text:
\chapter{Introduction}
\label{sect-1}

\paragraph{Asymptotics away from the boundary.}
\label{sect-1.1}
Let us start from the case of asymptotics when points $x$ and r $y$ are disjoint from the boundary. While sharp classical asymptotics of $e (x,x,\tau)$ for elliptic operators are known from
L.~H\"ormander \cite{hoermander:spectral}, and could be traced to B.~M.~Levitan \cite{levitan:asymp, levitan:spect} and V.~G.~Avakumovi{\v{c}} \cite{avak:eigen}, I could not find asymptotics of $e(x,y,\tau)$, despite it could be easily derived by the method of Fourier integral operators combined with Tauberian theorem.

\begin{theorem}
\label{thm-1.1}
Let $A=A^\mathsf{w} (x,hD,h)$ be a self-adjoint scalar pseudo-differential operator with $\mathscr{C}^K$-symbol ($K=K(d)$) which is $\xi$-microhyperbolic in $B(0,1)\subset X$  on the energy  level $\tau$\,\footnote{\label{foot-1} Which means that \begin{gather}
|a(x,\xi)-\tau|+ |\nabla_\xi a(x,\xi) |\ge \epsilon_0
\label{eqn-1.1}
\end{gather}
where $a(x,\xi)\coloneqq A(x,\xi,0)$.}.
Further, let  $\{\xi\colon a(x,\xi)=\tau\}$ be strongly convex. Then
\begin{gather}
e_h(x,y,\tau)=e_h^\mathsf{W} (x,y,\tau) +O(h^{1-d}) \qquad  \forall x,y\in B(0,\epsilon )
\label{eqn-1.2}\\
\shortintertext{where}
e_h^\mathsf{W} (x,y;\tau)=(2\pi h)^{-d}  \int _{\{a(\frac{1}{2}(x+y), \xi)<\tau\}} e^{ih^{-1}\langle x-y,\xi\rangle}\,d\xi
\label{eqn-1.3}
\end{gather}
is corresponding Weyl expression.
\end{theorem}

We will prove this theorem and discuss its generalizations and also more precise results in Section~\ref{sect-2}.

\paragraph{Asymptotics near the boundary.}
\label{sect-1.2}
Classical asymptotics of $e (x,x,\tau)$ for second order elliptic operators are known from  R.~Seeley \cite{seeley:sharp, seeley:boundary}, and sharp ones from V. Ivrii. The semiclassical version would carry remainder estimate $O(h^{1-d}{\nu}(x)^{-1/2})$ and $O(h^{1-d})$ respectively while the trivial remainder estimate would be $O(h^{1-d}{\nu}(x)^{-1})$ where ${\nu}(x)=\operatorname{dist}(x,\partial X)$.

Let
\begin{align*}
\ell^0(x,y)&\coloneqq |x-y|,\\
\ell\ (x,y) &\coloneqq |x-y|+{\nu}(x)+{\nu}(y) .
\end{align*}

\begin{theorem}\label{thm-1.2}
Let
\begin{gather}
A=\sum_{j,k} \bigl(hD_j-V_j(x)\bigr) g^{jk}(x)\bigl(hD_k-V_k(x)\bigr)+V(x),\qquad
g^{jk}=g^{kj},
\label{eqn-1.4}
\end{gather}
be a self-adjoint scalar operator  which is elliptic second order differential operator with $\mathscr{C}^K$-coefficients ($K=K(d,\delta )$), and it  is $\xi$-microhyperbolic in $B(0,1)$  on the energy  level $\tau$, with either Dirichlet or Neumann boundary condition  on $\partial X \cap B(0,1)$, and $\partial X\in \mathscr{C}^K$\,\footnote{\label{foot-2} More general boundary conditions could be also considered.}. Then
\begin{enumerate}[wide,label=(\roman*),  labelindent=0pt]
\item
For $d\ge 3$ asymptotics \textup{(\ref{eqn-1.2})} holds where now for Dirichlet or Neumann boundary condition
\begin{gather}
e_h^\mathsf{W} (x,y;\tau)= e_h^{0,\mathsf{W}} (x,y;\tau)\ [\mp] \ e_h^{0,\mathsf{W}}(x,\tilde{y};\tau)
\label{eqn-1.5}
\end{gather}
respectively, $e_h^{0,\mathsf{W}}(x,y,\tau)$ defined by \textup{(\ref{eqn-1.3})} and $\tilde{y}$ is a reflected point.
\item
For $d=2$ under one of the following assumptions
\begin{enumerate}[label=(\alph*), itemindent=10pt]
\item
$\ell (x,y) \le h^{\frac{1}{3}+\delta}$,
\item
$\ell (x,y) \ge h^{\frac{1}{3}-\delta}$,
\item
${\nu}(x) +{\nu}(y)\ge C_0(\ell^2 + h^{\frac{2}{3}-2\delta})$
\end{enumerate}
where here and below  $\delta>0$ is an arbitrarily small exponent, asymptotics
\begin{gather}
e_h(x,y,\tau)=e_h^\mathsf{W} (x,y,\tau) +e_{h,\mathsf{corr}} (x,y,\tau)+ O(h^{1-d}) \quad  \forall x,y\in B(0,\epsilon )
\label{eqn-1.6}
\end{gather}
holds with  correction term $e_{h,\mathsf{corr}} (x,y,\tau)$ will be defined by \textup{(\ref{eqn-5.53})} later.
\item
Finally, for $d=2$ if neither of condition (a)--(c) is fulfilled
\begin{gather}
e_h(x,y,\tau)= O(h^{-1-\delta}).
\label{eqn-1.7}
\end{gather}
\end{enumerate}
\end{theorem}

\begin{remark}\label{remark-1.3}
\begin{enumerate}[wide,label=(\roman*),  labelindent=0pt]
\item
According to  \cite{monsterbook}, Theorem~\ref{monsterbook-thm-8-1-6}  asymptotics (\ref{eqn-1.2}) holds for  $x=y$ in the coordinate system such that
$x_1$ is the distance from $x$ to $\partial X$ in the metrics $g^{jk}(\tau - V)^{-1}$. Such precision is needed for $d=2$ only.

\item
The leading term $e_h^\mathsf{W}(x,y,\tau)$ is $\asymp h^{-\frac{d-1}{2}}\ell ^{-\frac{d+1}{2}}$. In particular it is $O(h^{1-d})$ as
$\ell \gtrsim h^{\frac{d-1}{d+1}}$.

\item
The correction term (as $d=2$) is $O(h^{-\frac{3}{2}}\ell^{\frac{1}{2}})$ if $\ell \le h^{\frac{1}{2}}$ and $O(h^{-\frac{1}{2}}\ell^{-\frac{3}{2}})$ if
$h^{\frac{1}{2}}\le \ell \le \epsilon$.
\item
The trivial estimate (in much more general settings) is
\begin{gather}
|e_h(x,y;\tau)-e^\mathsf{W}_h( x,y,\tau)|\le Ch^{1-d} (1+\ell^{-1}(x,y)).
\label{eqn-1.8}
\end{gather}
\end{enumerate}
\end{remark}

\paragraph{Ideas of proofs. I.}
\label{sect-1.3}
Consider $x,y\in B(0,\frac{1}{2})$.  We already know that  under pretty general assumptions (which will be discussed later), in particular, in the frameworks of Theorems~\ref{thm-1.1} and~\ref{thm-1.2},   the following estimate holds:
\begin{gather}
e(x,x, \tau'+h)-e(x,x, \tau')\le Ch^{1-d} \qquad \text{for\ \ } |\tau'-\tau|\le \epsilon_0.
\label{eqn-1.9}\\
\shortintertext{Then}
|e(x,y;\tau)-e(x,y; \tau')|\le Ch^{-d}|\tau -\tau'|+ h^{1-d}
\label{eqn-1.10}\\
\intertext{and therefore due to Tauberian methods}
|e_h(x,y;\tau)-e_{T,h}^\mathsf{T} (x,y;\tau) |\le CT^{-1}h^{1-d}
\label{eqn-1.11}\\
\shortintertext{where}
e_{T,h}^\mathsf{T}(x,y;\tau)= h^{-1}\int_{-\infty}^\tau F_{t\to h^{-1}\tau'} \bigl(\bar{\chi}_T(t)u_h(x,y,t)\bigr) \,d\tau'
\label{eqn-1.12}
\end{gather}
is the \emph{Tauberian expression}, $u_h(x,y,t)$ is the Schwartz kernel of the propagator $e^{ih^{-1}At}$, ${h\le T\le \epsilon _1}$, $\epsilon_1$
is a small constant and $\bar{\chi}_T(t)= \bar{\chi}(t/T)$ and $\bar{\chi}\in \mathscr{C}_0^\infty ([-1,1])$, $\bar{\chi}(t)=1$ on $(-\frac{1}{2},\frac{1}{2})$.

Assume first that $B(0,1)\subset X$. Then for a small constant $T$ we can construct $u_h(x,y,t)$ as an oscillatory integral and we can construct $e_h^\mathsf{T}(x,y,\tau)$ even as $d=2$ or without strong convexity assumption. However the result could be simplified to $e^\mathsf{W}_h(x,y,\tau)$ in the framework of Theorem~\ref{thm-1.1}  (for scalar operators under strong convexity assumption).  This is done in Section~\ref{sect-2}.

\paragraph{Ideas of proofs. II.}
\label{sect-1.4}
Construction of the propagator is much more complicated close to the boundary, because there could be many generalized billiard rays with at least one reflection from $\partial X$,   from $x$ to $y$ on the energy level $\tau$. If
\begin{gather}
\ell(x,y) \coloneqq |x-y|+{\nu}(x)+{\nu}(y) \le \epsilon_1
\label{eqn-1.13}\\
\intertext{the length of all such rays is $\asymp \ell(x,y)$. Further, if}
{\nu}(x)+{\nu}(y)\ge \sigma \ell (x,y)
\label{eqn-1.14}
\end{gather}
with $\sigma \ge C_0\ell(x,y)$ there is exactly  one such ray and it has exactly one reflection, and the  reflection  angle $\ge \epsilon_1 ({\nu}(x)+{\nu}(y))/\ell (x,y)$. It allows us in the framework of Theorem~\ref{thm-1.2} under assumption (\ref{eqn-1.14}) with $\sigma = h^{\frac{1}{2}-\delta}\ell(x,y)^{-\frac{1}{2}} + C_0\ell (x,y)$
  to construct a reflected wave as an oscillatory integral and simplify it. It will be done in Section~\ref{sect-5}.

So, we should consider a case when (\ref{eqn-1.14}) is violated and we will do it in Section~\ref{sect-4}. As $|x-y|\le Ch^{\frac{1}{2}-\delta}$ the propagator could be constructed by the standard method of successive approximations with unperturbed constant coefficients operator.

Furthermore, in the framework of Theorem~\ref{thm-1.2} under certain assumptions, which are complementary to (\ref{eqn-1.14}) as $\ell\le h^{\frac{1}{3}+\delta}$, the reflected wave could be constructed also by  the method of the successive approximations, but with unperturbed the operator
\begin{gather}
\bar{A}(z,hD_z)= h^2D_1^2 + A (0,w',hD_z'),
\label{eqn-1.15}
\end{gather}
where $x=(x_1; x')= (x_1;x_2,\ldots, x_d)$ and  $X\cap B(0,1)= \{x\colon x_1>0\}\cap B(0,1)$ and we make a change of variables $w'=\frac{1}{2}(x'+y')$ and $z'=\frac{1}{2}(x'-y')$, leaving $x_1$ and $y_1$ unchanged.

\paragraph{Ideas of proofs. III.}
\label{sect-1.5}
Finally, to estimate the Tauberian expression rather than to find its asymptotics we apply in Section~\ref{sect-3} only methods of propagations. Let us illustrate this away from the boundary. Let $Q_{1,2}$ be  pseudodifferential operators with the symbols equal $1$  in $\rho$-vicinity of $\{\xi\in \Sigma (w,\tau)\colon \nabla _\xi a(w,\xi)\parallel \bar{x}-\bar{y}\}$; this set consists of just two points. Then
\begin{gather}
e_h^\mathsf{T} (\bar{x},\bar{y},\tau) = Q_{2x} e_h ^\mathsf{T}(x,y,\tau) \,^t\!Q_{1y} |_{x=\bar{x},y=\bar{y}}+ O(h^s)
\label{eqn-1.16}
\end{gather}
provided microlocal uncertainty principle $\rho\times \rho \ell \ge h^{1-\delta}$ is fulfilled. It allows us to upgrade  $|e^\mathsf{T}_h(x,y,\tau)|\le Ch^{1-d}\ell^{-1}(x,y)$ to
\begin{gather}
|e_h^\mathsf{T} (x,y,\tau)|\le C \rho^{d-1} h^{1-d}\ell^{-1}(x,y).
\label{eqn-1.17}
\end{gather}
 We need also $\rho \ge C_0\ell$.

\chapter{Asymptotics inside domain}
\label{sect-2}

\section{Tauberian asymptotics}
\label{sect-2.1}

As we already mentioned, asymptotics
\begin{gather}
e_h(x,x,\tau)=\kappa_0(x,\tau)h^{-d}+O(h^{1-d})
\label{eqn-2.1}
\end{gather}
is known under much more general assumptions. It holds for matrix operators under $\xi$-microhyperbolicity condition at point $x$ on the energy level $\tau$
and for Schr\"odinger operator without any condition in dimension $d\ge 3$ (see \cite{monsterbook}, Sections~\ref{monsterbook-sect-5-4}  and~\ref{monsterbook-sect-5-3} correspondingly).

Asymptotics (\ref{eqn-2.1}) implies (\ref{eqn-1.9}) and then (\ref{eqn-1.10}). Then the standard Tauberian arguments (see f.e. \cite{monsterbook}, Section~\ref{monsterbook-sect-5-2})  imply (\ref{eqn-1.11})--(\ref{eqn-1.12}) with  small constant $T$.

\section{Weyl asymptotics}
\label{sect-2.2}

While asymptotics (\ref{eqn-2.1}) holds in much more general assumptions, to construct $u_h(x,y,t)$ we need to impose some restrictions. Let $A$ be a scalar operator\footnote{\label{foot-3} Construction also works for matrix operators with characteristic roots of constant multiplicity.}. While construction of propagator as an oscillatory integral is possible even if $\xi$-microhyperbolicity condition is violated, we assume that it is fulfilled from the beginning, as we need it anyway.

Without any loss of the generality one can assume that
\begin{gather}
a(y,0)< 0.
\label{eqn-2.2}\\
\intertext{Indeed, otherwise we can achieve it by a corresponding gauge transformation. Then the strong convexity of $\Sigma (y,\tau)=\{\theta\colon a(y,\theta)=\tau\}$ implies that}
\langle \nabla _\theta a(y,\theta),\theta\rangle \ge \epsilon_0\qquad \forall \theta\in \Sigma (y,\tau).
\label{eqn-2.3}
\end{gather}

\begin{proposition}\label{prop-2.1}
Let  $A$ be a scalar operator, $\xi$-microhyperbolic in  $B(\bar{x}, 2\epsilon )\subset X$ on energy level $\tau$ and let $x,y\in B(\bar{x}, \epsilon)$. Let
\textup{(\ref{eqn-2.3})} be fulfilled. Then
\begin{gather}
u_h(x,y,t)\equiv  \int e^{ih^{-1}\Phi  (x,y,t,\theta)} B (x,y,t,\theta,h)\,d\theta
\label{eqn-2.4}\\
\intertext{modulo functions such that}
F_{t\to h^{-1}\tau'} \bar{\chi}_T(t)v_h(x,y,t) = O(h^s)\qquad \forall x, y\in B(0,\epsilon)
\label{eqn-2.5}\\
\intertext{as $T\le \epsilon_1$, $|\tau'-\tau|\le \epsilon_1$ where}
\Phi (x,y,t,\theta)  =\varphi  (x,y,\theta) +t  a(y,\theta),
\label{eqn-2.6}\\
\intertext{$\varphi  (x,y,\theta)$ satisfies stationary eikonal equation}
a(x, \nabla _x \varphi)= a(y,\theta)
\label{eqn-2.7}\\
\shortintertext{and}
\varphi  (x,y,\theta)|_{\langle x-y,\theta\rangle=0}=0,
\label{eqn-2.8}\\[2pt]
\nabla_x \varphi  (x,y,\theta)|_{x=y}= \theta
\label{eqn-2.9}\\
\shortintertext{and}
B(x,y,t,\theta,h)\sim \sum_{n\ge 0} B_{n} (x,y,t,\theta)h^n,
\label{eqn-2.10}
\end{gather}
and amplitudes $B_{jn}$ are defined from the Cauchy problems for transport equations; all functions are uniformly smooth; here and below $s$ is an arbitrarily large exponent.
\end{proposition}

\begin{proof}
The proof  is original L.~H\"ormander's construction (see M.~Shubin~\cite{shubin:spectral}, Theorem~20.1).
\end{proof}

While in \cite{hoermander:spectral,shubin:spectral} this decomposition was used to calculate $e^\mathsf{T}_h (x,y,\tau)$ as $x=y$ (actually, it's classical variant), we will use it
without this restriction. Due to $\xi$-microhyperbolicity
\begin{multline}
F_{t\to h^{-1}\tau'} \bigl(\bar{\chi}_T(t) u_h(x,y,t)\bigr)\\
\sim (2\pi h)^{-d} h \int_{\Sigma (y,\tau')}  e^{i h^{-1} \varphi (x,y,\theta) } B'(x,y,   \theta)\, d\theta:da(y,\theta)
\label{eqn-2.11}
\end{multline}
with $\Sigma (y,\tau)= \{\theta\colon a(y,\theta)=0\}$, $d\theta:da(y,\theta)$ a natural density on $\Sigma (y,\tau')$ and $B'(x,y,\theta)$ also allowing (\ref{eqn-2.10}) decomposition with uniformly smooth $B'_n$.

\begin{proposition}\label{prop-2.2}
Let in $B(\bar{x},2\epsilon_1)$ both $\xi$-microhyperbolicity and strong convexity conditions are fulfilled on energy level $\tau$. Then for  $|\tau'-\tau|\le \epsilon_1$ and $x\ne y$
$\varphi (x,y,\theta)$ has exactly two stationary points $\theta^*_{\pm}(x,y,\tau')$  on $\Sigma (y,\tau')$,  defined from
\begin{align}
\nabla_\theta \varphi (x,y,\theta)=- t_\pm \nabla_\theta a (y,\theta), && a(y,\theta)=\tau', && \pm t_\pm >0.
\label{eqn-2.12}
\end{align}
These points are non-degenerate and  $t_\pm\asymp |x-y|$ in \textup{(\ref{eqn-2.12})}.

