Propagation of Analytic Singularities for Short and Long Range Perturbations of the Free Schrödinger Equation

Abstract

We study the propagation of the analytic wave front set for solutions to the Schrödinger equation associated with perturbations of the free Laplacian.

1 Introduction

We are interested in the analytic singularities of the distributions $u=u(t,x)$ that are solutions in $\mathbb {R}\times \mathbb {R}^n$ to the Schrödinger equation,

where $P=P(x,D_x)$ is a second-order symmetric differential operator on $\mathbb {R}^n$ with analytic coefficients (typically a perturbation of the Laplace operator $P_0:=-\frac{1}{2}\Delta$), and $u_0$ is in $L^2(\mathbb {R}^n)$ or, more generally, in some Sobolev space.

For such a problem, it is quite natural to wonder if the analyticity of $u_0$ implies that of u(t) at time $t\not = 0$. But actually this is not true, as it can be seen from the example where $P=P_0$ and $u_0= (-2i\pi )^{-\frac{n}{2}}e^{-i|x|^2/2}$. In this case, using that the distributional kernel of $e^{-itP_0}$ is $(2i\pi t)^{-\frac{n}{2}}e^{i|x-y|^2/2t}$, one can see that u(t) just coincides with $v(t-1)$, where v solves the same Schrödinger equation with initial date $v(0)=\delta$ (the Dirac measure at $x=0$). In particular, $u(1)=\delta$ is singular, while u(0) is analytic. Such a phenomenon is called “infinite propagation speed of singularities”, and a question one may ask is: Is there any way to read the singularities of u(t) easily on $u_0$?

As we shall see, the answer is essentially yes, in the sense that (under some non-trapping conditions) the analytic wave front set of $e^{itP_0}u(t)$ propagates in a very precise way (while that of u(t) does not at all!).

As an example, in the particular case $P=P_0 +V$ where $V=V(x)$ is an analytic function tending to 0 at infinity (and thus, in that case, $u(t)=e^{-itP}u_0$), we will prove that, for all $t\in \mathbb {R}$, one has,

$$ WF_a(e^{itP_0}u(t))=WF_a(u_0) $$

or, equivalently,

$$ WF_a(u(t)) = WF_a(e^{-itP_0}u_0). $$

Here, $WF_a$ stands for the analytic wave front set, and the details of the proofs of the results we present here can be found in [7, 8] (see also [6] for related results).

2 Assumptions and Results

Let

$$ P= \frac{1}{2} \sum _{j,k=1}^n D_j a_{j,k}(x)D_k + \frac{1}{2}\sum _{j=1}^n (a_j(x)D_j+D_ja_j(x)) +a_0(x) $$

on $\mathcal {H}=L^2(\mathbb {R}^n)$, where $D_j = -i\partial _{x_j}$, and assume that the coefficients $\{a_\alpha (x)\}$ satisfy to the following hypothesis. For $\nu >0$ we denote

$$ \Gamma _\nu =\bigl \{z\in \mathbb {C}^n\bigm ||{\mathrm{Im}\;}z|<\nu \langle {\mathrm{Re}\;}z \rangle \bigr \}. $$

Assumption A

For each $\alpha$, $a_\alpha (x)\in C^\infty (\mathbb {R}^n)$ is real-valued and can be extended to a holomorphic function on $\Gamma _\nu$ with some $\nu >0$. Moreover, for $x\in \mathbb {R}^n$, the matrix $(a_{j,k}(x))_{1\le j, k\le n}$ is symmetric and positive definite, and there exists $\sigma >0$ such that,

$$\begin{aligned}&\bigl | a_{j,k}(x)-\delta _{j,k} \bigr | \le C_0 \langle x \rangle ^{-\sigma }, \quad j,k=1,\dots ,n, \\&\bigl | a_{j}(x) \bigr | \le C_0\langle x \rangle ^{1-\sigma } , \quad \qquad j=1,\dots ,n,\\&\bigl | a_0(x) \bigr | \le C_0 \langle x \rangle ^{2-\sigma }, \end{aligned}$$

for $x\in \Gamma _\nu$ and with some constant $C_0>0$.

The case $\sigma >1$ will be referred to as the short range case, while the case $\sigma \in (0,1]$ as the long range case.

We denote by $p(x,\xi ):=\frac{1}{2} \sum _{j,k=1}^n a_{j,k}(x)\xi _j\xi _k$ the principal symbol of P, and by $P_0:= -\frac{1}{2}\Delta$ the free Laplace operator. For any $(x,\xi )\in \mathbb {R}^{2n}$, we also denote by $(y(t;x,\xi ), \eta (t;x,\xi ))=\mathrm{exp}tH_p(x,\xi )$ the solution to the Hamilton system,

$$\begin{aligned} \frac{dy}{dt} =\frac{\partial p}{\partial \xi }(y,\eta ),\, \frac{d\eta }{dt} =-\frac{\partial p}{\partial x}(y,\eta ), \end{aligned}$$

(2.1)

with initial condition $(y(0),\eta (0))=(x,\xi )$.

We say that a point $(x,\xi )\in T^*\mathbb {R}^n\backslash 0$ is forward non-trapping (respectively backward non-trapping) when $|y(t,x,\xi )|\rightarrow \infty$ as $t\rightarrow +\infty$ (resp. as $t\rightarrow -\infty$).

