Wigner measures and observability for the Schrödinger equation on the disk

Abstract

We analyse the structure of semiclassical and microlocal Wigner measures for solutions to the linear Schrödinger equation on the disk, with Dirichlet boundary conditions. Our approach links the propagation of singularities beyond geometric optics with the completely integrable nature of the billiard in the disk. We prove a “structure theorem”, expressing the restriction of the Wigner measures on each invariant torus in terms of second-microlocal measures. They are obtained by performing a finer localization in phase space around each of these tori, at the limit of the uncertainty principle, and are shown to propagate according to Heisenberg equations on the circle. Our construction yields as corollaries (a) that the disintegration of the Wigner measures is absolutely continuous in the angular variable, which is an expression of the dispersive properties of the equation; (b) an observability inequality, saying that the $L^2$-norm of a solution on any open subset intersecting the boundary (resp. the $L^2$-norm of the Neumann trace on any nonempty open set of the boundary) controls its full $L^2$-norm (resp. $H^1$-norm). These results show in particular that the energy of solutions cannot concentrate on periodic trajectories of the billiard flow other than the boundary.

Appendices

Appendix A: Energy estimates, regularization of solutions, and localization on the characteristic set

In this appendix, we present general properties of the Schrödinger equation:

$$\begin{aligned} \left\{ \begin{array}{ll} \frac{1}{i}\frac{\partial u}{\partial t}= \left( -\frac{1}{2}\Delta + V \right) u + f, &{}\quad t\in (0,T), \quad z\in \mathbb {D},\\ u\rceil _{\partial \mathbb {D}} = 0, &{} \\ u\rceil _{t=0} = u^0,&{} \end{array} \right. \end{aligned}$$

(8.1)

that are used throughout the article. None of these properties are specific to the disk $\mathbb {D}$, and they all could be stated as well on any smooth manifold with boundary. We first recall basic energy estimates (and hidden regularity of the trace) for the solution u of (8.1). Second, we define an appropriate regularization operator. Third, we prove a localization property on the set $\{2H= |\xi |^2\}$ for solutions of (8.1). Finally, we give a proof of Lemma 2.13.

First recall that for every $V \in L^1(-T,T; L^\infty (\mathbb {D}; \mathbb {C}))$, $u^0 \in L^2(\mathbb {D})$ and $f \in L^1(-T,T; L^2(\mathbb {D}))$, there is a unique solution $u \in C^0([-T,T]; L^2(\mathbb {D}))$ to (8.1). Moreover, there exists $C_{T,V}>0$ such that for all such u, f and for all $t,s \in [-T,T]$ the following energy estimate holds:

$$\begin{aligned} \Vert u(t)\Vert _{L^2(\mathbb {D})} \le C_{T,V} \left( \Vert u(s)\Vert _{L^2(\mathbb {D})}+\int _{I(s,t)} \Vert f(\sigma )\Vert _{L^2(\mathbb {D})} d\sigma \right) . \end{aligned}$$

(8.2)

Above, I(s, t) denotes the interval of $\mathbb {R}$ whose endpoints are t, s. This estimate is obtained by taking the inner product of the equation with u, taking the real part and applying a Gronwall lemma, and using the fact that $u(-t)$ also solves a Schrödinger equation of the form (8.1) (with f, V replaced by their time-reversed counterparts).

Energy estimates at the $H^1$ level, though classical, are a little subtler. Assume now that $V \in L^1(-T,T; W^{1,\infty }(\mathbb {D}; \mathbb {C}))$ and let u be a smooth solution of (8.1). Using the equation, one obtains:

$$\begin{aligned} \frac{1}{2} \frac{d}{dt} \Vert \nabla u\Vert _{L^2(\mathbb {D})}^2&= \frac{d}{dt}\left\langle -\frac{\Delta }{2} u, u \right\rangle \\&= \left\langle \partial _t u, (D_t-V) u - f \right\rangle + \left\langle (D_t-V) u - f , \partial _t u \right\rangle \\&= \left\langle \partial _t u, -V u - f \right\rangle + \left\langle -V u - f , \partial _t u \right\rangle , \end{aligned}$$

which simplifies in:

$$\begin{aligned} \frac{1}{2} \frac{d}{dt} \Vert \nabla u\Vert _{L^2(\mathbb {D})}^2&= 2 \hbox {Im} \left\langle -\frac{\Delta }{2} u, V u + f \right\rangle \\&= \hbox {Im} \left( \left\langle \nabla u, V \nabla u \right\rangle +\left\langle \nabla u, u \nabla V \right\rangle + \left\langle \nabla u, \nabla f \right\rangle \right) . \end{aligned}$$

Gronwall’s lemma yields, for any $t,s \in (-T,T)$,

$$\begin{aligned}&\Vert \nabla u(t)\Vert _{L^2(\mathbb {D})} \le e^{\int _{I(s,t)} m(\sigma ) d\sigma } \Vert \nabla u(s)\Vert _{L^2(\mathbb {D})} \nonumber \\&\quad + 2\int _{I(s,t)} e^{\int _{I(\zeta ,t)} m(\sigma ) d\sigma } \Vert \nabla f(\zeta ) \Vert _{L^2(\mathbb {D})} d\zeta , \end{aligned}$$

(8.3)

where $m(\sigma ) = \Vert \mathrm{Im}~ V(\sigma )\Vert _{L^\infty (\mathbb {D})} + C_P \Vert \nabla V(\sigma )\Vert _{L^\infty (\mathbb {D})}$ and $C_P$ is the constant in the Poincaré inequality. This estimate implies the well-posedness of (8.1) in $C^0([-T,T]; H^1_0(\mathbb {D}))$ for data $u^0 \in H^1_0(\mathbb {D})$ and $f \in L^1(-T,T; H^1_0(\mathbb {D}))$. Of course, it is possible to relax the $L^1_t W^{1,\infty }_x$ regularity of the potential V; however, in the main part of the paper, much more regularity is required.

We also use the following classical “hidden regularity” estimate for the restriction to the boundary of normal derivatives of solutions of (8.1), whose proof can be found, for instance, in [39, p. 284] or [25, Lemma 2.1].

Proposition 8.1

For every $T>0$ there exists a constant $C>0$ such that, for every $u^0 \in H_{0}^{1}( \mathbb {D})$ and every $f \in L^1(-T,T ;H^1_0(\mathbb {D}))$, the solution $u \in C^0([-T,T]; H^1_0(\mathbb {D}))$ of (8.1) satisfies

$$\begin{aligned} \left\| \partial _{n} u \right\| _{L^{2}\left( (-T,T) \times \partial \mathbb {D}\right) }\le C\left( \Vert \nabla u^0\Vert _{L^{2}\left( \mathbb {D}\right) } + \Vert f\Vert _{L^1(-T,T ;H^1(\mathbb {D}))} \right) . \end{aligned}$$

(8.4)

These estimates will be used to derive properties of the quadratic expression

$$\begin{aligned} \langle u ,{\text {Op}}_1(a(z, \epsilon \xi , t, \epsilon ^2 H)) u \rangle , \end{aligned}$$

where u is the extension by zero outside $\mathbb {D}$ of a solution to (1.1) and a is smooth and compactly supported in all variables. We prove that, up to a small error in terms of $\epsilon$, we may truncate u in time t and in frequency H, so that the new function w is $\epsilon$-oscillating, and its corresponding quadratic expression is close to the original one. This type of result is rather straightforward in the case of a compact manifold without boundary and a time-independent potential.

We assume $u= U_V(t)u^0$ is the solution to (1.1) with initial datum $u^0$; take $g \in C^\infty _c(\mathbb {R})$, let $T,\delta >0$ and take $\chi _T \in C^\infty _c((-\delta -T, T+\delta ))$ equal to 1 in a neighborhood of $[-T,T]$. Let us define

$$\begin{aligned} w(t)= g(\epsilon ^2 D_t) \chi _T(t)U_V(t)u^0 , \quad \text {and} \quad w^0 = w\rceil _{t=0}. \end{aligned}$$

(8.5)

We have the following lemma concerning the map $u^0 \mapsto w^0$.

Lemma 8.2

The time T, the functions $\chi _T$ and g being fixed, and the functions w and $w^0$ being defined by (8.5), we have the following properties:

• (1) There is $C>0$ such that for all $u^0 \in L^2(\mathbb {D})$, and all $\epsilon \in (0,1]$, we have

  $$\begin{aligned} \Vert w^0\Vert _{L^2(\mathbb {D})} \le C \Vert u^0\Vert _{L^2(\mathbb {D})} , \qquad \Vert \epsilon \nabla w^0\Vert _{L^2(\mathbb {D})} \le C \Vert u^0\Vert _{L^2(\mathbb {D})}. \end{aligned}$$

• (2) For each $\epsilon >0$, the operator $u^0\mapsto w^0$ is compact on $L^2(\mathbb {D})$.

• (3) If $g=1$ in a neighborhood of zero, then $w^0 \rightarrow u^0$ in $L^2(\mathbb {D})$ as $\epsilon \rightarrow 0$.

• (4) For every $a \in C^\infty _c(T^*(\mathbb {R}^2 \times \mathbb {R}))$ such that $g=1$ in a neighborhood of the H-support of a, for any $\varphi \in C^\infty _c (-T,T)$, we have

  $$\begin{aligned} \left\| {\text {Op}}_1\left( a(x, \epsilon \xi , t, \epsilon ^2 H) \right) \varphi \left( U_V(t)u^0 \!-\! U_V(t) w^0\right) \right\| _{L^2(\mathbb {R}\times \mathbb {R}^2)} \!\le \! C \epsilon \Vert u^0\Vert _{L^2(\mathbb {D})}, \end{aligned}$$

  and

  $$\begin{aligned}&\left| \left\langle U_V(t) w^0, {\text {Op}}_1\left( a(x, \epsilon \xi , t, \epsilon ^2 H) \right) \varphi U_V(t)w^0\right\rangle _{L^{2}\left( \mathbb {R}^2 \times \mathbb {R}\right) }\right. \\&\left. \quad - \left\langle U_V(t) u^0, {\text {Op}}_1\left( a(x, \epsilon \xi , t, \epsilon ^2 H) \right) \varphi U_V(t)u^0\right\rangle _{L^{2}\left( \mathbb {R}^2 \times \mathbb {R}\right) } \right| \!\le \! C\epsilon ||u^0||^2_{L^2(\mathbb {D})}. \end{aligned}$$

In the context of this paper, the reader can think of $\epsilon$ as being h or $R^{-1}$. Note that, as a consequence of conclusion (4) in the above lemma, the restriction of semiclassical measures of the sequences $U_V(t) u^0_n$ and $U_V(t) w^0_n$ ($w^0_n$ being computed from $u^0_n$ according to (8.5)) to the set $t \in (-T,T)$, $H \in \{g=1\}$ are the same. When tested with compactly supported symbols, we may thus always assume that the sequence $u^0_n$ is $\epsilon$-oscillating.

Proof of Lemma 8.2

Using that u solves (1.1), the function w satisfies the equation

$$\begin{aligned} \left\{ \begin{array}{l} \frac{1}{i}\frac{\partial w}{\partial t}\!=\! \left( -\frac{1}{2}\Delta \!+\! V \right) w -i g(\epsilon ^2 D_t) \chi _T' u \!+\! [g(\epsilon ^2 D_t), V] \chi _T u,\quad t\in \mathbb {R}, \quad z\in \mathbb {D}, \\ w\rceil _{\partial \mathbb {D}} = 0, \\ w\rceil _{t=0} = w^0. \end{array} \right. \end{aligned}$$

(8.6)

Using the energy estimate (8.2) for w with $t=0$, and integrated over $s \in (-T,T)$, we obtain

$$\begin{aligned}&\Vert w^0\Vert ^2_{L^2(\mathbb {D})} \le C \int _{-T}^T \Vert w(t)\Vert _{L^2(\mathbb {D})}^2 dt + C \int _\mathbb {R}\Vert \tilde{\chi }_T g(\epsilon ^2 D_t) \chi _T' u\Vert _{L^2(\mathbb {D})}^2 dt\\&\quad + C \int _\mathbb {R}\Vert \tilde{\chi }_T [g(\epsilon ^2 D_t), V] \chi _T u\Vert _{L^2(\mathbb {D})}^2 dt, \end{aligned}$$

where $\tilde{\chi }_T \in C^\infty _c(\mathbb {R})$ such that $\tilde{\chi }_T =1$ in a neighborhood of $[-T,T]$ and $\chi _T =1$ on a neighborhood on ${\text {supp}}(\tilde{\chi }_T)$. Note moreover that $\tilde{\chi }_T g(\epsilon ^2 D_t) \chi _T' = O_{L^2 \rightarrow L^2}(\epsilon ^\infty )$ and $\tilde{\chi }_T [g(\epsilon ^2 D_t), V] \chi _T = O_{L^2 \rightarrow L^2}(\epsilon ^2)\Vert \partial _t V\Vert _{L^\infty }$. We hence obtain

$$\begin{aligned} \Vert w^0\Vert ^2_{L^2(\mathbb {D})} \le C \int _{-T}^T \Vert w(t)\Vert _{L^2(\mathbb {D})}^2 dt + C \epsilon ^2 \int _{-T-\delta }^{T+\delta }\Vert u(t)\Vert _{L^2(\mathbb {D})}^2 dt. \end{aligned}$$

We now notice that, by definition of w, we have

$$\begin{aligned} \int _{-T}^T \Vert w(t)\Vert _{L^2(\mathbb {D})}^2 dt \!\le \! \int _\mathbb {R}\Vert \chi _T g(\epsilon ^2 D_t) \chi _T u \Vert _{L^2(\mathbb {D})}^2 dt \!\le \! C \int _{-T-\delta }^{T+\delta }\Vert u(t)\Vert _{L^2(\mathbb {D})}^2 dt. \end{aligned}$$

(8.7)

Since u solves (1.1), the energy estimate (8.2) for u with $s=0$, integrated over $t \in (-T- \delta , T+\delta )$ then yields $\int _{-T-\delta }^{T+\delta }\Vert u(t)\Vert _{L^2(\mathbb {D})}^2 dt \le C \Vert u^0\Vert _{L^2(\mathbb {D})}^2$ and thus, combined with the two above estimates, proves the first inequality of Item (1).

