Decay of correlations for normally hyperbolic trapping

Abstract

We prove that for evolution problems with normally hyperbolic trapping in phase space, correlations decay exponentially in time. Normally hyperbolic trapping means that the trapped set is smooth and symplectic and that the flow is hyperbolic in directions transversal to it. Flows with this structure include contact Anosov flows, classical flows in molecular dynamics, and null geodesic flows for black holes metrics. The decay of correlations is a consequence of the existence of resonance free strips for Green’s functions (cut-off resolvents) and polynomial bounds on the growth of those functions in the semiclassical parameter.

Appendix: Evolution for the CAP-modified Hamiltonian

In the appendix we show some properties of the CAP-modified Hamiltonian, that is the Hamiltonian modified by adding a complex absorbing potential. At first we work under the general assumptions (1.9).

The semigroup $\exp ( - i t ( P - i W )/h ) : L^2 ( X ) \rightarrow L^2 ( X )$ is defined using the Hille-Yosida theorem: for $h$ small $P - i W - i$ is invertible as its symbol is elliptic in the semiclassical sense (see (1.11) and [55, Theorem 4.29]). Ellipticity assumption for large values of $\xi$ also shows that $P - i W$ is a Fredholm operator, and the comment about invertibility shows that it has index $0$. The estimate

$$\begin{aligned}&\Vert ( P - i W - z ) u \Vert \Vert u \Vert \ge - \hbox { Im } \langle ( P - i W - z ) u,u \rangle \\&\quad \ge \hbox { Im } z \Vert u \Vert ^2 , \ \ \ u \in H^m_h ( X ) , \end{aligned}$$

then shows invertibility for $\hbox { Im } z > 0$, with the bound

$$\begin{aligned} \Vert ( P - i W - z )^{-1} \Vert _{L^2 \rightarrow L^2 } \le \frac{1}{ \hbox { Im } z } , \ \ \hbox { Im } z > 0. \end{aligned}$$

Since the domain of $P - i W$ is given by $H^m ( X )$ which is dense in $L^2$, the hypotheses of the Hille-Yosida theorem are satisfied, and

$$\begin{aligned} \begin{aligned} \Vert e^{ - i t ( P - i W ) / h }\Vert _{ L^2 \rightarrow L^2 } \le 1 , \ \ \ t \ge 0 , \\ e^{ - i t ( P - i W ) / h } e^{ - i s ( P - i W ) / h } = e^{ - i (t + s ) ( P - i W ) / h }, \ \ t, s \ge 0. \end{aligned} \end{aligned}$$

(10.1)

Alternatively we can show the existence of the semigroup $\exp ( - it ( P \!-\! i W ) /h )$ using energy estimates, just as is done in the proof of [55, Theorem 10.3]. We get that for any $T>0$,

$$\begin{aligned} e^{- i t ( P - i W ) / h } \!\in \! C\big ( [ 0 , T ] ; {\mathcal L} ( H_h^s ( X ) , H_h^s ( X ) ) \big ) \cap C^1 \big ( [ 0 , T ]; { \mathcal L} ( H_h^{s} , H_h ^{s-m} ) \big ).\nonumber \\ \end{aligned}$$

(10.2)

Our final estimates will all be given for $L^2$ only and that is sufficient for our purposes.

The first result we state concerns propagation of semiclassical wave front sets. We recall the notation $\varphi _t=\exp (tH_p)$ for the Hamiltonian flow generated by $p(x,\xi )$.

Lemma 10.1

Suppose that $A \in \Psi ^\mathrm{comp} ( X )$. Then for any $T$ independent of $h$ there exists a smooth family of operators

$$\begin{aligned}{}[ 0, T ] \ni t \longmapsto Q ( t ) \in \Psi ^\mathrm{comp} ( X ) , \ \ \hbox {WF}_h( I - Q ( t ) ) \cap \varphi _t ( \mathrm{WF }_h ( A ) ) = \emptyset ,\nonumber \\ \end{aligned}$$

(10.3)

such that

$$\begin{aligned} ( I - Q ( t ) )\, e^{ - i t ( P - i W ) / h } A = {\mathcal O}( h^\infty ) _{L^2 \rightarrow L^2 }. \end{aligned}$$

(10.4)

In addition if $\hbox {WF}_h( A ) \subset w^{ -1} ( [ \epsilon _1, \infty ) )$, $\epsilon _1 > 0$, then for any fixed $t>0$,

$$\begin{aligned} e^{ - it ( P - i W )/h } A = {\mathcal O}( h^\infty )_{ L^2\rightarrow L^2} , \ \ \ \ A \,e^{ - it ( P - i W )/h } = {\mathcal O}( h^\infty )_{ L^2\rightarrow L^2}.\nonumber \\ \end{aligned}$$