Furthermore,
\begin{gather}
\theta^*_{\pm}(x,y,\tau')=\bar{\theta}_{\pm}(x,y,\tau')+O(|x-y|),
\label{eqn-2.13}\\
\intertext{where $\bar{\theta}_{\pm}(x,y,\tau')$ are defined from}
x-y=- t'_\pm \nabla_\theta a (y,\theta), \qquad a(y,\theta)=\tau', \quad \pm t'_\pm >0
\label{eqn-2.14}
\end{gather}
and also $t'_\pm =t_\pm + O(|x-y|^2)$.
\end{proposition}

\begin{proof}
Proof follows trivially from
\begin{gather}
\varphi (x,y,\theta)= \langle x-y, \theta\rangle + O(|x-y|^2),
\label{eqn-2.15}
\end{gather}
which follows from (\ref{eqn-2.8}), (\ref{eqn-2.9}).
\end{proof}

\begin{proposition}\label{prop-2.3}
In the framework of Proposition~\ref{prop-2.2} the following asymptotics hold
\begin{gather}
F_{t\to h^{-1}\tau} \bigl(\bar{\chi}_T(t) u_h(x,y,t)\bigr)\sim (2\pi )^{-d} h^{\frac{1-d}{2}}
\sum_{\varsigma = \pm}  e^{ih^{-1}S_\varsigma (x,y,\tau)} B'_\varsigma (x,y,\tau)
\label{eqn-2.16}\\
\intertext{and with $c\ell\le T\le \epsilon$ }
e^\mathsf{T}_{T,h}(x,y,\tau)\sim (2\pi )^{-d} h^{\frac{1-d}{2}}
\sum_{\varsigma = \pm}  e^{ih^{-1}S_\varsigma (x,y,\tau)} B''_\varsigma (x,y,\tau)
\label{eqn-2.17}\\
\shortintertext{with}
S_\varsigma (x,y,\tau)= \varphi (x,y,\theta ^*_\varsigma(x,y,\tau))
\label{eqn-2.18}
\end{gather}
and $B'_\varsigma$, $B''_\varsigma$   also decomposed into asymptotic series albeit with
$B'_{\varsigma n} \ell ^{n+\frac{d-1}{2}}$ and $B''_{\varsigma n} \ell ^{n+\frac{d+1}{2}}$uniformly smooth.
\end{proposition}

\begin{proof}
Asymptotics (\ref{eqn-2.17})  follows from Proposition~\ref{prop-2.2} and stationary phase principle with effective semiclassical parameter $\hbar =h\ell^{-1}$, $\ell =|x-y|+h$.

To prove (\ref{eqn-2.18}) we first rewrite (\ref{eqn-1.12}) as  the sum of two integrals with cut-off functions $\phi _1(\tau')$ and $\phi_2)(\tau')$;
$\phi_j\in \mathscr{C}_0^\infty$, $\operatorname{supp} (\phi_1 )\subset (\tau-\epsilon, \tau+\epsilon)$, $\operatorname{supp}(\phi_2)\subset (-\infty, \tau-\epsilon/2)$. In the integral with $\phi_1$ we plug (\ref{eqn-2.4}), observe that there are no stationary points in $\{\theta\colon \tau-\epsilon<a(y,\theta)<\tau\}$ and use Proposition~\ref{prop-2.2} and stationary phase principle with effective semiclassical parameter $\hbar =h\ell^{-1}$. In the integral with $\phi_2$ we simply observe that it has a complete asymptotics because it contains a mollification by $\tau'$ (see, f.e. Proposition~\ref{prop-3.5} below).
\end{proof}

\begin{proposition}\label{prop-2.4}
In the framework of Proposition~\ref{prop-2.2}
\begin{enumerate}[wide,label=(\roman*),  labelindent=0pt]
\item
The following estimates hold:
\begin{gather}
|F_{t\to h^{-1}\tau} \Bigl(\bar{\chi}_T (t)u_h(x,y,t)\Bigr) |\le Ch^{\frac{1-d}{2}}\ell^{\frac{1-d}{2}}
\label{eqn-2.19}
\intertext{and as $c\ell\le T\le \epsilon$ }
|e^\mathsf{T}_{T,h} (x,y,\tau)|\le Ch^{\frac{1-d}{2}}\ell^{\frac{-1-d}{2}}.
\label{eqn-2.20}
\end{gather}
\item
The following estimate holds as $\max(c\ell,\,h^{1-\delta})\le T\le \epsilon$:
\begin{gather}
|e^\mathsf{T}_{T,h} (x,y,\tau)-e^\mathsf{W}_h(x,y,\tau) |\le  Ch^{1-d}.
\label{eqn-2.21}
\end{gather}
\end{enumerate}
\end{proposition}

\begin{proof}
\begin{enumerate}[wide,label=(\alph*),  labelindent=0pt]
\item
Estimates (\ref{eqn-2.19})--(\ref{eqn-2.20}) follow from (\ref{eqn-2.16})--(\ref{eqn-2.17}).
\item
To prove Statement (ii)  for $d\ge 3$ one can observe that both $|e^\mathsf{T}_{T,h} (x,y,\tau|$ and $|e^W(x,y,\tau)|$ do not exceed $Ch^{1-d}$ as $h^{\frac{1}{2}}\le \ell\le \epsilon$,
while for $\ell\le h^{\frac{1}{2}}$, due to  (\ref{eqn-2.15}), replacing $\Phi (x,y,t,\theta)$ by
\begin{gather}
\bar{\Phi}(x,y,\theta)=\langle x-y,\theta\rangle  + t a(\frac{1}{2}(x+y),\theta)
\label{eqn-2.22}
\end{gather}
brings an error not exceeding
the right-hand expression of (\ref{eqn-2.20}) multiplied by $C\ell^2h^{-1}$, that is $Ch^{-\frac{1+d}{2}}\ell ^{\frac{3-d}{2}}\le Ch^{1-d}$.

Moreover, if we also replace $B(x,y,t,\theta,h)$ by $1$ then the error will not exceed the right-hand expression of (\ref{eqn-2.20}) multiplied by $C\ell $, that  is
$Ch^{\frac{1-d}{2}}\ell ^{\frac{1-d}{2}}\le Ch^{1-d}$.

Finally, with this choice of the phase function $\bar{\Phi}(x,y,t,\theta)$ and amplitude $B=1$,  we get $e^\mathsf{T}_{T,h}(x,y,\tau) =e^\mathsf{W}(x,y,\tau) + O(h^\infty)$.

\item
As $d=2$ we need more subtle arguments (which would also work for $d\ge 3$). Indeed, estimate (\ref{eqn-2.20}) implies that  $ |e^\mathsf{T}_{T,h} (x,y,\tau)|\le Ch^{-1}$ as
$h^{\frac{1}{3}}\le \ell\le \epsilon_1$ and  estimate (\ref{eqn-2.21}) leads to $O(h^{-1})$ error only as $\ell \asymp h$.

We refer to Proposition~\ref{prop-5.13}(i).
\end{enumerate}
\end{proof}

Finally, Theorem~\ref{thm-1.1}  follows from estimate (\ref{eqn-1.11}) and Proposition~\ref{prop-2.4}.

\section{Improvements and generalizations}
\label{sect-2.3}

\paragraph{Two-term asymptotics.}
We know that  if $A$ is a scalar operator on the closed manifold, $\xi$-microhyperbolic at point $x$ and if \emph{non-looping condition}
\begin{gather}
\upmu_{x,\tau} \{ \xi \in \Sigma (x,\tau) \colon \exists t\ne 0, \uppi _x e^{tH_a (x,\xi)} =x\}=0
\label{eqn-2.23}\\
\intertext{is fulfilled at point $x$ then two-term asymptotics}
e_h(x,x,\tau)=\kappa_0(x,\tau)h^{-d}+\kappa_1(x,\tau)h^{1-d} + o(h^{1-d})
\label{eqn-2.24}
\end{gather}
holds; here $\Sigma (x,\tau)= \{\xi\colon a(x,\xi)=\tau\}$, $ \upmu_{x,\tau}$   is a natural measure on $\Sigma (x,\tau)$, corresponding to a density $dxd\xi :d_\xi a(x,\xi)$, $H_{a}$ is a Hamiltonian field generated by $a$ and $e^{tH_a}$ is a Hamiltonian flow (see, f.e. \cite{monsterbook}, Section~\ref{monsterbook-sect-5-3}). Can we get a two-term  asymptotics for $e_h(x,y,\tau)$?
\enlargethispage{\baselineskip}

First of all, we need to improve Tauberian estimate (\ref{eqn-1.11}). Asymptotics (\ref{eqn-2.24}) at point $x$ implies
\begin{gather}
|e(x,x, \tau+T^{-1} h)-e(x,x, \tau')|\le CT^{-1}h^{1-d} +o_T(h^{1-d})
\label{eqn-2.25}
\end{gather}
and then
\begin{multline*}
|e(x,y;\tau )-e(x,y; \tau')|\\
\le C|\tau -\tau'| h^{-d}+C|\tau -\tau'|^{\frac{1}{2}}h^{\frac{1}{2}-d} + CT^{-\frac{1}{2}}h^{1-d}+o_T(h^{1-d})
\end{multline*}
with arbitrarily large $T$. Here condition (\ref{eqn-2.23}) is fulfilled at one of  points $x$, $y$. Then the standard Tauberian arguments imply
\begin{gather*}
|e_h(x,y;\tau)-e_{T,h}^\mathsf{T} (x,y;\tau) |\le CT^{-1/2}h^{1-d} + o_T(h^{1-d})
\end{gather*}
and propagation results combined with non-looping condition at one of points $x$, $y$ imply that in the left hand expression one can replace $T$ by  $T'=T(x,y)$ while still having arbitrary  $T$ in the right hand-expression, and then we arrive to
\begin{gather}
e_h(x,y;\tau) = e_{T,h}^\mathsf{T} (x,y;\tau) +o(h^{1-d})\qquad\text{with\ \ } T=T(x,y).
\label{eqn-2.26}
\end{gather}
Further, if we add another condition
\begin{gather}
\upmu_{x,\tau} \{ \xi \in \Sigma (x,\tau) \colon \exists t\ne 0, \uppi _x e^{tH_a (x,\xi)} =y\}=0
\label{eqn-2.27}
\end{gather}
or a similar condition, obtained by permutation of $x$ and $y$ (let's call it $(\ref{eqn-2.26})'$), then from (\ref{eqn-2.26}) and propagation results we obtain that
\begin{enumerate}[label =(\alph*), wide]
\item
If $\ell (x,y)\ge \epsilon$ then $e_h(x,y,\tau)=o(h^{1-d})$,
\item
If $\ell(x,y)\le \epsilon$, then we can take $T(x,y)=c\ell (x,y)$ in (\ref{eqn-2.26}).
\end{enumerate}

\paragraph{Weyl asymptotics.}
In particular, in the framework of Theorem~\ref{thm-1.1}, under extra assumptions (\ref{eqn-2.23}) and either (\ref{eqn-2.26}) or
$(\ref{eqn-2.26})'$, we can apply the machinery developed in Subsection~\ref{sect-2.2} (with the only exception $d=2$ and neither $\ell \gg h^{\frac{1}{3}}$ nor $\ell \ll h^{\frac{1}{3}}$). One can prove easily that then
\begin{gather}
\bigl(e(x,y,\tau)=e^\mathsf{W} (x,y,\tau)\bigr)+o(h^{1-d}) \qquad  \forall x,y\in B(0,\epsilon )
\label{eqn-2.28}
\end{gather}
where
$e_h^\mathsf{W} (x,y;\tau)$  is defined by (\ref{eqn-1.3}) with $a$ replaced by $a+ha_1$ where $a_1$ is the subprincipal symbol of $A$, that is with the main term defined by (\ref{eqn-1.3}) and with the second term
\begin{gather}
-(2\pi h)^{1-d}\int _{\Sigma (z, \tau)} a_{1}(z,\theta) e^{i\langle x-y,\theta\rangle}\,d\theta :d_{\theta} a(z,\theta),\qquad z=\frac{1}{2}(x+y).
\label{eqn-2.29}
\end{gather}

\paragraph{Generalizations.}
\begin{enumerate}[wide,label=(\roman*),  labelindent=0pt]
\item
We can consider the case when $a(x,\xi)$ has characteristic roots $\lambda_j(x,\xi)$ of constant multiplicity; and we need only to assume that it happens
as $|\lambda_j (x,\xi)-\tau|\le \epsilon$.
\item
If we consider only $y\colon y-x \in \Gamma$ where $\Gamma$ is a cone in $\mathbb{R}^d$ with a vertex at $0$, then in virtue of propagation results (see \cite{monsterbook}, Section~\ref{monsterbook-sect-2-2}) we need the previous assumption only for $\xi$ such that
\begin{gather}
\Gamma \cap (K (x,\xi) \cup -K(x,\xi))\ne \emptyset
\label{eqn-2.30}
\end{gather}
where $K(x,\xi)$ is a cone dual to $K'(x,\xi)$ which is a connected component of
\begin{gather*}
\{\eta \colon ((\eta \cdot \partial_\xi a(x,\xi))v,v)\ge 0\ ; \forall v\colon \|(a-\tau) v\|\le \epsilon \|v\|\}.
\end{gather*}
\item
Also, instead of  strong convexity we need to assume only that at points $\xi$ satisfying (\ref{eqn-2.30}), the matrix of the curvatures of the surface
$\{\xi \colon \lambda (x,\xi)=\tau\}$ has at least two eigenvalues, disjoint from $0$. Then it would be similar to the case $d\ge 3$ in Proposition~\ref{prop-2.4}. Probably, it would suffice to have just one eigenvalue, to recover more delicate arguments of the case $d=2$ in Proposition~\ref{prop-2.4}.
\end{enumerate}

\chapter{Microlocal methods}
\label{sect-3}

\section{Propagation}
\label{sect-3.1}
Recall that we consider Schr\"odinger operator (\ref{eqn-1.4}) and for such operator under $\xi$-microhyperbolicity near boundary condition on the energy level $\tau$ instead of on-diagonal asymptotics (\ref{eqn-2.1}) asymptotics with a boundary-layer type term and remainder estimate $O(h^{1-d})$ holds; see \cite{monsterbook}, Section~\ref{monsterbook-sect-8-1}. In particular, under Dirichlet or Neumann boundary conditions (and we consider only those for simplicity) asymptotics (\ref{eqn-1.5}) holds as $x=y$ and also (\ref{eqn-1.9}) holds. Then (\ref{eqn-1.10})--(\ref{eqn-1.12}) hold and all we need is to rewrite  Tauberian expression (\ref{eqn-1.12}) in more explicit terms.

Let us   decompose
\begin{gather}
u_h(x,y,t)= u_h^0 (x,y,t) + u_h^1 (x,y,t)
\label{eqn-3.1}\\
\intertext{where $u_h^0 (x,y,t)$ is the solution for the ``free space'' and $u_h^1(x,y,t)$ is a reflected wave, satisfying}
hD_t u_h^1 =A(x,hD,h)u_h ^1,
\label{eqn-3.2}\\
B(x,hD,h) u_h^1 |_{\partial X} = -B(x,hD,h) u_h^0 |_{\partial X},
\label{eqn-3.3}
\end{gather}
where $B(x,hD,h)$ is a boundary operator. Then
\begin{gather}
e^\mathsf{T}_h(x,y,\tau)= e^{0,\mathsf{T}}_h(x,y,\tau)+ e^{1,\mathsf{T}}_h(x,y,\tau),
\label{eqn-3.4}
\end{gather}
where $e^{0,\mathsf{T}}_h(x,y,\tau)$ and $e^{1,\mathsf{T}}_h(x,y,\tau)$ are derived by Tauberian expression  (\ref{eqn-1.12}) from  $u_h^0 (x,y,t) $ and $u^1 (x,y,t)$ correspondingly
with $T=\epsilon_1$.

While $u^0_h(x,y,t)$ can be constructed as an oscillatory integral, our purpose is to construct $u^1_h(x,y,t)$  in some way.
To do this, in this section  we apply the improved method of successive approximations near boundary, in the same way as the standard method was applied in \cite{monsterbook}, Section~\ref{monsterbook-sect-7-2} while in the next Section~\ref{sect-5} we use the geometric optics method.

Recall that now $A$ is a second order elliptic operator and
$X\cap B(0,1)= \{x\colon {\nu}(x)>0\}\cap B(0,1)$ with $|\nabla {\nu}|\ge \epsilon _0$. Without any loss of the generality one can assume that

\begin{claim}\label{eqn-3.5}
$X\cap B(0,1)=\{x\colon x_1>0\}\cap B(0,1)$ with  $V_1(x)=0$,  $g^{1k}=\updelta_{1k}$
\end{claim}
and therefore
\begin{gather}
{\nu}(x)= x_1,\qquad \ell(x,y)=|x'-y'|+x_1+y_1
\label{eqn-3.6}
\end{gather}
and  $A$ is $\xi$-microhyperbolic on the energy level $\tau$ at $x$ as long as
\begin{gather}
|V(x)-\tau|\ge \epsilon_0,
\label{eqn-3.7}
\end{gather}
and in this case $\{\xi\colon a(x,\xi)=\tau\}$ is strongly convex.

\begin{proposition}\label{prop-3.1}
\begin{enumerate}[wide,label=(\roman*),  labelindent=0pt]
\item
As $T\le \epsilon \ell(x,y)$ the following estimate holds:
\begin{gather}
|F_{t\to h^{-1}\tau}\Bigl(\bar{\chi}_T(t)u^1_h(x,y,t)\Bigr)|\le Ch^{-d}\Bigl(\frac{h}{T}\Bigr)^s.
\label{eqn-3.8}
\end{gather}
\item
Let condition \textup{(\ref{eqn-3.1})} be fulfilled. Then for $\max(C\ell(x,y),\, h^{1-\delta})\le T\le T'\le \epsilon _1$   the following estimate holds:
\begin{gather}
|F_{t\to h^{-1}\tau}\Bigl(\bigl(\bar{\chi}_{T'}(t)- \bar{\chi}_{T} (t)\bigr) u^1_h(x,y,t)\Bigr)|\le Ch^{-d}\Bigl(\frac{h}{T}\Bigr)^s.
\label{eqn-3.9}
\end{gather}
\end{enumerate}
\end{proposition}

\begin{proof}
Statement (i)  follows from the finite speed of propagation and Statement (ii) follows from the fact that the speed of propagation with respect to $x$ is disjoint from $0$
(either along $\partial X$, if $|\xi'|$ is disjoint from $0$ or in direction of $x_1$ if $\xi_1$ is disjoint from $0$)--see \cite{monsterbook}, Chapter~\ref{monsterbook-sect-3}.
\end{proof}

\begin{corollary}\label{coro-3.2}
Let $e^{1,\mathsf{T}}_h(x,y,\tau)$ be defined by \textup{(\ref{eqn-1.12})} with $u^1_h(x,y,t)$ instead of $u_h(x,y,t)$.
\begin{enumerate}[wide,label=(\roman*),  labelindent=0pt]
\item
Then in $e^{1,\mathsf{T}}_h(x,y,\tau)$ one can replace $T=\epsilon_1$ by $T= C\max(\ell(x,y),\, h^{1-\delta})$.
\item
Further, in the framework of Proposition~\ref{prop-3.1}(ii) and $\ell (x,y)\ge h^{1-\delta}$ one can replace $\bar{\chi}_{T}(t)$ with $T=\epsilon_1$
by $\bigl(\bar{\chi}_{T}(t)-\bar{\chi}_{T'}(t)\bigr)$ with $T=C_0\ell(x,y)$ and $T'=C_0^{-1}\ell(x,y)$.
\end{enumerate}
\end{corollary}

We need to be more precise about propagation especially with respect to $x_1$ and $\xi_1$ as $\ell ^0\asymp \ell $.
Let us $\bar{x}$, $\bar{y}$ and $\bar{w}$ be fixed final values of $x$, $y$ and $w$ respectively and we need to consider case $\ell \coloneqq \ell(\bar{x},\bar{y})\ge h^{1-\delta}$.

\begin{proposition}\label{prop-3.3}
Let $h^{1-\delta}\le T \le \epsilon_1$, $T=C_0\ell$ and
\begin{phantomequation}\label{eqn-3.10}\end{phantomequation}
\begin{gather}
\rho T \ge h^{1-\delta}, \qquad \rho \ge T^2.
\tag*{$\textup{(\ref*{eqn-3.10})}_{1,2} $}\label{eqn-3.10-*}
\end{gather}
Let   $b(\bar{y}',\bar{\xi}')=\tau$ and
\begin{gather}
|\nabla _{y'} b(y', \bar{\xi}  )_{y'=\bar{y'}}|\le \rho T^{-1}
\label{eqn-3.11}\\
\intertext{where here and below}
b(x',\xi')\coloneqq a(x,\xi)|_{x_1=\xi_1=0};
\label{eqn-3.12}
\end{gather}
and let $Q_1(y,\eta')$ have a symbol supported in $(T,   \rho)$-vicinity of $(\bar{y},\bar{\xi}')$.

\begin{enumerate}[wide,label=(\roman*),  labelindent=0pt]
\item
Let  $Q_2(x,\xi')$ have a symbol, equal $0$ in   $\{(x, \xi')\colon |\xi'-\bar{\xi}'|\le C_1\rho\}$. Then
\begin{gather}
F_{t\to h^{-1}\tau} \Bigl(\bar{\chi} _T(t) Q_{2x} u^1 _h(x,y,t) \,^t\!Q_{1y}\Bigr)=O(h^s).
\label{eqn-3.13}
\end{gather}
\item
Assume instead that  $Q_2(x,hD')$ has a symbol, equal $0$ as  $x_1\le C_1\sigma  T$ where here and below
\begin{gather}
\sigma = (\rho ^{\frac{1}{2}} +T).
\label{eqn-3.14}
\end{gather}
Then \textup{(\ref{eqn-3.13})} holds.