Let us now consider the second estimate of Item (1). Using the energy estimate (8.3) for w (satisfying (8.6)) with $t=0$, and integrated over $s \in (-T,T)$, we obtain (with $\tilde{\chi }_T$ defined above),

$$\begin{aligned} \Vert \nabla w^0\Vert ^2_{L^2(\mathbb {D})}&\le \ C \int _{-T}^T \Vert \nabla w(t)\Vert _{L^2(\mathbb {D})}^2 dt \\&\quad + C \int _\mathbb {R}\Vert \tilde{\chi }_T g(\epsilon ^2 D_t) \chi _T' \nabla u\Vert _{L^2(\mathbb {D})}^2 dt \\&\quad + C \int _\mathbb {R}\Vert \tilde{\chi }_T [g(\epsilon ^2 D_t), V] \chi _T \nabla u\Vert _{L^2(\mathbb {D})}^2 dt. \end{aligned}$$

With $B= \tilde{\chi }_T g(\epsilon ^2 D_t) \chi _T'$ or $B= \tilde{\chi }_T [g(\epsilon ^2 D_t), V] \chi _T$, we have, using that u solves (1.1),

$$\begin{aligned} \frac{1}{2} \Vert B u \Vert _{L^2(\mathbb {R}\times \mathbb {D})}^2= & {} \langle B \left( - \frac{\Delta }{2}\right) u, u \rangle _{L^2(\mathbb {R}\times \mathbb {D})} \\= & {} \langle B D_t u, u \rangle _{L^2(\mathbb {R}\times \mathbb {D})} - \langle B V u, u \rangle _{L^2(\mathbb {R}\times \mathbb {D})}. \end{aligned}$$

For both choices of B, we have $|\langle B V u, u \rangle _{L^2(\mathbb {R}\times \mathbb {D})}|\!\le \! C \epsilon ^2 \int _{-T-\delta }^{T+\delta }\Vert u(t)\Vert _{L^2(\mathbb {D})}^2 dt \le C \epsilon ^2 \Vert u^0\Vert _{L^2(\mathbb {D})}^2$, and, since g is compactly supported, $B D_t = O_{L^2 \rightarrow L^2}(\epsilon ^{-2})$, so that $| \langle B D_t u, u \rangle _{L^2(\mathbb {R}\times \mathbb {D})}|\le C \epsilon ^{-2} \Vert u^0\Vert _{L^2(\mathbb {D})}^2$. As a consequence, we obtain

$$\begin{aligned} \Vert \nabla w^0\Vert ^2_{L^2(\mathbb {D})} \le C \int _{-T}^T \Vert \nabla w(t)\Vert _{L^2(\mathbb {D})}^2 dt + C \epsilon ^{-2} \Vert u^0\Vert _{L^2(\mathbb {D})}^2. \end{aligned}$$

(8.8)

Integrating by parts, and using that w solves (8.6), we have

$$\begin{aligned}&\frac{1}{2} \int _{-T}^T \Vert \nabla w(t)\Vert _{L^2(\mathbb {D})}^2 dt = \int _{-T}^T \left\langle - \frac{\Delta }{2} w , w \right\rangle _{L^2(\mathbb {D})} dt\\&\quad = \int _{-T}^T \langle (D_t - V)w , w \rangle _{L^2(\mathbb {D})} dt \\&\quad +\int _{-T}^T \left\langle \bigg ( i g(\epsilon ^2 D_t) \chi _T' u - [g(\epsilon ^2 D_t), V] \chi _T u \bigg ),w \right\rangle _{L^2(\mathbb {D})} dt. \end{aligned}$$

We also have, as above,

$$\begin{aligned} \Vert D_t w \Vert _{L^2((-T,T)\times \mathbb {D})}^2= & {} \Vert D_t g(\epsilon ^2 D_t) \chi _T u \Vert _{L^2((-T,T)\times \mathbb {D})}^2 \\\le & {} C \epsilon ^{-4} \int _{-T-\delta }^{T+\delta }\Vert u(t)\Vert _{L^2(\mathbb {D})}^2 dt \le C \epsilon ^{-4} \Vert u^0\Vert _{L^2(\mathbb {D})}^2, \end{aligned}$$

together with the $L^2$ estimate (8.7) for w. This, together with above estimates and (8.8) finally implies $\Vert \nabla w^0\Vert ^2_{L^2(\mathbb {D})} \le C \epsilon ^{-2} \Vert u^0\Vert _{L^2(\mathbb {D})}^2$, which is the second inequality of Item (1).

Item (2) directly follows from the second estimate of Item (1). To prove Item (3), notice first that

$$\begin{aligned} \Vert u -w\Vert _{L^2((-T,T) \times \mathbb {D})} \le \Vert \tilde{\chi }_T (1 - g(\epsilon ^2 D_t) \chi _T)u\Vert _{L^2(\mathbb {R}\times \mathbb {D})} \rightarrow 0 , \quad \text {as }\; \epsilon \rightarrow 0, \end{aligned}$$

since $1 - g(\epsilon ^2H) =0$ on any compact set for $\epsilon$ sufficiently small. Then, since w solves (8.6) and u solves (1.1), the function $w-u$ also satisfies (8.6) with the same right hand-side, but with initial data $w^0-u^0$. Using the same estimates as for the proof of Item (1), we now have

$$\begin{aligned}&\Vert w^0 - u^0\Vert ^2_{L^2(\mathbb {D})} \le C \int _{-T}^T \Vert w(t) - u(t)\Vert _{L^2(\mathbb {D})}^2 dt \\&\quad \quad + C \int _\mathbb {R}\Vert \tilde{\chi }_T g(\epsilon ^2 D_t) \chi _T' u\Vert _{L^2(\mathbb {D})}^2 dt + C \int _\mathbb {R}\Vert \tilde{\chi }_T [g(\epsilon ^2 D_t), V] \chi _T u\Vert _{L^2(\mathbb {D})}^2 dt \\&\quad \le \Vert u -w\Vert _{L^2((-T,T) \times \mathbb {D})}^2 + C \epsilon ^2 \Vert u^0\Vert _{L^2(\mathbb {D})}^2 \rightarrow 0 , \quad \text {as }\; \epsilon \rightarrow 0, \end{aligned}$$

which proves Item (3).

To prove Item (4), we now write $A={\text {Op}}_1(a(x, \epsilon \xi , t, \epsilon ^2 H) )$, and compute

$$\begin{aligned} \left\| A \varphi \left( U_V(t)u^0 - U_V(t) w^0\right) \right\| _{L^2(\mathbb {R}\times \mathbb {R}^2)}&\le \left\| A\varphi \left( u- w \right) \right\| _{L^2((-T,T) \times \mathbb {R}^2)} \\&\quad + \left\| A \varphi \left( w(t) - U_V(t) w^0\right) \right\| _{L^2((-T,T) \times \mathbb {R}^2)}. \end{aligned}$$

Concerning the first term in the right hand-side, we have

$$\begin{aligned}&\Vert A\varphi \left( u- w \right) \Vert _{L^2((-T,T) \times \mathbb {R}^2)} = \Vert A \varphi (1 - g(\epsilon ^2 D_t) \chi _T )u\Vert _{L^2(\mathbb {R}\times \mathbb {R}^2)} \\&\quad \le C_N \epsilon ^N \Vert u\Vert _{L^2( (-\delta , T+\delta )\times \mathbb {D})} \le C_N \epsilon ^N \Vert u^0 \Vert _{L^2(\mathbb {D})}, \end{aligned}$$

since the supports of a and $1 - g$ are disjoint. Concerning the second term, we notice that $w(t) - U_V(t) w^0$ satisfies (8.6) with the same right hand-side, but with initial data 0. Thus, using the boundedness of A on $L^2$, and the same estimates as for the proof of Item (1), we have

$$\begin{aligned} \left\| A \varphi \left( w(t) - U_V(t) w^0\right) \right\| _{L^2(\mathbb {R}\times \mathbb {R}^2)}\le & {} C \Vert w(t) - U_V(t) w^0 \Vert _{L^2((-T,T) \times \mathbb {D})} \\\le & {} C \epsilon \Vert u^0 \Vert _{L^2(\mathbb {D})}. \end{aligned}$$

The last three estimates conclude the proof of the first estimate in Item (4). Finally, with $y = U_V(t) w^0$, we have

$$\begin{aligned}&\left| \left\langle y ,A \varphi y\right\rangle _{L^{2}\left( \mathbb {R}^2 \times \mathbb {R}\right) } - \left\langle u ,A \varphi u \right\rangle _{L^{2}\left( \mathbb {R}^2 \times \mathbb {R}\right) } \right| \\&\quad = \left| \left\langle y ,A\varphi (y-u)\right\rangle _{L^{2}\left( \mathbb {R}^2 \times \mathbb {R}\right) } - \left\langle y- u ,A \varphi u \right\rangle _{L^{2}\left( \mathbb {R}^2 \times \mathbb {R}\right) } \right| \\&\quad \le \Vert y \Vert _{L^{2}\left( \mathbb {R}^2 \times \mathbb {R}\right) } \Vert A \varphi (y-u)\Vert _{L^{2}\left( \mathbb {R}^2 \times \mathbb {R}\right) } \\&\quad \quad + \Vert u \Vert _{L^{2}\left( \mathbb {R}^2 \times \mathbb {R}\right) } \Vert \varphi A^* (y-u)\Vert _{L^{2}\left( \mathbb {R}^2 \times \mathbb {R}\right) }\\&\quad \le C\epsilon ||u^0||^2_{L^2(\mathbb {D})}, \end{aligned}$$

according to the first estimate in Item (4) (together with Item (1)) applied both to $A \varphi$ and $\varphi A^*$ (which, as well, is of the form $\varphi A^* \psi$ for some $\psi \in C^\infty _c(-T,T)$, since the t-support of a is contained in $(-T,T)$). This concludes the proof of the second estimate in Item (4), and hence, of the lemma. $\square$

The following proposition states that solutions of (1.1) are localized on the set $\{|\xi |^2 = 2H\}$ at high frequency.

Proposition 8.3

For all $s \in (1/2,1)$, for any $a \in C^\infty _c(T^*(\mathbb {R}^2 \times \mathbb {R}))$ supported away from $H=0$, respectively away from $\xi =0$, for all $\varphi \in C^\infty _c(\mathbb {R})$, there is $C>0$ such that for all solutions u to (1.1)–(1.2) with $u^0\in L^2(\mathbb {D})$, we have for $\epsilon \in (0,1]$,

$$\begin{aligned} \left| \left\langle u , {\text {Op}}_1\left( a(z, \epsilon \xi , t, \epsilon ^2 H)\left( \frac{|\xi |^2}{2H}-1\right) \right) \varphi u \right\rangle _{L^{2}(\mathbb {R}^2 \times \mathbb {R}) } \right| \le C \epsilon ^{1-s} ||u^0||^2_{L^{2}\left( \mathbb {D}\right) }, \end{aligned}$$

(8.9)

respectively,

$$\begin{aligned} \left| \left\langle u, {\text {Op}}_1\left( a(z, \epsilon \xi , t, \epsilon ^2 H)\left( \frac{2H}{|\xi |^2}-1\right) \right) \varphi u \right\rangle _{L^{2}\left( \mathbb {R}^2 \times \mathbb {R}\right) } \right| \le C \epsilon ^{1-s} ||u^0||^2_{L^{2}\left( \mathbb {D}\right) }. \end{aligned}$$

(8.10)

As everywhere in the paper, the notation u stands both for the function on $C^0([0,T] ; L^2(\mathbb {D}))$ and its extension by zero to the whole $\mathbb {R}^2$. Note that this holds for all $u^0\in L^2(\mathbb {D})$, without assuming a priori that $u^0$ is $\epsilon$-oscillating. This comes from the nice properties of the regularization (8.5) proved in Lemma 8.2.

