Propagation of Wave Packets for Systems Presenting Codimension One Crossings

Abstract

We analyze the propagation of wave packets through general Hamiltonian systems presenting codimension one eigenvalue crossings. The class of time-dependent Hamiltonians we consider is of general pseudodifferential form with subquadratic growth. It comprises Schrödinger operators with matrix-valued potential, as they occur in quantum molecular dynamics, but also covers matrix-valued models of solid state physics describing the motion of electrons in a crystal. We calculate precisely the non-adiabatic effects of the crossing in terms of a transition operator, whose action on coherent states can be spelled out explicitly.

1 Introduction

We consider systems of $N\ge 2$ equations of pseudodifferential form

$$\begin{aligned} i\varepsilon \partial _t \psi ^\varepsilon = {{\widehat{H}}}(t) \psi ^\varepsilon ,\;\;\psi ^\varepsilon _{|t=t_0}=\psi ^\varepsilon _0, \end{aligned}$$

(1)

where $(\psi ^\varepsilon _0)_{\varepsilon >0}$ is a bounded family in $L^2({{\mathbb {R}}}^d,{{\mathbb {C}}}^N)$. The Hamiltonian operator

$$\begin{aligned} {{\widehat{H}}}(t) = H(t,x,-i\varepsilon \nabla _x) \end{aligned}$$

is the semi-classical Weyl quantization of a time-dependent Hamiltonian

$$\begin{aligned} H: {{\mathbb {R}}}\times {{\mathbb {R}}}^{d}\times {{\mathbb {R}}}^d \rightarrow {{\mathbb {C}}}^{N\times N}, \quad (t,x,\xi )\mapsto H(t,x,\xi ), \end{aligned}$$

that is a smooth matrix-valued function and satisfies suitable growth conditions guaranteeing a well-defined and unique solution of the system. We denote with a “${{\widehat{\cdot }}}$” the semi-classical Weyl quantization, the definition of which is recalled in Sect. 2.1. Phase space variables are denoted by $z=(x,\xi )\in {{\mathbb {R}}}^{2d}$. The semi-classical parameter $\varepsilon >0$ is assumed to be small. The initial data are wave packets associated with one of the eigenspaces of the Hamiltonian matrix. That is,

$$\begin{aligned} \psi ^\varepsilon _0 = \widehat{\vec V_0}\,{\mathcal {WP}}^\varepsilon _{z_0}\varphi _0, \end{aligned}$$

(2)

where $\vec V_0(z)$ is a normalized eigenvector of the matrix $H(t_0,z)$ such that $\vec V_0:{{\mathbb {R}}}^{2d}\rightarrow {{\mathbb {C}}}^N$ is a smooth vector-valued function, and ${\mathcal {WP}}^\varepsilon _{z_0}\varphi _0$ denotes the wave packet transform of a Schwartz function $\varphi _0\in {{\mathcal {S}}}({{\mathbb {R}}}^d,{{\mathbb {C}}})$ for a phase space point $z_0=(x_0,\xi _0)\in {{\mathbb {R}}}^{2d}$,

$$\begin{aligned} {\mathcal {WP}}^{\varepsilon }_{z_0}\varphi _0(x)= \varepsilon ^{-d/4} \,{\mathrm{e}}^{i\xi _0\cdot (x-x_0)/\varepsilon } \varphi _0\!\left( \tfrac{x-x_0}{\sqrt{\varepsilon }}\right) . \end{aligned}$$

(3)

Such matrix systems arise from the analysis of scalar Schrödinger equations in an adiabatic limit, where higher energy levels are not taken into account due to a positive gap separating them from the part of the spectrum that corresponds to the eigenvalues of the Hamiltonian matrix H(t, z), see [28, 36] for example. Our aim is to describe the structure of the $N\times N$ system’s solutions in the case, when eigenvalues of the matrix H(t, z) coincide for some point $(t,z)\in {{\mathbb {R}}}\times {{\mathbb {R}}}^{2d}$, while all eigenvalues and eigenvectors retain their smoothness. The literature refers to them as codimension one crossings. In the presence of eigenvalue crossings the key assumption for space-adiabatic theory (the existence of a positive gap between eigenvalues) is violated, and the knowledge of the dynamics associated with one of the eigenvalues is not enough any more. Moreover, in addition to the necessity to include more than one eigenvalue for an effective dynamical description, also the non-adiabatic transitions between the coupled eigenspaces have to be properly resolved. These questions have already been addressed for special systems corresponding to the following physical settings: In his monograph [13, Chapter 5], G. Hagedorn investigated Schrödinger Hamiltonians with matrix-valued potential,

$$\begin{aligned} {{\widehat{H}}}_S = -\frac{\varepsilon ^2}{2}\Delta _x\, {\mathbb {I}}_{{{\mathbb {C}}}^N} + V(x),\quad V\in {{\mathcal {C}}}^\infty ({{\mathbb {R}}}^d,{{\mathbb {C}}}^{N\times N}). \end{aligned}$$

(4)

More recently, in [39], A. Watson and M. Weinstein studied models arising in solid state physics in the context of Bloch band decompositions,

$$\begin{aligned} {{\widehat{H}}} _A= A(-i\varepsilon \nabla _x) + W (x) {\mathbb {I}}_{{{\mathbb {C}}}^2},\quad A\in {{\mathcal {C}}}^\infty ({{\mathbb {R}}}^d,{{\mathbb {C}}}^{N\times N}),\quad W\in {\mathcal C}^\infty ({{\mathbb {R}}}^d,{{\mathbb {C}}}). \end{aligned}$$

(5)

In both settings, the eigenvalues of the matrices V(x), $x\in {{\mathbb {R}}}^d$, respectively $A(\xi )$, $\xi \in {{\mathbb {R}}}^d$, have a codimension one crossing of their eigenvalues.

We develop here a new analytical method which applies for general matrix-valued Hamiltonians with a codimension one crossings of eigenvalues, which might also have multiplicity larger than one. In particular, we give a general and unified computation of the transfer operator which describes the non-adiabatic interactions due to the crossing. The non-adiabatic transition formulae of Corollary 3.9 are explicit and derived in a self-contained and more accessible way than the previous ones in the literature. Due to their explicit form, they can directly be applied to numerical simulations based on thawed Gaussians that are currently investigated in chemical physics, see for example [2, 24, 25, 37] or the recent review [38].

As another byproduct of our method, we also obtain an effortless generalization of the semi-classical Herman–Kluk approximation to the case of systems with eigenvalue gaps (see Corollary 3.5 below). We expect that a refinement of the present error analysis is possible, such that our codimension one result can be extended to the Herman–Kluk framework as well. This is work in progress, which might also contribute to the algorithmical development of superpositions of surface-hopping approximations using frozen (or thawed) Gaussian wave packets in the spirit of [40], see also [21].

We assume that the matrix H(t, z) has a smooth eigenvalue $h_1(t,z)$, the eigenspace of which admits a smooth eigenprojector $\Pi _1(t,z)$, that is,

$$\begin{aligned} H(t,z) \Pi _1(t,z) = \Pi _1(t,z) H(t,z) = h_1(t,z) \Pi _1(t,z). \end{aligned}$$

We shall consider two situations, depending on whether the eigenvalue $h_1(t,z)$ crosses another smooth eigenvalue $h_2(t,z)$ or not. Because we assume the Hamiltonian matrix H(t, z) to be independent of $\varepsilon$, then, in the gap situation, the eigenvalue $h_1(t,z)$ is separated from $h_2(t,z)$ by a gap larger than some fixed positive real number $\delta _0>0$ that is of order one with respect to the semi-classical parameter $\varepsilon$. In the second case, the smooth crossing case, both eigenvalues are smooth and have smooth eigenprojectors. Note that it is not the case in general since eigenvalues may develop singularities at the crossing; however, we do not consider those situations here. We shall also assume that H(t, z) has no other eigenvalues since one can reduce to that case as soon as the set of these two eigenvalues is separated from the remainder of the spectrum of the matrix H(t, z) by a gap (uniformly in t and z).

The gap situation is well understood and corresponds to adiabatic situations that have been studied by several authors (see in particular the lecture notes [36] of S. Teufel or the memoirs [28] of A. Martinez and V. Sordoni and note that the thesis [3] is devoted to wave packets in the adiabatic situation). For avoided crossings, the coupling of the gap and the semi-classical parameter violates the key requirement for adiabatic decoupling. The resulting non-adiabatic dynamics have been studied for wave packets by G. Hagedorn and A. Joye in [14, 15] and for the Wigner function of general initial data in [22]. Smooth crossings have been less studied so far. Some results on the subject focus on the evolution at leading order in $\varepsilon$ of quadratic quantities of the wave function for initial data which are not necessarily wave packets (see [9, 20] and the references therein). The main results devoted to wave packet propagation through smooth eigenvalue crossings are the references [13] and [39] mentioned above. There, for the specific Hamiltonian operators (4) and (5), respectively, the authors gave rather explicit descriptions of the propagated wave packet, exhibiting non-adiabatic transitions that occur at the crossing between the two eigenvalues that are of order $\sqrt{\varepsilon }$. As in these contributions, we assume that the crossing set

$$\begin{aligned} \Upsilon =\{(t,z)\in {{\mathbb {R}}}^{2d+1},\; h_1(t,z)=h_2(t,z)\} \end{aligned}$$

(6)

of two smooth eigenvalues $h_1(t,z)$ and $h_2(t,z)$ is a codimension one manifold.

Our main result (Theorem 3.8 below) makes the following assumptions for the initial data $\psi ^\varepsilon _0$. Let $v^\varepsilon _0$ be a wave packet centered in a phase space point $z_0$, that is,

$$\begin{aligned} v^\varepsilon _0 = {\mathcal {WP}}_{z_0}^\varepsilon \varphi _0\quad \text {for some}\quad \varphi _0\in {{\mathcal {S}}}({{\mathbb {R}}}^d,{{\mathbb {C}}}). \end{aligned}$$

Let $\vec V_0(z)$ be a smooth normalized eigenvector of $H(t_0,z)$, that is, $\vec V_0\in {{\mathcal {C}}}^\infty ({{\mathbb {R}}}^{2d},{{\mathbb {C}}}^N)$ is a smooth vector-valued function that satisfies in a neighborhood U of $z_0$,

$$\begin{aligned} H(t_0,z)\vec V_0(z)=h_1(t_0,z)\vec V_0(z)\quad \text {for all}\quad z\in U. \end{aligned}$$

Then, we define the initial wave packet according to (2). Let $z_1(t)$ denote the classical trajectory associated with the eigenvalue $h_1(t,z)$ initiated in wave packet’s core $z_0$. Let $t^\flat >t_0$ be the first time, when the trajectory $z_1(t)$ meets the crossing set $\Upsilon$, and let $z_2(t)$ denote the classical trajectory associated with the second eigenvalue $h_2(t,z)$, that is initiated in the crossing point $z_1(t^\flat )$. That is,

$$\begin{aligned} \dot{z}_1(t)&= J \partial _z h_1(t,z_1(t)),\quad z_1(t_0)=z_0,\\ \dot{z}_2(t)&= J \partial _z h_2(t,z_2(t)),\quad z_2(t^\flat )=z_1(t^\flat ). \end{aligned}$$

Then, the solution of system (1) satisfies

$$\begin{aligned}\psi ^\varepsilon (t) = \widehat{ \vec V_1(t)} {\mathcal {WP}}^\varepsilon _{z_1(t)} (\varphi _1^0(t)+\sqrt{\varepsilon }\varphi _1^1(t)) + \sqrt{\varepsilon }\mathbf{1}_{t>t^\flat } \widehat{ \vec V_2(t)} {\mathcal {WP}}^\varepsilon _{z_2(t)} \varphi _2(t) +o(\sqrt{\varepsilon }),\end{aligned}$$

where the profiles of the wave packets

$$\begin{aligned} {\mathcal {WP}}^\varepsilon _{z_1(t)} (\varphi _1^0(t)+\sqrt{\varepsilon }\varphi _1^1(t))\quad \text {and}\quad {\mathcal {WP}}^\varepsilon _{z_2(t)} \varphi _2(t) \end{aligned}$$

are Schwartz functions $\varphi _1^0(t)$, $\varphi _1^1(t)$, and $\varphi _2(t)$, that solve $\varepsilon$-independent PDEs on $[t_0,t^\flat ]$ and $[t^\flat ,t_0+T]$, respectively, that are explicitly given in terms of the classical dynamics associated with the eigenvalues $h_1(t,z)$ and $h_2(t,z)$. The profile associated with the second eigenvalue is generated by the leading order profile of the first eigenvalue via

$$\begin{aligned} \varphi _2(t^\flat ) ={{\mathcal {T}}}^\flat \varphi _1^0(t^\flat ),\end{aligned}$$

where the non-adiabatic transfer operator ${{\mathcal {T}}}^\flat$ is a metaplectic transform (which implies that the structure of Gaussian states is preserved, see Corollary 3.9). The two families $\vec V_1(t,z)$ and $\vec V_2(t,z)$ are smooth normalized eigenvectors for $h_1(t,z)$ and $h_2(t,z)$, respectively, that are obtained by parallel transport.

We point out that, in the uniform gap case, an initial datum that is associated with one eigenvalue issues a solution at time t that is associated with the same eigenvalue up to terms of order $\varepsilon$, which is the standard order of the adiabatic approximation, while for smooth crossings a perturbative term of order $\sqrt{\varepsilon }$ associated with the other eigenvalue has to be taken into account for an order $\varepsilon$ approximation.

Before giving a more precise statement of the result, we mention that the propagation of wave packets was also studied for nonlinear systems in [6, 7, 16, 17], including situations with avoided crossings [16]. However, nonlinear systems with codimension one crossings have not yet been analysed. We expect that our result can be extended when imposing appropriate assumptions on the nonlinearity.

2 Preliminary Results

In this section, we introduce the relevant function spaces for the unitary propagation and also recall some known results on wave packets for scalar evolution equations.

2.1 Function spaces and quantization

Let $a\in {{\mathcal {C}}}^\infty ({{\mathbb {R}}}^{2d})$ be a smooth scalar-, vector- or matrix-valued function with adequate control on the growth of derivatives. Then, the Weyl operator ${{\widehat{a}}} = {\mathrm{op}}^w_\varepsilon (a)$ is defined by

$$\begin{aligned}{\mathrm{op}}^w_\varepsilon (a)f(x):= {{\widehat{a}}} f(x) := (2\pi \varepsilon )^{-d} \int _{{{\mathbb {R}}}^{2d}} a\!\left( {x+y\over 2}, \xi \right) {\mathrm{e}}^{i\xi \cdot (x-y)/\varepsilon } f(y) \,dy\, d\xi \end{aligned}$$

for all $f\in {{\mathcal {S}}}({{\mathbb {R}}}^d)$. According to [29], the unitary propagator ${{\mathcal {U}}}^\varepsilon _H(t,t_0)$ associated with the Hamiltonian operator ${{\widehat{H}}}(t)$,

$$\begin{aligned} i\varepsilon \,\partial _t\, {{\mathcal {U}}}^\varepsilon _H(t,t_0) = {{\widehat{H}}}(t)\, {{\mathcal {U}}}^\varepsilon _H(t,t_0),\quad {{\mathcal {U}}}^\varepsilon _H(t_0,t_0) = {\mathbb {I}}_{L^2({{\mathbb {R}}}^d)}, \end{aligned}$$

is well defined when the map $(t,z)\mapsto H(t,z)$ is in ${\mathcal C}^\infty ({{\mathbb {R}}}\times {{\mathbb {R}}}^{2d},{{\mathbb {C}}}^{N\times N})$, valued in the set of self-adjoint matrices and that it has subquadratic growth, i.e.

$$\begin{aligned} \forall \alpha \in {{\mathbb {N}}}^{2d} ,\;\;|\alpha |\ge 2,\;\; \exists C_\alpha >0,\;\;\sup _{(t,z)\in {{\mathbb {R}}}\times {{\mathbb {R}}}^{2d}}\Vert \partial ^\alpha _z H(t,z) \Vert _{{{\mathbb {C}}}^{N\times N}}\le C_\alpha . \end{aligned}$$

(7)

These assumptions guarantee the existence of solutions to equation (1) in $L^2({{\mathbb {R}}}^d,{{\mathbb {C}}}^N)$ and, more generally, in the functional spaces

$$\begin{aligned}\Sigma _\varepsilon ^k({{\mathbb {R}}}^d)=\{ f\in L^2({{\mathbb {R}}}^d),\;\;\forall \alpha ,\beta \in {{\mathbb {N}}}^d,\;\; |\alpha |+|\beta | \le k,\;\; x^\alpha (\varepsilon \partial _x)^\beta f\in L^2({{\mathbb {R}}}^d)\}\end{aligned}$$

endowed with the norm

$$\begin{aligned}\Vert f\Vert _{\Sigma ^k_\varepsilon } = \sup _{|\alpha |+|\beta | \le k}\Vert x^\alpha (\varepsilon \partial _x)^\beta f\Vert _{L^2}.\end{aligned}$$

We note that also with respect to the $\Sigma _\varepsilon ^k({{\mathbb {R}}}^d)$ spaces, the unitary propagator ${{\mathcal {U}}}^\varepsilon _H(t,t_0)$ is $\varepsilon$-uniformly-bounded in the sense, that for all $T>0$ there exists $C>0$ such that

$$\begin{aligned} \sup _{t\in [t_0,t_0+T]}\Vert {{\mathcal {U}}}^\varepsilon _H(t,t_0)\Vert _{{\mathcal L}(\Sigma ^k_\varepsilon )} \,\le \, C. \end{aligned}$$

Remark 2.1

The analysis below could apply to more general settings as long as the classical quantities are well-defined in finite time with some technical improvements that are not discussed here.

2.2 Scalar propagation and scalar classical quantities

The most interesting property of the coherent states is the stability of their structure through evolution, which can be described by means of classical quantities. Note that for all $z\in {{\mathbb {R}}}^{2d}$ and $k\in {{\mathbb {N}}}$, the operator $\varphi \mapsto \mathcal {WP}^\varepsilon _{z}\varphi$ is a unitary map in $L^2({{\mathbb {R}}}^d)$ which maps continuously $\Sigma ^1_k$ into $\Sigma ^\varepsilon _k$ with a continuous inverse. Other elementary properties of the wave packet transform are listed in Lemma A.1. We shall use the notation

$$\begin{aligned} J = \begin{pmatrix}0 &{} {\mathbb {I}}_{{{\mathbb {R}}}^d}\\ -{\mathbb {I}}_{{{\mathbb {R}}}^d} &{} 0\end{pmatrix}. \end{aligned}$$

(8)

For smooth functions $f,g\in {{\mathcal {C}}}^\infty ({{\mathbb {R}}}^{2d})$, that might be scalar-, vector- or matrix-valued, we denote the Poisson bracket by

$$\begin{aligned} \{f,g\}:= J\nabla f\cdot \nabla g= \sum _{j=1}^d \left( \partial _{\xi _j} f \partial _{x_j} g- \partial _{x_j} f \partial _{\xi _j} g\right) . \end{aligned}$$

Let $h:{{\mathbb {R}}}\times {{\mathbb {R}}}^{2d}\rightarrow {{\mathbb {R}}}$, $(t,z)\mapsto h(t,z)$ be a smooth function of subquadratic growth(7).We now review the main tools for the semi-classical propagation of wave-packets. We let $z(t) = (q(t),p(t))$ denote the classical Hamiltonian trajectory issued from a phase space point $z_0$ at time $t_0$, that is defined by the ordinary differential equation

$$\begin{aligned}\dot{z}(t) = J \partial _z h(t,z(t)),\;\; z(t_0)=z_0.\end{aligned}$$

The trajectory $z(t) = z(t,t_0,z_0)$ depends on the initial datum and defines via $\Phi _h^{t,t_0}(z_0) = z(t,t_0,z_0)$ the associated flow map $z\mapsto \Phi _h^{t,t_0}(z)$ of the Hamiltonian function h. We will also use the trajectory’s action integral

$$\begin{aligned} S(t,t_0,z_0) = \int _{t_0}^t \left( p(s)\cdot \dot{q}(s)-h(s,z(s)) \right) ds, \end{aligned}$$

(9)

and the Jacobian matrix of the flow map

$$\begin{aligned} F(t,t_0,z_0) = \partial _z \Phi _h^{t,t_0}(z_0). \end{aligned}$$

Note that $F(t,t_0,z_0)$ is a symplectic $2d\times 2d$ matrix, that satisfies the linearized flow equation

$$\begin{aligned} \partial _t F(t,t_0,z_0) = J {\mathrm{Hess}}_zh(t,z(t)) \, F(t,t_0,z_0),\;\;F(t_0,t_0,z_0) = {\mathbb {I}}_{{{\mathbb {R}}}^{2d}}. \end{aligned}$$

(10)

We denote its blocks by

$$\begin{aligned} F(t,t_0,z_0) = \begin{pmatrix} A(t,t_0,z_0) &{}B(t,t_0,z_0) \\ C(t,t_0,z_0) &{}D(t,t_0,z_0)\end{pmatrix}. \end{aligned}$$

(11)

In a last step, we define the corresponding unitary evolution operator, the metaplectic transformation, that acts on square integrable functions in $L^2({{\mathbb {R}}}^d)$.

Definition 2.2

(Metaplectic transformation) Let $h:{{\mathbb {R}}}\times {{\mathbb {R}}}^{2d}\rightarrow {{\mathbb {R}}}$ be a smooth function of subquadratic growth (7). Let $t,t_0\in {{\mathbb {R}}}$ and $z_0\in {{\mathbb {R}}}^{2d}$. Let $F(t,t_0,z_0)$ be the solution of the linearized flow equation (10) associated with the Hamiltonian function h(t). Then, we call the unitary operator

$$\begin{aligned} {{\mathcal {M}}}[F(t,t_0,z_0)] : \; \varphi _0\mapsto \varphi (t) \end{aligned}$$

that associates with an initial datum $\varphi _0$ the solution at time t of the Cauchy problem

$$\begin{aligned} i\partial _t \varphi = {\mathrm{op}}^w_1({\mathrm{Hess}}_z h(t, z(t))z\cdot z) \varphi ,\;\; \varphi (t_0)=\varphi _0, \end{aligned}$$

the metaplectic transformation associated with the matrix $F(t,t_0,z_0)$.

Using these three $\varepsilon$-independent building blocks – the classical trajectories, the action integrals, and the metaplectic transformations associated with the linearized flow map – we can approximate the action of the unitary propagator

$$\begin{aligned} i\varepsilon \partial _t\, {{\mathcal {U}}}_h^\varepsilon (t,t_0) = {\mathrm{op}}^w_\varepsilon (h(t))\, {{\mathcal {U}}}_h^\varepsilon (t,t_0),\quad {{\mathcal {U}}}_h^\varepsilon (t_0,t_0) = {\mathbb {I}}_{L^2({{\mathbb {R}}}^d)} \end{aligned}$$

on wave packets as follows.

Proposition 2.3

[8, §4.3] Consider a smooth scalar Hamiltonian h(t) of subquadratic growth (7). Let $T>0$, $k\ge 0$, $z_0\in {{\mathbb {R}}}^{2d}$, and $\varphi _0\in {{\mathcal {S}}}({{\mathbb {R}}}^d)$. Then, there exists a positive constant $C>0$ such that

$$\begin{aligned} \sup _{t\in [t_0,t_0+T]}\left\| {{\mathcal {U}}}_h^\varepsilon (t,t_0) \mathcal {WP}^\varepsilon _{z_0}\varphi _0 - {\mathrm{e}}^{\frac{i}{\varepsilon }S(t,t_0,z_0)} \mathcal {WP}^\varepsilon _{z(t)} \varphi ^\varepsilon (t)\right\| _{\Sigma _\varepsilon ^k} \le C\varepsilon , \end{aligned}$$

where the profile function $\varphi ^\varepsilon (t)$ is given by

$$\begin{aligned} \varphi ^\varepsilon (t) = {{\mathcal {M}}}[F(t,t_0,z_0)] \left( 1+\sqrt{\varepsilon }\, b_1(t,t_0,z_0)\right) \varphi _0, \end{aligned}$$

(12)

and the correction function $b_1(t,t_0,z_0)$ satisfies

$$\begin{aligned} b_1(t,t_0,z_0) \varphi _0 = \sum _{\vert \alpha \vert =3} \frac{1}{\alpha !} \frac{1}{i} \int _{t_0}^{t} \partial _z^\alpha h(s,z(s))\, {\mathrm{op}}_1^w[(F(s,t_0,z_0)z)^{\alpha }] \,\varphi _0\,ds. \end{aligned}$$

(13)

The constant $C=C(T,k,z_0,\varphi _0)>0$ is independent of $\varepsilon$ but depends on derivative bounds of the flow map $\Phi ^{t,t_0}_h(z_0)$ for $t\in [t_0,t_0+T]$ and the $\Sigma ^{k+3}_1$-norm of the initial profile $\varphi _0$.

Let us discuss the especially interesting case of initial Gaussian states. Gaussian states are wave packets with complex-valued Gaussian profiles, whose covariance matrix is taken in the Siegel half-space ${{\mathfrak {S}}}^ +(d)$ of $d\times d$ complex-valued symmetric matrices with positive imaginary part,

$$\begin{aligned} {{\mathfrak {S}}}^+(d) = \left\{ \Gamma \in {{\mathbb {C}}}^{d\times d},\ \Gamma =\Gamma ^\tau ,\ \text {Im}\Gamma >0\right\} . \end{aligned}$$

With $\Gamma \in {{\mathfrak {S}}}^+(d)$ we associate the Gaussian profile

$$\begin{aligned} g^\Gamma (x) := c_\Gamma \, {\mathrm{e}}^{\frac{i}{2}\Gamma x\cdot x},\quad x\in {{\mathbb {R}}}^d, \end{aligned}$$

(14)

where $c_\Gamma =\pi ^{-d/4} {\mathrm{det}}^{1/4}(\text {Im}\Gamma )$ is a normalization constant in $L^2({{\mathbb {R}}}^d)$. It is a non-zero complex number whose argument is determined by continuity according to the working environment. By Proposition 2.3, the Gaussian states remain Gaussian under the evolution by ${\mathcal U}^\varepsilon _h(t,t_0)$. Indeed, for $\Gamma _0\in {{\mathfrak {S}}}^+(d)$, we have

$$\begin{aligned} {{\mathcal {M}}}[F(t,t_0,z_0)] g^{\Gamma _0}= g^{\Gamma (t,t_0,z_0)}, \end{aligned}$$

where the width $\Gamma (t,t_0,z_0)\in {\mathfrak {S}}^+(d)$ and the corresponding normalization $c_{\Gamma (t,t_0,z_0)}$ are determined by the initial width $\Gamma _0$ and the Jacobian $F(t,t_0,z_0)$ according to

$$\begin{aligned} \Gamma (t,t_0,z_0)= & {} (C(t,t_0,z_0)+ D(t,t_0,z_0)\Gamma _0)(A(t,t_0,z_0) +B(t,t_0,z_0)\Gamma _0)^{-1}\nonumber \\ c_{\Gamma (t,t_0,z_0)}= & {} c_{\Gamma _0}\,{\mathrm{det}}^{-1/2}(A(t,t_0,z_0)+B(t,t_0,z_0)\Gamma _0). \end{aligned}$$

(15)

The branch of the square root in ${\mathrm{det}}^{-1/2}$ is determined by continuity in time.