(10.5)

Proof

We first construct $Q ( t)$ using a semiclassical adaptation of a standard microlocal procedure—see [30, §23.1]. For that, let $Q ( 0 ) \in \Psi ^{\mathrm{comp}} ( X)$ be an operator satisfying $\hbox {WF}_h( I - Q ( 0 ) ) \cap \hbox {WF}_h( A ) = \emptyset$, and with the principal symbol, $q_0 ( 0 )$, independent of $h$. Using the fact that the flow $\varphi _t$ is defined for all $t$ we put $q_0 ( t ) \mathop {=}\limits ^\mathrm{{def}}\varphi _{-t}^* q_0 (0)$. In terms of the Poisson bracket on the extended phase space $T^*( \mathbb {R}_t \times X )\ni ( t, x, \tau , \xi )$, this means that the function $q_0(t)$ satisfies the identity $\{ \tau + p , q_0 ( t ) \} = 0$. Consequently, at the quantum level we have

$$\begin{aligned}&\displaystyle [ h D_t + P , {\mathrm{Op }^{{w}}_h}( q_0 ( t ) ] = h R_1 ( t ) , \ \ R_1 ( t ) \in \Psi ^\mathrm{comp} ( X) , \\&\displaystyle {\mathrm{Op }^{{w}}_h}( q_0 ( 0 ) ) - Q ( 0 ) = h E_1 , \ \ E_1 \in \Psi ^\mathrm{comp} ( X ) , \end{aligned}$$

and the principal symbols of $R_1$, $E_1$, $r_1 , e_1 \in {\mathcal C}^\infty _\mathrm{{c}}( T^*X )$, are independent of $h$. If $p_1 = \sigma ( ( P - {\mathrm{Op }^{{w}}_h}( p ) )/ h$, we then solve (in the unknown $q_1(t)$) the equation

$$\begin{aligned} \{ \tau + p , q_1 ( t ) \} = r_1 - \{ p_1 , q_0 ( t ) \} , \ \ q_1 ( 0 ) = e_1. \end{aligned}$$

By iteration of this procedure we obtain $q_\ell \in {\mathcal C}^\infty ( T^*X )$ such that

$$\begin{aligned}{}[ h D_t + P , \sum _{ \ell =0}^{N-1} h^j {\mathrm{Op }^{{w}}_h}( q_\ell ( t )) ] = h^N R_N ( t ) , \ \ R_N ( t ) \in \Psi ^\mathrm{comp} ( X) , \\ \sum _{ \ell =1}^{N-1} h^\ell {\mathrm{Op }^{{w}}_h}( q_{\ell } ( 0 ) ) - Q ( 0 ) = h^N E_N , \ \ E_N \in \Psi ^\mathrm{comp} ( X ). \end{aligned}$$

By a standard Borel resummation we may construct $Q(t)\in \Psi ^{\mathrm{comp}}(X)$ such that $Q ( t ) \sim \sum _{\ell \ge 0}h^j {\mathrm{Op }^{{w}}_h}(q_\ell ( t ) )$.

For any $N>0$ we can iteratively construct a sequence of auxiliary operators $Q_j ( t )= Q_j ( t )^* \in \Psi ^\mathrm{comp} ( X )$, $0\le j\le N$, satisfying

$$\begin{aligned}&\hbox {WF}_h( I - Q_{ j+1} ( t ) ) \cap \hbox {WF}_h( Q_{j} ( t ) ) = \mathrm{WF }_h ( I - Q_{j} ( t ) ) \cap \varphi _t ( \mathrm{WF }_h ( A ) ) \nonumber \\&= \hbox {WF}_h( I - Q ( t ) ) \cap \hbox {WF}_h( Q_j ( t ) ) = \emptyset , \nonumber \\& [ Q_j ( t ) , h D_t + P ] \in {\mathcal C}^\infty \big ( [ 0 , T ]; h^\infty \Psi ^{\mathrm{comp} } ( X ) \big ). \end{aligned}$$

(10.6)

(These assumptions imply that $\varphi _t( \hbox {WF}_h( A ) ) \subset \mathrm{WF }( Q_j ( t ) ) \subset \mathrm{WF }( Q_{j+1} ( t ) ) \subset \mathrm{WF }( Q ( t ) )$.)

Let $v ( t ) \mathop {=}\limits ^\mathrm{{def}}e^{ - i t ( P - i W ) /h } A u$, $\Vert u \Vert _{ L^2 } = 1$. Our aim is to prove the following property:

$$\begin{aligned} w_j ( t) \mathop {=}\limits ^\mathrm{{def}}( I - Q_j ( t ) ) v ( t ) = {\mathcal O} ( h^{ j/ 2} ) _{ L^2 } , \ \ \text { for } j=0,\ldots ,N,\ \ 0 \le t \le T.\nonumber \\ \end{aligned}$$

(10.7)

Since $A \in \Psi ^\mathrm{comp}$, (10.2) shows that this property holds for $j = 0$. Let us now prove that, if true at the level $j$, it then holds at the level $j+1$.