Proof of Proposition 8.3

Note first that it suffices to prove the estimate

$$\begin{aligned}&\left| \left\langle u, {\text {Op}}_1\left( a(z, \epsilon \xi , t, \epsilon ^2 H)\left( \frac{\epsilon ^2|\xi |^2}{2}-\epsilon ^2 H\right) \right) \varphi u \right\rangle _{L^{2}\left( \mathbb {R}^2 \times \mathbb {R}\right) } \right| \nonumber \\&\quad \le C \epsilon ^{1-s} ||u^0||^2_{L^{2}\left( \mathbb {D}\right) }. \end{aligned}$$

(8.11)

Estimates (8.9) and (8.10) then follow by changing a into $\frac{a}{\epsilon ^2 H}$ and $\frac{a}{\epsilon ^2 |\xi |^2}$ respectively. According to Lemma 8.2 Item (4), (8.11) is equivalent to the same estimate with u replaced by $U_V(t)w^0$ (extended by zero outside $\mathbb {D}$), where $w^0$ is defined from $u^0$ by (8.5). Writing $A={\text {Op}}_1(a(z, \epsilon \xi , t, \epsilon ^2 H))$, we have ${\text {Op}}_1(a(z, \epsilon \xi , t, \epsilon ^2 H)(\frac{\epsilon ^2|\xi |^2}{2}-\epsilon ^2 H)) = A( -\frac{\epsilon ^2\Delta }{2} -\epsilon ^2D_t )$, where $\Delta$ is the Laplace operator on $\mathbb {R}^2$. We thus obtain

$$\begin{aligned}&{\text {Op}}_1\left( a(z, \epsilon \xi , t, \epsilon ^2 H)\left( \frac{\epsilon ^2|\xi |^2}{2}-\epsilon ^2 H\right) \right) \varphi U_V(t) w^0\\&\quad = A\varphi \left( -\frac{\epsilon ^2\Delta }{2} -\epsilon ^2D_t \right) U_V(t) w^0 + i \epsilon ^2 A \varphi ' U_V(t) w^0. \end{aligned}$$

Recall now that the extended function $U_V(t)w^0$ solves

$$\begin{aligned} \frac{\epsilon ^2}{2} \partial _{n}\left( U_V(t)w^0\right) \otimes \delta _{\partial \mathbb {D}}=\epsilon ^2\left( -\frac{\Delta }{2}+ V- D_t\right) U_V(t)w^0, \end{aligned}$$

so that

$$\begin{aligned}&{\text {Op}}_1\left( a(z, \epsilon \xi , t, \epsilon ^2 H)\left( \frac{\epsilon ^2|\xi |^2}{2}-\epsilon ^2 H\right) \right) \varphi U_V(t) w^0\nonumber \\&\quad = - \epsilon ^2 A\varphi V U_V(t) w^0 + \frac{\epsilon ^2}{2} A\varphi \left( \partial _{n}\left( U_V(t)w^0\right) \otimes \delta _{\partial \mathbb {D}} \right) + i \epsilon ^2A \varphi ' U_V(t) w^0.\nonumber \\ \end{aligned}$$

(8.12)

The operators $A \varphi$ and $A \varphi '$ being bounded on $L^2(-T,T;L^2(\mathbb {D}))$, and according to the energy estimate (8.2) for $U_V(t) w^0$ solution of (1.1), we have

$$\begin{aligned}&\Vert - \epsilon ^2 A\varphi V U_V(t) w^0 + i \epsilon ^2A \varphi ' U_V(t) w^0 \Vert _{L^2(\mathbb {R}^2 \times \mathbb {R})} \le C \epsilon ^2 \Vert w^0\Vert _{L^2(\mathbb {D})}\nonumber \\&\quad \le C \epsilon ^2 \Vert u^0\Vert _{L^2(\mathbb {D})}, \end{aligned}$$

(8.13)

after having used Lemma 8.2 Item (1), and it only remains to estimate

$$\begin{aligned} \epsilon ^2 \left\langle U_V(t)w^0, A\varphi \left( \partial _{n}\left( U_V(t)w^0\right) \otimes \delta _{\partial \mathbb {D}} \right) \right\rangle _{L^{2}\left( \mathbb {R}^2 \times \mathbb {R}\right) }. \end{aligned}$$

To this aim, we write, for every $s>0$,

$$\begin{aligned}&\left| \left\langle U_V(t)w^0, A \, \varphi \left( \partial _{n}\left( U_V(t)w^0\right) \otimes \delta _{\partial \mathbb {D}}\right) \right\rangle _{L^{2}\left( \mathbb {R}^2\times \mathbb {R}\right) } \right| \\&\quad \le ||A||_{ L^{2}\left( \left( -T,T\right) , H^{-s}(\mathbb {R}^2)\right) \rightarrow L^{2}\left( \left( -T,T\right) \times \mathbb {R}^2\right) } \, ||U_V(t) w^0||_{L^{2}\left( \left( -T,T\right) \times \mathbb {D}\right) }\\&\quad \quad \times || \partial _{n}U_V(t)w^0 \otimes \delta _{\partial \mathbb {D}}||_{ L^{2}\left( \left( -T,T\right) , H^{-s}(\mathbb {R}^2)\right) }. \end{aligned}$$

Moreover, for $s>$1/2, the standard trace estimates (see for instance [17, Chapter 2, Section 4]) imply that

$$\begin{aligned} || \partial _{n}U_V(t) w^0 \otimes \delta _{\partial \mathbb {D}}||_{ L^{2}\left( \left( -T,T\right) , H^{-s}(\mathbb {R}^2)\right) } \le C \left\| \partial _{n}\left( U_V(t)w^0\right) \right\| _{L^{2}\left( \left( -T,T\right) \times \partial \mathbb {D}\right) }, \end{aligned}$$

which, by Proposition 8.1, is bounded by $C\left\| \nabla w^0\right\| _{L^{2}\left( \mathbb {D}\right) }$. Using the fact that

$$\begin{aligned} ||A||_{ L^{2}\left( \left( -T,T\right) , H^{-s}(\mathbb {R}^2)\right) \rightarrow L^{2}\left( \left( -T,T\right) \times \mathbb {R}^2\right) }\le C\epsilon ^{-s}, \end{aligned}$$

we now obtain

$$\begin{aligned}&\left| \epsilon ^2 \left\langle U_V(t)w^0, A\varphi \left( \partial _{n}\left( U_V(t)w^0\right) \otimes \delta _{\partial \mathbb {D}}\right) \right\rangle _{L^{2}\left( \mathbb {R}^2 \times \mathbb {R}\right) } \right| \\&\quad \le C \epsilon ^{1-s} ||w^0||_{L^{2}\left( \mathbb {D}\right) }\left\| \epsilon \nabla w^0\right\| _{L^{2}\left( \mathbb {D}\right) } \le C \epsilon ^{1-s} ||u^0||_{L^{2}\left( \mathbb {D}\right) }^2, \end{aligned}$$

after having used Lemma 8.2 Item (1). This, together with (8.12) and (8.13), concludes the proof of the proposition. $\square$

Finally, we prove by dyadic decomposition a statement similar to that of (8.9)–(8.10) for homogeneous functions.

Proposition 8.4

Recall that $\chi \in C_{c}^{\infty }\left( \mathbb {R}\right)$ is a nonnegative cut-off function that is identically equal to one near the origin. For all $s\in (1/2, 1)$, all $a \in \mathcal {S}_0$ (see Definition 2.1) vanishing on the set $\{|\xi |^2= 2H\}$ and for all $\varphi \in C^\infty _c(\mathbb {R})$, there is $C>0$ such that for all $u^0 \in L^2(\mathbb {D})$ and R large enough, we have

$$\begin{aligned}&\left| \left\langle U_V(t) u^0, {\text {Op}}_1\left( a(z, \xi , t, H)\left( 1-\chi \left( \frac{|\xi |^2 +| H|}{R^2}\right) \right) \right) \varphi (t)U_V(t)u^0\right\rangle _{L^{2}\left( \mathbb {R}^2 \times \mathbb {R}\right) }\right| \nonumber \\&\quad \le C R^{s-1} ||u^0||^2_{L^{2}\left( \mathbb {D}\right) }. \end{aligned}$$

(8.14)

Proof

To see that, using the homogeneity of a for large R, we write the following decomposition:

$$\begin{aligned}&a(z, \xi , t, H)\left( 1-\chi \left( \frac{|\xi |^2 +| H|}{R^2}\right) \right) \\&\quad =\sum _{k=0}^\infty a(z, 2^{-k}R^{-1}\xi , t, 2^{-2k}R^{-2}H)\left( \chi \left( \frac{|\xi |^2 +| H|}{2^{2(k+1)}R^2}\right) -\chi \left( \frac{|\xi |^2 +| H|}{ 2^{2k}R^2}\right) \right) . \end{aligned}$$

For each k in the sum above, decompose further

$$\begin{aligned}&\chi \left( \frac{|\xi |^2 +| H|}{2^{2(k+1)}R^2}\right) -\chi \left( \frac{|\xi |^2 +| H|}{ 2^{2k}R^2}\right) =\left( \chi \left( \frac{|\xi |^2 +| H|}{2^{2(k+1)}R^2}\right) -\chi \left( \frac{|\xi |^2 +| H|}{ 2^{2k}R^2}\right) \right) \\&\quad \times \left( \chi \left( \frac{ |H|}{ 2^{2k-1}R^2}\right) + (1-\chi )\left( \frac{ |H|}{ 2^{2k-1}R^2}\right) \right) \end{aligned}$$

and note that we must have $|\xi |^2\ge 2^{2k-1}R^2$ or $|H|\ge 2^{2k-1}R^2$ on the support of this function.

If a vanishes on the set $\{|\xi |^2= 2H\}$, we can write

$$\begin{aligned} a(z, \xi , t, H)=b(z, \xi , t, H)\left( \frac{2H}{|\xi |^2}-1\right) \end{aligned}$$

where $|\xi |^2\ge 2^{2k-1}R^2$ and

$$\begin{aligned} a(z, \xi , t, H)= b(z, \xi , t, H)\left( \frac{|\xi |^2}{2H}-1\right) \end{aligned}$$

where $|H|\ge 2^{2k-1}R^2$. Applying (8.9) and (8.10) for each k (with $\epsilon =2^{-k}R^{-1}$), we finally obtain

$$\begin{aligned}&\left\langle U_V(t) u^0, {\text {Op}}_1\left( a(z, \xi , t, H)\left( 1-\chi \left( \frac{|\xi |^2 +| H|}{R^2}\right) \right) \right) U_V(t)u^0\right\rangle _{L^{2}(\mathbb {R}^2 \times \mathbb {R}) }\nonumber \\&\quad \le C \sum _{k=0}^{+\infty } R^{s-1}2^{k(s-1)} ||u^0||^2_{L^{2}\left( \mathbb {D}\right) }, \end{aligned}$$

(8.15)

which proves the proposition.

To conclude this section, we give a proof of Lemma 2.13.

Proof of Lemma 2.13

Note that operator $A(D_t)\varphi$ is bounded on $L^2(\mathbb {R}\times \mathbb {D})$. Moreover, we have

$$\begin{aligned} \Vert \nabla A(D_t)\varphi u\Vert _{L^2(\mathbb {R}\times \mathbb {D})}^2&= \langle - \Delta A(D_t)\varphi u , A(D_t)\varphi u \rangle _{L^2(\mathbb {R}\times \mathbb {D})} \nonumber \\&= \langle A(D_t)\varphi (- \Delta ) u , A(D_t)\varphi u \rangle _{L^2(\mathbb {R}\times \mathbb {D})} \nonumber \\&\quad + \langle [- \Delta , A(D_t)\varphi ] u , A(D_t)\varphi u \rangle _{L^2(\mathbb {R}\times \mathbb {D})}. \end{aligned}$$

(8.16)

One the one hand, we have

$$\begin{aligned}&\left| \langle [- \Delta , A(D_t)\varphi ] u , A(D_t)\varphi u \rangle _{L^2(\mathbb {R}\times \mathbb {D})} \right| \\&\quad = \left| - \langle 2 \nabla \varphi \cdot \nabla u+ u \Delta \varphi , A(D_t)^2 \varphi u \rangle _{L^2(\mathbb {R}\times \mathbb {D})} \right| \\&\quad \le 2 \left| \langle u , \mathrm{div} \left\{ \nabla \varphi \left( A(D_t)^2 \varphi u \right) \right\} \rangle _{L^2(\mathbb {R}\times \mathbb {D})} \right| + C \Vert \tilde{\varphi } u\Vert ^2_{L^2(\mathbb {R}\times \mathbb {D})} \\&\quad \le 2 \left| \langle u , \nabla \varphi \cdot \nabla \left( A(D_t)^2 \varphi u \right) \rangle _{L^2(\mathbb {R}\times \mathbb {D})} \right| + C \Vert \tilde{\varphi } u\Vert ^2_{L^2(\mathbb {R}\times \mathbb {D})} \\&\quad \le \varepsilon \Vert \nabla A(D_t)\varphi u\Vert _{L^2(\mathbb {R}\times \mathbb {D})}^2 + C(1+\varepsilon ^{-1}) \Vert \tilde{\varphi } u\Vert ^2_{L^2(\mathbb {R}\times \mathbb {D})} \end{aligned}$$

for some $\tilde{\varphi }$ equal to one on the support of $\varphi$, for all $\varepsilon >0$.

On the other hand, since u solves (1.1), we have

$$\begin{aligned}&\left| \langle A(D_t)\varphi (- \Delta ) u , A(D_t)\varphi u \rangle _{L^2(\mathbb {R}\times \mathbb {D})} \right| \\&\quad = \left| \langle A(D_t)\varphi (2 D_t - V ) u, A(D_t)\varphi u \rangle _{L^2(\mathbb {R}\times \mathbb {D})} \right| \\&\quad \le \left| \langle A(D_t)\varphi (2 D_t - V ) u, A(D_t)\varphi u \rangle _{L^2(\mathbb {R}\times \mathbb {D})} \right| \\&\quad \le 2 \left| \langle A(D_t)^2 \varphi D_t u, \varphi u \rangle _{L^2(\mathbb {R}\times \mathbb {D})} \right| + C \Vert \tilde{\varphi } u\Vert ^2_{L^2(\mathbb {R}\times \mathbb {D})} \\&\quad \le 2 \left| \langle A(D_t)^2 D_t \varphi u, \varphi u \rangle _{L^2(\mathbb {R}\times \mathbb {D})} \right| + C \Vert \tilde{\varphi } u\Vert ^2_{L^2(\mathbb {R}\times \mathbb {D})} \\&\quad \le C \Vert \tilde{\varphi } u\Vert ^2_{L^2(\mathbb {R}\times \mathbb {D})}, \end{aligned}$$

since $A(D_t)^2 D_t = 1/2 {\text {Op}}_1(\psi ^2(H))$ is bounded. Collecting these estimates in (8.16), recalling (8.2) that $\Vert \tilde{\varphi } u\Vert ^2_{L^2(\mathbb {R}\times \mathbb {D})}\le C\Vert u^0\Vert ^2_{L^2(\mathbb {D})}$, and taking $\varepsilon$ sufficiently small concludes the proof of Lemma 2.13. $\square$

Appendix B: Time regularity of Wigner measures

In this section we present a proof of the following (general) result on time regularity of semiclassical measures associated to solutions of the Schrödinger equation (1.1). Even if not stated here, its microlocal counterpart also holds.

Proposition 9.1

Let $\mu _{sc}$ be obtained as a limit (2.5). Then there exists $\mu \in L^{\infty }( \mathbb {R}_{t};\mathcal {M}_{+}(T^{*}\mathbb {R} ^{2}))$ such that, for every $a\in C_{c}^{\infty }( T^{*}\mathbb {R}^{2}\times T^{*}\mathbb {R})$ we have:

$$\begin{aligned}&\int _{T^{*}\mathbb {R}^{2}\times T^{*}\mathbb {R}}a\left( z,\xi ,t,H\right) \mu _{sc}\left( dz,d\xi ,dt,dH\right) \\&\quad =\int _{\mathbb {R}} \int _{T^{*}\mathbb {R}^{2}}a\left( z,\xi ,t,\frac{\left| \xi \right| ^{2}}{2}\right) \mu \left( t,dz,d\xi \right) dt. \end{aligned}$$

Note that if the potential V is complex valued, we only have $\mu \in L^{\infty }_{{{\mathrm{loc}}}}( \mathbb {R}_{t};\mathcal {M}_{+}(T^{*}\mathbb {R}^{2}))$.