The semiclassical wave packets used by G. Hagedorn in [11, 12] are Gaussian wave packets, which are multiplied with a specifically chosen complex-valued polynomial function, that depends on the Gaussian’s width matrix. If $A\in {\mathcal {C}}^\infty ({{\mathbb {R}}}^{2d},{{\mathbb {C}}})$ is an arbitrary polynomial function, then ${\mathrm{op}}^w_1(A) g^{\Gamma _0}$ is the product of a polynomial times a Gaussian, and we can again describe the action of the metaplectic transformation explictly. Indeed, by Egorov’s theorem (which is exact here),

$$\begin{aligned} {{\mathcal {M}}}[F(t,t_0,z_0)]({\mathrm{op}}_1^w(A)g^{\Gamma _0})&= {\mathrm{op}}_1^w(A\circ F(t,t_0,z_0)) {{\mathcal {M}}}[F(t,t_0,z_0)] g^{\Gamma _0} \\&={\mathrm{op}}_1^w(A\circ F(t,t_0,z_0)) g^{\Gamma (t,t_0,z_0)}. \end{aligned}$$

In particular, functions that are polynomials times a Gaussian remain of the same form under the evolution, even the polynomial degree is preserved.

3 Precise Statement of the Results

We now present our main results, that extend the previous theory of wave packet propagation for scalar evolution equations to systems associated with Hamiltonians that have smooth eigenvalues and eigenprojectors.

3.1 Vector-valued wave packets and parallel transport

We consider initial data that are vector-valued wave packets associated with a normalized eigenvector of the Hamiltonian matrix $H(t_0,z)$ as given in (2). The evolution of such a function also requires an appropriate evolution of its vector part, which we refer to as parallel transport. The following construction generalizes [6, Proposition 1.9], which was inspired by the work of G. Hagedorn, see [13, Proposition 3.1]. Let us denote the complementary orthogonal projector by $\Pi ^\perp (t,z) = {\mathbb {I}}_{{{\mathbb {C}}}^N}-\Pi (t,z)$ and assume that

$$\begin{aligned} H(t,z) = h(t,z)\Pi (t,z) + h^\perp (t,z)\Pi ^\perp (t,z) \end{aligned}$$

(16)

with the second eigenvalue given by $h^\perp (t,z) = \mathrm{tr}(H(t,z)) - h(t,z).$ The situation with more than two eigenvalues of constant multiplicity is a generalization of this case and can be treated similarly.

We introduce the auxiliary matrices

$$\begin{aligned} \Omega (t,z)&=-\tfrac{1}{2}\big (h(t,z)-h^\perp (t,z)\big )\Pi (t,z)\{\Pi ,\Pi \}(t,z)\Pi (t,z) , \end{aligned}$$

(17)

$$\begin{aligned} K(t,z)&= \Pi ^\perp (t,z)\left( \partial _t\Pi (t,z)+\{h,\Pi \}(t,z)\right) \Pi (t,z), \end{aligned}$$

(18)

$$\begin{aligned} \Theta (t,z)&= i\Omega (t,z) + i(K-K^*)(t,z), \end{aligned}$$

(19)

that are smooth and satisfy algebraic properties detailed in Lemma B.1 below. In particular, $\Omega$ is skew-symmetric and $\Theta$ is self-adjoint, $\Omega = -\Omega ^*\quad$ and $\Theta = \Theta ^*$. We note, that for the Schrödinger and the Bloch Hamiltonian,

$$\begin{aligned} H_S(z) = \tfrac{1}{2}|\xi |^2\,{\mathbb {I}}_{{{\mathbb {C}}}^N} + V(x)\quad \text {and}\quad H_A(z) = \begin{pmatrix} 0 &{} \xi _1+i\xi _2 \\ \xi _1-i\xi _2 &{} 0\end{pmatrix} + W(x){\mathbb {I}}_{{{\mathbb {C}}}^2}, \end{aligned}$$

the skew-symmetric $\Omega$-matrix vanishes, that is, $\Omega _S = 0$ and $\Omega _A = 0$. For Dirac Hamiltonians with electromagnetic potential or Hamiltonians that describe acoustic waves in elastic media, the $\Omega$-matrix need not vanish.

Proposition 3.1

Let H(t, z) be a smooth Hamiltonian with values in the set of self-adjoint $N\times N$ matrices that is of subquadratic growth (7) and has a smooth spectral decomposition (16). We assume that both eigenvalues are of subquadratic growth as well. We consider $\vec V_0\in {{\mathcal {C}}}_0^\infty ({{\mathbb {R}}}^{2d},{{\mathbb {C}}}^N)$ and $z_0\in {{\mathbb {R}}}^{2d}$ such that there exists a neighborhood U of $z_0$ such that for all $z\in U$

$$\begin{aligned}\vec V_0 (z)=\Pi (t_0,z)\vec V_0(z)\quad \text {and}\quad \Vert \vec V_0(z)\Vert _{{{\mathbb {C}}}^N} = 1.\end{aligned}$$

Then, there exists a smooth normalized vector-valued function $\vec V(t,t_0)$ satisfying

$$\begin{aligned}\vec V(t,t_0,z)= \Pi (t,z)\vec V(t,t_0,z)\quad \text {for all}\quad z\in \Phi _h^{t,t_0}(U),\end{aligned}$$

such that for all $t\in {{\mathbb {R}}}$ and $z\in \Phi _h^{t,t_0}(U)$,

$$\begin{aligned} \partial _t \vec V(t,t_0,z) + \{h, \vec V\} (t,t_0,z) = -i\Theta (t,z)\vec V (t,t_0,z),\;\; \vec V(t_0,t_0,z) = \vec V_0(z). \end{aligned}$$

(20)

Proposition 3.1 is proved in Appendix C. Note that it does not require any gap condition for the eigenvalues. We will use it in the crossing situation, with smooth eigenvalues and eigenprojectors.

The parallel transport is enough to describe at leading order the propagation of wave-packets associated with an eigenvalue h(t, z) of the matrix H(t, z), that is uniformly separated from the remainder of the spectrum in the sense that there exists $\delta >0$ such that for all $(t,z)\in {{\mathbb {R}}}\times {{\mathbb {R}}}^{2d}$,

$$\begin{aligned} {\mathrm{dist}}\left( h(t,z),\sigma (H(t,z))\setminus \{h(t,z)\}\right) > \delta . \end{aligned}$$

(21)

Note that, this gap assumption implies the existence of a Cauchy contour ${{\mathcal {C}}}$ in the complex plane, such that its interior only contains the eigenvalue h(t, z) and no other eigenvalues of H(t, z). Then, one can write the eigenprojector as $\Pi (t,z)=-{1\over 2\pi i}\oint _{{\mathcal {C}}} (H(t,z)-\zeta )^{-1} d\zeta ,$ which implies that the projector $\Pi (t,z)$ inherits the smoothness properties of the Hamiltonian H(t, z) in the presence of an eigenvalue gap. However, if the symbol $\Pi$ is of course of matrix norm 1, its derivatives may grow as |z| goes to infinity and we shall make assumption below (see (23)) in order to guarantee that this growth is at most polynomial. Since the pioneering work of T. Kato [23], numerous studies have been devoted to this adiabatic situation (see for example [28, 30, 31, 36] and references therein). One can derive from these results the following statement of adiabatic decoupling.

Theorem 3.2

[6, 28, 36] Let H(t, z) be a smooth Hamiltonian with values in the set of self-adjoint $N\times N$ matrices and h(t, z) a smooth eigenvalue of H(t, z). Assume that both H(t, z) and h(t, z) are of subquadratic growth (7) and that there exists an eigenvalue gap as in Assumption (21). Consider initial data $(\psi ^\varepsilon _0)_{\varepsilon >0}$ that are wave packets as in (2). Then, for all $T>0$, there exists $C>0$ such that $\psi ^\varepsilon (t) = {\mathcal {U}}^\varepsilon _H(t,t_0)\psi ^\varepsilon _0$ satisfies the estimate

$$\begin{aligned}\sup _{t\in [t_0,t_0+T]}\left( \left\| \widehat{\Pi ^\perp (t)}\psi ^\varepsilon (t) \right\| _{L^2({{\mathbb {R}}}^d)} + \left\| \psi ^\varepsilon (t) - \widehat{\vec V(t)} v^\varepsilon (t) \right\| _{L^2({{\mathbb {R}}}^d)} \right) \le C \varepsilon \end{aligned}$$

where $v^\varepsilon (t)= {{\mathcal {U}}}^\varepsilon _h(t,t_0) v^\varepsilon _0$ and $\vec V(t)$ is determined by Proposition 3.1. Besides, if there exists $k\in {{\mathbb {N}}}$ such that $(\psi ^\varepsilon _0)_{\varepsilon >0}$ is a bounded family in the space $\Sigma _\varepsilon ^k$, then the convergence above holds in $\Sigma ^k_\varepsilon$.

Theorem 3.2 is obtained as an intermediate result in the proof of Proposition 4.1, see Sect. 4 below. There, we perform a refined analysis of the adiabatic approximation that explicitly accounts for the size of the eigenvalue gap. We note that the estimate of Theorem 3.2 is unchanged, when allowing for perturbations of the initial data that are of order $\varepsilon$ in $L^2({{\mathbb {R}}}^d)$ or $\Sigma ^k_\varepsilon$, respectively. We also note, that in general the operator ${{\widehat{\Pi }}}(t)$ is not a projector, but coincides at order $\varepsilon$ with the superadiabatic operators constructed in [28, 36], which are projectors (see also Appendix B).

Remark 3.3

The result of Theorem 3.2 can be generalized by means of superadiabatic projectors, showing that $\psi ^\varepsilon (t)$ can be approximated at any order by an asymptotic sum of wave packets. The precise time evolution of coherent states was studied in the adiabatic setting in [3, 28, 32]. These results are obtained via an asymptotic quantum diagonalization, in the spirit of the construction of the superadiabatic projectors of [28, 36].

Theorem 3.2 allows a semi-classical description of the dynamics of an initial wave packet, that is associated with a gapped eigenvalue. The building blocks are the scalar classical quantities introduced in Sect. 2.2 and the parallel transport of eigenvectors given in Proposition 3.1. This is stated in the next Corollary; our aim is to derive a similar description for systems presenting a codimension one crossing.

Corollary 3.4

(Adiabatic wave packet) In the situation of Theorem 3.2, for any $T>0$, $k\in {{\mathbb {N}}}$, $z_0\in {{\mathbb {R}}}^{2d}$, and $\varphi _0\in {\mathcal S}({{\mathbb {R}}}^d,{{\mathbb {C}}})$, there exists a constant $C>0$ such

$$\begin{aligned} \sup _{t\in [t_0,t_0+T]}\left\| {{\mathcal {U}}}^\varepsilon _{H}(t,t_0)\, \widehat{\vec V_0}\, {\mathcal {WP}}^\varepsilon _{z_0}\varphi _0 - {\mathrm{e}}^{i S(t,t_0,z_0)/\varepsilon } \, \widehat{\vec V(t,t_0)}\, {\mathcal {WP}}^\varepsilon _{\Phi _{h}^{t,t_0}(z_0)}\varphi ^\varepsilon (t) \right\| _{\Sigma ^k_\varepsilon } \le C\varepsilon , \end{aligned}$$

where the profile $\varphi ^\varepsilon (t)$ is given by (12), and all the classical quantities are associated with the eigenvalue h(t).

We close this section devoted to gapped systems by formulating another semi-classical consequence of adiabatic theory using the Herman–Kluk propagator. This approximate propagator has first been proposed by M. Herman and E. Kluk in [18] for scalar Schrödinger equations and later used as a numerical method for quantum dynamics in the semi-classical regime, see for example [26] or more recently [5, 27] with references therein. The rigorous mathematical analysis of the Herman–Kluk propagator is due to [33, 35]. The starting point of this approximation is the wave packet inversion formula

$$\begin{aligned} \psi (x)= (2\pi \varepsilon )^{-d} \int _{z\in {{\mathbb {R}}}^{2d}}\langle g^\varepsilon _z,\psi \rangle g^\varepsilon _z (x) dz \end{aligned}$$

that allows to write any square integrable function $\psi \in L^2({{\mathbb {R}}}^d)$ as a continuous superposition of Gaussian wave packets of unit width,

$$\begin{aligned} g^\varepsilon _z(x) \ =\ \mathcal {WP}_z^\varepsilon (g^{i{\mathbb {I}}})(x) \ =\ (\pi \varepsilon )^{-d/4} {\mathrm{e}}^{-|x-q|^2/(2\varepsilon ) + i p\cdot (x-q)/\varepsilon }. \end{aligned}$$

The semi-classical description of unitary quantum dynamics within the framework of Gaussians of fixed unit width becomes possible due to a reweighting factor, the so-called Herman–Kluk prefactor,

$$\begin{aligned} a_h(t,t_0,z) = 2^{-d/2} \ {\mathrm{det}}^{1/2} \left( A(t,t_0,z)+D(t,t_0,z)+i(C(t,t_0,z)-B(t,t_0,z)) \right) , \end{aligned}$$

which is solely determined by the blocks of the Jacobian matrix of the classical flow map. The resulting propagator

$$\begin{aligned} \psi \,\mapsto \, {{\mathcal {I}}}_h^\varepsilon (t,t_0)\psi \,=\, (2\pi \varepsilon )^{-d} \int _{{{\mathbb {R}}}^{2d}} \langle g^\varepsilon _z,\psi \rangle a_h(t,t_0,z) {\mathrm{e}}^{i S(t,t_0,z)/\varepsilon } g^\varepsilon _{\Phi ^{t,t_0}_h(z)} dz \end{aligned}$$

provides an order $\varepsilon$ approximation to the scalar unitary propagator ${{\mathcal {U}}}^\varepsilon _h(t,t_0)$ in operator norm. Combining [35, Proposition 2 and Theorem 2] or [33, Theorem 1.2] with our previous results we obtain a Herman–Kluk approximation for gapped systems.

Corollary 3.5

(Adiabatic Herman–Kluk approximation) In the situation of Theorem 3.2, for all $T>0$ there exists a constant $C=C(T)>0$ such that

$$\begin{aligned} \sup _{t\in [t_0,t_0+T]}\left\| {{\mathcal {U}}}_H^\varepsilon (t,t_0)\psi ^\varepsilon _0 - {{\mathcal {I}}}_H^\varepsilon (t,t_0)\psi ^\varepsilon _0 \right\| _{L^2({{\mathbb {R}}}^d)} \le C\varepsilon , \end{aligned}$$

where the vector-valued Herman–Kluk propagator is defined by

$$\begin{aligned} {{\mathcal {I}}}_H^\varepsilon (t,t_0)\psi ^\varepsilon _0 = (2\pi \varepsilon )^{-d} \int _{{{\mathbb {R}}}^{2d}} \langle g^\varepsilon _z,v^\varepsilon _0\rangle \vec A(t,t_0,z) {\mathrm{e}}^{i S(t,t_0,z)/\varepsilon } g^\varepsilon _{\Phi ^{t,t_0}_h(z)} dz. \end{aligned}$$

The prefactor $\vec A(t,t_0,z)$ is given by $\vec A(t,t_0,z) = \vec V(t,t_0,z) a_h(t,t_0,z)$, where $a_h(t,t_0,z)$ is the Herman–Kluk prefactor associated with the eigenvalue h(t).

Theorem 3.2 formulates adiabatic decoupling for a single eigenvalue that is uniformly separated from the remainder of the spectrum. As it is well-known, adiabatic theory also extends to the situation where a subset of eigenvalues is isolated from the remainder of the spectrum. For this reason, in the next section, we reduce our analysis to the case of matrices with two eigenvalues that coincide on a hypersurface $\Upsilon$ of codimension one and differ away from it. We explicitly describe the dynamics of wave packets through this type of crossings, which is our main result.

3.2 Main result: propagation of wave packets through codimension one crossings

We assume that H(t, z) has two smooth eigenvalues $h_1$ and $h_2$ that cross on a hypersurface $\Upsilon$ and we write the Hamiltonian matrix H(t, z) as

$$\begin{aligned} H(t,z)= v(t,z) {\mathbb {I}}_{{{\mathbb {C}}}^N} + H_0(t,z),\;\; v(t,z)= \frac{1}{2}\left( h_1(t,z) + h_2(t,z)\right) \end{aligned}$$

(22)

where v(t, z) is a real number and $H_0(t,z)$ a self-adjoint $N\times N$ matrix, that is trace-free if $N=2$. Such a situation is called a codimension one crossing (see Hagedorn’s classification [13] for example). Let us formulate our assumptions on the crossing set more precisely.

Assumption 3.6

(Codimension one crossing) Let $H:{{\mathbb {R}}}^{2d+1}\rightarrow {{\mathbb {C}}}^{N\times N}$ be a smooth function with values in the set of self-adjoint $N\times N$ matrices that is of subquadratic growth (7). We assume:

Growth assumptions.

• a) The matrix H(t, z) has two smooth eigenvalues $h_1(t,z)$ and $h_2(t,z)$ that are of subquadratic growth (7).

• b) These eigenvalues satisfy a polynomial gap condition at infinity, in the sense that there exist constants $c_0,n_0,r_0>0$ such that

  $$\begin{aligned} |h_1(t,z)-h_2(t,z)| \ge c_0 \langle z\rangle ^{-n_0}\ \text {for all}\ (t,z)\ \text {with}\ |z|\ge r_0, \end{aligned}$$

  (23)

  where we denote $\langle z\rangle = (1+|z|^2)^{1/2}$.

Crossing assumptions.

• c) These eigenvalues cross on a hypersurface $\Upsilon$ of ${{\mathbb {R}}}^{2d+1}$ and differ outside of $\Upsilon$.

• d) The crossing is non-degenerate in a fixed point $(t^\flat ,z^\flat )\in \Upsilon$ in the sense, that the matrix $H_0(t,z)$ defined by the decomposition (22) satisfies

  $$\begin{aligned} d_{t,z} H_0(t^\flat ,z^\flat ) \not =0. \end{aligned}$$

• e) The crossing is transverse in $(t^\flat ,z^\flat )\in \Upsilon$ in the sense that for any smooth function f such that $f=0$ is a local equation of $\Upsilon$ close to $(t^\flat , z^\flat )$, then the scalar function $(t,z)\mapsto v(t,z)$ defined by the decomposition (22) satisfies

  $$\begin{aligned} (\partial _t f + \{v ,f\})(t^\flat ,z^\flat ) \not =0. \end{aligned}$$

  (24)

Example 3.7

Take $N=2$, $v,f\in {{\mathcal {C}}}^\infty ({{\mathbb {R}}}^{2d+1}, {{\mathbb {R}}})$ and $u\in {{\mathcal {C}}}^\infty ({{\mathbb {R}}}^{2d+1}, {{\mathbb {R}}}^3)$ with $|u(t,z)| = 1$ for all (t, z). Consider the Hamiltonian

$$\begin{aligned}H(t,z)=v(t,z)\mathrm{Id} + f(t,z) \begin{pmatrix} u_1(t,z) &{} u_2 (t,z) +iu_3(t,z) \\ u_2(t,z) -iu_3(t,z) &{}-u_1 (t,z) \end{pmatrix}.\end{aligned}$$

The smooth eigenvalues of H, $h_1 = v+f$ and $h_2 = v-f$, cross on the set $\Upsilon = \{f=0\}$, and H satisfies Assumption 3.6 as soon as the conditions (24) and (23) hold.

Assuming (d) and considering an equation $f=0$ of $\Upsilon$ in a neighborhood $\Omega$ of $(t^\flat ,z^\flat )$, we deduce from $H_0=0$ on $\Upsilon$ that we have $H_0= f {{\tilde{H}}}_0$ for some smooth matrix-valued map $(t,z) \mapsto {{\tilde{H}}}_0(t,z)$ defined on $\Omega$. Besides, the matrix ${{\tilde{H}}}_0(t,z)$ is invertible and its spectrum consists of two opposite distinct eigenvalues of constant multiplicity. We can then choose the function $f=\tfrac{1}{2}(h_1-h_2)$ as an equation of $\Upsilon$ and the eigenvalues of ${{\tilde{H}}}_0(t,z)$ are $+1$ and $-1$ in $\Omega$. In particular, the smooth eigenvalues of the matrix H(t, z) then satisfy

$$\begin{aligned} h_j(t,z)= v(t,z) - (-1)^j f(t,z) ,\quad j\in \{1,2\}, \end{aligned}$$

(25)

We shall choose f in that manner throughout the paper.

Note also that the condition (e) is satisfied as soon as it holds for one equation $f=0$ of $\Upsilon$. Besides, by restricting $\Omega$ if necessary, one can assume that $\partial _t v+\{f,v\}\not =0$ in $\Omega$. We will also use that condition (e) implies the transversality of the classical trajectories to the crossing set $\Upsilon$.

The gap condition at infinity (b) ensures that the derivatives of the eigenprojectors $\Pi _j(t)$, $j=1,2$, grow at most polynomially, in the sense that for all $\beta \in {{\mathbb {N}}}_0^{2d}$ there exists a constant $C_\beta >0$ such that

$$\begin{aligned} \Vert \partial _z^{\beta } \Pi _j(t,z)\Vert \le C_\beta \langle z\rangle ^{|\beta |(1+n_0)}\ \text {for all}\ (t,z)\ \text {with}\ |z|\ge r_0, \end{aligned}$$

(26)

see [6, Lemma B.2] for a proof of this estimate.

The growth condition (a) implies that the Hamiltonians $h_1$ and $h_2$ satisfy all the conditions of Sect. 2.2 and we associate with each eigenvalue $h_j$ the classical quantities introduced therein, that we index by j: $\Phi _j^{t,t_0}$, $S_j(t,t_0)$, $F_j(t,t_0)$, etc.

We consider initial data at time $t=t_0$ as in (2), where the coherent state is associated with the first eigenvalue $h_1$ and centered in a phase space point $z_0$ such that $(t_0,z_0)\notin \Upsilon$, while $z\mapsto \vec V_0(z)$ is a smooth map with $\Vert \vec V_0(z)\Vert =1$ for all z.

We assume that the Hamiltonian trajectory $z_1(t,t_0,z_0)= \Phi ^{t,t_0}_1(z_0)$ reaches $\Upsilon$ at time $t=t^\flat$ and point $z=z^\flat$. We denote by $S^\flat$ the corresponding action

$$\begin{aligned} S^\flat = S_1(t^\flat , t_0). \end{aligned}$$

(27)

We assume that (24) holds in $t^\flat ,z^\flat$. Therefore, $f(t,z)=0$ is a local equation of $\Upsilon$ in a neighborhood $\Omega$ of $(t^\flat ,z^\flat )$, and the assumption 3.6 implies

$$\begin{aligned}\frac{d}{dt}f(t, z_1(t,t_0))\ne 0\end{aligned}$$

close to $(t^\flat , z^\flat )$, and guarantees that the trajectory $z_1(t,t_0,z_0)$ passes through $\Upsilon$. The same holds for trajectories $\Phi ^{t,t_0}_1 (z)$ starting from z close enough to $z_0$.

We associate with $\vec V_0(z)$ the time-dependent eigenvector $(\vec V_1(t,z))_{t\ge t_0}$ constructed as in Proposition 3.1 for the eigenvalue $h_1(t,z)$ with initial data $\vec V_0(z)$ at time $t_0$.

We set

$$\begin{aligned} \gamma (t^\flat ,z) =\Vert \left( \partial _t \Pi _2+\{v,\Pi _2\}\right) \vec V_1(t^\flat ,z) \Vert _{{{\mathbb {C}}}^N} \end{aligned}$$

and when $\gamma (t^\flat ,z^\flat ) \not =0$, we consider the time-dependent eigenvector $(\vec V_2(t,z))_{t\ge t^\flat }$ constructed for $t\ge t^\flat$ and z in a neighborhood of $z^\flat$ as in Proposition 3.1 for the eigenvalue $h_2(t,z)$ and with initial data at time $t^\flat$

$$\begin{aligned} \vec V_2(t^\flat , z)= -\gamma (t^\flat ,z)^{-1} {\Pi _2(\partial _t \Pi _2 +\{v,\Pi _2\} )\vec V_1} (t^\flat ,z). \end{aligned}$$

(28)

Note that if $\gamma (t^\flat , z^\flat )=0$, there will be no transitions of order $\sqrt{\varepsilon }$ (see (31) below).

We introduce a family of transformations, which describes the non-adiabatic effects for a wave packet that passes the crossing. For $(\mu ,\alpha ,\beta )\in {{\mathbb {R}}}\times {{\mathbb {R}}}^{2d}$ and $\varphi \in {{\mathcal {S}}}({{\mathbb {R}}}^d)$, we set

$$\begin{aligned} {{\mathcal {T}}}_{\mu ,\alpha , \beta }\varphi (y) = \left( \int _{-\infty }^{+\infty }{\mathrm{e}}^{i\mu s^2}{\mathrm{e}}^{is(\beta \cdot y-\alpha \cdot D_y)}ds\right) \varphi (y). \end{aligned}$$

(29)

By the Baker-Campbell-Hausdorff formula, we have

$$\begin{aligned}{\mathrm{e}}^{is\beta \cdot y}{\mathrm{e}}^{-is\alpha \cdot D_y}= {\mathrm{e}}^{is\beta \cdot y-is\alpha \cdot D_y+is^2\alpha \cdot \beta /2},\end{aligned}$$

and we deduce the equivalent representation

$$\begin{aligned} {{\mathcal {T}}}_{\mu ,\alpha , \beta }\varphi (y) = \int _{-\infty }^{+\infty }{\mathrm{e}}^{i(\mu -\alpha \cdot \beta /2)s^2} {\mathrm{e}}^{is\beta \cdot y}\varphi (y-s\alpha )ds . \end{aligned}$$

(30)

We prove in Proposition E.1 below that this operator maps ${\mathcal {S}}({{\mathbb {R}}}^d)$ into itself if and only if $\mu \not =0$. Moreover, for $\mu \not =0$, it is a metaplectic transformation of the Hilbert space $L^2({{\mathbb {R}}}^d)$, multiplied by a complex number. In particular, for any Gaussian function $g^\Gamma$, the function ${\mathcal T}_{\mu ,\alpha , \beta }g^\Gamma$ is a Gaussian:

$$\begin{aligned} {{\mathcal {T}}}_{\mu ,\alpha , \beta }\,g^\Gamma = c_{\mu ,\alpha ,\beta ,\Gamma }\, g^{\Gamma _{\mu , \alpha ,\beta ,\Gamma }}, \end{aligned}$$

where $\Gamma _{\mu , \alpha ,\beta ,\Gamma }\in {\mathfrak {S}}^+(d)$ and $c_{\mu ,\alpha ,\beta ,\Gamma }\in {{\mathbb {C}}}$ are given in Proposition E.1.

Combining the parallel transport for the eigenvector and the metaplectic transformation for the non-adiabatic transitions, we obtain the following result.