\item
Assume instead that  $Q_2(x,hD')$ has a symbol, equal $0$  in
\begin{multline}
\{(x',\xi')\colon |x'-\bar{y}'  + t \nabla _{\xi'}b(\bar{y}', \xi')|_{\xi=\bar{\xi}'}|\le C_1\rho T\}\\
\forall t\colon \epsilon T \le \pm t \le C_0T.
\label{eqn-3.15}
\end{multline}
Assume also that
\begin{gather}
\rho^2 T\ge h^{1-\delta},\qquad \rho \ge C_0T.
\label{eqn-3.16}\\
\shortintertext{Then}
F_{t\to h^{-1}\tau} \Bigl(\chi^\pm  _T(t) Q_{2x} u^1 _h(x,y,t) \,^t\!Q_{1y}\Bigr)=O(h^s).
\label{eqn-3.17}
\end{gather}
where $\chi^\pm \in \mathscr{C}^\infty_0(\mathbb{R})$ is supported in $\{t\colon \epsilon \le \pm t\le 1\}$.
\end{enumerate}
\end{proposition}

\begin{proof}
\begin{enumerate}[wide,label=(\roman*),  labelindent=0pt]
\item
Observe that under assumptions $\textup{(\ref{eqn-3.10})}_{2}$ and (\ref{eqn-3.11}) the propagation speed with respect  to $\xi'$ does not exceed
$C_1 (\rho +T)$ in
\begin{gather*}
\{(x,\xi')\colon |x'-\bar{y}'|+|x_1|+|y_1|\le C_1T,\ |\xi'-\bar{\xi}' |\le C_1\rho\}.
\end{gather*}
This implies Statement~(i) by the standard methods of propagation; see Subsection~\ref{monsterbook-sect-5-1-2} of \cite{monsterbook}. Here  $\textup{(\ref*{eqn-3.10})}_{1}$  is the microlocal uncertainty principle).

\item
Due to Statement~(i) the propagation is confined to $\{(x,\xi')\colon |b(x',\xi')-\tau |\le C_0\sigma^2\}$ and thus to $|\xi_1|\le C\sigma $ and then to
$x_1 \le  C_1\sigma \ell$. The microlocal uncertainty principle with respect to $(x_1,\xi_1)$, that is
$\sigma \times\sigma \ell\ge h^{1-\delta}$ is fulfilled due to $\textup{(\ref{eqn-3.10})}_{1} $ and therefore we can consider  propagation with respect to $(x_1,\xi_1)$ as
$x_1\ge \sigma T$.

\item
Due to Statement~(i) the propagation in the time direction $\pm t>0$ is confined to the domain (\ref{eqn-3.15}) and here (\ref{eqn-3.16}) is a microlocal uncertainty principle.
\end{enumerate}
\end{proof}

\begin{remark}\label{remark-3.4}
\begin{enumerate}[wide,label=(\roman*),  labelindent=0pt]
\item
We can achieve (\ref{eqn-3.11})  by  gauge transformation, not affecting $x_1,\xi_1$. Further, if
\begin{gather}
\rho \ge T
\label{eqn-3.18}
\end{gather}
then (\ref{eqn-3.11})   is fulfilled automatically.

\item
Obviously, condition  (\ref{eqn-3.11})  is important only as $\rho \le T$, which is not the case as $T\le h^{\frac{1}{2}}$. Therefore we can cover also $\ell \le h^{1-\delta}$ with $T=h^{1-\delta}$.

\item
Both statements of Proposition~\ref{prop-3.3} hold for $\tau'\colon |\tau-\tau'|\le \sigma^2 =(\rho +\ell^2)$.
\end{enumerate}
\end{remark}

\section{Spectral estimates}
\label{sect-3.2}

The following proposition will be useful in the proof of Theorem~\ref{thm-1.2}:

\begin{proposition}\label{prop-3.5}
\begin{enumerate}[wide,label=(\roman*),  labelindent=0pt]
\item
Let $L \ell^0 (x,y)\ge h^{1-\delta}$. Let $\phi \in \mathscr{C}_0^\infty (\mathbb{R})$. Then
\begin{gather}
\int \phi ((\tau-\tau')/L ) F_{t\to h^{-1}\tau'} \Bigl((\bar{\chi}_T (t) u_h(x,y,t)\Bigr)\,d\tau'=O(h^s).
\label{eqn-3.19}
\end{gather}
\item
Let $L \ell (x,y)  \ge h^{1-\delta}$. Let $\phi \in \mathscr{C}_0^\infty (\mathbb{R})$. Then
\begin{gather}
\int \phi ((\tau-\tau')/L ) F_{t\to h^{-1}\tau'} \Bigl((\bar{\chi}_T (t) u^1 _h(x,y,t)\Bigr)\,d\tau'=O(h^s).
\label{eqn-3.20}
\end{gather}
\end{enumerate}
\end{proposition}

\begin{proof}
The left-hand expressions are just $\phi ((hD_t-\tau) /L) u_h(x,y,t)\bigr|_{t=0}$ and $\phi ((hD_t-\tau) /L) u^1_h(x,y,t)\bigr|_{t=0}$ correspondingly and we apply finite speed of propagation.
\end{proof}

\begin{proposition}\label{prop-3.6}
Under $\xi$-microhyperbolicity condition on the energy level $\tau$ estimates  \textup{(\ref{eqn-1.8})} and
\begin{gather}
|e_h(x,y,\tau)|\le Ch^{1-d} \bigl(1+\ell^{0\,-1}(x,y)\bigr)
\label{eqn-3.21}
\end{gather}
hold.
\end{proposition}

\begin{proof}
In virtue of Subsection~\ref{sect-2.1} for $h\le \mathsf{T}\le \epsilon$
\begin{multline*}
e_h(x,y,\tau) = e^{\mathsf{T}}_h(x,y,\tau) +O(T^{-1}h^{1-d}) \\
=e^{0,\mathsf{T}}_h(x,y,\tau) +e^{1,\mathsf{T}}_h(x,y,\tau) +O(T^{-1}h^{1-d})
\end{multline*}
and for $T=\epsilon_1 \ell(x,y)$
\begin{gather*}
e^{0,\mathsf{T}}_h(x,y,\tau) =e^{0,\mathsf{W}}(x,y,\tau)+O(h^{-d}(h\ell^{-1})^s)\\
\shortintertext{and}
e^{1,\mathsf{T}}_h(x,y,\tau) =O(h^s)
\end{gather*}
due to the finite speed of propagation, which implies (\ref{eqn-1.8}). Estimate  (\ref{eqn-3.21}) is proven in the same way.
\end{proof}

As $h^{1-\delta}\le \ell(\bar{x},\bar(y))\le \epsilon_1$ let us introduce $\bar{\xi}^{\prime\pm}$:
\begin{gather}
\nabla _{\xi'} b(\bar{w}', \xi')|_{\xi'=\bar{\xi}^{\prime\pm}}= t(\bar{y}-\bar{x}), \qquad  b(\bar{w}', \bar{\xi}^{\prime\pm})=\tau,\qquad \pm t>0
\label{eqn-3.22}
\end{gather}
with $w'= \frac{1}{2}(\bar{x}'+\bar{y}')$. Due to strong convexity $\bar{\xi}^{\prime\pm}$ are defined uniquely and $t \asymp \ell(\bar{x},\bar(y))$.

\begin{proposition}\label{prop-3.7}
Let $h^{1-\delta}\le T \le \epsilon_1$, $T=C_0\ell$,  \textup{(\ref{eqn-3.18})} and \textup{(\ref{eqn-3.16})} be fulfilled. Then
\begin{gather}
 e_h^{1,\mathsf{T} } (\bar{x},\bar{y},\tau) = Q_{2x}  e_h^{1,\mathsf{T}} (x,y,\tau) \,^t\!Q_{1y} |_{x=\bar{x},y=\bar{y}} +O(h^s)
 \label{eqn-3.23}
\end{gather}
where $Q_1$, $Q_2$ are operators with symbols equal $1$ in $(C_1\ell, C_1\rho)$ vicinities of $(\bar{w}', \bar{\xi}^{\prime\pm})$.
\end{proposition}

\begin{proof}
Due to Proposition~\ref{prop-3.5}
\begin{multline}
e_h^{1,\mathsf{T} } (x,y,\tau) \\
= h^{-1}\int_{-\infty}^\tau \bar{\phi } ((\tau -\tau')L^{-1}) F_{t\to h^{-1}\tau'} \bigl(\bar{\chi} _T(t)u^1 _h(x,y,t)\bigr) \,d\tau' +O(h^s)
\label{eqn-3.24}
\end{multline}
where $\bar{\phi } \in \mathscr{C}^\infty (\mathbb{R})$, $\bar{\phi } (\tau)=1$ as $\tau \le 1$ and $\bar{\phi } (\tau)=0$ as $\tau \ge 2$, $L\ell \ge h^{1-\delta}$.

Due to Proposition~\ref{prop-3.1}(ii) we can replace $\bar{\chi}_T(t)$ by $\chi_T(t)= \bar{\chi}_T(t)- \bar{\chi}_{\epsilon T}(t)$.
Then Proposition~\ref{prop-3.3}(iii)  implies that if $\psi _1, \psi_2\in \mathscr{C}_0^\infty $ are supported in $\epsilon\ell$-vicinities of $\bar{y}$ and $\bar{x}$ respectively and
$Q'=Q'(hD')$ is $0$ in $C\rho$-vicinities of $\bar{\xi}^{\prime\pm}$ then as $|\tau'-\tau |\le L = h^{1-\delta}L^{-1}$
\begin{gather*}
F_{t\to h^{-1}\tau'} \Bigl(\chi  _T(t) Q'_x\psi_2(x) u^1_h(x,y,t)\psi_1(y)\Bigr)=O(h^s), \\
F_{t\to h^{-1}\tau'} \Bigl(\chi _T(t) \psi_2(x) u^1_h(x,y,t)\psi_1(y)\,^t\!Q'_y\Bigr)=O(h^s)
\end{gather*}
Indeed condition now are stronger than those of Proposition~\ref{prop-3.3}(iii).

Then these equalities and (\ref{eqn-3.24}) imply (\ref{eqn-3.23}).
\end{proof}

\begin{proposition}\label{prop-3.8}
In the framework of Proposition~\ref{prop-3.7}
\begin{gather}
|e^{1,\mathsf{T}}_{T,h}(x,y,\tau)|\le Ch^{1-d} \ell^{-1}\sigma \rho^{d-2}
\label{eqn-3.25}
\end{gather}
with $\sigma = h^{\frac{1}{2}-\delta'}\ell^{-\frac{1}{2}}+\ell$.
\end{proposition}

\begin{proof}
Due to Propositions~\ref{prop-3.1}(ii) and~\ref{prop-3.7} we need to estimate the right-hand expression of (\ref{eqn-3.24}) with symbols of $Q_1$ and $Q_2$
supported in $\Omega$, the union of $(C_1 \ell, C_1 \rho)$-vicinities of $(\bar{w}',\bar{\xi}^{\prime+})$ and $(\bar{w}',\bar{\xi}^{\prime-})$ intersected with
\begin{gather*}
\{\xi \colon |b(x',\xi')-\tau|\le C\sigma^2\},
\end{gather*}
 and  with $\bar{\chi}_T(t)$
replaced by  $\chi_T(t)= \bar{\chi}_T(t)- \bar{\chi}_{\epsilon T}(t)$. Observe that modulo $O(h^s)$
\begin{multline*}
e_h^{1,\mathsf{T}} (x,y,\tau)  \equiv h^{-1} \int_{-\infty}^\tau F_{t\to h^{-1}\tau'} \Bigl (\chi_T(t) u^1_h (x,y,t) \Bigr)\,d\tau'\\
= T^{-1}  F_{t\to h^{-1}\tau} \Bigl (\beta_T(t) u^1_h (x,y,t) \Bigr)
\end{multline*}
with $\beta(t)=t^{-1}\chi(t)$.

Since $\ell \asymp \ell^0$ it sufficient to prove it for $e^{\mathsf{T}}_{T,h}(x,y,\tau)$.  Expressing $u_h(x,y,t)= \int e^{ih^{-1}t\tau} \,d_\tau e_{h}(x,y,\tau)$ we see that expression in question equals
\begin{gather*}
\int \hat{\beta}\bigl (\frac{(\tau-\tau')T}{h}\bigr) \,d_{\tau'} Q_{2x}e_h(x,y,\tau')\,^t\!Q_{1y}
\end{gather*}
which does not exceed
\begin{align*}
C\sup _{\tau'\colon |\tau'-\tau|\le L}& |Q_{2x} e_h(x,y,\tau'+ hT^{-1},\tau')\,^t\! Q_{1y}| +C'h^s.\\
\intertext{Since $e_h(x,y,\tau)$ is the Schwartz kernel of the orthogonal projector this does not exceed}
C\sup _{\tau'\colon |\tau'-\tau|\le L} & |Q_{2x} e_h(x,z,\tau'+ h T^{-1} ,\tau')\,^t\! Q_{2z}|_{z=x} |^{\frac{1}{2}}\\
\times& |Q_{1z} e_h(z,y,\tau'+ hT^{-1},\tau')\,^t\! Q_{1y}|_{z=y}|^{\frac{1}{2}} +C'h^s.
\end{align*}
Applying Tauberian estimate  for $x=y$ we see that the right-hand  expression does not exceed
\begin{gather*}
Ch^{1-d}\ell^{-1} \sup _{\tau'\colon |\tau'-\tau|\le L} \int _{\Sigma (x,\tau')\cap (\Omega  \times \mathbb{R}_{\xi_1})} \, d\xi:d_\xi a .
\end{gather*}
The integral in the right-hand expression does not exceed $C\sigma \rho^{d-2}$. Finally, recall that $T\asymp \ell $.
\end{proof}

Taking $\rho =\max ((h^{1-\delta'}\ell^{-1})^{\frac{1}{2}},\, \ell)$ to satisfy (\ref{eqn-3.16}) and (\ref{eqn-3.18}), and thus $\sigma=\rho$ we get
\begin{corollary}\label{cor-3.9}
The following estimates hold as $|x_1|+|y_1|\le C\sigma \ell$
\begin{gather}
e^{1,\mathsf{T}}_{h} (x,y,\tau)=O(h^{1-d}) \ \ \text{for\ \ }   \left\{\begin{aligned}
&\ell \ge h^{\frac{1}{2}+\delta} &&d\ge 4,\\
&\ell \ge h^{\frac{1}{2}-\delta} &&d=3,\\
&\ell \ge h^{\frac{1}{3}-\delta} &&d=2.
\end{aligned}\right.
\label{eqn-3.26}
\end{gather}
\end{corollary}

\begin{remark}\label{remark-3.10}
\begin{enumerate}[wide,label=(\roman*),  labelindent=0pt]
\item
Then $\sigma \ge h^{\frac{1}{3}-\delta'}$.
\item
The similar arguments work under assumption $|x'-y'|\ge \epsilon |x_1|+|y_1|$ instead of $|x_1|+|y_1|\le C\sigma \ell$.
\item
The similar but simpler arguments work for $e^{0,\mathsf{T}}_h(x,y,\tau)$  as $\rho^2 \ell ^0 \ge h^{1-\delta}$.
\end{enumerate}
\end{remark}

\chapter{Successive approximations}
\label{sect-4}

\section{Successive approximations inside domain}
\label{sect-4.1}

\subsection{Standard successive approximations}
\label{sect-4.1.1}
In this Section we are going to apply method of successive approximations to derive asymptotics of $e_h(x,y,\tau)$ near boundary. However we start from the partial proof of Theorem~\ref{thm-1.1} by this method away from the boundary. According to \cite{monsterbook}, Section~\ref{monsterbook-sect-5-3} (which is our standard reference here), we consider problem for propagator $u _h(x,y,t)$
\begin{align*}
&(hD_t-A) u_h =0, && u|_{t=0}=\updelta (x-y)
\end{align*}
with $A=A(x,hD_x,h)$ and rewrite  for $u^\pm_h(x,y,t)\coloneqq u_h(x,y,t)\uptheta(\pm t)$. We have equation
\begin{gather}
(hD_t-A)u_h^\pm = \mp ih\updelta (x-y)\updelta (t)
\label{eqn-4.1}
\end{gather}
which we are going to solve by the successive approximations with unperturbed operator $\bar{A}=a(y,hD_x)$.
Then (\ref{eqn-4.1}) yields the equality
\begin{gather}
(hD_t -\bar{A})u_h^\pm = \mp ih\updelta (x-y)\updelta (t)+Ru_h^\pm
\label{eqn-4.2}\\
\shortintertext{and hence}
u_h^\pm =\mp ih\bar{G}^\pm \updelta (x-y)\updelta (t)+\bar{G}^\pm Ru_h^\pm,
\label{eqn-4.3}
\end{gather}
where $\bar{G}^\pm$ and $G^\pm$ are parametrices of the problems
\begin{align}
&(hD_t-\bar{A})v=f, \qquad\operatorname{supp}(v)\subset\{\pm (t-t_0)\ge 0 \}
\label{eqn-4.4}\\
\shortintertext{and}
&(hD_t-{A})v=f, \qquad\operatorname{supp}(v)\subset \{\pm(t-t_0) \ge 0 \}
\label{eqn-4.5}
\end{align}
respectively with $\operatorname{supp}(f)\subset \{\pm (t-t_0) \ge 0\}$ for some $t_0 \in \mathbb{R}$; $R=A-\bar{A}$.

