

Theorem 6.5

Let $A_\varepsilon (\hbar )$ be a strongly $\hbar$-$\varepsilon$-admissible operator of regularity $\tau \ge 1$ which satisfies Assumption 4.1 and assume there exists a $\delta$ in (0, 1) such that $\varepsilon \ge \hbar ^{1-\delta }$. Suppose there exists a $\eta >0$ such that $a_{\varepsilon ,0}^{-1}((-\infty ,\eta ])$ is compact, where $a_{\varepsilon ,0}$ is the principal symbol of $A_\varepsilon (\hbar )$. Moreover we suppose that

$$\begin{aligned} |\nabla _p a_{\varepsilon ,0}(x,p)| \ge c \quad \text {for all } (x,p)\in a_{\varepsilon ,0}^{-1}(\{ 0 \}). \end{aligned}$$

(6.22)

Then we have that

$$\begin{aligned} |\text {Tr} [{\varvec{1}}_{(-\infty ,0]}(A_\varepsilon (\hbar ))] - \frac{1}{(2\pi \hbar )^d} \int _{{\mathbb {R}}^{2d}}{\varvec{1}}_{(-\infty ,0]}( a_{\varepsilon ,0}(x,p)) \,dx dp | \le C \hbar ^{1-d}, \end{aligned}$$

for all sufficiently small $\hbar$.

Proof

By the assumption in (6.22) there exists a $\nu >0$ such that

$$\begin{aligned} |\nabla _p a_{\varepsilon ,0}(x,p)| \ge \frac{c}{2} \quad \text {for all } (x,p)\in a_{\varepsilon ,0}^{-1}([-2\nu ,2\nu ]). \end{aligned}$$

Moreover, by Theorem 4.2 we have that the spectrum of $A_\varepsilon (\hbar )$ is bounded from below uniformly in $\hbar$. Let E denote a number with distance 1 to the bottom of the spectrums. We now take two functions $f_1$ and $f_2$ in $C_0^\infty ({\mathbb {R}})$ such that

$$\begin{aligned} f_1(t) + f_2(t) = 1, \end{aligned}$$

for every t in [E, 0], $\text {supp} (f_2)\subset [-\frac{\nu }{4},\frac{\nu }{4}]$, $f_2(t)=1$ for $t\in [-\frac{\nu }{8},\frac{\nu }{8}]$ and $f_2(t)=f_2(-t)$ for all t. With these functions we have

$$\begin{aligned} \text {Tr} [{\varvec{1}}_{(-\infty ,0]}(A_\varepsilon (\hbar )) ]= \text {Tr} [ f_1(A_\varepsilon (\hbar ))] + \text {Tr} [ f_2(A_\varepsilon (\hbar )){\varvec{1}}_{(-\infty ,0]}(A_\varepsilon (\hbar ))]. \end{aligned}$$

(6.23)

For the first term on the right hand side in the above equality we have by Theorem 4.13 that

$$\begin{aligned} \text {Tr} [ f_1(A_\varepsilon (\hbar ))] = \frac{1}{(2\pi \hbar )^d} \int _{{\mathbb {R}}^{2d}} f_1(a_{\varepsilon ,0}(x,p)) \,dxdp + {\mathcal {O}}(\hbar ^{1-d}). \end{aligned}$$

(6.24)

In order to calculate the second term on the right hand side in (6.23) we will study the function

$$\begin{aligned} \omega \rightarrow M(\omega ;\hbar ) = \text {Tr} [ f_2(A_\varepsilon (\hbar )){\varvec{1}}_{(-\infty ,\omega ]}(A_\varepsilon (\hbar ))]. \end{aligned}$$

(6.25)

We have that $M(\omega ;\hbar )$ satisfies the three first conditions in Theorem 6.4. In what follows, we will use the notation

$$\begin{aligned} {\mathcal {P}}=\text {supp} (f_2)\cap \text {spec} (A_\varepsilon (\hbar )), \end{aligned}$$

where $\text {spec} (A_\varepsilon (\hbar ))$ is the spectrum of the operator $A_\varepsilon (\hbar )$. The function M can be written in the following form

$$\begin{aligned} M(\omega ;\hbar ) = \sum _{e_j \in {\mathcal {P}}} f_2(e_j) {\varvec{1}}_{[e_j,\infty )}(\omega ), \end{aligned}$$

since $f_2$ is supported in the pure point spectrum of $A_\varepsilon (\hbar )$. This follows from Theorem 4.12. Let ${\hat{\chi }}_\hbar$ be defined as above. Then we will consider the convolution

$$\begin{aligned} (M(\cdot ;\hbar )*{\hat{\chi }}_\hbar )(\omega ) = \int _{\mathbb {R}}M(s;\hbar ) {\hat{\chi }}_\hbar (\omega -s) \,ds = \sum _{e_j \in {\mathcal {P}}} f_2(e_j) \int _{e_j}^\infty {\hat{\chi }}_\hbar (\omega -s) \,ds. \end{aligned}$$

If we take a derivative with respect to $\omega$ we get that

$$\begin{aligned} \begin{aligned} \partial _\omega (M(\cdot ;\hbar )*{\hat{\chi }}_\hbar )(\omega )&= \sum _{e_j \in {\mathcal {P}}} f_2(e_j) {\hat{\chi }}_\hbar (\omega -e_j) \\&= \frac{1}{2\pi \hbar } \int _{\mathbb {R}}\text {Tr} [f_2(A_\varepsilon (\hbar ))e^{it\hbar ^{-1} A_\varepsilon (\hbar )}] e^{-it\omega \hbar ^{-1}} \chi (t) \,dt, \end{aligned} \end{aligned}$$

by the definition of ${\hat{\chi }}_\hbar$. We get now by Theorem 6.1 the identity

$$\begin{aligned} \partial _\omega (M(\cdot ;\hbar )*{\hat{\chi }}_\hbar )(\omega ) = \frac{1}{(2\pi \hbar )^d} f_2(\omega ) \int _{\{a_{\varepsilon ,0}=\omega \}} \frac{1}{\left| \nabla a_{\varepsilon ,0} \right| } \,dS_\omega + {\mathcal {O}}(\hbar ^{1-d}), \end{aligned}$$

(6.26)

where the size of the error follows from the a priori estimates given in the Theorem 6.1. This verifies the fourth condition in Theorem 6.4 for $M(\cdot ;\hbar )$, hence Theorem 6.4 gives the identity

$$\begin{aligned} \text {Tr} [ f_2{} & {} (A_\varepsilon (\hbar )){\varvec{1}}_{(-\infty ,0]}(A_\varepsilon (\hbar ))] \nonumber \\{} & {} = \frac{1}{(2\pi \hbar )^d} \int _{-\infty }^0 f_2(\omega ) \int _{\{a_{\varepsilon ,0}=\omega \}} \frac{1}{\left| \nabla a_{\varepsilon ,0} \right| } \,dS_\omega d\omega + {\mathcal {O}}(\hbar ^{1-d}) \nonumber \\{} & {} =\frac{1}{(2\pi \hbar )^d} \int _{{\mathbb {R}}^{2d}} f_2(a_{\varepsilon ,0}(x,p)) {\varvec{1}}_{(-\infty ,0]}(a_{\varepsilon ,0}(x,p)) \,dxdp + {\mathcal {O}}(\hbar ^{1-d}). \end{aligned}$$

(6.27)

By combining (6.23), (6.24) and (6.27) we get that

$$\begin{aligned} \text {Tr} [{\varvec{1}}_{(-\infty ,0]}(A_\varepsilon (\hbar )) ] = \frac{1}{(2\pi \hbar )^d} \int _{{\mathbb {R}}^{2d}} {\varvec{1}}_{(-\infty ,0]}(a_{\varepsilon ,0}(x,p)) \,dxdp + {\mathcal {O}}(\hbar ^{1-d}). \end{aligned}$$

This is the desired estimate and this ends the proof. $\square$

6.2 Riesz means for rough pseudo-differential operators

Theorem 6.6

Let $A_\varepsilon (\hbar )$ be a strongly $\hbar$-$\varepsilon$-admissible operator of regularity $\tau \ge 2$ which satisfies Assumption 4.1 and assume there exists a $\delta$ in (0, 1) such that $\varepsilon \ge \hbar ^{1-\delta }$. Suppose there exists a $\eta >0$ such that $a_{\varepsilon ,0}^{-1}((-\infty ,\eta ])$ is compact, where $a_{\varepsilon ,0}$ is the principal symbol of $A_\varepsilon (\hbar )$. Moreover, we suppose that

$$\begin{aligned} |\nabla _p a_{\varepsilon ,0}(x,p)| \ge c \quad \text {for all } (x,p)\in a_{\varepsilon ,0}^{-1}(\{ 0 \}). \end{aligned}$$

(6.28)

Then for $1\ge \gamma >0$ we have that

$$\begin{aligned} \big |\text {Tr} [(A_\varepsilon (\hbar ))_{-}^\gamma ] - \frac{1}{(2\pi \hbar )^d}[ \Psi _0(\gamma ,A_\varepsilon ) + \hbar \Psi _1(\gamma ,A_\varepsilon )] \big | \le C \hbar ^{1+\gamma -d}, \end{aligned}$$

for all sufficiently small $\hbar$. The numbers $\Psi _j(\gamma ,A_\varepsilon )$ are given by

$$\begin{aligned} \begin{aligned} \Psi _0(\gamma ,A_\varepsilon )&= \int _{{\mathbb {R}}^{2d}} (a_{\varepsilon ,0}(x,p))_{-}^\gamma \,dxdp \\ \Psi _1(\gamma ,A_\varepsilon )&= \gamma \int _{{\mathbb {R}}^{2d}} a_{\varepsilon ,1}(x,p) (a_{\varepsilon ,0}(x,p))_{-}^{\gamma -1} \,dxdp. \end{aligned} \end{aligned}$$

(6.29)

Remark 6.7

The proof of this theorem is valid for any $\gamma >0$. For the case where $\gamma >1$ the expansion will have additional terms. These additional terms can also be found and calculated explicitly. But note that to ensure the error is of order $\hbar ^{1+\gamma -d}$ one will in general need to impose the restriction $\gamma \le \tau -1$ on $\gamma$ and the regularity $\tau$.