Proof

Let $u_{h}( \cdot ,t) :=U_{V}( t) u_{h}^{0}$ and note that the Wigner distributions:

$$\begin{aligned} \tilde{W}_{u_{h}}^{h}\left( t\right) :C_{c}^{\infty }\left( T^{*}\mathbb {R}^{2}\right) \ni l\longmapsto \left\langle U_{V}\left( t\right) u_{h}^{0},\hbox {Op}_{h}\left( l\right) U_{V}\left( t\right) u_{h}^{0}\right\rangle _{L^{2}(\mathbb {R}^2)}\in \mathbb {C} \end{aligned}$$

are uniformly bounded in $L^{\infty }( \mathbb {R}_{t};\mathcal {D} ^{\prime }(T^{*}\mathbb {R}^{2}))$. Hence, possibly after extracting a subsequence (and having used a diagonal extraction argument), we can assume that, for every $b\in C_{c}^{\infty }( T^{*}\mathbb {R}^{2}\times \mathbb {R})$:

$$\begin{aligned}&\lim _{h\rightarrow 0^{+}}\int _{\mathbb {R}}\left\langle U_{V}\left( t\right) u_{h}^{0},\hbox {Op}_{h}\left( b\left( \cdot ,t\right) \right) U_{V}\left( t\right) u_{h}^{0}\right\rangle _{L^{2}(\mathbb {R}^2 )} dt\\&\quad =\int _{\mathbb {R}}\int _{T^{*}\mathbb {R}^{2}}b\left( z,\xi ,t\right) \tilde{\mu }_{sc}\left( t,dz,d\xi \right) dt. \end{aligned}$$

Moreover, using the sharp Gårding inequality, we see that the limiting Wigner distribution is a nonnegative measure $\tilde{\mu } _{sc}\in L^{\infty }( \mathbb {R}_{t};\mathcal {M}_{+}(T^{*} \mathbb {R}^{2}))$. We next show that for any $b\in C_{c}^{\infty }( T^{*}\mathbb {R}^{2}\times \mathbb {R})$ with $b\ge 0$ one has:

$$\begin{aligned}&\int _{T^{*}\mathbb {R}^{2}\times T^{*}\mathbb {R}}b\left( z,\xi ,t\right) \mu _{sc}\left( dz,d\xi ,dt,dH\right) \nonumber \\&\quad \le \int _{\mathbb {R}}\int _{T^{*}\mathbb {R}^{2}}b\left( z,\xi ,t\right) \tilde{\mu }_{sc}\left( t,dz,d\xi \right) dt. \end{aligned}$$

(9.1)

To see this, let $\chi \in C_{c}^{\infty }( \mathbb {R})$ be a cut-off function satisfying $0\le \chi \le 1$, strictly positive in $( -3/2,3/2)$, vanishing outside that interval, and such that $\chi \rceil _{( -1,1) }\equiv 1$. Write, for $R>0$, $\chi _{R}:=\chi ( \cdot /R)$ and $\sigma _{R}:=\sqrt{1-\chi _{R}}$ (which we may also assume smooth). Then we have:

$$\begin{aligned}&\left\langle u_{h},\hbox {Op}_{h}\left( b\right) \chi _{R}\left( h^{2}D_{t}\right) u_{h}\right\rangle _{L^{2}(\mathbb {R}^2 \times \mathbb {R})}\nonumber \\&\quad =\left\langle u_{h},\hbox {Op}_{h}\left( b\right) u_{h}\right\rangle _{L^{2}(\mathbb {R}^2\times \mathbb {R})} +k_{h,R}\left( b\right) +O\left( h\right) , \end{aligned}$$

(9.2)

where:

$$\begin{aligned} k_{h,R}\left( b\right) :=\left\langle \sigma _{R}\left( h^{2}D_{t}\right) u_{h},\hbox {Op}_{h}\left( b\right) \sigma _{R}\left( h^{2}D_{t}\right) u_{h}\right\rangle _{L^{2}(\mathbb {R}^2\times \mathbb {R})}. \end{aligned}$$

Taking limits in (9.2) as $h\rightarrow 0^{+}$ we find that:

$$\begin{aligned}&\int _{T^{*}\mathbb {R}^{2}\times T^{*}\mathbb {R}}b\left( z,\xi ,t\right) \chi _{R}\left( H\right) \mu _{sc}\left( dz,d\xi ,dt,dH\right) \nonumber \\&\quad =\int _{\mathbb {R} }\int _{T^{*}\mathbb {R}^{2}}b\left( z,\xi ,t\right) \tilde{\mu }_{sc}\left( t,dz,d\xi \right) dt+\lim _{h\rightarrow 0^{+}}k_{h,R}\left( b\right) . \end{aligned}$$

(9.3)

But clearly, as $b\ge 0$, we always have

$$\begin{aligned} \lim _{h\rightarrow 0^{+}}k_{h,R}(b)=\lim _{h\rightarrow 0^{+}}\int _{\mathbb {R} }\tilde{W}_{\sigma _{R}\left( h^{2}D_{t}\right) u_{h}}^{h}\left( b\left( t,\cdot \right) \right) dt\ge 0, \end{aligned}$$

for every $R>0$. Taking this into account and letting $R\rightarrow \infty$ in (9.3) proves (9.1).

Now, as a consequence of (9.1) we have that the image of $\mu _{sc}$ under the projection onto the H-component is of the form $\mu ( t,\cdot ) dt$ for some $\mu \in L^{\infty }( \mathbb {R} _{t};\mathcal {M}_{+}(T^{*}\mathbb {R}^{2}))$. The disintegration theorem then ensures that $\mu _{sc}$ can be written as:

$$\begin{aligned} \mu _{sc}\left( dz, d\xi , dt, dH\right) =\mu _{z,\xi ,t}\left( dH\right) \mu \left( t,dz,d\xi \right) dt, \end{aligned}$$

where, for $\mu$-amost every $(z,\xi )$, $\mu _{z,\xi ,t}$ is a probability measure on $\mathbb {R}$. Since $\mu _{sc}\$is supported on the characteristic set $\vert \xi \vert ^{2}=2H$ (see Proposition 8.3), we conclude that $\mu _{z,\xi ,t}( dH) =\delta _{\vert \xi \vert ^{2}/2}( dH)$ and the result follows. $\square$

Appendix C: From action-angle coordinates to polar coordinates

Here we develop the technical calculations leading to the definitions of the operators $\mathcal {A}_{t, h^2D_t}(P)$ and $\widetilde{\mathcal {A}}_{t, h^2D_t}(P)$ used as a black-box in the paper. The point is that our “action-angle” coordinates $(s, \theta , E, J)$, well adapted to integrate the dynamics of the billiard flow, are not so convenient to express the Dirichlet boundary condition ($v(z)=0$ for $|z|=1$). Actually the best coordinates in which to write the boundary condition are the polar coordinates (which below will be written as $(x=-r\sin u, y=r\cos u)$) since the boundary is simply expressed as the set $\{r=1\}$.

Let $P(s, \theta , E, J)$ be a function expressed in the new coordinates and let $\mathscr {U}$ be the Fourier integral operator defined in (3.1). The technical calculations done below are aimed at understanding how $\mathscr {U}^* {\text {Op}}_h(P) \mathscr {U}$ acts in polar coordinates; in particular, under which conditions on the symbol P the boundary condition is preserved by $\mathscr {U}^* {\text {Op}}_h(P) \mathscr {U}$.

For our purposes we need to understand the operator $\mathscr {U}^* {\text {Op}}_h(P) \mathscr {U}$ modulo $O(h^2)$. Ideally we would like to separate it into a “tangential part” (involving only angular derivation $\frac{\partial }{\partial u}$) and a “radial part” involving the radial derivative $\frac{\partial }{\partial r}$ in a simple way. Below we calculate the action of the operator $\mathscr {U}^* {\text {Op}}_h(P) \mathscr {U}$ on a plane wave

$$\begin{aligned} e_\xi (z):=e^{i\frac{(\xi _x x+\xi _y y)}{h}} \end{aligned}$$

(where we use $z=(x, y)$, $\xi =(\xi _x, \xi _y)$ and $|\xi |^2= \xi _x^2 + \xi _y^2$) and apply the method of stationary phase. The length of the calculation comes from the fact that we explicitly need the term of order h in the expansion.

In this section, we shall assume that $P(s, \theta , E, J)$ satisfies the following properties.

Assumption 10.1

Assume first that $P(s, \theta , E, J)$ is a smooth compactly supported function (possibly depending on h), with support away from $\{E=0\}$ and inside $\{|J|<E\}$, and being $2\pi$-periodic in the variable $\theta$. We assume further that it satisfies the following estimates

$$\begin{aligned} ||\partial _s^{\alpha }\partial _\theta ^{\beta }\partial _E^{\gamma } \partial _J^{\delta }P||_\infty \le C_{\alpha , \beta , \gamma ,\delta } h^{-\delta }, \quad \text {for all }\; \alpha , \beta , \gamma ,\delta \in \mathbb {N}. \end{aligned}$$

The function P may also depend on the time variable t and its dual H, but, in this section, we omit them from the notation since they are transparent in the calculation. Typical symbols P for which the calculations below are needed can be found in (4.12) and (4.13).

Using the notation

$$\begin{aligned}&a(E) = \sqrt{E}, \quad (s_0, \theta _0, |\xi |, j_0) =\Phi ^{-1}(x, y,\xi _x, \xi _y),\\&\quad \text {and} \quad s(x, y, \theta )=-x\sin \theta +y\cos \theta , \end{aligned}$$

recalling the expressions of $\mathscr {U}$ and $\mathscr {U}^*$ in (3.2)–(3.3), and unfolding all the integrals, we write

$$\begin{aligned}&\mathscr {U}^* {\text {Op}}_h(P) \mathscr {U}e_\xi (x, y)\\&\quad = (2\pi h)^{-5}\int P\left( s,\theta , E', j\right) e^{\frac{ij(\theta -\theta ')}{h}}e^{\frac{iE'(s-s')}{h}} e^{-i\frac{S(x', y', \theta ', s', E)}{h}} e^{i\frac{(\xi _x x'+\xi _y y')}{h}} e^{i\frac{S(x, y, \theta , s, E'')}{h}}\\&\quad \quad \times a(E)a(E'') \, d\theta \, ds\, dE''\, dx'\, dy'\, dE \, d\theta '\, ds'\, dE' \, dj\\&\quad = (2\pi h)^{-3}\int P\left( s,\theta , E', j\right) e^{\frac{ij(\theta -\theta _0)}{h}}e^{\frac{iE'(s-s')}{h}} e^{i\frac{ s' |\xi |}{h}} e^{i\frac{S(x, y, \theta , s, E'')}{h}} \frac{a(|\xi |)}{|\xi |}\\&\quad \quad \times a(E'')\, d\theta \, ds\, dE'' \, ds'\, dE'\, dj\\&\quad = (2\pi h)^{-2}\int P\left( s,\theta , |\xi |, j\right) e^{\frac{ij(\theta -\theta _0)}{h}} e^{i\frac{ s |\xi |}{h}} e^{i\frac{S(x, y, \theta , s, E'')}{h}} \frac{a(|\xi |)}{|\xi |}a(E'')\, d\theta \, dE'' \, ds\, dj\\&\quad = (2\pi h)^{-1}\int \left( P\left( s(x, y, \theta ),\theta , |\xi |, j\right) a(|\xi |)\right. \\&\quad \quad \left. -ih\partial _s P\left( s(x, y, \theta ),\theta , |\xi |, j\right) a'(|\xi |) \right) \\&\quad \quad \times e^{\frac{ij(\theta -\theta _0)}{h}} e^{i\frac{-|\xi |x \sin \theta +|\xi |y \cos \theta }{h}} \frac{a(|\xi |)}{|\xi |}d\theta \, dj \quad +\frac{O(h^2)}{\inf _{P(s, \theta , E, J)\not =0}|E|^2}. \end{aligned}$$

By standard estimates on pseudodifferential operators, the remainder term will correspond to an estimate in the $L^2_{\mathrm{comp}}\rightarrow L^2_{{{\mathrm{loc}}}}$ topology of operators.

Letting $(x, y)=(-r\sin u, r\cos u)$, we have $r=\sqrt{x^2+y^2}$, $u=\arccos y/r$ and $s(x, y, \theta )=r\cos (\theta -u)$. Modulo $\frac{O(h^2)}{\inf _{P(s, \theta , E, J)\not =0}|E|^2}$, we are thus left with

$$\begin{aligned}&\mathscr {U}^* {\text {Op}}_h(P) \mathscr {U}e_\xi (x, y) \\&\quad = (2\pi h)^{-1} \frac{a(|\xi |)}{|\xi |}\int ( P\left( r\cos (\theta -u), \theta ,|\xi |, j\right) a(|\xi |)\\&\quad \quad -ih \partial _s P\left( r\cos (\theta -u), \theta ,|\xi |, j\right) a'(|\xi |))e^{\frac{ij(\theta -\theta _0)}{h}} e^{i\frac{|\xi |r\cos (\theta -u)}{h}}d\theta dj \\&\quad = (2\pi h)^{-1}\int ( P\left( r\cos (\theta -u), \theta ,|\xi |, j\right) \\&\quad \quad -\frac{ih}{2|\xi |} \partial _s P\left( r\cos (\theta -u), \theta ,|\xi |, j\right) e^{\frac{ij(\theta -\theta _0)}{h}} e^{i\frac{|\xi |r\cos (\theta -u)}{h}}d\theta dj. \end{aligned}$$

Remark 10.2

Note that in the above integral, the functions integrated are all $2\pi$-periodic in the variable $\theta$ except for the oscillating factor $e^{i \frac{j\theta }{h}}$. This integral has to be interpreted in one of the following two (equivalent) ways:

• either in the sense of oscillatory integrals [29, Section 7.8]: the integral over $\theta \in \mathbb {R}$ is the Fourier transform of the periodic function of $\theta$, seen as a tempered distribution. The result is a tempered distribution given by a linear combination of Dirac masses carried by $j\in h \mathbb {Z}$. The integration with respect to j has to be interpreted as a duality product;

• or in the sense of Fourier series: assuming $j \in h\mathbb {Z}$, the function of $\theta$ is $2\pi$-periodic and the integral over $\theta$ takes place on $\mathbb {R}/2\pi \mathbb {Z}$, i.e. on any period. The integral with respect to j has then to be understood as a discrete sum over $h\mathbb {Z}$.