In that case, one can prove the existence of $\eta _+(x,\xi )\in \mathbb {R}^n$ (resp. $\eta _-(x,\xi )$) such that $\eta (t,x,\xi )\rightarrow \eta _+(x,\xi )$ as $t\rightarrow +\infty$ (resp. $\eta (t,x,\xi )\rightarrow \eta _-(x,\xi )$ as $t\rightarrow -\infty$).

If in addition $\sigma >1$ (short range case), then one can also prove the existence of $y_{\pm }(x,\xi )\in \mathbb {R}^n$ such that,

$$ |y_+(x,\xi )+t \eta _+(x,\xi ) -y(t,x,\xi ) | \rightarrow 0\,\, \mathrm{as}\,\, t\rightarrow +\infty , $$

(resp. $|y_-(x,\xi )+t \xi _-(x,\xi ) -y(t,x,\xi ) | \rightarrow 0\,\, \mathrm{as}\,\, t\rightarrow -\infty$).

A proof of these two facts can be found, e.g., in [1], Lemma 2.2 (indeed, though only the short range case is treated, the proof given for the existence of $\eta _\pm (x,\xi )$ still works in the long range case).

Denoting by $NT^+$ (resp. $NT^-$) the set of forward (resp. backward) non-trapping points, we define the applications,

$$ S_{\pm }\, :\, NT^{\pm } \rightarrow \mathbb {R}^{2n} $$

by

$$ S_\pm (x,\xi ):= (y_\pm (x,\xi ), \eta _\pm (x,\xi )). $$

They respectively correspond to the forward and backward classical wave maps. For any distribution $u\in {\mathcal D}'(\mathbb {R}^n)$, we denote by $WF_a(u)$ the analytic wave front set of u (see, e.g., [13]), that can be described by introducing the FBI transform T defined by,

$$ Tu(z,h)=\int e^{-(z-y)^2/2h}u(y)dy, $$

where $z\in \mathbb {C}^n$ and $h>0$ is a small extra-parameter. Then, Tv belongs to the Sjöstrand space $H_{\Phi _0}^{loc}$ with $\Phi _0(z):= |{\mathrm{Im}\;}z|^2/2$ (see [13]), and a point $(x,\xi )$ is not in $W\!F_a(u)$ if and only if there exists some $\delta >0$ such that $Tu =\mathcal{O}(e^{(\Phi _0(z)-\delta )/h})$ uniformly for z close enough to $x-i\xi$ and $h>0$ small enough (in this case, we also use the notation: $Tu\sim 0$ in $H_{\Phi _0,x-i\xi }$). By Cauchy-formula, this is also equivalent to the existence of some $\delta '>0$ such that $\Vert e^{-\Phi _0/h}Tu\Vert _{L^2(\Omega )} =\mathcal{O}(e^{-\delta ' /h})$ for some complex neighborhood $\Omega$ of $x-i\xi$.

In the short range case, our main result is,

Theorem 2.1

Suppose Assumption A with $\sigma >1$, and let $u_0\in L^2(\mathbb {R}^n)$. Then,

• (i) For any $t<0$, one has,

  $$\begin{aligned} WF_a(e^{-itP}u_0)\cap NT^+ =S_+^{-1} (WF_a (e^{-itP_0}u_0)); \end{aligned}$$

  (2.2)

• (ii) For any $t>0$, one has,

  $$\begin{aligned} WF_a(e^{-itP}u_0)\cap NT^- =S_-^{-1} (WF_a (e^{-itP_0}u_0)). \end{aligned}$$

  (2.3)

Remark 2.2

In the particular case where the metric is globally non-trapping, this result gives a complete characterization of the analytic wave front set of u(t) in terms of that of $e^{-itP_0}u_0$.

Remark 2.3

By substituting $e^{itP}u_0$ to $u_0$, and $-t$ to t, this result implies that one has,

$$ \begin{aligned}&\forall t>0,\quad WF_a(e^{itP_0}u(t))=S_+(WF_a(u_0)\cap NT^+);\\&\forall t<0,\quad WF_a(e^{itP_0}u(t))=S_-(WF_a(u_0)\cap NT^-). \end{aligned} $$

In particular, this set does not depend on $t>0$ (resp. $t<0$).

In the important case where $a_{j,k}=\delta _{j,k}$, then one has $NT^{\pm }=\mathbb {R}^{2n}\backslash 0$ and $S_{\pm }=Id$, and we obtain the following immediate corollary:

Corollary 2.4

Suppose Assumption A with $\sigma >1$ and $a_{j,k}=\delta _{j,k}$ for all pair (j, k). Then, for all $t\in \mathbb {R}$ and all $u_0\in L^2(\mathbb {R}^n)$, one has,

$$ WF_a(e^{-itP}u_0) =WF_a(e^{-itP_0}u_0). $$

Remark 2.5

In the $C^\infty$ setting, analogous results have been obtained Hassell and Wunsch in [2]. They involve a notion of “scattering wave front set” in a more general context of manifolds. In the case of $\mathbb {R}^n$, this notion mainly coincides with that of $WF(e^{itP_0}u)$ (see also [3, 4, 9,10,11,12, 14] for related questions).