Moreover, equation (\ref{eqn-4.1}) yields that
\begin{equation}
u_h^\pm=\mp ihG^\pm \updelta (x-y)\updelta (t).
\label{eqn-4.6}
\end{equation}
Iterating (\ref{eqn-4.3}) $N$ times and then substituting (\ref{eqn-4.6}) we
obtain the equality
\begin{multline}
u_h^\pm = \mp ih \sum _{0 \le n \le N-1 }
(\bar{G}^\pm R)^n\bar{G}^\pm \updelta (x-y)\updelta (t) \mp\\
ih(\bar{G}^\pm R)^nG^\pm \updelta (x-y)\updelta (t).
\label{eqn-4.7}
\end{multline}
Finally, we apply $^t\!Q_y=\,^t\!Q(y, hD_y)$ to the right of (\ref{eqn-4.7}), where $Q$ is an operator with compactly supported symbol, equal $1$ as $a(y,\xi)\le \tau+\epsilon$.
After this cut-off, according to \cite{monsterbook}, Section~\ref{monsterbook-sect-5-3}, norms of the terms in (\ref{eqn-4.7}) in the strip $\{(x,t)\colon |t|\le T\}$ do not exceed $Ch^{-M-n} T^{2n}$ and
as $T\le h^{\frac{1}{2}+\delta}$ we can ignore the remainder term. Since we need to consider   $T\le c\ell(x,y)$ in the end we arrive to
\begin{gather}
\max(c\ell,\, h^{1-\delta}) \le T \le h^{\frac{1}{2}+\delta}.
\label{eqn-4.8}
\end{gather}
Then as it was shown (see \cite{monsterbook}, simplified (\ref{monsterbook-4-3-33}))
\begin{multline}
F_{t\to h^{-1}\tau} \Bigl(\bar{\chi}_T u_h^\pm(x,y,t)\,^t\!Q_y\Bigr) \\
\equiv
\mp i\sum_{0\le n\le N}  (2\pi)^{-d-1}h^{-d+n}\int e^{ih^{-1}\langle x-y,\xi\rangle} F_n(y,\xi,\tau ) [q_2(y,\xi,h)]\,d\xi,\quad \pm \Im \tau <0,
\label{eqn-4.9}
\end{multline}
where $F_n =\sum_{\beta\colon |\beta|\le n } F_{n,\beta} \partial_\xi^\beta $ are differential operators  applied to $q_2$, with coefficients $F_{n\beta}$ holomorphic with respect to $\xi,\tau$ as $\Im \tau \ne 0$, and have poles as $\tau \in \mathbb{R}$; denominators are $(\tau -a(y,\xi))^{2n+1+|\beta|}$. Then
\begin{multline}
F_{t\to h^{-1}\tau} \Bigl(\bar{\chi}_T u_h(x,y,t)\Bigr) \\
\equiv
\sum_{0\le n\le N}  (2\pi)^{-d-1}h^{-d+n}\int e^{ih^{-1}\langle x-y,\xi\rangle} \mathcal{F}_n(y,\xi,\tau ) \,d\xi,\qquad \tau\in \mathbb{R}
\label{eqn-4.10}
\end{multline}
where $ \mathcal{F}_n(y,\xi,\tau )=i\bigl( F_{n0} (y,\xi,\tau -i0)- F_{n0} (y,\xi,\tau +i0)\bigr)$. Then the principal term in $e^\mathsf{T}_{T,h}(x,y,\tau)$ can be rewritten as $e^\mathsf{W}_h(x,y,\tau)$ while all other terms as
\begin{gather}
h^{-d+n-|\alpha|  }  \sum_{\alpha\colon |\alpha|\le 2n-1} (x-y)^\alpha  \int _{\Sigma (y,\tau)} W_{n,\alpha} (y,\xi) e^{ih^{-1}\langle x-y,\xi\rangle}\,d\xi
\label{eqn-4.11}
\end{gather}
with  $n\ge 1$ and smooth $W_{n,\alpha} $.

Indeed, taking $\xi$-partition and using $\xi$-micro\-hyperbolicity, we can rewrite on each element $\mathcal{F}_n$ as a sum of  ${W'_{n,\alpha} (y,\xi)\partial_\xi^\alpha \updelta '(\tau-a(y,\xi))}$. Then, integrating by parts we will get (\ref{eqn-4.11}) for terms in $e^\mathsf{T}_{T,h} (x,y,\tau)$.

Now due to strong convexity and stationary phase principle these terms do not exceed
\begin{gather}
Ch^{-d+n-|\alpha|}  ({h}/{\ell})^{\frac{d-1}{2}} \ell^{|\alpha|} \le Ch^{-\frac{d+1}{2}+n -|\alpha| } \ell^{-\frac{d-1}{2} +|\alpha|},
\label{eqn-4.12}
\end{gather}
which is $O(h^{1-d})$ as $d\ge 3$ and $O(h^{-\frac{3}{2}}\ell^{\frac{1}{2}})$ as $d=2$. Recall that $\ell\le h^{\frac{1}{2}+\delta}$.

\subsection{Improved successive approximations}
\label{sect-4.1.2}
To improve this approach we need to freeze symbol at point $w=\frac{1}{2}(x+y)$ and to do this let us observe that \begin{align*}
&\bigl(hD_t-A(x,hD_x,h)\bigr) u_h(x,y,t)=0,\\
& \bigl(hD_t-  A(y,-hD_y,h)\bigr)u_h(x,y,t)=0, && u_h|_{t=0}=\updelta (x-y)
\end{align*}
and therefore denoting $v_h (w,z,t) = u_h (w+\frac{1}{2}z, w+\frac{1}{2}z, t)$ we have
\begin{gather}
\bigl(hD_t - \mathfrak{A} (w, z, hD_z, hD_w,h)\bigr)v_h(w,z,t)=0, \qquad v_h|_{t=0} =  \updelta (z)
\label{eqn-4.13}\\
\shortintertext{with}
\mathfrak{A}  = \frac{1}{2}\bigl(A(w +\frac{1}{2}z, hD_z+\frac{1}{2}hD_w,h) + A(w-\frac{1}{2}z, hD_z-\frac{1}{2}hD_w, h)\bigr).
\label{eqn-4.14}
\end{gather}
Taking as unperturbed operator
\begin{gather}
\bar{\mathfrak{A}}= A(w, hD_z,h)
\label{eqn-4.15}
\end{gather}
we get $R=\mathfrak{A}-\bar{\mathfrak{A}} $ a a sum of $z^\alpha (h\partial_w)^\beta$ with $|\alpha|+|\beta|\ge 2$.

With this modification we can write formula, similar to (\ref{eqn-4.7}).  Due to the propagation results we can add to $R$ factor $\bar{\chi}_{cT}(h\partial _w)$ and then each new term in this formula adds an extra factor $C(T^2+h) Th^{-1}$ instead of $CT^2h^{-1}$ and we can replace (\ref{eqn-4.8}) by a weaker condition
\begin{gather}
\max(c\ell,\, h^{1-\delta}) \le T \le h^{\frac{1}{3}+\delta}.
\label{eqn-4.16}
\end{gather}
Calculations, similar to those in the standard method lead us to the main term $e^\mathsf{W}(x,y,\tau)$ and to (\ref{eqn-4.11}) replaced by
\begin{gather}
h^{-d+n-|\alpha|  }  \sum_{\alpha\colon 2|\alpha|\le 3n-1} (x-y)^\alpha  \int _{\Sigma (w,\tau)} W_{n,\alpha} (w,\xi) e^{ih^{-1}\langle z,\xi\rangle}\,d\xi.
\label{eqn-4.17}
\end{gather}
and to the same estimate (\ref{eqn-4.12}) albeit with a restrictions $|\alpha|\le (3n-1)/2$, $\ell \le h^{\frac{1}{3}}$, and we get $O(h^{1-d})$ as $d\ge 1$.

\section{Successive approximations near boundary}
\label{sect-4.2}

\subsection{Standard successive approximations}
\label{sect-4.2.1}

We know (see \cite{monsterbook}, Section~\ref{monsterbook-sect-7-2}) that $u_h^0(x,y,t)$ and then $u_h^1(x,y,t)$ for $|t|\le T =h^{\frac{1}{2}+\delta}$, $x_1+y_1\le C_0T$  could be constructed by the method of successive approximations with unperturbed operator $\bar{A}=a(0,y'; hD_x)$ (so the principal part of $A$ is frozen at point $(0,y')$):  it leads us to the expression for $e^\mathsf{T}_h(x,y,\tau)$.

In this case unperturbed term would be as for   $\bar{A}$ in the half-space and it will not exceed $Ch^{-d}(h/\ell)^{{d+1}/2}$ and  and one can prove that the perturbed term would acquire factor $T^2/h$ with $T\asymp \ell$ and it does not exceed  (\ref{eqn-4.12}) (with redefined $\ell(x,y)$ now) and it is $O(h^{1-d})$ as $d\ge 3$ and
$Ch^{-\frac{3}{2}}\ell ^{\frac{1}{2}}$ as $d=2$.

Since we assumed that the boundary conditions are either Dirichlet or Neumann, we arrive to the expression (\ref{eqn-1.3}) for $e^{1,\mathsf{T}}_h(x,y,\tau)$ as described in Theorem~\ref{thm-1.2}.

\subsection{Improved successive approximations}
\label{sect-4.2.2}

Let us improve this construction in the same manner as we did inside domain. Consider problem for $u^1_h$:
\begin{align}
& (hD_t-A)u_h^1 =0,\qquad  u^1_h |_{t=0}=0,
\label{eqn-4.18}\\[3pt]
& \eth B u_h^1   =-\eth B u_h^0|_{x\in \partial X},
\label{eqn-4.19}
\end{align}
where $u^0_h(x,y,t)$ satisfies the same equation  in the ``whole space'' and $\eth$ is an operator restriction to $\partial X\ni x$.
Then
\begin{align}
& (hD_t-A)u_h^{1,\pm}  =0,
\label{eqn-4.20}\\[3pt]
& \eth B u_h^{1\,\pm}  =-\eth B u_h^{0\,\pm}\label{eqn-4.21}
\end{align}
where again $u^{j\,\pm}=\uptheta(\pm t)u^j_h (x,y,t)$ and
\begin{align}
&(hD_t -\bar{A})u_h^{1\,\pm} = Ru^{1\,\pm},
\label{eqn-4.22}\\
&\eth B u_h^{1\,\pm}  =- Bu_h^{0\,\pm},
\label{eqn-4.23}
\end{align}
where $u_h^{0,\pm}$ satisfies (\ref{eqn-4.2}); recall that $B=I$ or $B=D_1$ for Dirichlet and Neumann boundary conditions correspondingly\footnote{\label{foot-4} For more general boundary conditions we would need to replace $B$ by $\bar{B}+R_1$.}
Then like in \cite{monsterbook}, Section~\ref{monsterbook-sect-7-2}
\begin{gather}
u_h^{1\,\pm} =-\bar{G}^{\prime\,\pm} \eth B  u_h^{0\,\pm} + \bar{G}^\pm Ru_h^{1\,\pm},
\label{eqn-4.24}
\end{gather}
where  $\bar{G}^\pm$ are parametrices for the problems
\begin{align}
&(hD_t-\bar{A})v =f, && \eth Bv =0, && \operatorname{supp}(v)\subset \{\pm (t-t_0)\ge 0\}
\label{eqn-4.25}\\
\intertext{and $\bar{G}^{\prime\,\pm}$ are parametrices for the problems}
&(hD_t-\bar{A})v =0, && \eth Bv =f, && \operatorname{supp}(v)\subset \{\pm (t-t_0)\ge 0\}
\label{eqn-4.26}
\end{align}
respectively with $\operatorname{supp}(f)\subset \{\pm (t-t_0) \ge 0\}$ for some $t_0 \in \mathbb{R}$; $R= A-\bar{A}$. Moreover,
\begin{gather}
u_h^{1\,\pm} =- G ^{\prime\,\pm} \eth B  u_h^{0\,\pm}
\label{eqn-4.27}
\end{gather}
where ${G}^\pm$ and ${G}^{\prime\,\pm}$ are parametrices for the problems (\ref{eqn-4.25}) and (\ref{eqn-4.26}) but for operator $A$.
Iterating (\ref{eqn-4.24}) $N$ times and then substituting (\ref{eqn-4.27}) we arrive to formula similar to (\ref{eqn-4.7})
\begin{gather}
u_h^{1\,\pm} =-\sum_{0\le n\le N-1}
(\bar{G}^\pm R)^n \bar{G}^{\prime\,\pm} \eth B  u_h^{0\,\pm} - (\bar{G}^\pm R)^N  G ^{\prime\,\pm} \eth B  u_h^{0\,\pm}.
\label{eqn-4.28}
\end{gather}
What is more, we plug $u_h^{0\,\pm}$ given by (\ref{eqn-4.7}) with  $\bar{G}^{\pm}$ and $G^{\pm}$ replaced by $\bar{G}^{0\,\pm}$ and $G^{0\,\pm}$
which are parametrices for (\ref{eqn-4.4}) and (\ref{eqn-4.5}) in the ``whole space''.

Finally, we apply $^t\!Q_1=\,^t\!Q(y_1, z', hD'_z)$ to the right of (\ref{eqn-4.27}) and (\ref{eqn-4.7}), where $Q_1$ is an operator with the  symbol, supported in
\begin{gather}
\Omega_{ T, \rho,\sigma }\coloneqq \{(x_1, z',\zeta)\colon |z'|\le T,\ |\zeta- \bar{\xi}'| \le \rho, \ |x_1|\le \sigma T\}
\label{eqn-4.29}
\end{gather}
where $|a(0,\bar{x}', 0,\bar{\xi}')-\tau)|\le c\rho^2$. Then due to Proposition~\ref{prop-3.3}, under assumptions (\ref{eqn-3.11}) and $\textup{(\ref*{eqn-3.10})}_{1,2}$ we can insert
$Q_2 (x_1, hD'_z)$ to the right of each copy of $R$ with symbol supported in $\Omega_{3T, 3\rho, 3\sigma}$ and equal $1$ in $\Omega_{2T, 2\rho,2\sigma}$.

So far we we followed exactly \cite{monsterbook}, Section~\ref{monsterbook-sect-7-2}, except there we had $\rho=c$. However now instead of $\bar{A}=a(0,y'; hD_x)$ we take
\begin{gather}
\bar{A}=a(w,  hD_x), \qquad w =(0,w'),\qquad w'=\frac{1}{2}(x'+y').
\label{eqn-4.30}
\end{gather}

Then, the norm of $\bar{G}^{\pm}RQ_2$  does not exceed $C(T^2+\sigma T+h) Th^{-1}$ and we can replace (\ref{eqn-4.8}) by a weaker condition
$(T^2+\sigma T +h) T\le h^{1+\delta}$, which is equivalent to (\ref{eqn-4.16}) plus
\begin{gather}
\sigma \ell ^2 \le h^{1+\delta}.
\label{eqn-4.31}
\end{gather}

\begin{remark}\label{remark-4.1}
 Recall that $\sigma $ is defined by (\ref{eqn-3.14}) and with $\rho = h^{1-\delta}\ell^{-1}$ condition (\ref{eqn-4.31}) is also equivalent to (\ref{eqn-4.16}).
 However, for us it is more important which value of the reflection angle is allowed in this method.
\end{remark}

Thus we arrive to the following statement:
\begin{proposition} \label{prop-4.2}
Let $Q=Q(x_1,z',hD'_z)$ be an operator with the symbol $q(x_1,z',\zeta')$ supported in $\Omega_{T,\rho,\sigma}$ where condition  \textup{(\ref{eqn-3.11})}  is fulfilled  and let conditions \textup{(\ref{eqn-4.16})} and \textup{(\ref{eqn-4.31})} be also fulfilled.

Then we can skip  remainder terms in both   \textup{(\ref{eqn-4.27})} and modified \textup{(\ref{eqn-4.7})}, which  are $O(h^s)$,
leaving us only with $\bar{G}^\pm$, $\bar{G}^{\prime\,\pm}$ and $\bar{G}^{0\,\pm}$.
\end{proposition}

Thus (\ref{eqn-4.28}) without the last term becomes an asymptotic series. Now let us calculate $e_h^{1,\mathsf{T}} (x,y,\tau)$ under these assumptions.

\begin{proposition}\label{prop-4.3}
In the framework of Proposition~\ref{prop-4.2}
\begin{multline}
F_{t\to h^{-1}\tau} \Bigl(\bar{\chi}_T(t) u_h^{1\,\pm} \,^t\!Q_1 \Bigr)\\
\sim
\sum_{n\ge 0} h^{-d+n}  \int T\hat{\bar{\chi}} \bigl( \frac{(\tau-\tau')T}{h} \bigr) \,d\tau'  \int d\xi'\int _{\gamma_+} d\xi_1 \int_{\gamma_-} d\eta_1\\
\times
e^{ih^{-1}(x_1\xi_1- y_1\eta_1 +\langle x'-y',\xi'\rangle)}
F_{n}(w, \xi',\xi_1,\eta_1,\tau')[q (w,\xi')]
\label{eqn-4.32}
\end{multline}
for $ \tau \in \mathbb{C}_\mp $, where $\gamma_\pm$ are closed contours in $\mathbb{C}_\pm $ passing once
around the poles of $\bigl(\tau - a(z,\xi _1)\bigr)^{-1}$ lying in $\mathbb{C}_\pm$ and not passing around the poles lying in $\mathbb{C}_\mp$; for real $z$ and
$\Im \tau \ne 0$ there is no real pole, and $F_n$ are sums of the terms
\begin{multline}
\xi_1^j \eta_1^k W _{n,l,m,p, j,k}(w,\xi ',\tau )\\
\times
\bigl(\tau -a(w,\xi _1,\xi ')\bigr)^{-l}  \bigl(\tau -a(w,\eta_1,\xi ')\bigr)^{-m}  \bigl(\tau -b(w,\xi ')\bigr)^{-p}
\label{eqn-4.33}
\end{multline}
with
\begin{gather}
b(w,\xi')= a(w, 0, \xi'),
\label{eqn-4.34}\\[2pt]
l\ge 1, \; m\ge 1,\;  2(l+m +p )\le 3n +j+k+3, \; j\le l,\ k\le m
\label{eqn-4.35}
\end{gather}
and uniformly smooth $W _{n,l,m,p, j,k}$.
\end{proposition}

\begin{proof}
Proof follows the proof of  Theorem~\ref{monsterbook-thm-7-2-13} of \cite{monsterbook} with the some modifications, mainly introduced in Subsection~\ref{sect-4.1.2}. We rewrote $A$ as $\mathfrak{A}$ by (\ref{eqn-4.14}) and defined $\bar{\mathfrak{A}}$ by (\ref{eqn-4.15}) with the exception that $w'$ and $z'$ are now $(d-1)$-dimensional variables (see (\ref{eqn-4.30})) and we preserve $x_1$ and $y_1$.

The main part of the perturbation $R$ is  quadratic  with respect to $z', hD'_w$ and linear with respect to $x_1$.
 In comparison with (\ref{monsterbook-7-2-60}) of \cite{monsterbook} formula (\ref{eqn-4.33}) is simpler, because we have scalar factors here and products collapse into powers.
On the other hand, instead of claiming that $W$ of that proof are holomorphic satisfying (\ref{monsterbook-7-2-38}) of \cite{monsterbook} here we do it in much explicit way since now we need to follow very carefully what are relations between $j,l,l, m, p$ and $n$.

We have operators $a(w, hD_1, hD'_z, hD'_w)$ with symbols which do not depend on $x$. Let us make  $h$-Fourier transform by $x'$ and $t$.

