$U_{\sqrt{\rho}}^{\tau}(t)\simeq\Pi^{\tau}\circ
P^{\tau}_{t}\circ\Pi_{-t}^{\tau},$ (100)

where $\simeq$ means ‘equal modulo smoothing Toeplitz operators’. Furthermore,
the principal symbol $\sigma^{\tau}_{t,0}(x)$ is the inverse of the
$L^{2}$-pairing of two normalized Gaussian functions, related to complex
structures $J_{x}$ and ${J_{t}}_{x}$, respectively, where $J_{t}$ is the push-
forward of $J$ under the flow of $\upsilon_{\sqrt{\rho}}^{\tau}$ at time $t$;
in particular, $\sigma^{\tau}_{0,0}(x)=1$.

Up to smoothing terms, the Schwartz kernel of $\Pi_{-t}^{\tau}$ has the form

$\Pi_{-t}^{\tau}(x_{1},x_{2})\sim\int_{0}^{+\infty}e^{\imath\,u\,\psi^{\tau}\left(\Gamma^{\tau}_{-t}(x_{1}),x_{2}\right)}\,s^{\tau}\left(\Gamma^{\tau}_{-t}(x_{1}),x_{2},u\right)\,\mathrm{d}u,$
(101)

where $\psi^{\tau}$ and $s^{\tau}$ are as in (76). Using classical results on
the composition of pseudodifferential and Fourier integral operators ([Sh],
[T]), we reach the following conclusion.

######  Lemma 55.

Up to smoothing terms, the Schwartz kernel of the composition
$\mathcal{P}^{\tau}_{t}:=P^{\tau}_{t}\circ\Pi_{-t}^{\tau}$ has the form

$\mathcal{P}^{\tau}_{t}(x_{1},x_{2})\sim\int_{0}^{+\infty}e^{\imath\,u\,\psi^{\tau}\left(\Gamma^{\tau}_{-t}(x_{1}),x_{2}\right)}\,r^{\tau}_{t}\left(x_{1},x_{2},u\right)\,\mathrm{d}u,$
(102)

where

$r^{\tau}_{t}\left(x_{1},x_{2},u\right)\sim\sum_{j\geq
0}\,u^{d-1-j}\,r^{\tau}_{t\,j}(x_{1},x_{2}),\quad
r^{\tau}_{t\,0}(x_{1},x_{2})=\sigma^{\tau}_{t,0}\left(x_{1}\right)\cdot
s^{\tau}_{0}\left(\Gamma^{\tau}_{-t}(x_{1}),x_{2}\right).$

By (100), we have

$U_{\sqrt{\rho}}^{\tau}(t)\simeq\Pi^{\tau}\circ\mathcal{P}^{\tau}_{t}.$ (103)

##  4 Near-diagonal asymptotics for $\Pi^{\tau}_{\chi,\lambda}$

Before delving into the proof of Theorem 3, let us premise some notation.

The choice of a normal Heisenberg local chart $\varphi^{\tau}$ for $X^{\tau}$
at $x$ determines an isomorphism
$T_{x}X^{\tau}\cong\mathbb{R}\times\mathbb{R}^{2d-2}$; a general $\upsilon\in
T_{x}X^{\tau}$ will be written accordingly as a pair
$\upsilon=(\theta,\mathbf{u})$. The subspaces
$\mathbb{R}\times\\{\mathbf{0}\\}$ and $\\{0\\}\times\mathbb{R}^{2d-2}$
correspond, respectively, to $\mathcal{T}^{\tau}(x)$ and
$\mathcal{H}^{\tau}(x)$ (see (48)). By (63), the isomorphism
$\mathbb{C}^{d-1}\cong\\{0\\}\times\mathbb{R}^{2d-2}\rightarrow\mathcal{H}^{\tau}(x)$
is unitary, when $\mathcal{H}^{\tau}(x)$ is endowed with the Hermitian
structure associated to $\omega_{x}=\frac{1}{2}\,\Omega_{x}$.

With the notation of Corollary 45, let
$(a_{x},\mathbf{A}_{x})\in\mathbb{R}\times\mathbb{R}^{2d-2}$ be defined by

$\tau\,f(0,\vartheta,\mathbf{u})=\frac{1}{2\,\tau^{2}}\,a_{x}\,\vartheta+\langle\mathbf{A}_{x},\mathbf{u}\rangle+F_{2}^{\tau}(\vartheta,\mathbf{u}),$
(104)

where $F_{2}^{\tau}$ vanishes to second order at the origin. We may then
reformulate the conclusion of Corollary 44 writing

$\displaystyle\Gamma^{\tau}_{t}(y)$ $\displaystyle=$ $\displaystyle
x+\left(\vartheta-\tau\,t+t\,\left(\frac{1}{2\,\tau^{2}}\,a_{x}\,\vartheta+\langle\mathbf{A}_{x},\mathbf{u}\rangle\right)+R_{3}(\tau\,t,\vartheta,\mathbf{u}),\right.$
(105)
$\displaystyle\mathbf{u}+\mathbf{R}_{2}(\tau\,t,\vartheta,\mathbf{u})\Big{)}.$

We shall verify a posteriori that $(a_{x},\mathbf{A}_{x})=(0,\mathbf{0})$.

######  Definition 56.

With $(a_{x},\mathbf{A}_{x})$ as in (104), let us set

$\mathbf{a}_{x}(\theta_{1},\theta_{2}):=\frac{\theta_{1}-\theta_{2}}{\tau}\,\mathbf{A}_{x}.$

We further define

$\Psi_{2}:T_{x}X^{\tau}\times T_{x}X^{\tau}\rightarrow\mathbb{C}$

by setting, given $\upsilon_{j}=(\theta_{j},\mathbf{v}_{j})$,

$\displaystyle\Psi_{2}(\upsilon_{1},\upsilon_{2})$ $\displaystyle:=$
$\displaystyle-\imath\,\omega_{x}\left(\mathbf{v}_{1},\mathbf{v}_{2}\right)-\frac{\imath}{2}\,\omega_{x}\left(J_{x}\big{(}\mathbf{a}_{x}(\theta_{1},\theta_{2})\big{)},\mathbf{v}_{1}+\mathbf{v}_{2}\right)$
$\displaystyle-\frac{1}{4}\,\|\mathbf{v}_{1}-\mathbf{v}_{2}\|^{2}-\frac{1}{4}\,\left\|\mathbf{v}_{1}-\mathbf{v}_{2}+J_{x}\big{(}\mathbf{a}_{x}(\theta_{1},\theta_{2})\big{)}\right\|^{2}.$

###### Proof of Theorem 3.

By (3.6) and (100), we can rewrite (15) in the following form:

$\displaystyle\Pi^{\tau}_{\chi,\lambda}(x_{1},x_{2})$ $\displaystyle\sim$
$\displaystyle\frac{1}{\sqrt{2\,\pi}}\,\int_{X^{\tau}}\int_{-\epsilon}^{+\epsilon}\left[e^{-\imath\,\lambda\,t}\,\chi(t)\,\Pi^{\tau}(x_{1},y)\,\left[P^{\tau}_{t}\circ\Pi_{-t}^{\tau}\right](y,x_{2})\right]\,\mathrm{d}t\,\mathrm{d}V_{X^{\tau}}(y),$

where $\sim$ stands for ‘has the same asymptotics for
$\lambda\rightarrow+\infty$ as’.

The wave front set of $\Pi^{\tau}$ is given by (3.5). Therefore, given that
the geodesic flow preserves $\alpha$ (Lemma 11), the wave front of
$P^{\tau}_{t}\circ\Pi_{-t}^{\tau}$ is

$\displaystyle\mathrm{WF}\left(P^{\tau}_{t}\circ\Pi_{-t}^{\tau}\right)$
$\displaystyle=$
$\displaystyle\mathrm{WF}\left(\Pi_{-t}^{\tau}\right)=\left(\Gamma_{-t}^{\tau}\times\mathrm{id}_{X^{\tau}}\right)^{*}\big{(}\mathrm{WF}(\Pi^{\tau})\big{)}$
$\displaystyle=$
$\displaystyle\left\\{\left(\Gamma^{\tau}_{t}(x),r\alpha^{\tau}_{\Gamma^{\tau}_{t}(x)},x,-r\,\alpha^{\tau}_{x}\right)\,:\,x\in
X^{\tau},\,r>0\right\\}.$

Hence the singular supports of $\Pi^{\tau}$ and
$P^{\tau}_{t}\circ\Pi_{-t}^{\tau}$ are, respectively, the diagonal and the
graph of $\Gamma^{\tau}_{-t}$ in $X^{\tau}\times X^{\tau}$.

Let $c>0$ be such that

$\mathrm{dist}_{X^{\tau}}\left(\Gamma^{\tau}_{-t}(x),x\right)\leq
c\,|t|,\quad\forall\,x\in X^{\tau},\,t\in\mathbb{R}.$ (108)

######  Lemma 57.

Provided $\epsilon>0$ is sufficiently small, if
$\mathrm{dist}_{X^{\tau}}(x_{1},x_{2})\geq 3\,c\,\epsilon$ then

$\max\left\\{\mathrm{dist}_{X^{\tau}}\left(x_{1},y\right),\mathrm{dist}_{X^{\tau}}\left(\Gamma^{\tau}_{-t}(y),x_{2}\right)\right\\}\geq
c\,\epsilon,\quad\forall\,y\in X^{\tau},\,t\in\mathrm{supp}(\chi).$

###### Proof.

For any $y\in X^{\tau}$ and $t\in(-\epsilon,\epsilon)$

$\displaystyle 3\,c\,\delta$ $\displaystyle\leq$
$\displaystyle\mathrm{dist}_{X^{\tau}}\left(x_{1},x_{2}\right)$
$\displaystyle\leq$
$\displaystyle\mathrm{dist}_{X^{\tau}}\left(x_{1},y\right)+\mathrm{dist}_{X^{\tau}}\left(y,\Gamma^{\tau}_{-t}(y)\right)+\mathrm{dist}_{X^{\tau}}\left(\Gamma^{\tau}_{-t}(y),x_{2}\right)$
$\displaystyle\leq$
$\displaystyle\mathrm{dist}_{X^{\tau}}\left(x_{1},y\right)+c\,\epsilon+\mathrm{dist}_{X^{\tau}}\left(\Gamma^{\tau}_{-t}(y),x_{2}\right).$

Hence,

$\mathrm{dist}_{X^{\tau}}\left(x_{1},y\right)+\mathrm{dist}_{X^{\tau}}\left(\Gamma^{\tau}_{-t}(y),x_{2}\right)\geq
2\,c\,\epsilon\quad\forall\,t\in\mathrm{supp}(\chi).$

∎

######  Lemma 58.

For any $\epsilon\sim 0^{+}$, if
$\mathrm{supp}(\chi)\subset[-\epsilon,\epsilon]$ and
$\mathrm{dist}_{X^{\tau}}(x_{1},x_{2})\geq 3\,c\,\epsilon$, then
$\Pi^{\tau}_{\chi,\lambda}(x_{1},x_{2})=O\left(\lambda^{-\infty}\right)$.