If we, for the case $\gamma \in (0,1]$, had assumed $\tau =1$. Then the error obtained in Theorem 6.6 would have been of the order $\max (\hbar ^{1+\delta -d},\hbar ^{1+\gamma -d})$. Hence under the assumption that $\delta \ge \gamma$ we would get the desired order of the error. This choice of $\delta$ would be possible. However, in the proof of Theorem 1.5 we will also need to compare phase space integrals. That is we need to estimate the two differences

$$\begin{aligned} \Big |\frac{1}{(2\pi \hbar )^d} \int _{{\mathbb {R}}^{2d}} ( a_{\varepsilon ,0}^{\pm }(x,p))^{\gamma }_{-} \,dxdx - \frac{1}{(2\pi \hbar )^d}\int _{{\mathbb {R}}^{2d}} ( a_{0}(x,p))^{\gamma }_{-} \,dxdp\Big | \end{aligned}$$

and

$$\begin{aligned} \Big | \frac{1}{(2\pi \hbar )^d}\int _{{\mathbb {R}}^{2d}} a_{\varepsilon ,1}^{\pm }(x,p) ( a_{\varepsilon ,0}^{\pm }(x,p))^{\gamma -1}_{-} \,dx dp - \frac{1}{(2\pi \hbar )^d}\int _{{\mathbb {R}}^{2d}} a_{1}(x,p) ( a_{0}(x,p))^{\gamma -1}_{-} \,dx dp\Big |. \end{aligned}$$

The order of the error we can obtain here will be $\varepsilon ^{1+\mu }\hbar ^{-d}$ and $\varepsilon ^{\mu }\hbar ^{-d}$ respectively, where $\mu$ is the Hölder continuity parameter of the coeffcients. When choosing $\varepsilon =\hbar ^{1-\delta }$ we get that the order of the error from these two terms will be

$$\begin{aligned} \max \big ( \hbar ^{(1+\mu )(1-\delta ) - d}, \hbar ^{1+\mu (1-\delta ) - d}\big ). \end{aligned}$$

In order to ensure we get the sharp bound, we will need to have $\mu \in [0,1]$ and $\delta \in [\gamma ,1)$ such that

$$\begin{aligned} (1+\mu )(1-\delta ) \ge 1+\gamma \quad \text {and}\quad \mu (1-\delta ) \ge \gamma . \end{aligned}$$

Choosing $\delta =\gamma$ we get that for both inequalities to be true we will need $\mu$ to satisfies that

$$\begin{aligned} 1\ge \mu \ge \frac{2\gamma }{1-\gamma }. \end{aligned}$$

(6.30)

This relation is only possible to obtain for $\gamma \le \frac{1}{3}$. As for $\gamma >\frac{1}{3}$ the right hand side of (6.30) will be strictly greater than one.

To summarise, when considering Riesz means of order $\gamma \le \frac{1}{3}$, we can assume the coefficients are in $C^{1,\mu }({\mathbb {R}}^d)$, where $\mu$ satisfies (6.30).

Proof

By the assumption in (6.28) there exists a $\nu >0$ such that

$$\begin{aligned} |\nabla _p a_{\varepsilon ,0}(x,p)| \ge \frac{c}{2} \quad \text {for all } (x,p)\in a_{\varepsilon ,0}^{-1}([-2\nu ,2\nu ]). \end{aligned}$$

By Theorem 4.2 we have that the spectrum of $A_\varepsilon (\hbar )$ is bounded from below uniformly in $\hbar$. In the following, let E denote a number with distance 1 to the bottom of the spectrum. We now take two functions $f_1$ and $f_2$ in $C_0^\infty ({\mathbb {R}})$ such that

$$\begin{aligned} f_1(t) + f_2(t) = 1, \end{aligned}$$

for every t in [E, 0], $\text {supp} (f_2)\subset [-\frac{\nu }{4},\frac{\nu }{4}]$, $f_2(t)=1$ for t in $[-\frac{\nu }{8},\frac{\nu }{8}]$ and $f_2(t)=f_2(-t)$ for all t. We can now write

$$\begin{aligned} \text {Tr} [(A_\varepsilon (\hbar ))_{-}^\gamma ] = \text {Tr} [f_1(A_\varepsilon (\hbar ))(A_\varepsilon (\hbar ))_{-}^\gamma ] +\text {Tr} [f_2(A_\varepsilon (\hbar ))(A_\varepsilon (\hbar ))_{-}^\gamma ] \end{aligned}$$

(6.31)

The first term in (6.31) is now the trace of a smooth compactly supported function of our operator. Hence we can calculate the asymptotic using Theorem 4.13. Before applying this theorem we will study the second term in (6.31). Before we proceed we note that

$$\begin{aligned} (A_\varepsilon (\hbar ))_{-}^\gamma = (0-A_\varepsilon (\hbar ))_{+}^\gamma \end{aligned}$$

This form will be slightly more convenient to work with and we will introduce the notation $\varphi ^\gamma (t)=(t)_+^\gamma$. As we will use a smoothing procedure we will consider the expression

$$\begin{aligned} \begin{aligned} M_{f_2}^\gamma (\omega ;\hbar )&=\text {Tr} [f_2(A_\varepsilon (\hbar ))\varphi ^\gamma (\omega -A_\varepsilon (\hbar ))] \\&= \sum _{e_j(\hbar )\le \omega } f_2(e_j(\hbar ))\varphi ^\gamma (\omega -e_j(\hbar )) =(\varphi ^\gamma * dM_{f_2}^0(\cdot ;\hbar ))(\omega ), \end{aligned} \end{aligned}$$

(6.32)

where $dM_{f_2}^0$¹ is the measure induced by the function

$$\begin{aligned} M_{f_2}^0(\omega ;\hbar ) = \text {Tr} [f_2(A_\varepsilon (\hbar )) (\omega -A_\varepsilon (\hbar ))_{+}^0], \end{aligned}$$

as in Remark 6.2. Let ${\hat{\chi }}_\hbar (t)$ be as above and consider the convolution ${\hat{\chi }}_\hbar *M_{f_2}^\gamma$. Then we have that

$$\begin{aligned} \begin{aligned} {\hat{\chi }}_\hbar *M_{f_2}^\gamma (\omega ) = [ {\hat{\chi }}_\hbar *\varphi ^\gamma *dM_{f_2}^0(\cdot ;\hbar )] (\omega )=[ \varphi ^\gamma * {\hat{\chi }}_\hbar *dM_{f_2}^0(\cdot ;\hbar )] (\omega ). \end{aligned} \end{aligned}$$

From Remark 6.2 we have an asymptotic expansion of ${\hat{\chi }}_\hbar *dM_{f_2}^0(\cdot ;\hbar )(\omega )$. In order to see that we can use this expansion let g be in $C_0^\infty ([-\nu ,\nu ])$ such that $g(t)=1$ for all t in $[-\frac{\nu }{2},\frac{\nu }{2}]$. Then we have that

$$\begin{aligned}{}[\varphi ^\gamma * {\hat{\chi }}_\hbar *dM_{f_2}^0(\cdot ;\hbar )] (\omega ){} & {} ={}\int _{\mathbb {R}}\varphi ^\gamma (\omega -t) g(t) [{\hat{\chi }}_\hbar *dM_{f_2}^0(\cdot ;\hbar )](t) \,dt \nonumber \\{} & {} + \int _{\mathbb {R}}\varphi ^\gamma (\omega -t) (1-g(t)) [{\hat{\chi }}_\hbar *dM_{f_2}^0(\cdot ;\hbar )](t) \,dt.\nonumber \\ \end{aligned}$$

(6.33)

For the second term we have that

$$\begin{aligned} (1-g(t)) [{\hat{\chi }}_\hbar *dM_{f_2}^0(\cdot ;\hbar )](t) = \frac{ 1-g(t) }{2\pi \hbar } \sum _{e_j(\hbar )} f_2(e_j(\hbar )) {\hat{\chi }}\left( \tfrac{e_j(\hbar )-t}{\hbar }\right) . \end{aligned}$$

Since ${\hat{\chi }}$ is Schwarz class and $\left| t-e_j(\hbar ) \right| \ge \frac{\nu }{4}$ on the support of $(1-g(t))$ we get, by the support properties of $f_2$, that

$$\begin{aligned} \left| \int _{\mathbb {R}}\varphi ^\gamma (\omega -t) (1-g(t)) [{\hat{\chi }}_\hbar *dM_{f_2}^0(\cdot ;\hbar )](t) \,dt\right| \le C_N\hbar ^N, \end{aligned}$$

(6.34)

for all N in ${\mathbb {N}}$. We can use the expansion from Remark 6.2 on the first term in (6.33) due to the support properties of g and the estimate obtained in (6.34) for the second term in (6.33). This yields the estimate

$$\begin{aligned} \begin{aligned}&\Big | [\varphi ^\gamma * {\hat{\chi }}_\hbar *dM_{f_2}^0(\cdot ;\hbar )] (\omega ) - \frac{1}{(2\pi \hbar )^{d}}\sum _{j=0}^{1} \hbar ^j \int _{\mathbb {R}}\varphi ^\gamma (\omega -t) g(t) \xi _j(t) \,dt \Big | \le C\hbar ^{2-d}, \end{aligned} \end{aligned}$$

(6.35)

where we have also used Lemma 6.3 and the assumption that $\tau \ge 2$ to obtain the estimate. Using the definition of the functions $\xi _0(t)$ and $\xi _1(t)$ we get that

$$\begin{aligned} \begin{aligned}&\sum _{j=0}^{1} \hbar ^j \int _{\mathbb {R}}\varphi ^\gamma (\omega -t) g(t) \xi _j(t) \,dt \\&= \int _{{\mathbb {R}}^{2d}} f_2(a_{\varepsilon ,0})\varphi ^\gamma (\omega -a_{\varepsilon ,0}) \, dxdp + \hbar \int _{{\mathbb {R}}^{2d}} a_{\varepsilon ,1} \partial _t [f_2(t) \varphi ^\gamma (\omega -t)] \big |_{t=a_{\varepsilon ,0}} \, dxdp, \end{aligned} \end{aligned}$$

(6.36)

where we have also used that $g(t)=1$ for all $t\in \text {supp} (f_2)$. We now turn to the first term in (6.31). Here we apply Theorem 4.13 (as mentioned above) and get that

$$\begin{aligned} | \text {Tr} [f_1(A_\varepsilon (\hbar ))\varphi ^\gamma (-A_\varepsilon (\hbar ))] -\frac{1}{(2\pi \hbar )^d} \sum _{j=0}^1 \hbar ^j T_j(f_1\varphi ^\gamma ,A_\varepsilon (\hbar )) | \le C \hbar ^{2-d}, \end{aligned}$$

(6.37)

for all sufficiently small $\hbar$, where

$$\begin{aligned} \begin{aligned}&\sum _{j=0}^1 \hbar ^j T_j(f_1\varphi ^\gamma ,A_\varepsilon (\hbar )) \\&= \int _{{\mathbb {R}}^{2d}} f_1(a_{\varepsilon ,0})\varphi ^\gamma (-a_{\varepsilon ,0}) \, dxdp + \hbar \int _{{\mathbb {R}}^{2d}} a_{\varepsilon ,1} \partial _t [f_1(t) \varphi ^\gamma (-t)] \big |_{t=a_{\varepsilon ,0}} \, dxdp. \end{aligned} \end{aligned}$$

(6.38)

Combining (6.31), (6.32), (6.35), (6.37) we obtain that

$$\begin{aligned} \big |\text {Tr} [(A_\varepsilon (\hbar ))_{-}^\gamma ] - \frac{1}{(2\pi \hbar )^d}{} & {} [ \Psi _0(\gamma ,A_\varepsilon ) + \hbar \Psi _1(\gamma ,A_\varepsilon )] \big | \nonumber \\{} & {} \le \big |M_{f_2}^\gamma (0;\hbar ) - [{\hat{\chi }}_\hbar *M_{f_2}^\gamma (\cdot ;\hbar )](0) \big |+ C \hbar ^{2-d}, \end{aligned}$$