Remark 2.6

Using the FBI transform (see, e.g., [5, 13]) and the expression of the distributional kernel of $e^{-itP_0}$, one can see that a point $(x_0,\xi _0)\in \mathbb {R}^{2n}\backslash 0$ is not in $WF_a(e^{-itP_0}u_0)$ if and only if there exists some $\delta >0$ such that the quantity,

$$ {\mathcal T}u_0(x,\xi :h):=\int e^{i(x-hy)\xi /h -(x-hy)^2/2h} e^{iy^2/2t}u_0(y)dy, $$

is ${\mathcal O}(e^{-\delta /h})$, uniformly for $h>0$ small enough and $(x,\xi )$ in a neighborhood of $(-\frac{1}{t}\xi _0, \frac{1}{t}x_0)$.

In the long range case ($0<\sigma \le 1$), the maps $S_{\pm }$ are not defined anymore, and one need to modify the free evolution near infinity in order to be able to define similar maps.

For $h>0$ sufficiently small and $(x,\xi )\in \mathbb {R}^{2n}$, we denote by $\widetilde{p}(x,\xi ;h)$ the quantity,

$$ \widetilde{p}(x,\xi ):= \frac{1}{2}\sum _{j,k}a_{j,k}(x)\xi _j\xi _k + h\sum _{j}a_j(x)\xi _j + h^2a_0(x), $$

and by $(\widetilde{y}(t,x,\xi ;h), \widetilde{\eta }(t,x,\xi ;h)):=\exp tH_{\widetilde{p}}(x,\xi )$ the corresponding Hamilton flow. Then, we have the preliminary result,

Lemma 2.7

For any $\delta _0>0$, there exist two h-dependent smooth functions,

$$ W_\pm \, : \, \mathbb {R}_\pm \times \{ \xi \in \mathbb {R}^n\, ;\, |\xi |>\delta _0\} \rightarrow \mathbb {R}, $$

that are solutions to,

$$\begin{aligned} \frac{\partial W_\pm }{\partial t}(t,\xi ) = \widetilde{p}(\nabla _\xi W_\pm (t,\xi ), \xi ;h), \end{aligned}$$

(2.4)

and such that, for any $\pm t>0$ and $(x,\xi )\in NT^\pm$, the quantity,

$$\begin{aligned} \widetilde{y}(t/h,x,\xi )-\nabla _\xi W_\pm (t/h,\widetilde{\eta }(t/h,x,\xi ))+\nabla _\xi W_\pm (0, \eta _\pm (x,\xi )) \end{aligned}$$

(2.5)

admits a limit $\widetilde{y}_\pm (x,\xi )\in \mathbb {R}^n$ independent of t as $h\rightarrow 0_+$.

Remark 2.8

Actually, Eq. (2.4) must be satisfied up to short range terms only, in order to have (2.5). For instance, in the previous short range case, one can take $W_\pm (t,\xi ) = t\xi ^2/2$, that gives $\widetilde{y}_\pm (x,\xi )= y_\pm (x,\xi )$.

Using the notations of the previous lemma, we set,

$$\begin{aligned} \begin{aligned} \widetilde{S}_{\pm }(x,\xi )&:= (\widetilde{y}_\pm (x,\xi ),\eta _\pm (x,\xi )),\quad ((x,\xi )\in NT^{\pm });\\ z_\pm (x,\xi )&:= \widetilde{y}_\pm (x,\xi )-i\eta _\pm (x,\xi );\\ \widetilde{W}_\pm (t,\xi )&:= W_\pm (t,\xi ) - W_\pm (0,\xi ). \end{aligned} \end{aligned}$$

(2.6)

Then, the result for the long range case is,

Theorem 2.9

Suppose Assumption A with $0<\sigma \le 1$, and let $u_0\in L^2(\mathbb {R}^n)$. Then, with the notations (2.6), one has,

• (i) For any $t<0$ and $(x,\xi )\in NT^+$, one has the equivalence,

  $$ (x,\xi )\notin WF_a(e^{-itP}u_0)\iff e^{i\widetilde{W}_+(-t/h,hD_z)/h}Tu_0\sim 0 \text{ in } H_{\Phi _0, z_+(x,\xi )}; $$

• (ii) For any $t>0$ and $(x,\xi )\in NT^-$, one has the equivalence,

  $$ (x,\xi )\notin WF_a(e^{-itP}u_0)\iff e^{i\widetilde{W}_-(-t/h,hD_z)/h}Tu_0\sim 0 \text{ in } H_{\Phi _0, z_-(x,\xi )}; $$

Remark 2.10

Here, the operator $e^{i \widetilde{W}_\pm (-t/h, hD_z)/h}$ appearing in the statement is not defined by the Spectral Theorem, but rather as a Fourier integral operator acting on Sjöstrand’s spaces (see [8]).

Remark 2.11

Actually, $W_\pm$ can be constructed in such a way that the quantity $W_1^\pm (t,\xi ):= \widetilde{W}_\pm (-t/h,hD_x)/h$ does not depend on h, and in principle, the fact that $e^{i\widetilde{W}_\pm (-t/h,hD_z)/h}Tu_0\sim 0$ in $H_{\Phi _0, z_\pm (x,\xi )}$ essentially means that $\widetilde{S}_\pm (x,\xi ) \notin WF_a(e^{iW_1^\pm (-t,D_x)}u_0)$ (and in this sense, the result is very similar to that of the $C^\infty$ setting appearing in [11]). However, in order to define $e^{iW_1^\pm (-t,D_x)}$ properly one needs to extend $\widetilde{W}_\pm$ to all values of $\xi \in \mathbb {R}^n$, and this requires the use of cut-off functions. In the analytic setting, this introduces technical difficulties that can probably be overcome by the use of analytic pseudodifferential operators on the real domain (see [13]).