###### Proof.

Let us define

$\displaystyle U_{1}$ $\displaystyle:=$ $\displaystyle\left\\{y\in
X^{\tau}\,:\,\mathrm{dist}_{X^{\tau}}\left(x_{1},y\right)>\frac{c\,\epsilon}{2}\right\\},$
$\displaystyle U_{2}$ $\displaystyle:=$ $\displaystyle\left\\{y\in
X^{\tau}\,:\,\mathrm{dist}_{X^{\tau}}\left(\Gamma^{\tau}_{-t}(y),x_{2}\right)>\frac{c\,\epsilon}{2},\,\forall\,t\in\mathrm{supp}(\chi)\right\\}.$

By Lemma 57, $\mathcal{U}:=\\{U_{1},\,U_{2}\\}$ is an open cover of
$X^{\tau}$. Let $\gamma_{1}+\gamma_{2}=1$ be a partition of unity of
$X^{\tau}$ subordinate to $\mathcal{U}$. Then

$\Pi^{\tau}_{\chi,\lambda}(x_{1},x_{2})\sim\Pi^{\tau}_{\chi,\lambda}(x_{1},x_{2})_{1}+\Pi^{\tau}_{\chi,\lambda}(x_{1},x_{2})_{2},$

where

$\displaystyle\Pi^{\tau}_{\chi,\lambda}(x_{1},x_{2})_{j}$ $\displaystyle:=$
$\displaystyle\frac{1}{\sqrt{2\,\pi}}\,\int_{-\infty}^{+\infty}e^{-\imath\,\lambda\,t}\,\chi(t)\,\left[\int_{U_{j}}\gamma_{j}(y)\,\Pi^{\tau}(x_{1},y)\,\left[P^{\tau}_{t}\circ\Pi_{-t}^{\tau}\right](y,x_{2})\,\mathrm{d}V_{X^{\tau}}(y)\right]\,\mathrm{d}t.$

By definition of $U_{1}$, the function
$y\mapsto\gamma_{1}(y)\cdot\Pi^{\tau}(x_{1},y)$ is $\mathcal{C}^{\infty}$ (and
depends smoothly on $x_{1}$); therefore, for $j=1$ the inner integral in (4)
is the distributional kernel of the composition of a smoothing kernel with
$P^{\tau}_{t}$, whence it is itself a smoothing kernel. In other words, it a
smooth function of $(x_{1},x_{2},t)$, uniformly so under the assumption.
Therefore, $\Pi^{\tau}_{\chi,\lambda}(x_{1},x_{2})_{1}$ is the evaluation at
$\lambda$ of the Fourier transform of a compactly supported
$\mathcal{C}^{\infty}$ function of $t$, thus it is
$O\left(\lambda^{-\infty}\right)$.

Similarly, by definition of $U_{2}$, the function
$y\mapsto\gamma_{2}(y)\cdot\left[P^{\tau}_{t}\circ\Pi_{-t}^{\tau}\right](y,x_{2})$
is $\mathcal{C}^{\infty}$ (and depends smoothly on $x_{2}$), whence
$\Pi^{\tau}_{\chi,\lambda}(x_{1},x_{2})_{2}=O\left(\lambda^{-\infty}\right)$
by a similar argument.

∎

Assume from now on $\mathrm{dist}_{X^{\tau}}(x_{1},x_{2})<3\,c\,\epsilon$,
with $c$ as in (108). Let us define

$\displaystyle U^{\prime}_{1}$ $\displaystyle:=$ $\displaystyle\left\\{y\in
X^{\tau}\,:\,\mathrm{dist}_{X}(x_{1},y)<2\,\epsilon\right\\},$ (111)
$\displaystyle U^{\prime}_{2}$ $\displaystyle:=$ $\displaystyle\left\\{y\in
X^{\tau}\,:\,\mathrm{dist}_{X}(x_{1},y)>\epsilon\right\\}.$

Then $\mathcal{U}^{\prime}:=\\{U_{1}^{\prime},U_{2}^{\prime}\\}$ is an open
cover of $X^{\tau}$; let $\gamma_{1}^{\prime}+\gamma_{2}^{\prime}=1$ be a
partion of unity subordinate to $\mathcal{U}^{\prime}$. Thus

$\Pi^{\tau}_{\chi,\lambda}(x_{1},x_{2})\sim\Pi^{\tau}_{\chi,\lambda}(x_{1},x_{2})^{\prime}_{1}+\Pi^{\tau}_{\chi,\lambda}(x_{1},x_{2})^{\prime}_{2},$

where $\Pi^{\tau}_{\chi,\lambda}(x_{1},x_{2})^{\prime}_{j}$ is defined as in
(4), with $\gamma_{j}^{\prime}$ in place of $\gamma_{j}$. For $y\in
U_{2}^{\prime}$, the function $y\mapsto\Pi^{\tau}(x_{1},y)$ is
$\mathcal{C}^{\infty}$, and an adaptation of the previous argument implies
that
$\Pi^{\tau}_{\chi,\lambda}(x_{1},x_{2})^{\prime}_{2}=O\left(\lambda^{-\infty}\right)$.
Thus

$\Pi^{\tau}_{\chi,\lambda}(x_{1},x_{2})\sim\Pi^{\tau}_{\chi,\lambda}(x_{1},x_{2})^{\prime}_{1},$
(112)

where integration in $y$ is now over a small neighborhood of $x_{1}$.

We may assume that $x_{1},x_{2}$ and every $y$ in the support of the integrand
in $\Pi^{\tau}_{\chi,\lambda}(x_{1},x_{2})^{\prime}_{1}$ belong to a
Heisenberg coordinate neighborhood centered at some $x\in X^{\tau}$. Without
altering the asymptotics, $\Pi^{\tau}$ may be represented as a Fourier
integral operator (76), and apply Lemma 55 and (103). Then (4) may be
rewritten

$\displaystyle\Pi^{\tau}_{\chi,\lambda}(x_{1},x_{2})$ $\displaystyle\sim$
$\displaystyle\frac{1}{\sqrt{2\,\pi}}\,\int_{0}^{+\infty}\,\mathrm{d}u\,\int_{0}^{+\infty}\,\mathrm{d}v\,\int_{X^{\tau}}\,\mathrm{d}V_{X^{\tau}}(y)\int_{-\epsilon}^{+\epsilon}\,\mathrm{d}t$
$\displaystyle\left[e^{\imath\,[u\,\psi^{\tau}(x_{1},y)+v\,\psi^{\tau}(\Gamma^{\tau}_{-t}(y),x_{2})-\lambda\,t]}\,\gamma_{1}^{\prime}(y)\,\chi(t)\,s^{\tau}(x_{1},y,u)\,r^{\tau}_{t}\left(y,x_{2},v\right)\right].$

With the rescaling $u\rightsquigarrow\lambda\,u$ and
$v\rightsquigarrow\lambda\,v$, (4) becomes

$\displaystyle\Pi^{\tau}_{\chi,\lambda}(x_{1},x_{2})$ $\displaystyle\sim$
$\displaystyle\frac{\lambda^{2}}{\sqrt{2\,\pi}}\,\int_{0}^{+\infty}\,\mathrm{d}u\,\int_{0}^{+\infty}\,\mathrm{d}v\,\int_{X^{\tau}}\,\mathrm{d}V_{X^{\tau}}(y)\int_{-\epsilon}^{+\epsilon}\,\mathrm{d}t$
$\displaystyle\left[e^{\imath\,\lambda\,\Psi^{\tau}(x_{1},x_{2};u,v,t,y)}\,A_{\lambda}^{\tau}(x_{1},x_{2};u,v,t,y)\right],$

where

$\displaystyle\Psi^{\tau}(x_{1},x_{2};u,v,t,y)$ $\displaystyle:=$
$\displaystyle
u\,\psi^{\tau}(x_{1},y)+v\,\psi^{\tau}(\Gamma^{\tau}_{-t}(y),x_{2})-t,$ (115)
$\displaystyle A_{\lambda}^{\tau}(x_{1},x_{2};u,v,t,y)$ $\displaystyle:=$
$\displaystyle\chi(t)\,\gamma_{1}^{\prime}(y)\,s^{\tau}(x_{1},y,\lambda\,u)\,r^{\tau}_{t}\left(\Gamma^{\tau}_{-t}(y),x_{2},\lambda\,v\right).$
(116)

We shall let $y=x+(\theta,\mathbf{u})$ in Heisenberg local coordinates on
$X^{\tau}$ at $x$; then $\mathrm{d}V_{X^{\tau}}(y)$ by
$\mathcal{V}(\theta,\mathbf{u})\,\mathrm{d}\theta\,\mathrm{d}\mathbf{u}$, and
by (67)

$\mathcal{V}(0,\mathbf{0})=\frac{2^{d-1}}{\tau}.$ (117)

Integration in $(\theta,\mathbf{u},t)$ is compactly supported near the origin.
By (115), Proposition 33, Corollaries 44 and 48

$\partial_{\theta}\Psi^{\tau}=v-u+O(\epsilon),\quad\partial_{t}\Psi^{\tau}=\tau\,v-1+O(\epsilon).$
(118)

It follows by a standard ‘partial integration’ argument in the compactly
supported variables $(\theta,t)$ that the contribution of the locus where
$\|(u,v)\|\gg 0$ contributes negligibly to the asymtptotics. Similarly, if
$\epsilon\ll 1$, integration by parts in $t$ shows that the contribution of
the locus where $v\ll 1/\tau$ is also negligible. Finally, integration by
parts in $\theta$ yields a similar conclusion for $u$. We thus obtain the
following reduction.

######  Lemma 59.

There exists $D\gg 0$ such that the following holds. Let
$h\in\mathcal{C}^{\infty}_{0}\big{(}(1/(2D),2\,D)\big{)}$ be such that
$h\equiv 1$ on $(1/D,D)$. Then only a negligible contribution to the
asymptotics of (4) is lost, if the integrand is multiplied by $h(u)\cdot
h(v)$. Hence, integration in $(u,v)$ may be assumed to be compactly supported
in $(1/(2D),2\,D)^{2}$.

###### Proof of Statement 1) of Theorem 3.

There exist constants $0<a\leq A$ such that

$a\,\mathrm{dist}_{X^{\tau}}\left(\Gamma^{\tau}_{-t}(y),x\right)\leq\mathrm{dist}_{X^{\tau}}\left(y,\Gamma^{\tau}_{t}(x)\right)\leq
A\,\mathrm{dist}_{X^{\tau}}\left(\Gamma^{\tau}_{-t}(y),x\right)$

for every $t\in(-\epsilon,\epsilon)$ and $y,\,x\in X^{\tau}$.

Assume $\mathrm{dist}_{X^{\tau}}\left(x_{1},x_{2}^{\chi}\right)\geq
C\,\lambda^{\delta-1/2}$, and define

$\displaystyle U^{\lambda}_{1}$ $\displaystyle:=$ $\displaystyle\left\\{y\in
X^{\tau}\,:\,\mathrm{dist}_{X^{\tau}}(y,x_{1})<(C/2)\,\lambda^{\delta-1/2}\right\\},$
$\displaystyle U^{\lambda}_{2}$ $\displaystyle:=$ $\displaystyle\left\\{y\in
X^{\tau}\,:\,\mathrm{dist}_{X^{\tau}}(y,x_{1})>(C/3)\,\lambda^{\delta-1/2}\right\\}.$

Then $\mathcal{U}^{\lambda}:=\\{U^{\lambda}_{1},\,U^{\lambda}_{2}\\}$ is an
open cover of $X^{\tau}$.

If $y\in U^{\lambda}_{1}$, for any $t\in(-\epsilon,\epsilon)$ we have

$\displaystyle C\,\lambda^{\delta-1/2}$ $\displaystyle\leq$
$\displaystyle\mathrm{dist}_{X^{\tau}}\left(x_{1},,\Gamma^{\tau}_{t}(x_{2})\right)\leq\mathrm{dist}_{X^{\tau}}(x_{1},y)+\mathrm{dist}_{X^{\tau}}\left(y,\Gamma^{\tau}_{t}(x_{2})\right)$
$\displaystyle<$
$\displaystyle(C/2)\,\lambda^{\delta-1/2}+\mathrm{dist}_{X^{\tau}}\left(y,\Gamma^{\tau}_{t}(x_{2})\right)$
$\displaystyle\Rightarrow$
$\displaystyle\mathrm{dist}_{X^{\tau}}\left(\Gamma^{\tau}_{-t}(y),x_{2}\right)\geq\frac{1}{A}\,\mathrm{dist}_{X^{\tau}}\left(y,\Gamma^{\tau}_{t}(x_{2})\right)\geq\frac{C}{2\,A}\,\lambda^{\delta-1/2}.$

By Corollary 49, in the same range

$\left|\frac{\partial\Psi^{\tau}}{\partial
v}\right|=\left|\psi^{\tau}\left(\Gamma^{\tau}_{-t}(y),x_{2}\right)\right|\geq
D^{\tau}\,\left(\frac{C}{2\,A}\,\right)^{2}\lambda^{2\,\delta-1}.$

Integrating by parts in $v$, we obtain that the contribution of
$U_{1}^{\lambda}$ to the asymptotics of (4) is
$O\left(\lambda^{-\infty}\right)$.