(6.39)

where we have used (6.36), (6.38) and the definition of the functions $f_1$ and $f_2$ to obtain the terms $\Psi _0(\gamma ,A_\varepsilon )$ and $\Psi _1(\gamma ,A_\varepsilon )$. What remains is an Tauberian type argument to prove that

$$\begin{aligned} |M_{f_2}^\gamma (\omega ;\hbar ) - [{\hat{\chi }}_\hbar *M_{f_2}^\gamma (\cdot ;\hbar )](\omega ) | \le C \hbar ^{1+\gamma -d} \end{aligned}$$

(6.40)

uniformly in $\omega$. We have by definition of ${\hat{\chi }}_\hbar$ that

$$\begin{aligned} M_{f_2}^\gamma (\omega ;\hbar ) - [{\hat{\chi }}_\hbar *M_{f_2}^\gamma (\cdot ;\hbar )](\omega ) = \frac{1}{2\pi } \int _{\mathbb {R}}(M_{f_2}^\gamma (\omega ;\hbar ) - M_{f_2}^\gamma (\omega -t\hbar ;\hbar )){\hat{\chi }}(t) \,dt, \end{aligned}$$

(6.41)

where we have used that

$$\begin{aligned} \frac{1}{2\pi }\int _{\mathbb {R}}{\hat{\chi }}(t) \,dt = 1. \end{aligned}$$

The next step is to write the difference $M_{f_2}^\gamma (\omega ;\hbar ) - M_{f_2}^\gamma (\omega -t\hbar ;\hbar )$ as the inverse Fourier transform of the Fourier transform. However, before we do this we first find an expression for the Fourier transform of $M_{f_2}^\gamma (\omega ;\hbar )$. This function exists as a limit of a complex Fourier transform. We have that

$$\begin{aligned} \begin{aligned} {\mathcal {F}}[M_{f_2}^\gamma (\cdot ;\hbar )] (t - i\kappa )&=\int _{\mathbb {R}}e^{-i(t-i\kappa )s}\sum _{e_j(\hbar )} f_2(e_j(\hbar ))\varphi ^\gamma (s-e_j(\hbar )) \,ds \\&=\int _{\mathbb {R}}e^{-i(t-i\kappa )s}\varphi ^\gamma (s) \,ds \sum _{e_j(\hbar )} e^{-i(t-i\kappa )e_j(\hbar )} f_2(e_j(\hbar )) \\&= \int _0^\infty e^{(-it-\kappa )s}s^\gamma \, ds \sum _{e_j(\hbar )} e^{-i(t-i\kappa )e_j(\hbar )} f_2(e_j(\hbar )) \\&= \frac{\Gamma (\gamma +1)}{(t-i\kappa )^{\gamma +1}} i^{\gamma +1} \sum _{e_j(\hbar )} e^{-i(t-i\kappa )e_j(\hbar )} f_2(e_j(\hbar )), \end{aligned} \end{aligned}$$

where $\Gamma$ is the gamma-function. By taking $\kappa$ to zero we get that

$$\begin{aligned} {\mathcal {F}}[M_{f_2}^\gamma (\cdot ;\hbar )] (t ) = \frac{\Gamma (\gamma +1)}{(t+i0)^{\gamma +1}} e^{\frac{i\pi }{2}(\gamma +1)} \sum _{e_j(\hbar )} e^{-ite_j(\hbar )} f_2(e_j(\hbar )). \end{aligned}$$

With this established we write the difference $M_{f_2}^\gamma (\omega ;\hbar ) - M_{f_2}^\gamma (\omega -t\hbar ;\hbar )$ as the inverse Fourier transform of the Fourier transform and get from (6.41) that

$$\begin{aligned} \begin{aligned} M_{f_2}^\gamma (\omega ;\hbar ) - [{\hat{\chi }}_\hbar *M_{f_2}^\gamma (\cdot ;\hbar )](\omega )&= \frac{1}{2\pi } \int _{\mathbb {R}}(M_{f_2}^\gamma (\omega ;\hbar ) - M_{f_2}^\gamma (\omega -t\hbar ;\hbar )){\hat{\chi }}(t) \,dt \\&=\frac{1}{4\pi ^2} \int _{\mathbb {R}}\int _{\mathbb {R}}e^{is\omega } {\mathcal {F}}[M_{f_2}^\gamma (\cdot ;\hbar )] (s )(1-e^{-it\hbar s}){\hat{\chi }}(t) \, ds dt. \end{aligned} \end{aligned}$$

By reintroducing $\kappa$, using Fubini’s theorem and then take $\kappa$ to zero we get that

$$\begin{aligned}{} & {} M_{f_2}^\gamma (\omega ;\hbar ) - [{\hat{\chi }}_\hbar *M_{f_2}^\gamma (\cdot ;\hbar )](\omega ) \nonumber \\{} & {} =\lim _{\kappa \rightarrow 0} \frac{1}{4\pi ^2} \int _{{\mathbb {R}}^2} e^{is\omega } \frac{\Gamma (\gamma +1)}{(s-i\kappa )^{\gamma +1}} {i^{\gamma +1}} \sum _{e_j(\hbar )} e^{-i(s-i\kappa )e_j(\hbar )} f_2(e_j(\hbar ))) \nonumber \\{} & {} \quad \times (1-e^{-it\hbar s}){\hat{\chi }}(t) \, ds dt \nonumber \\{} & {} =\lim _{\kappa \rightarrow 0} \frac{1}{2\pi } \int _{\mathbb {R}}e^{is\omega } \frac{\Gamma (\gamma +1)}{(s-i\kappa )^{\gamma +1}} {i^{\gamma +1}} \sum _{e_j(\hbar )} e^{-i(s-i\kappa )e_j(\hbar )} f_2(e_j(\hbar ))(1-\chi (-\hbar s)) \, ds \nonumber \\{} & {} = C_\gamma \sum _{e_j(\hbar )} f_2(e_j(\hbar )) \hbar ^{-1} \lim _{\kappa \rightarrow 0} e^{-i\kappa e_j(\hbar )} \int _{\mathbb {R}}e^{is\hbar ^{-1}(\omega -e_j(\hbar ))} \frac{(1-\chi ( s))}{(s\hbar ^{-1}-i\kappa )^{\gamma +1}} \, ds \nonumber \\{} & {} = C_\gamma \sum _{e_j(\hbar )} f_2(e_j(\hbar )) \hbar ^{\gamma } \int _{\mathbb {R}}e^{-is\hbar ^{-1}(e_j(\hbar )-\omega )} \frac{(1-\chi ( s))}{s^{\gamma +1}} \, ds \nonumber \\{} & {} =C_\gamma \hbar ^{\gamma } \sum _{e_j(\hbar )} f_2(e_j(\hbar )){\hat{\psi }}\left( \tfrac{e_j(\hbar )-\omega }{\hbar } \right) , \end{aligned}$$

(6.42)

where $\psi (s) = (1-\chi ( s))s^{-\gamma -1}$. Now we just need to estimate the sum by a constant times $\hbar ^{1-d}$. Firstly, we note that

$$\begin{aligned} \begin{aligned} \sum _{e_j(\hbar )} f_2(e_j(\hbar )){\hat{\psi }}\big ( \tfrac{e_j(\hbar )-\omega }{\hbar } \big ) = \sum _{m\in {\mathbb {Z}}} \sum _{m\hbar < e_j(\hbar )-\omega \le (m+1)\hbar }f_2(e_j(\hbar )){\hat{\psi }}\big ( \tfrac{e_j(\hbar )-\omega }{\hbar } \big ). \end{aligned} \end{aligned}$$

(6.43)

Since $\chi (s)=1$ for all s in a neighbourhood of 0 and $\gamma >0$ it follows that $\psi \in L^1({\mathbb {R}})$. Moreover, for all $k\in {\mathbb {N}}$ we have that $\psi ^{(k)}\in L^1({\mathbb {R}})$, where $\psi ^{(k)}$ is the k’th derivative of $\psi$. This is due to the support properties of $\chi$ and that $\gamma >0$. Since we have that $\psi ^{(k)}\in L^1({\mathbb {R}})$ for all $k\in {\mathbb {N}}_0$, it follows that ${\hat{\psi }}$ is continuous and decays faster than any polynomial. This implies that

$$\begin{aligned} \Big |\Big (1+\big ( \tfrac{e_j(\hbar )-\omega }{\hbar } \big )^2\Big ){\hat{\psi }}\big ( \tfrac{e_j(\hbar )-\omega }{\hbar } \big )\Big | \le \sup _{t\in {\mathbb {R}}} (1+t^2)|{\hat{\psi }}(t)| \le C, \end{aligned}$$

(6.44)

for all values of $(e_j(\hbar )-\omega )\hbar ^{-1}$. Applying this bound we get that

$$\begin{aligned}{} & {} \Big | \sum _{m\in {\mathbb {Z}}} \sum _{m\hbar< e_j(\hbar )-\omega \le (m+1)\hbar }f_2(e_j(\hbar )){\hat{\psi }}\big ( \tfrac{e_j(\hbar )-\omega }{\hbar } \big ) \Big | \nonumber \\{} & {} \le \sum _{m\in {\mathbb {Z}}} \frac{C}{1+m^2} \sum _{m\hbar < e_j(\hbar )-\omega \le (m+1)\hbar }f_2(e_j(\hbar )). \end{aligned}$$

(6.45)

To estimate the sum we note that due to the support properties of $f_2$ we have that

$$\begin{aligned} \begin{aligned} \sum _{m\hbar < e_j(\hbar )-\omega \le (m+1)\hbar }f_2(e_j(\hbar )) \le \text {Tr} \big [ {\varvec{1}}_{[m\hbar +\omega , (m+1)\hbar +\omega ]\cap [-\nu /4,\nu /4]}(A_\varepsilon (\hbar )) \big ]. \end{aligned} \end{aligned}$$

(6.46)

For all m such that the intersection is empty we do not get any contributions. For the $m\in {\mathbb {Z}}$ such that the intersection is non-empty we will be in one of the following three cases

$$\begin{aligned} \begin{aligned} m\hbar +\omega< -\nu /4 \le (m+1)\hbar +\omega \\ [m\hbar +\omega , (m+1)\hbar +\omega ]\subset [-\nu /4,\nu /4] \\ m\hbar +\omega \le \nu /4 < (m+1)\hbar +\omega . \end{aligned} \end{aligned}$$

