LemmaTh Lemma PropTh Prop CorTh Cor conjTh conj DefTh Def ExTh Ex ConjTh Conj
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111A.R. is partially supported by NSF RTG grant DMS-1502632

# Scaling asymptotics for Szegő kernels on Grauert tubes

Robert Chang  Northeastern University, Boston, MA 02115
<EMAIL_ADDRESS>and  Abraham Rabinowitz  Northwestern University,
Evanston, IL 60208, USA<EMAIL_ADDRESS>
###### Abstract.

Let $M_{\tau}$ be the Grauert tube of radius $\tau$ of a closed, real analytic
Riemannian manifold $M$. Associated to the Grauert tube boundary is the
orthogonal projection $\Pi_{\tau}\colon L^{2}(\partial M_{\tau})\to
H^{2}(\partial M_{\tau})$, called the Szegő projector. Let $D_{\sqrt{\rho}}$
denote the Hamilton vector field of the Grauert tube function $\sqrt{\rho}$
acting as a differential operator. We prove scaling asymptotics for the
spectral localization kernel of the Toeplitz operator
$\Pi_{\tau}D_{\sqrt{\rho}}\Pi_{\tau}$. We also prove scaling asymptotics for
the smoothed spectral projection kernel
$P_{\chi,\lambda}(z,w)=\sum_{\lambda_{j}\leq\lambda}\chi(\lambda-\lambda_{j})e^{-2\tau\lambda_{j}}\varphi_{\lambda_{j}}^{\mathbb{C}}(z)\overline{\varphi_{\lambda_{j}}^{\mathbb{C}}(w)}$,
where $\varphi_{\lambda_{j}}^{\mathbb{C}}$ are CR holomorphic functions on the
Grauert tube boundary $\partial M_{\tau}$, which are obtained by analytically
continuing Laplace eigenfunctions on $M$.

###### Contents

  1. 1 Statement of main results
    1. 1.1 Comparison to the line bundle setting
    2. 1.2 Related results in the line bundle setting
  2. 2 Geometry of and analysis on Grauert tubes
    1. 2.1 Grauert tube and the cotangent bundle
    2. 2.2 Contact and CR structure on the boundary of a Grauert tube
    3. 2.3 The Szegő projector and the Boutet de Movel–Sjöstrand parametrix
    4. 2.4 Poisson wave operator and tempered spectral projection
    5. 2.5 The Toeplitz operator $\Pi_{\tau}D_{\sqrt{\rho}}\Pi_{\tau}$
  3. 3 Heisenberg coordinates on the boundary of a Grauert tube
    1. 3.1 Taylor expansions in Heisenberg coordinates
  4. 4 Proof of main results
    1. 4.1 Proof of Theorem 1.1
    2. 4.2 Proof of Theorem 1.2

##  1\. Statement of main results

Let $(M,g)$ be a closed, real analytic Riemannian manifold. Its Grauert tube
$M_{\tau}$ is a Kähler manifold with boundary that is diffeomorphic to the co-
ball bundle consisting of co-vectors of length at most $\tau$. The _Szegő
projector_ $\Pi_{\tau}$ associated to the boundary of a Grauert tube, that is,
the orthogonal projection

$\Pi_{\tau}\colon L^{2}(\partial M_{\tau})\to H^{2}(\partial M_{\tau})$ (1)

onto the space of CR holomorphic functions on the Grauert tube boundary that
are square integrable.

Fix a positive, even Schwartz function $\chi$ whose Fourier transform is
compactly supported with $\hat{\chi}(0)=1$. We study the smoothed out spectral
localizations

$\Pi_{\chi,\lambda}:=\Pi_{\tau}\chi(\Pi_{\tau}D_{\sqrt{\rho}}\Pi_{\tau}-\lambda)=\int_{\mathbb{R}}\hat{\chi}(t)e^{-it\lambda}\Pi_{\tau}e^{it\Pi_{\tau}D_{\sqrt{\rho}}\Pi_{\tau}}\,dt$
(2)

of the Toeplitz operator

$\Pi_{\tau}D_{\sqrt{\rho}}\Pi_{\tau}\colon H^{2}(\partial M_{\tau})\to
H^{2}(\partial M_{\tau}),$ (3)

where $D_{\sqrt{\rho}}=\frac{1}{i}\Xi_{\sqrt{\rho}}$ is a constant multiple of
the Hamilton vector field of the Grauert tube function $\sqrt{\rho}$ acting as
a differential operator; see subsection 2.1 for background on Grauert tubes.
As explained in subsection 1.1, the operator (2) is an analogue of the Fourier
components (33) of the Szegő kernel in the line bundle setting.

The scaling asymptotics is expressed in _Heisenberg coordinates_ on $\partial
M_{\tau}$ in the sense of [13, Section 14, 18]. We give the precise definition
of the coordinates here; additional geometric background is recalled in
section 3.

######  Definition .

By _Heisenberg coordinates_ (relative to the orthonormal frame
$Z_{1},\dotsc,Z_{m-1}$) at $T^{1,0}_{p}\partial M_{\tau}$, we mean holomorphic
coordinates $z_{0},\dotsc,z_{m-1}$ in a neighborhood of $p\in M_{\tau}$ with
the following properties.

  * (i)

If we set $\theta=\operatorname{Re}z_{0}$ and $z_{j}=x_{j}+iy_{j}$, then
$x_{1},\dotsc x_{m-1},y_{1},\dotsc,y_{m-1},\theta$ form a coordinate system in
a neighborhood of $p$ in $\partial M_{\tau}$.

  * (ii)

Moreover, we have

$\displaystyle Z_{j}$ $\displaystyle=\frac{\partial}{\partial
z_{j}}+i\overline{z}_{j}\frac{\partial}{\partial
t}+\sum_{k=1}^{m-1}O^{1}\frac{\partial}{\partial
z_{k}}+O^{2}\frac{\partial}{\partial\theta},$ (4) $\displaystyle T$
$\displaystyle=\frac{\partial}{\partial\theta}+\sum_{k=1}^{m-1}O^{1}\frac{\partial}{\partial
z_{k}}+\sum_{k=1}^{m-1}O^{1}\frac{\partial}{\partial\overline{z}_{k}}+O^{1}\frac{\partial}{\partial\theta}.$
(5)

Here, the Heisenberg-type order $O^{k}$ is defined inductively by

$\displaystyle f$ $\displaystyle=O^{1}\text{ if
$f(\eta)=O\left(\sum_{k=1}^{m-1}(\lvert x_{k}(\eta)\rvert+\lvert
y_{k}(\eta)\rvert)+\lvert\theta(\eta)\rvert^{\frac{1}{2}}\right)$ as $\eta\to
p$},$ (6) $\displaystyle f$ $\displaystyle=O^{k}\text{ if $f=O(O^{1}\cdot
O^{k-1})$}.$ (7)

######  Theorem 1.1 (Scaling asymptotics for $\Pi_{\chi,\lambda}$).

Let $\Pi_{\chi,\lambda}$ be the spectral projection (2) associated to the
Grauert tube boundary $\partial M_{\tau}$ of a closed, real analytic
Riemannian manifold $M$ of dimension $m$. Fix $p\in\partial M_{\tau}$. Then,
in Heisenberg coordinates centered at $p=0$, the distribution kernel has the
following scaling asymptotics:

$\displaystyle\Pi_{\chi,\lambda}$
$\displaystyle\left(\frac{\theta}{\lambda},\frac{u}{\sqrt{\lambda}};\frac{\phi}{\lambda},\frac{v}{\sqrt{\lambda}}\right)$
(8)
$\displaystyle=\frac{C_{m,M}}{\tau}\left(\frac{\lambda}{\tau}\right)^{m-1}e^{\frac{1}{\tau}\left(\frac{i}{2}\left(\theta-\phi\right)-\frac{|u|^{2}}{2}-\frac{|v|^{2}}{2}+v\cdot\overline{u}\right)}$
(9)
$\displaystyle\quad\times\left(1+\sum_{j=1}^{N}\lambda^{-\frac{j}{2}}P_{j}(p,u,v,\theta,\phi)+\lambda^{{}^{-\frac{N+1}{2}}}R_{N}\left(p,\theta,u,\phi,v,\lambda\right)\right),$
(10)

where $\lVert
R_{N}\left(p,\theta,u,\phi,v,\lambda\right)\rVert_{C^{j}\left(\\{\lvert\theta\rvert+\lvert\phi\rvert+|u|+|v|\leq\rho\\}\right)}\leq
C_{K,j,\rho}$ for $\rho>0,\ j=1,2,3,\dotsc$ and $P_{j}$ is a polynomial in
$u,v,\theta,\phi$.