On the other hand, if $y\in U^{\lambda}_{2}$ then

$\displaystyle\left|\frac{\partial\Psi^{\tau}}{\partial
u}\right|=\left|\psi^{\tau}\left(y,x_{1}\right)\right|\geq
D^{\tau}\,\mathrm{dist}_{X^{\tau}}(y,x_{1})^{2}>\frac{D^{\tau}\,C^{2}}{9}\,\lambda^{2\,\delta-1}.$

Integrating by parts in $u$, we obtain that the contribution of
$U_{2}^{\lambda}$ is also $O\left(\lambda^{-\infty}\right)$. ∎

We focus on statement 2. Let us set

$x_{j\lambda}:=x+\left(\frac{\theta_{j}}{\sqrt{\lambda}},\frac{\mathbf{v}_{j}}{\sqrt{\lambda}}\right).$
(119)

Under the assumptions,

$\mathrm{dist}_{X^{\tau}}(x,x_{j\lambda})\leq
2\,C\,\lambda^{\delta-\frac{1}{2}}.$ (120)

Let $h^{\tau}$ be a Riemannian metric on $X^{\tau}$ that in a sufficiently
small neighborhood of $x$ is given in Heisenberg local coordinates by

$h^{\tau}=\frac{1}{\tau^{2}}(\mathrm{d}\theta)^{2}+\mathrm{d}z^{\prime}\otimes\mathrm{d}\overline{z}^{\prime}+\mathrm{d}\overline{z}^{\prime}\otimes\mathrm{d}z^{\prime}.$

Thus $h^{\tau}_{x}=\hat{\kappa}^{\tau}_{x}$ by Corollary 34. Let
$\widetilde{\mathrm{dist}}_{X^{\tau}}:X^{\tau}\times
X^{\tau}\rightarrow\mathbb{R}$ be the Riemannian distance of $h^{\tau}$. By
the latter remark,

$\widetilde{\mathrm{dist}}_{X^{\tau}}(x,y)/\mathrm{dist}_{X^{\tau}}(x,y)\rightarrow
1\quad\text{for}\quad y\rightarrow x.$

For $r>0$, let $B_{x}(r)\subseteq X^{\tau}$ be the open ball centered at $x$
for $h^{\tau}$. Let $r$ be small enough that $B_{x}(r)\subset U^{\tau}$, and
consider the open cover of $X^{\tau}$

$\mathcal{B}:=\left\\{B_{x}(r),\overline{B_{x}(r/2)}^{c}\right\\}.$

Let $(b_{1},b_{2})$ be a partition of unity subordinate to $\mathcal{B}$. For
some fixed $R>0$ and any $\lambda\gg 1$, let
$b_{j}^{\lambda}\in\mathcal{C}^{\infty}(X^{\tau})$ be defined in Heisenberg
local coordinates by

$b_{j}^{\lambda}\left((\theta_{0},z^{\prime})\right):=b_{j}\left(\frac{r}{12\,C}\,\lambda^{1/2-\delta}\,(\theta_{0},z^{\prime})\right).$

Thus $\left\\{b_{1}^{\lambda},b_{2}^{\lambda}\right\\}$ is a partition of
unity subordinate to the rescaled open cover
$\mathcal{B}^{\lambda}:=\left\\{B_{x}\left(12\,C\,\lambda^{\delta-1/2}\right),\overline{B_{x}\left(6\,C\,\lambda^{\delta-1/2}\right)}^{c}\right\\}$.
For $y\in\mathrm{supp}\left(b_{2}^{\lambda}\right)$ and $\lambda\gg 0$,

$\mathrm{dist}_{X^{\tau}}\left(x,y\right)\geq\frac{1}{2}\,\widetilde{\mathrm{dist}}_{X^{\tau}}\left(x,y\right)\geq
3\,C\,\lambda^{\delta-1/2}.$ (121)

The asymptotics of $\Pi^{\tau}_{\chi,\lambda}(x_{1\lambda},x_{2\lambda})$ are
given by (4) with $(x_{1},x_{2})$ replaced by $(x_{1\lambda},x_{2\lambda})$.

######  Lemma 60.

Only a negligible contribution to the asymptotics of
$\Pi^{\tau}_{\chi,\lambda}(x_{1\lambda},x_{2\lambda})$ is lost, if the
integrand in (4) is multiplied by $b_{1}^{\lambda}(y)$.

###### Proof.

For $y\in\mathrm{supp}\left(b_{2}^{\lambda}\right)$ and $\lambda\gg 0$,

$\mathrm{dist}_{X^{\tau}}\left(x,y\right)\geq\frac{1}{2}\,\widetilde{\mathrm{dist}}_{X^{\tau}}\left(x,y\right)\geq
3\,C\,\lambda^{\delta-1/2}.$ (122)

Hence where $b_{1}^{\lambda}(y)=0$ in view of (120)

$\displaystyle\mathrm{dist}_{X^{\tau}}\left(x_{1\lambda},y\right)$
$\displaystyle\geq$
$\displaystyle\mathrm{dist}_{X^{\tau}}\left(x,y\right)-\mathrm{dist}_{X^{\tau}}\left(x,x_{1\lambda}\right)$
$\displaystyle\geq$ $\displaystyle
3\,C\,\lambda^{\delta-1/2}-2\,C\,\lambda^{\delta-1/2}=C\,\lambda^{\delta-1/2}.$

Recalling (115), with $x_{j}$ replaced by $x_{j\lambda}$, the claim follows
arguing as in the proof of statement 1. (integrate by parts in $u$). ∎

We conclude from (4) and Lemma 60 that

$\displaystyle\Pi^{\tau}_{\chi,\lambda}(x_{1\lambda},x_{2\lambda})$
$\displaystyle\sim$
$\displaystyle\frac{\lambda^{2}}{\sqrt{2\,\pi}}\,\int_{1/(2\,D)}^{2\,D}\,\mathrm{d}u\,\int_{1/(2\,D)}^{2\,D}\,\mathrm{d}v\,\int_{-\infty}^{+\infty}\,\mathrm{d}\theta\,\int_{\mathbb{R}^{2\,d}}\,\mathrm{d}\mathbf{u}\,\int_{-\epsilon}^{+\epsilon}\,\mathrm{d}t$
$\displaystyle\left[e^{\imath\,\lambda\,\Psi^{\tau}\big{(}x_{1\lambda},x_{2\lambda};u,v,t,y(\theta,\mathbf{u})\big{)}}\,b_{1}^{\lambda}\big{(}y(\theta,\mathbf{u})\big{)}\,A_{\lambda}^{\tau}\big{(}x_{1\lambda},x_{2\lambda};u,v,t,y(\theta,\mathbf{u})\big{)}\,\mathcal{V}(\theta,\mathbf{u})\right].$

On the domain of integration, $(\theta/\tau,\mathbf{u})$ ranges in a shrinking
ball of radius $O\left(\lambda^{\delta-1/2}\right)$ centered at the origin in
$\mathbb{R}^{2d+1}$.

Next we show that integration in $\mathrm{d}t$ may be localized to a shrinking
neighbourhood of the origin.

Let us fix a constant $C_{1}>0$ such that for all $\lambda\gg 0$ and
$y=y(\theta,\mathbf{u})$ on the support of
$b_{1}^{\lambda}\big{(}y(\theta,\mathbf{u})\big{)}$

$\displaystyle\mathrm{dist}_{X^{\tau}}\left(x_{2\lambda},y\right)\leq\mathrm{dist}_{X_{\tau}}(x_{2\lambda},x)+\mathrm{dist}_{X^{\tau}}(x,y)\leq
C_{1}\,\lambda^{\delta-\frac{1}{2}}.$ (124)

Let $\beta\in\mathcal{C}^{\infty}_{0}(\mathbb{R})$ be such that
$\beta(t)\equiv 1$ on $[-1,1]$, and define
$\beta^{\lambda}(t):=\beta\left(\frac{1}{3\,C_{1}}\,\lambda^{\frac{1}{2}-\delta}\,t\right)$.
Then $1-\beta^{\lambda}(t)=0$ if $|t|\leq
3\,C_{1}\,\lambda^{\delta-\frac{1}{2}}$.

We have

$\Pi^{\tau}_{\chi,\lambda}(x_{1\lambda},x_{2\lambda})\sim\Pi^{\tau}_{\chi,\lambda}(x_{1\lambda},x_{2\lambda})^{\prime}+\Pi^{\tau}_{\chi,\lambda}(x_{1\lambda},x_{2\lambda})^{\prime\prime},$

where $\Pi^{\tau}_{\chi,\lambda}(x_{1\lambda},x_{2\lambda})^{\prime}$ and
$\Pi^{\tau}_{\chi,\lambda}(x_{1\lambda},x_{2\lambda})^{\prime\prime}$ are as
in (4), but with the integrand multiplied, respectively, by
$\beta^{\lambda}(t)$ and $1-\beta^{\lambda}(t)$.

######  Lemma 61.

$\Pi^{\tau}_{\chi,\lambda}(x_{1\lambda},x_{2\lambda})^{\prime\prime}=O\left(\lambda^{-\infty}\right)$.

###### Proof.

Where $1-\beta^{\lambda}(t)\neq 0$, we have
$|t|>3\,C_{1}\,\lambda^{\delta-\frac{1}{2}}$. In view of (43), if $\epsilon$
is small enough and and
$|\epsilon|>|t|>3\,C_{1}\,\lambda^{\delta-\frac{1}{2}}$ then

$\mathrm{dist}_{X^{\tau}}\left(\Gamma^{\tau}_{-t}(y),y\right)\geq\frac{t}{2}\geq\frac{3}{2}\,C_{1}\,\lambda^{\delta-\frac{1}{2}}.$

Therefore, by (124) on the support of the integrand of
$\Pi^{\tau}_{\chi,\lambda}(x_{1\lambda},x_{2\lambda})^{\prime\prime}$ we have

$\displaystyle\mathrm{dist}_{X^{\tau}}\left(\Gamma^{\tau}_{-t}(y),x_{2\lambda}\right)$
$\displaystyle\geq$
$\displaystyle\mathrm{dist}_{X^{\tau}}\left(\Gamma^{\tau}_{-t}(y),y\right)-\mathrm{dist}_{X^{\tau}}\left(x_{2\lambda},y\right)\geq\frac{C_{1}}{2}\,\lambda^{\delta-\frac{1}{2}}.$

The claim follows again by the argument in the proof of statement 1, by
iterated integration by parts in $v$. ∎

With the rescalings $\theta\rightsquigarrow\theta/\sqrt{\lambda}$,
$\mathbf{u}\rightsquigarrow\mathbf{u}/\sqrt{\lambda}$, $t\rightsquigarrow
t/\sqrt{\lambda}$, (4) becomes

$\displaystyle\Pi^{\tau}_{\chi,\lambda}(x_{1\lambda},x_{2\lambda})$
$\displaystyle\sim$
$\displaystyle\frac{\lambda^{2-d}}{\sqrt{2\,\pi}}\,\int_{1/(2\,D)}^{2\,D}\,\,\mathrm{d}u\,\int_{1/(2\,D)}^{2\,D}\,\,\mathrm{d}v\,\int_{-\infty}^{+\infty}\,\mathrm{d}\theta\,\int_{\mathbb{R}^{2\,d}}\,\mathrm{d}\mathbf{u}\,\int_{-\infty}^{+\infty}\,\mathrm{d}t$
$\displaystyle\left[e^{\imath\,\lambda\,\Psi^{\tau}_{\lambda}\big{(}x_{1\lambda},x_{2\lambda};u,v,t,\theta,\mathbf{u}\big{)}}\,B_{\lambda}^{\tau}\big{(}x_{1\lambda},x_{2\lambda};u,v,t,\theta,\mathbf{u}\big{)}\right],$

where, in view of (115),

$\displaystyle\Psi^{\tau}_{\lambda}\big{(}x_{1\lambda},x_{2\lambda};u,v,t,\theta,\mathbf{u}\big{)}$
$\displaystyle:=$
$\displaystyle\Psi^{\tau}\left(x_{1\lambda},x_{2\lambda};u,v,\frac{t}{\sqrt{\lambda}},y\left(\frac{1}{\sqrt{\lambda}}\,(\theta,\mathbf{u})\right)\right),$
$\displaystyle=$ $\displaystyle
u\,\psi^{\tau}\left(x_{1\lambda},y\left(\frac{1}{\sqrt{\lambda}}\,(\theta,\mathbf{u})\right)\right)+v\,\psi^{\tau}\left(\Gamma^{\tau}_{-t/\sqrt{\lambda}}\left(y\left(\frac{1}{\sqrt{\lambda}}\,(\theta,\mathbf{u})\right)\right),x_{2\lambda}\right)$
$\displaystyle-\frac{t}{\sqrt{\lambda}},$ $\displaystyle
B_{\lambda}^{\tau}\big{(}x_{1\lambda},x_{2\lambda};u,v,t,\theta,\mathbf{u})\big{)}$
$\displaystyle:=$ $\displaystyle
b_{1}^{\lambda}\left(y\left(\frac{1}{\sqrt{\lambda}}\,(\theta,\mathbf{u}\right)\right)\,A_{\lambda}^{\tau}\left(x_{1\lambda},x_{2\lambda};u,v,\frac{t}{\sqrt{\lambda}},y\left(\frac{1}{\sqrt{\lambda}}\,(\theta,\mathbf{u})\right)\right)$
$\displaystyle\cdot\mathcal{V}\left(\frac{1}{\sqrt{\lambda}}\,(\theta,\mathbf{u})\right).$

Integration in $(t,\theta,\mathbf{v})$ in (4) is over an expanding ball
centered at the origin and radius $O\left(\lambda^{\delta}\right)$.