Using for instance the second approach of this remark, we now apply stationary phase w.r.t. $\theta$ (while $j\in h\mathbb {Z}$ is kept fixed, since our symbols may be rapidly oscillating in j). We start with the P term (the $ih\partial _s P$-term can be treated exactly the same way). Fixing j and looking at the $\theta$-integral, we let

$$\begin{aligned} \mathcal {I}=(2\pi h)^{-1/2} \int _{{ \mathbb {R}/2\pi \mathbb {Z}}} P\left( r\cos (\theta -u), \theta ,|\xi |, j\right) e^{\frac{ij(\theta -\theta _0)}{h}} e^{i\frac{|\xi |r\cos (\theta -u)}{h}}d\theta . \end{aligned}$$

(10.1)

The phase in $\mathcal {I}$ has 2 critical points $\theta =u+\theta _1,u+ \theta _2$, where $\theta _k$ are the solutions of $j- |\xi |r\sin \theta =0$. Since we are assuming that $P(s, \theta , E, j)$ is supported in $\{|j|< E\}$, these two solutions are distinct for r close to 1, and correspond to non-degenerate stationary points (in all that follows we consider that r is close to 1 since this calculation only serves to understand $\mathscr {U}^* {\text {Op}}_h(P) \mathscr {U}$ near the boundary of the disk). We will denote by $\theta _1(r, E, j), \theta _2(r, E, j)$ the solutions of $j-Er\sin \theta =0$. To fix ideas, $\theta _1$ will be the one with $\cos \theta _1>0$ and $\theta _2$ the one with $\cos \theta _2<0$ (that is, $\theta _1\in (-\pi /2, \pi /2), \theta _2\in (\pi /2, 3\pi /2)$).

We let $\chi _k$ be smoth cutoff functions such that $\chi _k=1$ on a neighborhood of $\theta _k$, for $k=1,2$ and such that ${\text {supp}}(\chi _1) \subset (-\pi /2, \pi /2)$ and ${\text {supp}}(\chi _2) \subset (\pi /2, 3\pi /2)$. Using the non-stationary phase lemma, we have modulo $O(h^\infty )$

$$\begin{aligned} \mathcal {I}&=(2\pi h)^{-1/2}\int \chi _1(\theta - u) P\left( r\cos (\theta -u), \theta ,|\xi |, j\right) e^{\frac{ij(\theta -\theta _0)}{h}} e^{i\frac{|\xi |r\cos (\theta -u)}{h}}d\theta \\&\quad + (2\pi h)^{-1/2}\int \chi _2(\theta - u) P\left( r\cos (\theta -u), \theta ,|\xi |, j\right) e^{\frac{ij(\theta -\theta _0)}{h}} e^{i\frac{|\xi |r\cos (\theta -u)}{h}} d\theta . \end{aligned}$$

Below, E will always take the value $E=|\xi |$.

We will work on the integral $\mathcal {I}$ by applying the following lemma (which follows from the method of stationary phase):

Lemma 10.3

Let

$$\begin{aligned} \mathcal {I}_f=\int \chi (\theta ) f(\theta )e^{i\frac{S(\theta )}{h}}d\theta , \end{aligned}$$

where $S:\mathbb {R}\rightarrow \mathbb {R}$ is a smooth function having only one critical point $\theta _c$, which is non degenerate; where $\chi$ is a smooth compactly supported function which is constant in a neighborhood of $\theta _c$; and where f is a smooth function. Then, letting $a= f(\theta _c)$, $b= \frac{ f'(\theta _c)}{S''(\theta _c)}$ and $c=\frac{1}{2}(\frac{f''(\theta _c)}{S''(\theta _c)^2}-\frac{S^{(3)}(\theta _c)f'(\theta _c)}{S''(\theta _c)^3})$, we have

• (i) $\mathcal {I}_f= a \int \chi (\theta ) e^{i\frac{S(\theta )}{h}}d\theta +b\int \chi (\theta ) S'(\theta ) e^{i\frac{S(\theta )}{h}}d\theta +c\int \chi (\theta ) S'(\theta )^2 e^{i\frac{S(\theta )}{h}}d\theta +O(h^{2+1/2}) ;$

• (ii) $\mathcal {I}_f=\left( a +ih c S''(\theta _c)\right) \int \chi (\theta ) e^{i\frac{S(\theta )}{h}}d\theta +O(h^{2+1/2}).$

Proof

(i) The functions f and $g: \theta \mapsto a +b S'(\theta ) +c S'(\theta )^2$ coincide up to order 2 at $\theta _c$. The method of stationary phase tells us that $\mathcal {I}_f$ and $\mathcal {I}_g$ coincide modulo $O(h^{2+1/2})=O(h^{5/2})$.

Item (ii) is obtained from (i) by integration by parts, noting that $S'(\theta ) e^{i\frac{S(\theta )}{h}}$ is the derivative of $\frac{h}{i} e^{i\frac{S(\theta )}{h}}$. $\square$

In what follows, this lemma will be applied with $S(\theta )= j(\theta -\theta _0) +Er \cos (\theta -u)$, $f(\theta )= P\left( r\cos (\theta -u), \theta ,E, j\right)$, $\chi (\theta )= \chi _k(\theta - u)$ ($k=1, 2$), $\theta _c= u+\theta _k$. Starting with

$$\begin{aligned} S'(\theta )=j- Er\sin (\theta -u)\sim Er \left[ -\cos \theta _k(\theta -u-\theta _k)+\frac{\sin \theta _k}{2} (\theta -u-\theta _k)^2\right] \nonumber \\ \end{aligned}$$

(10.2)

(modulo $O(\theta -u-\theta _k)^3$), we have

$$\begin{aligned}&P(r\cos (\theta \!-\!u), \theta )\!\sim \! P(r\cos \theta _k, u\!+\!\theta _k)-\frac{S'(\theta )}{Er\cos \theta _k}\frac{d}{d\theta }P(r\cos \theta _k, u\!+\!\theta _k)\nonumber \\&\quad +\frac{S'(\theta )^2}{(Er\cos \theta _k)^2} \left[ \frac{\sin \theta _k}{2\cos \theta _k}\frac{d}{d\theta }P(r\cos \theta _k, u+\theta _k)+\frac{1}{2}\frac{d^2}{d\theta ^2}P(r\cos \theta _k, u+\theta _k)\right] .\nonumber \\ \end{aligned}$$

(10.3)

We have momentarily dropped the j and E variables from the argument of P since they are fixed in the upcoming calculation.

We want to apply the method of Lemma 10.3 with $a=a_k=P(r\cos \theta _k, u+\theta _k)$ ($k=1, 2$),

$$\begin{aligned} c=c_k= & {} \frac{1}{(Er\cos \theta _k)^2} \left[ \frac{\sin \theta _k}{2\cos \theta _k}\frac{d}{d\theta }P(r\cos \theta _k, u+\theta _k)\right. \nonumber \\&\left. +\frac{1}{2}\frac{d^2}{d\theta ^2}P(r\cos \theta _k, u+\theta _k)\right] . \end{aligned}$$

The lemma yields that

$$\begin{aligned} \mathcal {I}= & {} (2\pi h)^{-1/2}(a_1+ih c_1 S''(u+\theta _1))\int \chi _1(\theta - u) e^{\frac{ij(\theta -\theta _0)}{h}} e^{i\frac{Er\cos (\theta -u)}{h}}d\theta \nonumber \\&+ (2\pi h)^{-1/2} (a_2+ih c_2 S''(u+\theta _2))\int \chi _2(\theta - u) e^{\frac{ij(\theta -\theta _0)}{h}} e^{i\frac{Er\cos (\theta -u)}{h}} d\theta +O(h^2).\nonumber \\ \end{aligned}$$

(10.4)

Remark 10.4

Denoting by $\partial _1=\partial _s, \partial _2=\partial _\theta$ (to avoid possible confusion), we have

$$\begin{aligned} \frac{d}{d\theta }P( r\cos \theta , u+\theta )=\partial _2 P ( r\cos \theta , u+\theta )-r\sin \theta \,\partial _1 P ( r\cos \theta , u+\theta ), \end{aligned}$$

and

$$\begin{aligned} \frac{d^2}{d\theta ^2}P( r\cos \theta , u+\theta )= & {} \partial ^2_2 P ( r\cos \theta , u+\theta )-r\cos \theta \,\partial _1 P ( r\cos \theta , u+\theta )\\&-r\sin \theta \,\partial _2\partial _1 P ( r\cos \theta , u+\theta )\\&+\,r^2\sin ^2\theta \,\partial ^2_1 P ( r\cos \theta , u+\theta ). \end{aligned}$$

Remark 10.5

Important remark about symmetry. We keep denoting $\theta _k$ for $\theta _k(r, E, j)$. We first note that $\theta _2=\pi -\theta _1$, $\cos \theta _1=-\cos \theta _2, \sin \theta _1=\sin \theta _2$.

Moreover, if P satisfies the symmetry condition (B) of Definition 4.1, we have for $r=1$ (restoring in our notation the dependence of P on the full set of variables)

$$\begin{aligned} P(\cos \theta _1, u+\theta _1, E, j)=P(\cos \theta _2, u+\theta _2, E, j). \end{aligned}$$

And similarly for all partial derivatives of P if we assume the stronger symmetry condition (C) (in Definition 4.1).

Here we don’t necessarily want to assume that P is symmetric; but, motivated by the previous remark, we introduce the functions $P^\sigma$ and $P^\alpha$, the symmetric and antisymmetric parts of P respectively:

$$\begin{aligned} P^\sigma (r, \theta , E, j):= & {} \frac{P(r\cos \theta _1, \theta \!+\!\theta _1, E, j)\!+ \!P(\!-\!r\cos \theta _1, \theta \!+\!\pi \!-\!\theta _1, E, j)}{2}\nonumber \\= & {} \frac{P(r\cos \theta _1, \theta +\theta _1, E, j)+ P(r\cos \theta _2, \theta +\theta _2, E, j)}{2},\nonumber \\ \end{aligned}$$

(10.5)

and

$$\begin{aligned} P^\alpha (r, \theta , E, j):= & {} \frac{P(r\cos \theta _1, \theta \!+\!\theta _1, E, j)\!-\! P(\!-\!r\cos \theta _1, \theta \!+\!\pi -\theta _1, E, j)}{2} \nonumber \\= & {} \frac{P(r\cos \theta _1, \theta +\theta _1, E, j)- P(r\cos \theta _2, \theta +\theta _2, E, j)}{2},\nonumber \\ \end{aligned}$$

(10.6)

for $\theta _1=\theta _1(r, E, j), \theta _2=\theta _2(r, E, j)$ defined previously, so that

$$\begin{aligned} P(r\cos \theta _1, \theta +\theta _1, E, j)= P^\sigma (r, \theta , E, j)+ P^\alpha (r, \theta , E, j), \end{aligned}$$

$$\begin{aligned} P(r\cos \theta _2, \theta +\theta _2, E, j) = P^\sigma (r, \theta , E, j)- P^\alpha (r, \theta , E, j). \end{aligned}$$

Working from the expression (10.4), the terms

$$\begin{aligned}&(2\pi h)^{-1/2}a_1 \int \chi _1(\theta - u) e^{\frac{ij(\theta -\theta _0)}{h}} e^{i\frac{Er\cos (\theta -u)}{h}}d\theta \\&\quad + (2\pi h)^{-1/2} a_2\int \chi _2(\theta - u) e^{\frac{ij(\theta -\theta _0)}{h}} e^{i\frac{Er\cos (\theta -u)}{h}} d\theta \end{aligned}$$

maybe grouped as follows:

$$\begin{aligned}&(2\pi h)^{-1/2} { \int \chi _1(\theta - u)} P(r\cos \theta _1, u+\theta _1, E, j) e^{\frac{ij(\theta -\theta _0)}{h}} e^{i\frac{Er\cos (\theta -u)}{h}}d\theta \nonumber \\&\quad \quad + (2\pi h)^{-1/2} { \int \chi _2(\theta - u)}P(r\cos \theta _2, u+\theta _2, E, j) e^{\frac{ij(\theta -\theta _0)}{h}} e^{i\frac{Er\cos (\theta -u)}{h}}d\theta \nonumber \\&\quad =( 2\pi h)^{-1/2} P^\sigma (r, u, E, j) \int _0^{2\pi } e^{\frac{ij(\theta -\theta _0)}{h}} e^{i\frac{Er\cos (\theta -u)}{h}}d\theta \nonumber \\&\quad \quad + ( 2\pi h)^{-1/2} P^\alpha (r, u, E, j)\left( \int \chi _1(\theta - u) e^{\frac{ij(\theta -\theta _0)}{h}} e^{i\frac{|\xi |r\cos (\theta -u)}{h}}d\theta \right. \nonumber \\&\quad \quad \left. -\int \chi _2(\theta - u) e^{\frac{ij(\theta -\theta _0)}{h}} e^{i\frac{|\xi |r\cos (\theta -u)}{h}} d\theta \right) . \end{aligned}$$

(10.7)