To give some context to our scaling asymptotics, recall the reduced Heisenberg
group $\mathbf{H}^{m-1}_{\mathrm{red}}=\mathbf{H}^{m-1}/\\{(0,2\pi
k):k\in\mathbb{Z}\\}=S^{1}\times\mathbb{C}^{m-1}$ of degree $m-1$, which is
obtained as a discrete quotient of the Heisenberg group
$\mathbf{H}^{m-1}=\mathbb{R}\times\mathbb{C}^{m-1}$. The group law on
$\mathbf{H}^{m-1}_{\mathrm{red}}$ is given by

$\displaystyle(e^{it},\zeta)\cdot(e^{is},\eta)=\Big{(}e^{i(t+s+\operatorname{Im}(\zeta\cdot\overline{\eta}))},\zeta+\eta\Big{)}.$
(11)

The level one Szegő projector $\Pi^{\mathbf{H}}_{1}\colon
L^{2}(\mathbf{H}^{m-1}_{\mathrm{red}})\to H^{2}_{1}$ is the orthogonal
projection onto the space $H^{2}_{1}$ of CR holomorphic functions square
integrable with respect to $e^{-|z|^{2}}$. Its Schwartz kernel has an exact
formula:

$\displaystyle\Pi_{1}^{\mathbf{H}}(\theta,z,\varphi,w)$
$\displaystyle=\frac{1}{\pi^{m-1}}e^{i\left(\theta-\varphi\right)}e^{z\cdot\overline{w}-\frac{1}{2}|z|^{2}-\frac{1}{2}|w|^{2}}.$
(12)

We refer to [2] for more details. It is known (cf (39)) that, in correctly
chosen coordinates, the Szegő kernel in the line bundle case is to leading
order some multiple of (12); see for example [2, Theorem 3.1] or [31, Theorem
2.1]. Theorem 1.1 shows that the same phenomenon holds in the Grauert tube
setting:

$\displaystyle\Pi_{\chi,\lambda}$
$\displaystyle\left(\frac{\theta}{\lambda},\frac{u}{\sqrt{\lambda}};\frac{\phi}{\lambda},\frac{v}{\sqrt{\lambda}}\right)=\frac{C_{m,M}}{\tau}\left(\frac{\lambda}{\tau}\right)^{m-1}\Pi^{\mathbf{H}}_{1}\left(\frac{\theta}{2\tau},\frac{u}{\sqrt{\tau}},\frac{\phi}{2\tau},\frac{v}{\sqrt{\tau}}\right)$
(13)
$\displaystyle\quad\times\left(1+\sum_{j=1}^{N}\lambda^{-\frac{j}{2}}P_{j}(p,u,v,\theta,\phi)+\lambda^{{}^{-\frac{N+1}{2}}}R_{N}\left(p,\theta,u,\phi,v,\lambda\right)\right)$
(14)

Another natural operator to study is the spectral projection kernel defined
using analytic continuation of Laplace eigenfunctions. Let

$\Delta=-\frac{1}{\sqrt{\lvert g\rvert}}\frac{\partial}{\partial
x_{j}}\bigg{(}\sqrt{\lvert g\rvert}g^{ij}\frac{\partial}{\partial
x_{k}}\bigg{)}$ (15)

be the (positive) metric Laplacian. It is well-known that $\Delta$ has a
discrete spectrum:

$\displaystyle
0=\lambda_{0}^{2}<\lambda_{1}^{2}\leq\dots\lambda_{j}^{2}\leq\dots\to\infty$
(16)

with associated $L^{2}$-normalized eigenfunctions

$\displaystyle\Delta\varphi_{\lambda_{j}}=\lambda_{j}^{2}\varphi_{\lambda_{j}},\quad\|\varphi_{\lambda_{j}}\|_{L^{2}(M)}=1.$
(17)

Our next result concerns the complex analogue of the partial spectral
projection kernel

$\displaystyle E_{\lambda}(x,y)$
$\displaystyle=\sum_{\lambda_{j}\leq\lambda}\varphi_{\lambda_{j}}(x)\overline{\varphi_{\lambda_{j}}(y)}.$
(18)

The Poisson wave operator is used to analytically continue eigenfunctions. One
starts with the half-wave operator $U(t)=e^{it\sqrt{\Delta}}\colon L^{2}(M)\to
L^{2}(M)$ whose Schwartz kernel can be expressed as a sum of eigenfunctions as
well as an oscillatory integral (for instance the Lax–Hörmander parametrix):

$U(t,x,y)=\sum_{j=0}^{\infty}e^{it\lambda_{j}}\varphi_{\lambda_{j}}(x)\overline{\varphi_{\lambda_{j}}(y)}=\int_{T^{*}_{y}M}A(t,x,y,\xi)e^{it\lvert\xi\rvert_{y}}e^{i\langle\xi,\exp_{y}^{-1}(x)\rangle}\,d\xi.$
(19)

Analytically continuing the oscillatory integral representation in the time
variable $t\mapsto t+i\tau\in\mathbb{C}$ and the spatial variable $x\to
z\in\partial M_{\tau}$ yields the Poisson wave kernel (see Theorem 2.2):

$U(i\tau)\colon L^{2}(M)\to\mathcal{O}(\partial M_{\tau}).$ (20)

Here, $\mathcal{O}(\partial M_{\tau})=\ker(\bar{\partial}_{b})$ denotes the
space of CR holomorphic functions on the Grauert tube boundary. Comparing (19)
with (20), we find

$U(i\tau,z,y)=\sum_{j=0}^{\infty}e^{-\tau\lambda_{j}}\varphi_{\lambda_{j}}^{\mathbb{C}}(z)\overline{\varphi_{\lambda_{j}}(y)}.$
(21)

Hence, the formula

$\varphi_{\lambda_{j}}^{\mathbb{C}}=e^{\tau\lambda_{j}}U(i\tau)\varphi_{\lambda_{j}}\in\mathcal{O}(\partial
M_{\tau})$ (22)

defines the analytic extension of Laplace eigenfunctions (17).

We define tempered spectral projections $P_{\lambda}$ by the eigenfunction
partial sums

$P_{\lambda}\colon\mathcal{O}(\partial M_{\tau})\to\mathcal{O}(\partial
M_{\tau}),\quad
P_{\lambda}(z,w)=\sum_{j:\lambda_{j}\leq\lambda}e^{-2\tau\lambda_{j}}\varphi_{\lambda_{j}}^{\mathbb{C}}(z)\overline{\varphi_{\lambda_{j}}^{\mathbb{C}}(w)}.$
(23)

The prefactor $e^{-2\tau\lambda_{j}}$ is introduced because of the exponential
growth estimate ([35, Corollary 3]) on complexified eigenfunctions

$\lambda_{j}^{-\frac{m-1}{2}}e^{\tau\lambda_{j}}\lesssim\sup_{\zeta\in
M_{\tau}}\lvert\varphi_{\lambda_{j}}^{\mathbb{C}}(z)\rvert\lesssim\lambda_{j}^{\frac{m-1}{2}}e^{\tau\lambda_{j}}.$
(24)

As before, fix a positive, even Schwartz function $\chi$ whose Fourier
transform is compactly supported with $\hat{\chi}(0)=1$. The techniques for
proving Theorem 1.1 are easily adapted to prove scaling asymptotics for the
smoothed sum

$\displaystyle P_{\chi,\lambda}(z,w)=\chi\ast
d_{\lambda}P_{\lambda}(z,w)=\sum_{j:\lambda_{j}\leq\lambda}\chi(\lambda-\lambda_{j})e^{-2\tau\lambda_{j}}\varphi^{\mathbb{C}}_{\lambda_{j}}(z)\overline{\varphi^{\mathbb{C}}_{\lambda_{j}}(w)}.$
(25)

######  Theorem 1.2 (Scaling asymptotics for $P^{\tau}_{\lambda}$).

Let $P_{\chi,\lambda}$ be the spectral projection (25) associated to the
Grauert tube boundary $\partial M_{\tau}$ of a closed, real analytic
Riemannian manifold $M$ of dimension $m$. Fix $p\in\partial M_{\tau}$. Then,
in Heisenberg coordinates centered at $p=0$, the distribution kernel has the
following scaling asymptotics:

$\displaystyle P_{\chi,\lambda}$
$\displaystyle\left(\frac{\theta}{\lambda},\frac{u}{\sqrt{\lambda}};\frac{\phi}{\lambda},\frac{v}{\sqrt{\lambda}}\right)$
(26)
$\displaystyle=\frac{C_{m,M}}{\tau^{m}}\lambda^{\frac{m-1}{2}}\Pi^{\mathbf{H}}_{1}\left(\frac{\theta}{2\tau},\frac{u}{\sqrt{\tau}},\frac{\phi}{2\tau},\frac{v}{\sqrt{\tau}}\right)$
(27)
$\displaystyle\quad\times\left(1+\sum_{j=1}^{N}\lambda^{-\frac{j}{2}}P_{j}(p,u,v,\theta,\phi)+\lambda^{{}^{-\frac{N+1}{2}}}R_{N}\left(p,\theta,u,\phi,v,\lambda\right)\right)$
(28)

where $\lVert
R_{N}\left(p,\theta,u,\phi,v,\lambda\right)\rVert_{C^{j}\left(\\{\lvert\theta\rvert+\lvert\phi\rvert+|u|+|v|\leq\rho\\}\right)}\leq
C_{K,j,\rho}$ for $\rho>0,\ j=1,2,3,\dots$ and $P_{j}$ is a polynomial in
$u,v,\theta,\phi$.

######  Remark .

The leading order asymptotics of Theorem 1.1 and Theorem 1.2 differ by a
factor of $\lambda^{-\frac{m-1}{2}}$ because the analytic extensions
$\varphi_{\lambda_{j}}^{\mathbb{C}}$ are not $L^{2}$-normalized on $\partial
M_{\tau}$.

Theorem 1.2 is the complex analogue of [6, Proposition 10], where it was shown
that the rescaled real spectral projection kernel
$E_{\lambda}(x+\frac{u}{\lambda},y+\frac{v}{\lambda})$ from (18) exhibits
Bessel-type scaling asymptotics as $\lambda\to\infty$. The Heisenberg-scaling
of the present paper is performed with parameter $\sqrt{\lambda}$ rather than
$\lambda$. The complex geometry of the Grauert tube win over the real
Riemannian geometry since as $\tau\to 0$ our scaling asymptotics do not
resemble Bessel asymptotics. It should also be mentioned that, unlike the
authors of [6], we do not study the sharp (i.e., without the smoothing $\chi$)
spectral projection in this paper.