We now make explicit the dependence of $\Psi^{\tau}_{\lambda}$ in (4) on the
rescaled variables. By Proposition 47,

$\displaystyle\imath\,\psi^{\tau}\left(x_{1\lambda},y\left(\frac{1}{\sqrt{\lambda}}\,(\theta,\mathbf{u})\right)\right)$
$\displaystyle=$
$\displaystyle\imath\,\frac{\theta_{1}-\theta}{\sqrt{\lambda}}-\frac{1}{4\,\tau^{2}\,\lambda}\,\left(\theta_{1}-\theta\right)^{2}+\frac{\psi_{2}^{\omega_{x}}\left(\mathbf{v}_{1},\mathbf{u}\right)}{\lambda}+R_{3}\left(\frac{\theta_{1}}{\sqrt{\lambda}},\frac{\mathbf{v}_{1}}{\sqrt{\lambda}},\frac{\theta}{\sqrt{\lambda}},\frac{\mathbf{u}}{\sqrt{\lambda}}\right).$

Furthermore, by Corollary 44 and (105),

$\displaystyle\Gamma^{\tau}_{-t/\sqrt{\lambda}}\left(y\left(\frac{1}{\sqrt{\lambda}}\,(\theta,\mathbf{u})\right)\right)$
$\displaystyle=$ $\displaystyle
x+\left(\frac{1}{\sqrt{\lambda}}\left(\theta+\tau\,t\right)-\frac{t}{\lambda}\,\left(\frac{1}{2\,\tau^{2}}\,a_{x}\,\theta+\langle\mathbf{A}_{x},\mathbf{u}\rangle\right)+R_{3}\left(\frac{\tau\,t}{\sqrt{\lambda}},\frac{\theta}{\sqrt{\lambda}},\frac{\mathbf{v}}{\sqrt{\lambda}}\right),\right.$
$\displaystyle\left.\frac{\mathbf{u}}{\sqrt{\lambda}}+\mathbf{R}_{2}\left(\frac{\tau\,t}{\sqrt{\lambda}},\frac{\theta}{\sqrt{\lambda}},\frac{\mathbf{u}}{\sqrt{\lambda}}\right)\right).$

Therefore, again by Proposition 47,

$\displaystyle\imath\,\psi^{\tau}\left(\Gamma^{\tau}_{-t/\sqrt{\lambda}}\left(y\left(\frac{1}{\sqrt{\lambda}}\,(\theta,\mathbf{u})\right)\right),x_{2\lambda}\right)$
$\displaystyle=$
$\displaystyle\frac{\imath}{\sqrt{\lambda}}\,(\theta+\tau\,t-\theta_{2})-\frac{\imath}{\lambda}\,t\,\left(\frac{1}{2\,\tau^{2}}\,a_{x}\,\theta+\langle\mathbf{A}_{x},\mathbf{u}\rangle\right)$
$\displaystyle-\frac{1}{4\,\tau^{2}\,\lambda}\,(\theta+\tau\,t-\theta_{2})^{2}+\frac{\psi_{2}^{\omega_{x}}\left(\mathbf{u},\mathbf{v}_{2}\right)}{\lambda}+R_{3}\left(\frac{\tau\,t}{\sqrt{\lambda}},\frac{\theta_{2}}{\sqrt{\lambda}},\frac{\mathbf{v}_{2}}{\sqrt{\lambda}},\frac{\theta}{\sqrt{\lambda}},\frac{\mathbf{u}}{\sqrt{\lambda}}\right).$

We finally obtain

$\displaystyle\imath\,\lambda\,\Psi^{\tau}_{\lambda}\big{(}x_{1\lambda},x_{2\lambda};u,v,t,\theta,\mathbf{u}\big{)}$
$\displaystyle=$
$\displaystyle\imath\,\sqrt{\lambda}\,\Upsilon^{\tau}(t,v,\theta,u)+S(\mathbf{u},t,v,\theta,u)+\lambda\,R_{3}\left(\frac{\tau\,t}{\sqrt{\lambda}},\frac{\theta_{j}}{\sqrt{\lambda}},\frac{\mathbf{v}_{j}}{\sqrt{\lambda}},\frac{\theta}{\sqrt{\lambda}},\frac{\mathbf{u}}{\sqrt{\lambda}}\right),$

where

$\displaystyle\Upsilon^{\tau}(t,v,\theta,u)$ $\displaystyle=$
$\displaystyle\Upsilon^{\tau}(\theta_{j},\mathbf{v}_{j};t,v,\theta,u)$
$\displaystyle:=$ $\displaystyle
u\,(\theta_{1}-\theta)+v\,(\theta+\tau\,t-\theta_{2})-t,$ $\displaystyle
S(\mathbf{u},t,v,\theta,u)=S_{\theta_{j},\mathbf{v}_{j}}(\mathbf{u},t,v,\theta,u)$
$\displaystyle:=$
$\displaystyle-\imath\,v\,t\,\left(\frac{1}{2\,\tau^{2}}\,a_{x}\,\,\theta+\langle\mathbf{A}_{x},\mathbf{u}\rangle\right)-\frac{1}{4\,\tau^{2}}\,u\,\left(\theta_{1}-\theta\right)^{2}-\frac{1}{4\,\tau^{2}}\,v\,(\theta+\tau\,t-\theta_{2})^{2}$
$\displaystyle+u\,\psi_{2}^{\omega_{x}}\left(\mathbf{v}_{1},\mathbf{u}\right)+v\,\psi_{2}^{\omega_{x}}\left(\mathbf{u},\mathbf{v}_{2}\right),$

and $R_{3}$ vanishes to third order at the origin. Notice that
$S(\mathbf{u},t,v,\theta,u)$ is homogenous of degree $2$ in the rescaled
variables.

If we set $\theta^{\prime}:=\theta-\theta_{1}$,
$t^{\prime}:=\theta+\tau\,t-\theta_{2}$, and
$\mathbf{u}^{\prime}:=\mathbf{u}-\mathbf{v}_{1}$, then

$\Re\big{(}S(\mathbf{u},t,v,\theta,u)\big{)}\leq-
b_{0}\,\|(t^{\prime},\theta^{\prime},\mathbf{u}^{\prime})\|^{2}$ (134)

for some $b_{0}>0$.

In light of (4), we can rewrite (4) as follows:

$\displaystyle\Pi^{\tau}_{\chi,\lambda}(x_{1\lambda},x_{2\lambda})\sim\frac{\lambda^{2-d}}{\sqrt{2\,\pi}}\,\int_{\mathbb{R}^{2\,d-2}}\,I_{\lambda}(\mathbf{u})\,\mathrm{d}\mathbf{u}$
(135)

where

$\displaystyle I_{\lambda}(\mathbf{u})$ $\displaystyle:=$
$\displaystyle\int_{-\infty}^{+\infty}\,\mathrm{d}t\,\int_{1/(2\,D)}^{2\,D}\,\mathrm{d}v\,\int_{-\infty}^{+\infty}\,\mathrm{d}\theta\,\int_{1/(2\,D)}^{2\,D}\,\,\mathrm{d}u$
$\displaystyle\left[e^{\imath\,\sqrt{\lambda}\,\Upsilon^{\tau}(t,v,\theta,u)}\,\tilde{B}_{\lambda}^{\tau}\big{(}x_{1\lambda},x_{2\lambda};\mathbf{u},t,v,\theta,u\big{)}\right]$

and, using (77), (116), (117), and (4)

$\displaystyle\tilde{B}_{\lambda}^{\tau}\big{(}x_{1\lambda},x_{2\lambda};\mathbf{u},t,v,\theta,u\big{)}$
$\displaystyle:=$ $\displaystyle
e^{S(\mathbf{u},t,v,\theta,u)+\lambda\,R_{3}\left(\frac{\tau\,t}{\sqrt{\lambda}},\frac{\theta_{j}}{\sqrt{\lambda}},\frac{\mathbf{v}_{j}}{\sqrt{\lambda}},\frac{\theta}{\sqrt{\lambda}},\frac{\mathbf{u}}{\sqrt{\lambda}}\right)}\,B_{\lambda}^{\tau}\big{(}x_{1\lambda},x_{2\lambda};\mathbf{u},t,v,\theta,u\big{)}.$

For every $N=1,2,\ldots$, Taylor expansion in the rescaled variables yields
for the third order remaninder $R_{3}$ in the exponent in (4)

$\displaystyle\lambda\,R_{3}\left(\frac{\tau\,t}{\sqrt{\lambda}},\frac{\theta_{j}}{\sqrt{\lambda}},\frac{\mathbf{v}_{j}}{\sqrt{\lambda}},\frac{\theta}{\sqrt{\lambda}},\frac{\mathbf{u}}{\sqrt{\lambda}}\right)$
$\displaystyle=$
$\displaystyle\sum_{s=0}^{N}\,\lambda^{1-\frac{3+s}{2}}\,P_{3+s}(\theta_{j},\mathbf{v}_{j},\theta,\mathbf{u})+\lambda\,R_{4+N}\left(\frac{\tau\,t}{\sqrt{\lambda}},\frac{\theta_{j}}{\sqrt{\lambda}},\frac{\mathbf{v}_{j}}{\sqrt{\lambda}},\frac{\theta}{\sqrt{\lambda}},\frac{\mathbf{u}}{\sqrt{\lambda}}\right),$

where $P_{l}$ denotes a homogeneous polynomial of degree $l$ in the argument.
On the domain of integration, the latter summand is bounded in absolute value
by $C_{N}\,\lambda^{1-\left(\frac{1}{2}-\delta\right)\,(4+N)}$ for some
constant $C_{N}>0$. Passing to the exponential, we therefore obtain an
asymptotic expansion

$\displaystyle
e^{\lambda\,R_{3}\left(\frac{\tau\,t}{\sqrt{\lambda}},\frac{\theta_{j}}{\sqrt{\lambda}},\frac{\mathbf{v}_{j}}{\sqrt{\lambda}},\frac{\theta}{\sqrt{\lambda}},\frac{\mathbf{u}}{\sqrt{\lambda}}\right)}=\sum_{k=0}^{+\infty}\frac{\lambda^{k}}{k!}\,R_{3}\left(\frac{\tau\,t}{\sqrt{\lambda}},\frac{\theta_{j}}{\sqrt{\lambda}},\frac{\mathbf{v}_{j}}{\sqrt{\lambda}},\frac{\theta}{\sqrt{\lambda}},\frac{\mathbf{u}}{\sqrt{\lambda}}\right)^{k}$
$\displaystyle\sim$
$\displaystyle\sum_{k=0}^{+\infty}\frac{1}{k!}\,\left[\sum_{s=0}^{N}\,\lambda^{1-\frac{3+s}{2}}\,P_{3+s}(\theta_{j},\mathbf{v}_{j},\theta,\mathbf{u})\right]^{k}=\sum_{k=0}^{+\infty}\,\lambda^{-\frac{k}{2}}\,Q_{k}(\theta_{j},\mathbf{v}_{j},\theta,\mathbf{u}),$

where $Q_{k}$ is a polynomial of degree $\leq 3\,k$.