In all three cases we will have that $[m\hbar +\omega , (m+1)\hbar +\omega ]\cap [-\nu /4,\nu /4]=[c_1,c_2]$ such that $|c_1-c_2|\le \hbar$ and the numbers $c_1$ and $c_2$ will be non-critical for our principal symbol. Hence we will in the following just consider a trace of the form $\text {Tr} [ {\varvec{1}}_{[c_1,c_2]}(A_\varepsilon (\hbar )) ]$, where $c_1$ and $c_2$ are non-critical values for the principal symbol. Since the numbers $c_1$ and $c_2$ are non-critical we get from applying Theorem 6.5 that

$$\begin{aligned} \text {Tr} [{\varvec{1}}_{(-\infty ,c_i]}(A_\varepsilon (\hbar ))] = \frac{1}{(2\pi \hbar )^d} \int _{{\mathbb {R}}^{2d}}{\varvec{1}}_{(-\infty ,c_i]}( a_{\varepsilon ,0}(x,p)) \,dx dp +{\mathcal {O}}\big ( \hbar ^{1-d}\big ) \end{aligned}$$

for $i=1,2$ and $\hbar$ sufficiently small. This gives us that

$$\begin{aligned} \text {Tr} [{\varvec{1}}_{[c_1,c_i]}(A_\varepsilon (\hbar ))] = \frac{1}{(2\pi \hbar )^d} \int _{{\mathbb {R}}^{2d}}{\varvec{1}}_{[c_1,c_2]}( a_{\varepsilon ,0}(x,p)) \,dx dp +{\mathcal {O}}\big ( \hbar ^{1-d}\big ). \end{aligned}$$

(6.47)

for all $\hbar$ succulently small. Moreover we have for $\hbar$ sufficiently small that $[c_1,c_2]\subset [-\nu ,\nu ]$. Hence all values in the interval $[c_1,c_2]$ will be non-critical. This gives us that we can apply the coarea formula and obtain that

$$\begin{aligned} \frac{1}{(2\pi \hbar )^d} \int _{{\mathbb {R}}^{2d}}{\varvec{1}}_{[c_1,c_2]}( a_{\varepsilon ,0}(x,p)) \,dx dp = \frac{1}{(2\pi \hbar )^d} \int _{c_1}^{c_2} \int _{\{a_\varepsilon (x,p) = E\}} \frac{1}{|\nabla a_\varepsilon (x,p)|} \, dS_E dE, \end{aligned}$$

(6.48)

where $S_E$ is the euclidean surface measure on the surface $\{a_\varepsilon (x,p) = E\}$. Due to the compactness assumption we have that

$$\begin{aligned} \sup _{E\in [-\nu ,\nu ]} \int _{\{a_\varepsilon (x,p) = E\}} \frac{1}{|\nabla a_\varepsilon (x,p)|} \, dS_E \le C, \end{aligned}$$

(6.49)

where the constant is independent of $c_1$ and $c_2$. combining the estimate in (6.48) and (6.49) we obtain that

$$\begin{aligned} \frac{1}{(2\pi \hbar )^d} \int _{{\mathbb {R}}^{2d}}{\varvec{1}}_{[c_1,c_2]}( a_{\varepsilon ,0}(x,p)) \,dx dp \le \frac{1}{(2\pi \hbar )^d} \int _{c_1}^{c_2} C \, dE\le C|c_1-c_2| \hbar ^{-d}, \end{aligned}$$

(6.50)

where the constant is independent of $c_1$ and $c_2$. Combining the estimates in (6.47) and (6.50) we obtain that

$$\begin{aligned} \text {Tr} [{\varvec{1}}_{[c_1,c_i]}(A_\varepsilon (\hbar ))] \le C\big ( |c_1-c_2| \hbar ^{-d} +\hbar ^{1-d}\big ). \end{aligned}$$

(6.51)

Since all the sets $[m\hbar +\omega , (m+1)\hbar +\omega ]\cap [-\nu /4,\nu /4]$ of interest are of the form $[c_1,c_2]$ such that $|c_1-c_2|\le \hbar$ and the numbers $c_1$ and $c_2$ are non-critical we get from (6.51) that

$$\begin{aligned} \text {Tr} \big [ {\varvec{1}}_{[m\hbar +\omega , (m+1)\hbar +\omega ]\cap [-\nu /4,\nu /4]}(A_\varepsilon (\hbar )) \big ] \le C\hbar ^{1-d}, \end{aligned}$$

(6.52)

where the constant is independent of $\omega$ and m. From combining the estimates in (6.43), (6.45), (6.46) and (6.52) we obtain that

$$\begin{aligned} \begin{aligned} \Big |\sum _{e_j(\hbar )} f_2(e_j(\hbar )){\hat{\psi }}\big ( \tfrac{e_j(\hbar )-\omega }{\hbar } \big ) \Big | \le C \hbar ^{1-d}, \end{aligned} \end{aligned}$$

(6.53)

where we have used that $\sum _{m\in {\mathbb {Z}}} \frac{1}{1+m^2} <\infty$. Finally from combining (6.42) and (6.53) we obtain (6.40) uniformly in $\omega$. This concludes the proof. $\square$

7 Proof of main theorems

Lemma 7.1

Let ${\mathcal {A}}_\hbar$ be a sesquilinear form which satisfies Assumption 1.2 with the numbers $(\tau ,\mu )$ in ${\mathbb {N}}\times [0,1]$. Suppose ${\mathcal {A}}_{\hbar ,\varepsilon }^{\pm }$ are the framing sesquilinear forms constructed in Proposition 2.5. Then for all $\varepsilon$ sufficiently small there exists two $\hbar$-$\varepsilon$-admissible operators of regularity $\tau$ $A^{\pm }_\varepsilon (\hbar )$ such that they satisfy Assumption 4.1. In particular they will be lower semi-bounded, selfajoint for all $\hbar$ sufficiently small and satisfies that

$$\begin{aligned} A^{\pm }_\varepsilon (\hbar )[\varphi ,\psi ] = {\mathcal {A}}_{\hbar ,\varepsilon }^{\pm }[\varphi ,\psi ] \qquad \text {for all }\varphi ,\psi \in {\mathcal {D}}( {\mathcal {A}}_{\hbar ,\varepsilon }^{\pm }). \end{aligned}$$

(7.1)

Remark 7.2

Assume we are in the setting of Lemma 7.1. Since we have that $A^{\pm }_\varepsilon (\hbar )$ both are $\hbar$-$\varepsilon$-admissible of regularity $\tau$ it follows from Theorem 3.18 that we can write the operators as a sum Weyl quantised symbols that is

$$\begin{aligned} A^{\pm }_\varepsilon (\hbar ) = \sum _{j=0}^{2m} \hbar ^j \text {Op} _\hbar ^{\text {w} }(a^{\pm }_{\varepsilon ,j}). \end{aligned}$$

(7.2)

Written in this form the operator still satisfies Assumption 4.1. In particular we have that

$$\begin{aligned} a^{\pm }_{\varepsilon ,0}(x,p) = \sum _{|\alpha |, |\beta |\le m}a_{\alpha \beta }^\varepsilon (x) p^{\alpha +\beta } \pm C_1 \varepsilon ^{\tau +\mu }\sum _{|\alpha |\le m} p^{2\alpha } \end{aligned}$$

(7.3)

and

$$\begin{aligned} a^{\pm }_{\varepsilon ,1}(x,p) = i \sum _{|\alpha |, |\beta |\le m}\sum _{j=1}^d \frac{\beta _j-\alpha _j}{2} \partial _{x_j} a_{\alpha \beta }^\varepsilon (x) p^{\alpha +\beta -\eta _j}, \end{aligned}$$

(7.4)

where $\eta _j$ is the multi-index with all zeroes except on the j’th coordinate, where it is 1. Note that from the assumption $a_{\alpha \beta }^\varepsilon (x)=\overline{a_{\beta \alpha }^\varepsilon (x)}$ it follows that $a^{\pm }_{\varepsilon ,1}(x,p)$ is real valued.

Proof

Assume $\varphi$ and $\psi$ are Schwartz functions and recall that the sesquilinear forms ${\mathcal {A}}_{\hbar ,\varepsilon }^{\pm }$ are given by

$$\begin{aligned} {\mathcal {A}}_{\hbar ,\varepsilon }^{\pm }[\varphi ,\psi ] = {}{} & {} \sum _{\left| \alpha \right| ,\left| \beta \right| \le m} \int _{{\mathbb {R}}^d} a_{\alpha \beta }^\varepsilon (x) (\hbar D_x)^\beta \varphi (x) \overline{(\hbar D_x)^\alpha \psi (x)} \, dx \nonumber \\{} & {} \pm C_1 \varepsilon ^{\tau +\mu } \sum _{\left| \alpha \right| \le m} \int _{{\mathbb {R}}^d} (\hbar D_x)^\alpha \varphi (x) \overline{(\hbar D_x)^\alpha \psi (x)} \, dx \nonumber \\{} & {} ={} \sum _{\left| \alpha \right| ,\left| \beta \right| \le m} \sum _{\alpha _1\le \alpha } (-i\hbar )^{|\alpha _1|} \left( {\begin{array}{c}\alpha \\ \alpha _1\end{array}}\right) \int _{{\mathbb {R}}^d} [\partial _x^{\alpha _1}a_{\alpha \beta }^\varepsilon ] (x) (\hbar D_x)^{\alpha +\beta -\alpha _1}\varphi (x) \overline{\psi (x)} \, dx \nonumber \\{} & {} \pm C_1 \varepsilon ^{\tau +\mu } \sum _{\left| \alpha \right| \le m} \int _{{\mathbb {R}}^d} (\hbar D_x)^{2\alpha } \varphi (x) \overline{\psi (x)} \, dx, \end{aligned}$$

(7.5)

where $a_{\alpha \beta }^\varepsilon (x)$ are smooth functions. In the identity we have made integration by parts. With this in mind we define the symbol $b_\varepsilon (x,p;\hbar )$ to be

$$\begin{aligned} b_\varepsilon (x,p)&= \sum _{j=0}^m (-i\hbar )^j \sum _{\begin{array}{c} j\le |\alpha | \le m \\ |\beta |\le m \end{array}} \sum _{\begin{array}{c} \alpha _1\le \alpha \\ |\alpha _1|= j \end{array} }\left( {\begin{array}{c}\alpha \\ \alpha _1\end{array}}\right) [\partial _x^{\alpha _1}a_{\alpha \beta }^\varepsilon ] (x) p^{\alpha +\beta -\alpha _1 } \pm C_1 \varepsilon ^{\tau +\mu }\sum _{|\alpha |\le m} p^{2\alpha } \nonumber \\&= \sum _{j=0}^m \hbar ^j b_{\varepsilon ,j}(x,p). \end{aligned}$$

(7.6)

Next we verify that there exists a $\zeta <0$ such that $b_{\varepsilon ,0}(x,p)-\zeta$ is a tempered weight and that $b_{\varepsilon ,j}$ is in $\Gamma ^{b_{\varepsilon ,0}-\zeta ,\tau -j}_{0,\varepsilon }({\mathbb {R}}_x^{d}\times {\mathbb {R}}_p^{d})$ for all j.

We have that the forms are elliptic for all $\varepsilon$ sufficiently small. Moreover, by assumption all coefficients are bounded below by a number $-\zeta _0$. This implies that

$$\begin{aligned} b_{\varepsilon ,0}(x,p)&= \sum _{|\alpha |, |\beta |\le m}a_{\alpha \beta }^\varepsilon (x) p^{\alpha +\beta } \pm C_1 \varepsilon ^{\tau +\mu }\sum _{|\alpha |\le m} p^{2\alpha } \nonumber \\&\ge C |p|^{2m} -\zeta _0 \sum _{|\alpha |, |\beta |< m}| p^{\alpha +\beta }| - C_1 \varepsilon ^{\tau +\mu }\sum _{|\alpha |< m} |p^{2\alpha }| \ge -C, \end{aligned}$$

(7.7)

for some positive constant C and for all (x, p). Here we have used that $a_{\alpha \beta }^\varepsilon (x)=\overline{a_{\beta \alpha }^\varepsilon (x)}$ which implies that $b_{\varepsilon ,0}(x,p)$ is real for all (x, p). Hence we can find $\zeta <0$ such that $b_{\varepsilon ,0}(x,p)-\zeta >0$ for all $(x,p)\in {\mathbb {R}}^d_x\times {\mathbb {R}}^d_p$.