Noting that

$$\begin{aligned} w_{ j+1} = ( I - Q_{j+1} ) w_j + {\mathcal O} ( h^\infty ) _{{\mathcal C}^\infty }, \end{aligned}$$

(10.8)

we have

$$\begin{aligned} ( h D_t + P - i W ) w_{j+1}&= ( I - Q_{j+1} ( t ) ) ( h D_t + P - iW ) w_{j} \\&- i [ W ,Q_{j+1} ] w_j + {\mathcal O} ( h^\infty ) _{L^2} \end{aligned}$$

Dividing by $h/i$, taking the inner product with $w_{j+1}$, taking real parts and integrating gives

$$\begin{aligned}&\displaystyle \Vert w_{ j+1} ( t ) \Vert _{L^2}^2 - \Vert w_{ j+1 } ( 0 ) \Vert ^2_{L^2} + 2 \int _0^t \langle W w_{j+1}( s) , w_{ j+1} (s) \rangle ds \nonumber \\&\displaystyle =\ \frac{2}{h} \int _0^t \mathrm {Re} \langle [ W, Q_{j+1} (s) ] w_j(s) , w_{j+1}( s) \rangle ds + {\mathcal O} ( h^\infty ) , \end{aligned}$$

(10.9)

Now,

$$\begin{aligned} ( I - Q_{ j+1}( s ) ) [ W, ( I - Q_{j+1} (s)) ] = i h B_{j+1} ( s) +h^2 C_{ j+1} ( s) , \\ B_{j+1} ( s ) , C_{j+1}( s ) \in \Psi ^\mathrm{comp} ( X ) , \ \ B_{ j+1}( s ) = B_{ j+1} ( s ) ^*. \end{aligned}$$

Hence, using (10.8) and the induction hypothesis (10.7), the right hand side of (10.9) becomes

$$\begin{aligned} 2 h \int _0^t \mathrm {Re} \langle C_{j+1} ( s) w_j ( s) , w_j ( s ) \rangle ds + {\mathcal O} ( h^\infty ) = {\mathcal O} (h^{j+1} ). \end{aligned}$$

Returning to (10.9) and using the non-negativity of $W$, we see that

$$\begin{aligned} \Vert w_{ j+1 } ( t ) \Vert _{L^2}^2 \le \Vert w_{j+1} ( 0 ) \Vert _{L^2 }^2 + C h^{ j + 1}. \end{aligned}$$

Since

$$\begin{aligned} w_{j+1} ( 0 ) = ( I - Q_{ j+1} ) A u = {\mathcal O} ( h^\infty ) _{ L^2}, \end{aligned}$$

we have established (10.7) with $j$ replaced by $j+1$.

The estimate (10.4) then follows from

$$\begin{aligned} ( I - Q ( t ) ) v ( t ) = ( I - Q ( t ) ) w_j ( t ) + {\mathcal O}_{L^2} ( h^\infty ), \end{aligned}$$

the estimate (10.7) at the level $j=N$, and the fact that $N$ could be taken arbitrary large.

To see (10.5) we note that if $A \in \Psi ^{\mathrm{comp}} ( X )$ then

$$\begin{aligned} \hbox {WF}_h( A ) \subset w^{-1} ( [ \epsilon _1 , \infty ) \ \!\Longrightarrow \! \ \varphi _t( \hbox {WF}_h( A )) \subset w^{-1} ( [ \epsilon _1 /2 , \infty ) \ \text { for } 0 \!\le \! t \!\le \! \delta . \end{aligned}$$

Hence, by (10.4),

$$\begin{aligned}&\mathrm{WF }_h ( v ( t ) ) \subset w^{-1} ( [ \epsilon _1 /2 , \infty ) , \ \ v ( t) \mathop {=}\limits ^\mathrm{{def}}e^{ - i t ( P - i W ) /h } A u ,\\&\quad \Vert u \Vert _{L^2} = 1, \ \ 0 \le t \le \delta . \end{aligned}$$

This means that we can modify $W$ into $W_1$, so that

$$\begin{aligned} \sigma ( W_1 ) ( x, \xi ) \ge \langle \xi \rangle ^k/C , \ \ \ W_1 \ge c_0 , \text { for } 0 < h < h_0 , \end{aligned}$$

while we have

$$\begin{aligned}&0 = ( h D_t + P - i W ) v ( t ) = ( h D_t + P - i W_1 ) v ( t) + {\mathcal O} ( h^\infty ) _{ {\mathcal C}^\infty }\\&\quad \quad \text {uniformly for }0 \le t \le \delta . \end{aligned}$$