On the other hand, the asymptotic expansion (77) yields

$\displaystyle s^{\tau}(x_{1\lambda},y,\lambda\,u)\sim\sum_{k\geq
0}\,(\lambda\,u)^{d-1-k}\,s^{\tau}_{j}(x_{1\lambda},y)$ $\displaystyle=$
$\displaystyle(\lambda\,u)^{d-1}\,s^{\tau}_{0}(x,x)+\sum_{k+l\geq
1}\lambda^{d-1-k-\frac{l}{2}}\,u^{d-1-k}\,S^{\tau}_{k,l}(\theta_{1},\mathbf{v}_{1},\theta,\mathbf{u}),$

where $S^{\tau}_{k,l}$ is homogenous of degree $l$. Similarly, the asymptotic
expansion in Lemma 55 yields

$\displaystyle
r^{\tau}_{t}\left(y,x_{2\,\lambda},\,\lambda\,v\right)\sim\sum_{k\geq
0}\,(\lambda\,u)^{d-1-k}\,r^{\tau}_{t\,j}(y,x_{2\,\lambda})$ $\displaystyle=$
$\displaystyle(\lambda\,u)^{d-1}\,s^{\tau}_{0}(x,x)+\sum_{k+l\geq
1}\lambda^{d-1-k-\frac{l}{2}}\,u^{d-1-k}\,R^{\tau}_{k,l}(\theta_{1},\mathbf{v}_{1},\theta,\mathbf{u}),$

where again $R^{\tau}_{k,l}$ is homogeneous of degree $l$, and we have used
that $r^{\tau}_{0,\,0}(x,x)=s^{\tau}_{0}(x,x)$.

Multiplying (4), (4), (4), and the Taylor expansion of $\chi$ and
$\mathcal{V}$ at the origin, we obtain an asymptotic expansion

$\displaystyle\tilde{B}_{\lambda}^{\tau}\big{(}x_{1\lambda},x_{2\lambda};\mathbf{u},t,v,\theta,u\big{)}$
$\displaystyle\sim$
$\displaystyle\lambda^{2d-2}\,\frac{2^{d-1}}{\tau}\,\chi(0)\,e^{S(\mathbf{u},t,v,\theta,u)}\,u^{d-1}\,v^{d-1}\,s^{\tau}_{0}(x,x)^{2}\cdot\beta\left(\lambda^{-\delta}\,\left(\mathbf{u},t,\theta\right)\right)$
$\displaystyle\cdot\left[1+\sum_{k\geq
1}\lambda^{-k/2}\,B_{k}(x;\mathbf{u},\theta_{j},\mathbf{v}_{j},t,v,\theta,u)\right],$

where $B_{k}(x;\cdot)$ is a polynomial of degree $\leq 3\,k$ in
$(\theta_{j},\mathbf{v}_{j},\mathbf{u},t,\theta)$, while $\beta$ is compactly
supported and identically equal to $1$ in a suitable neighborhood of the
origin. The latter is indeed an asymptotic expansion for $\delta\in(0,1/6)$.
Furthermore, fractional powers of $\lambda$ arise from Taylor expansion in
$(\theta_{j},\mathbf{v}_{j},\mathbf{u})$, while the asymptotic expansion for
the amplitude in the Szegö kernel parametrix is by descending integer powers.
Hence $P_{j}$ will be even for $j$ even (corresponding to integer powers of
$\lambda$), and odd for $j$ odd (corresponding to fractional powers).

Inserting (4) in (4), we obtain an asymptotic expansion for the integrand
which, in view of (134) or the previous remark on the domain of integration,
can be integrated term by term. Each term is an oscillatory integral with
phase (4) in the parameters $(t,v,\theta,u)$, and depending parametrically on
the other parameters.

Now we remark that the asymptotics of (135) are unaltered, if integration in
$(t,\theta)$ is restricted to a suitable compact set. In fact, since
$\partial_{u}\Upsilon^{\tau}=\theta_{1}-\theta$,
$\partial_{v}\Upsilon^{\tau}=\theta+\tau\,t-\theta_{2}$, integration by parts
in $(u,v)$ implies the following.

######  Lemma 62.

Only a negligible contribution to the asymptotics of (4) is lost, if the
amplitude (4) is multiplied by a compactly supported cut-off function in
$(\theta,t)$, identically equal to $1$ near
$\big{(}\theta_{1},(\theta_{2}-\theta_{1})/\tau\big{)}$.

We shall leave the latter cut-off implicit in the following, and simply assume
henceforth that integration in $(\theta,t)$ is over a compact neighborhood of
$\big{(}\theta_{1},(\theta_{2}-\theta_{1})/\tau\big{)}$.

The proof of the following is omitted.

######  Lemma 63.

$\Upsilon^{\tau}$ in (4) has a unique stationary point
$P_{s}=(t_{s},v_{s},\theta_{s},u_{s})$, given by

$P_{s}^{\tau}=\left(\frac{\theta_{2}-\theta_{1}}{\tau},\,\frac{1}{\tau},\,\theta_{1},\,\frac{1}{\tau}\right).$

The critical point is non-degenerate. The Hessian matrix and its inverse at
the critical point are

$H(\Upsilon^{\tau})_{P_{s}^{\tau}}=\begin{pmatrix}0&\tau&0&0\\\ \tau&0&1&0\\\
0&1&0&-1\\\ 0&0&-1&0\end{pmatrix},\quad
H(\Upsilon^{\tau})_{P_{s}^{\tau}}^{-1}=\begin{pmatrix}0&1/\tau&0&1/\tau\\\
1/\tau&0&0&0\\\ 0&0&0&-1\\\ 1/\tau&0&-1&0\end{pmatrix}.$

The Hessian determinant, the Hessian signature, and the critical value are

$\det
H(\Upsilon^{\tau})_{P_{s}^{\tau}}=\tau^{2},\quad\mathrm{sgn}H(\Upsilon^{\tau})_{P_{s}^{\tau}}=0,\quad\Upsilon^{\tau}(P_{s}^{\tau})=\frac{\theta_{1}-\theta_{2}}{\tau}.$

The third order remainder at $P_{s}$ is zero.

Let us set

$L:=\frac{1}{\tau}\,\left(\frac{\partial^{2}}{\partial t\partial
u}+\frac{\partial^{2}}{\partial t\partial
v}\right)-\frac{\partial^{2}}{\partial\theta\,\partial u},$ (143)

and apply the Lemma of Stationary Phase to (4) in correspondence to the $k$-th
summand in (4); we obtain an asymptotic expansion whose $r$-th summand (for
$r=0,1,2,\ldots$) is a multiple of

$\displaystyle(4\,\pi^{2}/\tau)\cdot
e^{\imath\,\sqrt{\lambda}\,\frac{\theta_{1}-\theta_{2}}{\tau}}\cdot\lambda^{2d-3-(r+k)/2}$
$\displaystyle\cdot\left.L^{r}\left(e^{S(\mathbf{u},t,v,\theta,u)}\,u^{d-1}\,v^{d-1}\,P_{k}(x;\theta_{j},\mathbf{v}_{j},\mathbf{u},t,v,\theta,u)\right)\right|_{P_{s}},$

where the dependence on $(\theta_{j},\mathbf{v}_{j})$ of the exponent is left
implicit for brevity. In view of (4), with the notation of Definition 56 the
value of $S(\mathbf{u},t,v,\theta,u)$ at the critical point is

$\displaystyle S_{c}(\theta_{j},\mathbf{v}_{j},\mathbf{u})$ $\displaystyle:=$
$\displaystyle
S_{\theta_{j},\mathbf{v}_{j}}(\mathbf{u},t_{s},v_{s},\theta_{s},u_{s})$
$\displaystyle=$
$\displaystyle\imath\cdot\left[\frac{(\theta_{1}-\theta_{2})\,\theta_{1}}{2\,\tau^{4}}\,a_{x}+\frac{\theta_{1}-\theta_{2}}{\tau^{2}}\cdot\langle\mathbf{A}_{x},\mathbf{u}\rangle\right]$
$\displaystyle+\frac{1}{\tau}\,\psi_{2}^{\omega_{x}}\left(\mathbf{v}_{1},\mathbf{u}\right)+\frac{1}{\tau}\,\psi_{2}^{\omega_{x}}\left(\mathbf{u},\mathbf{v}_{2}\right)$
$\displaystyle=$
$\displaystyle\imath\,\frac{(\theta_{1}-\theta_{2})\,\theta_{1}}{2\,\tau^{4}}\,a_{x}-\frac{\imath}{\tau}\cdot\omega_{x}\big{(}J_{x}\big{(}\mathbf{a}_{x}(\theta_{1},\theta_{2})\big{)},\mathbf{u}\big{)}$
$\displaystyle+\frac{1}{\tau}\,\psi_{2}^{\omega_{x}}\left(\mathbf{v}_{1},\mathbf{u}\right)+\frac{1}{\tau}\,\psi_{2}^{\omega_{x}}\left(\mathbf{u},\mathbf{v}_{2}\right).$

Thus

$\displaystyle S_{c}(\theta_{j},\mathbf{v}_{j},\mathbf{u})$ $\displaystyle=$
$\displaystyle\imath\,\frac{(\theta_{1}-\theta_{2})\,\theta_{1}}{2\,\tau^{4}}\,a_{x}$
$\displaystyle+\frac{1}{\tau}\left[-\imath\,\omega_{x}\big{(}\mathbf{v}_{1}-\mathbf{v}_{2}+J_{x}\big{(}\mathbf{a}_{x}(\theta_{1},\theta_{2})\big{)},\mathbf{u}\big{)}-\frac{1}{2}\,\left(\|\mathbf{v}_{1}\|^{2}+\|\mathbf{v}_{2}\|^{2}\right)\right.$
$\displaystyle\left.-\|\mathbf{u}\|^{2}+\langle\mathbf{u},\mathbf{v}_{1}+\mathbf{v}_{2}\rangle\right]$
$\displaystyle=$
$\displaystyle\imath\,\frac{(\theta_{1}-\theta_{2})\,\theta_{1}}{2\,\tau^{4}}\,a_{x}$
$\displaystyle+\frac{1}{\tau}\left[-\frac{1}{2}\,\left(\|\mathbf{v}_{1}\|^{2}+\|\mathbf{v}_{2}\|^{2}\right)+\frac{1}{4}\,\|\mathbf{v}_{1}+\mathbf{v}_{2}\|^{2}\right.$
$\displaystyle\left.-\imath\,\omega_{x}\big{(}\mathbf{v}_{1}-\mathbf{v}_{2}+J_{x}\big{(}\mathbf{a}_{x}(\theta_{1},\theta_{2})\big{)},\mathbf{u}\big{)}-\left\|\mathbf{u}-\frac{1}{2}\,(\mathbf{v}_{1}+\mathbf{v}_{2})\right\|^{2}\right].$

In particular, $S_{c}(\theta_{j},\mathbf{v}_{j},\mathbf{u})$ is homogenous of
degree $2$.

It follows from (4) and (143) that (4) is a linear combination of terms of the
form
$\lambda^{2d-3-k/2}\,P_{k}(x;\theta_{j},\mathbf{v}_{j},\mathbf{u})\,e^{S_{c}(\theta_{j},\mathbf{v}_{j},\mathbf{u})}$,
where $P_{k}(x;\cdot)$ is a polynomial of degree $\leq 3\,k$. Furthermore,
using that $S(\mathbf{u},t,v,\theta,u)$ is homogenous of degree $2$ in the
rescaled variables, the explicit expression (143) of $L$, and the linear
dependence of $\theta_{s}$ and $t_{s}$ on $(\theta_{1},\theta_{2})$, one
verifies that $P_{k}(x;\cdot)$ has parity $k$.

Putting all these asymptotic expansions together, we obtain an asymptotic
expansion for $I_{\lambda}(\mathbf{u})$ of the form

$\displaystyle I_{\lambda}(\mathbf{u})$ $\displaystyle\sim$ $\displaystyle
e^{\imath\,\sqrt{\lambda}\,\frac{\theta_{1}-\theta_{2}}{\tau}}\cdot\frac{\lambda^{2d-3}}{(2\,\pi^{2}\,\tau^{2})^{d-1}}\cdot\chi(0)\,e^{S_{c}(\theta_{j},\mathbf{v}_{j},\mathbf{u})}$
(147)
$\displaystyle\cdot\beta_{1}\left(\lambda^{-\delta}\,\mathbf{u},\right)\left[1+\sum_{k\geq
1}\lambda^{-k/2}\,F_{k}(x;\theta_{j},\mathbf{v}_{j},\mathbf{u})\right]$

where $F_{k}(x;\cdot)$ is a polynomial of degree $\leq 3\,k$ and parity $k$,
and $\beta_{1}$ is an appropriate cut-off function identically equal to one
near the origin.