Applying again Lemma 10.3 (this time with the function $f(\theta )=\cos (\theta -u)$), this expression can be rewritten modulo $O(h^2)$ as

$$\begin{aligned}&( 2\pi h)^{-1/2}\int _0^{2\pi } P^\sigma (r, u, E, j) e^{\frac{ij(\theta -\theta _0)}{h}} e^{i\frac{Er\cos (\theta -u)}{h}}d\theta \\&\quad +( 2\pi h)^{-1/2}\int _0^{2\pi } \frac{ P^\alpha (r, u, E, j)}{E\cos \theta _1(r, E, j)}\\&\quad \left( E\cos (\theta -u)-ih\frac{1}{2r\cos ^2\theta _1(r, E, j)}\right) \\&\quad \times e^{\frac{ij(\theta -\theta _0)}{h}} e^{i\frac{Er\cos (\theta -u)}{h}}d\theta . \end{aligned}$$

With the change of variable $\theta -\theta _0\rightsquigarrow u-\theta$, this may also be written as

$$\begin{aligned}&( 2\pi h)^{-1/2}\int _0^{2\pi } P^\sigma (r, u, E, j) e^{\frac{ij(u-\theta )}{h}} e^{i\frac{Er\cos (\theta -\theta _0)}{h}}d\theta \nonumber \\&\quad +( 2\pi h)^{-1/2}\int _0^{2\pi } \frac{ P^\alpha (r, u, E, j)}{E\cos \theta _1(r, E, j)}\nonumber \\&\quad \left( E\cos (\theta -\theta _0)-ih\frac{1}{2r\cos ^2\theta _1(r, E, j)}\right) \nonumber \\&\quad \times e^{\frac{ij(u-\theta )}{h}} e^{i\frac{Er\cos (\theta -\theta _0)}{h}}d\theta . \end{aligned}$$

(10.8)

We note that $e^{i\frac{Er\cos (\theta -\theta _0)}{h}}=e^{i\frac{(\xi _xx'+\xi _y y')}{h}} = e_\xi (x', y')$ if $(x', y')=(-r\sin \theta , r\cos \theta )$, and $E\cos (\theta -\theta _0) e^{i\frac{Er\cos (\theta -\theta _0)}{h}} =hD_r e^{i(\xi _xx'+\xi _y y')/h}$ where $D_r=\frac{1}{i} \partial _r$.

Terms of order h. Apart from the term of order h arising in the last line of (10.8), other terms of order h in (10.4) come from evaluation of the integrals

$$\begin{aligned}&(2\pi h)^{-1/2} ih c_1 S''(u+\theta _1) \int \chi _1(\theta - u) e^{\frac{ij(\theta -\theta _0)}{h}} e^{i\frac{Er\cos (\theta -u)}{h}}d\theta \\&\quad + (2\pi h)^{-1/2} ih c_2 S''(u+\theta _2)\int \chi _2(\theta - u) e^{\frac{ij(\theta -\theta _0)}{h}} e^{i\frac{Er\cos (\theta -u)}{h}} d\theta . \end{aligned}$$

This is equal to:

$$\begin{aligned}&\sum _{k=1,2} \bigg \{ (2\pi h)^{-1/2} ih { \int \chi _k(\theta - u)} \frac{1}{(Er\cos \theta _k)} \nonumber \\&\quad \times \left[ \frac{\sin \theta _k}{2\cos \theta _k}\frac{d}{d\theta }P(r\cos \theta _k,u+\theta _k)+\frac{1}{2}\frac{d^2}{d\theta ^2}P(r\cos \theta _k,u+\theta _k)\right] \nonumber \\&\quad \times e^{\frac{ij(\theta -\theta _0)}{h}}e^{i\frac{Er\cos (\theta -u)}{h}}d\theta \bigg \}. \end{aligned}$$

(10.9)

Here $\frac{d}{d\theta }P(r\cos \theta _k,u+\theta _k)$ and $\frac{d^2}{d\theta ^2}P(r\cos \theta _k,u+\theta _k)$ may be replaced by their expressions in terms of partial derivatives of P, as in Remark 10.4.

To summarize our computations, we need to introduce some notation.

Definition 10.6

Assume P satisfies Assumption 10.1. Then, we define by I(P), II(P), III(P), IV(P) the operators whose action on $e_\xi$ at the point $(x, y)=(-r\sin u, r\cos u)$ is given as follows: for $\xi =(\xi _x, \xi _y)$, $E=|{\xi }|$, we have (referring to Remark 10.2 for the meaning of the integrals)

$$\begin{aligned} I(P) e_\xi (x, y)&= \frac{1}{2\pi h}\int A(r, u, E, j) e^{\frac{ij(u-\theta )}{h}} e_\xi (-r\sin \theta , r\cos \theta )d\theta dj \\ II(P) e_\xi (x, y)&=\frac{1}{2\pi h}\int B(r, u, E, j) e^{\frac{ij(u-\theta )}{h}} hD_r e_\xi (-r\sin \theta , r\cos \theta )d\theta dj\\ III(P) e_\xi (x, y)&=\frac{1}{2\pi h}\int C(r, u, E, j) e^{\frac{ij(u-\theta )}{h}} e_\xi (-r\sin \theta , r\cos \theta )d\theta dj \\ IV(P) e_\xi (x, y)&\!=\!\frac{1}{2\pi h}\int D(r, u, E, j) e^{\frac{ij(u\!-\!\theta )}{h}} hD_r e_\xi (-r\sin \theta , r\cos \theta )d\theta dj, \end{aligned}$$

where

$$\begin{aligned} A(r, u, E, j)= & {} P^\sigma (r, u, E, j) ,\nonumber \\ B(r, u, E, j)= & {} \frac{P^\alpha (r, u, E, j)}{E\cos \theta _1(r, E, j)},\nonumber \\ C(r, u, E, j)= & {} - \frac{1}{2E}\partial _s P^\sigma (r, u, E, j) + c^\sigma (r, u, E, j) \nonumber \\&-\frac{1}{2r\cos ^2\theta _1(r, E, j)}\frac{P^\alpha (r, u, E, j)}{E\cos \theta _1(r, E, j)},\nonumber \\ D(r, u, E, j)= & {} -\frac{1}{2E}\frac{\partial _s P^\alpha (r, u, E, j)}{E\cos \theta _1(r, E, j)} +\frac{c^\alpha (r, u, E, j)}{E\cos \theta _1(r, E, j)},\nonumber \\ \end{aligned}$$

(10.10)

with the notation $P^\sigma , P^\alpha$ of (10.5), (10.6), and where, in addition

$$\begin{aligned}&c(s, \theta , E, j) =\frac{1}{(Es)} \bigg [\frac{j}{2Es} \left( \partial _2 P ( s, \theta , E, j)-\frac{j}{E} \partial _1 P ( s, \theta , E, j)\right) \nonumber \\&\quad +\frac{1}{2}\left( \partial ^2_2 P ( s, \theta , E, j)-s\partial _1 P ( s, \theta , E, j)-\frac{j}{E}\partial _2\partial _1 P ( s, \theta , E, j)\right. \nonumber \\&\quad \left. +\frac{j^2}{E^2}\partial ^2_1 P ( s, \theta , E, j)\right) \bigg ]. \end{aligned}$$

(10.11)

Remark first that the expression of $c(s, \theta , E, j)$ is calculated so that

$$\begin{aligned}&c(r\cos \theta _k,u+\theta _k, E, j)\\&\quad = \frac{1}{(Er\cos \theta _k)} \left[ \frac{\sin \theta _k}{2\cos \theta _k}\frac{d}{d\theta }P(r\cos \theta _k,u+\theta _k)\right. \nonumber \\&\quad \left. +\frac{1}{2}\frac{d^2}{d\theta ^2}P(r\cos \theta _k,u+\theta _k)\right] , \end{aligned}$$

which is the expression appearing in the last lines of (10.9).

Second, note that A, B, C, D are real-valued functions if P is.

With this notation in hand, we can now summarize our calculations in the following proposition.

Proposition 10.7

Assume P satisfies Assumption 10.1. Then, modulo a term of order $\frac{O(h^2)}{\inf _{P(s, \theta , E, J)\not =0}|E|^2}$ in the $L^2_{\mathrm{comp}}\rightarrow L^2_{{{\mathrm{loc}}}}$ topology of operators, $\mathscr {U}^* {\text {Op}}_h(P) \mathscr {U}$ satisfies

$$\begin{aligned} \mathscr {U}^* {\text {Op}}_h(P) \mathscr {U}= I(P)+II(P)+ih III(P)+ih IV(P). \end{aligned}$$

Let us now check that operators of the form I(P), II(P), III(P), IV(P) belong to a reasonable class of spatial pseudodifferential operators.

Lemma 10.8

Let M(r, u, E, j ) be a smooth (possibly h-dependent) function, compactly supported in r, E, j, $2\pi$-periodic in u, supported in $\{|j|<E\}$ and away from $\{r=0\}$ and $\{E=0\}$. Assume M satisfies estimates of the form

$$\begin{aligned} \sup _{h, r, u, E, j} h^{\gamma +\delta }|\partial _r^\alpha \partial _u^\beta \partial _E^{\gamma }\partial _j^{\delta } M | <+\infty , \quad \text {for all }\; \alpha , \beta , \gamma , \delta \in \mathbb {N}. \end{aligned}$$

(10.12)

Then the operators defined by their action on $e_\xi$ at $(x, y)=(-r\sin u, r\cos u)$ by

$$\begin{aligned} \hat{A} e_\xi (x, y) = \frac{1}{2\pi h}\int M (r, u, |\xi |, j) e^{\frac{ij(u-\theta )}{h}} e_\xi (-r\sin \theta , r\cos \theta )d\theta dj, \end{aligned}$$

and

$$\begin{aligned} \hat{B} e_\xi (x, y) =\frac{1}{2\pi h}\int M (r, u, |\xi |, j) e^{\frac{ij(u-\theta )}{h}} hD_r e_\xi (-r\sin \theta , r\cos \theta )d\theta dj \end{aligned}$$

are semiclassical pseudodifferential operators of the form $m_h(z,h D_z)$ where $m_h$ satisfies estimates of the form

$$\begin{aligned} \sup _{h, z, \xi } h^{|\beta |} |\partial _z^\alpha \partial _\xi ^\beta m_h| <+\infty ,\quad \text {for all }\; \alpha , \beta \in \mathbb {N}^2. \end{aligned}$$

(10.13)

In particular, these operators are bounded on $L^2(\mathbb {R}^2)$.

Proof

Let us first treat the case of $\hat{A}$. Define $\kappa (r, u)= (-r\sin u, r\cos u)$. The function $m_h$ is given by the formula

$$\begin{aligned} m_h(\kappa (r, u), h\xi )= \frac{1}{2\pi }\int M(r, u, h|\xi |, hj) e^{{ij(u-\theta )}} e^{i\xi \cdot (\kappa (r, \theta )-\kappa (r, u))}d\theta dj. \end{aligned}$$

The proof of [30, Theorem 18.1.17] applies to prove the desired estimate on $m_h$.

The operator $\hat{B}$ is an operator of the previous form, composed with $hD_r$. Since M is assumed to be compactly supported in E, the desired estimate also holds for $\hat{B}$. The bounded follows from the Calderón–Vaillancourt theorem [18].

Coming back to the operators defined in Definition 10.6, we have obtained the following corollary.

Corollary 10.9

Assume P satisfies Assumption 10.1. Then, the operators I(P), II(P), III(P), IV(P) of Definition 10.6 are semiclassical pseudodifferential operators of the form $m_h(z,h D_z)$ where $m_h$ satisfies estimates of the form (10.13). In particular, these operators are bounded on $L^2(\mathbb {R}^2)$.

Appendix D: The operators ${\mathcal {A}}_{t, h^2 D_t}(P)$ and $\tilde{\mathcal {A}}_{t, h^2 D_t}(P)$

We recall that the operators we manipulate are given by $\mathscr {U}^*{\text {Op}}_h( P(s, \theta ,E, J, t, hH))\mathscr {U}$ where the symbol $P(s, \theta ,E, J, t, H)$ is typically of the form (4.12) or (4.13), and thus satisfies Assumption 10.1 with respect to the space variable (or more precisely Assumption 11.1 below). The goal of this Appendix is to understand further (and up to order two in powers of h) how $\mathscr {U}^* {\text {Op}}_h(P) \mathscr {U}$ acts on functions vanishing on the boundary. The euclidean laplacian $\Delta _{\mathbb {R}^2}$ does not preserve the set of functions vanishing on the boundary. That is why we would like to eliminate the dependence of P on the variable E. We use the fact that the semiclassical measures associated with solutions of the Schrödinger equation are supported on $\{E^2=2H\}$, to replace E by $\sqrt{2H}$ in the calculations. This induces an additional error term, that will be shown to converge to 0 in Lemma 11.5 below.

This means, in particular, that we need to write explicitly the (t, H) dependence of the symbols (skipped in Appendix A above). As a consequence, the functions A, B, C, D defined from P in Definition 10.6 now also depend on t and H since P does.

The assumptions made on the symbol P in this section are similar to Assumption 10.1 above. We recall that typical symbols P under interest here are given by (4.12) and (4.13).

Assumption 11.1

Assume first that the function P is a smooth compactly supported function in s, E, j, t, H (possibly depending on h), with support in $\{|j|<E\}$ and away from $\{j=0, s=0\} \cup \{H=0\}$, and being $2\pi$-periodic in the variable $\theta$. We assume further that it satisfies the following estimates:

$$\begin{aligned} \sup _{h, s, u, E, j, t, H} h^{\nu } h^{\delta }|\partial _s^\alpha \partial _\theta ^\beta \partial _E^{\gamma }\partial _j^{\delta } \partial _t^\mu \partial _H^\nu P| <+\infty , \quad \text {for all }\; \alpha , \beta , \gamma , \delta , \nu , \mu \in \mathbb {N}. \end{aligned}$$

In the following formal calculations, it will be convenient to introduce the following notation.