Theorem 3.8

(Propagation through a codimension one crossing) Let Assumption 3.6 on the Hamiltonian matrix H(t) hold, and assume that the initial data $(\psi ^\varepsilon _0)_{\varepsilon >0}$ are wave packets as in (2). Let $T>0$ be such that the interval $[t_0,t^\flat ]$ is strictly included in the interval $[t_0,t_0+T]$. Then, for all $k\in {{\mathbb {N}}}$ there exists a constant $C>0$ such that for all $t\in [t_0,t^\flat )\cup (t^\flat ,t_0+T]$ and for all $\varepsilon \le [t-t^\flat |^{9/2}$,

$$\begin{aligned}\left\| \psi ^\varepsilon (t) - \widehat{\vec V}_1(t) v^\varepsilon _1(t) -\sqrt{\varepsilon }\mathbf{1}_{t>t^\flat } \widehat{\vec V}_2(t) v^\varepsilon _2(t) \right\| _{\Sigma ^k_\varepsilon }\le C \,\varepsilon ^{m},\end{aligned}$$

with an exponent $m\ge 5/9$. The components of the approximate solution are

$$\begin{aligned} v^\varepsilon _1(t) = {\mathcal {U}}_{h_1}^\varepsilon (t,t_0) v^\varepsilon _0\quad \text {and}\quad v^\varepsilon _2(t) ={\mathcal {U}}_{h_2}^\varepsilon (t,t^\flat )v^\varepsilon _2(t^\flat ) \end{aligned}$$

with

$$\begin{aligned} v^\varepsilon _2(t^\flat )= \gamma ^\flat {\mathrm{e}}^{iS^\flat /\varepsilon }\mathcal {WP}^\varepsilon _{z^\flat }{{\mathcal {T}}}^\flat \varphi _1(t^\flat ), \end{aligned}$$

(31)

where $\varphi _1(t) = {{\mathcal {M}}}[F_{h_1}(t,t_0,z_0)]\varphi _0$ is the leading order profile of the coherent state $v^\varepsilon _1(t)$ given by Proposition 2.3, and

$$\begin{aligned} \gamma ^\flat = \gamma (t^\flat ,z^\flat ) = \Vert \left( \{v,\Pi _2\}+\partial _t\Pi _2\right) \vec V_1(t^\flat ,z^\flat ) \Vert _{{{\mathbb {C}}}^N} . \end{aligned}$$

(32)

The transition operator

$$\begin{aligned} {\mathcal {T}}^\flat = {\mathcal {T}}_{ \mu ^\flat , \alpha ^\flat ,\beta ^\flat } \end{aligned}$$

(33)

is defined by the parameters

$$\begin{aligned} \mu ^\flat = \tfrac{1}{2}\left( \partial _t f+\{v,f\}\right) (t^\flat , z^\flat )\;\; \text {and}\;\; (\alpha ^\flat ,\beta ^\flat ) = Jd_zf(t^\flat , z^\flat ). \end{aligned}$$

(34)

The constant $C = C(T,k,z_0,\varphi _0)>0$ is $\varepsilon$-independent but depends on the Hamiltonian H(t, z), the final time T, and on the initial wave packet’s center $z_0$ and profile $\varphi _0$.

Theorem 3.8 approximates the action of the unitary propagator ${\mathcal {U}}^\varepsilon _H(t,t_0)$ on the initial wave packet $\psi ^\varepsilon _0$ by combining the two scalar evolutions ${\mathcal {U}}^\varepsilon _{h_1}(t,t_0)$ and ${\mathcal {U}}^\varepsilon _{h_2}(t,t^\flat )$ with transitions of order $\sqrt{\varepsilon }$. The approximation error is of order $\varepsilon ^m$ with $m\ge 5/9$. In particular, if the transition coefficient $\gamma ^\flat$ does not vanish, then the codimension one crossing clear reduces the usual adiabatic approximation error of order $\varepsilon$ that holds for systems with positive eigenvalue gap.

Note that by Assumption 3.6, $\mu ^\flat \not =0$, which guarantees that ${{\mathcal {T}}}^\flat \varphi _1(t^\flat )$ is Schwartz class. Besides, if the Hamiltonian is time-independent, then Assumption 3.6 also implies that $(\alpha ^\flat ,\beta ^\flat )\not =(0,0)$. The coefficient $\gamma ^\flat$ quantitatively describes the distortion of the projector $\Pi _1$ during its evolution along the flow generated by $h_1(t)$. In particular, we have

$$\begin{aligned}\gamma ^\flat =\Vert \left( \{v,\Pi _2\}+\partial _t\Pi _2\right) \vec V_1(t^\flat ,z^\flat ) \Vert _{{{\mathbb {C}}}^N} =\Vert \left( \{v,\Pi _1\}+\partial _t\Pi _1\right) \vec V_1(t^\flat ,z^\flat )\Vert _{{{\mathbb {C}}}^N}.\end{aligned}$$

Moreover, if the matrix H is diagonal (or diagonalizes in a fixed orthonormal basis that is (t, z)-independent), then $\gamma ^\flat =0$: the equations are decoupled (or can be decoupled), and one can then apply the result for a system of two independent equations with a scalar Hamiltonian and, of course, there is no interaction between the modes. As an opposite situation, $\gamma ^\flat$ is non-zero in the simple examples (4) and (5) for which the eigenprojectors of H are non constant.

The proof uses two types of arguments, one of them applying away from the crossing set $\Upsilon$, and the other one in a boundary layer of $\Upsilon$. The boundary layer is taken of size $\delta >0$, and we have to balance the two estimates: an error of order $\varepsilon \delta ^{-2}$ which comes from the adiabatic propagation of wave packets outside the boundary layer, and an additional error of order $\delta \varepsilon ^{1/3}$ generated by the passage through the boundary. The choice of $\delta =\varepsilon ^{2/9}$ optimizes the combined estimate and yields convergence of order $\varepsilon ^m$ with $m\ge 5/9$. We also want to emphasize that the method of proof we propose here allows to systematically avoid the impressive computations, which appear in [13] pages 65 to 72, and are also present in [39] via the reference [46] to which the authors refer therein.

The wave packet that makes the transition to the other eigenspace can be described even more explicitly for the special case that the initial wave packet is a Gaussian state. The following corollary is proved in Proposition E.1.

Corollary 3.9

(Transitions for Gaussian wave packets) We consider the situation of Theorem 3.8 and in particular the transition operator ${\mathcal {T}}^\flat$ defined by the parameters $\mu ^\flat \ne 0$ and $(\alpha ^\flat ,\beta ^\flat )\in {{\mathbb {R}}}^{2d}$.

• (1) If $v_0^\varepsilon =\mathcal {WP}^\varepsilon _{z_0} (g^{\Gamma _0})$ is a Gaussian state with width matrix $\Gamma _0\in {{\mathfrak {S}}}^+(d)$, then

  $$\begin{aligned} v^\varepsilon _2(t^\flat )= \gamma ^\flat \sqrt{2i\pi \over \mu ^\flat }\, {\mathrm{e}}^{iS^\flat /\varepsilon }\, \mathcal {WP}^\varepsilon _{z^\flat }( g^{\Gamma ^\flat }) \end{aligned}$$
  $$\begin{aligned} \text{ with } \;\; \Gamma ^\flat = \Gamma _1(t^\flat ,t_0,z_0) -\frac{(\beta ^\flat -\Gamma _1(t^\flat ,t_0,z_0)\alpha ^\flat )\otimes (\beta ^\flat -\Gamma _1(t^\flat ,t_0,z_0)\alpha ^\flat ) }{2\mu ^\flat -\alpha ^\flat \cdot \beta ^\flat +\alpha ^\flat \cdot \Gamma _1(t^\flat ,t_0,z_0)\alpha ^\flat }\end{aligned}$$

  and $\Gamma _1(t^\flat ,t_0,z_0)$ is the image of $\Gamma _0$ by the flow map associated with $h_1(t,z)$ by (15).

• (2) If $P\in {\mathcal {C}}^\infty ({{\mathbb {R}}}^{2d})$ is a polynomial function and $v_0^\varepsilon =\mathcal {WP}^\varepsilon _{z_0} ({\mathrm{op}}^w_1(P)g^{\Gamma _0})$, then

  $$\begin{aligned} v^\varepsilon _2(t^\flat )= \gamma ^\flat \sqrt{2i\pi \over \mu ^\flat }\, {\mathrm{e}}^{iS^\flat /\varepsilon }\, \mathcal {WP}^\varepsilon _{z^\flat }\!\left( {\mathrm{op}}^w_1(P^\flat ) g^{\Gamma ^\flat }\right) \end{aligned}$$

  with $P^\flat = P\circ \Phi _{\alpha ^\flat ,\beta ^\flat }((4\mu ^\flat )^{-1})$ where $\Phi _{\alpha ^\flat ,\beta ^\flat }(t)$ is the symplectic $2d\times 2d$ matrix given by

  $$\begin{aligned}&\qquad \qquad \Phi _{\alpha ^\flat ,\beta ^\flat }(t) =\begin{pmatrix} {\mathbb {I}}-2t\beta ^\flat \otimes \alpha ^\flat &{}2t\alpha ^\flat \otimes \alpha ^\flat , \\ -2t\beta ^\flat \otimes \beta ^\flat &{} {\mathbb {I}}+2t\alpha ^\flat \otimes \beta ^\flat \end{pmatrix}. \end{aligned}$$

  (35)

As a concluding remark of this section, we want to emphasize that our results indeed generalize those of [13, 39].

• (1) In the Schrödinger example (4), denoting by $E_A(x)$ and $E_B(x)$ the two eigenvalues of the potential matrix V(x) as in [13], one has

  $$\begin{aligned} \alpha ^\flat _S=0 ,\;\;\beta ^\flat _S= \nabla (E_A-E_B)(q^\flat ),\;\; \mu ^\flat _S= p^\flat \cdot \nabla (E_A-E_B)(q^\flat ). \end{aligned}$$

  These coefficients appear in equation (5.3) of [13]. There, the initial states are Gaussian wave packets that are multiplied with a polynomial function. Thus, the second part of Corollary 3.9 reproduces these results.

• (2) For the Bloch example (5), we obtain

  $$\begin{aligned}\alpha ^\flat _A=\nabla (E_+-E_-)(p^\flat ) ,\;\;\beta ^\flat _A= 0,\;\;\mu ^\flat _A = -\frac{1}{2} \nabla W(q^\flat ) \cdot \nabla (E_+-E_-)(p^\flat ), \end{aligned}$$

  where $E_\pm (\xi )$ are the eigenvalues of $A(\xi )$ as in equation (3.41) of [39]. The result of [39, Theorem 3.20 (via Definition 3.18)] is therefore a special case of ours.

We notice, that for these special examples either one of the coefficients $\alpha ^\flat$ or $\beta ^\flat$ is 0. This need not be the case for more general Hamiltonians that have position and momentum variables mixed in the matrix part of the Hamiltonian. Actually, for Dirac Hamiltonians with electromagnetic potential (V, A), the function $\xi -A(t,x)$ appears in the coefficients of the matrix. Also for the propagation of acoustical waves in elastic media the Hamiltonian is of the form $\rho (x){\mathbb {I}}_{{{\mathbb {C}}}^N} -\Gamma (x,\xi )$,where $\rho (x)>0$ is the density and $\Gamma (x,\xi )$ the elastic tensor.

Remark 3.10

A straightforward extension of our result to the case of Hamiltonians of the more general form ${{\widehat{H}}} + \varepsilon \widehat{H}_1$ does not seem possible. A sub-principal operator ${{\widehat{H}}}_1$ contributes both to the adiabatic decoupling outside and the transition analysis inside the crossing region. We believe that higher order asymptotic expansions with respect to $\varepsilon$ and the distance parameter $\delta$, that will be introduced below, are needed to reveil the effect of such a sub-principal contribution. We will comment on the possibility of such higher order expansions alongside the proof.

3.3 Organization of the paper

The proof of Theorem 3.8 is decomposed into two steps: an analysis outside the crossing region in Sect. 4 and an analysis in the crossing region in Sect. 5, that allows to conclude the proof in Sect. 5.4, together with the one of Corollary 3.9. Finally, we gather in four Appendices various results about wave packets, algebraic properties of the projectors and parallel transport, analysis of the transfer operators ${\mathcal {T}}_{\mu ,\alpha ,\beta }$, and technical computations.

4 Adiabatic Decoupling Outside the Crossing Region

In this section, we consider a family of solutions to equation (1) in the case where the Hamiltonian H(t, z) satisfies Assumption 3.6 and with an initial datum which is a coherent state as in (2). We focus here on regions where the classical trajectories associated with the coherent state do not touch the crossing set $\Upsilon$ but are close enough. We prove the next adiabatic result.

Proposition 4.1

Let $k\in {{\mathbb {N}}}$, $\delta = \delta (\varepsilon )$ be such that $\sqrt{\varepsilon }\ll \delta \le 1$. Let $f(t,z)=0$ be an equation of $\Upsilon$ in an open set $\Omega \subset {{\mathbb {R}}}\times {{\mathbb {R}}}^{2d}$. Assume that for $j\in \{1,2\}$,

$$\begin{aligned}u^\varepsilon _j=\mathcal {WP}^\varepsilon _{{{\widetilde{z}}}_j} ({{\widetilde{\varphi }}}_j),\end{aligned}$$

where ${{\widetilde{\varphi }}}_1,\, {{\widetilde{\varphi }}}_2\in \mathcal S({{\mathbb {R}}}^d)$, ${{\widetilde{z}}}_1,\,{{\widetilde{z}}}_2\in {{\mathbb {R}}}^d$ are such that there exist $s_1,s_2\in {{\mathbb {R}}}$, $c,C>0$ such that for all $j\in \{1,2\}$ and $t\in [s_1,s_2]$, $z_j(t):=\Phi _{j}^{t,s_1}({{\widetilde{z}}}_j)\in \Omega$ with $|f(z_j(t))|>c\delta$ and

$$\begin{aligned}\left\| \psi ^\varepsilon (s_1)-\widehat{\vec V_1(s_1)} u^\varepsilon _1 - \widehat{\vec V_2(s_1)} u^\varepsilon _2\right\| _{\Sigma ^k_\varepsilon }\le C\varepsilon .\end{aligned}$$

Then, there exists $C_k>0$ such that for all $j\in \{1,2\}$,

$$\begin{aligned}\sup _{t\in [s_1,s_2]} \left\| {{\widehat{\Pi }}}_j \psi ^\varepsilon (t) - \widehat{\vec V_j(t)} {{\mathcal {U}}}^\varepsilon _{h_j}(t,s_1) u^\varepsilon _j \right\| _{\Sigma ^k_\varepsilon } \le C_k\,\varepsilon \, \delta ^{-2}.\end{aligned}$$

The constant $C_k$ does not depend on $\delta$ and $\varepsilon$.

For fixed $\delta$, that is independent of $\varepsilon$, this Proposition implies Theorem 3.8 for $t\in [0,t^\flat [$. We shall choose later $\delta =\varepsilon ^{1/3}$ for obtaining a global a priori estimate in Sect. 4.2 below. Finally with $\delta =\varepsilon ^{2/9}$, we will prove Theorem 3.8 in Sect. 5.4 by using the Proposition 4.1 for propagation times $t \in [t_0, t^\flat -\delta ]$ and $t\in [t^\flat +\delta , t_0 +T]$ with initial data at times $t=t_0$ and $t=t^\flat +\delta$ respectively.

Remark 4.2

Pushing the construction of superadiabatic projectors of Appendix B, we would obtain that $\psi ^\varepsilon (t)$ can be approximated by an asymptotic sum of wave packets up to order $\varepsilon ^N \delta ^{p(N)}$ for some $p(N)\le N$ to be computed precisely.

4.1 Proof of the adiabatic decoupling

We prove here Proposition 4.1.

Proof

Because of the linearity of the equation, it is enough to assume that the contribution of $\psi ^\varepsilon (s_1)$ on one of the modes is negligible at the initial time $s_1$. The roles of the two modes being symmetric, we can choose equivalently one or the other one. Therefore, without loss of generality, we assume $\psi ^\varepsilon (s_1) = \widehat{\vec V_1(s_1)} u^\varepsilon _1$, and we focus on

$$\begin{aligned}\psi ^\varepsilon _{1,\mathrm{app}}(t) := \widehat{\vec V_1(t)} {\mathcal {U}}^\varepsilon _{h_1} (t,s_1) u^\varepsilon _1\end{aligned}$$

Then, using the parallel transport equation (20) associated with the eigenvalue $h_1$,

$$\begin{aligned} i\varepsilon \partial _t \psi ^\varepsilon _{1,\mathrm{app}}(t)&= {{\widehat{h}}}_1 \psi ^\varepsilon _{1,\mathrm{app}}(t) + \left( \left[ \widehat{\vec V_1(t)} , {{\widehat{h}}}_1 \right] +i\varepsilon \partial _t \widehat{ \vec V_1(t)}\right) {\mathcal {U}}^\varepsilon _{h_1} (t,s_1) u^\varepsilon _1 \nonumber \\&=({{\widehat{h}}}_1\mathrm{Id} + \varepsilon {{\widehat{\Theta }}}_1) \psi ^\varepsilon _{1,\mathrm{app}}(t) + \varepsilon ^2\widehat{r(t)}\,{\mathcal {U}}^\varepsilon _{h_1} (t,s_1) u^\varepsilon _1, \end{aligned}$$

(36)

where the remainder r(t) depends on second order derivatives of $h_1$ and $\vec V_1$. Since $u^\varepsilon _1$ is a wave packet with a Schwartz function amplitude, we obtain

$$\begin{aligned} i\varepsilon \partial _t \psi ^\varepsilon _{1,\mathrm{app}}(t) = ({{\widehat{h}}}_1\mathrm{Id} + \varepsilon {{\widehat{\Theta }}}_1) \psi ^\varepsilon _{1,\mathrm{app}}(t) + O(\varepsilon ^2) \end{aligned}$$

(37)

in $\Sigma ^k_\varepsilon$ for all $k\in {{\mathbb {N}}}$.

We now use the superadiabatic correctors of $\Pi _1$ and $\Pi _2$ defined in Definition B.3 (see also [28, 36]) that we denote by ${\mathbb {P}}_1$ and $\mathbb P_2$, respectively, and the associated correctors $\Theta _1$ and $\Theta _2$ of the Hamiltonian H. Since ${\mathbb {P}}_1$ and $\mathbb P_2$ are singular on $\Upsilon$, we use cut-off functions that follow the flows arriving at time $s_2$ in $\Phi ^{s_2,s_1} _{h_1}({{\tilde{z}}}_1)$. We introduce two sets of cut-off functions, one for each mode. Let I be an interval containing $[s_1,s_2]$ and for $j\in \{1,2\}$ let the cut-off functions $\chi ^\delta _j ,{\tilde{\chi }}^\delta _j\in {\mathcal {C}}(I,{{\mathcal {C}}}_0^\infty ({{\mathbb {R}}}^{2d}))$ satisfy as in Lemma B.5:

• (1) For any $t\in I$ and any z in the support of $\chi ^\delta _j(t)$ and ${{\tilde{\chi }}}^\delta _j(t)$ we have $|f(t,z)|>\delta$.

• (2) The functions $\chi ^\delta _j$ and ${{\tilde{\chi }}}^\delta _j$ are identically equal to 1 close to a trajectory $\Phi ^{t,s_1}_{j}({{\tilde{z}}}_1)$ for all $t\in I$ and they satisfy

  $$\begin{aligned}\partial _t \chi ^\delta _j + \left\{ h_j, \chi ^\delta \right\} =0,\;\;\partial _t {{\tilde{\chi }}}^\delta _j + \left\{ h_j, {{\tilde{\chi }}}^\delta _j \right\} =0.\end{aligned}$$

• (3) The functions ${{\tilde{\chi }}}^\delta _j$ are supported in $\{\chi ^\delta _j=1\}$.

• (4) Finally, we require $\chi _1^\delta (s_2)=\chi _2^\delta (s_2)$ and ${\tilde{\chi }}_1^\delta (s_2)={{\tilde{\chi }}}_2^\delta (s_2).$

We set for $t\in [s_1,s_2]$

$$\begin{aligned}w^\varepsilon _1 (t)= \widehat{{{\tilde{\chi }}}^\delta _1}(\widehat{\chi ^\delta _1 \Pi ^\varepsilon _1}\psi ^\varepsilon (t)- \psi ^\varepsilon _{1,\mathrm{app}}(t)) ,\;\; w^\varepsilon _2(t)= \widehat{{\tilde{\chi }}^\delta _2}\widehat{\chi ^\delta _2 \Pi ^\varepsilon _{2}}\psi ^\varepsilon (t),\end{aligned}$$

$$\begin{aligned}\text{ where } \;\;\Pi ^\varepsilon _j(t,z)=\Pi _j(t,z)+\varepsilon {\mathbb {P}}_j(t,z),\;\;\forall z\in {{\mathbb {R}}}^{2d}\setminus \Upsilon ,\;\; t\in I, \;\;j\in \{1,2\}.\end{aligned}$$

Then, as a consequence of (37) and of Lemma B.5, we have for $j\in \{1,2\}$ and in $\Sigma ^k_\varepsilon$,

$$\begin{aligned}i\varepsilon \partial _t w^\varepsilon _j(t)= ({{\hat{h}}}_j + \varepsilon {{\widehat{\Theta }}}_j) w^\varepsilon _j(t) +O(\varepsilon ^2\delta ^{-2}).\end{aligned}$$

For the initial data at time $t=s_1$, we have in $\Sigma ^k_\varepsilon$,

$$\begin{aligned} w^\varepsilon _1(s_1) = \widehat{{{\tilde{\chi }}}^\delta _1} \big (\widehat{\chi ^\delta _1 \Pi ^\varepsilon _1}\widehat{\vec V_1}-\widehat{\vec V_1}\big )u^\varepsilon _1 = O(\varepsilon \delta ^{-1}), \quad w^\varepsilon _2(s_1) = \widehat{{{\tilde{\chi }}}^\delta _2}\, \widehat{\chi ^\delta _2 \Pi ^\varepsilon _2}\,\widehat{\vec V_1}u^\varepsilon _1 = O(\varepsilon \delta ^{-1}). \end{aligned}$$

We deduce that for any $k\in {{\mathbb {N}}}$, $j\in \{1,2\}$ and $t\in [s_1,s_2]$, we have in $\Sigma ^k_\varepsilon$, $w^\varepsilon _j(t)=O( \varepsilon \delta ^{-2}).$ When $t=s_2$, we have

$$\begin{aligned} w_1^\varepsilon (s_2) + w_2^\varepsilon (s_2)&= \widehat{{{\tilde{\chi }}}^\delta _1} \left( {\mathrm{op}}_\varepsilon (\chi ^\delta _1 (\Pi ^\varepsilon _1+\Pi ^\varepsilon _2))\psi ^\varepsilon (s_2) - \psi ^\varepsilon _{1,\mathrm app}(s_2)\right) \\&= \widehat{{{\tilde{\chi }}}^\delta _1}(\psi ^\varepsilon (s_2)- \psi ^\varepsilon _{1,\mathrm app}(s_2)) + O(\varepsilon \delta ^{-1}) \end{aligned}$$

and thus $\displaystyle { \widehat{{\tilde{\chi }}^\delta _1}(s_2)\psi ^\varepsilon (s_2)= \widehat{{\tilde{\chi }}^\delta _1}(s_2)\psi ^\varepsilon _{1,\mathrm app}(s_2) +O(\varepsilon \delta ^{-2}). }$ Because of the localisation of the wave packet $\psi ^\varepsilon _{1,\mathrm app} (s_2)$, as stated in Remark A.2, we have in $\Sigma ^k_\varepsilon$ for any $N\in {{\mathbb {N}}}$,

$$\begin{aligned} \widehat{{{\tilde{\chi }}}^\delta _1}(s_2)\psi ^\varepsilon _{1,\mathrm app}(s_2) = \psi ^\varepsilon _{1,\mathrm app}(s_2) + O(\varepsilon ^{N/2} \delta ^{-N}). \end{aligned}$$

Hence, choosing $N=2$, we obtain

$$\begin{aligned}\widehat{{{\tilde{\chi }}}^\delta _1}(s_2)\psi ^\varepsilon (s_2)= \psi ^\varepsilon _{1,\mathrm app}(s_2) +O(\varepsilon \delta ^{-2}),\end{aligned}$$

and it only remains to study $(1-\widehat{{\tilde{\chi }}^\delta _1}(s_2))\psi ^\varepsilon (s_2)$. Before that, some remarks are in order. Note that the arguments developed above do not depend on the choice of $s_2$ and could have been developed for any $s\in [s_1,s_2]$. They are also independent of the choices of the functions $\chi ^\delta _j$ and ${{\widetilde{\chi }}}^\delta _j$ as long as they satisfy the properties stated above. Therefore, we have actually obtained a more general result, namely that for any function $\theta$ supported in $\{|f|>\delta \}$ and equal to 1 close to $\Phi _{1}^{t,s_1}({{\widetilde{z}}}_1)$, we have for $t\in [s_1,s_2]$,

$$\begin{aligned} {\widehat{\theta }}\psi ^\varepsilon (t)= {{\widehat{\theta }}} \psi ^\varepsilon _{1,\mathrm app}(t) +O(\varepsilon \delta ^{-2}). \end{aligned}$$

(38)

We can now study $(1-\widehat{{\tilde{\chi }}^\delta _1}(s_2))\psi ^\varepsilon (s_2)$. We set for $s\in [s_1,s_2]$, $w^\varepsilon (s)= (1-\widehat{{{\tilde{\chi }}}^\delta _1}(s))\psi ^\varepsilon (s).$ We have

$$\begin{aligned} i\varepsilon \partial _s w^\varepsilon (s)&= {{\widehat{H}}}(s) w^\varepsilon (s) -\left[ \widehat{{{\tilde{\chi }}}^\delta _1}(s), {{\widehat{H}}}(s)\right] \psi ^\varepsilon (s) -i\varepsilon \widehat{\partial _s{{\tilde{\chi }}}^\delta _1(s)}\psi ^\varepsilon (s)\\&= {{\widehat{H}}}(s) w^\varepsilon (s) -\varepsilon \widehat{r^\varepsilon _\delta (s)} \psi ^\varepsilon (s) +O(\varepsilon ^2\delta ^{-2}) \end{aligned}$$

where $r^\varepsilon _\delta (s)$ depends linearly on $d{\tilde{\chi }}^\delta _1(s)$, and thus is compactly supported close to the trajectory $\Phi _{1}^{t,s_1}({{\widetilde{z}}}_1)$ and equal to 0 very close to it. Therefore, by (38) and Remark A.2,

$$\begin{aligned}\varepsilon \,\widehat{r^\varepsilon _\delta (s)} \psi ^\varepsilon (s)=\varepsilon \,\widehat{r^\varepsilon _\delta (s)} \psi ^\varepsilon _{1,\mathrm app}(s)= O(\varepsilon ^{N/2+1} \delta ^{-N-1})\end{aligned}$$

for any $N\in {{\mathbb {N}}}$. Choosing $N=1$, we deduce $w^\varepsilon (s_2)=O(\varepsilon \delta ^{-2})$. $\square$

4.2 A global a priori estimate

In this section, we prove the following a priori estimate.

Lemma 4.3

Let $k\in {{\mathbb {N}}}$ and $T>0$ such that $[t_0,t^\flat ]$ is strictly included in $[t_0,t_0+T]$. Then there exists a constant $C_k>0$ such that

$$\begin{aligned} \sup _{t\in [t_0,t_0+T]} \Vert \psi ^\varepsilon (t)- \widehat{ \vec V_1}(t) v^\varepsilon _1(t)\Vert _{\Sigma ^k_\varepsilon } \le C_k \,\varepsilon ^{1/3}, \end{aligned}$$

(39)

where $v^\varepsilon _1(t) = {{\mathcal {U}}}_{h_1}^\varepsilon (t,t_0)v^\varepsilon _0$ for all $t\in [t_0,t_0+T]$.