Taking the imaginary part of the inner product of the above expression with $v ( t )$ gives

$$\begin{aligned} \frac{h}{2} \partial _t \Vert v ( t ) \Vert ^2_{L^2} = - \langle W_1 v ( t ) , v ( t ) \rangle + {\mathcal O} ( h^\infty ) \le -c_0 \Vert v ( t) \Vert ^2 + {\mathcal O} ( h^\infty ) , \end{aligned}$$

and hence

$$\begin{aligned} \Vert v ( t) \Vert _{L^2}^2 = {\mathcal O} ( h^\infty ) \quad \text {uniformly for } \delta /2\le t \le \delta . \end{aligned}$$

This proves the first part of (10.5). The second part follows by taking a conjugate: $A \,e^{ - i t ( P - i W ) /h } = \left( e^{ - it ( - P - i W ) / h } A^* \right) ^*$, and all the arguments remain valid for $P$ replaced by $- P$. $\square$

The next lemma is needed in Sect. 7 and follows immediately from Lemma 10.1:

Proposition 10.2

Suppose that $A \in \Psi ^\mathrm{comp} ( X )$ satisfies

$$\begin{aligned} \hbox {WF}_h( A ) \subset p^{ -1} ( ( - \delta , \delta )) \cap w^{-1} ( [ 0 , \epsilon _1 ) ), \end{aligned}$$

(10.10)

for some $\epsilon _1 > 0$ and that $T$ is independent of $h$.

Then there exists $B \in \Psi ^\mathrm{comp} ( X )$ for which (10.10) holds with $B$ in place of $A$, and

$$\begin{aligned} e^{ -i t ( P - i W ) /h } A \!=\! B e^{ - i t ( P - i W ) /h } A + {\mathcal O} ( h^\infty ) _{L^2 \rightarrow L^2} , \ \ 0 \le t \le T.\qquad \qquad \end{aligned}$$

(10.11)

Proof

Using again the operator $Q ( t )$ constructed in the proof of Lemma 10.1, we take a compact set $L$ containing $\hbox {WF}_h( Q ( t ) )$ for all $0 \le t \le T$. By taking $\hbox {WF}_h( Q ( 0 ) ) \subset p^{-1} ( ( - \delta , \delta ) )$ (which is possible due the assumptions on $A$) we see that we can assume $L \subset p^{-1} ( ( - \delta , \delta ) )$. We can now choose $B \in \Psi ^\mathrm{comp} ( X )$ such that

$$\begin{aligned}&\hbox {WF}_h( I - B ) \cap L \cap w^{-1} ( [0, \epsilon _1/3] ) = \emptyset ,\\&\quad \hbox {WF}_h( B ) \subset p^{-1} ( (- \delta , \delta ) ) \cap w^{-1} ( [ 0, \epsilon _1/2 ). \end{aligned}$$

This implies that $( I - B ) Q ( t ) = C ( t )$, where $\hbox {WF}_h( C( t )) \subset w^{-1} ( [ \epsilon _1/3, \infty ) )$, and hence, by (10.4) and (10.5),

$$\begin{aligned}&( I - B ) e^{ - i t ( P - i W ) / h} A=\\&\quad \big ( C ( t ) + ( I - B ) ( I - Q ( t) \big ) e^{ - i t ( P - i W ) / h} A= {\mathcal O} ( h^\infty ) _{L^2 \rightarrow L^2}, \end{aligned}$$

proving (10.11). $\square$

Finally we present a modification of [38, Lemma A.1]. The modification lies in slightly different assumptions on $P$ and $W$, and the proof also corrects a mistake in the proof given in [38]. From now on we work under the extra assumption (1.10) on the CAP. We remark that in [38] we only needed Lemma 10.1 and hence the assumption (1.10) was not required.

Proposition 10.3

Suppose that $X$ is a compact manifold, $P$ is a self-adjoint operator, $P \in \Psi ^m ( X)$, $W \in \Psi ^k ( X )$, $W \ge 0$, and that (1.9) and (1.10) hold. Then for any $t$ independent of $h$, for $A \in \Psi ^\mathrm{comp} ( X )$ satisfying (10.10), we may write

$$\begin{aligned} e^{ i t P/h } e^{ - i t ( P - i W ) / h } A = V_A ( t ) + {\mathcal O} ( h^\infty ) _{ L^2 \rightarrow L^2} , \end{aligned}$$

where

$$\begin{aligned} V_A (t)&\in \Psi _{\gamma }^{\mathrm{comp} } ( X ), \ \ \hbox {WF}_h( V_A ( t ) ) \subset \bigcap _{ 0 \le s \le t } ( \varphi _{-s} (w^{-1} (0))) \cap \hbox {WF}_h( A ), \nonumber \\ \sigma ( V_A ( t ) )&= \exp \left( - \frac{1}{h} \int _0^t \varphi _s^* W ds \right) \sigma (A). \end{aligned}$$

(10.12)

The class of operators $\Psi _{\gamma }^{\mathrm{comp}}$ was introduced in Sect. 3.2.