Consider first $u^0_h(x,y,t)$.
We claim that
\begin{multline}
F_{t\to h^{-1}\tau} \Bigl(\bar{\chi}_T(t) \eth (hD_{x_1})^r u_h^{0\,\pm} \,^t\!Q_1 \Bigr)\\
\sim
\sum_{n\ge 0} h^{1-d+n}  \int T\hat{\bar{\chi}} \bigl( \frac{(\tau-\tau')T}{h} \bigr) \,d\tau'  \int d\xi'\int_{\gamma_-} d\eta_1\\
\times
e^{ih^{-1}(- y_1\eta_1 +\langle x'-y',\xi'\rangle)}
F^0_{n}(w, \xi', \eta_1,\tau')[q (w,\xi')]
\label{eqn-4.36}
\end{multline}
for $ \tau \in \mathbb{C}_\mp$, $r=1$  and $F^0_n$ are sums of the terms
\begin{gather}
\eta_1^k W^0 _{n,m, j,k}(w,\xi ',\tau ) \bigl(\tau -a(w,\xi _1,\xi ')\bigr)^{-l}  \bigl(\tau -a(w,\eta_1,\xi ')\bigr)^{-m}
\label{eqn-4.37}
\end{gather}
with $2m \le 3n +k+2-r$ and $ k< m $ with the only exception $n=0$, $m=1$, $k=r=1$.

\begin{enumerate}[wide,label=(\alph*),  labelindent=0pt]
\item
Like in that proof of \cite{monsterbook},  each factor $z_r$ and $hD_{w_r}$  with $r =2,\ldots, d$ is moved to the right towards $\updelta (z')$,  using commutator relations
\begin{align}
&z_r\bar{G}^{0\,\pm} = \bar{G}^{0\,\pm}  z_r -  \bar{G}^{0\,\pm} [ \bar{\mathfrak{A}},z_r ]\bar{G}^{0\,\pm}
\label{eqn-4.38}
\end{align}
which also hold for  $hD_{w_r}$. So, each such commuting adds factor $h$ and increases either $m$ or $p$ by $1$. However, each pair of  those factors are accompanied by  $\bar{G}^{0\,\pm}$, and therefore in this process  increment  of the power of $h$ by $1$ is ``paid'' by  increment   of $m$ with the factor $3/2$\,\footnote{\label{foot-5} Compare with the standard method, when the factor was $2$.}. On the other hand, commuting with $^t\!Q_1$, each increment  power of $h$ by $1$ is ``paid'' by increment of $(m+p)$ with the factor $1$.

\item
Also recall that  each factor $x_1$   is moved to the left, towards $\eth B$, also using   (\ref{eqn-4.38}) but for $x_1$.
In this commutator factor $h\eta_1$ is also gained and $m$ is increased by $1$   and also $\bar{G}^{0\,\pm}$ is applied so in the end  $m$ is increased by $2$. When $x_1$ reaches $\eth B$ the term  disappears for Dirichlet boundary condition, or this factor cancels with $B=h\partial_{x_1}$ for Neumann boundary condition (and factor $h$ is gained). process each increment  power of $h\eta_1$ by $1$ is ``paid'' by  increment   of $m$ with the factor $2$. Therefore (\ref{eqn-4.36})--(\ref{eqn-4.37}) has been proven.

\item
Then for
\begin{gather}
F_{t\to h^{-1}\tau} \Bigl(\bar{\chi}_T(t) \bar{G}^{\prime\,\pm}\eth   B  u_h^{0\,\pm} \,^t\!Q_1 \Bigr)
\label{eqn-4.39}
\end{gather}
formula (\ref{eqn-4.33}) holds with $l=1$, $p=0$  and $j=1,0$ for Dirichlet and Neumann boundary conditions respectively.

\item
Again  each factor $z_r$ and $hD_{w_r}$  with $r =2,\ldots, d$ is moved to the right towards $\updelta (z')$,  using commutator relations
(\ref{eqn-4.38}) for $\bar{G}^\pm$ rather than for $\bar{G}^{0\,\pm}$ and also
\begin{align}
&z_r \bar{G}^{\prime\,\pm} = \bar{G}^{\prime\,\pm} z_r -  \bar{G}^\pm [ \bar{\mathfrak{A}},z_r]\bar{G}^{\prime\,\pm}
\label{eqn-4.40}
\end{align}
which  holds also for $hD_{w_r}$. Observe that  that $p$ can increase but  again as in (a) increment  of the power of $h$ by $1$ is ``paid'' by  increment   of $m+p$ with the factor $3/2$\,\footnote{\label{foot-6} Indeed, it is sufficient to consider boundary value problem for ODE $(\tau-D^2)v=f$ with $f(x)= \int_{\gamma_+}\xi ^j (\tau-\xi^2)^{-l}e^{ix \xi }\,d\xi$. One can see easily that
\begin{gather*}
v(x)= \int_{\gamma_+}\Bigl( \xi ^j (\tau-\xi^2)^{-l-1} - \kappa \xi^{j'} \tau ^{(j-j')/2 -l } (\tau- \xi^2)^{-1} \Bigl) e^{ix \xi }\,d\xi
\end{gather*}
satisfies this equation and $v(0)=0$ for some constant $\kappa = \kappa_{\mathsf{D} jl}$ and  $j'\equiv j \mod 2$ and $j'=0,1$. Also $v'(0)=0$ with some other constant
$\kappa = \kappa_{\mathsf{N} jl}$.}.

\item
Also recall that  each factor $x_1$ which is to the left from $\bar{G}^{\prime\,\pm}\eth B$  is moved to the right, towards $\bar{G}^{\prime\,\pm}\eth B$ but instead of
(\ref{eqn-4.38}) we use
\begin{align}
&x_1\bar{G}_* ^\pm\ = \bar{G}_\mathsf{D}^\pm x_1 -  \bar{G}_\mathsf{D}^\pm [ \bar{\mathfrak{A}},x_1]\bar{G}_*^\pm,
\label{eqn-4.41}\\
&x_1\bar{G}_*^{\prime\,\pm} =   -  \bar{G}_\mathsf{D}^\pm [ \bar{\mathfrak{A}},x_1]\bar{G}_*^{\prime\,\pm}
\label{eqn-4.42}
\end{align}
with $*=\mathsf{D},\mathsf{N}$ (in particular, for $\bar{G}_\mathsf{D}$ (\ref{eqn-4.38}) holds).

Again, similarly to  (c) either $l+p$ is increased by $2$ and $j$ does not change, or $j$ is decreased by $1$ and $l+p$ is increased by $1$.

\item
Then we have $n=n'+j+k$ with $l+p=l' +2j$, $m=m'+2k$ and $l'+m' \le \frac{3}{2}n'+2$.
Those are ``the worst case  scenarios'' when we consider the main part of perturbation $R$, quadratic by $(x'-y')$ and linear by $x_1$. For the rest of perturbation increments of $l+p,m$ are relatively smaller. In the end we arrive to (\ref{eqn-4.35}).

\item
One can see easily that for the main term we have $l=m=1$, $p=0$ and $j+k=1$ with $j=0,1$ for Dirichlet and Neumann problems respectively and $W_{*}=\mathsf{const} $.
\end{enumerate}
\vspace{-\baselineskip}
\end{proof}

\begin{proposition}\label{prop-4.4}
In the framework of Proposition~\ref{prop-4.2}
\begin{multline}
F_{t\to h^{-1}\tau} \Bigl(\bar{\chi}_T(t) u_h^{1\,\pm} \,^t\!Q_1 \Bigr)\\
\sim
\sum_{n\ge 0, j\ge 0, k\ge 0} h^{-d+n}  \int T\hat{\bar{\chi}} \bigl( \frac{(\tau-\tau')T}{h} \bigr) \,d\tau'  \int d\xi'\int _{\gamma_+} d\xi_1 \\
\times
e^{ih^{-1}((x_1+y_1)\xi_1+\langle x'-y',\xi'\rangle)}
F_{n,j,k}(w, \xi',\xi_1,\tau')[q (w,\xi')]
\label{eqn-4.43}
\end{multline}
for $\tau\in \mathbb{C}_\mp$  where $F_{n,j,k}$ are sums of the terms
\begin{gather}
\xi_1^j \eta_1^k W _{n,l,j,k}(w,\xi ',\tau ) \bigl(\tau -a(y',\xi _1,\xi ')\bigr)^{-l}
\label{eqn-4.44}
\end{gather}
with $1\le l\le \frac{3}{2}n +j+k+1$ and uniformly smooth $W _{n,l,j,k}$ and also
\begin{multline}
F_{t\to h^{-1}\tau} \Bigl(\bar{\chi}_T(t) u_h^{1} \,^t\!Q_1 \Bigr)\\
\sim
\sum_{n\ge 0, j\ge 0, k\ge 0} h^{-d+n}  \int T\hat{\bar{\chi}} \bigl( \frac{(\tau-\tau')T}{h} \bigr) \,d\tau'  \int d\xi\\
\times
e^{ih^{-1}((x_1+y_1)\xi_1+\langle x'-y',\xi'\rangle)}
\mathcal{F}_{n,j,k}(w, \xi',\xi_1,\tau')[q (w,\xi')],
\label{eqn-4.45}
\end{multline}
where $\mathcal{F}_{n,j,k}(w, \xi',\xi_1,\tau') = F_{n,j,k}(w, \xi',\xi_1,\tau'-i0) - F_{n,j,k}(w, \xi',\xi_1,\tau'+i0)$.
\end{proposition}

\begin{proof}
(\ref{eqn-4.43})--(\ref{eqn-4.44})  follows  from Proposition~\ref{prop-4.3}  by calculating residues at zeroes of $(\tau -a(x,\xi_1,\xi'))$ and  $(\tau -a(x,\eta_1,\xi'))$ in (\ref{eqn-4.33}) and (\ref{eqn-4.43}).

Next we observe that in (\ref{eqn-4.43}) we can replace integral along $\gamma_+$ by integral along $\mathbb{R}$. Then (\ref{eqn-4.45}) follows trivially.
\end{proof}

\begin{proposition}\label{prop-4.5}
Let $\xi$-microhyperbolicity condition be fulfilled at energy level $\tau$ and let
\begin{gather}
{\nu}(x)+{\nu}(y) \le \sigma \ell
\label{eqn-4.46}\\
\shortintertext{with}
\ell(x,y)\le  h^{1/3+\delta}, \quad \sigma = h^{\frac{1}{2}+\delta }\ell^{-\frac{1}{2}}.
\label{eqn-4.47}
\end{gather}
Let $T=T^* \ge  C_0\ell (x,y)$. Then for any $T\colon T^*\le T\le \epsilon$
\begin{enumerate}[wide,label=(\roman*),  labelindent=0pt]
\item
As $d\ge 3$ estimate
\begin{gather}
|e^{1,\mathsf{T}}_{T,h} (x,y,\tau) -[\mp]  e_h^{0,\mathsf{W}}(x,\tilde{y},\tau)|\le Ch^{1-d}
\label{eqn-4.48}
\end{gather}
with sign $[\mp ]$ for Dirichlet and Neumann boundary conditions respectively.
\item
As $d=2$ estimate
\begin{gather}
|e^{1,\mathsf{T}}_{T,h} (x,y,\tau) -[\mp]  e_h^{0,\mathsf{W}}(x,\tilde{y},\tau)- e_{h,\mathsf{corr}} (x,y,\tau)|\le Ch^{-1}
\label{eqn-4.49}
\end{gather}
with $e_{h,\mathsf{corr}} (x,y,\tau)$ given by \textup{(\ref{eqn-4.54})} below.
\end{enumerate}
\end{proposition}

\begin{proof}
\begin{enumerate}[wide,label=(\alph*),  labelindent=0pt]
\item
Consider first $\ell \le h^{\frac{1}{2}-\delta}$. Let us take $\rho =\min ( (h/\ell)^{\frac{1}{2}}h^{-\delta} ,\, C_2)$.  Then $\rho\ge C_0\ell$,  $\sigma=\rho^{\frac{1}{2}}$ and
\begin{gather}
\sigma \ell^2 \le h^{1+\delta} \qquad \text{as\ \ } \ell\le h^{\frac{3}{7}+\delta}
\label{eqn-4.50}
\end{gather}
and the method of successive approximations works. Let us take take $\rho$-admissible partition of unity by $\zeta'$ in $\rho$-vicinity of $\Sigma(w,\tau)$.
In this case $\rho \ge \ell$ and $\sigma=\rho^{\frac{1}{2}}$.
Moreover, as $\rho=C_2$ it covers the whole domain
$\{\zeta'\colon b(w,\zeta')<2\tau\}$ and as $\rho < C_2$ and therefore $\ell \ge h^{1-2\delta}$ and also $\rho^2 \ell \ge h^{ 1-2\delta}$ we can use the fact that

\begin{claim}\label{eqn-4.51}
If $\rho \ell \ge h^{1-\delta}$, $\rho \ge C_0\ell^2$, (\ref{eqn-3.11}) is fulfilled  and $Q_1$ is an operator with the symbol equal $0$ in $\rho$-vicinity of $\Sigma(w,\tau)$ and $\ell\ge h^{1-\delta}$ then
\begin{gather*}
e^{1,\mathsf{T}}_h(x,y,\tau) \,^t\!Q_{1,z} =O(h^s)
\end{gather*}
\end{claim}
\vspace{-\baselineskip}
which follows from Section~\ref{sect-3}. Therefore we can take cut-off operator $Q_1=I$.

Then the main term of the final expression equals to $e^{0,\mathsf{W}}_h(x,\tilde{y},\tau)$ and is $\asymp h^{-d}(h\ell^{-1})^{(d+1)/2}$ and $n$-th term does not exceed
\begin{gather}
Ch^{-d-\delta'}\bigl(\frac{h}{\ell}\bigr)^{(d+1)/2} \times (\rho^{\frac{1}{2}}\ell^2 h^{-1})^{n-1}
\label{eqn-4.52}
\end{gather}
due to the same microlocal arguments as above. One can see easily that expression  (\ref{eqn-4.52}) is $O(h^{1-d+\delta''})$ as $d\ge 2$ and $n\ge 3$ and $\ell \le h^{\frac{1}{3}+\delta}$. Furthermore, if we consider terms $O(\ell^2)$ in $R$ then the corresponding second term does not exceed
\begin{gather}
Ch^{-d-\delta'}\bigl(\frac{h}{\ell}\bigr)^{(d+1)/2} \times  \ell^3 h^{-1})^{n-1} = O(h^{1-d+\delta''}).
\label{eqn-4.53}
\end{gather}
Therefore we need to consider only the second term in the successive approximations. It is equal to
\begin{multline}
 [\mp] e_{h,\mathsf{corr}} (x,y,\tau)\\
\coloneqq  -\frac{1}{2}(2\pi h)^{-d} \int _{\Sigma (w,\tau)} \lambda (w',\xi') (x_1+y_1) e^{ih^{-1}\langle x-\tilde{y},\xi \rangle}\,d\xi :d_\xi a(w,\xi)
\label{eqn-4.54}
\end{multline}
with
\begin{gather}
\lambda(w',\xi')= a_{x_1} (x_1,w',\xi)|_{x_1=\xi_1=0}.
\label{eqn-4.55}
\end{gather}
Indeed, one can see easily that the second term in the successive approximation for
$[\mp] F_{t\to h^{-1}\tau} \bigl(\bar{\chi}_T(t)u^{1\,\pm}_h (x,y,t) \bigr)$ is equal to
\begin{multline*}
\pm 2 h^3 (2\pi h)^{-1-d}  \int _{\gamma_+\times \mathbb{R}} \lambda (w',\xi ') \xi_1 (\tau -a(x,\xi))^{-3} e^{i\langle x-\tilde{y},\xi \rangle} \, d\xi \\
= \mp \frac{i}{2}h^2  (2\pi h)^{-1-d}  \int _{\gamma_+\times \mathbb{R}} \lambda (w',\xi ') (x_1+y_1)  (\tau -a(x,\xi))^{-2} e^{i\langle x-\tilde{y},\xi \rangle}  \qquad \mp  \Im \tau <0;
\end{multline*}
then the second term in decomposition for $[\mp] F_{t\to h^{-1}\tau} \bigl(\bar{\chi}_T(t)u^{1}_h (x,y,t) \bigr)$
is equal to
\begin{gather*}
-\frac{1}{2}h(2\pi h)^{-d} \int \lambda (w',\xi') (x_1+y_1) e^{ih^{-1}\langle x-\tilde{y},\xi \rangle}\updelta '(\tau-a(w,\xi))\,d\xi
\end{gather*}
which implies (\ref{eqn-4.55}).

\item
Due  to stationary phase integral  in \ref{eqn-4.54}) does not exceed $Ch^{-d} \bigl(\frac{h}{\ell}\bigr)^{(d-1)/2}$,
and  expression (\ref{eqn-4.54}) is $O(h^{1-d})$ as $d\ge 3$, $x_1+y_1\le \sigma \ell$.

Due to Corollary~\ref{cor-3.9} as $d\ge 3$ we need to consider only this case $\ell\le h^{\frac{1}{2}-\delta}$. Therefore Statement~(i)  has been proven.

\item
Let  $d=2$. Then  we need to consider also $h^{\frac{1}{2}-\delta}\le \ell \le h^{\frac{1}{3}+\delta}$ and with selected $\rho = (h/\ell)^{\frac{1}{2}}h^{-\delta}$
condition $\sigma \ell^2 \le h^{1+\delta}$ may fail. We need to consider lesser $\rho$. Let us pick up $\rho = h^{1-\delta}\ell^{-1}$.
Then $\rho\ge C_0\ell^2$ and $\sigma \ell^2 \le h^{1+\delta}$ as $\ell\le h^{\frac{1}{3}-\delta}$.

However condition  $\rho \ge C_0\ell$ may fail and then we need to take care of (\ref{eqn-3.11})  at each point   of $ \Sigma (w,\tau)\cap \{\xi_1=0\}$.

 The good news is that this set  consists of two points $\bar{\xi}^{\prime\pm}$ and we do not need $\rho^2 \ell \ge h^{1-\delta}$ because we can deal with each of these two points separately. Therefore we can replace  $\rho$-admissible operator $Q_1$ with the symbol, equal $1$ in $\rho$-vicinity of $\bar{\xi}^{\prime\pm}$ by operator with the symbol equal $1$ in $\epsilon$-vicinity of this point. Observing that the required gauge transformations are
 $\xi_2\mapsto \xi_2 - h^{-1}\alpha_\pm z_2$, that are multiplications by $e^{ih^{-1}\alpha_\pm z_2^2/2}$ which do not affect $e(x,y,\tau)$ as $z_2=\frac{1}{2}(x_2-y_2)$. Therefore again we can take $Q_2=I$.