###  1.1. Comparison to the line bundle setting

Let $(L,h)\to(M,\omega)$ be a positive Hermitian line bundle over a closed
Kähler manifold. The _Bergman projections_ are orthogonal projections

$\Pi_{h^{k}}\colon L^{2}(M,L^{k})\to H^{0}(M,L^{k})$ (29)

from the space of $L^{2}$ sections of the $k$th tensor power of the line
bundle onto the space of square integrable holomorphic sections.

The Bergman projections are related to the Szegő projection

$\Pi_{h}\colon L^{2}(\partial D)\to H^{2}(\partial D),$ (30)

where $\partial D=\\{\ell\in L^{*}:\lVert\ell\rVert_{h^{*}}=1\\}$ is the unit
co-circle bundle. Indeed, let $r_{\theta}$ denote the circle action on
$\partial D$, then the map

$\displaystyle H^{0}(M,L^{k})$ $\displaystyle\to\\{f\in H^{2}(\partial
D):f(r_{\theta}x)=e^{ik\theta}f(x)\\}$ (31) $\displaystyle s_{k}(z)$
$\displaystyle\mapsto f(x)=(\ell^{\otimes k},s_{k}(z))\quad\text{where
$x=(\ell,z)\in\pi^{-1}(z)\times M$}$ (32)

defines a unitary equivalence between the space of holomorphic sections of
$L^{k}$ and the subspace of equivariant functions in $H^{2}(\partial D)$.
Since the circle action on $\partial D$ commutes with
$\overline{\partial}_{b}$, we conclude that the kernel of (29) are Fourier
coefficients of the kernel of (30), that is,

$\Pi_{h^{k}}(x,y)=\frac{1}{2\pi}\int_{0}^{2\pi}e^{-ik\theta}\Pi_{h}(r_{\theta}x,y)\,d\theta=\left(\frac{1}{2\pi}\int_{0}^{2\pi}e^{-ik\theta}r_{\theta}^{*}\Pi_{h}\,d\theta\right)(x,y).$
(33)

Note that the Fourier decomposition of $\Pi_{h}$ coincides with the spectral
decomposition of $D_{\theta}=\frac{1}{i}\frac{\partial}{\partial\theta}$ on
$\partial D$.

In the Grauert tube setting, $\partial M_{\tau}$ plays the role of the
strongly pseudoconvex CR hypersurface $\partial D\subseteq L^{*}$ and the
Hamilton vector field $\Xi_{\sqrt{\rho}}$ of the Grauert tube function on
$\partial M_{\tau}$ plays the role of the Reeb vector field
$\partial/\partial\theta$ on $\partial D$. With this analogy in mind, the
direct analogue of (33) in the Grauert tube setting is

$\int_{\mathbb{R}}\hat{\chi}(t)e^{-i\lambda
t}\Pi_{\tau}(G^{t}_{\tau}(z),w)\,dt=\bigg{(}\int_{\mathbb{R}}\hat{\chi}(t)e^{-i\lambda
t}(G^{t}_{\tau})^{*}\Pi_{\tau}\,dt\bigg{)}(z,w),$ (34)

in which

  * •

integration against (the Fourier transform of) a suitably chosen Schwartz
function $\hat{\chi}$ replaces integration on the circle in the line bundle
setting;

  * •

pullback by the Hamilton flow (51) on the Grauert tube boundary replaces that
of the circle action in the line bundle setting;

  * •

the orthogonal projection (58) onto the Grauert tube boundary replaces the
Szegő projection in the line bundle setting.

It might appear that (34) is the correct object to study, but this is not
quite right. Unlike the circle action on $\partial D$ generated by
$\frac{\partial}{\partial\theta}$, the geodesic flow on a Grauert tube is
never holomorphic, nor does its orbits form a fiber bundle over a quotient
space. In particular, $G^{t}_{\tau}$ need not commute with $\Pi_{\tau}$, so
(34) fails to be CR holomorphic in the $z$ variable. This issue of holomorphy
can be fixed: It is shown in [32, Proposition 5.3] (see subsection 2.5 for the
statement) that there exists a polyhomogeneous pseudodifferential operator
$\hat{\sigma}$ on $\partial M_{\tau}$ so that

$\Pi_{\tau}e^{it\Pi_{\tau}D_{\sqrt{\rho}}\Pi_{\tau}}=\Pi_{\tau}\hat{\sigma}(G^{t}_{\tau})^{*}\Pi_{\tau}\quad\text{modulo
smoothing Toeplitz operators}.$ (35)

Evidently, the kernel of (35) is CR holomorphic in both variables. Thus, we
are led to the CR holomorphic analogue of (33) that is

$\int_{\mathbb{R}}\hat{\chi}(t)e^{-i\lambda
t}\Pi_{\tau}\hat{\sigma}(G^{t}_{\tau})^{*}\Pi_{\tau}\,dt=\int_{\mathbb{R}}\hat{\chi}(t)e^{-i\lambda
t}\Pi_{\tau}e^{it\Pi_{\tau}D_{\sqrt{\rho}}\Pi_{\tau}}\,dt\quad\text{to leading
order}.$ (36)

######  Remark .

We note that (36) is essentially the spectral decomposition of the elliptic
Toeplitz operator $\Pi_{\tau}D_{\sqrt{\rho}}\Pi_{\tau}$ introduced in (3). An
interesting problem is to compare the spectrum of the Toeplitz operator with
that of $\sqrt{\Delta}$, as

$\displaystyle
U({i\tau})\sqrt{\Delta}^{\frac{m+1}{2}}U({i\tau})^{*}=\Pi_{\tau}D_{\sqrt{\rho}}\Pi_{\tau}\quad\text{modulo
zeroth order Toeplitz operator}$ (37)

where $U(i\tau)$ is the Poisson-wave operator discussed in subsection 2.4. We
include further discussions in this direction in a subsequent paper [7,
Section 5].

###  1.2. Related results in the line bundle setting

Scaling asymptotics in the line bundle setting have been proved in varying
degrees of generality; see [2, 31, 26, 27, 25, 17]. The simplest setup is a
closed Kähler manifold $M$ polarized by an ample line bundle $L$. Endow $L$
with a Hermitian metric $h$ so that the curvature two-form $\Omega$ is
positive, and let $\omega=\frac{1}{2}\Omega$ be the Kähler form on $M$.

Choose coordinates $z=(z_{1},\dotsc,z_{m})$ in a neighborhood $U$ centered at
$p\in M$ so that the Kähler potential is locally of the form
$\varphi(z)=\lvert z\rvert^{2}+O(\lvert z\rvert^{3})$. We identify the Bergman
kernel with the Fourier components of the Szegő kernel using (30) and write

$\Pi_{h^{k}}(x,y)=\Pi_{h^{k}}(z,\theta,w,\varphi),$ (38)

where $x=(z,\theta)$ are linear coordinates on $T_{z}M\times S^{1}$. Then the
Bergman kernel scaling asymptotics is of the form

$\displaystyle
k^{-m}\Pi_{h^{k}}\bigg{(}\frac{z}{\sqrt{k}},\frac{\theta}{k},\frac{w}{\sqrt{k}},\frac{\varphi}{k}\bigg{)}$
$\displaystyle=\frac{1}{\pi^{m}}e^{i(\theta-\varphi)+i\Im(z\cdot\bar{w})-\frac{1}{2}\lvert
z-w\rvert^{2}}$ (39)
$\displaystyle\quad\times\bigg{[}1+\sum_{n=1}^{N}k^{-\frac{n}{2}}b_{k}(z,w)+k^{-\frac{N+1}{2}}R_{N}(z,w)\bigg{]}$

with

$\lVert R_{N}\rVert_{C^{j}(\\{\lvert z\rvert+\lvert w\rvert\leq\delta\\})}\leq
C(N,j,\delta).$ (40)

The remainder estimate implies Gaussian decay of the Bergman kernel in a
neighborhood of the diagonal:

$\Big{\lvert}\Pi_{h^{k}}(z,\theta,w,\varphi)\Big{\rvert}\leq\frac{k^{m}}{\pi^{m}}e^{\frac{-kD(z,w)}{2}}\bigg{(}1+O\bigg{(}\frac{1}{k}\bigg{)}\bigg{)}\quad\text{whenever
$d(z,w)\lesssim\sqrt{\frac{\log k}{k}}$}.$ (41)

Here, $D(z,w)$ is the Calabi diastasis (66), which is bounded above and below
by the square $d^{2}(z,w)$ of the Riemannian distance function.

A complete asymptotic expansion far away from the diagonal is available thanks
to the recent papers [30, 10, 19, 11] on _analytic Bergman kernel_. For
instance, [18, Theorem 1.1] shows (under analyticity assumption of the Kähler
potential) that

$\Big{\lvert}\Pi_{h^{k}}(z,\theta,w,\varphi)\Big{\rvert}=\frac{k^{m}}{\pi^{m}}e^{\frac{-kD(z,w)}{2}}\bigg{(}1+O\bigg{(}\frac{1}{k}\bigg{)}\bigg{)}\quad\text{whenever
$d(z,w)\lesssim\frac{1}{k}$}.$ (42)

In more general settings (including when the Kähler potential is no longer
assumed to be analytic), global Agmon-type estimates

$\Big{\lvert}\Pi_{h^{k}}(z,\theta,w,\varphi)\Big{\rvert}\leq
Ce^{-c\sqrt{k}d(z,w)}\quad\text{for all $z,w\in M$}$ (43)

have been established in [8, 12, 24, 1, 28, 37, 9].