3 Sketch of Proof

We explain the proof for the forward non-trapping case only (the backward non-trapping case being similar), and we start by considering the short range case with a flat metric (that is, $a_{j,k}=\delta _{j,k}$ for all j, k, and thus $S_\pm (x,\xi )=(x,\xi )$).

Replacing $u_0$ by $e^{itP}u_0$, and then changing t to $-t$, we see that we have to prove that for any $t>0$, one has

$$ WF_a(u_0)=WF_a(e^{itP_0}e^{-itP}u_0). $$

Following [10], we set $v(t):=e^{itP_0}e^{-itP}u_0$, that solves the system,

$$\begin{aligned} i\frac{\partial v}{\partial t} = L(t)v\quad ; \quad v(0)=u_0. \end{aligned}$$

(3.1)

Here,

$$\begin{aligned} L(t)= e^{itP_0}(P-P_0)e^{-itP_0}= L_2(t)+L_1(t)+L_0(t), \end{aligned}$$

(3.2)

with,

$$\begin{aligned}&L_2(t):= \frac{1}{2} \sum _{j,k=1}^n D_j (a_{j,k}^W(x+tD_x)-\delta _{j,k})D_k \\&L_1(t):= \frac{1}{2}\sum _{\ell =1}^n (a_\ell ^W(x+tD_x)D_\ell +D_\ell a_\ell ^W(x+tD_x)) \\&L_0(t):=a_0^W(x+tD_x), \end{aligned}$$

where we have denoted by $a^W(x,D_x)$ the usual Weyl-quantization of a symbol $a(x,\xi )$, defined by,

$$ a^W(x,D_x)u(x) =\frac{1}{(2\pi )^n}\int e^{i(x-y)\xi }a((x+y)/2,\xi )u(y)dyd\xi . $$

Observe that, in the flat case, one has $L_2(t)=0$. The expressions for $L_j(t)$, $0\le j\le 2$ can be proved directly (using the fact that $e^{\pm iP_0}$ is just the multiplication by $e^{\pm i\xi ^2/2}$ in the Fourier variables), but they also result from the standard Egorov theorem (that becomes exact in this case).

Since the FBI transform T is a convolution operator, we immediately observe that $TD_{x_j} = D_{z_j}T$. However, in order to study the action of L(t) after transformation by T, we need the following key-lemma that will allow us to enter the framework of Sjöstrand’s microlocal analytic theory. Mainly, this lemma tells us that, if f is holomorphic near $\Gamma _\nu$, then, the operator $\tilde{T}:= T\circ f^W (x+thD_x)$ is a FBI transform with the same phase as T, but with some symbol $\tilde{f} (t,z,x;h)$.

Lemma 3.1

([7], Lemma 3.1) Let f be a holomorphic function on $\Gamma _\nu$, verifying $f(x)=\mathcal{O}(\langle x \rangle ^\rho )$ for some $\rho \in \mathbb {R}$, uniformly on $\Gamma _\nu$. Let also $K_1$ and $K_2$ be two compact subsets of $\mathbb {R}^n$, with $0\notin K_2$. Then, there exists a function $\tilde{f}(t,z, x;h)$ of the form,

$$\begin{aligned} \tilde{f}(t,z, x;h) = \sum _{k=0}^{1/Ch}h^k f_k(t,z,x), \end{aligned}$$

(3.3)

where $f_k$ is defined, smooth with respect to t and holomorphic with respect to (z, x) near $\Sigma := \mathbb {R}_t \times \{ (z,x)\, ;\, {\mathrm{Re}\;}z\in K_1,\, |{\mathrm{Re}\;}(z -x)| + |{\mathrm{Im}\;}x| \le \delta _0,\, {\mathrm{Im}\;}z\in K_2\}$ with $\delta _0 >0$ small enough, and such that, for any $u\in L^2(\mathbb {R}^n)$, one has,

$$\begin{aligned} Tf^W(x+thD_x) u (z,h)= & {} \int _{|x-{\mathrm{Re}\;}z| < \delta _0} e^{-(z-x)^2/2h}\tilde{f}(t,z, x,h)u(x)dx \\&\qquad +\mathcal{O}(\langle t \rangle ^{\rho _+} e^{(\Phi _0(z)-\varepsilon )/h}), \end{aligned}$$

for some $\varepsilon =\varepsilon (u)>0$ and uniformly with respect to $h>0$ small enough, z in a small enough neighborhood of $K:= K_1+iK_2$, and $t\in \mathbb {R}$. (Here, we have set ${\rho _+}=\max (\rho , 0)$.)