 Then  $n$-th term does not exceed (\ref{eqn-4.52}) which is $O(h^{-1+\delta''})$ as $n\ge 3$ in current settings. Again, if we consider $O(\ell^2)$  terms  in $R$ then the corresponding second term does not exceed (\ref{eqn-4.53}). Thus again we are left with expression (\ref{eqn-4.54}) but this time we cannot claim that it is $O(h^{-1})$ even for
 $\ell \le h^{\frac{1}{2}+\delta}$ and $\rho = h^{1-\delta'}\ell^{-1}$. Therefore Statement~(ii)  has been proven.
\end{enumerate}
\end{proof}

\chapter{Geometric optics method}
\label{sect-5}

\section{Constructing solution}
\label{sect-5.1}

\begin{proposition}\label{prop-5.1}
Let
\begin{gather}
{\nu}(x)+{\nu}(y)\ge C_0 \ell^2(x,y)
\label{eqn-5.1}
\end{gather}
and $\ell(x,y)\le \epsilon$. Then there exists a single generalized Hamiltonian billiard  of the length $\le c\epsilon$ with at least one reflection from $\partial X$, connecting
$y$ and $x$; it has exactly one reflection and the incidence angle is $\asymp \bigl({\nu}(x)+{\nu}(y)\bigr)\ell^{-1}(x,y)$ (so, it is a standard  Hamiltonian billiard).
\end{proposition}

\begin{proof}
The proof is obvious.
\end{proof}

\enlargethispage{\baselineskip}

\begin{remark}\label{remark-5.2}
Assumption (\ref{eqn-5.1})  is essential. Indeed, if $\partial X$ is strongly concave\footnote{\label{foot-7} With respect to Hamiltonian trajectories.} then for ${\nu}(x)+{\nu}(y)\le C_0 \ell^2(x,y)$ there could be multiple reflections and if $\partial X$ is strongly convex\footref{foot-7} then the incidence angle may be $0$. In the former case there could be also multiple  reflections  and multiple billiard rays from $x$ to $y$ on the same energy level $\tau$ while in the latter case there could be no rays at all.
\end{remark}

\begin{proposition}\label{prop-5.3}
\begin{enumerate}[wide,label=(\roman*),  labelindent=0pt]
\item
Let $A$ be $\xi$-microhyperbolic on the energy level ${\tau=a(y,\theta)}$ and
\begin{gather*}
a(x,0)<\tau -\epsilon.
\end{gather*}
Let $\varphi ^0(x,y,\theta)$ be a solution of the stationary eikonal equation   \textup{(\ref{eqn-2.7})} satisfying
\textup{(\ref{eqn-2.8})} and \textup{(\ref{eqn-2.9})}. Then
\begin{multline}
|\partial_{x} ^\alpha \partial_y^\beta \partial_\theta^\gamma \varphi^0(x,y,\theta) |\le C_{\alpha\beta\gamma}
\left\{\begin{aligned}
&T &&\alpha=\beta =0,\\
&1 && |\alpha|+|\beta|\ge 1
\end{aligned}\right.\\
\forall x,y \colon |x-y| \le T
\label{eqn-5.2}
\end{multline}
provided $T\le \epsilon$ with sufficiently small constant $\epsilon>0$.

\item
Let $\varphi  (x,y) $ be another solution of the same equation satisfying
\begin{gather}
\varphi  |_{x_1=0}= \varphi^0 |_{x_1=0},\qquad \partial_{x_1}\varphi  |_{x_1=0}= - \partial_{x_1}\varphi^0 |_{x_1=0}.
\label{eqn-5.3}\\
\shortintertext{Assume that }
\sigma = |  \partial_{x_1}\varphi  |_{x_1=0}|\ge C_0T.
\label{eqn-5.4}
\end{gather}
Then as $|x-y|\le T$ the following inequalities hold:
\begin{gather}
|\partial_x ^{\alpha}\partial_y^\beta \partial_\theta^\gamma \varphi  |\le C_{\alpha\beta\gamma}
\left\{\begin{aligned}
&T &&\alpha=\beta =0,\\
&1 && \alpha_1\le 1,\\
&\sigma ^{3-\alpha_1-|\alpha|-|\beta|-|\gamma|}\qquad &&\alpha_1\ge 2.
\end{aligned}\right.
\label{eqn-5.5}
\end{gather}
\item
Furthermore,
\begin{gather}
|\partial_{x}^{\alpha}\partial_y^\beta \partial_\theta^\gamma \varphi  |\le C_{\alpha\beta\gamma}
\sigma ^{4-\alpha_1-|\alpha|-|\beta|-|\gamma|}\qquad \alpha_1\ge 2,\ |\gamma'|\ge 1.
\label{eqn-5.6}
\end{gather}
\end{enumerate}
\end{proposition}

\begin{proof}
\begin{enumerate}[wide,label=(\alph*),  labelindent=0pt]
\item
Consider first $\varphi=\varphi^0$. Then the left-hand expression of (\ref{eqn-5.2}) does not exceed $C_{\alpha\beta\gamma}$ and we need to consider only
$\alpha=\beta=0$. Applying $\partial _\theta^\gamma$ to  eikonal equation we see that  $\frac{d}{dt}\partial^\gamma_\theta \varphi$ is bounded
where here and below
\begin{gather}
\dfrac{d\ }{dt} \coloneqq \bigl(\partial _t -\sum_k a^{(k)}(x, \nabla_x\varphi )\bigr).
\label{eqn-5.7}
\end{gather}
Since $|\partial^\gamma_\theta \varphi^0(x,y,t,\theta)|\le C_\gamma T$ as $t=0$ and $|x-y|\le T$ we conclude that the same is true for $|t|\le T$.

\item
First, we provide this estimate as $x_1=0$; from  (\ref{eqn-5.2}) and (\ref{eqn-5.3})  we conclude that (\ref{eqn-5.5})  holds as $\alpha_1=0,1$.
Consider
\begin{multline}
\varphi_{x_1}^2 = a(y,\theta)+ b(x, \nabla_{x'}\varphi),\\
b(x, \nabla_{x'}\varphi) \coloneqq  -V -\sum_{j\ge 2} g^{jk}(\varphi _{x_j} -V_j)(\varphi _{x_k} -V_k).
\label{eqn-5.8}
\end{multline}
Differentiating once by $x_1$ we get
\begin{gather}
2\varphi_{x_1}\varphi_{x_1x_1} = b_{x_1} (x, \nabla_{x'}\varphi) + \sum_{k\ge 2} b^{(k)}(x,\nabla_{x'} )\varphi_{x_kx_1}
\label{eqn-5.9}
\end{gather}
which in virtue of (\ref{eqn-5.5}) with $\alpha_1\le 1$ does not exceed $C$ and therefore
\begin{gather}
|\varphi_{x_1x_1}| \le C\sigma^{-1}.
\label{eqn-5.10}
\end{gather}
Further, applying to (\ref{eqn-5.8}) $\partial_{x}^{\alpha-\upvarepsilon}\partial_y^\beta \partial_\theta^\gamma$ with $\upvarepsilon =(1,0,\ldots,0)$ and $\alpha_1=2$ and plugging $x_1=0$ we see that
$2\varphi_{x_1}\varphi_{\alpha\beta\gamma}$\,\footnote{\label{foot-8} Here and below we use notation  $\varphi_{\alpha\beta\gamma}= \partial_{x,t}^{\alpha}\partial_y^\beta \partial_\theta^\gamma \varphi$.} is a linear combination with bounded coefficients of
$\varphi_{\alpha^1\beta^1\gamma^1}$ with $|\alpha^1|+|\beta^1|+|\gamma^1|<|\alpha|+|\beta|+|\gamma|$ plus bounded terms, and then by induction by
$|\alpha|+|\beta|+|\gamma|$ we arrive to (\ref{eqn-5.5}) for $\alpha_1=2$.

Consider induction by $\alpha_1$. Assume that (\ref{eqn-5.5}) is proven for lesser $\alpha_1$  and for  arbitrary $\alpha ',\beta,\gamma $.

Applying to (\ref{eqn-5.8}) $\partial_{x,t}^{\alpha-\upvarepsilon}\partial_y^\beta \partial_\theta^\gamma$ and plugging $x_1=0$ we get
linear combinations with bounded coefficients of products
\begin{gather}
\prod _{1\le j\le m} \varphi_{\alpha^j\beta^j\gamma^j}
\label{eqn-5.11}
\end{gather}
where $\sum_j |\alpha^j|\le |\alpha| +m-1$, $\sum_j \beta^j\le \beta$, $\sum_j\gamma^j\le \gamma$. These terms appear when
\begin{enumerate}[ label=(\Alph*),  labelindent=0pt]
\item
We differentiate $b(x,\nabla_{x'}\varphi)$, in this case $\sum_j \alpha^j_1<\alpha_1$, $\sum_j |\alpha^j|\le |\alpha|$ and these terms due to induction assumption are
$O(\sigma^{4-\alpha_1 -|\alpha|-|\beta|-|\gamma})$.

\item
We differentiate $\varphi_{x_1}^2$ and  $m=2$, $|\alpha^1|+|\alpha^2|=|\alpha|+1$, $\alpha^1_1+\alpha_1^2=\alpha_1+1$, $\alpha_1^j<\alpha$ and these terms due to induction also are  $O(\sigma^{4-\alpha_1 -|\alpha|-|\beta|-|\gamma|})$.
\item
We differentiate $\varphi_{x_1}^2$ and  $m=1$, $\alpha^1_1 = \alpha_1$, $\alpha_1^1<\alpha_1$. Then we apply induction by $|\alpha|+|\beta|+|\gamma|$, and these terms due to induction assumption are $O(\sigma^{4-\alpha_1 -|\alpha|-|\beta|-|\gamma|})$ as well.
\end{enumerate}

Then dividing by $\varphi_{x_1}$ and we get  $O(\sigma^{4-\alpha_1 -|\alpha|-|\beta|-|\gamma|})$. We need base of the induction in (C). However as $|\alpha|=\alpha_1$,
$\beta=\gamma=0$ terms of type (C) do not appear.

\item
We need to expand these estimates to $x_1>0$. Applying to equation (\ref{eqn-5.8}) $\partial_{x,t}^{\alpha}\partial_y^\beta \partial_\theta^\gamma$ with $|\alpha|+|\beta|+|\gamma|=1$
we see that $\dfrac{d\ }{dt}\varphi_{\alpha\beta\gamma}$ is bounded and therefore in this case we extend   (\ref{eqn-5.5}) from $x_1=0$ to $x_1>0$.

Next, for $|\alpha|+|\beta|+|\gamma|=2$ we get that $\dfrac{d\ }{dt}\varphi_{\alpha\beta\gamma}$ does not exceed $C(S+1)$ with
\begin{gather}
S=\sum_{\alpha,\beta,\gamma\coloneqq |\alpha|+|\beta|+|\gamma|=2} |\varphi_{\alpha\beta\gamma}|^2
\label{eqn-5.12}
\end{gather}
and therefore $|\dfrac{dS^{1/2}}{dt}|\le C(S+1)$; since $S|_{x_1=0}=O(\sigma^{-1})$ due to (\ref{eqn-5.10}) we conclude that for $T\le \epsilon \sigma$ estimate $S^{1/2}\le C\sigma ^{-1}$ holds. Then
\begin{gather}
|\varphi_{\alpha\beta\gamma}|\le C\sigma^{-1}\qquad \text{for\ \ } |\alpha|+|\beta|+|\gamma|=2.
\label{eqn-5.13}
\end{gather}
But then, taking $\alpha_1\le 1$ we get that $\dfrac{d\ }{dt}\varphi_{\alpha\beta\gamma}$ does not exceed $C\sigma^{-1}(S^{1/2}+1)$ with $S$ defined by
(\ref{eqn-5.12}) with summation restricted to $\alpha_1\le 1$ and then  due to (\ref{eqn-5.5}) for $x_1=0$ we conclude that
\begin{gather}
|\varphi_{\alpha\beta\gamma}|\le C\qquad \text{for\ \ } |\alpha|+|\beta|+|\gamma|=2,\ \alpha_1\le 1.
\label{eqn-5.14}
\end{gather}
And then, taking $\alpha_1=0$ we get that $\dfrac{d\ }{dt}\varphi_{\alpha\beta\gamma}$ does not exceed $C$  and then  due to (\ref{eqn-5.5}) for $x_1=0$  we conclude that
\begin{gather}
|\varphi_{\alpha\beta\gamma}|\le CT\qquad \text{for\ \ }\alpha=\beta=0, \ |\gamma|=2.
\label{eqn-5.15}
\end{gather}
So, (\ref{eqn-5.5}) holds as $|\alpha|+|\beta|+|\gamma|\le 2$, $x_1>0$.

\item
Assume that for $|\alpha|+|\beta|+|\gamma|< p$ estimate (\ref{eqn-5.5}) has been proven. Applying to equation (\ref{eqn-5.8}) $\partial_{x,t}^{\alpha}\partial_y^\beta \partial_\theta^\gamma$ with $|\alpha|+|\beta|+|\gamma|=p\ge 3$ we again get a linear combination with bounded coefficients of (\ref{eqn-5.11}) products (plus bounded terms) where this time $\sum_j |\alpha^j|\le |\alpha|+m$. Then, as $\alpha_1=q$, $|\alpha|+|\beta|+|\gamma|=p$ we see from the same analysis as in (ii) that
\begin{gather*}
|\frac{d\ }{dt}\varphi_{\alpha\beta\gamma}|\le C\sigma^{-1}S ^{1/2} + C\sigma^{3-p-q}+C
\end{gather*}
and then (\ref{eqn-5.5}) is extended from $x_1=0$ to $x_1>0$, but for $\alpha=\beta=0$, when we proved so far that $\partial_\theta^\gamma=O(1)$. But then
$|\frac{d\ }{dt}\varphi_{\gamma}|\le C$ and (\ref{eqn-5.5}) is again extended from $x_1=0$ to $x_1>0$.

\item
To prove Statement (iii) we again start from estimates at $x_1=0$. Then, exactly like in (b) we prove see that
\begin{gather*}
\varphi_{x_1}\varphi_{x_1x_1\theta_j} + \varphi_{x_1\theta_j}\varphi_{x_1x_1} =O(1)
\end{gather*}
and since $ \varphi_{x_1\theta_j}=- \varphi^0_{x_1\theta_j}=O(\sigma)$ as $j\ge 2$, and we know that  $\varphi_{x_1x_1}=O(\sigma ^{-1})$, we conclude that
$\varphi_{x_1x_1\theta_j} =O(\sigma^{-1})$. Again, like in (b), using double induction by $|\alpha|+|\beta|+|\gamma|$ and $\alpha_1$ we prove (\ref{eqn-5.6}) as $x_1=0$.

Finally, like in (c) and (d) we expand this estimate to $x_1>0$. We leave easy details to the reader.
\end{enumerate}
\vspace{-\baselineskip}
\end{proof}

\begin{proposition}\label{prop-5.4}
Let $\varphi ^0(x,y,\theta)$ and $\varphi  (x,y, \theta)$ be defined in Proposition~\ref{prop-5.3}(i) and (ii) respectively.
Consider  asymptotic solution
\begin{gather}
u^0_h(x,y,t)\sim (2\pi h)^{-d}\int e^{i\Phi ^0(x,y,t,\theta)} \sum_{n\ge 0}  B^0_n(x,y,t,\theta)h^n\,d\theta
\label{eqn-5.16}\\
\intertext{where
$\Phi^0 (x,y,t,\theta)= \varphi^0 (x,y,\theta)+ta(y,\theta)$ and}
\Phi  (x,y,t,\theta)= \varphi  (x,y,\theta)+ta(y,\theta),
\label{eqn-5.17}
\end{gather}
and $B^0_n$ are uniformly smooth and satisfying corresponding transport equations with initial conditions such that
$u^0_h(x,y,0) = \updelta (x-y)$:
\begin{gather}
|\partial_{x,t} ^\alpha \partial_y^\beta \partial_\theta^\gamma B^0_n (x,y,t, \theta) |\le C_{\alpha\beta\gamma} \quad
\forall x,y, t\colon |x-y|+|t|\le T\  \forall \alpha,\beta,\gamma.
\label{eqn-5.18}
\end{gather}
Consider  \underline{formal} asymptotic solution
\begin{gather}
U^1_h(x,y,t)=(2\pi h)^{-d}\int e^{i\varphi (x,y,t,\theta)} \sum_{n\ge 0}  B_n(x,y,t,\theta)h^n \,d\theta
\label{eqn-5.19}
\end{gather}
where $B_n$ satisfy corresponding transport equations and \underline{one} of the boundary conditions
\begin{align}
&B_n = -B^0_n    &&\text{as\ \ } x_1=0,
\label{eqn-5.20}\\[3pt]
&\varphi_{x_1} B_n -i\partial_{x_1} B_{n-1} =  \varphi_{x_1} B^0_n +i\partial_{x_1} B^0_{n-1}=0
&&\text{as\ \ } x_1=0,
\label{eqn-5.21}
\end{align}
corresponding to $u^1_h=-u^0_h$ as $x_1=0$ and $\partial_{x_1}u^1_h=-\partial_{x_1} u^0_h$  as $x_1=0$ respectively.
Then the following inequalities hold:
\begin{gather}
|\partial_{x,t}^{\alpha}\partial_y^\beta \partial_\theta^\gamma B_n |\le C_{n\alpha\beta\gamma}
\left\{\begin{aligned}
&1 + T\sigma^{-1-|\alpha|- |\beta|-|\gamma|} &&  \alpha_1=n=0,\\
&\sigma^{-3n-\alpha_1-|\alpha|-|\beta|-|\gamma|}\qquad &&\alpha_1+n\ge 1,
\end{aligned}\right.
\label{eqn-5.22}\\
\intertext{and if $|\gamma|\ge 1$}
|\partial_{x,t}^{\alpha}\partial_y^\beta \partial_\theta^\gamma B_n |\le C_{n\alpha\beta\gamma}
\left\{\begin{aligned}
&1 + T\sigma^{-|\alpha|- |\beta|-|\gamma|} &&  \alpha_1=n=0,\\
&\sigma^{1-3n-\alpha_1-|\alpha|-|\beta|-|\gamma|}\qquad &&\alpha_1+n\ge 1.
\end{aligned}\right.
\label{eqn-5.23}
\end{gather}
\end{proposition}

\begin{proof}
\begin{enumerate}[wide,label=(\alph*),  labelindent=0pt]
\item
Observe that the transport equation is
\begin{gather}
\Bigl(\frac{d\ }{dt} +f\Bigr) B_n =\mathcal{L} B_{n-1},
\label{eqn-5.24}
\end{gather}
where $f$ is a linear combination with the smooth coefficients of second derivatives of $\Phi$,
$\mathcal{L}=\mathcal{L}(x,D_x)$ is a second order differential operator with the smooth coefficients, and with the coefficient $1$ at $D_{x_1}^2$, and
$B_{-1}=0$. In virtue of (\ref{eqn-5.5})
\begin{gather}
|D^\alpha_{x,t} D^\beta_yD_\theta^\gamma f|\le \sigma^{-1-\alpha_1-|\alpha|-|\beta|-|\gamma|}.
\label{eqn-5.25}
\end{gather}

Consider first $n=0$. Then equation (\ref{eqn-5.24}) has a right-hand expression $0$ and boundary condition (\ref{eqn-5.21}) becomes
$B_0 =\mp B_0^0$ as $x_1=0$. Let us establish first (\ref{eqn-5.22})  as $x_1=0$. As $\alpha_1=0$ these estimates follow from the above boundary condition  and
(\ref{eqn-5.18}). Consider transport equation (\ref{eqn-5.24}):
\begin{gather}
\varphi_{x_1} B_{0,x_1}+ \sum_{k\ge 2}b ^{(k)}(x,\nabla_{x'} \varphi) B_{0,x_k} -\frac{1}{2} B_{0,t} +\frac{1}{2} f B_0 =0
\label{eqn-5.26}
\end{gather}
and set $x_1=0$. Then all terms in (\ref{eqn-5.26}) are smooth, except the first and the last one, and the latter satisfies
\begin{gather*}
|\partial^{\alpha -\upvarepsilon}_{x,t}\partial^\beta_y\partial^\gamma_\theta (f\, B_0)|\le C\sigma^{-|\alpha| -|\beta|-|\gamma|}\qquad \alpha_1=1
\end{gather*}
due to (\ref{eqn-5.21}) and smoothness of $B_0$ at $x_1=0$.  Sinnce $\partial^{\alpha }_{x,t}\partial^\beta_y\partial^\gamma_\theta \varphi$ is obtained by division by $\varphi_{x_1}$ we arrive to (\ref{eqn-5.22}) as $n=0$, $\alpha_1=0$.