In the presence of a real Hamiltonian $f\in C^{\infty}(M)$, there is a
Hamiltonian flow $\Phi^{M}_{\tau}$ and its lift $\Phi^{L}_{\tau}$ to $L$ in
addition to the circle action. In [29], the phase space scaling asymptotics is
computed near a fixed point $m$ and time $\tau$ in the graph of
$\Phi_{\tau}^{M}$ for the dynamical Toeplitz operators

$\displaystyle U_{\tau}=R_{\tau}\circ\Pi\circ(\Phi^{L}_{-\tau}\circ\Pi),$ (44)

where $R_{\tau}$ is a $0$th order Toeplitz operator. These asymptotics have an
additional dependence on the derivative of $\Phi^{M}_{-\tau}$ at $m$. The
nature of the computation in our proof is similar to the one in [29].

## Acknowledgment

The authors would like to thank their thesis advisor Steve Zelditch for many
helpful discussions and guiding their attention towards Szegő kernels and
Grauert tubes. The referee, whose careful reading of the manuscript led to
various corrections and improvement, is also much appreciated.

##  2\. Geometry of and analysis on Grauert tubes

We assume throughout that $(M,g)$ is a closed, real analytic Riemannian
manifold of dimension $m$. Grauert tubes were first studied by [15, 16, 22,
23] in relation to the complex Monge–Ampère equation and complexified
geodesics. The Poisson wave operator and the tempered spectral projection
(76), as well as other related operators, have been studied in [34, 35, 32].

###  2.1. Grauert tube and the cotangent bundle

Bruhat–Whitney proved that a real analytic manifold $M$ admits a
complexification $M_{\mathbb{C}}$ such that $M\subseteq M_{\mathbb{C}}$ is a
totally real submanifold, that is, $T_{p}M\cap JT_{p}M=\\{0\\}$ for all $p\in
M$.

In a neighborhood of $M$ in $M_{\mathbb{C}}$, there exists a unique strictly
plurisubharmonic function $\rho\colon U\subseteq M_{\mathbb{C}}\to[0,\infty)$
whose square root is the solution of the Monge–Ampère equation

$\det\bigg{(}\frac{\partial^{2}\sqrt{\rho}}{\partial
z_{j}\partial\overline{z}_{k}}\bigg{)}=0\quad\text{on $U\setminus M$},$ (45)

with the initial condition that the metric induced by the Kähler form
$i\partial\overline{\partial}\rho$ restricts to the Riemannian metric $g$ on
$M$. In fact, $\sqrt{\rho}$ is given by

$\sqrt{\rho}\colon U\subseteq
M_{\mathbb{C}}\to\mathbb{R},\quad\sqrt{\rho}(z)=\frac{1}{2i}\sqrt{r^{2}_{\mathbb{C}}(z,\overline{z})},$
(46)

where $r^{2}_{\mathbb{C}}(z,\overline{w})$ is the analytic extension of the
square of the Riemannian distance function $r\colon M\times M\to\mathbb{R}$ to
a neighborhood of the diagonal in
$M_{\mathbb{C}}\times\overline{M}_{\mathbb{C}}$. We call (46) the _Grauert
tube function_. For each $0<\tau<\tau_{\mathrm{max}}$, the sublevel set

$M_{\tau}=\\{z\in M_{\mathbb{C}}:\sqrt{\rho}(z)<\tau\\}$ (47)

is called the _Grauert tube of radius $\tau$_.

The complexified exponential map can be used to identify (47) with the co-ball
bundle of radius $\tau$:

$B^{*}_{\tau}M=\\{(x,\xi):\lvert\xi\rvert<\tau\\}.$ (48)

To explain this, we introduce notation for geodesic flows.

  * •

The (geometer’s) geodesic flow $g^{t}\colon T^{*}M\to T^{*}M$ is the Hamilton
flow of the metric norm squared function $H(x,\xi)=\lvert\xi\rvert^{2}_{x}$ .

  * •

The homogeneous geodesic flow $G^{t}\colon T^{*}M-0\to T^{*}M-0$ is the
Hamilton flow of the metric norm function
$\sqrt{H}(x,\xi)=\lvert\xi\rvert_{x}$. Note that
$G^{t}(x,\varepsilon\xi)=\varepsilon G^{t}(x,\xi)$.

Let $\pi\colon T^{*}M\to M$ be the natural projection. The exponential map
$\exp\colon T^{*}M\to M$ is defined by $\exp_{x}\xi=\pi g^{1}(x,\xi)$.
Analyticity of the metric allows us to complexify $t\mapsto t+i\tau$ in the
time variable. For $0<\tau<\tau_{\mathrm{max}}$, the imaginary time geodesic
flow $E$ is a diffeomorphism ([36, Lemma 1.1])

$E\colon B^{*}_{\tau}M\to M_{\tau},\quad E(x,\xi)=\exp_{x}^{\mathbb{C}}i\xi.$
(49)

With $\sqrt{H}(x,\xi)=\lvert\xi\rvert$ and $\sqrt{\rho}$ the Grauert tube
function, we have

$H=\rho\circ E\quad\text{and}\quad\sqrt{H}=\sqrt{\rho}\circ E.$ (50)

It follows that $E^{-1}$ conjugates the geodesic flow $G^{t}$ (resp. $g^{t}$)
on the cotangent bundle to the Hamilton flow of Hamiltonian vector field
$\Xi_{\sqrt{\rho}}$ (resp. $\Xi_{\rho}$) on the Grauert tube. In particular,
the transfer

$G^{t}_{\tau}\colon\partial M_{\tau}\to\partial M_{\tau},\quad
G^{t}_{\tau}=E\circ G^{t}\circ E^{-1}|_{\partial M_{\tau}}$ (51)

of the geodesic flow from the energy surface $\partial B_{\tau}^{*}M$ to the
Grauert tube boundary $\partial M_{\tau}$ coincides with the restriction of
the Hamilton flow of $\Xi_{\sqrt{\rho}}$ to $\partial M_{\tau}$.

Finally, if we write $\alpha_{T^{*}M}=\xi\,dx$ and $\omega_{T^{*}M}=d\xi\wedge
dx$ for the canonical 1-form and the symplectic form on the cotangent bundle,
then we also have

$(E^{-1})^{*}\alpha_{T^{*}M}=\operatorname{Im}\partial\rho=d^{c}\sqrt{\rho}\quad\text{and}\quad(E^{-1})^{*}\omega_{T^{*}M}=-i\partial\bar{\partial}\rho.$
(52)

Note that the right-hand sides of the two equalities are the canonical 1-form
and the Kähler form on the Grauert tube $M_{\tau}$.

######  Remark .

It is useful to think of the Grauert tube as the co-ball bundle endowed with
an _adapted complex structure_ $J=J_{g}$ induced from the Riemannian metric on
$M$. This complex structure is characterized as follows. Let
$\gamma\colon\mathbb{R}\to M$ be a geodesic (i.e., a $g^{t}$ orbit) and let
$\gamma_{\mathbb{C}}\colon\\{t+i\tau\in\mathbb{C}:\tau<\tau_{\mathrm{max}}\\}\to
M_{\mathbb{C}}$ be its analytic continuation to a strip. Then $J_{g}$ is the
complex structure on $M_{\tau}=B^{*}_{\tau}M$ so that $\gamma_{\mathbb{C}}$ is
a holomorphic map; see [22].

###  2.2. Contact and CR structure on the boundary of a Grauert tube

The boundary of a Grauert tube, being a level set of the strictly
plurisubharmonic function $\rho$, is strongly pseudoconvex and real analytic.
We endow $\partial M_{\tau}$ with the volume form $d\mu_{\tau}$ obtained by
pulling back the standard Liouville form $\alpha\wedge\omega_{T^{*}M}^{m-1}$
on $\partial B_{\tau}^{*}M_{\tau}$ under the symplectic diffeomorphism (49):

$d\mu_{\tau}=(E^{-1})^{*}(\alpha\wedge\omega_{T^{*}M}^{m-1})\Big{|}_{\partial
M_{\tau}}.$ (53)

Let $J=J_{g}$ denote the adapted complex structure on a Grauert tube (see
subsection 2.1). The boundary $\partial M_{\tau}$, being a real hypersurface
of the complex manifold $M_{\tau}$, carries a CR structure. The tangent space
admits the decomposition

$T\partial M_{\tau}=H\oplus\mathbb{R}T,$ (54)

where $H=\ker\alpha=JT\partial M_{\tau}\cap T\partial M_{\tau}$ is a real,
$J$-invariant hyperplane bundle and $T=\Xi_{\sqrt{\rho}}$, called the _Reeb
vector field_ , is the Hamilton vector field of the Grauert tube function.
Equivalently, $JT=\nabla\sqrt{\rho}$.

Complexifying the tangent space yields

$\displaystyle T^{\mathbb{C}}\partial M_{\tau}=H^{(1,0)}\partial
M_{\tau}\oplus H^{(0,1)}\partial M_{\tau}\oplus\mathbb{C}T$ (55)

where $H^{(1,0)}$ and $H^{(0,1)}$ are the $J$-holomorphic and
$J$-antiholomorphic subspaces. The boundary Cauchy–Riemann operators are
defined by

$\displaystyle\partial_{b}f=df|_{H^{(1,0)}}\quad\text{and}\quad\overline{\partial}_{b}f=df|_{H^{(0,1)}}.$
(56)

###  2.3. The Szegő projector and the Boutet de Movel–Sjöstrand parametrix

The Szegő projector is a complex Fourier integral operator (FIO) with a
positive complex canonical relation. We recall the Boutet de Monvel–Sjöstrand
parametrix construction for the Szegő kernel in the context of Grauert tubes,
but the same construction holds with $\partial M_{\tau}$ replaced by any
bounded, strongly pseudoconvex domain.