In the next section, we shall improve this estimate to go beyond this approximation and exhibits elements of order $\sqrt{\varepsilon }$. However, we shall use this a priori estimate, together with elements developed in this section.

Proof

Of course, in view of the results of the preceding section, we choose $\delta >0$ and we focus on the time interval $[t^\flat -\delta ,t^\flat +\delta ]$, taking into account that for times $t\in [t_0, t^\flat -\delta ]$, we have

$$\begin{aligned} \Vert \psi ^\varepsilon (t)- \widehat{ \vec V_1}(t) v^\varepsilon _1(t)\Vert _{\Sigma ^k_\varepsilon } \le C_k\, \varepsilon \delta ^{-2}\end{aligned}$$

for some constant $C_k>0$, and that for $t\in [t^\flat +\delta , t_0+T]$ we can use the same kind of transport estimate since the trajectory does not meet again the crossing set. It is thus enough to pass from $t^\flat -\delta$ to $t^\flat +\delta$ and analyze $\psi ^\varepsilon (t^\flat +\delta )$. Between the times $t^\flat -\delta$ and $t^\flat +\delta$, we cannot use the super-adiabatic corrections to the projectors $\Pi _1$ and $\Pi _2$, because they become singular when the eigenvalue gap closes. We thus simply work with the projectors $\Pi _1$ and $\Pi _2$. We define the families $w^\varepsilon (t)=(w^\varepsilon _1(t),w^\varepsilon _2(t))$ by

$$\begin{aligned} w_1^\varepsilon = {{\widehat{\Pi }}}_1 \psi ^\varepsilon - \widehat{\vec V_1} v^\varepsilon _1,\;\; w^\varepsilon _2 = {{\widehat{\Pi }}}_2 \psi ^\varepsilon . \end{aligned}$$

(40)

Since $\psi ^\varepsilon (t)$ and $\widehat{\vec V_1} v^\varepsilon _1(t)$ are in all spaces $\Sigma ^{\ell }_\varepsilon ({{\mathbb {R}}}^d)$ for $\ell \in {{\mathbb {N}}}$ and $t\in [t_0,t_0+T]$, the same is true for $w_1^\varepsilon (t)$ and $w_2^\varepsilon (t)$. We now use our former observations, that is, the evolution equation (36) for the approximate wave packet and the relation (53) of Appendix B, which gives that $w^\varepsilon (t)$ satisfies the following system:

$$\begin{aligned}\left\{ \begin{array}{rl} i\varepsilon \partial _t w^\varepsilon _1 &{}= {{\widehat{h}}}_1w^\varepsilon _1 + i\varepsilon f^\varepsilon _1,\\ i\varepsilon \partial _t w^\varepsilon _2 &{}= {{\widehat{h}}}_2 w^\varepsilon _2 + \tfrac{i\varepsilon }{2} \widehat{B_2\Pi _1} \widehat{\vec V_1} v_1 + i\varepsilon f^\varepsilon _2 \end{array}\right. \end{aligned}$$

with

$$\begin{aligned} f^\varepsilon _1 = -i{{\widehat{\Theta }}}_1 w^\varepsilon _1 + \tfrac{1}{2} \widehat{B_1\Pi _2} w^\varepsilon _2 + \varepsilon r^\varepsilon _1\quad \text {and}\quad f^\varepsilon _2 = -i{{\widehat{\Theta }}}_2 w^\varepsilon _2 + \tfrac{1}{2} \widehat{B_2\Pi _1} w^\varepsilon _1 + \varepsilon r^\varepsilon _2. \end{aligned}$$

(41)

The matrices $B_1$ and $B_2$ are defined according to

$$\begin{aligned} B_j = -2\partial _t\Pi _j - \{h_j,\Pi _j\} + \{\Pi _j,H\},\qquad j=1,2, \end{aligned}$$

and the sequences $(r^\varepsilon _1(t))_{\varepsilon >0}$ and $(r^\varepsilon _2(t))_{\varepsilon >0}$ are uniformly bounded in $\Sigma ^k_\varepsilon ({{\mathbb {R}}}^d)$ due to the polynomial growth estimate (26) for the eigenprojectors. We immediately deduce that for all $t\in [t_0,t_0+T]$,

$$\begin{aligned} w^\varepsilon _1(t)&=\ {{\mathcal {U}}}_{h_1}^\varepsilon (t,t^\flat -\delta ) w^\varepsilon _1(t^\flat -\delta ) + \int _{t^\flat -\delta }^t {\mathcal U}_{h_1}^\varepsilon (t,\sigma ) f^\varepsilon _1(\sigma ) d\sigma , \nonumber \\ w^\varepsilon _2(t)&=\ {{\mathcal {U}}}_{h_2}^\varepsilon (t,t^\flat -\delta ) w^\varepsilon _2(t^\flat -\delta ) + \int _{t^\flat -\delta }^t {\mathcal U}_{h_2}^\varepsilon (t,\sigma ) f^\varepsilon _2(\sigma ) d\sigma \nonumber \\&\quad +{1\over 2} \int _{t^\flat -\delta }^t {\mathcal U}_{h_2}^\varepsilon (t,\sigma ) \widehat{B_2 \Pi _1} \widehat{\vec V_1}(\sigma ) v_1^\varepsilon (\sigma )d\sigma . \end{aligned}$$

(42)

Therefore, in $\Sigma ^k_\varepsilon ({{\mathbb {R}}}^d)$, for all times $t\in [t^\flat -\delta ,t^\flat +\delta ]$ and $j\in \{1,2\}$, $w^\varepsilon _j(t)= O ( \varepsilon \delta ^{-2}) +O(\delta ) .$ Choosing $\delta =\varepsilon ^{1/3}$, we obtain (39). $\square$

5 Analysis in the Crossing Region

We now want to pass through the crossing and derive a more precise estimate on the function $\psi ^\varepsilon (t^\flat +\delta )$. We prove the following result.

Proposition 5.1

Assume $\sqrt{\varepsilon }\ll \delta \ll 1$. Then, for all $k\in {{\mathbb {N}}}$, there exists a constant $C_k>0$ such that

$$\begin{aligned} \left\| \psi ^\varepsilon (t^\flat +\delta ) - \widehat{\vec V_1}(t^\flat +\delta ) v^\varepsilon _1(t^\flat +\delta ) - \sqrt{\varepsilon }\widehat{\vec V_2}(t^\flat +\delta ) v^\varepsilon _2(t^\flat +\delta ) \right\| _{\Sigma ^k_\varepsilon } \le C_k(\varepsilon \delta ^{-2} + \varepsilon ^{1/3}\delta ), \end{aligned}$$

where $v^\varepsilon _1(t) = {\mathcal {U}}^\varepsilon _{h_1}(t,t_0)v^\varepsilon _0$ and $v^\varepsilon _2(t)={\mathcal {U}}^\varepsilon _{h_2}(t,t^\flat )v^\varepsilon _2(t^\flat )$ are as in Theorem 3.8.

Proof

We split the proof in several steps. In Lemma 5.2 we use the a priori estimate of Lemma 4.3 to simplify the approximation of $\psi ^\varepsilon (t^\flat + \delta )$ and exhibit the contribution of order $\sqrt{\varepsilon }$ according to

$$\begin{aligned} \psi ^\varepsilon (t^\flat +\delta )= \widehat{\vec V_1}(t^\flat +\delta )v^\varepsilon _1(t^\flat +\delta ) + {\mathrm{e}}^{iS^\flat /\varepsilon }\, {\mathcal {U}}_{h_2}^\varepsilon (t^\flat +\delta ,t^\flat )\, A_\varepsilon + O(\varepsilon \delta ^{-2}) +O(\varepsilon ^{1/3} \delta ). \end{aligned}$$

Then, we carefully analyze the contribution $A_\varepsilon$ and construct a preliminary transfer operator ${\mathcal {T}}^\varepsilon$ satisfying

$$\begin{aligned} A_\varepsilon = \mathcal {WP}^\varepsilon _{z^\flat } {{\mathcal {T}}}^\varepsilon \varphi _1(t^\flat ) + O(\sqrt{\varepsilon }\delta ), \end{aligned}$$

see Lemma 5.3. As the third step, Lemma 5.5 establishes the relation to the transfer operator ${\mathcal {T}}^\flat$ according to

$$\begin{aligned} {{\mathcal {T}}}^\varepsilon = \sqrt{\varepsilon }\,{\mathcal {Q}}^\varepsilon (0) {\mathcal {T}}^\flat + O(\sqrt{\varepsilon }\delta ) + O(\varepsilon \delta ^{-1}) \end{aligned}$$

with ${{\mathcal {Q}}}^\varepsilon (0) = {\mathrm{op}}_1^w((\gamma \vec V_2)(t^\flat ,z^\flat + \sqrt{\varepsilon }\bullet ))$. The wave packet relation (50) in combination with symbolic calculus implies for all $\varphi \in {\mathcal {S}}({{\mathbb {R}}}^d)$ that

$$\begin{aligned} \mathcal {WP} ^\varepsilon _{z^\flat } {{\mathcal {Q}}}^\varepsilon (0)\varphi&= \widehat{\gamma \vec V_2}(t^\flat ) \mathcal {WP} ^\varepsilon _{z^\flat }\varphi \\&= \widehat{\vec V_2}(t^\flat ) {\widehat{\gamma }}(t^\flat )\mathcal {WP} ^\varepsilon _{z^\flat }\varphi + O(\varepsilon ) = \widehat{\vec V_2}(t^\flat ) \gamma ^\flat \,\mathcal {WP} ^\varepsilon _{z^\flat }\varphi + O(\sqrt{\varepsilon }). \end{aligned}$$

Hence, we have proven that

$$\begin{aligned}&{\mathrm{e}}^{iS^\flat /\varepsilon }\, {\mathcal {U}}_{h_2}^\varepsilon (t^\flat +\delta ,t^\flat )\, A_\varepsilon \\&\quad = \sqrt{\varepsilon }\, \gamma ^\flat \,{\mathrm{e}}^{iS^\flat /\varepsilon }\, {\mathcal {U}}_{h_2}^\varepsilon (t^\flat +\delta ,t^\flat ) \widehat{\vec V_2}(t^\flat ) \mathcal {WP}^\varepsilon _{z^\flat } {{\mathcal {T}}}^\flat \varphi _1(t^\flat ) + O(\varepsilon \delta ^{-2}) +O(\varepsilon ^{1/3} \delta )\\&\quad =\sqrt{\varepsilon }\, {\mathcal {U}}_{h_2}^\varepsilon (t^\flat +\delta ,t^\flat )\, \widehat{\vec V_2}(t^\flat ) v^\varepsilon _2(t^\flat ) + O(\varepsilon \delta ^{-2}) +O(\varepsilon ^{1/3} \delta ). \end{aligned}$$

It remains to analyze the function $\omega (t) = \vec V_2(t){\mathcal {U}}^\varepsilon _{h_2}(t,t^\flat )-{\mathcal {U}}^\varepsilon _{h_2}(t,t^\flat )\vec V_2(t^\flat )$. An analogous calculation to the one at the beginning of the proof of Proposition 4.1 yields that

$$\begin{aligned} i\varepsilon \partial _t\omega = {{\widehat{h}}}_2\omega + O(\varepsilon ). \end{aligned}$$

Since $\omega (t^\flat ) = 0$, the Duhamel principle implies that $\omega (t^\flat +\delta ) = O(\delta )$ and

$$\begin{aligned} {\mathrm{e}}^{iS^\flat /\varepsilon }\, {\mathcal {U}}_{h_2}^\varepsilon (t^\flat +\delta ,t^\flat )\, A_\varepsilon = \sqrt{\varepsilon }\, \widehat{\vec V_2}(t^\flat +\delta ) {\mathcal {U}}_{h_2}^\varepsilon (t^\flat +\delta ,t^\flat )\, v^\varepsilon _2(t^\flat ) + O(\varepsilon \delta ^{-2}) +O(\varepsilon ^{1/3} \delta ). \end{aligned}$$

$\square$

5.1 Using the a priori estimate

We start describing the part of the wave packet that has been transferred at the crossing and identify its main contribution.

Lemma 5.2

Let $k\in {{\mathbb {N}}}$. With the assumptions of Proposition 5.1, we have in $\Sigma ^k_\varepsilon ({{\mathbb {R}}}^d)$,

$$\begin{aligned}\psi ^\varepsilon (t^\flat +\delta )= \widehat{\vec V_1}(t^\flat +\delta )v^\varepsilon _1(t^\flat +\delta ) + {\mathrm{e}}^{iS^\flat /\varepsilon }\, {\mathcal {U}}_{h_2}^\varepsilon (t^\flat +\delta ,t^\flat )\, A_\varepsilon + O(\varepsilon \delta ^{-2}) +O(\varepsilon ^{1/3} \delta )\end{aligned}$$

$$\begin{aligned} \text{ with }\;\;A_\varepsilon = \int _{t^\flat -\delta }^{t^\flat +\delta } {\mathcal {U}}_{h_2}^\varepsilon (t^\flat ,\sigma ) \widehat{\gamma \vec V_2}(\sigma )\, {\mathcal {U}}_{h_1}^\varepsilon (\sigma , t^\flat ) {\mathcal {WP}}^\varepsilon _{z^\flat } \varphi _1(t^\flat ) d\sigma , \end{aligned}$$

(43)

where the eigenvector $\vec V_2$ is defined in (28) and the Schwartz function $\varphi _1(t^\flat )$ is associated with the profile $\varphi _0$ of the initial wave packet according to Proposition 2.3.

Proof

We again analyse the functions $w^\varepsilon _1(t)$ and $w^\varepsilon _2(t)$ introduced in (40), that are of order $\varepsilon \delta ^{-2}$ at time $t=t^\flat -\delta$. By the a priori estimate of Lemma 4.3, the remainder terms $f^\varepsilon _1(t)$ and $f^\varepsilon _2(t)$, which appear in (41), are of order $\varepsilon ^{1/3}$. Therefore, the relation (42) gives for all times $t\in [t^\flat -\delta ,t^\flat +\delta ]$ and in $\Sigma ^k_\varepsilon ({{\mathbb {R}}}^d)$,

$$\begin{aligned} w^\varepsilon _1(t)&= O(\varepsilon \delta ^{-2}) + O(\delta \varepsilon ^{1/3}) ,\\ w^\varepsilon _2(t)&= O(\varepsilon \delta ^{-2}) + O(\delta \varepsilon ^{1/3}) + {1\over 2} \int _{t^\flat -\delta }^t {\mathcal U}_{h_2}^\varepsilon (t,\sigma ) \widehat{B_2 \Pi _1} \widehat{\vec V_1}(\sigma ) v_1^\varepsilon (\sigma )d\sigma . \end{aligned}$$

At this stage of the proof, we write $B_2\Pi _1 = \Pi _1B_2\Pi _1 + \Pi _2B_2\Pi _1$ and take advantage of $\displaystyle {\Pi _1B_2\Pi _1=(h_2 -h_1)\Pi _1 \{\Pi _1,\Pi _1\} \Pi _1}$ (see Lemma B.1) to write

$$\begin{aligned}&\int _{t^\flat -\delta }^t {{\mathcal {U}}}_{h_2}^\varepsilon (t,\sigma ) \widehat{\Pi _1 B_2 \Pi _1} \widehat{\vec V_1}(\sigma )\, {{\mathcal {U}}}^\varepsilon _{h_1}(\sigma ,t^\flat -\delta ) v^\varepsilon _1(t^\flat -\delta ) d\sigma \\&\quad = i\varepsilon \int _{t^\flat -\delta }^t {d\over d\sigma } \left( {{\mathcal {U}}}_{h_2}^\varepsilon (t,\sigma )\, {\mathrm{op}}_\varepsilon ^w\!\left( \Pi _1 \{\Pi _1,\Pi _1\} \Pi _1\vec V_1(\sigma )\right) {\mathcal U}_{h_1}^\varepsilon (\sigma ,t^\flat -\delta )\right) \\&\quad v_1^\varepsilon (t^\flat -\delta )\,d\sigma + \varepsilon \rho ^\varepsilon (t) \\&\quad = \varepsilon {{\tilde{\rho }}}^\varepsilon (t), \end{aligned}$$

where both families $(\rho ^\varepsilon (t))_{\varepsilon >0}$ and $({{\tilde{\rho }}}^\varepsilon (t))_{\varepsilon >0}$ are uniformly bounded in $\Sigma ^k_\varepsilon ({{\mathbb {R}}}^d)$. Therefore,

$$\begin{aligned} \int _{t^\flat -\delta }^t {{\mathcal {U}}}_{h_2}^\varepsilon (t,\sigma ) \widehat{B_2 \Pi _1} \widehat{\vec V_1}(\sigma ) v_1^\varepsilon (\sigma )d\sigma = \int _{t^\flat -\delta }^t {{\mathcal {U}}}_{h_2}^\varepsilon (t,\sigma ) \widehat{\Pi _2 B_2\Pi _1}\widehat{\vec V_1}(\sigma ) v_1^\varepsilon (\sigma )d\sigma + O(\varepsilon ). \end{aligned}$$

By Lemma B.1 and the definition of the eigenvector $\vec V_2$ (see (28))

$$\begin{aligned} \tfrac{1}{2} \Pi _2 B_2 \vec V_1 = \Pi _2(-\partial _t\Pi _2 -\{v,\Pi _2\})\vec V_1 = \gamma \vec V_2. \end{aligned}$$

According to Proposition 2.3, we have for the wave packet

$$\begin{aligned} v^\varepsilon _1(\sigma )&= {\mathcal {U}}_{h_1}(\sigma ,t^\flat ) v^\varepsilon _1 (t^\flat ) = {\mathcal {U}}_{h_1}(\sigma ,t^\flat ) {\mathrm{e}}^{iS^\flat /\varepsilon } {\mathcal {WP}}^\varepsilon _{z^\flat } \varphi _1(t^\flat ) +O(\varepsilon ). \end{aligned}$$

Therefore,

$$\begin{aligned}&{1\over 2} \int _{t^\flat -\delta }^t {\mathcal U}_{h_2}^\varepsilon (t,\sigma ) \widehat{B_2 \Pi _1} \widehat{\vec V_1}(\sigma ) v_1^\varepsilon (\sigma )d\sigma \\&\quad = {\mathrm{e}}^{iS^\flat /\varepsilon }\, {\mathcal U}_{h_2}^\varepsilon (t,t^\flat ) \int _{t^\flat -\delta }^t {\mathcal U}_{h_2}^\varepsilon (t^\flat ,\sigma ) \,\widehat{\gamma \vec V_2}(\sigma )\, {\mathcal {U}}_{h_1}(\sigma ,t^\flat ) {\mathcal {WP}}^\varepsilon _{z^\flat } \varphi _1(t^\flat )d\sigma + O(\varepsilon ), \end{aligned}$$

and, in terms of the function $A_\varepsilon$ is defined in (43), we are left at time $t=t^\flat +\delta$ with

$$\begin{aligned} w^\varepsilon _1(t^\flat +\delta )&=O(\varepsilon \delta ^{-2} ) + O(\delta \varepsilon ^{1/3}) ,\\ w^\varepsilon _2(t^\flat +\delta )&= {\mathrm{e}}^{iS^\flat /\varepsilon }\, {\mathcal {U}}_{h_2}^\varepsilon (t^\flat +\delta ,t^\flat )\, A_\varepsilon +O(\varepsilon \delta ^{-2} ) + O(\delta \varepsilon ^{1/3}). \end{aligned}$$

$\square$

5.2 Constructing the transfer operator

Next, we relate the transition term $A_\varepsilon$ to an integral operator that is defined in terms of the crossing parameters $\mu ^\flat$ and $(\alpha ^\flat ,\beta ^\flat )$ introduced in Theorem 3.8.

Lemma 5.3

Let $k\in {{\mathbb {N}}}$. With the assumptions of Proposition 5.1, there exist

• a smooth real-valued map $\sigma \mapsto \Lambda (\sigma )$ with $\Lambda (0) = 0$, ${{\dot{\Lambda }}}(0) = 0$, ${\ddot{\Lambda }}(0) = 2\mu ^\flat + \alpha ^\flat \cdot \beta ^\flat$,

• a smooth vector-valued map $\sigma \mapsto z(\sigma ) = (q(\sigma ),p(\sigma ))$ with $z(0) = 0$, $\dot{z}(0)= (\alpha ^\flat ,\beta ^\flat )$,

• a smooth map $\sigma \mapsto {{\mathcal {Q}}}^\varepsilon (\sigma )$ of operators, that map Schwartz functions to Schwartz functions, with ${\mathcal Q}^\varepsilon (0) = {\mathrm{op}}_1^w(\gamma \vec V_2(t^\flat ,z^\flat + \sqrt{\varepsilon }\bullet ))$,

such that the transition quantity $A_\varepsilon$ defined in Lemma 5.2 satisfies

$$\begin{aligned} A_\varepsilon = \mathcal {WP}^\varepsilon _{z^\flat } {{\mathcal {T}}}^\varepsilon \varphi _1(t^\flat ) + O(\sqrt{\varepsilon }\delta ) \end{aligned}$$

(44)

in $\Sigma ^k_\varepsilon ({{\mathbb {R}}}^d)$ for the integral operator ${\mathcal T}^\varepsilon$ defined by

$$\begin{aligned} {{\mathcal {T}}}^\varepsilon \varphi (y) = \int _{-\delta }^{+\delta } {\mathrm{e}}^{\frac{i}{\varepsilon } \Lambda (\sigma )} {{\mathcal {Q}}}^\varepsilon (\sigma ) {\mathrm{e}}^{i p_\varepsilon (\sigma )\cdot (y-q_\varepsilon (\sigma ))} \varphi (y-q_\varepsilon (\sigma )) \,d\sigma ,\quad \varphi \in {\mathcal S}({{\mathbb {R}}}^d), \end{aligned}$$

where we have used the scaling notation $z_\varepsilon (\sigma ) = z(\sigma )/\sqrt{\varepsilon }$.

Proof

We use Egorov’s semi-classical theorem [8, Theorem 12] (see also [4]) and obtain that in $\Sigma ^k_\varepsilon ({{\mathbb {R}}}^d)$,

$$\begin{aligned} {\mathcal {U}}_{h_2}^\varepsilon (t^\flat ,\sigma ) \widehat{\gamma \vec V_2}(\sigma )f = {\mathrm{op}}^w_\varepsilon ( (\gamma \vec V_2)(\sigma )\circ \Phi ^{\sigma ,t^\flat }_2 )\,{\mathcal {U}}_{h_2}^\varepsilon (t^\flat ,\sigma )f+ O(\varepsilon ) \end{aligned}$$

for all $f\in \bigcap _{\ell \ge k}\Sigma ^\ell _\varepsilon ({{\mathbb {R}}}^d)$. Hence,

$$\begin{aligned} A_\varepsilon = \int _{t^\flat -\delta }^{t^\flat +\delta } {\mathrm{op}}^w_\varepsilon ((\gamma \vec V_2)(\sigma )\circ \Phi ^{\sigma ,t^\flat }_2)\, {\mathcal {U}}_{h_2}^\varepsilon (t^\flat , \sigma )\,{\mathcal {U}}_{h_1}^\varepsilon (\sigma ,t^\flat ) \,\mathcal {WP}_{z^\flat }^\varepsilon \varphi _1(t^\flat )d\sigma +O(\delta \varepsilon ). \end{aligned}$$

We set $\vec Q_2(\sigma )= (\gamma \vec V_2)(t^\flat +\sigma )\circ \Phi ^{\sigma +t^\flat ,t^\flat }_2,$ and note that $\vec Q_2(0) = (\gamma \vec V_2)(t^\flat )$. We get after a change of variables

$$\begin{aligned} A_\varepsilon&= \int _{-\delta }^{\delta } \widehat{\vec Q_2}(\sigma )\, {\mathcal {U}}_{h_2}^\varepsilon (t^\flat , t^\flat +\sigma ){\mathcal {U}}_{h_1}^\varepsilon (t^\flat +\sigma ,t^\flat ) \mathcal {WP}_{z^\flat }^\varepsilon \varphi _1(t^\flat )d\sigma +O(\delta \varepsilon ). \end{aligned}$$

Now we apply successively Proposition 2.3 to the evolutions ${{\mathcal {U}}}_{h_1}^\varepsilon$ and ${{\mathcal {U}}}_{h_2}^\varepsilon$ without encorporating the first amplitude correction, that is, for a basic approximation of order $\sqrt{\varepsilon }$. We obtain

$$\begin{aligned} {\mathcal {U}}_{h_2}^\varepsilon (t^\flat , t^\flat +\sigma ){\mathcal {U}}_{h_1}^\varepsilon (t^\flat +\sigma ,t^\flat ) \mathcal {WP}_{z^\flat }^\varepsilon \varphi _1(t^\flat ) = {\mathrm{e}}^{\frac{i}{\varepsilon }S(\sigma )} \mathcal {WP}_{\zeta (\sigma )}^\varepsilon {\mathcal M}(\sigma )\varphi _1(t^\flat ) + O(\sqrt{\varepsilon }), \end{aligned}$$

where we denoted the combined center, phase and metaplectic transform by

$$\begin{aligned} \zeta (\sigma )&=\Phi _2^{t^\flat ,t^\flat + \sigma }\big (\Phi _1^{t^\flat +\sigma ,t^\flat }(z^\flat )\big ),\\ S(\sigma )&= S_1(t^\flat +\sigma ,t^\flat , z^\flat ) +S_2(t^\flat ,t^\flat +\sigma ,\Phi _1^{t^\flat +\sigma , t^\flat }(z^\flat )),\\ {{\mathcal {M}}}(\sigma )&= {{\mathcal {M}}}[F_2(t^\flat ,t^\flat +\sigma ,\Phi _1^{t^\flat +\sigma , t^\flat }(z^\flat ))] \,{{\mathcal {M}}}[F_1(t^\flat +\sigma ,t^\flat ,z^\flat )]. \end{aligned}$$

This implies

$$\begin{aligned} A_\varepsilon = \int _{-\delta }^{+\delta } \widehat{\vec Q_2}(\sigma ) {\mathrm{e}}^{\frac{i}{\varepsilon }S(\sigma )} \mathcal {WP}_{\zeta (\sigma )}^\varepsilon {\mathcal M}(\sigma )\varphi _1(t^\flat ) d\sigma + O(\delta \sqrt{\varepsilon }). \end{aligned}$$

We observe that

$$\begin{aligned} \zeta (0) = z^\flat ,\quad S(0) = 0, \quad {{\mathcal {M}}}(0) = {\mathbb {I}}, \end{aligned}$$

and write $\zeta (\sigma ) = z^\flat + z(\sigma )$ with $z(0) = 0$. By Lemma D.1,

$$\begin{aligned} \dot{z}(0) = (\alpha ^\flat ,\beta ^\flat ), \quad \dot{S}(0) = p^\flat \cdot \alpha ^\flat . \end{aligned}$$