The asymptotic expansion (147) may be integrated term by term. In view of the
rapidly decreasing exponential
$e^{S_{c}(\theta_{j},\mathbf{v}_{j},\mathbf{u})}$, we obtain

$\displaystyle\Pi^{\tau}_{\chi,\lambda}(x_{1\lambda},x_{2\lambda})=\frac{\lambda^{2-d}}{\sqrt{2\,\pi}}\,\int_{\mathbb{R}^{2\,d-2}}\,I_{\lambda}(\mathbf{u})\,\mathrm{d}\mathbf{u}$
$\displaystyle\sim$ $\displaystyle
e^{\imath\,\sqrt{\lambda}\,\frac{\theta_{1}-\theta_{2}}{\tau}}\cdot\frac{\lambda^{d-1}}{\sqrt{2\,\pi}}\cdot\frac{1}{(\sqrt{2}\,\pi\,\tau)^{2\,(d-1)}}\cdot\chi(0)\,\int_{\mathbb{R}^{2\,d-2}}\,e^{S_{c}(\theta_{j},\mathbf{v}_{j},\mathbf{u})}\mathrm{d}\mathbf{u}$
$\displaystyle\cdot\left[1+\sum_{k\geq
1}\lambda^{-k/2}\,\int_{\mathbb{R}^{2\,d-2}}\,e^{S_{c}(\theta_{j},\mathbf{v}_{j},\mathbf{u})}F_{k}(x;\theta_{j},\mathbf{v}_{j},\mathbf{u})\,\mathrm{d}\mathbf{u}\right].$

We compute the leading order term using (4). With the change of variable

$\mathbf{w}:=\mathbf{u}-\frac{1}{2}\,(\mathbf{v}_{1}+\mathbf{v}_{2}),$

we have

$\displaystyle S_{c}(\theta_{j},\mathbf{v}_{j},\mathbf{u})$ $\displaystyle=$
$\displaystyle\imath\,\frac{(\theta_{1}-\theta_{2})\,\theta_{1}}{2\,\tau^{4}}\,a_{x}$
$\displaystyle+\frac{1}{\tau}\left[-\imath\,\omega_{x}\left(\mathbf{v}_{1}-\mathbf{v}_{2}+J_{x}(\mathbf{a}),\frac{1}{2}\,(\mathbf{v}_{1}+\mathbf{v}_{2})\right)-\frac{1}{2}\,\left(\|\mathbf{v}_{1}\|^{2}+\|\mathbf{v}_{2}\|^{2}\right)+\frac{1}{4}\,\|\mathbf{v}_{1}+\mathbf{v}_{2}\|^{2}\right.$
$\displaystyle\left.-\imath\,\omega_{x}\left(\mathbf{v}_{1}-\mathbf{v}_{2}+J_{x}\big{(}\mathbf{a}_{x}(\theta_{1},\theta_{2})\big{)},\mathbf{w}\right)-\left\|\mathbf{w}\right\|^{2}\right]$
$\displaystyle=$
$\displaystyle\imath\,\frac{(\theta_{1}-\theta_{2})\,\theta_{1}}{2\,\tau^{4}}\,a_{x}$
$\displaystyle+\frac{1}{\tau}\left[-\imath\,\omega_{x}\left(\mathbf{v}_{1},\mathbf{v}_{2}\right)-\imath\,\omega_{x}\left(J_{x}\big{(}\mathbf{a}_{x}(\theta_{1},\theta_{2})\big{)},\frac{1}{2}\,(\mathbf{v}_{1}+\mathbf{v}_{2})\right)-\frac{1}{2}\,\left(\|\mathbf{v}_{1}\|^{2}+\|\mathbf{v}_{2}\|^{2}\right)\right.$
$\displaystyle\left.+\frac{1}{4}\,\|\mathbf{v}_{1}+\mathbf{v}_{2}\|^{2}-\imath\,\omega_{x}\left(\mathbf{v}_{1}-\mathbf{v}_{2}+J_{x}\big{(}\mathbf{a}_{x}(\theta_{1},\theta_{2})\big{)},\mathbf{w}\right)-\left\|\mathbf{w}\right\|^{2}\right].$

Let us set $\mathbf{v}_{j}^{\prime}:=\frac{1}{\sqrt{\tau}}\,\mathbf{v}_{j}$,
$\mathbf{a}^{\prime}:=\frac{1}{\sqrt{\tau}}\,\mathbf{a}_{x}(\theta_{1},\theta_{2})$.
With the further change of variable
$\mathbf{w}=\frac{1}{\sqrt{2}}\,\mathbf{r}$, we obtain

$\displaystyle\int_{\mathbb{R}^{2\,d-2}}\,e^{\frac{1}{\tau}\,\big{[}-\imath\,\omega_{x}\left(\mathbf{v}_{1}-\mathbf{v}_{2}+J_{x}\big{(}\mathbf{a}_{x}(\theta_{1},\theta_{2})\big{)},\mathbf{w}\right)-\left\|\mathbf{w}\right\|^{2}\big{]}}\mathrm{d}\mathbf{w}$
$\displaystyle=$
$\displaystyle\frac{1}{2^{d-1}}\,\int_{\mathbb{R}^{2\,d-2}}\,e^{\frac{1}{\tau}\,\left[-\imath\,\omega_{x}\left(\frac{1}{\sqrt{2}}\big{(}\mathbf{v}_{1}-\mathbf{v}_{2}+J_{x}\big{(}\mathbf{a}_{x}(\theta_{1},\theta_{2})\big{)}\big{)},\mathbf{r}\right)-\frac{1}{2}\,\left\|\mathbf{r}\right\|^{2}\right]}\mathrm{d}\mathbf{r}$
$\displaystyle=$
$\displaystyle\frac{1}{2^{d-1}}\,\int_{\mathbb{R}^{2\,d-2}}\,e^{-\imath\,\omega_{x}\left(\frac{1}{\sqrt{2\,\tau}}\big{(}\mathbf{v}_{1}^{\prime}-\mathbf{v}_{2}^{\prime}+J_{x}(\mathbf{a}^{\prime})\big{)},\frac{\mathbf{r}}{\sqrt{\tau}}\right)-\frac{1}{2}\,\left\|\frac{\mathbf{r}}{\sqrt{\tau}}\right\|^{2}}\mathrm{d}\mathbf{r}$
$\displaystyle=$
$\displaystyle\left(\frac{\tau}{2}\right)^{d-1}\,\int_{\mathbb{R}^{2\,d-2}}\,e^{-\imath\,\omega_{x}\left(\frac{1}{\sqrt{2\,\tau}}\big{(}\mathbf{v}_{1}^{\prime}-\mathbf{v}_{2}^{\prime}+J_{x}(\mathbf{a}^{\prime})\big{)},\mathbf{s}\right)-\frac{1}{2}\,\left\|\mathbf{s}\right\|^{2}}\mathrm{d}\mathbf{s}$
$\displaystyle=$
$\displaystyle\left(\frac{\tau}{2}\right)^{d-1}\,(2\,\pi)^{d-1}e^{-\frac{1}{4\,\tau}\,\left\|\mathbf{v}_{1}-\mathbf{v}_{2}+J_{x}\big{(}\mathbf{a}_{x}(\theta_{1},\theta_{2})\big{)}\right\|^{2}}$
$\displaystyle=$ $\displaystyle\left(\tau\,\pi\right)^{d-1}\cdot
e^{-\frac{1}{4\,\tau}\,\left\|\mathbf{v}_{1}-\mathbf{v}_{2}\right\|^{2}-\frac{1}{4\,\tau}\,\left\|J_{x}\big{(}\mathbf{a}_{x}(\theta_{1},\theta_{2})\big{)}\right\|^{2}-\frac{1}{2\,\tau}\,\left\langle\mathbf{v}_{1}-\mathbf{v}_{2},J_{x}\big{(}\mathbf{a}_{x}(\theta_{1},\theta_{2})\big{)}\right\rangle}.$

We have

$\displaystyle-\frac{1}{2\,\tau}\,\left(\|\mathbf{v}_{1}\|^{2}+\|\mathbf{v}_{2}\|^{2}\right)+\frac{1}{4\,\tau}\,\|\mathbf{v}_{1}+\mathbf{v}_{2}\|^{2}-\frac{1}{4\,\tau}\,\left\|\mathbf{v}_{1}-\mathbf{v}_{2}\right\|^{2}$
$\displaystyle=$
$\displaystyle-\frac{1}{2\,\tau}\,\left(\|\mathbf{v}_{1}\|^{2}+\|\mathbf{v}_{2}\|^{2}\right)+\frac{1}{\tau}\,\langle\mathbf{v}_{1},\mathbf{v}_{2}\rangle=-\frac{1}{2\,\tau}\,\|\mathbf{v}_{1}-\mathbf{v}_{2}\|^{2}.$

Hence

$\displaystyle\int_{\mathbb{R}^{2\,d-2}}\,e^{S_{c}(\theta_{j},\mathbf{v}_{j},\mathbf{u})}\,\mathrm{d}\mathbf{u}$
$\displaystyle=$ $\displaystyle
e^{\imath\,\frac{(\theta_{1}-\theta_{2})\,\theta_{1}}{2\,\tau^{4}}\,a_{x}}\cdot\left(\tau\,\pi\right)^{d-1}\,e^{\frac{1}{\tau}\,\left[-\imath\,\omega_{x}\left(\mathbf{v}_{1},\mathbf{v}_{2}\right)-\frac{1}{2}\,\|\mathbf{v}_{1}-\mathbf{v}_{2}\|^{2}-\frac{\imath}{2}\,\omega_{x}\left(J_{x}(\mathbf{a}),\mathbf{v}_{1}+\mathbf{v}_{2}\right)\right]}$
$\displaystyle\cdot
e^{-\frac{1}{4}\,\left\|J_{x}(\mathbf{a})\right\|^{2}-\frac{1}{2}\,\langle\mathbf{v}_{1}-\mathbf{v}_{2},J_{x}(\mathbf{a})\rangle}$
$\displaystyle=$ $\displaystyle
e^{\imath\,\frac{(\theta_{1}-\theta_{2})\,\theta_{1}}{2\,\tau^{4}}\,a_{x}}\cdot\left(\tau\,\pi\right)^{d-1}$
$\displaystyle\cdot
e^{\frac{1}{\tau}\,\left[-\imath\,\omega_{x}\left(\mathbf{v}_{1},\mathbf{v}_{2}\right)-\frac{1}{4}\,\|\mathbf{v}_{1}-\mathbf{v}_{2}\|^{2}-\frac{1}{4}\,\|\mathbf{v}_{1}-\mathbf{v}_{2}+J_{x}(\mathbf{a})\|^{2}-\frac{\imath}{2}\,\omega_{x}\left(J_{x}(\mathbf{a}),\mathbf{v}_{1}+\mathbf{v}_{2}\right)\right]}.$

Recalling Definition 56, we conclude

$\int_{\mathbb{R}^{2\,d-2}}\,e^{S_{c}(\theta_{j},\mathbf{v}_{j},\mathbf{u})}\,\mathrm{d}\mathbf{u}=e^{\imath\,\frac{(\theta_{1}-\theta_{2})\,\theta_{1}}{2\,\tau^{4}}\,a_{x}}\cdot\left(\tau\,\pi\right)^{d-1}\,e^{\frac{1}{\tau}\,\Psi_{2}\big{(}(\theta_{1},\mathbf{v}_{1}),(\theta_{2},\mathbf{v}_{2})\big{)}}.$
(153)

Hence the leading order term in the asymptotic expansion (4) is

$\displaystyle\frac{1}{\sqrt{2\,\pi}}\cdot\left(\frac{\lambda}{2\,\pi\tau}\right)^{d-1}\,e^{\imath\,\sqrt{\lambda}\,\frac{\theta_{1}-\theta_{2}}{\tau}}\,e^{\imath\,\frac{(\theta_{1}-\theta_{2})\,\theta_{1}}{2\,\tau^{4}}\,a_{x}}\,\chi(0)\,e^{\frac{1}{\tau}\,\Psi_{2}\big{(}(\theta_{1},\mathbf{v}_{1}),(\theta_{2},\mathbf{v}_{2})\big{)}}.$
(154)

######  Lemma 64.

$a_{x}=0$.