To see that this is a tempered weight we first recall how the new coefficients $a_{\alpha \beta }^\varepsilon (x)$ was constructed. They are given by

$$\begin{aligned} a_{\alpha \beta }^\varepsilon (x) = \int _{{\mathbb {R}}^d} a_{\alpha \beta }(x-\varepsilon y)\omega (y) \,dy, \end{aligned}$$

where $\omega (y)$ is a real Schwarz function integrating to 1. By assumption there exists a constant c such that $a_{\alpha \beta }^\varepsilon (x)+ c>0$ and we have that

$$\begin{aligned} a_{\alpha \beta }^\varepsilon (x)+ c{} & {} = \int _{{\mathbb {R}}^d} (a_{\alpha \beta }(x-\varepsilon y)+c)\omega (y) \,dy \nonumber \\{} & {} \le C_1(a_{\alpha \beta }(z)+c) \int _{{\mathbb {R}}^d}(1+|x-\varepsilon y-z|)^M \omega (y) \,dy \nonumber \\{} & {} \le C (a_{\alpha \beta }^\varepsilon (z) + a_{\alpha \beta }(z) - a_{\alpha \beta }^\varepsilon (z) +c) (1+|x-z|)^M \nonumber \\{} & {} \le {\tilde{C}} (a_{\alpha \beta }^\varepsilon (z) + c) (1+|x-z|)^M, \end{aligned}$$

(7.8)

where we have used that $\varepsilon \le 1$, some of the assumptions on $a_{\alpha \beta }(x)$ and that

$$\begin{aligned} |a_{\alpha \beta }(z) - a_{\alpha \beta }^\varepsilon (z)| \le c\varepsilon ^{1+\mu }, \end{aligned}$$

(7.9)

by Proposition 2.4. This gives us that the coefficients $a_{\alpha \beta }^\varepsilon (x)$ have at most polynomial growth. Due to the polynomial structure of $b_{\varepsilon ,0}(x,p)$ and the at most polynomial growth of $a_{\alpha \beta }^\varepsilon (x)$ there exist constants C and N independent of $\varepsilon$ in (0, 1] such that

$$\begin{aligned} b_{\varepsilon ,0}(x,p) - \zeta \le C (b_{\varepsilon ,0}(y,q) - \zeta )\lambda (x-y,p-q)^N \qquad \forall (x,p)\in {\mathbb {R}}^d_x\times {\mathbb {R}}^d_p. \end{aligned}$$

(7.10)

This shows that $b_{\varepsilon ,0}(x,p)$ is a tempered weight. To see that $b_{\varepsilon ,j}$ is in $\Gamma ^{b_{\varepsilon ,0}-\zeta ,\tau -j}_{0,\varepsilon }({\mathbb {R}}_x^{d}\times {\mathbb {R}}_p^{d})$ for all j we first note that for $\eta$ in ${\mathbb {N}}^d_0$ with $|\eta |\le \tau$ we have that

$$\begin{aligned} \left| \partial _{x}^\eta a_{\alpha \beta }^\varepsilon (x) \right|{} & {} = | \int _{{\mathbb {R}}^d} \partial _{x}^\eta a_{\alpha \beta }(x-\varepsilon y)\omega (y) \,dy| \nonumber \\{} & {} \le \int _{{\mathbb {R}}^d} | \partial _{x}^\eta a_{\alpha \beta }(x-\varepsilon y)\omega (y) | \,dy \nonumber \\{} & {} \le \int _{{\mathbb {R}}^d} c_\eta (a_{\alpha \beta }(x-\varepsilon y) + c) |\omega (y) | \,dy \nonumber \\{} & {} \le C(a_{\alpha \beta }(x) + c) \int _{{\mathbb {R}}^d} (1+\varepsilon |y|)^M |\omega (y)| \,dy \nonumber \\{} & {} \le C_\eta (a_{\alpha \beta }^\varepsilon (x) + c), \end{aligned}$$

(7.11)

where we again have used (7.9). The calculation also shows that $C_j$ is uniform for $\varepsilon$ in (0, 1]. For any $\eta$ in ${\mathbb {N}}_0^d$ with $\left| \eta \right| \ge \tau$ we have that

$$\begin{aligned} \left| \partial _x^\eta a_{\alpha \beta }^\varepsilon (x) \right| \le c \varepsilon ^{\tau +\mu - \left| \eta \right| } \le C_\eta \varepsilon ^{\tau -\left| \eta \right| }(a_{\alpha \beta }^\varepsilon (x) + c), \end{aligned}$$

(7.12)

by Proposition 2.4 with a constant which is uniform for $\varepsilon$ in (0, 1]. From combining (7.11) and (7.12) it follows that we for all $\eta$ and $\delta$ in ${\mathbb {N}}_0^d$ can find constants $C_{\eta \delta }$ such that

$$\begin{aligned} | \partial _x^\eta \partial _p^\delta b_{\varepsilon ,j}(x,p) | \le C_{\eta \delta } \varepsilon ^{-(\tau -j-|\eta |)_{-}} (b_{\varepsilon ,0}(x,p)-\zeta ) \qquad \forall (x,p)\in {\mathbb {R}}^d_x\times {\mathbb {R}}^d_p. \end{aligned}$$

(7.13)

This estimate gives us that $b_{\varepsilon ,j}$ is in $\Gamma ^{b_{\varepsilon ,0}-\zeta ,\tau -j}_{0,\varepsilon }({\mathbb {R}}_x^{d}\times {\mathbb {R}}_p^{d})$ for all j.

Define the operators $A^{\pm }_\varepsilon (\hbar )$ to be

$$\begin{aligned} A^{\pm }_\varepsilon (\hbar ) = \text {Op} _{\hbar ,0}(b_\varepsilon (\hbar )). \end{aligned}$$

(7.14)

From the above estimates we have that $A^{\pm }_\varepsilon (\hbar )$ is a $\hbar$-$\varepsilon$-admissible of regularity $\tau$. That the regularity is $\tau$ follows directly from Proposition 2.4. Moreover, we already have that the operator $A^{\pm }_\varepsilon (\hbar )$ satisfies 2 and 3 from Assumption 4.1. From (7.5) we have that $A^{\pm }_\varepsilon (\hbar )$ also satisfies 1 of Assumption 4.1. Theorem 4.2 now gives us that $A^{\pm }_\varepsilon (\hbar )$ are lower semi-bounded and essentially self-adjoint for all $\hbar$ sufficiently small. Denote also by $A^{\pm }_\varepsilon (\hbar )$ the self-adjoint exstension. From (7.5) we also get that

$$\begin{aligned} A^{\pm }_\varepsilon (\hbar )[\varphi ,\psi ] = {\mathcal {A}}_{\hbar ,\varepsilon }^{\pm }[\varphi ,\psi ] \qquad \text {for all }\varphi ,\psi \in {\mathcal {D}}( {\mathcal {A}}_{\hbar ,\varepsilon }^{\pm }). \end{aligned}$$

(7.15)

This concludes the proof. $\square$

Lemma 7.3

Let ${\mathcal {A}}_\hbar$ be a sesquilinear form which satisfies Assumption 1.2 with the numbers $(\tau ,\mu )$ in ${\mathbb {N}}\times [0,1]$. Then the form is symmetric, lower semi-bounded, densely defined and closed.

Proof

Recall that the sesquilinear form ${\mathcal {A}}_\hbar$ is given by

$$\begin{aligned} {\mathcal {A}}_\hbar [\varphi ,\psi ] = \sum _{\left| \alpha \right| ,\left| \beta \right| \le m} \int _{{\mathbb {R}}^d} a_{\alpha \beta }(x) (\hbar D_x)^\beta \varphi (x) \overline{(\hbar D_x)^\alpha \psi (x)} \, dx, \qquad \varphi ,\psi \in {\mathcal {D}}( {\mathcal {A}}_\hbar ). \end{aligned}$$

(7.16)

We have by assumption that $a_{\alpha \beta }(x)=\overline{a_{\beta \alpha }(x)}$ for all $\alpha$ and $\beta$. This gives us that the form is symmetric. Moreover, we also have that the coefficients $a_{\alpha \beta }(x)$ at most grow polynomially, hence it follows that the Schwartz functions are contained in the form domain. This gives that the form is densely defined.

From Proposition 2.5 we get the two forms ${\mathcal {A}}_{\hbar ,\varepsilon }^{\pm }$ such that

$$\begin{aligned} {\mathcal {A}}_{\hbar ,\varepsilon }^{-}[\varphi ,\varphi ]\le {\mathcal {A}}_{\hbar }[\varphi ,\varphi ]\le {\mathcal {A}}_{\hbar ,\varepsilon }^{+}[\varphi ,\varphi ] \end{aligned}$$

(7.17)

for all $\varphi$ in ${\mathcal {D}}( {\mathcal {A}}_\hbar )$. Moreover, the result in Lemma 7.1 implies that the forms ${\mathcal {A}}_{\hbar ,\varepsilon }^{\pm }$ are closed. Hence ${\mathcal {A}}_{\hbar }$ will also be closed. This concludes the proof. $\square$

7.1 Proof of Theorem 1.3

Recall that we assume ${\mathcal {A}}_\hbar$ is sesquilinear form which satisfies Assumption 1.2 with the numbers $(1,\mu )$ with $\mu >0$. Then for the Friedrichs extension of ${\mathcal {A}}_\hbar$ we have

$$\begin{aligned} \big |\text {Tr} [{\varvec{1}}_{(-\infty ,0]}(A(\hbar ))] - \frac{1}{(2\pi \hbar )^d} \int _{{\mathbb {R}}^{2d}} {\varvec{1}}_{(-\infty ,0]}( a_{0}(x,p)) \,dx dp \big | \le C \hbar ^{1-d}, \end{aligned}$$

Proof of Theorem 1.3

From Lemma 7.3 we get the existence of the operator $A(\hbar )$ as a Friedrichs extension. Combining Proposition 2.5 and Lemma 7.1 we get the existence of two $\hbar$-$\varepsilon$-admissible operators $A^{\pm }_\varepsilon (\hbar )$ of regularity 1 satisfying Assumption 4.1 for all $\varepsilon$ sufficiently small such that

$$\begin{aligned} A_\varepsilon ^{-}(\hbar ) \le A(\hbar ) \le A_\varepsilon ^{+}(\hbar ), \end{aligned}$$

in the sense of quadratic forms. Moreover the operators $A^{\pm }_\varepsilon (\hbar )$ have the following expansions

$$\begin{aligned} A^{\pm }_\varepsilon (\hbar ) = \sum _{j=0}^{2m} \hbar ^j \text {Op} _\hbar ^{\text {w} }(a^{\pm }_{\varepsilon ,j}). \end{aligned}$$