Moreover, using the group and translation properties of the wave packet transform (49) and (48), we have

$$\begin{aligned} \mathcal {WP}^\varepsilon _{\zeta (\sigma )}&= {\mathrm{e}}^{-\frac{i}{\varepsilon } p^\flat \cdot q(\sigma )} \mathcal {WP}^\varepsilon _{z^\flat }\Lambda _\varepsilon ^{-1} \mathcal {WP}^\varepsilon _{z(\sigma )}\\&= {\mathrm{e}}^{-\frac{i}{\varepsilon } p^\flat \cdot q(\sigma )} {\mathrm{e}}^{-\frac{i}{2\varepsilon }p(\sigma )\cdot q(\sigma )} \mathcal {WP}^\varepsilon _{z^\flat } \Lambda _\varepsilon ^{-1} {{\widehat{T}}}^\varepsilon (z(\sigma )) \Lambda _\varepsilon \\*[1ex]&= {\mathrm{e}}^{-\frac{i}{\varepsilon } p^\flat \cdot q(\sigma )} {\mathrm{e}}^{-\frac{i}{2\varepsilon }p(\sigma )\cdot q(\sigma )} \mathcal {WP}^\varepsilon _{z^\flat } {{\widehat{T}}}^1(z_\varepsilon (\sigma )), \end{aligned}$$

By the translation properties of the metaplectic transform [8, Section 3.3], we have

$$\begin{aligned} {{\widehat{T}}}^1(z_\varepsilon (\sigma )) {{\mathcal {M}}}(\sigma )= {\mathcal M}(\sigma ){{\widehat{T}}}^1({{\widetilde{z}}}_\varepsilon (\sigma )) \end{aligned}$$

with new center

$$\begin{aligned} {{\widetilde{z}}}(\sigma ) = F_1(t^\flat +\sigma ,t^\flat ,z^\flat )^{-1} F_2(t^\flat ,t^\flat +\sigma ,\Phi _1^{t^\flat +\sigma , t^\flat }(z^\flat ))^{-1} z(\sigma ) \end{aligned}$$

We observe that

$$\begin{aligned} {{\widetilde{z}}}(0) = z(0) = 0,\quad \dot{{{\widetilde{z}}}}(0) = \dot{z}(0) = (\alpha ^\flat ,\beta ^\flat ). \end{aligned}$$

Moreover, in view of the relation (50),

$$\begin{aligned} \widehat{\vec Q_2}(\sigma )\mathcal {WP}_{\zeta (\sigma )}^\varepsilon {\mathcal M}(\sigma )&= {\mathrm{e}}^{-\frac{i}{\varepsilon } p^\flat \cdot q(\sigma )} {\mathrm{e}}^{-\frac{i}{2\varepsilon }p(\sigma )\cdot q(\sigma )} \widehat{\vec Q_2}(\sigma ) \mathcal {WP}^\varepsilon _{z^\flat } {{\mathcal {M}}}(\sigma )\widehat{T}^1({{\widetilde{z}}}_\varepsilon (\sigma ))\\&= {\mathrm{e}}^{-\frac{i}{\varepsilon } p^\flat \cdot q(\sigma )} {\mathrm{e}}^{-\frac{i}{2\varepsilon }p(\sigma )\cdot q(\sigma )} \mathcal {WP}^\varepsilon _{z^\flat } {\mathrm{op}}_1^w(\vec Q_2(\sigma ,z^\flat +\sqrt{\varepsilon }\bullet ))\\&{{\mathcal {M}}}(\sigma )\widehat{T}^1({{\widetilde{z}}}_\varepsilon (\sigma )). \end{aligned}$$

Since

$$\begin{aligned} {{\widehat{T}}}^1({{\widetilde{z}}}_\varepsilon (\sigma ))\varphi _1(t^\flat ,y) = {\mathrm{e}}^{\frac{i}{2}{{\widetilde{q}}}_\varepsilon (\sigma )\cdot \widetilde{p}_\varepsilon (\sigma )} {\mathrm{e}}^{i {{\widetilde{p}}}_\varepsilon (\sigma )\cdot (y-{{\widetilde{q}}}_\varepsilon (\sigma ))} \varphi _1(t^\flat ,y-\widetilde{q}_\varepsilon (\sigma )), \end{aligned}$$

we may introduce the phase ${{\widetilde{\Lambda }}}(\sigma )$ and the operator ${{\mathcal {Q}}}^\varepsilon (\sigma )$ acccording to

$$\begin{aligned} {{\widetilde{\Lambda }}}(\sigma )&= S(\sigma ) - p^\flat \cdot q(\sigma ) - p(\sigma )\cdot q(\sigma ) + {{\widetilde{p}}}(\sigma )\cdot {{\widetilde{q}}}(\sigma ),\nonumber \\ {{\mathcal {Q}}}^\varepsilon (\sigma )&= {\mathrm{op}}_1^w(\vec Q_2(\sigma ,z^\flat +\sqrt{\varepsilon }\bullet )) {{\mathcal {M}}}(\sigma ), \end{aligned}$$

(45)

to obtain the approximation

$$\begin{aligned} A_\varepsilon = \mathcal {WP}^\varepsilon _{z^\flat } \int _{-\delta }^{+\delta } {\mathrm{e}}^{\frac{i}{\varepsilon } {{\widetilde{\Lambda }}}(\sigma )} {\mathcal Q}^\varepsilon (\sigma ) {\mathrm{e}}^{i {{\widetilde{p}}}_\varepsilon (\sigma )\cdot (y-{{\widetilde{q}}}_\varepsilon (\sigma ))} \varphi _1(t^\flat ,y-\widetilde{q}_\varepsilon (\sigma )) \,d\sigma + O(\delta \sqrt{\varepsilon }). \end{aligned}$$

We clearly have ${{\widetilde{\Lambda }}}(0) = \dot{{{\widetilde{\Lambda }}}}(0) = 0$ and ${{\mathcal {Q}}}^\varepsilon (0) = {\mathrm{op}}_1^w((\gamma \vec V_2)(t^\flat ,z^\flat +\sqrt{\varepsilon }\bullet ))$, whereas, by Lemma D.1,

$$\begin{aligned} \ddot{{{\widetilde{\Lambda }}}}(0) = {\ddot{S}}(0) - p^\flat \cdot \ddot{q}(0) = 2\mu ^\flat + \alpha ^\flat \cdot \beta ^\flat . \end{aligned}$$

$\square$

Remark 5.4

Note that the first step of the proof of Lemma 5.3 can be performed at any order in $\varepsilon$ with a remainder of the form $O(\delta \varepsilon ^N)$: pushing the Egorov theorem at higher order, we obtain

$$\begin{aligned}A_\varepsilon= & {} \int _{t^\flat -\delta }^{t^\flat +\delta } \widehat{\vec Q^{\varepsilon ,N}_2}(\sigma )\, {\mathcal {U}}_{h_2}^\varepsilon (t^\flat , \sigma ){\mathcal {U}}_{h_1}^\varepsilon (\sigma ,t^\flat ) \mathcal {WP}_{z^\flat }^\varepsilon \varphi _1(t^\flat )d\sigma +O(\delta \varepsilon ^{N+1})\\&\quad \quad \text{ with }\;\;\vec Q_2^{\varepsilon ,N}= \vec Q_2 + \varepsilon \vec Q_{2}^{(1)}+\cdots +\varepsilon ^N \vec Q_{2}^{(N)}.\end{aligned}$$

Similarly, also Proposition 2.3 can be generalized at any order in $\varepsilon$, which then implies

$$\begin{aligned}A_\varepsilon = \mathcal {WP}^\varepsilon _{z^\flat } \,{{\mathcal {T}}}^{\varepsilon ,N} \varphi _1^\varepsilon (t^\flat ) +O(\varepsilon ^{N/2+1} \delta )\end{aligned}$$

where $\displaystyle {\varphi _1^\varepsilon =\varphi _1+\sqrt{\varepsilon }\varphi _1^{(1)} + \cdots +\varepsilon ^{N/2} \varphi _1^{(N)}}$ and

$$\begin{aligned}{{\mathcal {T}}}^{\varepsilon ,N}\varphi (y)=\int _{-\delta }^{+\delta }{\mathrm{e}}^{\frac{i}{\varepsilon } \Lambda (\sigma )} {{\mathcal {Q}}}^{\varepsilon ,N}(\sigma ) {\mathrm{e}}^{ i (y-q_\varepsilon (\sigma )))\cdot p_\varepsilon (\sigma )} \varphi (y-q_\varepsilon (\sigma )) d\sigma \end{aligned}$$

for all $\varphi \in {{\mathcal {S}}}({{\mathbb {R}}}^d)$. The phase function $\Lambda (\sigma )$ and the phase space center $z(\sigma )$ stay the same as in Lemma 5.3, while the operator ${\mathcal Q}^{\varepsilon ,N}(\sigma )$ is associated with $\vec Q_2^{\varepsilon , N}(\sigma )$ according to (45) by selecting terms up to order $\varepsilon ^{N/2}$ in its definition.

5.3 The transfer operator

Consider the family of operators

$$\begin{aligned} {{\mathcal {T}}}^\varepsilon \varphi (y)= & {} \int _{-\delta }^{+\delta }{\mathrm{e}}^{\frac{i}{\varepsilon } \Lambda (\sigma )} {{\mathcal {Q}}}^\varepsilon (\sigma ) {\mathrm{e}}^{ i (y-q_\varepsilon (\sigma )))\cdot p_\varepsilon (\sigma )} \varphi (y-q_\varepsilon (\sigma )) d\sigma ,\quad \varphi \in {\mathcal S}({{\mathbb {R}}}^d), \end{aligned}$$

as introduced in Lemma 5.3. We next describe such an operator ${{\mathcal {T}}}^\varepsilon$, when $\varepsilon$ goes to 0, with control in the norm of $\Sigma ^k_\varepsilon ({{\mathbb {R}}}^d)$ for $k=0$. In Appendix 5.4 we slightly extend the construction of the transfer operator to gain control also for the general case $k\ge 0$.

Lemma 5.5

Let $k\in {{\mathbb {N}}}$. If $\sqrt{\varepsilon }\ll \delta \ll 1$, then for all $\varphi \in {{\mathcal {S}}}({{\mathbb {R}}}^d)$,

$$\begin{aligned} {{\mathcal {T}}}^\varepsilon \varphi = \sqrt{\varepsilon }\,{\mathcal {Q}}^\varepsilon (0) \mathcal T^\flat \varphi + O(\sqrt{\varepsilon }\delta ) + O(\varepsilon \delta ^{-1}) \end{aligned}$$

(46)

in $L^2({{\mathbb {R}}}^d)$ with $\displaystyle { {\mathcal {T}}^\flat = \int _{-\infty }^{+\infty } {\mathrm{e}}^{i\mu ^\flat s^2 } {\mathrm{e}}^{is(\beta ^\flat \cdot y-\alpha ^\flat \cdot D_y)} \,ds. }$

Proof

The proof relies on the analysis of the integrand close to $\sigma =0$. We write

$$\begin{aligned} {{\mathcal {T}}}^\varepsilon =\sqrt{\varepsilon }\int _{-\delta /\sqrt{\varepsilon }}^{+\delta /\sqrt{\varepsilon }}{\mathrm{e}}^{\frac{i}{\varepsilon } \Lambda (\sqrt{\varepsilon }s) -\frac{i}{2} q_\varepsilon (s\sqrt{\varepsilon }) \cdot p_\varepsilon (s\sqrt{\varepsilon }) } {{\mathcal {Q}}}^\varepsilon (s\sqrt{\varepsilon }) {\mathrm{e}}^{i L^\varepsilon (s) }ds \end{aligned}$$

where $\displaystyle {L^\varepsilon (s) := p_\varepsilon (s\sqrt{\varepsilon })\cdot y -q_\varepsilon (s\sqrt{\varepsilon }) D_y }$ defines a family of self-adjoint operators $s \mapsto L^\varepsilon (s)$ mappping ${\mathcal {S}}({{\mathbb {R}}}^d)$ into itself. Recall that the functions $s\mapsto p_\varepsilon (s\sqrt{\varepsilon })$ and $s\mapsto q_\varepsilon (s\sqrt{\varepsilon })$ are uniformly bounded with respect to $\varepsilon$, and that $q(0) = p(0) = 0$, while

$$\begin{aligned} \mu ^\flat =\frac{1}{2} \left( {\ddot{\Lambda }}(0)- \dot{q}(0) \cdot \dot{p}(0)\right) ,\;\;\alpha ^\flat =\dot{q}(0),\;\;\beta ^\flat =\dot{p}(0). \end{aligned}$$

(47)

We set $L= \beta ^\flat \cdot y-\alpha ^\flat \cdot D_y.$ Using Taylor expansion in $s=0$, we obtain

$$\begin{aligned}\frac{1}{\varepsilon }\Lambda (\sqrt{\varepsilon }s) -\frac{1}{2} q_\varepsilon (s\sqrt{\varepsilon })\cdot p_\varepsilon (s\sqrt{\varepsilon }) = \mu ^\flat s^2 + \sqrt{\varepsilon }s^3 f_1(s\sqrt{\varepsilon }) \end{aligned}$$

with $\sigma \mapsto f_1(\sigma )$ bounded, together with its derivatives, for $\sigma \in [t_0,t_0+T]$. In the following, the notation $f_j$ will denote functions that have the same property. We also have

$$\begin{aligned}L^\varepsilon (s) = sL +\sqrt{\varepsilon }s^2 L^\varepsilon _1(s\sqrt{\varepsilon })\end{aligned}$$

where the family of operator $\sigma \mapsto L^\varepsilon _1(\sigma )$ maps ${\mathcal {S}}({{\mathbb {R}}}^d)$ into itself, for $\sigma \in [t_0,t_0+T]$. Besides, the commutator $[ L, L_1(s\sqrt{\varepsilon })]$ is a scalar, and we set

$$\begin{aligned}\frac{1}{2}[ L, L_1(s\sqrt{\varepsilon })] = f_2(s\sqrt{\varepsilon })\end{aligned}$$

with the notation we have just introduced. Therefore, by Baker-Campbell-Hausdorff formula

$$\begin{aligned}{\mathrm{e}}^{iL^\varepsilon (s)} = {\mathrm{e}}^{isL} {\mathrm{e}}^{is^2\sqrt{\varepsilon }L_1(s\sqrt{\varepsilon })}{ \mathrm e} ^{i \sqrt{\varepsilon }s^3 f_2(s\sqrt{\varepsilon })}.\end{aligned}$$

Besides,

$$\begin{aligned}{\mathrm{e}}^{i\sqrt{\varepsilon }s^2 L_1(s\sqrt{\varepsilon })} =\mathrm{Id} + \sqrt{\varepsilon }s^2 \Theta (s\sqrt{\varepsilon })\end{aligned}$$

where the operator-valued map $\sigma \mapsto \Theta (\sigma )$ is smooth and such that for all $\sigma \in [t_0,t_0+T]$, the operator $\Theta (\sigma )$ and its derivatives maps ${\mathcal {S}}({{\mathbb {R}}}^d)$ into itself. Setting $f_3=f_1+f_2$, we deduce that ${{\mathcal {T}}}^\varepsilon$ writes

$$\begin{aligned}&{{\mathcal {T}}}^\varepsilon =\sqrt{\varepsilon }\int _{-\delta /\sqrt{\varepsilon }}^{+\delta /\sqrt{\varepsilon }} {\mathrm{e}}^{i\mu ^\flat s^2 + \sqrt{\varepsilon }s^3 f_3(s\sqrt{\varepsilon })} {{\mathcal {Q}}}^\varepsilon (s\sqrt{\varepsilon }) {\mathrm{e}}^{i sL } ds + R^{\varepsilon ,\delta }\\ \text{ with }\;\;&R^{\varepsilon ,\delta }= \varepsilon \int _{-\delta /\sqrt{\varepsilon }}^{+\delta /\sqrt{\varepsilon }} {\mathrm{e}}^{i\mu ^\flat s^2 + \sqrt{\varepsilon }s^3 f_3(s\sqrt{\varepsilon })} {{\mathcal {Q}}}^\varepsilon (s\sqrt{\varepsilon }) {\mathrm{e}}^{i sL } s^2 \Theta ^\varepsilon (s\sqrt{\varepsilon })ds. \end{aligned}$$

Let us analyze $R^{\varepsilon ,\delta }$. For this, we perform an integration by parts. Indeed,

$$\begin{aligned}\partial _s ( \mu ^\flat s^2 +\sqrt{\varepsilon }s^3 f_3(s\sqrt{\varepsilon }))= 2 \mu ^\flat s ( 1+ s\sqrt{\varepsilon }f_4(s\sqrt{\varepsilon }))\end{aligned}$$

for some smooth bounded function $f_4$ with bounded derivatives. Moreover, since $\delta$ is small, we have $1+s\sqrt{\varepsilon }f_4(s\sqrt{\varepsilon }) > 1/2$ for all $s\in ]-\delta /\sqrt{\varepsilon },+\delta /\sqrt{\varepsilon }[$. Therefore, we can write

$$\begin{aligned} R^{\varepsilon ,\delta } =&\left[ \frac{\varepsilon s}{2i\mu ^\flat ( 1+ s\sqrt{\varepsilon }f_4(s\sqrt{\varepsilon }))} {\mathrm{e}}^{i\mu ^\flat s^2 +i\sqrt{\varepsilon }s^3 f_3(s\sqrt{\varepsilon })} {{\mathcal {Q}}}^\varepsilon (s\sqrt{\varepsilon }) {\mathrm{e}}^{isL} \right] _{-\frac{\delta }{\sqrt{\varepsilon }}}^{+\frac{\delta }{\sqrt{\varepsilon }}} \\&- \frac{\varepsilon }{2i\mu ^\flat } \int _{-\frac{\delta }{\sqrt{\varepsilon }}}^{+\frac{\delta }{\sqrt{\varepsilon }}} {\mathrm{e}}^{i\mu ^\flat s^2+i\sqrt{\varepsilon }s^3 f_3(s\sqrt{\varepsilon })} \frac{d}{ds}\left( \frac{s}{1+ s\sqrt{\varepsilon }f_4(s\sqrt{\varepsilon })} {{\mathcal {Q}}}^\varepsilon (s\sqrt{\varepsilon }) {\mathrm{e}}^{isL} \right) ds, \end{aligned}$$

where $\mu ^\flat \ne 0$ by the transversality condition (24). We deduce that for all $k\in {{\mathbb {N}}}$ and $\varphi \in {{\mathcal {S}}}({{\mathbb {R}}}^d)$, we have in $L^2({{\mathbb {R}}}^d)$ that $R^{\varepsilon ,\delta } \varphi = O(\sqrt{\varepsilon }\delta ) + R^{\varepsilon ,\delta }_1\varphi$ with

$$\begin{aligned}R^{\varepsilon ,\delta }_1\varphi = - \frac{\varepsilon }{2i\mu ^\flat } \int _{-\frac{\delta }{\sqrt{\varepsilon }}}^{+\frac{\delta }{\sqrt{\varepsilon }}} {\mathrm{e}}^{i\mu ^\flat s^2+i\sqrt{\varepsilon }s^3 f_3(s\sqrt{\varepsilon })} \left( \frac{s}{1+ s\sqrt{\varepsilon }f_4(s\sqrt{\varepsilon })} {{\mathcal {Q}}}^\varepsilon (s\sqrt{\varepsilon }) {\mathrm{e}}^{isL} L\varphi \right) ds. \end{aligned}$$

We then need another integration by parts to obtain that $R^{\varepsilon ,\delta }_1 \varphi = O(\sqrt{\varepsilon }\delta )$. Note that this additional integration by parts is required by the presence of a s without a coefficient $\sqrt{\varepsilon }$ in the integrand. We write

$$\begin{aligned} R^{\varepsilon ,\delta }_1\varphi&= - \frac{\varepsilon }{(2i\mu ^\flat )^2} \left[ {\mathrm{e}}^{i\mu ^\flat s^2+i\sqrt{\varepsilon }s^3 f_3(s\sqrt{\varepsilon })} \left( \frac{1}{(1+ s\sqrt{\varepsilon }f_4(s\sqrt{\varepsilon }))^2} {{\mathcal {Q}}}^\varepsilon (s\sqrt{\varepsilon }) {\mathrm{e}}^{isL} L\varphi \right) \right] _{-\frac{\delta }{\sqrt{\varepsilon }}}^{+\frac{\delta }{\sqrt{\varepsilon }}} \\&\quad + \frac{\varepsilon }{(2i\mu ^\flat )^2} \int _{-\frac{\delta }{\sqrt{\varepsilon }}}^{+\frac{\delta }{\sqrt{\varepsilon }}} {\mathrm{e}}^{i\mu ^\flat s^2+i\sqrt{\varepsilon }s^3 f_3(s\sqrt{\varepsilon })} \frac{d}{ds}\left( \frac{1}{(1+ s\sqrt{\varepsilon }f_4(s\sqrt{\varepsilon }))^2} {\mathcal Q}^\varepsilon (s\sqrt{\varepsilon }) {\mathrm{e}}^{isL} L\varphi \right) ds\\&= O(\delta \sqrt{\varepsilon }) \end{aligned}$$

Therefore, we are left with

$$\begin{aligned} {{\mathcal {T}}}^\varepsilon = \sqrt{\varepsilon }\int _{-\frac{\delta }{\sqrt{\varepsilon }}}^{+\frac{\delta }{\sqrt{\varepsilon }}} {\mathrm{e}}^{i\mu ^\flat s^2 +i\sqrt{\varepsilon }s^3 f_3(s\sqrt{\varepsilon })} {\mathcal Q}^\varepsilon (s\sqrt{\varepsilon }) {\mathrm{e}}^{isL} \,ds + O(\sqrt{\varepsilon }\delta ) . \end{aligned}$$

We perform the change of variable

$$\begin{aligned} z= s(1+\sqrt{\varepsilon }s f_3(s\sqrt{\varepsilon })/\mu ^\flat )^{1/2} \end{aligned}$$

and observe that $s=z(1+\sqrt{\varepsilon }z g_1(z\sqrt{\varepsilon }) ) \;\;\text{ and } \;\; \partial _s z = 1+\sqrt{\varepsilon }z g_2( z\sqrt{\varepsilon })$ for some smooth bounded functions $g_1$ and $g_2$ with bounded derivatives. Note that, here again, we have used that $s\sqrt{\varepsilon }$ is small in the domain of the integral. Besides, there exists a family of operator $\widetilde{{\mathcal {Q}}}^\varepsilon (z)$ such that ${{\mathcal {Q}}}^\varepsilon (s\sqrt{\varepsilon }) = \widetilde{{\mathcal {Q}}}^\varepsilon (z\sqrt{\varepsilon })$ with $\widetilde{{\mathcal {Q}}}^\varepsilon (0)= {{\mathcal {Q}}}^\varepsilon (0)$. We deduce that there exists a bounded function of $\delta$ denoted by $b(\delta )$ such that

$$\begin{aligned} {\mathcal {T}}^\varepsilon = \sqrt{\varepsilon }\int _{-\frac{\delta }{\sqrt{\varepsilon }}}^{+\frac{\delta }{\sqrt{\varepsilon }} b(\delta )} {\mathrm{e}}^{i\mu ^\flat z^2 } \widetilde{{\mathcal {Q}}}^\varepsilon (z\sqrt{\varepsilon }) {\mathrm{e}}^{iz(1+\sqrt{\varepsilon }z g_1(z\sqrt{\varepsilon }))\, L} \frac{dz}{1+\sqrt{\varepsilon }z g_2( z\sqrt{\varepsilon })}. \end{aligned}$$

A Taylor expansion allows to write

$$\begin{aligned} \widetilde{{\mathcal {Q}}}^\varepsilon (z\sqrt{\varepsilon }) {\mathrm{e}}^{iz(1+\sqrt{\varepsilon }z g_1(z\sqrt{\varepsilon }))\, L} \frac{1}{1+\sqrt{\varepsilon }z g_2( z\sqrt{\varepsilon })}&= \widetilde{{\mathcal {Q}}}^\varepsilon (0)+ \sqrt{\varepsilon }z (\widetilde{\mathcal Q}^\varepsilon _1(z\sqrt{\varepsilon })+ z \widetilde{\mathcal Q}^\varepsilon _2(z\sqrt{\varepsilon }))\\&= {{\mathcal {Q}}}^\varepsilon (0)+ \sqrt{\varepsilon }z (\widetilde{\mathcal Q}^\varepsilon _1(z\sqrt{\varepsilon })+ z \widetilde{{\mathcal {Q}}}^\varepsilon _2(z\sqrt{\varepsilon })) \end{aligned}$$

for some smooth operator-valued maps $z\mapsto \widetilde{\mathcal Q}^\varepsilon _j(z\sqrt{\varepsilon })$ mapping ${\mathcal {S}}({{\mathbb {R}}}^d)$ into itself, such that for all $\varphi \in {{\mathcal {S}}}({{\mathbb {R}}}^d)$ the family $({\widetilde{{\mathcal {Q}}}^\varepsilon _j(z\sqrt{\varepsilon })}\varphi )_{\varepsilon >0}$ is bounded in $L^2({{\mathbb {R}}}^d)$. We obtain

$$\begin{aligned}&{\mathcal {T}}^\varepsilon = \sqrt{\varepsilon }\; {{\mathcal {Q}}}^\varepsilon (0) \int _{-\frac{\delta }{\sqrt{\varepsilon }}}^{+\frac{\delta }{\sqrt{\varepsilon }} b(\delta )} {\mathrm{e}}^{i\mu ^\flat z^2 } {\mathrm{e}}^{izL} dz + \tilde{R}^{\varepsilon ,\delta }\\ \text{ with }\;\;&{{\tilde{R}}}^{\varepsilon ,\delta } = \varepsilon \int _{-\frac{\delta }{\sqrt{\varepsilon }}}^{+\frac{\delta }{\sqrt{\varepsilon }} b(\delta )} z \, {\mathrm{e}}^{i\mu ^\flat z^2 } (\widetilde{{\mathcal {Q}}}^\varepsilon _1 (z\sqrt{\varepsilon }) +z\widetilde{{\mathcal {Q}}}^\varepsilon _2(z\sqrt{\varepsilon }) )\, dz. \end{aligned}$$

Arguing by integration by parts as previously, we obtain

$$\begin{aligned} {{\tilde{R}}}^{\varepsilon ,\delta } =&\varepsilon \left[ \frac{1}{2i\mu ^\flat } {\mathrm{e}}^{i\mu ^\flat z^2 } (\widetilde{{\mathcal {Q}}}^\varepsilon _1 (z\sqrt{\varepsilon }) +z\widetilde{{\mathcal {Q}}}^\varepsilon _2(z\sqrt{\varepsilon }) )\right] _{-\frac{\delta }{\sqrt{\varepsilon }}}^{+\frac{\delta }{\sqrt{\varepsilon }} b(\delta )} \\&\;\; - \frac{\varepsilon }{2i\mu ^\flat } \int _{-\frac{\delta }{\sqrt{\varepsilon }}}^{+\frac{\delta }{\sqrt{\varepsilon }} b(\delta )} \, {\mathrm{e}}^{i\mu ^\flat z^2 }\frac{d}{dz} (\widetilde{{\mathcal {Q}}}^\varepsilon _1 (z\sqrt{\varepsilon }) +z\widetilde{{\mathcal {Q}}}^\varepsilon _2(z\sqrt{\varepsilon }) )\, dz =O(\sqrt{\varepsilon }\delta ) . \end{aligned}$$

We deduce $\displaystyle { {{\mathcal {T}}}^\varepsilon = \sqrt{\varepsilon }\, {\mathcal Q}^\varepsilon (0 )\, \int _{-\frac{\delta }{\sqrt{\varepsilon }} b(\delta )}^{+\frac{\delta }{\sqrt{\varepsilon }}b(\delta )} {\mathrm{e}}^{i\mu ^\flat s^2 } {\mathrm{e}}^{isL} \,ds + O(\sqrt{\varepsilon }\delta ) }$ and it remains to pass to infinity in the domain of the integral. For this, we set $m_\varepsilon =\frac{\delta }{\sqrt{\varepsilon }}b(\delta )$ and consider for $\varphi \in {{\mathcal {S}}}({{\mathbb {R}}}^d)$,

$$\begin{aligned}{\mathcal {G}}_0^\varepsilon \varphi = \int _{m_\varepsilon }^{+\infty } {\mathrm{e}}^{i\mu ^\flat s^2 } {\mathrm{e}}^{isL} \varphi \,ds.\end{aligned}$$

We make two successive integration by parts. We write in $L^2({{\mathbb {R}}}^d)$,

$$\begin{aligned} {\mathcal {G}}_0^\varepsilon \varphi&= \left[ (2is\mu ^\flat )^{-1} {\mathrm{e}}^{i\mu ^\flat s^2} {\mathrm{e}}^{ isL} \varphi \right] _{m_\varepsilon }^{+\infty } - \int _{m_\varepsilon }^{+\infty } {\mathrm{e}}^{i\mu ^\flat s^2} \frac{d}{ds} \left( \frac{{\mathrm{e}}^{isL} \varphi }{2is\mu ^\flat }\right) ds\\&= O( m_\varepsilon ^{-1}) \Vert \varphi \Vert _{\Sigma ^{k}} - \int _{m_\varepsilon }^{+\infty } {\mathrm{e}}^{i\mu ^\flat s^2} \frac{i{\mathrm{e}}^{isL} L\varphi }{2is\mu ^\flat } ds +\int _{m_\varepsilon }^{+\infty } {\mathrm{e}}^{i\mu ^\flat s^2} \frac{{\mathrm{e}}^{isL} \varphi }{2i\mu ^\flat s^2 }ds\\&= O( m_\varepsilon ^{-1}) \Vert \varphi \Vert _{\Sigma ^{k}} {-} \left[ (2is\mu ^\flat )^{-2} {\mathrm{e}}^{i\mu ^\flat s^2} i{\mathrm{e}}^{ isL} L\varphi \right] _{m_\varepsilon }^{+\infty } {+} \int _{m_\varepsilon }^{+\infty } {\mathrm{e}}^{i\mu ^\flat s^2} \frac{d}{ds}\left( \frac{i{\mathrm{e}}^{isL} L\varphi }{(2is\mu ^\flat )^2 }\right) ds\\*[1ex]&= O( m_\varepsilon ^{-1}) \left( \Vert \varphi \Vert _{\Sigma ^{k}} + \Vert L\varphi \Vert _{\Sigma ^{k}} + \Vert L^2\varphi \Vert _{\Sigma ^{k}}\right) . \end{aligned}$$

We deduce that $\displaystyle { {{\mathcal {T}}}^\varepsilon = \sqrt{\varepsilon }\,{{\mathcal {Q}}}^\varepsilon (0)\, \int _{-\infty }^{+\infty } {\mathrm{e}}^{i\mu ^\flat s^2 } {\mathrm{e}}^{isL} \,ds + O(\sqrt{\varepsilon }\delta ) +O(\varepsilon \delta ^{-1}). }$ $\square$

Remark 5.6

Note that the previous remainder terms could again be transformed by integration by parts. This implies that ${\mathcal {T}}^\varepsilon \varphi$ has an asymptotic expansion in $\sqrt{\varepsilon }$ and $\delta$ at any order and each term of the expansion is a Schwartz function.