Moreover, the $f_k's$ verify,

$$\begin{aligned}&f_0(t,z,x) = f(x+it(z-x)) \, ;\\&|\partial _{z,x}^\alpha f_k (t,z,x)| \le C^{k+|\alpha | +1}(k+|\alpha |)! \langle t \rangle ^\rho , \end{aligned}$$

for some constant $C>0$, and uniformly with respect to $k\in \mathbb {Z}_+$, $\alpha \in \mathbb {Z}_+^{2n}$, and $(t,z,x)\in \Sigma$.

Thanks to this lemma, and using again Sjöstrand’s theory of microlocal analytic singularities [13], we deduce the existence of an analytic second-order (that is, with a symbol ${\mathcal O}(h^{-2})$) pseudodifferential operator Q(t, h) on $H_{\Phi _0}^{loc}(\mathbb {C}^n\backslash \{ {\mathrm{Im}\;}z =0\})$, such that,

$$ TL(t) =Q(t,h)T. $$

Moreover, in the flat case, Q(t, h) becomes of the first order, and its symbol is mainly given by,

$$ q(t,h; z,\zeta ) \sim h^{-1}\sum _{\ell =1}^n a_\ell (z+i\zeta +th^{-1}\zeta )\zeta _\ell + a_0 (z+i\zeta +th^{-1}\zeta ). $$

Actually, using Lemma 3.1, an exact formula can be obtained for the symbol of Q(t, h), that coincides with the previous expression up to ${\mathcal O}(1)$-terms as $h\rightarrow 0_+$. We refer to [7], Sect. 4, for more details.

Then, applying T to (3.1), multiplying it by $h^2$, and changing the time-scale by setting $s:=t/h$, we obtain the new evolution equation,

$$\begin{aligned} ih\frac{\partial Tv}{\partial s} = B(s,h)Tv\quad ; \quad Tv(0)=Tu_0, \end{aligned}$$

(3.4)

where B(s, h) is an analytic pseudodifferential operator of order -1 (still in the sense of [13]), acting on $H_{\Phi _0}^{loc}(\mathbb {C}^n\backslash \{ {\mathrm{Im}\;}z =0\})$, with symbol b(s, h) verifying,

$$ b(s,h)\sim \sum _{k\ge 1} h^kb_k(s) $$

(in the sense of analytic symbols), with

$$\begin{aligned}&b_1(s; z,\zeta ) =\mathcal{O} (\langle s\rangle ^{1-\sigma });\nonumber \\&b_k(s; z,\zeta ) =\mathcal{O} (\langle s\rangle ^{2-\sigma }) \text{ for } k\ge 2, \end{aligned}$$

(3.5)

uniformly with respect to $s\,{>}\,0$, and locally uniformly with respect to $z\in \mathbb {C}^n\backslash \{ {\mathrm{Im}\;}z\,{=}\,0\}$ and $\zeta$ close enough to $-{\mathrm{Im}\;}z$ (note that, in particular, for $k\ge 2$ and $s=\mathcal{O}(h^{-1})$, one also has: $hb_k ={\mathcal O}(\langle s\rangle ^{1-\sigma })$.)

Let us recall from [13] that the quantization of such a symbol $b(s,h;z,\zeta )$ on $H_{\Phi _0}^{loc}$ is given by,

$$ B(s,h)w(z ;h)=\frac{1}{(2\pi h)^n}\int _{\gamma (z)}e^{i(z-y)\zeta /h}b(s,h;z,\zeta )w(y)dyd\zeta , $$

where $\gamma (z)$ is a complex contour of the form,

$$ \gamma (z)\, :\, \zeta =-{\mathrm{Im}\;}z +iR(\overline{z-y})\,\, ;\, \, |y-z| <r, $$

with $R>0$ is fixed large enough, and $r>0$ can be taken arbitrarily small. In particular, we deduce from (3.5) that B(s, h) can be written as,

$$ B(s,h) = hB_1(s,h), $$

where $B_1(s,h)$ admit a symbol uniformly $\mathcal{O} (\langle s\rangle ^{1-\sigma }+h\langle s\rangle ^{2-\sigma })$, for $s>0$, z in a compact subset of $\mathbb {C}^n\backslash \{ {\mathrm{Im}\;}z =0\}$, and $(y,\zeta )\in \gamma (z)$.

Then, for $z_0\in \mathbb {C}^n\backslash \{ {\mathrm{Im}\;}z =0\}$ and $\varepsilon _0 >0$, if we set,

$$ L^2_{ \Phi _0}(z_0,\varepsilon _0):= L^2(\{ |z-z_0|<\varepsilon _0\}; e^{-2 \Phi _0/h}d{\mathrm{Re}\;}z d {\mathrm{Im}\;}z)\cap H_{\Phi _0}(|z-z_0|<\varepsilon _0), $$

we see that $B_1(s,h)$ is a bounded operator from $L^2_{ \Phi _0}(z_0,\varepsilon _0)$ to $L^2_{\tilde{\Phi }_0}(z_0,\varepsilon _0/2)$, and its norm can be easily estimated in terms of the supremum of its symbol. Thus, here we obtain,

$$\begin{aligned} \Vert B_1(s)\Vert _{\mathcal{L}\left( L^2_{ \Phi _0}(z_0,\varepsilon _0); L^2_{ \Phi _0}(z_0,\varepsilon _0/2)\right) }=\mathcal{O}(\langle s \rangle ^{1-\sigma } + h\langle s \rangle ^{2-\sigma })=\mathcal{O}(\langle s \rangle ^{1-\sigma }), \end{aligned}$$

(3.6)

uniformly with respect to $h>0$ small enough and $|s|\le T_0/h$ ($T_0>0$ fixed arbitrarily).