(7.18)

In particular we have that

$$\begin{aligned} a^{\pm }_{\varepsilon ,0}(x,p) = \sum _{|\alpha |, |\beta |\le m}a_{\alpha \beta }^\varepsilon (x) p^{\alpha +\beta } \pm C_1 \varepsilon ^{1+\mu }\sum _{|\alpha |\le m} p^{2\alpha }. \end{aligned}$$

(7.19)

Moreover, from Proposition 2.5 we also get that 0 is a non-critical value for the operators $A^{\pm }_\varepsilon (\hbar )$. What remains in order to be able to apply Theorem 6.5 for the two operators $A^{\pm }_\varepsilon (\hbar )$ is the existence of a ${\tilde{\nu }}>0$ such that the preimage of $(-\infty ,{\tilde{\nu }}]$ under $a^{\pm }_{\varepsilon ,0}$ is compact. By the ellipticity of the operators we have that the preimage is compact in p. Let ${\tilde{\nu }}=\frac{\nu }{2}$ and note that as in the proof of Proposition 2.5 we have the estimate

$$\begin{aligned} | \sum _{\left| \alpha \right| ,\left| \beta \right| \le m} (a_{\alpha \beta }^\varepsilon (x) - a_{\alpha \beta }(x))p^{\alpha +\beta } \pm C_1 \varepsilon ^{1+\mu }(1+p^2)^m |\le C \varepsilon ^{1+\mu } , \end{aligned}$$

(7.20)

since we can assume p to be in a compact set. This estimate implies the inclusion

$$\begin{aligned} \{ (x,p) \in {\mathbb {R}}^{2d} \, |\, a_{\varepsilon ,0}^{\pm }(x,p) \le \frac{\nu }{2} \} \subseteq \{ (x,p) \in {\mathbb {R}}^{2d} \, |\, a_{0}(x,p) \le \frac{\nu }{2} + C \varepsilon ^{1+\mu } \}. \end{aligned}$$

Hence for a sufficiently small $\varepsilon$ we have, that $\{ (x,p) \in {\mathbb {R}}^{2d} \, |\, a_{\varepsilon ,0}^{\pm }(x,p) \le \frac{\nu }{2} \}$ is compact due to our assumptions. Now by Theorem 6.5 we get for sufficiently small $\hbar$ and $\varepsilon \ge \hbar ^{1-\delta }$ for a positive $\delta <1$ that

$$\begin{aligned} \big |\text {Tr} [{\varvec{1}}_{(-\infty ,0]}(A_\varepsilon ^{\pm }(\hbar ))] - \frac{1}{(2\pi \hbar )^d} \int _{{\mathbb {R}}^{2d}} {\varvec{1}}_{(-\infty ,0]}( a_{\varepsilon ,0}^\pm (x,p)) \,dx dp \big | \le C \hbar ^{1-d}. \end{aligned}$$

(7.21)

Here we choose $\delta =\frac{\mu }{1+\mu }$. Now if we consider the following difference between integrals

$$\begin{aligned} \big | \int _{{\mathbb {R}}^{2d}} {\varvec{1}}_{(-\infty ,0]}( a_{\varepsilon ,0}^\pm (x,p)) \,dx dp - \int _{{\mathbb {R}}^{2d}} {\varvec{1}}_{(-\infty ,0]}( a_{0}(x,p)) \,dx dp \big |. \end{aligned}$$

Then for $\varepsilon$ and hence $\hbar$ sufficiently small we obtain that

$$\begin{aligned}{} & {} \big | \int _{{\mathbb {R}}^{2d}} {\varvec{1}}_{(-\infty ,0]}( a_{\varepsilon ,0}^\pm (x,p)) \,dx dp - \int _{{\mathbb {R}}^{2d}} {\varvec{1}}_{(-\infty ,0]}( a_{0}(x,p)) \,dx dp \big | \nonumber \\{} & {} \le \int _{{\mathbb {R}}^{2d}} {\varvec{1}}_{[-C\varepsilon ^{1+\mu },C\varepsilon ^{1+\mu }]}( a_{\varepsilon ,0}(x,p)) \,dx dp \le {\tilde{C}}\varepsilon ^{1+\mu }, \end{aligned}$$

(7.22)

where we in the last inequality have used the non-critical condition. By combining (7.21) and (7.22) we get that

$$\begin{aligned} \big |\text {Tr} [{\varvec{1}}_{(-\infty ,0]}(A_\varepsilon ^{\pm }(\hbar ))] - \frac{1}{(2\pi \hbar )^d} \int _{{\mathbb {R}}^{2d}} {\varvec{1}}_{(-\infty ,0]}( a_{\varepsilon ,0}(x,p)) \,dx dp \big | \le C \hbar ^{1-d} + {\tilde{C}}\varepsilon ^{1+\mu } \hbar ^{-d}. \end{aligned}$$

(7.23)

If we take $\varepsilon = \hbar ^{1-\delta }$ we have that

$$\begin{aligned} \varepsilon ^{1+\mu } = \hbar ^{(1+\mu )(1-\delta )}=\hbar . \end{aligned}$$

Hence (7.23) with this choice of $\delta$ and $\varepsilon$ gives the estimate

$$\begin{aligned} \big |\text {Tr} [{\varvec{1}}_{(-\infty ,0]}(A_\varepsilon ^{\pm }(\hbar ))] - \frac{1}{(2\pi \hbar )^d} \int _{{\mathbb {R}}^{2d}} {\varvec{1}}_{(-\infty ,0]}( a_{\varepsilon ,0}(x,p)) \,dx dp \big | \le C \hbar ^{1-d}. \end{aligned}$$

(7.24)

Now as the framing operators satisfied the relation

$$\begin{aligned} A_\varepsilon ^{-}(\hbar ) \le A(\hbar ) \le A_\varepsilon ^{+}(\hbar ), \end{aligned}$$

in the sense of quadratic forms. We get by the min-max theorem the relation

$$\begin{aligned} \text {Tr} [{\varvec{1}}_{(-\infty ,0]}(A_\varepsilon ^{+}(\hbar ))] \le \text {Tr} [{\varvec{1}}_{(-\infty ,0]}(A(\hbar )) ] \le \text {Tr} [{\varvec{1}}_{(-\infty ,0]}(A_\varepsilon ^{-}(\hbar ))]. \end{aligned}$$

Combining this with (7.24) we get the estimate

$$\begin{aligned} \big |\text {Tr} [{\varvec{1}}_{(-\infty ,0]}(A(\hbar ))] - \frac{1}{(2\pi \hbar )^d}\int _{{\mathbb {R}}^{2d}} {\varvec{1}}_{(-\infty ,0]}( a_{\varepsilon ,0}(x,p)) \,dx dp \big | \le C \hbar ^{1-d}. \end{aligned}$$

(7.25)

This is the desired estimate and this ends the proof. $\square$

7.2 Proof of Theorem 1.5

Recall that we have given a number $\gamma$ in (0, 1] and a sesquilinear form ${\mathcal {A}}_\hbar$ which satisfies Assumption 1.2 with the numbers $(2,\mu )$, where we suppose $\mu =0$ if $\gamma <1$ and $\mu >0$ if $\gamma =1$. Then for the Friedrichs extension of ${\mathcal {A}}_\hbar$ we have that

$$\begin{aligned} \big |\text {Tr} [(A(\hbar ))^\gamma _{-}] - \frac{1}{(2\pi \hbar )^d} \int _{{\mathbb {R}}^{2d}} ( a_{0}(x,p))^{\gamma }_{-} +\hbar \gamma a_1(x,p) ( a_{0}(x,p))^{\gamma -1}_{-} \,dx dp \big | \le C \hbar ^{1+\gamma -d}, \end{aligned}$$

for all sufficiently small $\hbar$, where the symbol $a_1(x,p)$ is defined as

$$\begin{aligned} a_{1}(x,p) = i \sum _{|\alpha |, |\beta |\le m}\sum _{j=1}^d \frac{\beta _j-\alpha _j}{2} \partial _{x_j} a_{\alpha \beta } (x) p^{\alpha +\beta -\eta _j}, \end{aligned}$$

(7.26)

where $\eta _j$ is the multi-index with all entries zero except the j’th which is one.

Proof of Theorem 1.5

The start of the proof is analogous to the proof of Theorem 1.3. Again from Lemma 7.3 we get the existence of the operator $A(\hbar )$ and by combining Proposition 2.5 and Lemma 7.1 we get the existence of two $\hbar$-$\varepsilon$-admissible operators $A^{\pm }_\varepsilon (\hbar )$ of regularity 2 satisfying Assumption 4.1 for all $\varepsilon$ sufficiently small such that

$$\begin{aligned} A_\varepsilon ^{-}(\hbar ) \le A(\hbar ) \le A_\varepsilon ^{+}(\hbar ), \end{aligned}$$

in the sense of quadratic forms. Moreover, the operators $A^{\pm }_\varepsilon (\hbar )$ have the following expansions

$$\begin{aligned} A^{\pm }_\varepsilon (\hbar ) = \sum _{j=0}^{2m} \hbar ^j \text {Op} _\hbar ^{\text {w} }(a^{\pm }_{\varepsilon ,j}). \end{aligned}$$

(7.27)

In particular we have that

$$\begin{aligned} a^{\pm }_{\varepsilon ,0}(x,p) = \sum _{|\alpha |, |\beta |\le m}a_{\alpha \beta }^\varepsilon (x) p^{\alpha +\beta } \pm C_1 \varepsilon ^{2+\mu }\sum _{|\alpha |\le m} p^{2\alpha } \end{aligned}$$

(7.28)

and

$$\begin{aligned} a^{\pm }_{\varepsilon ,1}(x,p) = i \sum _{|\alpha |, |\beta |\le m}\sum _{j=1}^d \frac{\beta _j-\alpha _j}{2} \partial _{x_j} a_{\alpha \beta }^\varepsilon (x) p^{\alpha +\beta -\eta _j}. \end{aligned}$$

(7.29)

Moreover, from Proposition 2.5 we also get that 0 is a non-critical value for the operators $A^{\pm }_\varepsilon (\hbar )$. Just as before we also get that there exist ${\tilde{\nu }}$ such that $a_{\varepsilon ,0}^{-1}((-\infty ,{\tilde{\nu }}])$ is compact for all sufficiently small $\varepsilon$. Hence we are in a situation where we can apply Theorem 6.6 for the operators $A^{\pm }_\varepsilon (\hbar )$. This gives us that

$$\begin{aligned} \begin{aligned}&\big |\text {Tr} [(A^{\pm }_\varepsilon (\hbar ))^\gamma _{-}] - \frac{1}{(2\pi \hbar )^d} \int _{{\mathbb {R}}^{2d}} ( a_{\varepsilon ,0}^{\pm }(x,p))^{\gamma }_{-} +\hbar \gamma a_{\varepsilon ,1}^{\pm }(x,p) ( a_{\varepsilon ,0}^{\pm }(x,p))^{\gamma -1}_{-} \,dx dp \big | \\&\le C \hbar ^{1+\gamma -d}. \end{aligned} \end{aligned}$$