5.4 Proof of Theorem 3.8 and Corollary 3.9.

We now complete the proof of Theorem 3.8. We choose $\delta =\varepsilon ^{2/9}$, and $\varepsilon$ is small enough so that $\varepsilon \le |t-t^\flat |^{9/2}$. Then, one has $|t-t^\flat |\ge \delta$. If $t\in [t_0, t^\flat -\delta ]$, then Proposition 4.1 gives the result. If $t\in [t^\flat +\delta , t_0+T]$, then one combines Proposition 4.1 between times $s_1=t^\flat +\delta$ and $s_2=t$ with Proposition 5.1. In summary, we obtain an error estimate of order $\varepsilon \delta ^{-2} = \varepsilon ^{1/3}\delta = \varepsilon ^{5/9}$.

Corollary 3.9 comes from Theorem 3.8 and point (3) of Proposition E.1.

Appendices

Appendix A: The Wave Packet Transform

We discuss here useful properties of the wave-packet transform. We define the Weyl translation operator ${{\widehat{T}}}^\varepsilon$

$$\begin{aligned} {{\widehat{T}}}^\varepsilon (z) = {\mathrm{e}}^{\frac{i}{\varepsilon }(p\cdot \widehat{x}-q\cdot {{\widehat{\xi }}})}, \;\; z=(q,p)\in {{\mathbb {R}}}^{2d}, \end{aligned}$$

the semi-classical scaling operator $\Lambda _\varepsilon$

$$\begin{aligned} \Lambda _\varepsilon \varphi (x) = \varepsilon ^{-d/4}\varphi \!\left( \tfrac{x}{\sqrt{\varepsilon }}\right) , \qquad \varphi \in {{\mathcal {S}}}({{\mathbb {R}}}^d), \end{aligned}$$

and we denote by $a_{\varepsilon ,z}\in {{\mathcal {C}}}^\infty ({{\mathbb {R}}}^{2d})$ the function $a_{\varepsilon ,z}(w) = a(\sqrt{\varepsilon }w+z)$, $w\in {{\mathbb {R}}}^{2d}$.

Lemma A.1

The wave packet transform satisfies for all points $z,z'\in {{\mathbb {R}}}^{2d}$ and all smooth functions $a\in {{\mathcal {C}}}^\infty ({{\mathbb {R}}}^{2d})$

$$\begin{aligned} \mathcal {WP}^\varepsilon _{z}&= {\mathrm{e}}^{-\frac{i}{2\varepsilon }p\cdot q} \, {{\hat{T}}}^\varepsilon (z)\, \Lambda _\varepsilon , \end{aligned}$$

(48)

$$\begin{aligned} \mathcal {WP}^\varepsilon _{z+z^\prime }&= {\mathrm{e}}^{-\frac{i}{\varepsilon } p\cdot q^\prime }\,\mathcal {WP}^\varepsilon _z\,\Lambda _\varepsilon ^{-1} \mathcal {WP}^\varepsilon _{z^\prime }, \end{aligned}$$

(49)

$$\begin{aligned} {\mathrm{op}}_\varepsilon ^w(a) \mathcal {WP}^\varepsilon _z&= \mathcal {WP}^\varepsilon _z\, {\mathrm{op}}_1^w(a_{\varepsilon ,z}), \end{aligned}$$

(50)

Proof

We consider $\varphi \in {{\mathcal {S}}}({{\mathbb {R}}}^d)$. Then $\widehat{T}^\varepsilon (z)\varphi$ is the solution at time $t=1$ of the initial value problem

$$\begin{aligned} i\varepsilon \partial _t\psi = (q\cdot {{\widehat{\xi }}}-p\cdot \widehat{x})\psi ,\qquad \psi (0) = \varphi . \end{aligned}$$

The explicit form of this solution

$$\begin{aligned} \psi (t,x) = {\mathrm{e}}^{-\frac{i}{2\varepsilon }t^2q\cdot p}\, {\mathrm{e}}^{\frac{i}{\varepsilon }t p\cdot x} \varphi (x-tq) \end{aligned}$$

implies for the action of the Weyl translation that

$$\begin{aligned} {{\widehat{T}}}^\varepsilon (z)\varphi (x) = {\mathrm{e}}^{-\frac{i}{2\varepsilon }q\cdot p}\, {\mathrm{e}}^{\frac{i}{\varepsilon }p\cdot x}\varphi (x-q). \end{aligned}$$

This yields

$$\begin{aligned} {\mathrm{e}}^{-\frac{i}{2\varepsilon }p\cdot q} \, {{\hat{T}}}^\varepsilon (z)\, \Lambda _\varepsilon \varphi (x) = \varepsilon ^{-d/4}\,{\mathrm{e}}^{-\frac{i}{\varepsilon }p\cdot q} \,{\mathrm{e}}^{\frac{i}{\varepsilon }p\cdot x}\varphi (\tfrac{x-q}{\sqrt{\varepsilon }})= \mathcal {WP}^\varepsilon _z\varphi (x). \end{aligned}$$

For the commutation property we compute

$$\begin{aligned} {\mathrm{e}}^{-\frac{i}{\varepsilon } p\cdot q^\prime }\mathcal {WP}^\varepsilon _z\Lambda _\varepsilon ^{-1} \mathcal {WP}^\varepsilon _{z^\prime }\varphi (x)&= {\mathrm{e}}^{-\frac{i}{\varepsilon } p\cdot q^\prime }\mathcal {WP}^\varepsilon _z {\mathrm{e}}^{\frac{i}{\varepsilon }p'\cdot (\sqrt{\varepsilon }x-q')}\varphi \!\left( \tfrac{\sqrt{\varepsilon }x-q'}{\sqrt{\varepsilon }}\right) \\&= {\mathrm{e}}^{-\frac{i}{\varepsilon } p\cdot q^\prime }\varepsilon ^{-d/4} {\mathrm{e}}^{\frac{i}{\varepsilon }p\cdot (x-q)} {\mathrm{e}}^{\frac{i}{\varepsilon }p'\cdot (x-q-q')}\varphi \!\left( \tfrac{x-q-q'}{\sqrt{\varepsilon }}\right) \\&= \mathcal {WP}^\varepsilon _{z+z^\prime }\varphi (x). \end{aligned}$$

Moreover,

$$\begin{aligned}&\mathcal {WP}^\varepsilon _z\,{\mathrm{op}}_1^w(a_{\varepsilon ,z})\varphi (x) \\&\quad = \varepsilon ^{-d/4} {\mathrm{e}}^{\frac{i}{\varepsilon }p\cdot (x-q)}(2\pi )^{-d} \int _{{{\mathbb {R}}}^{2d}} a\!\left( \tfrac{\sqrt{\varepsilon }}{2}\left( \tfrac{x-q}{\sqrt{\varepsilon }}+y\right) +q,\sqrt{\varepsilon }\xi +p\right) ) {\mathrm{e}}^{i\xi \cdot ((x-q)/\sqrt{\varepsilon }-y)} \varphi (y) \,dy \,d\xi \\&\quad = \varepsilon ^{-d/4} {\mathrm{e}}^{\frac{i}{\varepsilon }p\cdot (x-q)}(2\pi \varepsilon )^{-d} \int _{{{\mathbb {R}}}^{2d}} a\!\left( \tfrac{1}{2}(x+y')+q,\xi '\right) {\mathrm{e}}^{\frac{i}{\varepsilon }(\xi '-p)\cdot (x-y')} \varphi \!\left( \tfrac{y'-q}{\sqrt{\varepsilon }}\right) \,dy' \,d\xi '\\&\quad = {\mathrm{op}}_\varepsilon ^w(a)\,\mathcal {WP}^\varepsilon _z\varphi (x). \end{aligned}$$

$\square$

The intertwining property (50), that relates the wave packet transform with Weyl quantization, allows to describe the localisation properties of wave packets as follows.

Remark A.2

(Localisation on scale $\sqrt{\varepsilon }$) Let $\chi \in {\mathcal {C}}_0^\infty ({{\mathbb {R}}}^{2d})$ be a cut-off function such that $\chi =1$ close to 0 and $0\le \chi \le 1$. Define for $R>0$, $\chi _R(z)=\chi (R^{-1} z)$ for all $z\in {{\mathbb {R}}}^{2d}$. Then, for any $k,N\in {{\mathbb {N}}}$ and any Schwartz function $\varphi \in {{\mathcal {S}}}({{\mathbb {R}}}^d)$

$$\begin{aligned} \left\| {\mathrm{op}}_1^w(1-\chi _R)\varphi \right\| _{\Sigma ^k_1} \le C R^{-N}, \end{aligned}$$

where the constant $C>0$ depends on k, N and the norm of $\varphi$ in $\Sigma ^{k+N}_1$. Decomposing a wave packet as

$$\begin{aligned} \mathcal {WP}^\varepsilon _0\varphi = \mathcal {WP}^\varepsilon _0\,{\mathrm{op}}_1^w(\chi _R)\varphi + \mathcal {WP}^\varepsilon _0\,{\mathrm{op}}_1^w(1-\chi _R)\varphi , \end{aligned}$$

the combination of the above estimate with equation (50) and the continuity of the wave packet transform as a mapping from $\Sigma ^k_1$ to $\Sigma ^k_\varepsilon$ yields

$$\begin{aligned} \left\| \mathcal {WP}^\varepsilon _0\varphi - {\mathrm{op}}_\varepsilon ^w(\chi _{R\sqrt{\varepsilon }})\mathcal {WP}^\varepsilon _0\varphi \right\| _{\Sigma ^k_\varepsilon } \le C R^{-N}. \end{aligned}$$

(51)

Appendix B: Algebraic Properties of the Eigenprojectors

We consider a smooth eigenvalue h(t, z) of a matrix-valued Hamiltonian H(t, z), associated with a smooth eigenprojector $\Pi (t,z)$ so that $H=h\Pi +h^\perp \Pi ^\perp$. We emphasize that, in this section, we just assume smoothness of the projector and make no gap assumption. Let us project the solution of the Hamiltonian system (1) to the eigenspace and consider the function ${{\tilde{w}}}^\varepsilon (t) = {{\widehat{\Pi }}}\psi ^\varepsilon (t)$. We have

$$\begin{aligned} i\varepsilon \partial _t{{\tilde{w}}}^\varepsilon (t) = \left( i\varepsilon \widehat{\partial _t\Pi } + {{\widehat{\Pi }}}{{\widehat{H}}}\right) \psi ^\varepsilon (t), \end{aligned}$$

and by symbolic calculus

$$\begin{aligned} {{\widehat{\Pi }}}{{\widehat{H}}}&= \widehat{h\Pi } + \frac{\varepsilon }{2i}\widehat{\{\Pi ,H\}} + O(\varepsilon ^2)= {\widehat{h}}{\widehat{\Pi }} -\frac{\varepsilon }{2i}\widehat{\{h,\Pi \}}+ \frac{\varepsilon }{2i}\widehat{\{\Pi ,H\}} + O(\varepsilon ^2), \end{aligned}$$

where the order $\varepsilon ^2$ remainder will be given a precise meaning in Lemma B.2 and Lemma B.5 below. Therefore, if we introduce the matrix

$$\begin{aligned} B = -2\partial _t \Pi - \{h,\Pi \}+\{\Pi , H\}, \end{aligned}$$

(52)

then we may write

$$\begin{aligned} i\varepsilon \partial _t {{\tilde{w}}}^\varepsilon (t) = {{\widehat{h}}} {{\tilde{w}}}^\varepsilon (t) + \frac{\varepsilon }{2i} {{\widehat{B}}} \psi ^\varepsilon (t) + O(\varepsilon ^2). \end{aligned}$$

Let us examine the algebraic properties of the first order contribution B in more detail.

Lemma B.1

Consider a Hermitian matrix $H = h\Pi +h^\perp \Pi ^\perp$ with eigenvalues $h, h^\perp$ and corresponding eigenprojectors $\Pi ,\Pi ^\perp$. Then, the matrix $\{\Pi ,\Pi \}$ is skew-symmetric and diagonal,

$$\begin{aligned} \Pi ^\perp \{\Pi ,\Pi \}\Pi = \Pi \{\Pi ,\Pi \}\Pi ^\perp = 0. \end{aligned}$$

The matrix B defined in (52) satisfies

$$\begin{aligned} B\Pi = -2(\Omega +K) = 2i\Theta \Pi \;\; \text{ and }\;\; \Pi ^\perp B\Pi ^\perp = (h-h^\perp )\Pi ^\perp \{\Pi ,\Pi \}\Pi ^\perp , \end{aligned}$$

where the matrices $\Omega$, K, and $\Theta$ have been introduced in (17), (18), and (19). Moreover, the matrix $\Omega$ is skew-symmetric and $\Theta$ self-adjoint.

Proof

We use the relation $\{{{\mathcal {A}}},{{\mathcal {B}}}{{\mathcal {C}}}\} -\{{{\mathcal {A}}}{{\mathcal {B}}},{{\mathcal {C}}}\} = \{{\mathcal A},{{\mathcal {B}}}\} {{\mathcal {C}}}-{{\mathcal {A}}}\{{{\mathcal {B}}},{\mathcal C}\}$. and apply it to ${{\mathcal {A}}}={{\mathcal {B}}}={{\mathcal {C}}}=\Pi$. Since $\Pi ^2 = \Pi$, we obtain $0=\{\Pi ,\Pi \}\Pi -\Pi \{\Pi ,\Pi \}$ and therefore

$$\begin{aligned} \Pi ^\perp \{\Pi ,\Pi \} \Pi =\Pi \{\Pi ,\Pi \} \Pi ^\perp =0. \end{aligned}$$

Besides, by the definition of the Poisson bracket, we have $\{\Pi ,\Pi \}^* = -\{\Pi ,\Pi \}$, so that $\{\Pi ,\Pi \}$ and $\Omega = -\frac{1}{2}(h-h^\perp )\Pi \{\Pi ,\Pi \}\Pi$ are skew-symmetric. In view of

$$\begin{aligned} \{\Pi ,H\}&= (h-h^\perp )\{\Pi ,\Pi \} - \{ h, \Pi \}\Pi -\{h^\perp ,\Pi \}\Pi ^\perp ,\\ \{ h,\Pi \}&= \{h,\Pi \} \Pi + \{ h,\Pi \} \Pi ^\perp , \end{aligned}$$

we obtain that

$$\begin{aligned} B&= -2\partial _t\Pi -\{ h,\Pi \} +\{\Pi ,H\}\\&= -2\partial _t\Pi + (h-h^\perp )\{\Pi ,\Pi \} - 2 \{ h,\Pi \} \Pi - \{h+h^\perp ,\Pi \}\Pi ^\perp . \end{aligned}$$

Hence,

$$\begin{aligned} B\Pi&= -2\Pi ^\perp (\partial _t\Pi +\{h,\Pi \})\Pi + (h-h^\perp )\Pi \{\Pi ,\Pi \}\Pi = -2(K+\Omega ) \end{aligned}$$

$$\begin{aligned} \text{ and }\;\; \Pi ^\perp B\Pi ^\perp = (h-h^\perp )\Pi ^\perp \{\Pi ,\Pi \}\Pi ^\perp . \end{aligned}$$

The matrix $\Theta = i\Omega +i(K-K^*)$ is hermitian, since $\Theta ^* = -i\Omega ^* - i(K^*-K) = \Theta$. It also satisfies $2i\Theta \Pi = 2i(i\Omega + iK) \Pi = B\Pi$. $\square$

Decomposing the matrix $B = B\Pi +B\Pi ^\perp$, we may view the contribution associated with the projector $\Pi$ as an effective dynamical correction to the eigenvalue h. We obtain the following:

Lemma B.2

Let $H = h\Pi +h^\perp \Pi ^\perp$ be a smooth matrix-valued Hamiltonian with smooth eigenvalues $h,h^\perp$ and smooth eigenprojectors $\Pi ,\Pi ^\perp$. Then, there exists a smooth matrix-valued symbol $R^\varepsilon$ such that

$$\begin{aligned} {{\widehat{\Pi }}} (i\varepsilon \partial _t -{{\widehat{H}}}) =&(i\varepsilon \partial _t - {{\widehat{h}}} -\varepsilon {{\widehat{\Theta }}}){{\widehat{\Pi }}} +\frac{\varepsilon }{2i} \widehat{B\Pi ^\perp } {{\widehat{\Pi }}}^\perp +\varepsilon ^2 {{\widehat{R}}}^\varepsilon , \end{aligned}$$

(53)

where the matrices B and $\Theta$ have been defined in (52) and (19), respectively. If the Hamiltonian and its eigenvalues are of subquadratic growth (7), while the projectors grow at most polynomially (26), then for all $k\in {{\mathbb {N}}}$ there exist $C_k>0$ and $\ell \in {{\mathbb {N}}}$ such that

$$\begin{aligned} \sup _{t\in [t_0,t_0+T]}\Vert {{\widehat{R}}}^\varepsilon (t)\varphi \Vert _{\Sigma ^k_\varepsilon } \le C_k \Vert \varphi \Vert _{\Sigma ^\ell _\varepsilon }\;\; \forall \varphi \in \Sigma ^k_\varepsilon ({{\mathbb {R}}}^d). \end{aligned}$$

Proof

We write

$$\begin{aligned} {{\widehat{\Pi }}} (i\varepsilon \partial _t -{{\widehat{H}}}) =&(i\varepsilon \partial _t -{{\widehat{h}}}){{\widehat{\Pi }}} -i\varepsilon \widehat{\partial _t\Pi } + {{\widehat{h}}} {{\widehat{\Pi }}} - {{\widehat{\Pi }}}{{\widehat{H}}}. \end{aligned}$$

The symbolic calculus gives

$$\begin{aligned} {{\widehat{h}}} {{\widehat{\Pi }}} - {{\widehat{\Pi }}}{{\widehat{H}}}= \frac{\varepsilon }{2i} \left( \widehat{ \{ h,\Pi \} }-\widehat{\{\Pi ,H\}}\right) + \varepsilon ^2 R^\varepsilon , \end{aligned}$$

where the remainder ${{\widehat{R}}}^\varepsilon (t)$ satisfies the claimed estimate due to the growth assumptions on the symbols h, H and $\Pi$. In view of Lemma B.1, we have

$$\begin{aligned} -i\partial _t \Pi + \frac{1}{2i}\left( \{ h,\Pi \} -\{\Pi ,H\}\right) = \Theta \Pi + \frac{1}{2i} B\Pi ^\perp , \end{aligned}$$

which concludes our proof. $\square$

We note that for the projected solution ${{\tilde{w}}}^\varepsilon (t) = {{\widehat{\Pi }}} \psi ^\varepsilon (t)$, equation (53) implies an evolution equation of the form

$$\begin{aligned} i\varepsilon \partial _t {{\tilde{w}}}^\varepsilon (t) = ({{\widehat{h}}} + \varepsilon {{\widehat{\Theta }}}){{\tilde{w}}}^\varepsilon (t) - \frac{\varepsilon }{2i} \widehat{B\Pi ^\perp } {{\widehat{\Pi }}}^\perp \psi ^\varepsilon (t) + O(\varepsilon ^2). \end{aligned}$$

In a next step we use the matrix B for introducing the first order super-adiabatic correction of the eigenprojector $\Pi$, following ideas from [3, 10, 28, 30, 31, 34, 36].

Definition B.3

We assume that H is a smooth Hermitian matrix that has two smooth eigenvalues h and $h^\perp$ and smooth eigenprojectors $\Pi$ and $\Pi ^\perp$, that is, $H = h\Pi + h^\perp \Pi ^\perp$. The first super-adiabatic corrector of $\Pi$ is the hermitian matrix $\mathbb P = {\mathbb {P}}^*$ defined by

$$\begin{aligned} \Pi {\mathbb {P}} \Pi ^\perp&= \frac{i}{h-h^\perp }\Pi \left( \partial _t \Pi +\frac{1}{2} \{h+h^\perp ,\Pi \}\right) \Pi ^\perp ,\\ \Pi ^\perp {\mathbb {P}} \Pi&= -\frac{i}{h-h^\perp }\Pi ^\perp \left( \partial _t \Pi +\frac{1}{2} \{h+h^\perp ,\Pi \}\right) \Pi ,\\ \Pi {\mathbb {P}}\Pi&= -\frac{1}{2i}\Pi \{\Pi ,\Pi \}\Pi ,\;\;\;\;\;\; \Pi ^\perp {\mathbb {P}}\Pi ^\perp = \frac{1}{2i}\Pi ^\perp \{\Pi ,\Pi \}\Pi ^\perp . \end{aligned}$$

Note that one has $\displaystyle { {\mathbb {P}} \Pi ^\perp = \frac{1}{2i}(h-h^\perp )^{-1} B\Pi ^\perp \;\;\text{ and }\;\; \mathbb P\Pi = -\frac{1}{2i}(h-h^\perp )^{-1} B\Pi }$.

Note that the diagonal part of the matrix ${\mathbb {P}}$ is smooth, while the off-diagonal part of ${\mathbb {P}}$ is singular on the crossing set $\Upsilon = \{f = 0\}$. Besides, for all $\beta \in {{\mathbb {N}}}^{2d}$ and $R>0$,

$$\begin{aligned} \exists C_{\beta ,R}>0,\;\; \forall z\in B(0,R) \cap \{f(t,z)>\delta \},\;\;\forall t\in {{\mathbb {R}}},\;\; \Vert \partial ^\beta _z{\mathbb {P}}(t,z)\Vert \le C_{\beta ,R} \,\delta ^{|\beta |+1}. \end{aligned}$$

(54)

The main interest in the corrector ${\mathbb {P}}$ comes from the following relations:

Lemma B.4

With the assumptions of Definition B.3, the corrector matrix ${\mathbb {P}}$ satisfies

$$\begin{aligned}{}[H,{{\mathbb {P}}}]= i\partial _t \Pi -\frac{1}{2i} (\{H,\Pi \}-\{\Pi ,H\} )\quad \text{ and }\quad {\mathbb {P}} \Pi +\Pi {\mathbb {P}} ={\mathbb {P}} - \frac{1}{2i} \{\Pi ,\Pi \}, \end{aligned}$$

(55)

as well as

$$\begin{aligned} i\partial _t\Pi + {\mathbb {P}}(H-h) + \frac{1}{2i} \{\Pi , H+h\} = \Theta \Pi \quad \text{ and }\quad i(\partial _t\Pi + \{h,\Pi \}) = [\Theta ,\Pi ], \end{aligned}$$

where the matrix $\Theta$ is given by (19).