Next we apply a double induction by $\alpha_1$ and $|\alpha|+|\beta|+|\gamma|$ exactly as in the proof of Proposition~\ref{prop-5.3}, Part (b) with the following  modifications:
\begin{enumerate}[label=(\Alph*)]
\item
We use transport equation (\ref{eqn-5.24}) rather than eikonal equation.
\item
We observe that $\partial^{\alpha -\upvarepsilon}_{x,t}\partial^\beta_y\partial^\gamma_\theta $, applied to the first  term in (\ref{eqn-5.26}) equals to
$\varphi_{x_1} \varphi_{\alpha\beta\gamma}$ plus a linear combination of
\begin{gather}
\prod_ {1\le j\le m} \varphi_{\alpha^j \beta^j\gamma^j} \partial^{\alpha^0}_{x,t}\partial^{\beta^0}_y\partial^{\gamma^0}_\theta B_0
\label{eqn-5.27}
\end{gather}
with $m=1$ and
$\sum_j \alpha^j =\alpha+ \upvarepsilon$, $\sum_j  \beta^j= \beta$, $\sum_j \gamma^j =\gamma$\,\footnote{\label{foot-9} With summation over  $0\le j\le m$} and\\ $|\alpha^0|+|\beta^0|+|\gamma^0|<|\alpha|+|\beta|+|\gamma|$.

\item
We  observe that $\partial^{\alpha -\upvarepsilon}_{x,t}\partial^\beta_y\partial^\gamma_\theta $, applied to the second term in (\ref{eqn-5.26}) also is a linear combination  with bounded coefficients of (\ref{eqn-5.27}) with $\sum_j \beta^j=\beta$, $\sum_j \gamma^j=\gamma$, $\sum_j  |\alpha^j| \le  |\alpha|+m$, $\sum_j  \alpha^j_1 <\alpha_1$\,\footref{foot-9} and $|\alpha^j|+|\beta^j|+|\gamma^j| \ge 2$ for $j\ge 1$; it is possible that $m=0$ here.
\end{enumerate}
We leave simple but tedious arguments to the reader.

\item
Then we extend these estimates to $x_1>0$, $T\le \sigma$. We apply arguments of Parts (c), (d) of the proof of Proposition~\ref{prop-5.3}. First, transport equation (\ref{eqn-5.24}) and boundary condition $B_0=B^0_0$ as $x_1=0$ imply  that $|B_0|\le C$ for $T\le C_0^{-1}\sigma$.

Then applying $\partial_{x,t}^\alpha \partial_y^\beta\partial_\theta^\gamma$ with $|\alpha|+|\beta|+|\gamma|=1$ to transport equation  (\ref{eqn-5.24})  we get
\begin{gather}
|\frac{d\ }{dt} B_{0,\alpha\beta\gamma}|\le C\sigma^{-1}S^{\frac{1}{2}} +C \sigma^{-3}, \qquad S\coloneqq \Bigl(\sum_{\alpha,\beta,\gamma} |B_{0,\alpha\beta\gamma}|^2\Bigr)^{\frac{1}{2}}
\label{eqn-5.28}
\end{gather}
due to estimates $|\partial_{x,y,\theta} ^2 \varphi|\le C\sigma^{-1}$ and $|\partial_{x,t, y,\theta} f|\le C\sigma^{-3}$
($B_{n,\alpha\beta}\coloneqq \partial_{x,t}^\alpha \partial_y^\beta\partial_\theta^\gamma B_n$) and then
for $T\le C_0^{-1}\sigma$ we get from this estimate and estimate $|B_{0,\alpha\beta\gamma}|\le C\sigma^{-2}$ at $x_1=0$ to estimate $ |B_{0,\alpha\beta\gamma}|\le C\sigma^{-2}$. So far the only restriction is $|\alpha|+|\beta|+|\gamma|=1$--here and in summation in the definition of $S$.

However, if we add an extra restriction $\alpha_1=0$, then due to this estimate (without restriction) and estimates $|\partial_{x,y, \theta} ^2 \varphi|\le C$ and
$|\partial_{x,y,t,\theta} f|\le C\sigma^{-2}$ if there is only one derivative with respect to $x_1$, we arrive to
\begin{gather*}
|\frac{d\ }{dt} B_{0,\alpha\beta\gamma}|\le C\sigma^{-1}S^{\frac{1}{2}} +C \sigma^{-2}
\end{gather*}
where $S$ is defined by (\ref{eqn-5.28}) with summation under the same restriction. Then due to this estimate and estimate $|B_{0,\alpha\beta\gamma}|\le C$ as $x_1=0$ and $\alpha_1=0$ we arrive to the estimate $|B_{0,\alpha\beta\gamma}|\le C + CT \sigma^{-2}$.

We apply induction by $q\coloneqq |\alpha|+|\beta|+|\gamma|$. Assuming that for lesser values it is proven, we due to this induction assumption and estimates (\ref{eqn-5.5}) and (\ref{eqn-5.25}) arrive to
\begin{gather}
|\frac{d\ }{dt} B_{0,\alpha\beta\gamma}|\le C\sigma^{-1}S^{\frac{1}{2}} +C \sigma^{p-1-2|\alpha|+|\beta|+|\gamma|}, \quad S\coloneqq \Bigl(\sum_{\alpha,\beta,\gamma} |B_{0,\alpha\beta\gamma}|^2\Bigr)^{\frac{1}{2}}
\label{eqn-5.29}\\
\intertext{under restrictions}
|\alpha|+|\beta|+|\gamma|=q,\quad \alpha_1\le |\alpha|-p
\label{eqn-5.30}\\
\intertext{with $p=0$ and then we recover estimate}
|B_{0,\alpha\beta\gamma}|\le C\sigma^{p-2|\alpha|-|\beta|-|\gamma|}
\label{eqn-5.31}
\end{gather}
under these  restrictions. Based on it we arrive to (\ref{eqn-5.29})  then to (\ref{eqn-5.31}) now under restriction (\ref{eqn-5.30}) with $p=1$; again due to (\ref{eqn-5.31}) at $x_1=0$, and so on for $p=2,\ldots$ until we reach $p=|\alpha|$ but in the latter case we use $|B_{0,\alpha\beta\gamma}|\le C_{\alpha\beta\gamma}$ at $x_1=0$ as $\alpha_1=0$ and thus we achieve some improvement over (\ref{eqn-5.31}): namely we get  (\ref{eqn-5.22}) as $n=\alpha_1=0$. We leave easy but tedious details to the reader.

\item
Next we apply induction by $n\ge 1$ to estimate $B_{n,\alpha\beta\gamma}$ at $x_1=0$.  However,  first we consider $n=1$ to establish the pattern. Observe first that under condition (\ref{eqn-5.20}) $B_n=-B^0_n$ and therefore
\begin{gather}
|B_{n,\alpha\beta\gamma}|\le C_{\alpha\beta\gamma}\qquad \text{at\ \ } x_1=0 \text{\ \ as\ \ } \alpha_1=0,
\label{eqn-5.32}\\
\shortintertext{ but under condition (\ref{eqn-5.21}) }
B_n=-B_n^0 +i \varphi_{x_1}^{-1}\bigr(B_{n-1,x_1}+B^0_{n-1,x_1}\bigr)\qquad \text{at\ \ } x_1=0
\notag
\end{gather}
and therefore estimate
(\ref{eqn-5.22}) holds as $n=1$, $\alpha_1=0$, $x_1=0$. It follows from estimates for $|\partial_{x_1} B_{0,\alpha\beta\gamma}|\le C_{\alpha\beta\gamma} \sigma^{-2-|\alpha|-|\beta|-|\gamma|}$ as $\alpha_1=0$, $x_1=0$ (but we divide by $\varphi_{x_1}$ which brings an extra factor $\sigma^{-1}$).

Recall that the transport equation is
\begin{gather}
(\frac{d\ }{dt} +f )B_n= G_n,\qquad G_n\coloneqq \mathcal{L} B_{n-1},
\label{eqn-5.33}\\
\intertext{and due to results of Part (b)}
|G_{n,(\alpha-\upvarepsilon)\beta\gamma}|\le C\sigma^{1-3n -\alpha_1 -|\alpha|-|\beta|-|\gamma|}
\label{eqn-5.34}
\end{gather}
and therefore in both cases estimate (\ref{eqn-5.22}) holds (as $n=1$, $\alpha_1=1$ and $x_1=0$).

Applying the same argument as in Part (b) we can extend  (\ref{eqn-5.34}) and   (\ref{eqn-5.22}) to arbitrary  $\alpha_1$ as $n=1$, and using induction by $n$ to arbitrary $n$ as well. Again, we leave easy but tedious details to the reader.

\item
Using the same arguments as in Part (e) of the proof of Proposition~\ref{prop-5.3}, we can improve these estimates to (\ref{eqn-5.23}) as $|\gamma |\ge 1$.
\end{enumerate}\vspace{-\baselineskip}
\end{proof}

\begin{remark}\label{remark-5.5}
Under assumption (\ref{eqn-5.20}) we can further improve estimates (\ref{eqn-5.22}) and (\ref{eqn-5.23}) for $n\ge 1$. However it would not improve our final result.
\end{remark}

\begin{remark}\label{remark-5.6}
One can wonder how sharp are our estimates.
\begin{enumerate}[wide,label=(\roman*),  labelindent=0pt]
\item
Due to
(\ref{eqn-5.2}) and (\ref{eqn-5.3}) $\varphi_{x_1x_1}\asymp \sigma^{-1}$ as $b_{x_1} (x, \nabla_{x'}\varphi) \asymp 1$. Indeed
\begin{align*}
&2\varphi^0_{x_1}\varphi^0_{x_1x_1} + \sum_{k\ge 2}b^{(k)}(x,\nabla_{x'} \varphi^0) \varphi^0_{x_kx_1}-\varphi^0_{x_1t} +b_{x_1}(x,\nabla_{x'} \varphi^0)=0,\\
&2\varphi_{x_1}\varphi_{x_1x_1} + \sum_{k\ge 2}b^{(k)}(x,\nabla_{x'} \varphi ) \varphi _{x_kx_1}-\varphi _{x_1t} +b_{x_1}(x,\nabla_{x'} \varphi )=0,
\end{align*}
and since $\varphi^0$ is smooth function, we conclude that
\begin{gather*}
\sum_{k\ge 2}b^{(k)}(x,\nabla_{x'} \varphi^0) \varphi^0_{x_kx_1}-\varphi^0_{x_1t} +b_{x_1}(x,\nabla_{x'} \varphi^0)=O(\sigma)\\
\intertext{and then due to (\ref{eqn-5.3}) that}
\varphi_{x_1}\varphi_{x_1x_1} + b_{x_1}(x,\nabla_{x'} \varphi )=O(\sigma)\qquad \text{as\ \ } x_1=0.
\end{gather*}
Therefore estimates (\ref{eqn-5.5}) cannot be improved, at least for derivatives only with respect to $x_1$.

\item
Repeating construction of Proposition~\ref{prop-5.3}, we conclude that  under the same assumption $b_{x_1} (x, \nabla_{x'}\varphi) \asymp 1$,
$\partial_{x_1}^k \varphi \asymp \sigma^{3-2k}$ for all $k\ge 2$.
Then from transport equation (\ref{eqn-5.24}) we  we conclude that $B_{0,x_1}\asymp \sigma^{-2}$ and repeating construction of Proposition~\ref{prop-5.4} we conclude that
$\partial_{x_1}^k B_0 \asymp \sigma^{-2k}$ for all $k\ge 1$ and finally
$\partial_{x_1}^k B_n \asymp \sigma^{-3n-2k}$ for all $n\ge 0$, $k\ge 1$.
\end{enumerate}
\end{remark}

\begin{proposition}\label{prop-5.7}
\begin{enumerate}[wide,label=(\roman*),  labelindent=0pt]
\item
In the framework of Proposition~\ref{prop-5.3}(i) as $|\alpha|\le 1$
\begin{align}
&|\partial_{x,t}^\alpha \partial_\theta^\gamma\bigl(\Phi^0-\bar{\Phi}^0 \bigr)|\le C_\gamma T^{2-|\alpha|} &&\text{with\ \ } \bar{\Phi}^0\coloneqq \langle x-y,\theta \rangle + t a(y,\theta).
\label{eqn-5.35}
\end{align}
\item
In the framework of Proposition~\ref{prop-5.3}(ii) as $|\alpha|\le 1$, $\alpha_1=0$
\begin{multline}
|\partial_{x,t}^\alpha \partial_\theta^\gamma\bigl(\Phi\ -\bar{\Phi}\  \bigr)|\le C_\gamma \bigl(\sigma^{1-|\alpha|-|\gamma|}T^2 + T^{2-|\alpha|}\bigr)\\
\text{with\ \ }
\bar{\Phi}\ \coloneqq \langle x-\tilde{y},\theta \rangle + t a(0,y',\theta),
\label{eqn-5.36}
\end{multline}
where $\tilde{y}=(-y_1,y')$ as $y=(y_1,y')$.
\end{enumerate}
\end{proposition}

\begin{proof}
Proof of both statements  follows from decomposition of  $\partial ^\alpha _{x,t}\partial_\theta^\gamma \Phi^0$ into Taylor series with quadratic error.  First we prove (i) using estimates (\ref{eqn-5.2}) and then (ii) using estimates (\ref{eqn-5.5}).

For $d=2$ we will need a better approximation; it will be proven later.
\end{proof}

Then we have immediately
\begin{corollary}\label{corollary-4.8}
\begin{enumerate}[wide,label=(\roman*),  labelindent=0pt]
\item
In the framework of Proposition~\ref{prop-5.3}(i) for $Ch\le \ell^0(x,y)+|t|\le \epsilon$
\begin{gather}
a(x,\nabla_x \varphi^0 )=\tau,\quad  \nabla_\theta \Phi^0 (x,y,t,\theta) =0
\label{eqn-5.37}
\end{gather}
has  no more than a single solution $\theta$ and if it has, then
$|t|\asymp \ell^0$ and $\nabla^2_\theta \Phi^0 $ is a positive definite matrix. This solution $\theta=\bar{\theta}+O(T)$ where
$\bar{\theta}$ is a solution to $t\nabla_\theta a(x,\theta) =y-x$.

\item
In the framework of Proposition~\ref{prop-5.3}(ii) for $Ch\le \ell(x,y) +|t|\le \epsilon$
\begin{gather}
a(x,\nabla_x \varphi )=\tau,\quad  \nabla_\theta \Phi (x,y,t,\theta) =0
\label{eqn-5.38}
\end{gather}
has has  no more than  a single solution $\theta$ and if it has then
$|t| \asymp \ell$ and $\nabla^2_\theta \Phi $ is a positive definite matrix.  This solution $\theta=\bar{\theta}+O(T)$ where
$\bar{\theta}$ is a solution to $t\nabla_\theta a(x,\theta) =\tilde{y}-x$.
\end{enumerate}
\end{corollary}

\begin{proposition}\label{prop-5.9}
In the framework of Proposition~\ref{prop-5.3}(ii) under extra assumption
\begin{gather}
\sigma^2\ell \ge h^{1-\delta}
\label{eqn-5.39}\\
\intertext{the following estimate holds}
|F_{t\to h^{-1}\tau } \Bigl(\bar{\chi}_{T'}(t) u^1_h (x,y,t)- \bar{\chi}_{T}(t) U^1_h(x,y,t)\Bigr)|\le Ch^s
\label{eqn-5.40}
\end{gather}
with  $U^1_h(x,y,t)$ defined by \textup{(\ref{eqn-5.19})} with $B_n(x,y,t,\theta)$ described above and multiplied  by $\phi(\theta)$ supported  in $2\epsilon_3\rho $-vicinity and equal $1$ in $\epsilon_3\rho $-vicinity  of $\theta$ described in Corollary~\ref{corollary-4.8}(ii) and with $Ch\le T\le T' \le \epsilon$, $T=c \ell$, $\rho=\sigma^2$.
\end{proposition}

\begin{proof}
Let us fix $x=\bar{x}$ and $y=\bar{y}$  in (\ref{eqn-5.40}) and use $x$ as a variable. Due to Proposition~\ref{prop-5.1}  there is a single generalized billiard of the length
$\le \epsilon$ with reflections at $\partial X$ from $\bar{x}$ to $\bar{y}$.  It has the length $\asymp \ell$, exactly one reflection and a reflection angle $\asymp \rho = ({\nu}(\bar{x})+{\nu}(\bar{y}))\ell^{-1}\gtrsim h^{\frac{1}{3}-\delta}$.

Further, Proposition~\ref{prop-5.4}(ii) implies that  for $\sigma \ge h^{\frac{1}{3}-\delta}$ (which is due to (\ref{eqn-5.39}) and $\sigma \ge C_0\ell$)
\textup{(\ref{eqn-5.19})} is a proper asymptotic series as $|t|\le T$ and we  define $U^1_h(x,y,t)$ through it (with cut-off).

Furthermore, Corollary~\ref{corollary-4.8} implies that then $U^1_h (x,y,t)$ is negligible outside $\Omega_{2T,3\rho,3\sigma}$
(by $(x_1,x',\xi')$) of this billiard  while the standard propagation results  imply that  $u^1_h(x,y,t) \,^t\!Q_y$ is also negligible outside of this vicinity provided
symbol of $Q$ is supported in the similar vicinity of $(\bar{y},\bar{\theta}')$, $\bar{\theta}'$ corresponds to this billiard.
Let  symbol of $Q$ be  $1$ in the similar vicinity of $(\bar{y},\bar{\xi}')$. Then  $ u^1_h(x,y,t) \,^t\!Q_y \equiv U^1_h (x,y,t)$ for $-T \le t\le  T$
with $0<T=C_0 \ell$.

On the other hand, in this case  $u^1_h(x,y,t) (I-^t\!Q_y)\equiv 0$ in $(\epsilon \rho \ell, \epsilon \ell )$-vicinity of $\bar{x}$ as $-T\le t\le \epsilon T$.
Case of  $-\epsilon T \le t\le T$ is considered in the same way, albeit with the billiard from $\bar{x}$ to $\bar{y}$. Then we arrive to
(\ref{eqn-5.40}) with $T'=T$.

Finally, the standard propagation results imply that
\begin{gather}
|F_{t\to h^{-1}\tau } \Bigl(\bigl(\bar{\chi}_{T'}(t) - \bar{\chi}_{T}(t)\bigr) u^1_h(x,y,t)\Bigr)|\le Ch^s
\label{eqn-5.41}
\end{gather}
as $T\le T'\le \epsilon$. Then (\ref{eqn-5.40}) expands to $T'\colon T\le T'\le \epsilon$.
\end{proof}

\section{Spectral asymptotics}
\label{sect-5.2}

\begin{proposition}\label{prop-5.10}
In the framework of Proposition~\ref{prop-5.3}(ii) under extra assumption
\begin{gather}
\sigma^2\ell \ge h^{1-\delta},\qquad \ell \ge h^{\frac{3}{5}}
\label{eqn-5.42}\\
\intertext{the following asymptotics holds}
e^{1,\mathsf{T}}_h(x,y,\tau) =(2\pi h)^{-d}\int _{a(y,\theta)<\tau } e^{ih^{-1}\varphi (x,y,\theta)}\,d\theta +O(h^{1-d}).
\label{eqn-5.43}
\end{gather}
\end{proposition}

\begin{proof}
Let us   plug $U^1_h$ instead of $u^1_h$ into Tauberian expression $e^{1,\mathsf{T}}_h(x,y,\tau)$ with $T=\epsilon$.
Let us decompose $B_n (x,y,t,\theta)$ into asymptotic series by $t^{k}$.

\begin{enumerate}[wide,label=(\alph*),  labelindent=0pt]
\item
Then terms with $k=0$ become
\begin{gather}
(2\pi h)^{-d} \int _{a(y,\theta)< \tau} \sum_{n\ge 0} e^{ih^{-1}\varphi (x,y,\theta)} B_n (x,y,0, \theta) h^n \,d\theta
\label{eqn-5.44}
\end{gather}
and we claim that these terms do not exceed $C\sigma^{-3n} h^{-d+n} (h\ell^{-1})^{(d+1)/2}$. Indeed, we know from Proposition~\ref{prop-5.3}(ii) that
$\varphi$ is uniformly infinitely smooth by $\theta$ and from Corollary~\ref{corollary-4.8} that there are no stationary points by $\theta$ and restriction of $\varphi (x,y,\theta)$ to $\Sigma (y,\tau)$ has two non-degenerate critical points. The leading terms would be of this magnitude, while all other terms would have extra factors not exceeding $h/\sigma^2\ell$ due to (\ref{eqn-5.22}).