###### Proof.

Let us consider the special case where
$\mathbf{v}_{1}=\mathbf{v}_{2}=\mathbf{0}$ and $\theta_{2}=0$, and set
$x_{\lambda}:=x+(\theta/\sqrt{\lambda},\mathbf{0})$. By (154), the leading
order term for the asymptotic expansion of $\Pi_{\chi,\lambda}(x_{\lambda},x)$
is

$\frac{1}{\sqrt{2\,\pi}}\cdot\left(\frac{\lambda}{2\,\pi\tau}\right)^{d-1}\,e^{\imath\,\sqrt{\lambda}\,\frac{\theta}{\tau}}\,e^{\imath\,\frac{\theta^{2}}{\tau^{4}}\,a_{x}}\,\chi(0)\,e^{-\frac{1}{4\,\tau}\,\|J_{x}(\mathbf{a})\|^{2}},$

while the one for $\Pi_{\chi,\lambda}(x,x_{\lambda})$ is

$\frac{1}{\sqrt{2\,\pi}}\cdot\left(\frac{\lambda}{2\,\pi\tau}\right)^{d-1}\,e^{-\imath\,\sqrt{\lambda}\,\frac{\theta}{\tau}}\,\chi(0)\,e^{-\frac{1}{4\,\tau}\,\|J_{x}(\mathbf{a})\|^{2}}.$

Since
$\Pi_{\chi,\lambda}(x,x_{\lambda})=\overline{\Pi_{\chi,\lambda}(x_{\lambda},x)}$
for any $\theta$, $a_{x}=0$. ∎

######  Lemma 65.

$\mathbf{A}_{x}=0$.

###### Proof.

Given Lemma 42 and Corollary 44,

$\Gamma^{\tau}_{\theta/\sqrt{\lambda}}(x)=x+\left(-\frac{\tau\,\theta}{\sqrt{\lambda}}+R_{3}\left(\frac{\tau\,\theta}{\sqrt{\lambda}}\right),\mathbf{R}_{2}\left(\frac{\tau\,\theta}{\sqrt{\lambda}}\right)\right).$

Hence, in view of Lemma 64, (4), and (154)

$\displaystyle\Pi^{\tau}_{\chi,\lambda}\left(\Gamma^{\tau}_{\theta/\sqrt{\lambda}}(x),x\right)$
$\displaystyle\asymp$
$\displaystyle\frac{1}{\sqrt{2\,\pi}}\cdot\left(\frac{\lambda}{2\,\pi\tau}\right)^{d-1}\,e^{-\imath\,\sqrt{\lambda}\,\theta}\,\chi(0)\,e^{\frac{1}{\tau}\,\Psi_{2}\big{(}(-\tau\,\theta,\mathbf{0}),(0,\mathbf{0})\big{)}}$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2\,\pi}}\cdot\left(\frac{\lambda}{2\,\pi\tau}\right)^{d-1}\,e^{-\imath\,\sqrt{\lambda}\,\theta}\,\chi(0)\,e^{-\frac{\theta^{2}}{4\,\tau}\,\|\mathbf{A}_{x}\|^{2}},$

where $\asymp$ means ‘asymptotic to leading order’. The estimate holds
uniformly for $|\theta|\leq C\,\lambda^{\delta}$.

On the other hand, by Theorem 1.1 of [CR2] in the same range we also have

$\displaystyle\left|\Pi^{\tau}_{\chi,\lambda}\left(\Gamma^{\tau}_{\theta/\sqrt{\lambda}}(x),x\right)\right|$
$\displaystyle\asymp$ $\displaystyle
C_{d,\tau}^{\prime}\,\lambda^{d-1}\,\chi(0),$

for some constant $C^{\prime}_{d,\tau}$, and the claim follows. ∎

Let us consider the lower order terms. The coefficient of
$\lambda^{d-1-\frac{k}{2}}$ in the expansion (4) is a linear combination of
Gaussian integrals of the form

$\theta_{j}^{l}\cdot\mathbf{v}_{j}^{L}\int_{\mathbb{R}^{2\,d-2}}\,e^{S_{c}(\theta_{j},\mathbf{v}_{j},\mathbf{u})}\mathbf{u}^{L^{\prime}}\,\mathrm{d}\mathbf{u},$
(155)

where $L$ and $L^{\prime}$ are multi-indexes, $l+|L|+|L^{\prime}|\leq 3\,j$,
and $l+|L|+|L^{\prime}|$ has the same parity as $j$. In turn, in view of (4),
the Gaussian integral in (155) may be written as a linear combination of
summands of the form

$\theta_{j}^{a}\cdot\mathbf{v}_{j}^{B}\,D^{(1)}_{\theta_{j},\mathbf{v}_{j}}\circ\cdots\circ
D^{(|C|)}_{\theta_{j},\mathbf{v}_{j}}\left[\int_{\mathbb{R}^{2\,d-2}}\,e^{S_{c}(\theta_{j},\mathbf{v}_{j},\mathbf{u})}\,\mathrm{d}\mathbf{u}\right],$
(156)

where each $D^{(a)}_{\theta_{j},\mathbf{v}_{j}}$ a first order differential
operator with constant coefficients and no zeroth order term in
$(\theta_{j},\mathbf{v}_{j})$, with $a+|B|+|C|=l+|L|+|L^{\prime}|$. It then
follows from (56) and (153) that the coefficient of
$\lambda^{d-1-\frac{k}{2}}$ has the form

$P_{k}(x;\mathbf{v}_{j},\theta_{j})\,e^{\frac{1}{\tau}\,\Psi_{2}\big{(}(\theta_{1},\mathbf{v}_{1}),(\theta_{2},\mathbf{v}_{2})\big{)}},$

where again $P_{k}$ is a polynomial of degree $\leq 3\,k$ and parity $k$.

∎

##  5 Proof of Theorem 6

###  5.1 Preliminaries for Theorem 6

In addition to the general setting of Theorem 3, we need the description, also
due to Zelditch, of the wave group in the complex domain as a dynamical
Toeplitz operator ([Z-2010], Proposition 7.1; [Z-2012], especially §8-9;
[Z-2014], §4; see also the discussions in [CR1] and [CR2]). This is the
analogue of the description of $U_{\sqrt{\rho}}^{\tau}(t)$ as a dynamical
Toeplitz operator recalled in §3.6.

For $t\in\mathbb{R}$ and $\tau>0$ sufficiently small, Zelditch considers the
complexified Poisson wave kernel

$U_{\mathbb{C}}(t+2\,\imath\,\tau)=P^{\tau}\circ U(t)\circ{P^{\tau}}^{*},$
(157)

where $P^{\tau}$ is as in (19). The distributional kernel of (157) admits the
expansion

$U_{\mathbb{C}}(t+2\,\imath\,\tau,x,y)=\sum_{j}e^{(-2\,\tau+\imath\,t)\,\mu_{j}}\,\tilde{\varphi}_{j}(x)\,\overline{\tilde{\varphi}_{j}(y)}.$
(158)

Arguing as for (3.6), one may then rewrite (22) as

$P_{\chi,\lambda}(x,y)=\frac{1}{\sqrt{2\,\pi}}\,\int_{-\infty}^{+\infty}\chi(t)\,e^{-\imath\,\lambda\,t}\,U_{\mathbb{C}}(t+2\,\imath\,\tau,x,y)\,\mathrm{d}t.$
(159)

Alternatively, $U_{\mathbb{C}}(t+2\,\imath\,\tau)$ is a Fourier integral
operator with complex phase of positive type on $X^{\tau}$, of degree
$-(d-1)/2$; in the terminology of [BdM-G], it is in fact a Fourier-Hermite
operator adapted to the symplectomorphism
$\Sigma^{\tau}\rightarrow\Sigma^{\tau}$ induced by the homogeneous geodesic
flow at time $t$. More precisely, for every $t\in\mathbb{R}$ the real locus of
the (complex) canonical relation of $U_{\mathbb{C}}(t+2\,\imath\,\tau)$ is as
follows.

Let $\gamma_{t}:T^{*}M\setminus(0)\rightarrow T^{*}M\setminus(0)$ denote the
(genuine) homogeneous geodesic flow; thus $\gamma_{t}$ gets intertwined by
$E^{\epsilon}$ with Hamiltonian flow of $\sqrt{\rho}$ on $\tilde{M}\setminus
M$. Furthermore, recall that $\Gamma^{\tau}_{t}:X^{\tau}\rightarrow X^{\tau}$
is the restriction of the latter flow of $\upsilon_{\sqrt{\rho}}$ on
$\tilde{M}\setminus M$, and $\alpha^{\tau}$ is invariant under
$\Gamma^{\tau}_{t}$. Therefore, the cotangent lift of $\Gamma^{\tau}_{t}$
restricts on $\Sigma^{\tau}\subset T^{*}X^{\tau}\setminus(0)$ to a flow
$\Sigma^{\tau}\rightarrow\Sigma^{\tau}$ by homogeneous symplectomorphisms,
that by abuse of language we shall also denote $\Gamma^{\tau}_{t}$
($\Sigma^{\tau}$ as in (10)).

Let $\iota_{\tau}:T^{*}M\setminus(0)\rightarrow\Sigma_{\tau}$ be given by

$\iota_{\tau}(m,\beta):=\left(E^{\epsilon}\left(m,\tau\,\frac{\beta}{\|\beta\|}\right),\|\beta\|\,\alpha_{E^{\epsilon}\left(x,\tau\,\frac{\beta}{\|\beta\|}\right)}\right).$
(160)

Then $\iota_{\tau}$ is a homogeneous symplectomorphism, and intertwines the
$\gamma_{t}:T^{*}M\setminus(0)$ with $\Gamma^{\tau}_{t}$. Then the wave front
of $P^{\tau}$ is (see e.g. [Z-2010])

$\mathrm{WF}^{\prime}\left(P^{\tau}\right)=\left\\{\big{(}\iota_{\tau}(m,\beta),\,(m,\beta)\big{)}\,:\,(m,\beta)\in
T^{*}M\setminus(0)\right\\}.$ (161)

On the other hand, the wave front of $U(t)$ is (see e.g. [DG])

$\mathrm{WF}^{\prime}\big{(}U(t)\big{)}=\left\\{\big{(}\gamma_{t}(m,\beta),\,(m,\beta)\big{)}\,:\,(m,\beta)\in
T^{*}M\setminus(0)\right\\}.$ (162)

Given (157) and the composition law for wave fronts we conclude

$\displaystyle\mathrm{WF}^{\prime}\big{(}U_{\mathbb{C}}(t+2\,\imath\,\tau)\big{)}$
$\displaystyle=$
$\displaystyle\left\\{\left(\Gamma^{\tau}_{t}\left(x,\xi\right),(x,\xi)\right)\,:\,(x,\xi)\in\Sigma^{\tau}\right\\}.$
(163)

This is however the same wave front of the composition
$\Pi^{\tau}\circ\Pi^{\tau}_{-t}$, whence of any composition $\Pi\circ
Q\circ\Pi^{\tau}_{-t}$, with $Q$ a pseudodifferential operator which is
elliptic on a conic neighourhood of $\Sigma^{\tau}$. Hence, for any such $Q$,
$U_{\mathbb{C}}(t+2\,\imath\,\tau)$ and $\Pi\circ Q\circ\Pi^{\tau}_{-t}$ are
both Fourier integral operators of Hermite type with the same real canonical
relation; it follows that they are associated to the same complex Lagrangian.
This is the basis for the following analogue of Theorem 54 (notation is as in
§3.6).

######  Theorem 66.

(Zelditch) There exist a polyhomogeneous classical symbol on
$X^{\tau}\times\mathbb{R}_{+}$ of the form

$\gamma^{\tau}_{t}(x,r)\sim\sum_{j\geq
0}\gamma_{t,j}^{\tau}(x)\,r^{-\frac{d-1}{2}-j},$

and a pseudo-differential operator
$Q^{\tau}_{t}\sim\gamma^{\tau}_{t}(x,D_{\sqrt{\rho}}^{\tau})$ of degree
$-(d-1)/2$ on $X^{\tau}$, such that up to smoothing operators

$U_{\mathbb{C}}(t+2\,\imath\,\tau)\sim\Pi^{\tau}\circ
Q^{\tau}_{t}\circ\Pi^{\tau}_{-t}=\Pi^{\tau}\circ\mathcal{Q}^{\tau}_{t},$ (164)

where $\mathcal{Q}^{\tau}_{t}:=Q^{\tau}_{t}\circ\Pi_{-t}^{\tau}$.