Definition 11.2

If P depends on t and H, we write $I_{t,H}(P):= I(P)$, $II_{t,H}(P):= II(P)$, $III_{t,H}(P):= III(P)$, $IV_{t,H}(P):= IV(P)$: they are t, H-families of operators defined in Definition 10.6. We then denote by ${\mathcal {A}}_{t,H}(P)$ the family of operators given by

$$\begin{aligned} {\mathcal {A}}_{t,H}(P) = I_{t,H}(P)+II_{t,H}(P)+ih III_{t,H}(P)+ih IV_{t,H}(P). \end{aligned}$$

(11.1)

We have shown (see Proposition 10.7) that for any given (t, H), $\mathcal {A}_{t, H}(P)$ coincides with $\mathscr {U}^* {\text {Op}}_h (P(\cdot , t,H)) \mathscr {U}$ (where the quantification is only in the variables $(s, \theta ,E, J)$) modulo $\frac{O(h^2)}{\inf _{P(s, \theta , E, J)\not =0}|E|^2}$ in the $L^2_{\mathrm{comp}}\rightarrow L^2_{{{\mathrm{loc}}}}$ topology of operators.

We now define a modified operator $\widetilde{\mathcal {A}}_{t, H}(P)$ whose action on functions vanishing on $\partial \mathbb {D}$ is easier to understand.

Definition 11.3

We denote by $\widetilde{\mathcal {A}}_{t, H}(P)$ the family of operators

$$\begin{aligned} \widetilde{\mathcal {A}}_{t, H}(P) = \widetilde{I}_{t, H}(P)+\widetilde{II}_{t, H}(P)+ih\widetilde{ III}_{t, H}(P)+ih\widetilde{ IV}_{t, H}(P), \end{aligned}$$

(11.2)

where the four operators involved are defined by their action on $e_\xi$ at the point $(x, y)=(-r\sin u, r\cos u)$ is given by

$$\begin{aligned}&\widetilde{I}_{t, H}(P)(e_\xi )(x, y)\\&\quad = \frac{1}{2\pi h}\int A(r, u, \sqrt{2H}, j,t,H) e^{\frac{ij(u-\theta )}{h}} e_\xi (-r\sin \theta , r\cos \theta )d\theta dj ,\\&\widetilde{II}_{t, H}(P)(e_\xi )(x, y)\\&\quad = \frac{1}{2\pi h}\int B(r, u, \sqrt{2H}, j,t,H) e^{\frac{ij(u-\theta )}{h}} hD_r e_\xi (-r\sin \theta , r\cos \theta )d\theta dj,\\&\widetilde{III}_{t, H}(P)(e_\xi )(x, y)\\&\quad = \frac{1}{2\pi h}\int C(r, u, \sqrt{2H}, j,t,H) e^{\frac{ij(u-\theta )}{h}} e_\xi (-r\sin \theta , r\cos \theta )d\theta dj, \\&\widetilde{IV}_{t, H}(P)(e_\xi )(x, y)\\&\quad = \frac{1}{2\pi h}\int D(r, u, \sqrt{2H}, j,t,H) e^{\frac{ij(u-\theta )}{h}} hD_r e_\xi (-r\sin \theta , r\cos \theta )d\theta dj, \end{aligned}$$

where A, B, C, D are defined (as functions of P) in Definition 10.6.

In other words, in the definition of $\mathcal {A}_{t, H}(P)$ we have replaced $|\xi |$ by $\sqrt{2H}$ in the symbols. For us, $\widetilde{\mathcal {A}}_{t, H}(P)$ is a very convenient operator to study the Dirichlet boundary problem, since we have

$$\begin{aligned}&\widetilde{I}_{t, H}(P)=A(r, u, \sqrt{2H}, hD_u, t, H), \\&\widetilde{ III}_{t, H}(P)=C(r, u,h \sqrt{2H}, hD_u, t, H) \end{aligned}$$

(so that they do not involve any derivative w.r.t. r) and

$$\begin{aligned}&\widetilde{II}_{t, H}=B(r, u, \sqrt{2H}, hD_u, t, H)\circ hD_r,\\&\widetilde{ IV}_{t, H}(P)=D(r, u,\sqrt{2H}, hD_u, t, H)\circ hD_r \end{aligned}$$

which are only of degree 1 w.r.t. the variable r. We define the operators

$$\begin{aligned} {\mathcal {A}}_{t, h^2 D_t}(P) := {\text {Op}}_{h^2} \left( {\mathcal {A}}_{t, H}(P) \right) , \quad \text {and} \quad \widetilde{\mathcal {A}}_{t, h^2 D_t}(P):= {\text {Op}}_{h^2} \left( \widetilde{\mathcal {A}}_{t, H}(P) \right) , \end{aligned}$$

where the quantification only concerns the variables (t, H). We have the analogue of Corollary 10.9 stating that these operators are proper pseudodifferential operators.

Corollary 11.4

Assume P satisfies Assumption 11.1. Then, the operators $I_{t, h^2 D_t}(P)$, $II_{t, h^2 D_t}(P)$, $III_{t, h^2 D_t}(P)$, $IV_{t, h^2 D_t}(P)$, ${\mathcal {A}}_{t, h^2 D_t}(P)$ and the operators $\widetilde{I}_{t, h^2 D_t}(P)$, $\widetilde{II}_{t, h^2 D_t}(P)$, $\widetilde{III}_{t, h^2 D_t}(P)$, $\widetilde{IV}_{t, h^2 D_t}(P)$, $\widetilde{\mathcal {A}}_{t, h^2 D_t}(P)$ are semiclassical pseudodifferential operators of the form $m_h(z,t,h D_z, h^2 D_t)$ where $m_h$ satisfies estimates of the form:

$$\begin{aligned} \sup _{h, z, \xi ,t,H} h^{|\beta |}h^{\nu } |\partial _z^\alpha \partial _\xi ^\beta \partial _t^\mu \partial _H^\nu m_h| <+\infty ,\quad \text {for all }\; \alpha , \beta \in \mathbb {N}^2, \mu , \nu \in \mathbb {N}. \end{aligned}$$

In particular, these operators are bounded on $L^2(\mathbb {R}^2 \times \mathbb {R})$.

Now, we want to replace everywhere ${\mathcal {A}}_{t, h^2 D_t}(P)$ by $\widetilde{\mathcal {A}}_{t, h^2 D_t}(P)$. This is possible thanks to the fact that our semiclassical measures are supported by the set $\{E^2=2H\}$; a precise statement is given in Lemma 11.5 below.

Lemma 11.5

If $u_h$ is a solution to the Schrödinger equation (1.1) satisfying in addition the assumptions of Remark 2.4, then, for any P satisfying Assumption 11.1, we have

$$\begin{aligned}&\langle u_h, {\mathcal {A}}_{t, h^2 D_t}(P) u_h\rangle _{L^2(\mathbb {R}^2 \times \mathbb {R})}-\langle u_h, \widetilde{\mathcal {A}}_{t, h^2 D_t}(P) u_h\rangle _{L^2(\mathbb {R}^2 \times \mathbb {R})}\\&\quad = O_\epsilon \left( h^{1/2-\epsilon }\right) ||u^0_h||_{L^2(\mathbb {D})}^2. \end{aligned}$$

Proof

We write the decompositions (11.1) and (11.2) of the operators $\mathcal {A}_{t, h^2 D_t }(P)$ and $\widetilde{\mathcal {A}}_{t, h^2 D_t }(P)$. Terms coming from $III_{t, h^2 D_t}(P), \widetilde{III}_{t, h^2 D_t}(P), IV_{t, h^2 D_t}(P), \widetilde{IV}_{t, h^2 D_t}(P)$ are of order h, and it suffices to treat the term $I_{t, h^2 D_t}(P) - \widetilde{I}_{t, h^2 D_t}(P)$ (the term $II_{t, h^2 D_t}(P) - \widetilde{II}_{t, h^2 D_t}(P)$ is treated similarly).

First, for E restricted to a compact set, we can divide $A(r, u, E, j, t, H)-A(r, u, \sqrt{2H}, j, t, H)$ by $E^2-2H$ thanks to the following Taylor formula:

$$\begin{aligned}&A(r, u, E, j, t, H)-A(r, u, \sqrt{2H}, j, t, H)\\&\quad \!=\!(E\!+\!\sqrt{2H})^{-1} \int _{0}^1 \frac{\partial A}{\partial E} (r, u, \sqrt{2H}\!+\! l(E\!-\!\sqrt{2H}), j, t, H) dl\, (E^2\!-\!2H). \end{aligned}$$

The function $(E+\sqrt{2H})^{-1} \int _{0}^1 \frac{\partial A}{\partial E} (r, u, \sqrt{2H}+ l(E-\sqrt{2H}), j, t, H) dl$, restricted to a compact set in E, satisfies the estimate of Assumption 11.1. We can apply Corollary 11.4 to see that, for any compactly supported $\chi$, the operator

$$\begin{aligned} \left( I_{t, h^2 D_t}(P) - \widetilde{I}_{t, h^2 D_t}(P) \right) \chi (-h^2 \Delta _{\mathbb {R}^2}) \end{aligned}$$

is of the form $\tilde{a}_h(z, hD_z, t, h^2 D_t)$ where $\tilde{a}_h$ is of the form

$$\begin{aligned} \tilde{a}_h(z, \xi , t, H)= a_h(z, \xi , t, H)(E^2-2H) \end{aligned}$$

and $a_h$ is compactly supported in $\xi , H$ and satisfies

$$\begin{aligned} \sup _{h, z, \xi ,t,H} h^{|\beta |}h^{2\nu } |\partial _z^\alpha \partial _\xi ^\beta \partial _t^\mu \partial _H^\nu a_h| <+\infty ,\quad \text {for all }\; \alpha , \beta \in \mathbb {N}^2, \mu , \nu \in \mathbb {N}. \end{aligned}$$

We then apply (8.9) to conclude.

Second, since A is compactly supported in the variable H, for sufficiently large E it is clear that we may divide $A(r, u, E, j, t, H)-A(r, u, \sqrt{2H}, j, t, H)$ by $E^2-2H$:

$$\begin{aligned}&A(r, u, E, j, t, H)-A(r, u, \sqrt{2H}, j, t, H)\\&\quad \!=\!\left( A(r, u, E, j, t, H)\!-\!A(r, u, \sqrt{2H}, j, t, H) \right) (E^2-2H)^{-1} (E^2-2H). \end{aligned}$$

The conclusion of the lemma is obtained for large E by an argument similar to those developed in the proof of Proposition 8.4 (we now need a dyadic decomposition only in the variable E, since $A(r, u, E, j, t, H)-A(r, u, \sqrt{2H}, j, t, H)$ is compactly supported in H but not in E). $\square$

Appendix E: Commutators

The goal of this section is to calculate explicitly (in terms of P) the expression of the commutator $[\Delta , \widetilde{\mathcal {A}}_{t, h^2D_t}(P)]$, where $\Delta$ is the laplacian on $\mathbb {R}^2$. This could, in principle, be done by brutal calculation, using the expression of the laplacian in polar coordinates ($\Delta _{r, u}=\frac{\partial ^2}{\partial r^2}+\frac{1}{r} \frac{\partial }{\partial r }+\frac{1}{r^2}\frac{\partial ^2}{\partial u^2}$). But this is too cumbersome and we try a less frontal approach. We want to use the fact that $[\Delta , \mathcal {A}_{t, h^2D_t}(P)]$ is known (from the exact Egorov theorem, Eq. (12.1) below) and to see how the calculus is modified when we replace $\mathcal {A}_{t, h^2D_t}(P)$ by $\widetilde{\mathcal {A}}_{t, h^2D_t}(P)$.

Recall from Lemma 3.1 and formula (2.10) that we have the exact formula (without remainder term)

$$\begin{aligned} \left[ -\frac{ih\Delta }{2}, \mathscr {U}^* {\text {Op}}_h(P) \mathscr {U}\right] = \mathscr {U}^* {\text {Op}}_h\left( E\partial _1 P-\frac{ih}{2}\partial _1^2 P\right) \mathscr {U}, \end{aligned}$$

(12.1)

where $\Delta$ is the Laplacian on $\mathbb {R}^2$.

When doing this commutator analysis, the time variables are completely transparent and (t, H) are frozen parameters. In particular, in the following, ${\text {Op}}_h$ denotes the quantization with respect to space variables only.

1.1 E.1 Formal calculation of $[\Delta , \mathcal {A}_{t, H}(P)]$

We use the expression of $\nabla$ in polar coordinates: $\nabla =( \partial _r , r^{-1}\partial _u)$ in the orthonormal frame $(e_r, e_u)$. We also use the formula $\Delta (fg)=f\Delta g +2\nabla f\cdot \nabla g+g\Delta f$. We obtain the following expression of $[\Delta , I_{t, H}(P)]$ applied to $e_\xi$ at $(x, y)=(-r\sin u, r\cos u)$:

$$\begin{aligned}&[\Delta , I_{t, H}(P)]e_\xi (x, y) \!=\! (2\pi h)^{-1}\int \Delta _{r, u}A(r, u, E, j) e^{\frac{ij(\theta -\theta _0)}{h}} e^{i\frac{Er\cos (\theta -u)}{h}}d\theta dj\nonumber \\&\quad +\frac{2i}{h}(2\pi h)^{-1}\int \partial _r A(r, u, E, j)E\cos (\theta -u) e^{\frac{ij(\theta -\theta _0)}{h}} e^{i\frac{Er\cos (\theta -u)}{h}}d\theta dj\nonumber \\&\quad +\frac{2i}{h} (2\pi h)^{-1}\int r^{-2}\partial _u A(r, u, E, j)j e^{\frac{ij(\theta -\theta _0)}{h}} e^{i\frac{Er\cos (\theta -u)}{h}}d\theta dj \end{aligned}$$

(12.2)

Note that the details of the calculations are actually not important, we only need to know “what the calculations look like” at a formal level (in particular, small errors of calculation are harmless).