Now, let us denote by $\tilde{\Phi }_0 =\tilde{\Phi }_0(z,\overline{z})$ a smooth real-valued function defined near $z=z_0$, such that $|\tilde{\Phi }_0 -\Phi _0|$ and $|\nabla _{(z,\overline{z})}(\tilde{\Phi }_0 -\Phi _0 )|$ are small enough, and verifying,

$$\begin{aligned}&\tilde{\Phi }_0\ge \Phi _0 \text{ in } \{ |z-z_0|\le \varepsilon _0\};\end{aligned}$$

(3.7)

$$\begin{aligned}&\tilde{\Phi }_0 = \Phi _0 \text{ in } \{ |z-z_0|\le \varepsilon _0/4\};\end{aligned}$$

(3.8)

$$\begin{aligned}&\tilde{\Phi }_0>\Phi _0 +\varepsilon _1 \text{ in } \{ |z-z_0|\ge \varepsilon _0/2\}, \end{aligned}$$

(3.9)

for some $\varepsilon _1>0$. By modifying the contour defining $B_1(s)$ (see [13], Remarque 4.4), we know that $B_1(s)$ is also bounded from $L^2_{\tilde{\Phi }_0}(z_0,\varepsilon _0)$ to $L^2_{\tilde{\Phi }_0}(z_0,\varepsilon _0/2)$, and its norm on these space verifies the same estimate (3.6) as on $L^2_{\Phi _0}$.

Setting $w=Tv$, Eq. (3.4) gives,

$$\begin{aligned} i\partial _s w(s) = B_1(s,h)w(s)\, \text{ in } H_{\Phi _0}(|z-z_0|<\varepsilon _0), \end{aligned}$$

(3.10)

with $\varepsilon _0 >0$ fixed small enough, and thus,

$$\begin{aligned} \partial _s\Vert w(s)\Vert _{L^2_{\tilde{\Phi }_0}(z_0,\varepsilon _0/2)}^2 =2{\mathrm{Im}\;}\langle B_1(s)w(s) ,w(s) \rangle _{L^2_{\tilde{\Phi }_0}(z_0,\varepsilon _0/2)}. \end{aligned}$$

Using Cauchy–Schwarz inequality and (3.6), we obtain,

$$\begin{aligned} \left| \partial _s\Vert w(s)\Vert _{L^2_{\tilde{\Phi }_0}(z_0,\varepsilon _0/2)}^2 \right| =\mathcal{O}(\langle s \rangle ^{1-\sigma })\Vert w(s)\Vert _{L^2_{\tilde{\Phi }_0}(z_0,\varepsilon _0)}^2. \end{aligned}$$

(3.11)

On the other hand, using (3.9) and the fact that $\Vert v(t)\Vert _{L^2} = \Vert u_0\Vert _{L^2}$ does not depend on t, we also have the estimate,

$$ \Vert w(s)\Vert _{L^2_{\tilde{\Phi }_0}(z_0,\varepsilon _0)}^2 = \Vert w(s)\Vert _{L^2_{\tilde{\Phi }_0}(z_0,\varepsilon _0/2)}^2 +\mathcal{O}(e^{-\varepsilon _1/h}), $$

that, inserted into (3.11), gives,

$$ \left| \partial _s\Vert w(s)\Vert _{L^2_{\tilde{\Phi }_0}(z_0,\varepsilon _0/2)}^2\right| \le C\langle s \rangle ^{1-\sigma }\Vert w(s)\Vert _{L^2_{\tilde{\Phi }_0}(z_0,\varepsilon _0/2)}^2 + Ce^{-\varepsilon _1/h}, $$

with some constant $C>0$. Setting $g(s):= C\int _0^s\langle s' \rangle ^{1-\sigma }ds'$, and using Gronwall’s lemma, we finally obtain,

$$\begin{aligned}&\Vert w(s)\Vert _{L^2_{\tilde{\Phi }_0}(z_0,\varepsilon _0/2)}^2 \le e^{g(s)}\Vert w(0)\Vert _{L^2_{\tilde{\Phi }_0}(z_0,\varepsilon _0/2)}^2 +C\int _0^se^{g(s)-g(s')-\varepsilon _1/h}ds';\\&\Vert w(0)\Vert _{L^2_{\tilde{\Phi }_0}(z_0,\varepsilon _0/2)}^2 \le e^{g(s)}\Vert w(s)\Vert _{L^2_{\tilde{\Phi }_0}(z_0,\varepsilon _0/2)}^2 +C\int _0^se^{g(s')-\varepsilon _1/h}ds'. \end{aligned}$$

Then, replacing s by t / h and observing that $g(s)=\mathcal{O}(\langle s \rangle ^{2-\sigma }) = \mathcal{O}(h^{\sigma -2})=o(h^{-1})$, the equivalence $(x_0,\xi _0)\notin WF_a(u_0)\Longleftrightarrow (x_0,\xi _0)\notin WF_a(u(t))$ follows immediately, and the result is proved in this case.