######  Definition .

The Szegő projector $\Pi_{\tau}$ is the orthogonal projection

$\displaystyle\Pi_{\tau}\colon L^{2}(\partial M_{\tau},d\mu_{\tau})\to
H^{2}(\partial M_{\tau},d\mu_{\tau})$ (57)

onto the Hardy space consisting of boundary values of holomorphic functions in
$M_{\tau}$ that are square integrable with respect to the volume form (53).
Its distributional kernel $\Pi_{\tau}(z,w)$ is defined by the relation

$\displaystyle\Pi_{\tau}f(z)=\int_{\partial
M_{\tau}}\Pi_{\tau}(z,w)f(w)\,d\mu_{\tau}(w)\quad\text{for all $f\in
L^{2}(\partial M_{\tau})$}.$ (58)

The Szegő projector is a Fourier integral operator with a positive complex
canonical relation. The real points of this canonical relation is the graph of
the identity map on the symplectic cone $\Sigma_{\tau}$ spanned by the contact
form $\alpha:=d^{c}\sqrt{\rho}|_{T*\partial M_{\tau}}$, i.e.,

$\Sigma_{\tau}=\\{(\zeta,r\alpha_{\zeta}):r\in\mathbb{R}_{+}\\}\subseteq
T^{*}(\partial M_{\tau}).$ (59)

Using the imaginary time geodesic flow $E$ in (49), we can construct the
symplectic equivalence $\iota_{\tau}$ between the cotangent bundle and the
symplectic cone:

$\iota_{\tau}\colon
T^{*}M-0\to\Sigma_{\tau},\quad\iota_{\tau}(x,\xi)=\bigg{(}E\Big{(}x,\tau\frac{\xi}{\lvert\xi\rvert}\Big{)},\lvert\xi\rvert\alpha_{E(x,\tau\frac{\xi}{\lvert\xi\rvert})}\bigg{)}.$
(60)

We now briefly describe the symbol of the Szegő projector. Details can be
found in [3, Theorem 11.2] or [33, Section 3]. Let
$\Sigma_{\tau}^{\perp}\otimes\mathbb{C}$ be the complexified normal bundle of
(59). The symbol $\sigma(\Pi_{\tau})$ of $\Pi_{\tau}$ is a rank one projection
onto a ground state $e_{\Lambda_{\tau}}$, which is annihilated by a Lagrangian
system of Cauchy–Riemann equations corresponding to a Lagrangian subspace
$\Lambda_{\tau}\subseteq\Sigma_{\tau}^{\perp}\otimes\mathbb{C}$. The time
evolution $\Pi_{\tau}\mapsto G^{-t}_{\tau}\Pi_{\tau}G^{t}_{\tau}$ under the
Hamilton flow (51) yields another rank one projection onto some time-dependent
ground state $e_{\Lambda_{\tau}^{t}}$, where $\Lambda_{\tau}^{t}$ is the
pushforward of $\Lambda_{\tau}$ under the flow. The quantity

$\sigma_{t,\tau,0}=\langle
e_{\Lambda_{\tau}^{t}},e_{\Lambda_{\tau}}\rangle^{-1}$ (61)

appears in subsection 2.4 and subsection 2.5. It was shown in [33] that
$\langle e_{\Lambda_{\tau}^{t}},e_{\Lambda_{\tau}}\rangle$ is the $L^{2}$
inner product of two Gaussians, hence nowhere vanishing.

Finally, we discuss an oscillatory integral representation of the Szegő
kernel. Recall from subsection 2.1 that $\rho$ is the real analytic Kähler
potential on the Grauert tube. We introduce the defining function for the
boundary $\partial M_{\tau}$ of a Grauert tube:

$\varphi_{\tau}\colon
M_{\tau_{\mathrm{max}}}\to[0,\infty),\quad\varphi_{\tau}(z):=\rho(z)-\tau^{2}$
(62)

Let $\varphi_{\tau}(z,\overline{w})$ be the analytic extension of
$\varphi_{\tau}(z)=\varphi_{\tau}(z,\overline{z})$ to
$M_{\tau}\times\overline{M}_{\tau}$ obtained by polarization. Put (c.f. the
formula (46) for the Grauert tube function)

$\displaystyle\psi_{\tau}(z,w)=\frac{1}{i}\varphi_{\tau}(z,\overline{w})=\frac{1}{i}\left(-\frac{1}{4}r^{2}_{\mathbb{C}}(z,\overline{w})-\tau^{2}\right).$
(63)

By construction, $\psi$ is homolomorphic in $z$, antiholomorphic in $w$, and
satisfies $\psi(z,w)=-\overline{\psi(z,w)}$.

######  Theorem 2.1 (The Boutet de Monvel–Sjöstrand parametrix, [5, Theorem
1.5]).

With $\psi$ as in (63), there exists a classical symbol

$s\in S^{m-1}(\partial M_{\tau}\times\partial
M_{\tau}\times\mathbb{R}^{+})\quad\text{with}\quad
s(z,w,\sigma)\sim\sum_{k=0}^{\infty}\sigma^{m-1-k}s_{k}(z,w)$ (64)

so that the Szegő kernel (58) has the oscillatory integral representation

$\displaystyle\Pi_{\tau}(z,w)=\int_{0}^{\infty}e^{i\sigma\psi_{\tau}(z,w)}s(z,w,\sigma)\,d\sigma\quad\text{modulo
a smoothing kernel}.$ (65)

We record a key estimate for $\varphi_{\tau}$ (or equivalently for the phase
function $\psi$). Define the _Calabi diastatis function_ by

$D(z,w)=\varphi_{\tau}(z,\overline{z})+\varphi_{\tau}(w,\overline{w})-\varphi_{\tau}(z,\overline{w})-\varphi_{\tau}(w,\overline{z}).$
(66)

On $\partial M_{\tau}$, the diastatis simplifies to

$\displaystyle
D(z,w)=-\varphi_{\tau}(z,\overline{w})-\varphi_{\tau}(w,\overline{z})=-2\operatorname{Re}\varphi_{\tau}(z,\overline{w})\quad\text{for
$z,w\in\partial M_{\tau}$}.$ (67)

In the closure of the Grauert tube, [5, Corollary 1.3] gives the lower bound

$\displaystyle D(z,w)\geq C\big{(}d(z,\partial M_{\tau})+d(w,\partial
M_{\tau})+d(z,w)^{2}\big{)}\quad\text{for $z,w\in\overline{M_{\tau}}$}.$ (68)

###  2.4. Poisson wave operator and tempered spectral projection

Recall the eigenequation (17) for the Laplacian on $M$. The eigenfunction
expansion of the Schwartz kernel of the half-wave operator
$U(t):=e^{it\sqrt{\Delta}}$ is given by

$\displaystyle
U(t,x,y)=\sum_{j=0}^{\infty}e^{it\lambda_{j}}\varphi_{\lambda_{j}}(x)\overline{\varphi_{\lambda_{j}}(y)}.$
(69)

It is well-known (see for instance [16, Theorem]) that for
$0\leq\tau\leq\tau_{\mathrm{max}}$, the Schwartz kernel $U(t,x,y)$ admits an
analytic extension $U(t+i\tau,x,y)$ in the time variable $t\mapsto
t+i\tau\in\mathbb{C}$. Note that the corresponding operator
$U(i\tau)=e^{-\tau\sqrt{\Delta}}$ is the Poisson operator. Moreover, for $y$
and $\tau$ fixed, the Poisson kernel can be further analytically extended in
the spacial variable $x\mapsto z\in M_{\tau}$. The resulting operator is a
Fourier integral operator of complex type. More precisely, denote by
$\mathcal{O}(\partial M_{\tau})$ the space of CR holomorphic functions on the
Grauert tube boundary.

######  Theorem 2.2 ([4, 16, 14, 35]).

For $0<\tau<\tau_{\mathrm{max}}$, the Poisson operator

$U(i\tau)=e^{-\tau\sqrt{\Delta}}\colon L^{2}(M)\to\mathcal{O}(\partial
M_{\tau})$ (70)

whose Schwartz kernel $U(i\tau,z,y)$ is obtained from analytically continuing
the half-wave kernel $U(t,x,y)$ of (69) is a Fourier integral operator of
order $-(m-1)/4$ with complex phase associated to the canonical relation

$\\{(y,\eta,\iota_{\tau}(y,\eta)\\}\subseteq T^{*}M\times\Sigma_{\tau},$ (71)

where $\iota_{\tau}$ and $\Sigma_{\tau}$ are defined in (60) and (59).