Proof

Since H is acting as a scalar on $\mathrm{Ran}\Pi$ and $\mathrm{Ran}\Pi ^\perp$, we have

$$\begin{aligned} {[}H,{\mathbb {P}}]&= [H,\Pi {\mathbb {P}}\Pi ^\perp + \Pi ^\perp {\mathbb {P}} \Pi ] = (h-h^\perp )\Pi {\mathbb {P}}\Pi ^\perp + (h^\perp -h)\Pi ^\perp {\mathbb {P}} \Pi \\&= i(\partial _t \Pi + \tfrac{1}{2} \{h+h^\perp ,\Pi \}). \end{aligned}$$

Since

$$\begin{aligned}\{H,\Pi \} - \{\Pi ,H\} = \Pi \{h,\Pi \} + \Pi ^\perp \{h^\perp ,\Pi \} - \{\Pi ,h\}\Pi - \{\Pi ,h^\perp \}\Pi ^\perp = \{h+h^\perp ,\Pi \},\end{aligned}$$

we have proven the first equation. For the second equation, we calculate

$$\begin{aligned} {\mathbb {P}}\Pi + \Pi {\mathbb {P}} = 2\Pi {\mathbb {P}} \Pi + \Pi ^\perp {\mathbb {P}} \Pi + \Pi {\mathbb {P}} \Pi ^\perp = {\mathbb {P}} + \Pi {\mathbb {P}} \Pi - \Pi ^\perp {\mathbb {P}} \Pi ^\perp = {\mathbb {P}} - \tfrac{1}{2i}\{\Pi ,\Pi \},\end{aligned}$$

where we have used that $\{\Pi ,\Pi \}$ is diagonal. For the first relation with $\Theta$, we write $H-h = (h^\perp -h)\Pi ^\perp$ and obtain

$$\begin{aligned} {\mathbb {P}}(H-h) = (h^\perp -h){\mathbb {P}}\Pi ^\perp = -\frac{1}{2i}B\Pi ^\perp . \end{aligned}$$

Therefore, by Lemma B.1,

$$\begin{aligned} i\partial _t\Pi + {\mathbb {P}}(H-h) + \frac{1}{2i} \{\Pi , H+h\} = \frac{1}{2i}B - \frac{1}{2i}B\Pi ^\perp = \Theta \Pi . \end{aligned}$$

For the commutator of $\Theta$ and $\Pi$, we have

$$\begin{aligned} {[}\Theta ,\Pi ]&= i[\Omega ,\Pi ]+ i[K,\Pi ] - i[K^*,\Pi ] \\&= i\Pi ^\perp (\partial _t\Pi + \{h,\Pi \})\Pi + i\Pi (\partial _t\Pi ^\perp + \{h,\Pi \})\Pi =i(\partial _t\Pi + \{h,\Pi \}). \end{aligned}$$

$\square$

If the crossing set $\Upsilon$ were empty and all the symbols in consideration were bounded, the relations of Lemma B.4 would imply that setting $\Pi ^\varepsilon = \Pi + \varepsilon {\mathbb {P}},$ then ${{\widehat{\Pi }}}^\varepsilon$ would be “better” than ${{\widehat{\Pi }}}$ in terms of being an eigenprojector of ${{\widehat{H}}}$: in $\mathcal L(L^2({{\mathbb {R}}}^d))$,

$$\begin{aligned}\widehat{\Pi ^\varepsilon } \widehat{\Pi ^\varepsilon } =\widehat{\Pi ^\varepsilon } +O(\varepsilon ^2) \quad \text {and}\quad \widehat{\Pi ^\varepsilon }(-i\varepsilon \partial _t+\widehat{H}) = (-i\varepsilon \partial _t+{{\widehat{h}}}+\varepsilon {{\widehat{\Theta }}})\widehat{\Pi ^\varepsilon } +O(\varepsilon ^2),\end{aligned}$$

while the estimate would be only $O(\varepsilon )$ when using the uncorrected ${{\widehat{\Pi }}}$. However, because the symbols we consider are smooth only outside $\Upsilon$, we need to use cut-off functions to correctly state such properties.

Lemma B.5

Let I be an interval of ${{\mathbb {R}}}$ and $\chi ^\delta ,{\tilde{\chi }}^\delta \in {\mathcal {C}}(I,{{\mathcal {C}}}_0^\infty ({{\mathbb {R}}}^{2d}))$ be two cut-off functions that satisfy:

• (1) For any $t\in I$ and z in the support of $\chi ^\delta (t)$ and ${{\tilde{\chi }}}^\delta (t)$ we have $|(h-h^\perp )(t,z)|>\delta$.

• (2) The functions $\chi ^\delta$ and ${{\tilde{\chi }}}^\delta$ satisfy

  $$\begin{aligned}\partial _t \chi ^\delta + \left\{ h, \chi ^\delta \right\} =0,\;\;\partial _t {{\tilde{\chi }}}^\delta + \left\{ h, {{\tilde{\chi }}}^\delta \right\} =0.\end{aligned}$$

• (3) The functions ${{\tilde{\chi }}}^\delta$ are supported in $\{\chi ^\delta =1\}$.

Let $k\in {{\mathbb {N}}}$. Then, we have for all $t\in I$ in $\Sigma ^k_\varepsilon$,

$$\begin{aligned} \widehat{{{\tilde{\chi }}}^\delta }\left( -i\varepsilon \partial _t +({{\hat{h}}} + \varepsilon {{\widehat{\Theta }}})\right) \widehat{\chi ^\delta \Pi ^\varepsilon }&= \widehat{{{\tilde{\chi }}}^\delta }\widehat{\chi ^\delta \Pi ^\varepsilon }(-i\varepsilon \partial _t + {{\widehat{H}}} )+ O(\varepsilon ^2 \delta ^{-2}). \end{aligned}$$

In particular, the function $w^\varepsilon (t) = \widehat{{\tilde{\chi }}^\delta }\widehat{\chi ^\delta \Pi ^\varepsilon }\psi ^\varepsilon (t)$ satisfies for all $t\in I$ in $\Sigma ^k_\varepsilon$,

$$\begin{aligned}i\varepsilon \partial _t w^\varepsilon (t)= ({{\hat{h}}} + \varepsilon {{\widehat{\Theta }}}) w^\varepsilon (t) +O(\varepsilon ^2\delta ^{-2}).\end{aligned}$$

Proof

We write

$$\begin{aligned} \widehat{\chi ^\delta \Pi ^\varepsilon }(-i\varepsilon \partial _t {+} {{\widehat{H}}})&= {\mathrm{op}}_\varepsilon ( \chi ^\delta h \Pi ) {+} \varepsilon {\mathrm{op}}_\varepsilon ( \chi ^\delta {\mathbb {P}} H {+} \frac{1}{2i} \{ \chi ^\delta \Pi , H\}) {-} \widehat{\chi ^\delta \Pi ^\varepsilon }(i\varepsilon \partial _t) + \varepsilon ^2 \widehat{R^\delta (t)}, \end{aligned}$$

where the remainder $R^\delta (t)$ depends on first order derivatives of $\chi ^\delta {\mathbb {P}}$ and H as well as second order derivatives of $\chi ^\delta \Pi$ and H. Hence, $\widehat{R}^\delta (t) = O(\delta ^{-2})$. Next, we write

$$\begin{aligned} {\mathrm{op}}_\varepsilon ( \chi ^\delta h \Pi ) - \widehat{\chi ^\delta \Pi ^\varepsilon }(i\varepsilon \partial _t)&= {\mathrm{op}}_\varepsilon (h) {\mathrm{op}}_\varepsilon ( \chi ^\delta \Pi ) -\frac{\varepsilon }{2i} {\mathrm{op}}_\varepsilon (\{h, \chi ^\delta \Pi \})\\&\quad -(i\varepsilon \partial _t) \widehat{\chi ^\delta \Pi ^\varepsilon }+ i\varepsilon {\mathrm{op}}_\varepsilon ( \partial _t(\chi ^\delta \Pi ^\varepsilon )) + \varepsilon ^2\widehat{\rho _2^\delta (t)}, \end{aligned}$$

where $\rho _2^\delta (t)$ depends linearly on second derivatives of $\chi ^\delta \Pi$ and h. By Lemma B.4, one part of the first order contributions can be combined according to

$$\begin{aligned} i\partial _t\Pi + {\mathbb {P}}(H-h) + \frac{1}{2i} \{\Pi , H+h\} = \Theta \Pi . \end{aligned}$$

All this implies

$$\begin{aligned} \widehat{\chi ^\delta \Pi ^\varepsilon }(-i\varepsilon \partial _t + {{\widehat{H}}})&=(- i\varepsilon \partial _t + {\mathrm{op}}_\varepsilon (h +\varepsilon \Theta )) {\mathrm{op}}_\varepsilon ( \chi ^\delta \Pi ^\varepsilon ) \\&\quad + \varepsilon {\mathrm{op}}_\varepsilon (\rho _1^\delta (t)) + \varepsilon ^2{\mathrm{op}}_\varepsilon (\rho _2^\delta (t) + R^\delta (t)), \end{aligned}$$

where the remainder is given by

$$\begin{aligned} \rho _2^\delta (t)&= \Pi (i\partial _t\chi ^\delta + \frac{1}{2i}\{\chi ^\delta ,H+h\})\\&= \Pi (i\partial _t\chi ^\delta + \frac{1}{i}\{\chi ^\delta ,h\} + \frac{1}{2i}(h-h^\perp )\{\chi ^\delta ,\Pi \} )\\&= \frac{1}{2i}(h-h^\perp )\Pi \{\chi ^\delta ,\Pi \}\Pi ^\perp , \end{aligned}$$

since $\partial _t\chi ^\delta +\{h,\chi ^\delta \}=0$. We note that $\rho _1^\delta (t)$ and $\rho _2^\delta (t)$ are smooth symbols, depending linearly on derivatives of $\chi ^\delta$ and thus are 0 on the support of ${{\tilde{\chi }}}^\delta$. The latter observation implies the first result. For the function $w^\varepsilon (t)$ we then have

$$\begin{aligned}&(i\varepsilon \partial _t -{{\hat{h}}} - \varepsilon {{\widehat{\Theta }}})w^\varepsilon (t)\\&\quad = \widehat{{{\tilde{\chi }}}^\delta }(i\varepsilon \partial _t -{{\hat{h}}} - \varepsilon {{\widehat{\Theta }}})\widehat{\chi ^\delta \Pi ^\varepsilon }\psi ^\varepsilon (t) + [ (i\varepsilon \partial _t -{{\hat{h}}} - \varepsilon {{\widehat{\Theta }}}), \widehat{{\tilde{\chi }}^\delta }]\widehat{\chi ^\delta \Pi ^\varepsilon } \psi ^\varepsilon (t)\\&\quad = \widehat{{{\tilde{\chi }}}^\delta }\widehat{\chi ^\delta \Pi ^\varepsilon }(i\varepsilon \partial _t -{{\widehat{H}}}) \psi ^\varepsilon (t) + [ (i\varepsilon \partial _t -{{\hat{h}}} - \varepsilon {{\widehat{\Theta }}}), \widehat{{\tilde{\chi }}^\delta }] \widehat{\chi ^\delta \Pi ^\varepsilon } \psi ^\varepsilon (t)+\varepsilon ^2\widehat{R^\delta (t)}. \end{aligned}$$

Moreover, since $\partial _t {{\tilde{\chi }}}^\delta + \{ h,{{\tilde{\chi }}}^\delta \}=0$, we have

$$\begin{aligned}= \varepsilon ^3 \widehat{r^\delta _3(t)} + \varepsilon ^2 \widehat{r^\delta _1(t)},\end{aligned}$$

where the first part of the remainder $r^\delta _3(t)$ depends on third derivatives of h and ${{\tilde{\chi }}}^\delta$, while the second part $r^\delta _1(t)$ depends on first derivatives of $\Theta$ and ${{\tilde{\chi }}}^\delta$. Since $\delta \gg \sqrt{\varepsilon }$, we have

$$\begin{aligned} {[}(i\varepsilon \partial _t -{{\hat{h}}} - \varepsilon {{\widehat{\Theta }}}), \widehat{{\tilde{\chi }}^\delta }] = O(\varepsilon ^2\delta ^{-1}). \end{aligned}$$

Using $(i\varepsilon \partial _t -{{\widehat{H}}}) \psi ^\varepsilon (t)=0$, we obtain the equation for $w^\varepsilon (t)$. $\square$

Appendix C: Parallel Transport

We prove here Proposition 3.1 that provides the time-dependent eigenvector $\vec V(t,z)$ defined by parallel transport. We adapt the proof of [6, Proposition C.1] to account for the matrix $\Omega (t,z)$, noting that we only require that $\Omega (t,z)$ is a skew-symmetric matrix mapping into the range of $\Pi (t,z)$.

Proof

We consider the solution $\vec V(t,z)$ of the parallel transport equation and set $Y(t,z) = \vec V(t,\Phi _h^{t,t_0}(z))$. We observe that Y(t, z) solves the equation

$$\begin{aligned} \partial _t Y(t,z)= & {} \partial _t \vec V(t,\Phi _h^{t,t_0}(z)) +J\partial _zh(\Phi _h^{t,t_0}(z)) V(t,\Phi _h^{t,t_0}(z))\nonumber \\= & {} \Omega (t,\Phi ^{t,t_0}_h(z))Y(t,z) + K(t,\Phi ^{t,t_0}_h(z))Y(t,z). \end{aligned}$$

(56)

In particular, since $\Omega (t,z)$ maps into the range of $\Pi (t,z)$,

$$\begin{aligned} \Pi ^\perp (t,\Phi _h^{t,t_0}(z))\ \partial _t Y(t,z) = K(t,\Phi ^{t,t_0}_h(z))Y(t,z). \end{aligned}$$

We now start proving that for $z\in U$, $\Pi (t,\Phi _h^{t,t_0}(z))Y(t,z)=Y(t,z)$, or equivalently that

$$\begin{aligned}Z(t,z)=\Pi ^\perp (t,\Phi _h^{t,t_0}(z))Y(t,z)\end{aligned}$$

is constant and equal to 0. We compute

$$\begin{aligned} \partial _t Z(t,z) =\left( -\partial _t\Pi (t,\Phi _h^{t,t_0}(z))- J\partial _zh(\Phi ^{t,t_0}_h(z))\partial _z\Pi (t,\Phi ^{t,t_0}_h(z)) + K(t,\Phi ^{t,t_0}_h(z))\right) Y(t,z). \end{aligned}$$

We recall that $K = ({\mathbb {I}}-\Pi )(\partial _t\Pi + \{h,\Pi \})\Pi$. Since all derivatives of the projector are off-diagonal, we have

$$\begin{aligned} -\partial _t\Pi - \{h,\Pi \} + K = - \Pi \left( \partial _t\Pi + \{h,\Pi \}\right) \Pi ^\perp \end{aligned}$$

and therefore

$$\begin{aligned} \partial _t Z(t,z) =- \Pi (t,\Phi _h^{t,t_0}(z)) \left( \partial _t\Pi (t,\Phi _h^{t,t_0}(z)) + J\partial _zh(\Phi ^{t,t_0}_h(z))\partial _z\Pi (t,\Phi ^{t,t_0}_h(z)) \right) Z(t,z). \end{aligned}$$

In particular, $\partial _t Z(t,z)$ is an element of the range of $\Pi (t,\Phi ^{t,t_0}_h(z))$ and thus orthogonal to Z(t, z). Hence, its norm is constant, $Z(t,z)=0$ and $Y(t,z)\in \mathrm{Ran}\, \Pi (t,\Phi ^{t,t_0}_h(z))$.

Besides, we have for any $z\in {{\mathbb {R}}}^{2d}$

$$\begin{aligned} \partial _t Y(t,z) \cdot Y(t,z) = \Omega (t,\Phi ^{t,t_0}_h(z))Y(t,z)\cdot Y(t,z) + K(t,\Phi ^{t,t_0}_h(z))Y(t,z)\cdot Y(t,z) = 0, \end{aligned}$$

because

$$\begin{aligned} \Omega (t,z)^*=-\Omega (t,z)\ \text {and}\ K(t,z)=\Pi ^\perp (t,z)K(t,z). \end{aligned}$$

Therefore, $\Vert Y(t,z)\Vert _{{{\mathbb {C}}}^N}=1$. $\square$

Remark C.1

(Polynomial growth of the eigenvector) The above proof shows that the time-evolution of $Y(t,z) = \vec V(t,\Phi ^{t,t_0}_h(z))$ is generated by a norm-conserving evolution operator, that is, $Y(t) = {{\mathcal {L}}}(t,t_0) Y(t_0).$ This observation allows to literally repeat the inductive argument in the proof of [6, Proposition C.1] for inferring from a polynomial bound on the projector $\Pi (t,z)$ a polynomial bound for the eigenvector $\vec V(t,z)$. Indeed, if (26) holds for $\Pi (t,z)$, then for all $T>0$ and $\beta \in {{\mathbb {N}}}_0^{2d+1}$ there exists a constant $c_{\beta ,T}>0$ such that

$$\begin{aligned} \sup _{t\in [t_0,t_0+T],|z|\ge r_0}\Vert \partial _{t,z}^{\beta } \vec V(t,z)\Vert \le c_{\beta ,T} \langle z\rangle ^{|\beta |(1+n_0)}. \end{aligned}$$

Appendix D: The Phase $\Lambda (\sigma )$ and the Function $\zeta (\sigma )$

Lemma D.1

Let $\Lambda$ and $\zeta$ be defined as

$$\begin{aligned} \zeta (\sigma )&=\Phi _2^{t^\flat ,t^\flat + \sigma }\big (\Phi _1^{t^\flat +\sigma ,t^\flat }(z^\flat )\big ),\nonumber \\ \Lambda (\sigma )&= S_1(t^\flat +\sigma ,t^\flat , z^\flat ) +S_2(t^\flat ,t^\flat +\sigma ,\Phi _1^{t^\flat +\sigma , t^\flat }(z^\flat )) - q(\sigma )\cdot p^\flat . \end{aligned}$$

(57)

We have

$$\begin{aligned} \zeta (0)&=(q(0),p(0))=z^\flat ,\;\;{{\dot{\zeta }}}(0)= (\dot{q}(0),\dot{p}(0))=J\partial _z (h_1-h_2) (t^\flat , z^\flat ) \end{aligned}$$

(58)

$$\begin{aligned} \Lambda (0)&={{\dot{\Lambda }}}(0)= 0, \end{aligned}$$

(59)

$$\begin{aligned} {\ddot{\Lambda }} (0)&= \partial _t (h_2-h_1)-\partial _q h_2\cdot \partial _p(h_2-h_1)+\partial _p h_1 \cdot \partial _q (h_2-h_1) \end{aligned}$$

(60)

In particular, we have

$$\begin{aligned}&\frac{1}{2}({\ddot{\Lambda }}(0) -{\dot{p}(0)\cdot \dot{q}(0)})\\&\quad = \frac{1}{2}\left( \partial _t(h_2{-}h_1) {-} \partial _q h_2\cdot \partial _p (h_2{-}h_1){+}\partial _p h_1\cdot \partial _q(h_2{-}h_1) {+}\partial _p(h_2{-}h_1) \cdot \partial _q (h_2{-}h_1)\right) \\&\quad = \frac{1}{2}\left( \partial _t(h_2-h_1) - \partial _q h_1\cdot \partial _p (h_2-h_1)+\partial _p h_1\cdot \partial _q(h_2-h_1)\right) \\&\quad = \frac{1}{2}\left( \partial _t(h_2-h_1) +\left\{ \frac{h_1+h_2}{2}, h_2-h_1\right\} \right) \end{aligned}$$

which yields that (47) is consistent with (34).

Proof

We begin with the function $\zeta$ and we compute the Taylor expansion at the order 2 for $(q(\sigma ),p(\sigma ))=\zeta (\sigma )-z^\flat$ at $\sigma =0$. Let be $h=h_1, h_2$. We have :

$$\begin{aligned} \Phi _h^{t,t_0}(z) = z&+(t-t_0)J\partial _z h(t_0,z) + \frac{(t-t_0)^2}{2}\left( J\partial ^2_{t,z}h(t_0,z)+ J\partial ^2_{z,z}h(t_0,z)J\partial _zh(t_0,z)\right) \nonumber \\&+ O(|t-t_0|^3). \end{aligned}$$

(61)

Applying this formula, we obtain (omitting the argument $(t^\flat , z^\flat )$ in the functions $h_1$, $h_2$ and their derivatives)

$$\begin{aligned}&\Phi _1^{t^\flat +\sigma ,t^\flat }(z^\flat )= z^\flat +\sigma J\partial _z h_1 +\frac{\sigma ^2}{2} \left( J\partial ^2_{t,z}h_1+ J\partial ^2_{z,z}h_1J\partial _zh_1\right) + O(|\sigma |^3),\\&\zeta (t) = \Phi _1^{t^\flat +\sigma ,t^\flat }(z^\flat ) -\sigma J\partial _z h_2(t^\flat +\sigma ,\Phi _1^{t^\flat +\sigma ,t^\flat }(z^\flat )) +\frac{\sigma ^2}{2} \left( J\partial ^2_{t,z}h_2+ J\partial ^2_{z,z}h_2J\partial _zh_2\right) + O(|\sigma |^3). \end{aligned}$$

We deduce

$$\begin{aligned} \zeta (t)&= z^\flat +\sigma J\partial _z (h_1-h_2) + O(|\sigma |^3)\\&\quad +\frac{\sigma ^2}{2} \left( J\partial ^2_{t,z}(h_1-h_2)+ J\partial ^2_{z,z}(h_1-h_2)J\partial _zh_1 +J\partial ^2_{z,z} h_2 J\partial _z (h_2-h_1)\right) , \end{aligned}$$

and, for further use, the relation

$$\begin{aligned} -p^\flat \dot{q}(0) =&-p^\flat \cdot \partial _q(h_1-h_2), \end{aligned}$$

(62)

$$\begin{aligned} -p^\flat \cdot \ddot{q}(0)=&-p^\flat \cdot (\partial ^2_{t,p}(h_1-h_2)+ \partial ^2_{z,p}(h_1-h_2)J\partial _zh_1 +\partial ^2_{z,p} h_2 J\partial _z (h_2-h_1)) \end{aligned}$$

(63)

We continue with the function $\Lambda$ (defined in (57)) and we use Taylor expansion of the actions for general Hamiltonian h. In view of (9) and (61), we have (omitting the argument $(t_0,z_0)$ in the terms of the form $\partial ^\alpha h(t_0,z_0)$)

$$\begin{aligned} S(t,t_0,z_0)&= \int _{t_0}^t (p_0-(s-t_0)\partial _qh)\cdot (\partial _p h+(s-t_0) (\partial ^2_{t,p}h +\partial ^2_{z,p}hJ\partial _z h) )ds\\&\quad -\int _{t_0}^{t} (h+(s-t_0)\partial _t h ) ds +O((t-t_0)^3)\\&= (p_0\cdot \partial _p h-h)(t-t_0) -\frac{(t-t_0)^2}{2} (\partial _t h+\partial _qh \cdot \partial _p h -p_0\cdot (\partial _{t,p}^2 h+ \partial ^2_{z,p}hJ\partial _z h))\\&\quad +O((t-t_0) ^3). \end{aligned}$$

We first apply the formula with $h=h_1$, $t=t^\flat +\sigma$, $t=t^\flat$ and $z=z^\flat$, which gives (when the arguments of the functions are omitted, they are fixed to $(t^\flat ,z^\flat )$)

$$\begin{aligned} S_1(t^\flat +\sigma ,t^\flat ,z^\flat ) =&\, \sigma (p\cdot \partial _p h_1-h_1)\\&-\frac{\sigma ^2}{2} (\partial _t h_1+\partial _qh_1 \cdot \partial _p h_1 -p\cdot (\partial _{t,p}^2 h_1+ \partial ^2_{z,p}h_1J\partial _z h_1)) +O(\sigma ^3). \end{aligned}$$

We now use the same formula with $h=h_2$, $t=t^\flat$, $t_0=t^\flat +\sigma$, $z_0=\Phi _1^{t^\flat +\sigma , t^\flat }(z^\flat )$. We obtain

$$\begin{aligned}&S_2(t^\flat ,t^\flat +\sigma ,\Phi _1^{t^\flat +\sigma ,t^\flat }(z^\flat )) \\&\quad =- \sigma (p_1(t^\flat +\sigma ,t^\flat ,z^\flat )\cdot \partial _p h_2(t^\flat +\sigma ,\Phi _1^{t^\flat +\sigma ,t^\flat }(z^\flat )) -h_2(t^\flat +\sigma ,\Phi _1^{t^\flat +\sigma ,t^\flat }(z^\flat )) )\\&\qquad -\frac{\sigma ^2}{2} (\partial _t h_2+\partial _qh_2 \cdot \partial _p h_2 -p\cdot (\partial _{t,p}^2 h_2+ \partial ^2_{z,p}h_2J\partial _z h_2)) +O(\sigma ^3) \end{aligned}$$

Note that the treatment of the term of order $\sigma$ has to be performed carefully in the case of $S_2(t^\flat ,t^\flat +\sigma ,\Phi _1^{t^\flat +\sigma ,t^\flat }(z^\flat ))$. We obtain

$$\begin{aligned}&S_2(t^\flat ,t^\flat +\sigma ,\Phi _1^{t^\flat +\sigma ,t^\flat }(z^\flat ))\\&\quad = -\sigma (p\cdot \partial _p h_2 -h_2) \\&\qquad -\sigma ^2(-\partial _t h_2-\partial _q h_2\cdot \partial _p h_1+p\cdot (\partial ^2_{t,p} h_2 +\partial ^2_{z,p} h_2 J\partial _z h_2))\\&\qquad -\frac{\sigma ^2}{2} (\partial _t h_2+\partial _qh_2 \cdot \partial _p h_2 -p\cdot (\partial _{t,p}^2 h_2+ \partial ^2_{z,p}h_2J\partial _z h_1)) +O(\sigma ^3)\\&\quad = \, (p\cdot \partial _p h_2 -h_2)\sigma \\&\qquad +\frac{\sigma ^2}{2} (\partial _t h_2+\partial _qh_2 \cdot \partial _p (2h_1-h_2) -p\cdot (\partial _{t,p}^2 h_2+ \partial ^2_{z,p}h_2J\partial _z (2h_2-h_1)\\&\qquad +O(\sigma ^3) \end{aligned}$$

As a consequence,

$$\begin{aligned}&S_1(t^\flat +\sigma ,t^\flat ,z^\flat ) + S_2(t^\flat ,t^\flat +\sigma ,\Phi _1^{t^\flat +\sigma ,t^\flat }(z^\flat )) =\sigma \, p\cdot \partial _p (h_1-h_2) + \frac{\sigma ^2}{2}(\partial _t (h_2-h_1)\\&\qquad -\partial _q h_2\cdot \partial _p(h_2-h_1)+\partial _p h_1 \cdot \partial _q (h_2-h_1) \\&\qquad +p\cdot (\partial _{t,p}^2(h_1-h_2)+\partial _{z,p}^2 (h_1-h_2)J\partial _z h_1+\partial _{z,p}^2 h_2 J\partial _z(h_1-h_2)))+O(\sigma ^3). \end{aligned}$$

Combining with (63), we obtain

$$\begin{aligned}\Lambda (\sigma )= \frac{\sigma ^2}{2}(\partial _t (h_2-h_1)-\partial _q h_2\cdot \partial _p(h_2-h_1)+\partial _p h_1 \cdot \partial _q (h_2-h_1)) +O(\sigma ^3),\end{aligned}$$

whence (60). $\square$

Appendix E: The Operators ${\mathcal {T}}_{\mu ,\alpha ,\beta }$

We study here the operators ${\mathcal {T}}_{\mu ,\alpha ,\beta }$ that are defined in (29) for $(\mu ,\alpha ,\beta )\in {{\mathbb {R}}}^{2d+1}$. An explicit computation gives the following useful connection with the Fourier transform

$$\begin{aligned} {{\mathcal {F}}}{{\mathcal {T}}}_{\mu ,\alpha , \beta }= {\mathcal T}_{\mu +\alpha \cdot \beta , \beta , -\alpha }{{\mathcal {F}}}. \end{aligned}$$

(64)

The next proposition sums up the main information that we will use about these operators.

Proposition E.1

Let $(\mu ,\alpha ,\beta )\in {{\mathbb {R}}}^{2d+1}$.

• (1) The operator ${\mathcal {T}}_{\mu ,\alpha ,\beta }$ maps ${{\mathcal {S}}}({{\mathbb {R}}}^d)$ into itself if and only if $\mu \not =0$.