Now, let us still consider the case where the perturbation is short range, but the metric is not necessarily flat anymore. Then, the result we have to prove is the following: for any $t>0$ and $(x_0,\xi _0)\in NT^+$, one has the equivalence,

$$ (x_0,\xi _0)\in WF_a(u_0) \Longleftrightarrow S_+(x_0,\xi _0)\in WF_a(e^{itP_0}e^{-itP}u_0). $$

Proceeding as in the flat case, we arrive again at Eq. (3.4), but this time B(s, h) is of order 0, and can be written as,

$$ B(s,h) =B_0(s,h) + hB_1(s,h), $$

where $B_1$ is as before, and the symbol of $B_0$ is,

$$ b_0(s; z,\zeta ) =\frac{1}{2}\sum _{j,k=1}^n (a_{j,k}(z+i\zeta +s\zeta )-\delta _{j,k})\zeta _j\zeta _k. $$

Then, in order to get rid of $B_0(s)$, we construct a Fourier integral operator F(s, h) on $H_{\Phi _0, z_0}$, verifying,

$$ \left\{ \begin{array}{l} ih\partial _s F(s,h)-B_0(s,h)F(s,h) \sim \mathcal{O}(h);\\ F\left| _{s=0}\right. = I. \end{array}\right. $$

More precisely, we look for F(s, h) of the form,

$$\begin{aligned} F(s)v (z) =\frac{1}{(2\pi h)^n}\int _{\gamma _s (z)} e^{i(\psi (s,z,\eta ) -y\eta )/h}v(y)dyd\eta , \end{aligned}$$

(3.12)

where $\gamma _s (z)$ is a convenient contour and $\psi$ is a holomorphic function that must solve the system (eikonal equation),

$$\begin{aligned} \left\{ \begin{array}{l} \partial _s\psi + b_0(s,z,\nabla _z\psi ) =0;\\ \psi \left| _{s=0}\right. = z.\eta . \end{array}\right. \end{aligned}$$

(3.13)

The construction of $\psi (s)$ for small s just follows from standard Hamilton-Jacobi theory, and the extension to larger values of s can be made by using the classical flow $R_s$ of $b_0(s)$, that is related to the Hamilton flow of p through the formula,

$$\begin{aligned} R_{s}= \kappa \circ \exp (-sH_{p_0}) \circ \exp sH_p\circ \kappa ^{-1}, \end{aligned}$$

(3.14)

where $\kappa (x, \xi ) =(x-i\xi ,\xi )$ is the complex canonical transformation associated with T. We refer to [7], Sect. 6, for the detailed construction.

In that way, we find a solution $\psi (s,\zeta ,\eta )$ of (3.13), defined for $s\in \mathbb {R}$, z close to $z_0:=x_0-i\xi _0$ (where $(x_0,\xi _0)\in NT^+$ is fixed arbitrarily), and $\eta$ close to $\xi _0$. One also has the relation,

$$\begin{aligned} (z,\nabla _z\psi (s,z,\eta )) = R_s(\nabla _\eta \psi (s,z,\eta ), \eta ), \end{aligned}$$

(3.15)

which means that $\psi$ is a generating function of the complex canonical transformation $R_s$. In other words, the operator F(s, h) defined by (3.12) quantizes the canonical relation $R_s$, and, setting $z_s:=\pi _zR_s(z_0,\xi _0)$ (where $\pi _z : (z,\zeta ) \mapsto z$), one can show that for any $\varepsilon _0 >0$ small, F(s, h) acts as,

$$\begin{aligned} F(s)\, :\, H_{\Phi _0}(|z-z_0|<\varepsilon _0) \rightarrow H_{\Phi _0}(|z-z_s|<\varepsilon _1), \end{aligned}$$

(3.16)

for some $\varepsilon _1=\varepsilon _1(\varepsilon _0)>0$. A priori, $\varepsilon _1$ also depends on s, but as a matter of fact, since $R_s$ tends to $R_\infty := \kappa \circ S_+\circ \kappa ^{-1}$ on a neighborhood of $(z_0,\xi _0)$ as $s\rightarrow +\infty$, one can prove that F(s; h) admits a limit $F_\infty (h)$ that is a FIO quantizing $R_\infty$. Then, the action (3.16) remains valid for $0\le s\le +\infty$ (with $z_\infty :=\pi _zR_\infty (z_0,\xi _0)$), $\varepsilon _1$ can be taken independent of s, and the norm of F(s) is uniformly bounded both with respect to h and $s\ge 0$.

Now, by construction, for $s\in \mathbb {R}$, F(s) verifies,

$$ ih\partial _sF(s) -B_0(s)F(s) = hF_1(s), $$

where $F_1(s)\, :\, H_{\Phi _0}(|z-z_0|<\varepsilon _0) \rightarrow H_{\Phi _0}(|z-z_s|<\varepsilon _1)$ is of the form,

$$ F_1(s)v (z) =\frac{1}{(2\pi h)^n}\int _{\gamma _s (z)} e^{i(\psi (s,z,\eta ) -y\eta )/h}f_1(s,z,\eta ;h)v(y)dyd\eta , $$

with $f_1$ is an analytic symbol that is $\mathcal{O}(\langle s \rangle ^{-1-\sigma })$ as $s\rightarrow \infty$.