We want to estimate all extra terms by $Ch^{1-d}$. One can see easily, that $d=2$ is the worst case but even then therms with $n\ge 1$ are $O(h^{1-d})$ and terms with $h/\sigma^2\ell$ appear when we differentiate $B_0 (x,y,0,\theta)$ by $\theta$. Then due to (\ref{eqn-5.23}) these terms do not exceed
\begin{gather*}
C h^{-d} (h\ell^{-1})^{\frac{d+3}{2}} (1 + \ell \sigma^{-2} )
\end{gather*}
and this does not exceed $Ch^{1-d}$ under assumption (\ref{eqn-5.32}). \enlargethispage{\baselineskip}

Finally, since $B_0(x,y,0,\theta)= 1+O(\ell\sigma^{-1})$ we can replace $B_0(x,y,0,\theta)$ by $1$ resulting in the main part of asymptotics (\ref{eqn-5.43}).

\item
Consider now terms with $k\ge 1$ in the decomposition of $B_n(x,y,t,\theta)$. We can rewrite these terms as
\begin{align}
&(2\pi h)^{1-d} \partial_\tau^{k-1}\int_{\Sigma  (y,\tau)} B'_{n,k} (x,y,\theta) h^{n+k-1} \,d\theta:d_\theta a(y,\theta)\notag\\
\intertext{and using $\xi$-microhyperbolicity rewrite it as}
&(2\pi h)^{1-d} \int_{\Sigma (y,\tau)} B''_{n,k} (x,y,\theta) h^{n} (h\ell^{-1})^k  \,d\theta:d_\theta a(y,\theta).
\label{eqn-5.45}
\end{align}
with $B''_{n,k}$  coming from $B_n$ with  $k$ derivatives by $t$ and no more than $(k-1)$ by $\theta$.  Then using (\ref{eqn-5.22})--(\ref{eqn-5.23}) we can estimate these terms by $Ch^{-\frac{d-1}{2}}\ell^{-\frac{d-1}{2}}\le Ch^{1-d}$.
\end{enumerate}
\end{proof}

\begin{remark}\label{remark-5.11}
The similar asymptotics
\begin{gather}
e^{0,\mathsf{T}}_h(x,y,\tau) =(2\pi h)^{-d}\int _{a(y,\theta)<\tau } e^{ih^{-1}\varphi^0 (x,y,\theta)}\,d\theta +O(h^{1-d})
\label{eqn-5.46}
\end{gather}
is well-known.
\end{remark}

\begin{remark}\label{remark-5.12}
Consider in the framework of Propositions~\ref{prop-5.3}(i), (ii)   integrals $I_h(\varphi^0)$ and $I_h(\varphi^0)$ in the right-hand expressions of (\ref{eqn-5.46}) and  (\ref{eqn-5.43}) correspondingly:
\begin{gather}
I_h (\varphi)\coloneqq (2\pi h)^{-d}\int _{a(y,\theta)<\tau } e^{ih^{-1}\varphi  (x,y,\theta)}\,d\theta.
\label{eqn-5.47}
\end{gather}
\begin{enumerate}[wide,label=(\roman*),  labelindent=0pt]
\item
Let   $d\ge 3$. Then  $I_h (\varphi^0) = I _h (\bar{\varphi}^0)+O(h^{1-d})$ and  $I_h (\varphi) = I _h (\bar{\varphi})+O(h^{1-d})$
with  $\bar{\varphi}^0(x,y,\theta)= \langle x-y,\theta \rangle$, $\bar{\varphi}^0(x,y,\theta) =\langle x-\tilde{y},\theta \rangle$.
\item
Let $d=2$. Then  $I_h(\varphi^0) \equiv I_h(\bar{\varphi}^0)$ with an error $O(h^{-1})$ as $\ell ^0\ge h^{\frac{1}{3}}$,
$O(h^{-1})$ as $\ell ^0(x,y)\ge h^{\frac{1}{3}}$, $O(h^{-\frac{1}{2}}\ell^{-\frac{3}{2}})$ as $h^{\frac{1}{2}}\le \ell^0(x,y)\le h^{\frac{1}{3}}$ and  $O(h^{-\frac{3}{2}}\ell^{\frac{1}{2}})$ as $h\le \ell ^0(x,y)\le h^{\frac{1}{2}}$.

Also $I_h(\varphi) \equiv I_h(\bar{\varphi}^)$ with an error $O(h^{-1})$ as $\ell ^0\ge h^{\frac{1}{3}}$, $O(h^{-1})$ as $\ell ^0(x,y)\ge h^{\frac{1}{3}}$, $O(h^{-\frac{1}{2}}\ell^{-\frac{3}{2}})$ as $h^{\frac{1}{2}}\le \ell (x,y)\le h^{\frac{1}{3}}$ and  $O(h^{-\frac{3}{2}}\ell^{\frac{1}{2}})$ as $h\le \ell (x,y)\le h^{\frac{1}{2}}$.
\end{enumerate}
\end{remark}

\begin{proof}
Observe first that due to the stationary phase principle $I_h (\varphi) = Ch^{-d}(h\ell^{ -1})^{(d+1)/2}$  and the same is true for $I_h (\bar{\varphi})$.
Therefore for $\ell \ge h^{\frac{1}{2}}$ we have simply estimates rather than the error estimates.

On the other hand due to proposition~\ref{prop-5.7}  $\varphi (x,y,\theta)= \langle x-\tilde{y},\theta \rangle + O(\ell^2)$, for $\ell \le h^{\frac{1}{2}}$ the error does not exceed
$Ch^{-d}(h\ell^{ -1})^{(d+1)/2}\times \ell^2h^{-1}$ which is $O(h^{1-d})$.

The same is true for $I_h (\varphi^0)$ and $I_h (\bar{\varphi}^0)$ albeith with $\ell^0$ rather than $\ell$.
\end{proof}

Now we need to deal with $d=2$ and $\ell^0\le h^{\frac{1}{3}}$ or  $\ell \le h^{\frac{1}{3}}$.

\begin{proposition}\label{prop-5.13}
Let $d=2$. Then
\begin{enumerate}[wide,label=(\roman*),  labelindent=0pt]
\item
In the framework of Proposition~\ref{prop-5.3}(i) for $h^{1-\delta}\le \ell^0(x,y)\le h^{\frac{1}{3}}$
\begin{gather}
\int_{a(y,\theta)<\tau} e^{ih^{-1}\varphi^0(x,y,\theta)}\,d\theta = \int_{a(w,\xi )<\tau} e^{ih^{-1}\langle x-y,\xi\rangle }\,d\xi +O(h)
\label{eqn-5.48}
\end{gather}
with  $w=\frac{1}{2}(x+y)$.
\item
In the framework of Proposition~\ref{prop-5.3}(ii) for $h^{1-\delta}\le \ell (x,y)\le h^{\frac{1}{3}}$
\begin{multline}
\int_{a(y,\theta)<\tau} e^{ih^{-1}\varphi (x,y,\theta)}\,d\theta \\
= \int_{a(w,\xi )<\tau} e^{ih^{-1}(\langle x-\tilde{y},\xi\rangle + \frac{1}{2}\varkappa (w',\xi',\tau)(x_1^2+y_1^2)) }\,d\xi +O(h)
\label{eqn-5.49}
\end{multline}
with $w=(0,\frac{1}{2}(x'+y'))$ and with
\begin{gather}
\varkappa (x',\xi',\tau)=(\tau-b(x_2,\xi_2))^{-\frac{1}{2}}a_{x_1}(x,\xi)|_{x_1=\xi_1=0};
\label{eqn-5.50}
\end{gather}
recall that $b(w_2,\xi_2)=a(x,\xi)|_{x_1=\xi_1=0}$.
\end{enumerate}
\end{proposition}

\begin{proof}
\begin{enumerate}[wide,label=(\roman*),  labelindent=0pt]
\item
Consider $(z,\bar{\xi})\colon a(\bar{w},\bar{\xi})=\tau$ and Hamiltonian trajectory $\Psi_t (\bar{w},\bar{\xi})$ of
$a(x,\xi)$, $\Psi_0 (\bar{w},\bar{\xi})=(\bar{w},\bar{\xi})$. Then as  $(y(t), \theta(t))=\Psi_t(\bar{w},\bar{\xi})$ and $(x(t),.)=\Psi_{-t}(\bar{w},\bar{\xi})$, $|t|\le C_0\ell$  we have
\begin{gather}
\varphi^0 (x(t),y(t),\theta(t))= \langle x(t)-y(t), \bar{\xi} \rangle + O(\ell^3)
\label{eqn-5.51}
\end{gather}
where  in this part of the proof for simplicity we write $\ell$ rather than $\ell^0$.

Observe that as $\ell \ge h^{1-\delta} $ we can reduce  integrals in (\ref{eqn-5.48}) to $\Sigma (y,\tau)$ and $\Sigma (\bar{w},\tau)$ respectively, gaining factor $h\ell^{-1}$. Consider first those reduced integrals over $\epsilon$-vicinities of $\theta(t)$ and $\bar{\xi}$. We see that there is a diffeomorphism of these vicinities and
\begin{gather*}
\varphi^0(x(t),y(t), \theta (\bar{w}, \xi(s), t)) =  \langle x(t)-y(t),  \xi  \rangle +  O(\ell^3+s^2\ell )
\end{gather*}
provided $a_\theta \parallel \theta$ at $(\bar{w},\bar{\xi})$ (we can assume it without any loss of the generality), $s$ is an angle parameter on $\Sigma(\bar{w},\tau)$, $\xi(0)=\bar{\xi}$. Making change of variables we estimate  an error by $C(h\ell^{-1})^{\frac{3}{2}}\times (\ell^3h^{-1}+\varepsilon^2 \ell h^{-1} + \ell)$ when we integrate over $|s|\le \varepsilon$, and when we integrate over $|s|\ge \varepsilon$, we can multiply it by an arbitrary large power of $h/\varepsilon ^2\ell$. Taking $\varepsilon =(h\ell^{-1})^{\frac{1}{2}}$ we make both errors  $O(h)$.

On the other hand, fix $\bar{x}$ and $\bar{y}$ and find corresponding $\bar{w}$, $\bar{\xi}$ and $t$. One can see easily that $\bar{w}=w+O(\ell^2)$ and the error in
the integral in the right hand expression of (\ref{eqn-5.48}) when we redefine $w$ does not exceed $C(h\ell)^{\frac{1}{2}}\ell^2 =O(h)$.

\item
Making change of variables $\mathsf{x}_1= x_1+\frac{1}{2}\varkappa (w_2,\xi',\tau)x_1^2$ and therefore  $\mathsf{y}_1= y_1+\frac{1}{2}\varkappa (w_2,\xi',\tau)y_1^2$ we would arrive to operator with the symbol which, after division by $(a-\tau)$, modulo  $O(\ell^2)$ does not depend on $\mathsf{x}_1$  and therefore with the corresponding phase function
\begin{gather}
\varphi   = (\mathsf{x}_1+\mathsf{y}_1)\xi_1 + (x_2-y_2)\xi_2 + O(\ell^3).
\notag
\intertext{Then we can use the the method of reflection and after this, Statement~(i). So we get}
\int_{a(w,\xi )<\tau} e^{ih^{-1}(  (\mathsf{x}_1+\mathsf{y}_1)\xi_1  + (x_2-y_2)\xi_2}\,d\xi +O(h)
\label{eqn-5.52}
\end{gather}
with $w =(0, \frac{1}{2}(x_2+y_2))$; which is exactly the right-hand expression of (\ref{eqn-5.49}).
\end{enumerate}
\end{proof}

\begin{remark}\label{remark-5.14}
 \begin{enumerate}[wide,label=(\roman*),  labelindent=0pt]
 \item
 Therefore in Theorem~\ref{thm-1.2}
\begin{multline}
e_{h,\mathsf{corr}} (x,y,\tau)\\
 =[\mp] (2\pi h)^{-1}\int _{\{a(\frac{1}{2}(x+y),\xi)<\tau\}} e^{ih^{-1}\langle x-\tilde{y},\xi\rangle}
\Bigl(e^{ih^{-1} \frac{1}{2}\varkappa (\frac{1}{2}(x_2+y_2),\xi_2,\tau)}-1\Bigr)\,d\xi
\label{eqn-5.53}
\end{multline}
with $\varkappa (x_2,\xi_2,\tau)$ defined by (\ref{eqn-5.50}).
\item
One can see easily that in the framework of Proposition~\ref{prop-4.5} expressions (\ref{eqn-5.53}) and (\ref{eqn-4.49}) coincide modulo $O(h^{-1})$.
\end{enumerate}
\end{remark}

\chapter{Synthesis and final remarks}
\label{sect-6}
\begin{proof}[Proof of Theorems~\ref{thm-1.1} and~\ref{thm-1.2}]
We know that for $\ell(x,y)\le \epsilon$ in the frameworks of Theorems~\ref{thm-1.1} and\ref{thm-1.2} estimate (\ref{eqn-1.8}) holds.
 \begin{gather*}
 e_h(x,y,\tau) = e^\mathsf{T} _{T,h} (x,y,\tau)+O(h^{1-d}).
 \end{gather*}
  \begin{enumerate}[wide,label=(\roman*),  labelindent=0pt]
 \item
We also know from Section~\ref{sect-2} that in the framework of Theorem~\ref{thm-1.1} $e^\mathsf{T} _{T,h} (x,y,\tau)$ can be defined as an oscillatory integral and Proposition~\ref{prop-5.13}(i) implies that modulo $O(h^{1-d})$ this oscillatory integral can be rewritten as $e^\mathsf{W}_h(x,y,\tau)$ defined by (\ref{eqn-1.3}). This proves
Theorem~\ref{thm-1.1}.

\item
In the framework of Theorem~\ref{thm-1.2} we established in Section~\ref{sect-3}
\begin{gather*}
e^{1,\mathsf{T}} _{T,h} (x,y,\tau)=O(h^{1-d})
\end{gather*}
 as $d\ge 3$ and $\ell(x,y)\ge h^{\frac{1}{2}-\delta}$ and as $d=2$ and  $ \ell(x,y)\ge h^{\frac{1}{3}-\delta}$.
Further, we proved there that $e^{1,\mathsf{T}} _{T,h} (x,y,\tau)=O(h^{1-d-\delta})$ as $d=2$ and $\ell (x,y)\ge h^{\frac{1}{3}+\delta}$. In both cases we also assume there that $|x'-y'| \ge \epsilon \ell(x,y)$. However it follows from Section~\ref{sect-5} that this latter condition is not necessary. Combined with Remark~\ref{remark-1.3}(ii) it proves Theorem~\ref{thm-1.2} in this case.

Let $d\ge 3$.
In Section~\ref{sect-4}  we proved   asymptotics
\begin{gather}
e^{1,\mathsf{T}}_{T,h}(x,y)= [\mp] e^{0,\mathsf{W}}_h(x,\tilde{y},\tau)+O(h^{1-d})
\label{eqn-6.1}
\end{gather}
if either  $\ell (x,y)\le h^{\frac{1}{2} +\delta}$ \underline{or}  $ h^{\frac{1}{2}+\delta}\le \ell(x,y)  \le h^{\frac{1}{2} -\delta}$ and
\begin{gather}
{\nu}(x)+{\nu}(y)\le \sigma \ell (x,y)
\label{eqn-6.2}
\end{gather}
with $ \sigma = h^{2\delta}$. In Section~\ref{sect-5}  we proved the same asymptotics as
 \begin{gather}
{\nu}(x)+{\nu}(y)\ge \sigma \ell (x,y)
\label{eqn-6.3}
\end{gather}
with $\sigma= h^{\frac{1}{2}-\delta}\ell^{-\frac{1}{2}}+C_0 \ell$. These domains overlap and cover all values of $\ell \le h^{\frac{1}{2}-\delta}$  and $\sigma $.

\item
Let $d=2$. Then instead of asymptotics  (\ref{eqn-6.1})  we have
\begin{gather}
e^{1,\mathsf{T}}_{T,h}(x,y)= [\mp] e^{0,\mathsf{W}}_h(x,\tilde{y},\tau)+e_{h,\mathsf{corr}}(x,y,\tau)+ O(h^{1-d})
\label{eqn-6.4}
\end{gather}
 and Sections~\ref{sect-4} and~\ref{sect-5} cover cases (\ref{eqn-6.2}) with $\sigma =h^{1+\delta}\ell^{-2}$ and (\ref{eqn-6.3}) with
 $\sigma =h^{\frac{1}{3}-\delta}$, $\ell(x,y)\le h^{\frac{1}{3}+\delta}$.  These domains overlap and cover all values  $\ell \le h^{\frac{1}{3}+\delta}$ and $\sigma$.
Here we are left with domain $h^{\frac{1}{3}+\delta }\le \ell\le h^{\frac{1}{3}-\delta}$, $\sigma =h^{\frac{1}{3}-\delta}$ where we have less precise estimate
$e^{1,\mathsf{T}} _{T,h} (x,y,\tau)=O(h^{1-d-\delta})$.
\end{enumerate}
\end{proof}
\enlargethispage{\baselineskip}

\begin{remark}\label{remark-6.1}
\begin{enumerate}[wide,label=(\roman*),  labelindent=0pt]
\item
We would like to derive remainder estimate $O(h^{-1})$ as $d=2$ and $\ell(x,y)\le \epsilon$, thus removing the gap  $h^{\frac{1}{3}+\delta} \le \ell \le h^{\frac{1}{3}-\delta}$, ${\nu} (x)+{\nu}(y)\le h^{|frac{1}{3}-\delta}$.

\item
For Schr\"odinger operator (\ref{eqn-1.4})  can get rid of $\xi$-microhyperboloicity assumption by rescaling method \cite{monsterbook}, Subsection~\ref{monsterbook-sect-5-3-1},   Subsubsection~\nameref{monsterbook-sect-5-3-1-3}.
Then as $d\ge 3$ we have a remainder estimate $O(h^{1-d})$ and as $d=2$ the remainder estimate is $O(h^{-\frac{4}{3}})$ as $\ell(x,y)\le \epsilon |V(x)-\tau|$. We would like to extend it to  $\ell(x,y)\le \epsilon$.
\item
As $d=2$ and $x=y$ away from $\partial X$ we can get a remainder estimate $O(h^{-1})$ provided $|V(x)-\tau|+|\nabla V|\ge \epsilon_0$ (see \cite{monsterbook}, Subsection~\ref{monsterbook-sect-5-3-1},   Subsubsection~\nameref{monsterbook-sect-5-3-1-4}),  but with the correction term. The reason is that $V(x)=\tau$ is a kind of ``soft boundary'' and there are two trajectories from $x$ to $y$, one of them is of the length $0$ and another bounces once from $V(x)=\tau$. We would like to extend it to $\ell(x,y)\le \epsilon |V(x)-\tau|$ and then to $\ell(x,y)\le \epsilon$.
\end{enumerate}
\end{remark}

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