The proof is based on the following lemma inspired by the pseudodifferential approach to constructing parametrices for parabolic equations presented in [33].

Lemma 10.4

Suppose that $t \mapsto p( t, z , h)$, $p ( t, \bullet , h ) \in {\mathcal C}^\infty _\mathrm{{c}}( \mathbb {R}^{2n}; \mathbb {R})$, is a family of functions satisfying

$$\begin{aligned} \partial _t^k \partial _z^\alpha p ( t, z , h )&= \mathcal O _{ k, \alpha } ( 1 ), \ \ \ p \ge -C h , \ \ \ 0 < h < h_0 , \nonumber \\ | \partial _z^\alpha p ( t, z, h ) |&= {\mathcal O}_\alpha ( p^{ 1 - \delta } ), \ \ 0 < \delta < {\frac{1}{2}}. \end{aligned}$$

(10.13)

Then, for $0 \le s \le t$ there exists $E ( t, s ) \in \Psi _{\delta } ( \mathbb {R}^n)$ such that

$$\begin{aligned} ( h \partial _t + p^w ( t , x , h D_x , h ) ) E ( t , s ) = 0, \ \ t \ge s \ge 0 , \ \ E ( s, s ) = { I}.\end{aligned}$$

Moreover, $E ( t, s ) = e^w ( t , s, x, h D_x , h )$ where $e(t,s) \in S_{\delta } ( \mathbb {R}^{2n} )$ has an explicit expansion given in (10.27) below.

Proof

Replacing $p$ by $p + (C+1)h$, gives $p \ge h$ and $p(t, \bullet ,h ) \in (C+1)h + {\mathcal C}^\infty _\mathrm{{c}}(\mathbb {R}^{2n}_z )$. The multiplicative factor $e^{ ( C + 1 ) (t-s) }$ in the evolution equation is irrelevant to our estimates.

For any $N \ge 0$ we try to approximate the symbol $e(t,s,x,\xi ,h)$ by an expansion of the form

$$\begin{aligned} f_N ( t , s, z , h ) \mathop {=}\limits ^\mathrm{{def}}\sum _{ j=0}^N h^j e_j ( t, s , z , h ) . \end{aligned}$$

(10.14)

The symbol of the operator $h\partial _t f_N^w + p^w f_N^w$ can be expanded using the standard notation $a^w \circ b^w = ( a \# b )^w$ and the product formula (see for instance [55, Theorem 4.12]):

$$\begin{aligned}&h \partial _t f_N ( t, s ) + \left[ p ( t ) \# f_N ( t, s) \right] \nonumber \\&\quad \!=\!\sum _{ j=0}^N h^j \left( h \partial _t e_j (t,s)\right. \nonumber \\&\quad \quad \left. \!+\! \sum _{ k=0}^{N-j -1 } \frac{1}{ k!} \left( {\frac{1}{2}} i h \omega ( D_z , D_w ) \right) ^k p (t, z) e_j (t,s, w ) |_{ z = w }+ h^{N-j} r_{N,j} \right) \nonumber \\&\quad =\sum _{ j=0}^N h^j \left( ( h \partial _t + p ( t ))e_j ( t, s)\right. \nonumber \\&\qquad \left. + \sum _{ \ell =0}^{j -1 } \frac{1}{ (j-\ell )!} \left( {\frac{1}{2}} i \omega ( D_z , D_w ) \right) ^{j-\ell } \,p (t, z) e_\ell (t,s, w ) |_{ z = w } \right) + h^{N} r_N ( t, s , z ) , \nonumber \\&\qquad \qquad \ \ \ \ \ \ \ \ \ r_N ( t, s, z ) \mathop {=}\limits ^\mathrm{{def}}\sum _{ j=0}^{N-1} r_{N,j} ( t, s, z ). \end{aligned}$$

(10.15)