The coefficients $\gamma_{t,j}^{\tau}$ depend on the choice of volume form. We
shall determine $\gamma_{0,0}^{\tau}$ a posteriori by deriving a local Weyl
law for the complexified eigenfunctions and comparing it to the one in
Proposition 3.8 of [Z-2014].

######  Remark 67.

Up to smoothing operators that do not affect the asymptotics, the composition
(164) only depends on the behaviour of $Q^{\tau}_{t}$ in a small conic
neighborhood of $\Sigma^{\tau}$, where $D_{\sqrt{\rho}}^{\tau}$ may be assumed
to be microlocally elliptic.

Using known results on the composition of pseudodifferential and Fourier
integral operators, in view of (74) and Corollary 48 one obtains from Theorem
66 the following analogue of Lemma 55.

######  Lemma 68.

Up to smoothing terms, the Schwartz kernel of $\mathcal{Q}^{\tau}_{t}$ has the
form

$\mathcal{Q}^{\tau}_{t}(x_{1},x_{2})\sim\int_{0}^{+\infty}e^{\imath\,u\,\psi^{\tau}\left(\Gamma^{\tau}_{-t}(x_{1}),x_{2}\right)}\,q^{\tau}_{t}\left(x_{1},x_{2},u\right)\,\mathrm{d}u,$
(165)

where

$q^{\tau}_{t}\left(x_{1},x_{2},u\right)\sim\sum_{j\geq
0}\,u^{\frac{d-1}{2}-j}\,q^{\tau}_{t\,j}(x_{1},x_{2}),$

with

$q^{\tau}_{0\,0}(x_{1},x_{2})=\frac{1}{\tau^{\frac{d-1}{2}}}\cdot\gamma_{0,0}^{\tau}(x)\,s^{\tau}_{0}\left(\Gamma^{\tau}_{t}(x_{1}),x_{2}\right).$

###  5.2 Proof of Theorem 6

###### Proof.

By the considerations in §5.1, the analysis of
$P^{\tau}_{\chi,\lambda}(x_{1\lambda},x_{2\lambda})$ (notation as in (119))
parallels the one for $\Pi^{\tau}_{\chi,\lambda}(x_{1\lambda},x_{2\lambda})$
in Theorem 3, with the following change.

The leading order term of the amplitude has been multiplied by a factor
$\gamma_{0,0}^{\tau}(x)\cdot(u\,\tau)^{-\frac{d-1}{2}}$. In view of the
rescaling $u\mapsto\lambda\,u$ in (4), this change entails an additional
factor $\gamma_{0,0}^{\tau}(x)\cdot\lambda^{-\frac{d-1}{2}}$ in front of the
resulting asymptotic expansion. Furthermore, by Lemma 63 at the critical point
$u\,\tau=1$.

Hence, the leading order term of the asymptotic expansion for
$P_{\chi,\lambda}(x_{1\lambda},x_{2\lambda})$ only differs from the one of
$\Pi^{\tau}_{\chi,\lambda}(x_{1\lambda},x_{2\lambda})$ by the factor
$\gamma_{0,0}^{\tau}(x)\cdot\lambda^{-\frac{d-1}{2}}$. ∎

###  5.3 The pointwise Weyl law

###### Proof of Proposition 9.

We shall prove (26); the same argument, with obvious adaptations, also proves
(27).

Let us choose
$\chi\in\mathcal{C}^{\infty}_{c}\big{(}(-\epsilon,\epsilon)\big{)}$ with
$\hat{\chi}>0$. Define $f:\mathbb{R}\times\mathbb{R}\rightarrow[0,+\infty)$ by

$f(t,s):=\hat{\chi}(t)\,H(\lambda-s-t),$

where $H$ is the Heaviside function.

Let us consider the following positive measures $\mathcal{L}$ and
$\mathcal{T}_{x}^{\tau}$ on $\mathbb{R}$. First, $\mathcal{L}$ is the Lebesgue
measure. Second,

$\mathcal{T}_{x}^{\tau}:=\sum_{j\geq
1}\,\Pi_{j}^{\tau}(x,x)\,\delta_{\lambda_{j}}.$ (166)

Let us endow $\mathbb{R}\times\mathbb{R}$ with the product measure
$\mathcal{L}\times\mathcal{T}$.

By the Fubini Theorem (see e.g. Ch. 8 of [R]),

$\int_{\mathbb{R}}\,\mathrm{d}\mathcal{T}_{x}(s)\,\left[\int_{\mathbb{R}}\,f(t,s)\,\,\mathrm{d}\mathcal{L}(t)\right]=\int_{\mathbb{R}}\,\mathrm{d}\mathcal{L}(t)\,\left[\int_{\mathbb{R}}\,f(t,s)\,\,\mathrm{d}\mathcal{T}_{x}(s)\right];$
(167)

the claim will follow by comparing both sides of (167).

The former integral in (167) is

$\displaystyle\int_{\mathbb{R}}\,\mathrm{d}\mathcal{T}_{x}(s)\,\left[\int_{\mathbb{R}}\,f(t,s)\,\mathrm{d}\mathcal{L}(t)\right]=\int_{\mathbb{R}}\,\mathrm{d}\mathcal{T}_{x}(s)\,\left[\int_{-\infty}^{\lambda-s}\,\hat{\chi}(t)\,\mathrm{d}t\right]$
(168) $\displaystyle=$
$\displaystyle\int_{\mathbb{R}}\,\mathrm{d}\mathcal{T}_{x}(s)\,\left[\int_{-\infty}^{\lambda}\,\hat{\chi}(t-s)\,\mathrm{d}t\right]=\sum_{j\geq
1}\,\Pi_{j}^{\tau}(x,x)\cdot\int_{-\infty}^{\lambda}\,\hat{\chi}(t-\lambda_{j})\,\mathrm{d}\,t$
$\displaystyle=$ $\displaystyle\int_{-\infty}^{\lambda}\,\left[\sum_{j\geq
1}\,\Pi_{j}^{\tau}(x,x)\cdot\hat{\chi}(t-\lambda_{j})\right]\,\mathrm{d}\,t=\int_{-\infty}^{\lambda}\,\Pi^{\tau}_{\chi,t}(x,x)\,\mathrm{d}\,t;$

on the last line, we have made use of (15) and Theorem 1.27 of [R]. In view of
Corollary 4, (168) implies that as $\lambda\rightarrow+\infty$

$\displaystyle\int_{\mathbb{R}}\,\mathrm{d}\mathcal{T}_{x}(s)\,\left[\int_{\mathbb{R}}\,f(t,s)\,\mathrm{d}\mathcal{L}(t)\right]=\frac{\sqrt{2\,\pi}}{d\cdot(2\,\pi)^{d}}\cdot\frac{\lambda^{d}}{\tau^{d-1}}\cdot\chi(0)+O\left(\lambda^{d-1}\right).$
(169)

On the other hand, the latter integral in (167) is

$\displaystyle\int_{\mathbb{R}}\,\mathrm{d}\mathcal{L}(t)\,\left[\int_{\mathbb{R}}\,f(t,s)\,\,\mathrm{d}\mathcal{T}_{x}(s)\right]=\int_{-\infty}^{+\infty}\left[\sum_{j}\,\Pi_{j}^{\tau}(x,x)\,f(t,\lambda_{j})\right]\,\mathrm{d}\,t$
$\displaystyle=$
$\displaystyle\int_{-\infty}^{+\infty}\left[\sum_{j}\,\Pi_{j}^{\tau}(x,x)\,H(\lambda-t-\lambda_{j})\right]\,\hat{\chi}(t)\,\mathrm{d}\,t$
$\displaystyle=$
$\displaystyle\int_{-\infty}^{+\infty}\,\mathcal{W}_{x}(\lambda-\tau)\,\hat{\chi}(t)\,\mathrm{d}\,t$
$\displaystyle=$
$\displaystyle\mathcal{W}_{x}(\lambda)\,\int_{-\infty}^{+\infty}\,\hat{\chi}(t)\,\mathrm{d}\,t+\int_{-\infty}^{+\infty}\,\big{[}\mathcal{W}_{x}(\lambda-\tau)-\mathcal{W}_{x}(\lambda)\big{]}\,\hat{\chi}(t)\,\mathrm{d}\,t$
$\displaystyle=$
$\displaystyle\sqrt{2\,\pi}\,\chi(0)\cdot\mathcal{W}_{x}(\lambda)+\int_{-\infty}^{+\infty}\,\big{[}\mathcal{W}_{x}(\lambda-\tau)-\mathcal{W}_{x}(\lambda)\big{]}\,\hat{\chi}(t)\,\mathrm{d}\,t.$

######  Lemma 69.

For $\lambda\rightarrow+\infty$, we have

$\int_{-\infty}^{+\infty}\,\big{[}\mathcal{W}_{x}(\lambda-\tau)-\mathcal{W}_{x}(\lambda)\big{]}\,\hat{\chi}(t)\,\mathrm{d}\,t=O\left(\lambda^{d-1}\right).$

###### Proof.

It follows from Corollary 4 that for $\lambda\gg 0$

$\mathcal{W}_{x}(\lambda+1)-\mathcal{W}_{x}(\lambda)=\sum_{\lambda\leq\lambda_{j}\leq\lambda+1}\,\Pi_{j}(x,x)=O\left(\lambda^{d-1}\right).$

Hence there exist $C_{0},\,C_{1}>0$ such that

$\mathcal{W}_{x}(\lambda+1)-\mathcal{W}_{x}(\lambda)\leq
C_{1}\,|\lambda|^{d-1}+C_{0}\qquad\forall\,\lambda\in\mathbb{R}.$

Thus for suitable $C^{\prime},\,C^{\prime\prime}>0$ for any
$\lambda,\,t\in\mathbb{R}$

$\big{|}\mathcal{W}_{x}(\lambda-t)-\mathcal{W}_{x}(t)\big{|}\leq
C^{\prime}\cdot|t|\,\left[|\lambda|^{d-1}+|\lambda-t|^{d-1}\right]+C^{\prime\prime}.$

Therefore for $\lambda\gg 0$

$\displaystyle\left|\int_{-\infty}^{+\infty}\,\big{[}W_{x}(\lambda-\tau)-W_{x}(\lambda)\big{]}\,\hat{\chi}(t)\,\mathrm{d}\,t\right|$
$\displaystyle\leq$ $\displaystyle
C^{\prime}\,\int_{-\infty}^{+\infty}\,|t|\,\left[|\lambda|^{d-1}+|\lambda-t|^{d-1}\right]\,\hat{\chi}(t)\,\mathrm{d}\,t+C^{\prime\prime\prime}\leq
A\,\lambda^{d-1}+B,$

for appropriate constants $A,B>0$.

∎

Comparing (168) and (5.3), we conclude that

$\sqrt{2\,\pi}\,\chi(0)\cdot
W_{x}(\lambda)=\frac{\sqrt{2\,\pi}}{d\cdot(2\,\pi)^{d}}\cdot\frac{\lambda^{d}}{\tau^{d-1}}\cdot\chi(0)+O\left(\lambda^{d-1}\right),$

whence

$W_{x}(\lambda)=\frac{\tau}{d}\cdot\left(\frac{\lambda}{2\,\pi\,\tau}\right)^{d}+O\left(\lambda^{d-1}\right).$

This establishes (26). To obtain (27), we need only run over the same
argument, with the following changes: first, in (166) replace
$\Pi_{j}^{\tau}(x,x)$ by $P^{\tau}_{\chi,\lambda}\left(x,x\right)$; second, in
the derivation of (169), invoke Corollary 8 in place of Corollary 4. ∎

The author states that there is no conflict of interest.

Acknowledgments. The author is a member of GNSAGA (Gruppo Nazionale per le
Strutture Algebriche, Geometriche e le loro Applicazioni) of INdAM (Istituto
Nazionale di Alta Matematica ”Francesco Severi”), and thanks the group for its
support.

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