(7.30)

What remains is to compare the phase space integrals. We let $\kappa >0$ be a number such that $a_{0}(x,p)$ is non-critical on $a_{0}^{-1}(-2\kappa ,2\kappa )$. We then do the following splitting of integrals

$$\begin{aligned}{} & {} \big | \int _{{\mathbb {R}}^{2d}} ( a_{\varepsilon ,0}^{\pm }(x,p))^{\gamma }_{-} \,dxdx - \int _{{\mathbb {R}}^{2d}} ( a_{0}(x,p))^{\gamma }_{-} \,dxdp\big | \nonumber \\{} & {} \le {} \big | \int _{{\mathbb {R}}^{2d}}{\varvec{1}}_{(-c,-\kappa ]}(a_{0}(x,p)) \big [ ( a_{\varepsilon ,0}^{\pm }(x,p))^{\gamma }_{-} - ( a_{0}(x,p))^{\gamma }_{-}\big ] \,dxdp\big | \nonumber \\{} & {} \quad + \big | \int _{{\mathbb {R}}^{2d}}{\varvec{1}}_{(-\kappa ,0]}(a_{0}(x,p)) \big [ ( a_{\varepsilon ,0}^{\pm }(x,p))^{\gamma }_{-} - ( a_{0}(x,p))^{\gamma }_{-}\big ] \,dxdp\big |, \end{aligned}$$

(7.31)

where $c>0$ is a number such that $a_{0}(x,p)-1 \ge -c$ for all (x, p). On the preimage $a_{0}^{-1}((-\infty ,-\kappa ])$ we have that $a_{0,\varepsilon }^{\pm }$ will be bounded from above by $-\frac{\kappa }{2}$ for all $\varepsilon$ sufficiently small. Hence using that the map $t\mapsto (t)^\gamma _{-}$ is globally Lipschitz continuous when restricted to any interval of the form $(-\infty ,-\frac{\kappa }{2})$ we get that

$$\begin{aligned}{} & {} \big | \int _{{\mathbb {R}}^{2d}}{\varvec{1}}_{(-c,-\kappa ]}(a_{0}(x,p)) \big [ ( a_{\varepsilon ,0}^{\pm }(x,p))^{\gamma }_{-} - ( a_{0}(x,p))^{\gamma }_{-}\big ] \,dxdp\big | \nonumber \\{} & {} \le C \int _{{\mathbb {R}}^{2d}}{\varvec{1}}_{(-c,-\kappa ]}(a_{0}(x,p)) | a_{\varepsilon ,0}^{\pm }(x,p) - a_{0}(x,p) | \,dxdp \le C \varepsilon ^{2+\mu }, \end{aligned}$$

(7.32)

where we in the last inequality have used estimates similar to the ones used in the proof of Proposition 2.5. Before we estimate the second term in (7.31) we note that for all $\gamma$ in (0, 1] and $t\le 0$ we have that

$$\begin{aligned} (t)_{-}^\gamma = \gamma \int _{t}^0 (-s)^{\gamma -1} \, ds. \end{aligned}$$

Using this observation we get that

$$\begin{aligned}&\big | \int _{{\mathbb {R}}^{2d}}{\varvec{1}}_{(-\kappa ,0]}(a_{0}(x,p)) \big [ ( a_{\varepsilon ,0}^{\pm }(x,p))^{\gamma }_{-} - ( a_{0}(x,p))^{\gamma }_{-}\big ] \,dxdp\big | \nonumber \\&= \big | \gamma \int _{-\frac{3\kappa }{2}}^0 (-s)^{\gamma -1} \int _{{\mathbb {R}}^{2d}} {\varvec{1}}_{(-\kappa ,0]}(a_{0}(x,p))\big [ {\varvec{1}}_{( a_{\varepsilon ,0}^{\pm }(x,p),0]}(s) - {\varvec{1}}_{( a_{0}(x,p),0]}(s)\big ] \,dxdp ds\big | \nonumber \\&\le \gamma \int _{-\frac{3\kappa }{2}}^0 (-s)^{\gamma -1} \int _{{\mathbb {R}}^{2d}} {\varvec{1}}_{(-\kappa ,0]}(a_{0}(x,p)) {\varvec{1}}_{[s-{\tilde{c}}\varepsilon ^{2+\mu },s+{\tilde{c}}\varepsilon ^{2+\mu }]}(a_{0}(x,p)) \,dxdp ds \nonumber \\&\le C \varepsilon ^{2+\mu }, \end{aligned}$$

(7.33)

where we in the last inequality have used the non-critical assumption. Combining (7.31), (7.32) and (7.33) we get that

$$\begin{aligned} \begin{aligned} \Big | \int _{{\mathbb {R}}^{2d}} ( a_{\varepsilon ,0}^{\pm }(x,p))^{\gamma }_{-} \,dxdx - \int _{{\mathbb {R}}^{2d}} ( a_{0}(x,p))^{\gamma }_{-} \,dxdp\Big | \le C \varepsilon ^{2+\mu }. \end{aligned} \end{aligned}$$

(7.34)

Next we consider the difference

$$\begin{aligned} \Big | \int _{{\mathbb {R}}^{2d}} a_{\varepsilon ,1}^{\pm }(x,p) ( a_{\varepsilon ,0}^{\pm }(x,p))^{\gamma -1}_{-} \,dx dp - \int _{{\mathbb {R}}^{2d}} a_{1}(x,p) ( a_{0}(x,p))^{\gamma -1}_{-} \,dx dp\Big | \end{aligned}$$

(7.35)

Firstly we notice that from Proposition 2.5 we get that

$$\begin{aligned} \Big | \int _{{\mathbb {R}}^{2d}} a_{\varepsilon ,1}^{\pm }(x,p) ( a_{\varepsilon ,0}^{\pm }(x,p))^{\gamma -1}_{-} \,dx dp - \int _{{\mathbb {R}}^{2d}} a_{1}(x,p) ( a_{\varepsilon ,0}^{\pm }(x,p))^{\gamma -1}_{-} \,dx dp\Big | \le C\varepsilon ^{1+\mu }. \end{aligned}$$

(7.36)

Moreover, we have that

$$\begin{aligned}{} & {} \Big | \int _{{\mathbb {R}}^{2d}} a_{1}(x,p) ( a_{\varepsilon ,0}^{\pm }(x,p))^{\gamma -1}_{-} \,dx dp - \int _{{\mathbb {R}}^{2d}} a_{1}(x,p) ( a_{0}(x,p))^{\gamma -1}_{-} \,dx dp\Big | \nonumber \\{} & {} \le {} \Big | \int _{{\mathbb {R}}^{2d}} a_{1}(x,p) \big [ {\varvec{1}}_{(-\kappa ,0]}(a_{\varepsilon ,0}^{\pm }(x,p)) (- a_{\varepsilon ,0}^{\pm }(x,p))^{\gamma -1} \nonumber \\{} & {} \quad - {\varvec{1}}_{(-\kappa ,0]}(a_{0}(x,p)) ( -a_{0}(x,p))^{\gamma -1} \big ] \,dx dp \Big | +C\varepsilon ^{2+\mu }, \end{aligned}$$

(7.37)

where the number $\kappa >0$ is the same as above. In this estimate we have used that

$$\begin{aligned}{} & {} \Big | \int _{{\mathbb {R}}^{2d}} a_{1}(x,p) \big [ {\varvec{1}}_{(-c,-\kappa ]}(a_{\varepsilon ,0}^{\pm }(x,p)) (- a_{\varepsilon ,0}^{\pm }(x,p))^{\gamma -1} \nonumber \\{} & {} \quad - {\varvec{1}}_{(-c,-\kappa ]}(a_{0}(x,p)) ( -a_{0}(x,p))^{\gamma -1} \big ] \,dx dp \Big | \le C\varepsilon ^{2+\mu }, \end{aligned}$$

(7.38)

where the arguments are analogous to the ones used for estimating the first difference of phase space integrals after recalling that the map $t\rightarrow (t)^{\gamma -1}_{-}$ is Lipschitz continuous when restricted to a bounded interval of the form $(-a,-b)$, where $0<b<a$.

On the pre-image $a_{0}^{-1}((-\kappa ,0])$ we have that

$$\begin{aligned} |\nabla _p a_{0}(x,p) | \ge c_0 >0. \end{aligned}$$

(7.39)

Then by using a partition of unity we may assume that $|\partial _{p_1}a_{0}(x,p)|\ge c>0$ on $a_{0}^{-1}((-\kappa ,0])$. From the construction of the approximating symbols it follows that for $\varepsilon$ sufficiently small we get that $|\partial _{p_1}a_{\varepsilon ,0}^{\pm }(x,p)|\ge {\tilde{c}}$ when $a_{\varepsilon ,0}^{\pm }(x,p)\in (-\kappa ,0]$. As in the proof of Theorem 6.1 we now define the maps $F^{\pm }$ and F by

$$\begin{aligned} F^\pm :(x,p) \rightarrow (x_1,\dots ,x_d,a_{\varepsilon ,0}^{\pm }(x,p),p_2,\dots ,p_d) \end{aligned}$$

and the map F is defined similarly with $a_{\varepsilon ,0}^{\pm }(x,p)$ replaced by $a_{0}(x,p)$. The determinants for the Jacobian matrices are given by

$$\begin{aligned} \det (DF^{\pm })= \partial _{p_1}a_{\varepsilon ,0}^{\pm } \quad \text {and}\quad \det (DF)= \partial _{p_1}a_{0}. \end{aligned}$$

This gives us that the inverse maps ($G^{\pm }$, G) exists and they will be close in the sense that

$$\begin{aligned} |G^{\pm }(y,q) - G(y,q)| \le C\varepsilon ^{2+\mu } \end{aligned}$$

(7.40)

on the domain, where both functions are defined. To see this note that they will only be different in one coordinate and note that these coordinates will be determined by the equation $a_0(y,(p_1,q_2,\dots ,q_d))=q_1$ and similarly for the two other functions. With a change of variables we get that

$$\begin{aligned} \begin{aligned}&\Big | \int _{{\mathbb {R}}^{2d}} a_{1}(x,p) \big [ {\varvec{1}}_{(-\kappa ,0]}(a_{\varepsilon ,0}^{\pm }(x,p)) (- a_{\varepsilon ,0}^{\pm }(x,p))^{\gamma -1} \\&\quad - {\varvec{1}}_{(-\kappa ,0]}(a_{0}(x,p)) ( -a_{0}(x,p))^{\gamma -1} \big ] \,dx dp \Big | \\&\le {} \Big | \int _{\Omega } \frac{a_{1}( G^{\pm }(x,p)) {\varvec{1}}_{(-\kappa ,0]}(p_1) (- p_1)^{\gamma -1}}{\partial _{p_1}a_{\varepsilon ,0}^{\pm }(G^{\pm }(x,p))} -\frac{a_{1}( G(x,p)) {\varvec{1}}_{(-\kappa ,0]}(p_1) (- p_1)^{\gamma -1}}{\partial _{p_1}a_{0}(G(x,p))} \,dx dp \Big | \\&\le {} \int _{\Omega } \Big | \frac{a_{1}( G^{\pm }(x,p)) }{\partial _{p_1}a_{\varepsilon ,0}^{\pm }(G^{\pm }(x,p))} -\frac{a_{1}( G(x,p))}{\partial _{p_1}a_{0}(G(x,p))} \Big | {\varvec{1}}_{(-\kappa ,0]}(p_1) (- p_1)^{\gamma -1} \,dx dp \le C \varepsilon ^{1+\mu }, \end{aligned} \end{aligned}$$