The remainders satisfy the following bounds (see for instance [43, (3.12)]):

$$\begin{aligned}&\sup _{z} | \partial _z^\alpha r_{N,j} ( t, s, z ) | \nonumber \\&\quad \le C_{\alpha , N , j} \sum _{ \alpha _1 + \alpha _2 = \alpha } \sup _{ z, w } \sup _{ | \beta | \le M , \beta \in \mathbb {N}^{4n} }\left| ( h^{\frac{1}{2}} \partial _{z,w})^\beta ( \sigma ( D_z, D_w) )^{N-j} \partial _z^{\alpha _1} p ( z ) \partial _w^{\alpha _2 } e_j ( w ) \right| .\nonumber \\ \end{aligned}$$

(10.16)

The standard strategy is now to iteratively construct the symbols $e_j$ so that each term in the above expansion vanishes. The term $j=0$ simply reads $(h\partial _t + p)e_0=0$. From the initial condition $e_0(s,s)\equiv 1$, it is solved by

$$\begin{aligned} e_0 ( t , s , z , h ) = \exp \left( - {\frac{1}{h}} \int _s^t p ( s' , z , h ) ds' \right) . \end{aligned}$$

(10.17)

For $j \ge 1$, the symbol $e_j$ is obtained iteratively by solving

$$\begin{aligned}&e_j ( t, s , z ) \mathop {=}\limits ^\mathrm{{def}}\frac{1}{h} \int _s^t e_0 ( t, s' , z ) q_j ( s', s , z ) ds',\nonumber \\&\quad e_j ( t, s , \bullet ) \in {\mathcal C}^\infty _\mathrm{{c}}( \mathbb {R}^{2n} ) ,\nonumber \\&q_j ( t, s , z ) \mathop {=}\limits ^\mathrm{{def}}\nonumber \\&\quad - \sum _{ \ell =0}^{j -1 } \frac{1}{ (j-\ell )!} \left( {\frac{1}{2}} i \omega ( D_z , D_w ) \right) ^{j-\ell } p (t, z) e_\ell (t,s, w ) |_{ z = w } \in {\mathcal C}^\infty _\mathrm{{c}}( \mathbb {R}^{2n}_z).\nonumber \\ \end{aligned}$$

(10.18)

This construction formally leads to an approximate solution:

$$\begin{aligned} h \partial _t f_N ( t, s , z ) + \left[ p ( t, \bullet ) \# f_N ( t, s, \bullet ) \right] ( z ) = h^N r_N ( t,s, z ). \end{aligned}$$

(10.19)

To make the approximation effective, we now need to check that the sum (10.14) is indeed an expansion in power of $h$. We thus need to estimate the $e_j$’s and thereby the remainders $r_{N,j}$’s.

We will prove the following estimate by induction:

$$\begin{aligned} | \partial _z^\alpha e_j ( t, s , z ) | \!\le \! C_{\alpha , j } h^{- 2 \delta j \!-\! \delta |\alpha | } \left( 1 \!+\! \left( {\frac{1}{h}} \int _s^t p ( s', z ) ds' \right) ^{ 2 j + | \alpha | } \right) e_0 ( t, s , z).\nonumber \\ \end{aligned}$$

(10.20)

For that we first note that, as $p \ge h$, and $| \partial ^\alpha p | \le C_\alpha p^{1-\delta }$, we have

$$\begin{aligned} | \partial ^\alpha p | \le C_\alpha h^{ - \delta } p. \end{aligned}$$

(10.21)

Consequently, for $j = 0$ we have

$$\begin{aligned} |\partial _z^\alpha e_0 ( t, s , z ) |&\le \sum _{ \sum _{\ell =1}^k \alpha _\ell = \alpha } \prod _{\ell =1}^k \left( {\frac{1}{h}} \int _s^t |\partial ^{\alpha _\ell } p ( s' , z )| ds' \right) e_0 ( t, s, z )\nonumber \\&\le C_\alpha \sum _{ \sum _{\ell =1}^k \alpha _\ell = \alpha } \prod _{\ell =1}^k \left( h^{ -\delta } {\frac{1}{h}} \int _s^t p ( s' , z ) ds' \right) e_0 ( t, s, z ) \nonumber \\&\le C_\alpha ' h^{ - \delta | \alpha | } \left( 1 + \left( {\frac{1}{h}} \int _s^t p ( s' , z ) ds' \right) ^{|\alpha | }\right) e_0 ( t, s , z ) ,\qquad \qquad \end{aligned}$$

(10.22)

Here we used the fact that $k \le | \alpha |$ and that

$$\begin{aligned} A^{k} \le c_\alpha ( 1 + A^{|\alpha | } ) , \ \ \ A = {\frac{1}{h}} \int _s^t p ( s' , z ) ds' \ge 0. \end{aligned}$$

This gives (10.20) for $j = 0$.