In the same way, for any y close enough to $z_0$, we can define a Fourier integral operator $\tilde{F}(s)$ of the form,

$$ \tilde{F}(s)v(y):= \frac{1}{(2\pi h)^n}\int _{\tilde{\gamma }_s (y)} e^{i(y\eta -\psi (s,z,\eta ) )/h}v(z)dzd\eta , $$

(where $\tilde{\gamma }_s (y)$ is again a convenient contour), such that $\tilde{F}(s)$ maps $H_{\Phi _0}(|z-z_s|<\varepsilon _0)$ into $H_{\Phi _0}(|z-z_0|<\varepsilon _1)$, and verifies,

$$\begin{aligned} ih\partial _s\tilde{F}(s) +\tilde{F}(s)B = h\tilde{F}_1(s), \end{aligned}$$

(3.17)

where $\tilde{F}_1(s)\, :\, H_{\Phi _0}(|z-z_s|<\varepsilon _0) \rightarrow H_{\Phi _0}(|z-z_0|<\varepsilon _1)$ is a FIO with same phase as $\tilde{F}(s)$ and symbol $\tilde{f}_1 =\mathcal{O}(\langle s \rangle ^{-1-\sigma })$.

Now, setting,

$$ \tilde{w}(s) = \tilde{F}(s)T u (hs) \in H_{\Phi _0}(|z-z_0|<\varepsilon _1), $$

by (3.4) and (3.17), we see that $\tilde{w}$ verifies,

$$ i\partial _s\tilde{w}(s) = \left[ \tilde{F}(s)B_1(s)+\tilde{F}_1(s)\right] Tu(hs). $$

Moreover, since $A(s):=F(s)\tilde{F}(s)$ is an elliptic pseudodifferential operator on $H_{\Phi _0,z_s}$, by taking a parametrix $\tilde{A}(s)$, we have,

$$\begin{aligned} T u (hs) = \tilde{A}(s)F(s)w(s)\, \text{ in } H_{\Phi _0}(|z-z_s|<\varepsilon ), \end{aligned}$$

(3.18)

(for some $\varepsilon >0$ independent of s), and thus, we obtain,

$$\begin{aligned} i\partial _s\tilde{w}(s) = \tilde{B}_1(s)\tilde{w}(s). \end{aligned}$$

(3.19)

in $H_{\Phi _0}(|z-z_0|<\varepsilon ')$, where $\tilde{B}_1(s):= \left[ \tilde{F}(s)B_1(s)+\tilde{F}_1(s)\right] \tilde{A}(s)F(s)$ is a pseudodifferential operator on $H_{\Phi _0}(|z-z_0|<\varepsilon ')$ with the same properties as $B_1(s)$ when $s\rightarrow +\infty$.

Thus, we are reduced to a situation completely similar to that of the flat case, and, if for instance $(x_0,\xi _0)\notin W\!F_a(u_0)$, the same arguments show that,

$$ \Vert w(s)\Vert _{L^2_{\Phi _0}(z_0,\delta )}\le Ce^{-\delta /h}, $$

for some positive constant $\delta$ independent of $h>0$ small enough and $s\in [0, T/h]$. As a consequence, using (3.18) and the fact that $\tilde{A}(s)F(s)$ is uniformly bounded from $L^2_{\Phi _0}(z_0,\delta )$ to $L^2_{\Phi _0}(z_s,\delta ')$ for some $\delta '>0$, we obtain (with some new constant $C>0$),

$$ \Vert Tu(hs)\Vert _{L^2_{\Phi _0}(z_s,\delta ')}\le Ce^{-\delta /h}. $$

Replacing s by t / h with $t>0$ fixed, and observing that $z_{t/h}$ tends to $\kappa \circ S_+(x_0,\xi _0)$ as $h\rightarrow 0_+$, we conclude that $S_+(x_0,\xi _0)\notin W\!F_a(u(t))$. The converse can be seen in the same way, and thus Theorem 2.1 is proved.

In the long range case, the construction of $W_\pm$ results from standard Hamilton-Jacobi theory, and the proof is very similar, except that we now have to handle expressions like

$$ e^{i\widetilde{W}_\pm (s,hD_z)/h}v(z;h):= \int _{\gamma (s,z)}e^{i(z-y)\zeta /h +i\widetilde{W}_\pm (s,\zeta )/h}v(y)dydz, $$

where $\gamma (s,z)$ is a good contour in the sense of [13], with some uniformity as $s\rightarrow \infty$. Then, one can show that $e^{i\widetilde{W}_\pm (s,hD_z)/h}$ is a Fourier integral operator acting of Sjöstrand’s spaces $H_{\Phi _0}$, in the sense that one has,

$$ e^{i\widetilde{W}_\pm (s,hD_z)/h}\, :\, H_{\Phi _0}(\Omega _s(z_0,\varepsilon _1))\rightarrow H_{\Phi _0}(\Omega _s(Z_s(z_0),\varepsilon _2)), $$

with $\varepsilon _1, \varepsilon _2 >0$ small enough, and where we have set,

$$ \Omega _s(Z, \varepsilon ):=\{ z\in \mathbb {C}^n\,;\, \langle s\rangle ^{-1}|{\mathrm{Re}\;}(z-Z)| + |{\mathrm{Im}\;}(z-Z)| <\varepsilon \}. $$

We refer to [8] for more details.