Similarly, $[\Delta , II_{t, H}(P)]$ has the expression

$$\begin{aligned}&[\Delta , II_{t, H}(P)] e_\xi (x,y)\nonumber \\&\quad = (2\pi h)^{-1}\int \Delta _{r, u}B(r, u, E, j)E\cos (\theta -u) e^{\frac{ij(\theta -\theta _0)}{h}} e^{i\frac{Er\cos (\theta -u)}{h}}d\theta dj\nonumber \\&\quad \quad +\frac{2i}{h}(2\pi h)^{-1}\int \partial _r B(r, u, E, j)(E\cos (\theta -u))^2 e^{\frac{ij(\theta -\theta _0)}{h}} e^{i\frac{Er\cos (\theta -u)}{h}}d\theta dj\nonumber \\&\quad \quad +\frac{2i}{h} (2\pi h)^{-1}\int r^{-2}\partial _u B(r, u, E, j)j E\cos (\theta -u)e^{\frac{ij(\theta -\theta _0)}{h}} e^{i\frac{Er\cos (\theta -u)}{h}}d\theta dj\nonumber \\&\quad =(2\pi h)^{-1}\int \Delta _{r, u}B(r, u, E, j)E\cos (\theta -u) e^{\frac{ij(\theta -\theta _0)}{h}} e^{i\frac{Er\cos (\theta -u)}{h}}d\theta dj\nonumber \\&\quad \quad +\frac{2i}{h}(2\pi h)^{-1}\int \partial _r B(r, u, E, j)\nonumber \\&\qquad \left[ (E\cos (\theta _1))^2+ih\frac{\cos \theta _1}{(Er)^2}\right] e^{\frac{ij(\theta -\theta _0)}{h}} e^{i\frac{Er\cos (\theta -u)}{h}}d\theta dj\nonumber \\&\quad \quad +\frac{2i}{h} (2\pi h)^{-1}\int r^{-2}\partial _u B(r, u, E, j)j\nonumber \\&\qquad E\cos (\theta -u)e^{\frac{ij(\theta -\theta _0)}{h}} e^{i\frac{Er\cos (\theta -u)}{h}}d\theta dj +O(h^2) \end{aligned}$$

(12.3)

Similar calculations can be done for $[\Delta , III_{t, H}(P)]$ and $[\Delta , IV_{t, H}(P)]$. We do not need the explicit expressions, but need only to note that it gives a final expression modulo $O(h^2)$ of $[-ih\Delta /2, \mathcal {A}_{t, H}(P)]$ applied to $e_\xi$ at $(x,y)=(-r\sin u, r\cos u)$ in the form:

$$\begin{aligned}&[-ih\Delta /2, \mathcal {A}_{t, H}(P)] e_\xi (x,y) \!=\! \frac{1}{2\pi h}\int K(r, u, E, j) e^{\frac{ij(u-\theta )}{h}} e^{i\frac{Er\cos (\theta -\theta _0)}{h}}d\theta dj \\&\quad +\frac{1}{2\pi h}\int L(r, u, E, j) E\cos (\theta -\theta _0)e^{\frac{ij(u-\theta )}{h}} e^{i\frac{Er\cos (\theta -\theta _0)}{h}}d\theta dj\\&\quad + \frac{ih}{2\pi h}\int M(r, u, E, j) e^{\frac{ij(u-\theta )}{h}} e^{i\frac{Er\cos (\theta -\theta _0)}{h}}d\theta dj \\&\quad +\frac{ih}{2\pi h}\int N(r, u, E, j) E\cos (\theta -\theta _0)e^{\frac{ij(u-\theta )}{h}} e^{i\frac{Er\cos (\theta -\theta _0)}{h}}d\theta dj\\&\quad + \frac{1}{2\pi h}\int \partial _r B(r, u, E, j)\left[ (E\cos (\theta _1))^2+ih\frac{\cos \theta _1}{(Er)^2}\right] e^{\frac{ij(\theta -\theta _0)}{h}} e^{i\frac{Er\cos (\theta -u)}{h}}d\theta dj\\&\quad + \frac{1}{2\pi h}\int ih \partial _r D(r, u, E, j)(E\cos (\theta _1))^2 e^{\frac{ij(\theta -\theta _0)}{h}} e^{i\frac{Er\cos (\theta -u)}{h}}d\theta dj. \end{aligned}$$

Note that the last two lines may obviously be incorporated into the previous terms; but we shall see later why it is convenient to keep them separate.

The functions K, L, M, N are partial differential operators applied to A, B, C, D, and could in principle be expressed explicitly in terms of P, but we actually do not need these expressions.

1.2 E.2 Identification

We know from (12.1) that

$$\begin{aligned} \left[ -\frac{ih\Delta }{2}, \mathscr {U}^* {\text {Op}}_h(P) \mathscr {U}\right]= & {} \mathscr {U}^* {\text {Op}}_h\left( E\partial _1 P-\frac{ih}{2}\partial _1^2 P\right) \mathscr {U}\\= & {} \mathcal {A}_{t, H}\left( E\partial _1 P-\frac{ih}{2}\partial _1^2 P\right) +O(h^2). \end{aligned}$$

Using the identification Lemma 12.1 below, this leads directly to the identifications:

$$\begin{aligned}&K(r, u, E, j) +\partial _r B(r, u, E, j)(E\cos (\theta _1))^2= A_{E\partial _1 P}\\&L(r, u, E, j)=B_{E\partial _1 P}\\&M(r, u, E, j)+ \partial _r B(r, u, E, j) \frac{\cos \theta _1}{(Er)^2}+ \partial _r D(r, u, E, j)(E\cos (\theta _1))^2\\&\quad = C_{E\partial _1 P}-\frac{1}{2}A_{\partial _1^2 P}\\&N(r, u, E, j)=D_{E\partial _1 P}-\frac{1}{2} B_{\partial _1^2 P} \end{aligned}$$

where $\theta _1=\theta _1(r, E, j)$ denotes as before the solution in $[-\pi /2, \pi /2)$ of $\sin \theta _1=j/Er$. On the right-hand sides, notation such as $A_{E\partial _1 P}$, $B_{E\partial _1 P}$ etc. means “the functions A, B etc. associated to $E\partial _1P$ by the formulas of Definition 10.6”.

To justify these identifications we are using the following:

Lemma 12.1

Let A and B be two smooth real-valued functions. Then the values of

$$\begin{aligned}&\frac{1}{2\pi h}\int A(r, u, E, j) e^{\frac{ij(u-\theta )}{h}} e^{i\frac{Er\cos (\theta -\theta _0)}{h}}d\theta dj \nonumber \\&\quad +\frac{1}{2\pi h}\int B(r, u, E, j) \cos (\theta -\theta _0)e^{\frac{ij(u-\theta )}{h}} e^{i\frac{Er\cos (\theta -\theta _0)}{h}}d\theta dj \end{aligned}$$

(12.4)

for all $r, u, \theta _0, E$ determine A and B uniquely.

Proof

Integrating (12.4) along $e^{in\theta _0}d\theta _0$ ($\theta _0\in [0, 2\pi ]$, n an arbitrary integer) yields the value

$$\begin{aligned}&\int A(r, u, E, nh) e^{{in(u-\theta )}} e^{i\frac{Er\cos (\theta -\theta _0)}{h}}d\theta \nonumber \\&\quad +\int B(r, u, E, nh) \cos (\theta -\theta _0)e^{{in(u-\theta )}} e^{i\frac{Er\cos (\theta -\theta _0)}{h}}d\theta . \end{aligned}$$

(12.5)

If we take $n=n(h)$ a family of even integers growing like 1 / h, application of the method of stationary phase yields that this is (up to O(h))

$$\begin{aligned}&2e^{inu}(2\pi h)^{1/2}[\sin ^{1/2}\theta _1\, A(r, u, E, hn(h))\cos (-n\theta _1\!+\!Erh^{-1}\cos \theta _1\!+\!\pi /4)\nonumber \\&\quad +iB(r, u, E, hn(h))\sin (-n\theta _1+Erh^{-1}\cos \theta _1+\pi /4)] \end{aligned}$$

(12.6)

where $\theta _1$ is the solution in $[-\pi /2, \pi /2)$ of $\sin \theta _1=\frac{h n(h)}{Er}$. If A and B are continuous and real-valued then (12.6) suffices to determine A and B. $\square$

1.3 E.3 Formal calculation of $[\Delta , \tilde{\mathcal {A}}_{t, H}(P)]$

We want to use the previous identities to find the formal expression of $[\Delta , \widetilde{\mathcal {A}}_{t, H}(P)]$. Remember that $\widetilde{\mathcal {A}}_{t, H}(P)$ is the operator we want to use in all our proofs, because it comes naturally into a “tangential” part and a “radial” part of degree 1.

If we compare the formal calculations leading to the expressions of $[\Delta , \mathcal {A}_{t, H}(P)]$ and $[\Delta , \widetilde{\mathcal {A}}_{t, H}(P)]$, we see that they are identical and thus $[-ih\Delta /2, \widetilde{\mathcal {A}}_{t, H}(P)]$ applied to $e_\xi$ at $(-r\sin u, r\cos u)$ has the form

$$\begin{aligned}&\frac{1}{2\pi h}\int K(r, u, \sqrt{2H}, j) e^{\frac{ij(u-\theta )}{h}} e^{i\frac{Er\cos (\theta -\theta _0)}{h}}d\theta dj \\&\quad +\frac{1}{2\pi h}\int L(r, u, \sqrt{2H}, j) E\cos (\theta -\theta _0)e^{\frac{ij(u-\theta )}{h}} e^{i\frac{Er\cos (\theta -\theta _0)}{h}}d\theta dj\\&\quad + \frac{ih}{2\pi h}\int M(r, u, \sqrt{2H}, j) e^{\frac{ij(u-\theta )}{h}} e^{i\frac{Er\cos (\theta -\theta _0)}{h}}d\theta dj \\&\quad +\frac{ih}{2\pi h}\int N(r, u, \sqrt{2H}, j) E\cos (\theta -\theta _0)e^{\frac{ij(u-\theta )}{h}} e^{i\frac{Er\cos (\theta -\theta _0)}{h}}d\theta dj\\&\quad + (2\pi h)^{-1}\int \partial _r B(r, u, \sqrt{2H}, j)\\&\quad \times \left[ (E\cos (\theta _1))^2+ih\frac{\cos \theta _1}{(Er)^2}\right] e^{\frac{ij(\theta -\theta _0)}{h}} e^{i\frac{Er\cos (\theta -u)}{h}}d\theta dj\\&\quad + (2\pi h)^{-1}\int ih \partial _r D(r, u, \sqrt{2H}, j)(E\cos (\theta _1))^2 e^{\frac{ij(\theta -\theta _0)}{h}} e^{i\frac{Er\cos (\theta -u)}{h}}d\theta dj \end{aligned}$$

Note that $\theta _1=\theta _1(r, E, j)$ and that the symbol in the last two lines still depends on E (this is why we treat it separately). Everywhere else in the symbol, E has been replaced by $\sqrt{2H}$. Note also that $(E\cos (\theta _1))^2=E^2-\frac{j^2}{r^2}$.

From this and from the identifications of Sect. 5.2, we deduce the following final formula.

Proposition 12.2

There exists a function $R(r, u, E, \sqrt{2H}, j)$ such that

$$\begin{aligned}&[-ih\Delta /2, \widetilde{\mathcal {A}}_{t, h^2D_t}(P)] =\widetilde{\mathcal {A}}_{t, h^2D_t}(E\partial _1 P)-\frac{ih}{2}\widetilde{\mathcal {A}}_{t, h^2D_t}(\partial _1^2 P)+O(h^2)\nonumber \\&\quad + \partial _r B(r, u, { h\sqrt{2D_t}}, hD_u)\circ (-h^2\Delta -2{ h^2 D_t})\nonumber \\&\quad + ih R(r, u, \sqrt{-h^2\Delta }, { h\sqrt{2D_t}}, hD_u)\circ (-h^2\Delta -2{ h^2 D_t}), \end{aligned}$$

(12.7)

where B is the function given from P by Definition 10.6.

Proof

Indeed, the identifications of Sect. 5.2 yield

$$\begin{aligned}&[-ih\Delta /2, \widetilde{\mathcal {A}}_{t, h^2D_t}(P)]= \widetilde{I}_{t, h^2D_t}(E\partial _1 P)+\widetilde{II}_{t, h^2D_t}( E\partial _1 P) \nonumber \\&\quad + ih\left( \widetilde{III}_{t, h^2D_t}(E\partial _1 P)-1/2 \widetilde{I}_{t, h^2D_t}(\partial _1^2 P)\right) \nonumber \\&\quad +ih\left( \widetilde{IV}_{t, h^2D_t}(E\partial _1 P) -1/2 \widetilde{II}_{t, h^2D_t}(\partial _1^2 P) \right) \nonumber \\&\quad + \partial _r B(r, u, { h\sqrt{2D_t}}, hD_u)\circ (-h^2\Delta -2{ h^2 D_t}) \nonumber \\&\quad + ih R(r, u, \sqrt{-h^2\Delta }, { h\sqrt{2D_t}}, hD_u)\circ (-h^2\Delta -2{ h^2 D_t}) \end{aligned}$$

(12.8)

where the function R is defined by the identity

$$\begin{aligned}&R(r, u, E, \sqrt{2H}, j) (E^2-2H)\\&\quad = \partial _r B(r, u, \sqrt{2H}, j)\left[ \frac{\cos \theta _1(r, E, j)}{(Er)^2}- \frac{\cos \theta _1(r, \sqrt{2H}, j)}{2Hr^2}\right] . \end{aligned}$$

Indeed, we can apply a simple division lemma (actually the Taylor integral formula) to write

$$\begin{aligned} \frac{\cos \theta _1(r, E, j)}{(Er)^2}- \frac{\cos \theta _1(r, \sqrt{2H}, j)}{2Hr^2}= S(r, u, E, j, \sqrt{2H}) (E^2-2H), \end{aligned}$$

and thus

$$\begin{aligned} R(r, u, E, \sqrt{2H}, j)=\partial _r B(r, u, \sqrt{2H}, j)S(r, u, E, j, \sqrt{2H}). \end{aligned}$$

$\square$