To proceed with the induction we put

$$\begin{aligned} a_{j , \alpha } ( t, s , z )&\mathop {=}\limits ^\mathrm{{def}}&{ \partial ^\alpha _z e_j ( t, s , z ) } / { e_0 ( t, s, z )} ,\nonumber \\ \quad b_{ j , \alpha } ( t,s , z )&\mathop {=}\limits ^\mathrm{{def}}&{ \partial ^\alpha _z q_j ( t, s , z ) } / { e_0 ( t, s, z )} , \end{aligned}$$

noting that, for some coefficients, $c_\bullet$,

$$\begin{aligned} b_{j, \alpha }( t, s, z )&= \sum _{ \ell =0}^{j-1} \sum _{ \beta _1 + \beta _2 = \alpha } c_{ \beta _1, \beta _2, \ell , j} \omega ( D_z, D_w ) ^{ j - \ell } \partial _z^{\beta _1 } p ( t, z ) a_{ \ell , \beta _2 } ( t ,s,w ) |_{z = w } , \nonumber \\ a_{ j, \alpha } (t ,s, z)&= \frac{1}{h} \sum _{ \beta _1 + \beta _2 = \alpha } c_{ \beta _1, \beta _2, j} \int _s^t a_{ 0 , \beta _1 } ( t, s' ,z ) b_{ j, \beta _2 } ( s', s) d s', \end{aligned}$$

(10.23)

where the last equality follows from $e_0 ( t, s' ,z ) e_0 ( s', s, z ) = e_0( t, s, z )$, $s \le s' \le t$.

Our aim is to show

$$\begin{aligned} | b_{j, \alpha } ( t, s , z) | \!\le \! C_{\alpha , j } h^{- 2 \delta j -\delta |\alpha | } p ( t , z ) \left( 1 \!+\! \left( {\frac{1}{h}} \int _s^t p ( s', z ) ds' \right) ^{2 j + | \alpha | -1 } \right) ,\nonumber \\ \end{aligned}$$

(10.24)

and

$$\begin{aligned} | a_{j, \alpha } ( t, s , z) | \le C'_{\alpha , j } h^{-2 \delta j - \delta |\alpha | } \left( 1 + \left( {\frac{1}{h}} \int _s^t p ( s', z ) ds' \right) ^{ 2j + | \alpha | } \right) ,\nonumber \\ \end{aligned}$$

(10.25)

assuming the statements are true for $j$ replaced by smaller values.

We note that the case of $j =0$ has been shown in (10.22), and since $b_{0, \alpha } \equiv 0$.

The first estimate (10.24) follows immediately from the inductive hypothesis on $a_{\ell , \alpha }$, $0 \le \ell \le j-1$ and the estimates on $p$ in (10.21). The second estimate (10.25) follows from (10.22), (10.24) and the obvious fact that $\int _{s_1}^{s_2} p (s') ds' \le \int _s^t p (s') ds'$, $s\le s_1 \le s_2 \le t$.

We note that (10.20) and the definition of $e_0$ given in (10.17) imply that

$$\begin{aligned} \partial ^\alpha _z e_j (t,s, z ) = {\mathcal O} ( h^{ - \delta |\alpha | - 2 \delta j }),\quad j\ge 0. \end{aligned}$$

so from (10.14) we see that the symbol $f_N(t,s)\in S_{\delta }(\mathbb {R}^{2n})$.

The bounds (10.16) then show that the remainders satisfy

$$\begin{aligned} | \partial ^\alpha r_N (t, s , z ) | \le C_{N, \alpha } h^{ - 2 \delta N - \delta | \alpha | }. \end{aligned}$$

Going back to (10.19) we get the expression

$$\begin{aligned} E( t, s ) = f_N^w ( t,s , x , h D_x ) + h^{N-1} \int _s^t E ( t, s' ) r_N^w ( s', s, x , h D_x ).\quad \quad \end{aligned}$$

(10.26)

(We note that, since $p^w ( t, x,h D_x ) \ge - Ch$ by the sharp Gårding inequality [55, Theorem 4.32], and since $p^w$ is bounded on $L^2$, the operator $E ( t, s)$ exists and is bounded on $L^2$, uniformly in $h$.) Since operators in $\Psi _{\delta }$ are uniformly bounded on $L^2$ [55, Theorem 4.23], it follows that

$$\begin{aligned} E ( t , s ) = f^w_N ( t, s ,x , h D_x ) + {\mathcal O}( h^{( 1 - 2 \delta ) N } ) _{L^2 \rightarrow L^2}. \end{aligned}$$