• (2) Moreover, if $\mu \not =0$, $\mathcal T_{\mu ,\alpha ,\beta }$ is a metaplectic transformation in the Hilbert space $L^2({{\mathbb {R}}}^d)$ multiplied by a complex number:

  $$\begin{aligned} {{\mathcal {T}}}_{\mu ,\alpha , \beta } = \sqrt{\frac{2i\pi }{\mu }}{\mathrm{e}}^{-\frac{i}{4\mu }(\beta \cdot y -\alpha \cdot D_y)^2}. \end{aligned}$$

  (65)

• (3) If $\mu \not =0$, $\Gamma \in {{\mathfrak {S}}}^+(d)$ and $P\in {\mathcal C}^\infty ({{\mathbb {R}}}^{2d})$ is a polynomial function then there exists $\Gamma _{\mu ,\alpha ,\beta ,\Gamma }\in {{\mathfrak {S}}}^ +(d)$ such that

  $$\begin{aligned} {{\mathcal {T}}}_{\mu ,\alpha ,\beta } ({\mathrm{op}}^w_1(P) g^\Gamma ) = \sqrt{\frac{2i\pi }{\mu }} {\mathrm{op}}^w_1(P\circ \Phi _{\alpha ,\beta } ((4\mu )^{-1} )g^{\Gamma _{\mu ,\alpha ,\beta ,\Gamma }} \end{aligned}$$

  where $\Phi _{\alpha ,\beta }$ satisfies (35) and

  $$\begin{aligned} \Gamma _{\mu ,\alpha ,\beta ,\Gamma } = \Gamma -\frac{(\beta -\Gamma \alpha )\otimes (\beta -\Gamma \alpha ) }{2\mu -\alpha \cdot \beta +\alpha \cdot \Gamma \alpha }. \end{aligned}$$

  (66)

Remark E.2

The matrix $\Gamma _{\mu ,\alpha ,\beta ,\Gamma }$ is in $\mathfrak S^+(d)$ since $g^{\Gamma _{\mu ,\alpha ,\beta ,\Gamma }}$ is proved to be Schwartz class. It is also important to notice that $2\mu -\alpha \cdot \beta +\alpha \cdot \Gamma \alpha$ is non zero because its imaginary part is non zero.

Proof

Point (1) is linked with Point (2) and comes from the formula (30) and (29). Indeed, when $\mu \not =0$, equation (65) is an application of relation (29) and of functional calculus on the self-adjoint operator $(\beta \cdot y -\alpha \cdot D_y)^2$ and the Fourier-transform formula of complex Gaussian functions:

$$\begin{aligned} \int _{-\infty }^{+\infty }{\mathrm{e}}^{is^2\mu }{\mathrm{e}}^{is\tau }ds =\sqrt{\frac{2i\pi }{\mu }}{\mathrm{e}}^{\frac{\tau ^2}{4i\mu }} \;\mathrm{with}\;\arg (i\mu )\in ]-\pi , \pi [. \end{aligned}$$

(67)

It remains to analyze the case where $\mu =0$. The computations are different whether $\alpha \cdot \beta =0$ or not. We assume $\alpha \not =0$ and we set

$$\begin{aligned}{{\hat{\alpha }}}=\frac{\alpha }{|\alpha |},\;\; y= (y\cdot {{\hat{\alpha }}}){{\hat{\alpha }}} +y_\perp .\end{aligned}$$

Similar formulas can be obtained when $\beta \not =0$ using (64). Let us first assume $\alpha \cdot \beta =0$.

$$\begin{aligned} {\mathcal {T}}_{0,\alpha ,\beta } \varphi (y)&= \int {\mathrm{e}}^{is\beta \cdot y_\perp } \varphi ( y\cdot {{\hat{\alpha }}} {{\hat{\alpha }}} -s\alpha +y_\perp ) ds\\&=|\alpha |^{-1} \int {\mathrm{e}}^{i|\alpha |^{-1} (y\cdot {{\hat{\alpha }}}-\sigma ) (\beta \cdot y_\perp )} \varphi (\sigma {{\hat{\alpha }}} +y_\perp )d\sigma \\&= |\alpha |^{-1} {\mathrm{e}}^{i |\alpha |^{-1} (y\cdot {{\hat{\alpha }}}) (\beta \cdot y_\perp )} {\mathcal {F}}_{\alpha } \varphi \left( \frac{\beta \cdot y_\perp }{|\alpha |} ,y_\perp \right) \end{aligned}$$

where $\varphi \in {{\mathcal {S}}}({{\mathbb {R}}}^d)$, $y_\perp =y-{{{\hat{\alpha }}}\cdot y} {{\hat{\alpha }}}$ and ${\mathcal {F}}_\alpha$ is the partial Fourier transform in the direction $\alpha$.

In the case where $\alpha \cdot \beta \not =0$, we write

$$\begin{aligned} {\mathcal {T}}_{0,\alpha ,\beta } \varphi (y)&= (2\pi )^{-1} \int _{{{\mathbb {R}}}^2} {\mathrm{e}}^{-is^2 \frac{\alpha \cdot \beta }{2}+is\beta \cdot y+i\eta (y\cdot {{\hat{\alpha }}} -s|\alpha |) }{\mathcal {F}}_\alpha \varphi (\eta ,y_\perp ) d\eta ds\\&= \sqrt{\frac{1}{i\pi \beta \cdot \alpha }} \int {\mathrm{e}}^{ i\frac{(\beta \cdot y -\eta |\alpha |)^2 }{2\alpha \cdot \beta } +i\eta y\cdot {{\hat{\alpha }}}} {\mathcal {F}}_\alpha \varphi (\eta ,y_\perp ) d\eta \\&= \sqrt{\frac{1}{i\pi \beta \cdot \alpha }} {\mathrm{e}}^{i \frac{(\beta \cdot y)^2}{ 2\beta \cdot \alpha }} \int {\mathrm{e}}^{-i\eta \frac{\beta _\perp \cdot y_\perp }{\beta \cdot {{\hat{\alpha }}}} e^{i \frac{\eta ^2 |\alpha |^2}{2\beta \cdot \alpha }} \mathcal F_\alpha \varphi (\eta ,y_\perp ) d\eta } \\&= \sqrt{\frac{1}{i\pi \beta \cdot \alpha }} {\mathrm{e}}^{i \frac{(\beta \cdot y)^2}{ 2\beta \cdot \alpha }} \int {\mathrm{e}}^{-i\eta \frac{\beta _\perp \cdot y_\perp }{\beta \cdot {{\hat{\alpha }}}} {\mathcal {F}}_\alpha \left( e^{i \frac{(D_{y}\cdot \alpha )^2}{2\beta \cdot \alpha } } \varphi \right) (\eta ,y_\perp ) d\eta }\\&= \sqrt{\frac{4\pi }{i\beta \cdot \alpha }} {\mathrm{e}}^{i \frac{(\beta \cdot y)^2}{ 2\beta \cdot \alpha }} \left( e^{i \frac{(D_{y}\cdot \alpha )^2}{2\beta \cdot \alpha }} \varphi \right) \left( -\frac{\beta _\perp \cdot y_\perp }{\beta \cdot {{\hat{\alpha }}}}{{\hat{\alpha }}}+ y_\perp \right) \end{aligned}$$

This concludes the proof of Points (1) and (2).

Point (3) derives from the formulation of $\mathcal T_{\mu ,\alpha ,\beta }$ as a metaplectic transform. We use general results concerning the action of a metaplectic transformation on Gaussian $g^\Gamma$ (for details see [8], Chapter 3). With the quadratic Hamiltonian $K(y,\eta ) = (\beta \cdot y-\alpha \cdot \eta )^2$, one associates the linear flow $\Phi _{\alpha ,\beta }(t)=(\Phi _{ij}(t))_{1\le i,j\le 2}$ (in a $d\times d$ block form) given by (35). Besides, the Egorov theorem and the classical propagation of the Gaussian are both exact: we have for P a smooth polynomial function

$$\begin{aligned} {\mathrm{e}}^{-it{{\hat{K}}}}({\mathrm{op}}_1^w(P)g^\Gamma ) = {\mathrm{op}}_1^w(P\circ \Phi _{\alpha ,\beta }(t)){\mathrm{e}}^{-it{{\hat{K}}}} g^\Gamma = ({\mathrm{op}}_1^w(P\circ \Phi _{\alpha ,\beta }(t)) g^{\Gamma _t}, \end{aligned}$$

where,in view of (15), the matrix $\Gamma _t\in {\mathfrak {S}}^+(d)$ is given by

$$\begin{aligned} \Gamma _t = (\Phi _{21}(t) +\Phi _{22}(t)\Gamma )(\Phi _{11}(t)+\Phi _{12}(t)\Gamma )^{-1} ,\;\; c_{\Gamma _t} = c_\Gamma \, {\mathrm{det}}^{-1/2}(\Phi _{11}(t) +\Phi _{12}(t)\Gamma ), \end{aligned}$$

We deduce that if $\mu \not =0$,

$$\begin{aligned}{\mathcal {T}}_{\mu ,\alpha ,\beta } g^\Gamma =\sqrt{\frac{2i\pi }{\mu }} {\mathrm{e}}^{-\frac{i}{4\mu } {{\hat{K}}}} g^\Gamma = \sqrt{\frac{2i\pi }{\mu }} g^{\Gamma _{(4\mu )^{-1}}}.\end{aligned}$$

This induces the existence of the matrix $\Gamma _{\mu ,\alpha ,\beta ,\Gamma }\in {\mathfrak {S}}^+(d)$ of Point (3) of the Proposition. It remains to prove the formula (66). We use that if $\varphi =g^\Gamma$, we have

$$\begin{aligned} {{\mathcal {T}}}_{\mu ,\alpha ,\beta } g^\Gamma (y) = c_\Gamma \int _{-\infty }^{+\infty }{\mathrm{e}}^{is^2(\mu -\alpha \cdot \beta /2)}\mathrm{e }^{is\beta \cdot y} {\mathrm{e}}^{\frac{i}{2}(y-s\alpha )\cdot (\Gamma (y-s\alpha ))}ds. \end{aligned}$$

Applying again (67) we get,

$$\begin{aligned} {{\mathcal {T}}}_{\mu ,\alpha ,\beta } g^\Gamma (y) = c_\Gamma \sqrt{\frac{4i\pi }{2\mu -\alpha \cdot \beta +\alpha \cdot \Gamma \alpha }}\; {\mathrm{e}}^{\frac{i}{2}\left( y\cdot \Gamma y -\frac{(y\cdot (\beta -\Gamma \alpha ))^2}{2\mu -\alpha \cdot \beta +\alpha \cdot \Gamma \alpha }\right) } , \end{aligned}$$

which gives (66). $\square$

Appendix F: k-Norms Estimates

Here we give the mathematical details for proving our results on the approximation of the transfer operator

$$\begin{aligned} {{\mathcal {T}}}^\varepsilon =\sqrt{\varepsilon }\int _{-\delta /\sqrt{\varepsilon }}^{+\delta /\sqrt{\varepsilon }}{\mathrm{e}}^{\frac{i}{\varepsilon } \Lambda (\sqrt{\varepsilon }s) -\frac{i}{2} q_\varepsilon (s\sqrt{\varepsilon }) \cdot p_\varepsilon (s\sqrt{\varepsilon }) } {{\mathcal {Q}}}^\varepsilon (s\sqrt{\varepsilon }) {\mathrm{e}}^{i L^\varepsilon (s) }ds ,\quad \varphi \in {{\mathcal {S}}}({{\mathbb {R}}}^d), \end{aligned}$$

in $\Sigma ^k_\varepsilon$-norms with $k>0$. We recall that the proof of Lemma 5.5 works with the phase space translated function ${\mathrm{e}}^{i L^\varepsilon (s) }\varphi$ whose $\Sigma ^k_\varepsilon$-norm may grow as $s^k$. Therefore, integration by parts with respect to s introduces uncontrolled boundary terms, if $k>0$. The first step towards a proof avoiding such terms is to revisit the equations (41) that initiate the construction of the transfer operator. The second step then works with the estimates around the transfer operator.

1.1 The system of equations (41)

We recall that the function $w^\varepsilon (t) = (w^\varepsilon _1(t),w^\varepsilon _2(t))$, that describes the approximation error, satisfies the system

$$\begin{aligned}\left\{ \begin{array}{rl} i\varepsilon \partial _t w^\varepsilon _1 &{}= {{\widehat{h}}}_1 w^\varepsilon _1 + i\varepsilon F^\varepsilon _1,\\ i\varepsilon \partial _t w^\varepsilon _2 &{}= {{\widehat{h}}}_2 w^\varepsilon _2 + i\varepsilon F^\varepsilon _2 \end{array}\right. \end{aligned}$$

with

$$\begin{aligned} F_1^\varepsilon = f_1^\varepsilon ,\quad F_2^\varepsilon = \frac{1}{2}\widehat{B_2\Pi _1}\widehat{\vec V_1}v_1 + f_2^\varepsilon , \end{aligned}$$

where the function $f^\varepsilon (t) = (f^\varepsilon _1(t),f^\varepsilon _2(t))$ is defined in (41). Analysing the evolution for the time-interval $[t^\flat -\delta , t^\flat +\delta ]$, we replace the characteristic function ${\mathbb {I}}_{[t^\flat -\delta , t^\flat +\delta ]}$ by a smooth version in a $\delta$-neighborhood of $t^\flat$. For that we introduce a partition of unity on the time interval under consideration $[t_0, t_0+T]$. It is convenient to use a Littlewood-Paley decomposition defined as follows (see [1, p. 91]): Let $\chi _0\in C_0^\infty ({{\mathbb {R}}})$ be such that $\chi _0(u) = 1$ if $\vert u\vert \le 1/2$ and $\chi _0(u) =0$ if $\vert u\vert \ge 1$. Define $\chi _1(u) = \chi _0(u/2) - \chi _0(u)$ (supported in $\{1/2\le \vert u\vert \le 2\}$). Then we have

$$\begin{aligned} 1 = \chi _0(u) +\sum _{j\ge 0}\chi _1(2^{-j}u),\,\, \forall u\in {{\mathbb {R}}}. \end{aligned}$$

We set $\chi _0^c=1-\chi _0$ and define for $j=1,2$,

$$\begin{aligned} F_j^\varepsilon (t) = \chi _{0,\delta }(t)F_j^\varepsilon (t) + \chi _{0,\delta }^c(t)F_j^\varepsilon (t) \end{aligned}$$

where

$$\begin{aligned} \chi _{0,\delta }(t) = \chi _0\left( \frac{t-t^\flat }{\delta }\right) , \;\; \chi ^c_{0,\delta }(t) = \chi ^c_0\left( \frac{t-t^\flat }{\delta }\right) . \end{aligned}$$

We split the above system into two systems with the two input functions $\chi _{0,\delta }F_{j}^\varepsilon$ and $\chi _{0,\delta }^c F_{j}^{\varepsilon }$,

$$\begin{aligned}\left\{ \begin{array}{rl} i\varepsilon \partial _t {\tilde{w}}^\varepsilon _1 &{}= {{\widehat{h}}}_1{\tilde{w}}^\varepsilon _1 + i\varepsilon \chi _{0,\delta }F^\varepsilon _{1},\\ i\varepsilon \partial _t {\tilde{w}}^\varepsilon _2 &{}= {{\widehat{h}}}_2 {\tilde{w}}^\varepsilon _2 + i\varepsilon \chi _{0,\delta } F^\varepsilon _{2} \end{array}\right. \end{aligned}$$

and

$$\begin{aligned}\left\{ \begin{array}{rl} i\varepsilon \partial _t{\tilde{w}}^{\varepsilon ,c}_1 &{}= \widehat{h}_1{\tilde{w}}^{\varepsilon ,c} _1 + i\varepsilon \chi _{0}^c F^{\varepsilon }_{1},\\ i\varepsilon \partial _t {\tilde{w}}^{\varepsilon ,c}_2 &{}= {{\widehat{h}}}_2 {\tilde{w}}^{\varepsilon ,c} _2 + i\varepsilon \chi _{0}^c F^{\varepsilon }_{2}. \end{array}\right. \end{aligned}$$

We choose the initial values ($j=1,2$)

$$\begin{aligned} {\tilde{w}}_j^\varepsilon (t^\flat -\delta ) = w_j^\varepsilon (t^\flat -\delta ),\;\; {\tilde{w}}_j^{\varepsilon ,c}(t^\flat -\delta ) =0. \end{aligned}$$

By the uniqueness of the solution for Cauchy problems we get that

$$\begin{aligned} w_j^\varepsilon (t) = {\tilde{w}}_j^\varepsilon (t) + {\tilde{w}}_j^{\varepsilon ,c}(t) ,\; \forall t\in [t_0, t_0+T]. \end{aligned}$$

Moreover we also have the Duhamel formulas

$$\begin{aligned} {\tilde{w}}_j^\varepsilon (t)= & {} {\mathcal {U}}_{h_j}(t,t^\flat -\delta ){\tilde{w}}_j^\varepsilon (t^\flat -\delta ) + \int _{t^\flat -\delta }^t{\mathcal {U}}_{h_j}(t,\sigma )\chi _{0,\delta }(\sigma )F_j^\varepsilon (\sigma )d\sigma \nonumber \\ {\tilde{w}}_j^{\varepsilon ,c}(t)= & {} \int _{t^\flat -\delta }^t{\mathcal {U}}_{h_j}(t,\sigma )\chi ^c_{0,\delta }(\sigma )F_j^\varepsilon (\sigma )d\sigma . \end{aligned}$$

(68)

Recall that supp($\chi _{0,\delta })\subseteq ]t^\flat -\delta , t^\flat +\delta [$ and that $\chi ^c_{0,\delta }(t)=0$ for $\vert t-t^\flat \vert \le \delta /2$. So for computing the transfer operator ${{\mathcal {T}}}^\varepsilon$ we only consider the contribution of ${\tilde{w}}_j^\varepsilon (t)$, while the contribution of ${\tilde{w}}_j^{\varepsilon ,c}(t)$ falls into the remainder estimates.

1.2 Reduction to a quadratic phase

Revisiting the arguments of Sect. 5.3 we have to estimate

$$\begin{aligned} \tilde{\mathcal T}^\varepsilon= & {} \sqrt{\varepsilon }\int _{-\infty }^{+\infty }\chi _0\left( \frac{s}{\lambda }\right) {\mathrm{e}}^{i\mu ^\flat s^2 + \sqrt{\varepsilon }s^3 f_3(\sqrt{\varepsilon }s)} {{\mathcal {Q}}}^\varepsilon (\sqrt{\varepsilon }s ) {\mathrm{e}}^{isL} ds+ {{\tilde{R}}}^{\varepsilon ,\delta }\\ \text{ with }\;\;{{\tilde{R}}}^{\varepsilon ,\delta }= & {} \varepsilon \int _{-\infty }^{+\infty } \chi _0\left( \frac{s}{\lambda }\right) \mathrm{e}^{{i}\mu ^\flat s^2 + \sqrt{\varepsilon }s^3 f_3(\sqrt{\varepsilon }s)} {{\mathcal {Q}}}^\varepsilon (\sqrt{\varepsilon }s) {\mathrm{e}}^{is L } s^2 \Theta ^\varepsilon (\sqrt{\varepsilon }s)ds, \end{aligned}$$

where $f_3$ is a smooth function, that is bounded together with its derivatives, and ${{\mathcal {Q}}}^\varepsilon$ is a bounded operator built from the ${\mathrm{op}}_1^w$ quantisation of a smooth function with polynomially controlled derivatives and metaplectic transformations. Note that $\lambda := \frac{\delta }{\sqrt{\varepsilon }}$ is a large parameter. Recall that $\chi _0\left( \frac{s}{\lambda }\right)$ is smooth and supported in $s\in ]-\lambda , +\lambda [$. To estimate the remainder ${{\tilde{R}}}^{\varepsilon ,\delta }$ we integrate by parts as in Sect. 5.3. But here no boundary terms appear, so we easily get k-norms estimates. We detail the argument for $\tilde{{\mathcal {T}}}^\varepsilon$. The first step is to reduce the integral with the quadratic phase $\mu ^\flat \sigma ^2$. To do that we use a deformation argument like in [19, Section 7.7] . Let us denote

$$\begin{aligned} \tilde{{\mathcal {T}}}^\varepsilon (\theta ) = \int _{-\infty }^{+\infty }{\mathrm{e}}^{i\phi _\theta ^\varepsilon (s)}{{\mathcal {A}}}^{\varepsilon ,\delta }(s)ds,\;\; \theta \in [0, 1], \end{aligned}$$

where

$$\begin{aligned} \phi ^\varepsilon _\theta (s) = \mu ^\flat s^2 + \theta \sqrt{\varepsilon }s^3 f_3(\sqrt{\varepsilon }s),\;\; {{\mathcal {A}}}^{\varepsilon ,\delta }(s) = \sqrt{\varepsilon }\,\chi _0\left( \frac{s}{\lambda }\right) {\mathcal Q}^\varepsilon (\sqrt{\varepsilon }s) {\mathrm{e}}^{isL }. \end{aligned}$$

Differentiating 2j times in $\theta$ we get

$$\begin{aligned} \tilde{{\mathcal {T}}}^{\varepsilon ,(2j)}(\theta )=(i\sqrt{\varepsilon })^{2j}\int _{-\infty }^{\infty }{\mathrm{e}}^{i\phi ^\varepsilon _\theta (s)}(s^3f_3(\sqrt{\varepsilon }s))^{2j}{\mathcal A}^{\varepsilon ,\delta }(s)ds \end{aligned}$$

(69)

There exists $\delta _0>0$ such that for $\lambda \sqrt{\varepsilon }=\delta \le \delta _0$ we have

$$\begin{aligned} \vert \partial _s\phi ^\varepsilon _\theta (s)\vert \ge \mu ^\flat \vert s\vert ,\; \forall s \in [-\lambda , \lambda ]. \end{aligned}$$

Hence we can integrate by parts to eliminate powers of s. These occur also in ${{\mathcal {A}}}^{\varepsilon ,\delta }(s)$, since $\Vert {\mathrm{e}}^{i Ls}\varphi \Vert _k = O_\varphi (|s|^k)$ and for any $\varphi \in {\mathcal S}({{\mathbb {R}}}^d)$. Furthermore we can transform the oscillating integral (69) into a convergent integral. To get that we can use and iterate the following relation, for $s\ne 0$,

$$\begin{aligned} \frac{1}{s}\left( \frac{s}{i\partial _s\phi ^\varepsilon _\theta (s)}\right) \partial _s {\mathrm{e}}^{i\phi ^\varepsilon _\theta (s)} = {\mathrm{e}}^{i\phi ^\varepsilon _\theta (s)}, \end{aligned}$$

see also the proof of Lemma F.1 below. Now let us apply the Taylor formula in the deformation parameter $\theta$, for any $\varphi \in {{\mathcal {S}}}({{\mathbb {R}}}^d)$ and any k-norm with $k\ge 0$,

$$\begin{aligned} \Vert \tilde{{\mathcal {T}}}^{\varepsilon }(1)\varphi -\sum _{0\le \ell <2j} \tilde{{\mathcal {T}}}^{\varepsilon ,(\ell )}(0)\varphi \Vert _k \le \sup _{0\le \theta \le 1}\Vert \tilde{\mathcal T}^{\varepsilon ,(2j)}(\theta )\varphi \Vert _k \end{aligned}$$

And from the above estimates and arguments we get

$$\begin{aligned} \Vert \tilde{{\mathcal {T}}}^{\varepsilon }(1)\varphi -\sum _{0\le \ell <2j} \tilde{{\mathcal {T}}}^{\varepsilon ,(\ell )}(0)\varphi \Vert _k \le C(\varphi )\varepsilon ^{j}. \end{aligned}$$

where $C(\varphi )$ depends on semi-norms of $\varphi$ in ${\mathcal S}({{\mathbb {R}}}^d)$. Hence it is enough to consider $\tilde{\mathcal T}^{\varepsilon }(0)\varphi$ where the phase is purely quadratic.

1.3 k-estimates for the transfer operator

A Taylor expansion of ${{\mathcal {Q}}}^\varepsilon (\sqrt{\varepsilon }s)$ around $s=0$ leads us to the final step of the proof, where we consider the operators

$$\begin{aligned} \tilde{{\mathcal {T}}}_{0,\lambda }^\varepsilon :=\sqrt{\varepsilon }\int _{-\infty }^{+\infty }\chi _0\left( \frac{s}{\lambda }\right) {\mathrm{e}}^{i\mu ^\flat s^2 } {\mathrm{e}}^{isL} ds \end{aligned}$$

and

$$\begin{aligned} \tilde{{\mathcal {T}}}_{0,\infty }^\varepsilon :=\sqrt{\varepsilon }\int _{-\infty }^{+\infty } {\mathrm{e}}^{i\mu ^\flat s^2 } {\mathrm{e}}^{isL} ds \end{aligned}$$

Lemma F.1

For any $\varphi$ in ${{\mathcal {S}}}({{\mathbb {R}}}^d)$, any $k\ge 0$, and any $N\ge 1$, there exists a constant $C_{k,N}(\varphi )>0$ such that for all $\lambda \ge 1$,

$$\begin{aligned} \Vert \tilde{{\mathcal {T}}}_{0,\infty }^\varepsilon - \tilde{\mathcal T}_{0,\lambda }^\varepsilon \Vert _k \le C_{k, N}(\varphi )\lambda ^{-N}. \end{aligned}$$

(70)

Proof

Using the Littlewood-Paley decomposition introduced above we have

$$\begin{aligned} \tilde{{\mathcal {T}}}_{0,\infty }^\varepsilon \varphi - \tilde{\mathcal T}_{0,\lambda }^\varepsilon \varphi = \sum _{j\ge 0}\int \chi _1\left( \frac{s}{2^j\lambda }\right) {\mathrm{e}}^{i\mu ^\flat s^2 } {\mathrm{e}}^{isL}\varphi ds. \end{aligned}$$

Then we integrate by parts using the differential operator $P:=\frac{1}{2i\mu ^\flat s}\frac{\partial }{\partial s}$. We have

$$\begin{aligned} \int \chi _1\left( \frac{s}{2^j\lambda }\right) {\mathrm{e}}^{i\mu ^\flat s^2 } {\mathrm{e}}^{isL}\varphi ds = \int {\mathrm{e}}^{i\mu ^\flat s^2 } (P^\top )^M\left( \chi _1\left( \frac{s}{2^j\lambda }\right) {\mathrm{e}}^{isL}\right) \varphi ds. \end{aligned}$$

Besides, for any $M\in {{\mathbb {R}}}$, there exists a constant $C_M>0$ such that if $f\in {\mathcal {C}}^\infty _0({{\mathbb {R}}})$,

$$\begin{aligned}| (P^\top )^M f(s)| \le C_M \langle s\rangle ^M \sup _{1\le p\le M} |f^{(p)} (s)|.\end{aligned}$$

Therefore, using that $\Vert {\mathrm{e}}^{isL}\varphi \Vert _k = O_\varphi (\vert s\vert ^k)$, and noticing that $\vert s\vert$ is of order $2^j\lambda$ on the support of the amplitude, we obtain

$$\begin{aligned} \left\| \int \chi _1\left( \frac{s}{2^j\lambda }\right) {\mathrm{e}}^{i\mu ^\flat s^2 } {\mathrm{e}}^{isL}\varphi ds \right\| _{\Sigma ^k} \le C_\varphi \int _{ |s| \sim 2^j \lambda } |s| ^{M-k} ds \le C'_k (2^j\lambda )^{-M+k+ 2} \int _{1/2}^2 \frac{ds}{s^2}. \end{aligned}$$

Therefore, taking M large enough, the series is convergent and we have (70).$\square$