To introduce the complexified spectral projection kernels, we need to further
continue $U(i\tau,z,y)$ anti-holomorphically in the $y$ variable. Consider the
operator

$U^{\mathbb{C}}(t+2i\tau)=U(i\tau)U(t)U(i\tau)^{*}\colon\mathcal{O}(\partial
M_{\tau})\to\mathcal{O}(\partial M_{\tau})$ (72)

with Schwartz kernel

$U^{\mathbb{C}}(t+2i\tau,z,w)=\sum_{j=0}^{\infty}e^{i(t+2i\tau)j}\varphi^{\mathbb{C}}_{\lambda_{j}}(z)\overline{\varphi^{\mathbb{C}}_{\lambda_{j}}(w)}.$
(73)

######  Proposition  ([32, Proposition 7.1]).

Let $(G^{t}_{\tau})^{*}$ denote the pullback by the Hamilton flow (51) of
$\Xi_{\sqrt{\rho}}$ on $\partial M_{\tau}$. There exists a classical
polyhomogeneous pseudodifferential operator
$\hat{\sigma}_{t,\tau}(w,D_{\sqrt{\rho}})$ on $\partial M_{\tau}$ so that

$\displaystyle
U^{\mathbb{C}}(t+2i\tau)=\Pi_{\tau}\hat{\sigma}_{t,\tau}\left(G^{t}_{\tau}\right)^{*}\Pi_{\tau}\quad\text{modulo
a smoothing Toeplitz operator}.$ (74)

The symbol $\sigma_{t,\tau}$ of $\hat{\sigma}_{t,\tau}$ admits a complete
asymptotic expansion

$\displaystyle\sigma_{t,\tau}\left(w,r\right)\sim\sum_{j=0}^{\infty}\sigma_{t,\tau,j}\left(w\right)r^{-\frac{m-1}{2}-j},$
(75)

in which $\sigma_{t,\tau,0}=\langle
e_{\Lambda_{\tau}^{t}},e_{\Lambda_{\tau}}\rangle^{-1}$ is the reciprocal of
the overlap of two Gaussians, as in (61).

Finally, we define the tempered spectral projection kernel by

$P_{\lambda}(z,w)=\sum_{j:\lambda_{j}\leq\lambda}e^{-2\tau\lambda_{j}}\varphi^{\mathbb{C}}_{\lambda_{j}}(z)\overline{\varphi^{\mathbb{C}}_{\lambda_{j}}(w)}.$
(76)

Fix $\varepsilon>0$ and let $\chi$ be a positive even Schwartz function such
that $\hat{\chi}\left(0\right)=1$ and
$\operatorname{supp}\hat{\chi}\subseteq[-\varepsilon,\varepsilon]$. Then

$\displaystyle P_{\chi,\lambda}(z,w)=\chi\ast
d_{\lambda}P_{\lambda}(z,w)=\int_{\mathbb{R}}\hat{\chi}(t)e^{-it\lambda}U^{\mathbb{C}}(t+2i\tau,z,w)\,dt$
(77)

######  Remark .

It was shown in [35] that the complexified spectral function

$\sum_{\lambda_{j}\leq\lambda}\lvert\varphi_{\lambda_{j}}^{\mathbb{C}}(\zeta)\rvert^{2}\qquad(\zeta\in
M_{\tau})$ (78)

grows exponentially at a rate of $e^{2\lambda\sqrt{\rho}(\zeta)}$. We
introduce the exponentially damping prefactor to obtain polynomial growth in
the tempered version (76).

###  2.5. The Toeplitz operator $\Pi_{\tau}D_{\sqrt{\rho}}\Pi_{\tau}$

Let $D_{\sqrt{\rho}}=\frac{1}{i}\Xi_{\sqrt{\rho}}$ denote the Hamilton vector
field of the Grauert tube function acting as a differential operator. The
symbol of $D_{\sqrt{\rho}}$ is nowhere vanishing on $\Sigma_{\tau}-0$, where
$\Sigma_{\tau}$ is the symplectic cone (59). Thus,
$\Pi_{\tau}D_{\sqrt{\rho}}\Pi_{\tau}$ is an elliptic Toeplitz operator with
discrete spectrum.

As discussed in subsection 1.1, the Grauert tube analogue of Fourier
coefficients of the Szegő kernel are given by the spectral localization
operator

$\displaystyle\Pi_{\chi,\lambda}=\Pi_{\tau}\chi(\Pi_{\tau}D_{\sqrt{\rho}}\Pi_{\tau}-\lambda)=\int_{\mathbb{R}}\hat{\chi}(t)e^{-it\lambda}\Pi_{\tau}e^{it\Pi_{\tau}D_{\sqrt{\rho}}\Pi_{\tau}}\,dt$
(79)

######  Proposition  ([32, Proposition 5.3]).

Let $(G^{t}_{\tau})^{*}$ denote the pullback by the Hamilton flow (51) of
$\Xi_{\sqrt{\rho}}$ on $\partial M_{\tau}$. There exists a classical
polyhomogeneous pseudodifferential operator
$\hat{\sigma}_{t,\tau}(w,D_{\sqrt{\rho}})$ on $\partial M_{\tau}$ so that

$\displaystyle\Pi_{\tau}e^{it\Pi_{\tau}D_{\sqrt{\rho}}\Pi_{\tau}}=\Pi_{\tau}\hat{\sigma}_{t,\tau}\left(G^{t}_{\tau}\right)^{*}\Pi_{\tau}\
\quad\text{modulo a smoothing Toeplitz operator}.$ (80)

The symbol $\sigma_{t,\tau}$ of $\hat{\sigma}_{t,\tau}$ admits a complete
asymptotic expansion

$\displaystyle\sigma_{t,\tau}(w,r)\sim\sum_{j=0}^{\infty}\sigma_{t,\tau,j}(w)r^{-j},$
(81)

in which $\sigma_{t,\tau,0}=\langle
e_{\Lambda_{\tau}^{t}},e_{\Lambda_{\tau}}\rangle^{-1}$ is the reciprocal of
the overlap of two Gaussians, as in (61).

######  Remark .

The difference between subsection 2.5 and subsection 2.4 is that the symbol
(75) is of order zero and (81) is of order $-(m-1)/2$. This is because the
partial sums (76) are not divided through by $\lVert
e^{-\tau\lambda}\varphi_{\lambda}\rVert_{L^{2}(\partial
M_{\tau})}^{2}\sim\lambda^{-\frac{m-1}{2}}$. For an in-depth discussion of
complexified eigenfunctions and dynamical Toeplitz operators see [21, 35, 32].

##  3\. Heisenberg coordinates on the boundary of a Grauert tube

This section reviews the construction and properties of Heisenberg coordinates
following [13, Section 18]. Roughly speaking, in these coordinates, the
strongly pseudoconvex boundary $\partial M_{\tau}\subseteq M_{\mathbb{C}}$ is,
to a first approximation, the Heisenberg group viewed as a hypersurface in
complex Euclidean space. We briefly recall the CR structure on the Heisenberg
group.

The Heisenberg group $\mathbf{H}^{m-1}$ of degree $m-1$ is the Lie group
$\mathbb{R}\times\mathbb{C}^{m-1}$ with group law

$(\theta,z)\cdot(\phi,w)=(\theta+\phi+2\operatorname{Im}(z\cdot w),z+w).$ (82)

######  Remark .

It is more common to use $t$ for the $\mathbb{R}$-coordinate but, since $t$ is
already reserved for the time parameter of the geodesic or Hamiltonian flow,
we use $\theta$ instead.

For $1\leq j\leq m-1$, the vector fields

$Z_{j}=\frac{\partial}{\partial
z_{j}}+i\overline{z}_{j}\frac{\partial}{\partial\theta},\qquad\overline{Z}_{j}=\frac{\partial}{\partial\overline{z}_{j}}+iz_{j}\frac{\partial}{\partial\theta},\qquad
T=\frac{\partial}{\partial\theta}$ (83)

satisfy the commutation relations

$[Z_{j},\overline{Z}_{k}]=-2i\delta_{jk}T\quad\text{and}\quad[Z_{j},Z_{k}]=[\overline{Z}_{j},\overline{Z}_{k}]=[Z_{j},T]=[\overline{Z}_{j},T]=0.$
(84)

Furthermore, the $1$-form

$\alpha=d\theta-i\sum_{j=1}^{m-1}(\overline{z}_{j}\,dz_{j}-z_{j}\,d\overline{z}_{j})$
(85)

annihilates $Z_{j}$ and $\overline{Z}_{j}$ for all $1\leq j\leq m-1$. Hence,
the complexified tangent space of the Heisenberg group admits the
decomposition

$T^{\mathbb{C}}\mathbf{H}^{m-1}=H^{(1,0)}\oplus H^{(0,1)}\oplus\mathbb{C}T,$
(86)

in which $H^{(1,0)}$ and $H^{(0,1)}$ are spanned by the holomorphic vector
fields $Z_{j}$ and the anti-holomorphic vector fields $\overline{Z}_{j}$,
respectively. Direct verification shows that the subspace $H^{(1,0)}$ defines
a CR structure on $\mathbf{H}^{m-1}$.

The Levi form $\langle\,,\,\rangle_{L}$ is the Hermitian form on $H^{(1.0)}$
defined by

$\langle Z,W\rangle_{L}=-i\langle
d\alpha,Z\wedge\overline{W}\rangle\quad\text{for all $Z,W\in H^{(1,0)}$}.$
(87)

Direct computation shows that the Levi form with respect to the basis $Z_{j}$
is the identity. Thus, $\mathbf{H}^{m-1}$ is strongly pseudoconvex.