(7.41)

where $\Omega$ is a compact subset of ${\mathbb {R}}^{d}_x \times {\mathbb {R}}^{d}_p$. The last estimate follows similarly to the previous estimates. From combining (7.35), (7.36), (7.37) and (7.41) we get that

$$\begin{aligned} \Big | \int _{{\mathbb {R}}^{2d}} a_{\varepsilon ,1}^{\pm }(x,p) ( a_{\varepsilon ,0}^{\pm }(x,p))^{\gamma -1}_{-} \,dx dp - \int _{{\mathbb {R}}^{2d}} a_{1}(x,p) ( a_{0}(x,p))^{\gamma -1}_{-} \,dx dp\Big | \le C\varepsilon ^{1+\mu }. \end{aligned}$$

(7.42)

Now by choosing $\delta = 1- \frac{1+\gamma }{2+\mu }$ and combining (7.30), (7.34) (7.42) we obtain that

$$\begin{aligned}&\big |\text {Tr} [(A^{\pm }_\varepsilon (\hbar ))^\gamma _{-}] - \frac{1}{(2\pi \hbar )^d} \int _{{\mathbb {R}}^{2d}} ( a_{0}(x,p))^{\gamma }_{-} +\hbar \gamma a_{1}(x,p) ( a_{0}(x,p))^{\gamma -1}_{-} \,dx dp \big | \nonumber \\&\le C_1 \hbar ^{1+\gamma -d} + C_2\hbar ^{-d}\varepsilon ^{2+\mu } +C_3\hbar ^{1-d}\varepsilon ^{1+\mu } \le C\hbar ^{1+\gamma -d}. \end{aligned}$$

(7.43)

Now as the framing operators satisfied the relation

$$\begin{aligned} A_\varepsilon ^{-}(\hbar ) \le A(\hbar ) \le A_\varepsilon ^{+}(\hbar ) \end{aligned}$$

in the sense of quadratic forms. We get by the min-max-theorem the relation

$$\begin{aligned} \text {Tr} [(A_\varepsilon ^{+}(\hbar ))^\gamma _{-}] \le \text {Tr} [(A(\hbar ) )^\gamma _{-}] \le \text {Tr} [(A_\varepsilon ^{-}(\hbar ))^\gamma _{-}]. \end{aligned}$$

These inequalities combined with (7.43) give the desired estimates and this concludes the proof. $\square$

7.3 Proof of Theorem 1.9 and Theorem 1.10

Recall that we assume $B(\hbar )$ is an admissible operator and that $B(\hbar )$ satisfies Assumption 1.8 with $\hbar$ in $(0,\hbar _0]$. Moreover we assume that ${\mathcal {A}}_\hbar$ is a sesquilinear form which satisfies Assumption 1.2 with the numbers $(k,\mu )$, where the value of the two numbers k and $\mu$ depend on which Theorem one consider. We defined the symbol ${\tilde{b}}_0(x,p)$ to be

$$\begin{aligned} {\tilde{b}}_0(x,p) = b_0(x,p) + \sum _{\left| \alpha \right| ,\left| \beta \right| \le m} a_{\alpha \beta }(x)p^{\alpha +\beta }. \end{aligned}$$

(7.44)

Finally we supposed that there is a $\nu$ such that ${\tilde{b}}_0^{-1}((-\infty ,\nu ])$ is compact and there is $c>0$ such that

$$\begin{aligned} |\nabla _p {\tilde{b}}_0(x,p)| \ge c \quad \text {for all } (x,p)\in {\tilde{b}}_0^{-1}(\{0\}). \end{aligned}$$

(7.45)

Proof of Theorem 1.9 and Theorem 1.10

From Lemma 7.3 we get the existence of the self-adjoint operator $A(\hbar )$ as the Friedrichs extension ${\mathcal {A}}_\hbar$. Since the Schwartz functions are in the domain of the operators $A(\hbar )$ and $B(\hbar )$ we can define their form sum ${\tilde{B}}(\hbar ) =B(\hbar )+ A(\hbar )$ as a selfadjoint, lower semi-bounded operator see e.g. [30, Proposition 10.22]. Using Proposition 2.5 and by arguing as in Lemma 7.1 we get the existence of two $\hbar$-$\varepsilon$-admissible operators ${\tilde{B}}^{\pm }_\varepsilon (\hbar )$ of regularity 1 or 2 satisfying Assumption 4.1 for all $\varepsilon$ sufficiently small such that

$$\begin{aligned} {\tilde{B}}_\varepsilon ^{-}(\hbar ) \le {\tilde{B}}(\hbar ) \le {\tilde{B}}_\varepsilon ^{+}(\hbar ). \end{aligned}$$

In the sense of quadratic forms. In fact we will have that ${\tilde{B}}^{\pm }_\varepsilon (\hbar ) = B(\hbar )+ A^{\pm }_\varepsilon (\hbar )$, where $A^{\pm }_\varepsilon (\hbar )$ is the operator from Lemma 7.1. From here the proof is either analogous to that of Theorem 1.3 if it is the proof of Theorem 1.9. If it is the proof of Theorem 1.10 it will be analogous to the proof of Theorem 1.5 from here. $\square$

Appendix A Faà di Bruno formula

In this appendix we will recall some results about multivariate differentiation. In particular we will recall the Faà di Bruno formula, which gives a multivariate chain rule for any number of derivatives:

Theorem A.1

(Faà di Bruno formula) Let f be a function from $C^\infty ({\mathbb {R}})$ and g a function from $C^\infty ({\mathbb {R}}^d)$. Then for all multi-indices $\alpha$ with $\left| \alpha \right| \ge 1$ the following formula holds:

$$\begin{aligned} \partial _x^\alpha f(g(x)) = \sum _{k=1}^{\left| \alpha \right| } f^{(k)}(g(x)) \sum _{\begin{array}{c} \alpha _1 + \cdots + \alpha _k = \alpha \\ \left| \alpha _j \right| >0 \end{array}} c_{\alpha _1\cdots \alpha _k}\partial _x^{\alpha _1} g(x) \cdots \partial _x^{\alpha _k}g(x), \end{aligned}$$

where $f^{(k)}$ is the k’th derivative of f. The second sum should be understood as a sum over all ways to split the multi-index $\alpha$ in k non-trivial parts. The numbers $c_{\alpha _1\cdots \alpha _k}$’s are combinatorial constants independent of the functions.

A proof of the Faà di Bruno formula can be found in [32], where they prove the formula in greater generality then stated here. It is also possible to find the constants from their proof, but for our purpose here the exact values of the constants are not important. The next Corollary is the Faà di Bruno formula in the case of ${\mathbb {R}}^{2d}$ instead of just ${\mathbb {R}}^d$. But we need to control the exact number of derivatives in the first d components hence it is stated separately.

Corollary A.2

Let f be a function from $C^\infty ({\mathbb {R}})$ and a(x, p) a function from $C^\infty ({\mathbb {R}}_x^d\times {\mathbb {R}}^d_p)$. Then for all multi-indices $\alpha$ and $\beta$ with $\left| \alpha \right| +\left| \beta \right| \ge 1$ the following formula holds:

$$\begin{aligned} \partial _p^\beta \partial _x^\alpha f(a(x,p)) =\!\!\! \sum _{k=1}^{\left| \alpha \right| +\left| \beta \right| } f^{(k)} (a(x,p)) \!\!\! \sum _{{\mathcal {I}}_k(\alpha ,\beta )}c_{\alpha _1\cdots \alpha _k}^{\beta _1\cdots \beta _k} \partial _p^{\beta _1} \partial _x^{\alpha _1} a(x,p) \cdots \partial _p^{\beta _k} \partial _x^{\alpha _k} a(x,p), \end{aligned}$$

where the set ${\mathcal {I}}_k(\alpha ,\beta )$ is defined by

$$\begin{aligned} \begin{aligned} {\mathcal {I}}_k(\alpha ,\beta ) = \{(\alpha _1,\dots ,\alpha _k,&\beta _1,\dots ,\beta _k) \in {\mathbb {N}}^{2kd} \\&| \ \sum _{l=1}^k \alpha _l=\alpha , \, \sum _{l=1}^k \beta _l=\beta , \, \max (\left| \alpha _l \right| ,\left| \beta _l \right| ) \ge 1 \, \forall l \}. \end{aligned} \end{aligned}$$

The second sum is a sum over all elements in the set ${\mathcal {I}}_k(\alpha ,\beta )$, the constants $c_{\alpha _1\cdots \alpha _k}^{\beta _1\cdots \beta _k}$ are combinatorial constants independent of the functions and $f^{(k)}$ is the k’th derivative of the function f.

We will only give a short sketch of the proof of this corollary.

Sketch of proof

We have by the Faà di Bruno formula (Theoram A.1) the identity

$$\begin{aligned} \partial _x^\alpha f(a(x,p)) = \sum _{k=1}^{\left| \alpha \right| } f^{(k)}(a(x,p)) \sum _{\begin{array}{c} \alpha _1 + \cdots + \alpha _k = \alpha \\ \left| \alpha _j \right| >0 \end{array}} c_{\alpha _1\cdots \alpha _k}\partial _x^{\alpha _1} a(x,p) \cdots \partial _x^{\alpha _k}a(x,p) \end{aligned}$$

(A1)

By Leibniz’s formula we have

$$\begin{aligned} \begin{aligned} \partial _p^\beta&\partial _x^\alpha f(a(x,p)) \\&= \sum _{\gamma \le \beta } \left( {\begin{array}{c}\beta \\ \gamma \end{array}}\right) \sum _{k=1}^{\left| \alpha \right| } \partial _p^{\gamma } f^{(k)}(a(x,p)) \sum _{\begin{array}{c} \alpha _1 + \cdots + \alpha _k = \alpha \\ \left| \alpha _j \right| >0 \end{array}} c_{\alpha _1\cdots \alpha _k} \partial _p^{\beta -\gamma } [\partial _x^{\alpha _1} a(x,p) \cdots \partial _x^{\alpha _k}a(x,p)]. \end{aligned} \end{aligned}$$

(A2)

In order to obtain the form stated in the corollary we need to use the Faà di Bruno formula on the terms

$$\begin{aligned} \partial _p^{\gamma } f^{(k)}(a(x,p)), \end{aligned}$$

(A3)

and we need to use Leibniz’s formula (multiple times) on the terms

$$\begin{aligned} \partial _p^{\beta -\gamma } [\partial _x^{\alpha _1} a(x,p) \cdots \partial _x^{\alpha _k}a(x,p)]. \end{aligned}$$

(A4)

If this is done, then by using some algebra the stated form can be obtained. The particular form of the index set ${\mathcal {I}}_k(\alpha ,\beta )$ also follows from the above calculations. $\square$