To show that $E ( t , s ) - e^w_0 ( s, t, x , h D_x) \in \Psi ^\mathrm{comp}_{\delta } ( \mathbb {R}^n )$, we use (10.26) and Beals’s lemma in the form given in [43, Lemma 3.5, $\tilde{h} =1$]: $\ell _j$ are linear functions on $\mathbb {R}^{2n}$, $\ell _j^w = \ell _j^w ( x, h D)$, then

$$\begin{aligned}&\mathrm{ad }_{\ell _1^w } \cdots \mathrm{ad }_{ \ell _J^w } E ( s, t ) \\&=\mathrm{ad }_{\ell _1^w } \cdots \mathrm{ad }_{ \ell _J^w } f_N^w ( s, t, x , h D_x) + h^{N-1} \int _s^t \mathrm{ad }_{\ell _1^w } \cdots \mathrm{ad }_{ \ell _J^w }\\&\left( E ( s, s' ) r^w_N ( s', s, x, h D_x ) \right) ds' \\&= {\mathcal O} ( h^{ ( 1 - 2 \delta ) J} ) _{ L^2 \rightarrow L^2 } + {\mathcal O} ( h^{ ( 1 - 2 \delta ) N } ) _{ L^2 \rightarrow L^2 } = \mathcal O ( h^{( 1 - 2 \delta ) J } ) _{ L^2 \rightarrow L^2 }, \end{aligned}$$

if $N$ is large enough. Here we used the fact that $f_N, r_N \in S_{\delta }$ and that $\mathrm{ad }_{\ell _1^w } \cdots \mathrm{ad }_{ \ell _J^w } E ( s, t )= {\mathcal O} (1 ) _{L^2 \rightarrow L^2 }$, which follows from considering the evolutions equation for the operator on the left hand side.

In conclusion we have shown that $E ( t, s ) = e^w ( t, s , x , h D_x)$, where $e \in S_{\delta } ( \mathbb {R}^n )$ admits the expansion

$$\begin{aligned} e( t, s , z , h ) \sim \sum _{ j \ge 0 } h^j e_j ( t , s, z, h ), \ \ \quad e_j(t,s) \in h^{- 2 \delta j} S^\mathrm{comp}_{\delta } (\mathbb {R}^{2n}), \ \ j \ge 1,\nonumber \\ \end{aligned}$$

(10.27)

with $e_0$ given by (10.17). $\square$

Proof of Proposition 10.3 We first observe that Lemma 10.1 (applied both to propagators for $P - i W$ and for $P$) shows that for $B \in \Psi ^\mathrm{comp} ( X )$ satisfying $\hbox {WF}_h( I - B ) \cap \hbox {WF}_h( A ) = \emptyset$,

$$\begin{aligned} e^{ i t P/h } e^{ - i t ( P - i W ) / h } A = B e^{ i t P/h } e^{ - i t ( P - i W ) / h } A + {\mathcal O} (h^\infty ) _{L^2 \rightarrow L^2}. \end{aligned}$$

We can choose $B = B^*$. Since

$$\begin{aligned} h \partial _t \left( B e^{ i t P/h } e^{ - i t ( P - i W ) / h } A \right)&= - B e^{it P/h } W e^{ -i t P/h } e^{ i t P/h } e^{ - i t ( P - i W ) / h } A \\&= - \left( B e^{it P/h } W e^{ -i t P/h } B \right) \\&\quad \times \left( B e^{ i t P/h } e^{ - i t ( P - i W ) / h } A\right) + {\mathcal O} (h^\infty ) _{L^2 \rightarrow L^2}, \end{aligned}$$

it follows that

$$\begin{aligned} B\,e^{ i t P/h } e^{ - i t ( P - i W ) / h } A = V^{B} ( t ) + {\mathcal O} (h^\infty ) _{L^2 \rightarrow L^2}, \end{aligned}$$

(10.28)

where

$$\begin{aligned} h \partial _t V^B ( t) = - W_B ( t) V^B ( t ) , \quad W_B ( t ) \mathop {=}\limits ^\mathrm{{def}}B e^{i t P/h } W e^{- it P/h} B.\quad \quad \end{aligned}$$

(10.29)

We note that $W_B ( t ) \in \Psi ^\mathrm{comp} ( X)$, $\hbox {WF}_h( W_B ( t ) ) \subset \hbox {WF}_h( B )$, and that $W_B(t)\ge 0$. Hence $V^B ( t ) = {\mathcal O}( 1 ) _{L^2 \rightarrow L^2}$ and (10.28) follows from Duhamel’s formula.

By decomposing $A$ as a sum of operators, we can assume that $\hbox {WF}_h( A )$ is supported in a neighbourhood of a fiber of a point in $X$. Hence, by choosing $B$ with a sufficiently small wave front set, we only need to prove that $V^B ( t ) \in \Psi _{\delta }$ for $X = \mathbb {R}^n$; that follows from Lemma 10.4, since the symbol of $W_B(t)$ satisfies the assumptions (10.13). The second and third properties in (10.12) follows from (10.5) and (10.27). $\square$