Set

$\displaystyle D_{m}$
$\displaystyle=\bigg{\\{}\zeta=(\zeta_{0},\dotsc,\zeta_{m-1})\in\mathbb{C}^{m}:\sum_{j=1}^{m-1}\lvert\zeta_{j}\rvert^{2}<\operatorname{Im}\zeta_{0}\bigg{\\}}.$
(88) $\displaystyle\partial D_{m}$
$\displaystyle=\bigg{\\{}\zeta=(\zeta_{0},\dotsc,\zeta_{m-1})\in\mathbb{C}^{m}:\sum_{j=1}^{m-1}\lvert\zeta_{j}\rvert^{2}=\operatorname{Im}\zeta_{0}\bigg{\\}}.$
(89)

The Heisenberg group $\mathbf{H}^{m-1}$ acts on $\mathbb{C}^{m}$ by
holomorphic affine transformations; given
$(\theta,z)=(\theta,z_{1},\dotsc,z_{m-1})\in\mathbf{H}^{m-1}$ and
$\zeta=(\zeta_{0},\dotsc,\zeta_{m-1})\in\mathbb{C}^{m}$, we have
$(\theta,z)\cdot\zeta=(\zeta_{0}^{\prime},\dotsc,\zeta_{m-1}^{\prime})$, where

$\displaystyle\zeta_{0}^{\prime}$ $\displaystyle=\zeta_{0}+\theta+i\lvert
z\rvert^{2}+2i\sum_{j=1}^{m-1}\zeta_{j}\overline{z}_{j},$ (90)
$\displaystyle\zeta_{j}^{\prime}$ $\displaystyle=\zeta_{j}+z_{j}\quad\text{for
$1\leq j\leq m-1$}.$

Note that the group action preserves $D_{m}$ (which is biholomorphic to the
unit ball in $\mathbb{C}^{m}$) and $\partial D_{m}$. Indeed,
$\mathbf{H}^{m-1}$ acts simply and transitively on $\partial D_{m}$. Thus,
comparing (82) and (90), we find that the Heisenberg group may be identified
to $\partial D_{m}$ via the correspondence

$H_{m-1}\ni(\theta,z)\leftrightarrow(\theta,z)\cdot 0=(\theta+i\lvert
z\rvert^{2},z_{1},\dotsc,z_{m-1})\in\partial D_{m}.$ (91)

On any nondegenerate CR manifold (in particular the boundary of a Grauert
tube), there exists a Heisenberg-like coordinate system
$(\theta,z_{1},\dotsc,z_{m-1})$ so that $Z_{j}=(\partial/\partial
z_{j})+i\overline{z}_{j}(\partial/\partial\theta)$ and
$T=(\partial/\partial\theta)$ up to lower order terms. The precise statement,
section 1, is already given in the introduction, so we do not reproduce it
here. An explicit construction of such “normal coordinates” was presented in
[13]. We call the same coordinates “Heisenberg coordinates” in this paper to
emphasize the osculating Heisenberg structure (see subsection 3.1).

######  Remark .

If $f$ is smooth, then

$\displaystyle f$ $\displaystyle=O^{1}\text{ if and only if
$f(\eta)=O\left(\sum_{k=1}^{m-1}(\lvert x_{k}(\eta)\rvert+\lvert
y_{k}(\eta)\rvert)+\lvert\theta(\eta)\rvert\right)$},$ (92) $\displaystyle f$
$\displaystyle=O^{2}\text{ if and only if
$f(\eta)=O\left(\sum_{k=1}^{m-1}(\lvert x_{k}(\eta)\rvert^{2}+\lvert
y_{k}(\eta)\rvert^{2})+\lvert\theta(\eta)\rvert\right)$}.$ (93)

###  3.1. Taylor expansions in Heisenberg coordinates

We fix notation. Let $p\in\partial M_{\tau}$ and let $z,w$ be two points in
$M_{\tau_{\mathrm{max}}}$ that lie in a Heisenberg coordinate patch (see
section 1) containing $p$. We write

$\displaystyle z$ $\displaystyle=(z_{0},\dotsc,z_{m-1}),\quad$
$\displaystyle\theta=\operatorname{Re}z_{0},\quad$ $\displaystyle
u=(z_{1},\dotsc,z_{m-1}),$ (94) $\displaystyle w$
$\displaystyle=(w_{0},\dotsc,w_{m-1}),\quad$
$\displaystyle\phi=\operatorname{Re}w_{0},\quad$ $\displaystyle
v=(w_{1},\dotsc,w_{m-1}).$

Note that if $z\in\partial M_{\tau}$, then $z=(\theta,u)$.

We record the Taylor expansions of the boundary defining function (62) and the
Hamilton flow (51) on the boundary in these coordinates. Detailed computations
in the setting of a real hypersurface in a complex manifold whose induced CR
structure is strongly pseudoconvex (as is the case of $\partial
M_{\tau}\subseteq M_{\mathbb{C}}$) are found in [13, Section 18].

######  Lemma .

Let $\varphi_{\tau}$ be the defining function (62) of the Grauert tube
boundary. Denote by $\varphi_{\tau}(z,\overline{w})$ be its analytic extension
obtained by polarization. Then, in Heisenberg coordinates near $p$ in
$M_{\tau}$, we have

$\displaystyle\varphi_{\tau}(z,\overline{w})=\frac{i}{2}(z_{0}-\overline{w}_{0})+\sum_{j=1}^{m-1}z_{j}\overline{w}_{j}+R(z,\overline{w}).$
(95)

The remainder $R$ takes the form

$\displaystyle
R(z,\overline{w})=R^{\varphi_{\tau}}(z_{0},u,\overline{w}_{0},\overline{v})=R_{2}(z_{0},\overline{w}_{0})+R_{2}(z_{0},u,\overline{w}_{0},\overline{v})+R_{3}(u,\overline{v}),$
(96)

where

  * •

$R_{2}(z_{0},\overline{w}_{0})$ contains only of mixed terms of the form
$z_{0}^{\alpha}\overline{w}_{0}^{\beta}$ with $|\alpha|+|\beta|\geq 2$.

  * •

$R_{2}(z_{0},u,\overline{w}_{0},\overline{v})$ contains only of mixed terms of
the form $z_{0}^{\alpha}\overline{v}^{\beta}$ or
$\overline{w}_{0}^{\alpha}u^{\beta}$ with $|\alpha|+|\beta|\geq 2$.

  * •

$R_{3}(u,\overline{v})$ contains only of mixed terms of the form
$u^{\alpha}\overline{v}^{\beta}$ with $|\alpha|+|\beta|\geq 3$.

###### Proof.

Direct computation (see [13, (18.4)]) yields

$\displaystyle\varphi_{\tau}(z)=\varphi_{\tau}(z_{0},u)=-\operatorname{Im}z_{0}+\lvert
u\rvert^{2}+O(\lvert z_{0}\rvert\lvert u\rvert+\lvert z_{0}\rvert^{2}+\lvert
u\rvert^{3}),$ (97)

so

$\displaystyle
d\varphi_{\tau}\big{|}_{p}=-\operatorname{Im}dz_{0}\big{|}_{p}\quad\text{and}\quad\frac{\partial^{2}\varphi_{\tau}}{\partial
z_{j}\partial\overline{z}_{k}}\bigg{|}_{p}=\delta_{jk}\quad(1\leq j,k\leq
m-1).$ (98)

Combined with the near-diagonal Taylor expansion

$\displaystyle\varphi_{\tau}(p+h,p+k)=\sum_{\alpha,\beta}\frac{\partial^{\alpha+\beta}\varphi_{\tau}}{\partial
z^{\alpha}\partial\bar{z}^{\beta}}(p)\frac{h^{\alpha}}{\alpha!}\frac{\overline{k}^{\beta}}{\beta!}$
(99)

of $\varphi_{\tau}$, we obtain the desired result. ∎

######  Remark .

Note that the formula (97) for the defining function implies $\partial
M_{\tau}$ is highly tangent at $p$ to the hypersurface
$\operatorname{Im}z_{0}=\sum_{k=1}^{m-1}\lvert z_{k}\rvert^{2}$, which is the
geometric model (89) for the Heisenberg group.

######  Lemma .

Let $G^{t}_{\tau}$ be the Hamilton flow (51) of the Grauert tube function on
$\partial M_{\tau}$. Then, in Heisenberg coordinates near $p$ in $\partial
M_{\tau}$, we have

$G^{t}_{\tau}(z)=G^{t}_{\tau}(\theta,u)=\Big{(}\theta+2\tau t+t\cdot
O^{1}+O(t^{2}),u+t\cdot O^{1}+O(t^{2})\Big{)},$ (100)

where $O^{1}$ denotes the Heisenberg-type order (6).

###### Proof.

By [13, Theorem 18.5], the Reeb vector field is of the form

$\displaystyle\frac{1}{2\tau}H_{\sqrt{\rho}}=\frac{\partial}{\partial\theta}+\sum_{j=1}^{m-1}O^{1}\frac{\partial}{\partial
z_{j}}+\sum_{j=1}^{m-1}O^{1}\frac{\partial}{\partial\overline{z}_{j}}+O^{1}\frac{\partial}{\partial\theta}.$
(101)

The Lemma then follows by Taylor expanding $G^{t}_{\tau}$ near $t=0$. ∎

##  4\. Proof of main results

We briefly outline the proof of Theorem 1.1 before delving into the
computations. As indicated in a subsequent Section, the same techniques work
almost verbatim for proving Theorem 1.2.

Since we are interested in scaling asymptotics, variables are rescaled
according to their Heisenberg-type order (cf (6)). Namely, given
$(\theta,u)\in\partial M_{\tau}$ in Heisenberg coordinates near $p\in\partial
M_{\tau}$ (recall section 1 and (94)), we consider

$(\theta,u)\mapsto\bigg{(}\frac{\theta}{\lambda},\frac{u}{\sqrt{\lambda}}\bigg{)}\in\partial
M_{\tau}.$ (102)

We rewrite the (rescaled) spectral localization kernel of (2) in terms of the
kernel of a “dynamical Toeplitz operator” using subsection 2.5. Then,
substituting the Boutet de Monvel–Sjöstrand parametrix (65) for the Szegő
projectors, we are left with an oscillatory integral. Taylor expanding the
phase function using subsection 3.1, subsection 3.1, and then applying the
method of stationary phase complete the proof.

###  4.1. Proof of Theorem 1.1

From (2) and subsection 2.5, we have

$\Pi_{\chi,\lambda}=\int_{\mathbb{R}}\hat{\chi}(t)\Pi_{\tau}\hat{\sigma}_{t,\tau}(G^{t}_{\tau})^{*}\Pi_{\tau}\,dt\quad\text{modulo
a smoothing Toeplitz operator}.$ (103)

Fix a point $p\in\partial M_{\tau}$ and a Heisenberg coordinate chart centered
at $p=0$. Fix two points $(\theta,u),(\phi,v)\in\partial M_{\tau}$ in
Heisenberg coordinates. Then, substituting the parametrix (65) for each
instance of $\Pi_{\tau}$ above and composing the resulting kernels, we arrive
at the oscillatory integral representation

$\displaystyle\Pi_{\chi,\lambda}\bigg{(}\frac{\theta}{\lambda},\frac{u}{\sqrt{\lambda}},\frac{\phi}{\lambda},\frac{v}{\sqrt{\lambda}}\bigg{)}\sim\int_{\mathbb{R}\times\partial
M_{\tau}\times\mathbb{R}^{+}\times\mathbb{R}^{+}}e^{i\lambda\Psi}A\,d\sigma_{1}d\sigma_{2}d\mu_{\tau}(w)dt,$
(104)

in which the phase $\Psi$ and the amplitude $A$ are given by

$\displaystyle\Psi$
$\displaystyle=-t+\frac{1}{\lambda}\sigma_{2}\psi_{\tau}\bigg{(}\bigg{(}\frac{\theta}{\lambda},\frac{u}{\sqrt{\lambda}}\bigg{)},w\bigg{)}+\frac{1}{\lambda}\sigma_{1}\psi_{\tau}\bigg{(}G^{t}_{\tau}(w),\bigg{(}\frac{\phi}{\lambda},\frac{v}{\sqrt{\lambda}}\bigg{)}\bigg{)},$
(105) $\displaystyle A$
$\displaystyle=\hat{\chi}(t)s\bigg{(}\frac{\theta}{\lambda},\frac{u}{\sqrt{\lambda}},w,\sigma_{2}\bigg{)}s\bigg{(}G^{t}_{\tau}(w),\frac{\phi}{\lambda},\frac{v}{\sqrt{\lambda}},\sigma_{1}\bigg{)}\sigma_{t,\tau}(w,\sigma_{1}).$

######  Remark .

Recall $\sigma_{t,\tau}$ is the unitarization symbol of order zero from
subsection 2.5, $s=s(z,w,\sigma)$ is the symbol of order $m-1$ in the Boutet
de Monvel–Sjöstrand parametrix (65), and $\sigma=\sigma_{j}\in\mathbb{R}^{+}$
are parameters of integration in the parametrix. We do not yet put the spatial
variable $x\in\partial M_{\tau}$ of integration in Heisenberg coordinates.

Make the change-of-variables $\sigma_{j}\mapsto\lambda\sigma_{j}$. Homogeneity
of the symbols implies

$\displaystyle\Pi_{\chi,\lambda}\bigg{(}\frac{\theta}{\lambda},\frac{u}{\sqrt{\lambda}},\frac{\phi}{\lambda},\frac{v}{\sqrt{\lambda}}\bigg{)}\sim\lambda^{2m}\int_{\mathbb{R}\times\partial
M_{\tau}\times\mathbb{R}^{+}\times\mathbb{R}^{+}}e^{i\lambda\tilde{\Psi}}\tilde{A}\,d\sigma_{1}d\sigma_{2}d\mu_{\tau}(w)dt,$
(106)

in which the phase $\tilde{\Psi}$ and the amplitude $\tilde{A}$ are given by

$\displaystyle\tilde{\Psi}$
$\displaystyle=-t+\sigma_{2}\psi_{\tau}\bigg{(}\bigg{(}\frac{\theta}{\lambda},\frac{u}{\sqrt{\lambda}}\bigg{)},w\bigg{)}+\sigma_{1}\psi_{\tau}\bigg{(}G^{t}_{\tau}(w),\bigg{(}\frac{\phi}{\lambda},\frac{v}{\sqrt{\lambda}}\bigg{)}\bigg{)},$
(107) $\displaystyle\tilde{A}$ $\displaystyle=\lambda^{-2m+2}A.$

######  Remark .

It follows from subsection 4.1 that $\tilde{A}$ is a symbol of order $0$.

We begin by localizing in $(w,t)\in\partial M_{\tau}\times\mathbb{R}$. Fix
$C>0$ and $0<\delta\ll 1$. Set

$\displaystyle V_{\lambda}$
$\displaystyle=\left\\{(w,t):\textstyle\max\Big{\\{}d\big{(}w,\big{(}\frac{\theta}{\lambda},\frac{u}{\sqrt{\lambda}}\big{)}\big{)},d\big{(}G^{t}_{\tau}(w),\big{(}\frac{\phi}{\lambda},\frac{v}{\sqrt{\lambda}}\big{)}\big{)}\Big{\\}}<\frac{2C}{3}\lambda^{\delta-\frac{1}{2}}\right\\},$
(108) $\displaystyle W_{\lambda}$
$\displaystyle=\left\\{(w,t):\textstyle\max\Big{\\{}d\big{(}w,\big{(}\frac{\theta}{\lambda},\frac{u}{\sqrt{\lambda}}\big{)}\big{)},d\big{(}G^{t}_{\tau}(w),\big{(}\frac{\phi}{\lambda},\frac{v}{\sqrt{\lambda}}\big{)}\big{)}\Big{\\}}>\frac{C}{2}\lambda^{\delta-\frac{1}{2}}\right\\}.$

Let $\\{\varrho_{\lambda},1-\varrho_{\lambda}\\}$ be a partition of unity
subordinate to the cover $\\{V_{\lambda},W_{\lambda}\\}$ and decompose the
integral (106) into

$\displaystyle\Pi_{\chi,\lambda}\bigg{(}\frac{\theta}{\lambda},\frac{u}{\sqrt{\lambda}},\frac{\phi}{\lambda},\frac{v}{\sqrt{\lambda}}\bigg{)}$
$\displaystyle\sim I_{1}+I_{2},$ (109) $\displaystyle I_{1}$
$\displaystyle=\lambda^{2m}\int
e^{i\lambda\tilde{\Psi}}\varrho_{\lambda}(t,w)\tilde{A}\,d\sigma_{1}d\sigma_{2}d\mu_{\tau}(w)dt,$
(110) $\displaystyle I_{2}$ $\displaystyle=\lambda^{2m}\int
e^{i\lambda\tilde{\Psi}}(1-\varrho_{\lambda}(t,w))\tilde{A}\,d\sigma_{1}d\sigma_{2}d\mu_{\tau}(w)dt.$
(111)

######  Lemma .

We have $I_{2}=O(\lambda^{-\infty})$.

###### Proof.

By definition of $W_{\lambda}$, on the support of $1-\varrho_{\lambda}$ either

$\displaystyle\lvert d_{\sigma_{2}}\tilde{\Psi}\rvert$
$\displaystyle=\left\lvert\psi_{\tau}\bigg{(}\bigg{(}\frac{\theta}{\lambda},\frac{u}{\sqrt{\lambda}}\bigg{)},w\bigg{)}\right\rvert\geq
2D\bigg{(}\bigg{(}\frac{\theta}{\lambda},\frac{u}{\sqrt{\lambda}}\bigg{)},w\bigg{)}\geq
C^{\prime}\lambda^{2\delta-1}$ (112)

or

$\displaystyle\lvert d_{\sigma_{1}}\tilde{\Psi}\rvert$
$\displaystyle=\left\lvert\psi_{\tau}\bigg{(}\bigg{(}G^{t}_{\tau}(w),\frac{\varphi}{\lambda},\frac{v}{\sqrt{\lambda}}\bigg{)}\bigg{)}\right\rvert\geq
2D\bigg{(}\bigg{(}G^{t}_{\tau}(w),\frac{\varphi}{\lambda},\frac{v}{\sqrt{\lambda}}\bigg{)}\bigg{)}\geq
C^{\prime}\lambda^{2\delta-1}$ (113)

where $D$ is the Calabi diastasis (66) and the inequalities follow from (68).
Repeated integration by parts in $\sigma_{1}$ or $\sigma_{2}$ as appropriate
completes the proof. ∎

So far, we have shown

$\Pi_{\chi,\lambda}\bigg{(}\frac{\theta}{\lambda},\frac{u}{\sqrt{\lambda}},\frac{\phi}{\lambda},\frac{v}{\sqrt{\lambda}}\bigg{)}\sim\lambda^{2m}\int
e^{i\lambda\tilde{\Psi}}\varrho_{\lambda}\tilde{A}\,d\sigma_{1}d\sigma_{2}d\mu_{\tau}(w)dt,$
(114)

with $\tilde{\Psi},\tilde{A}$ as in (107) and $\varrho_{\lambda}$ a smooth
cutoff to a neighborhood of $V_{\lambda}$ as in (108). The next step is to
Taylor expand the phase function and absorb the remainder into the amplitude.
Since the expansion (97) is stated in terms of ambient coordinates on the
Grauert tube (as opposed to coordinates on the boundary), we need to introduce
some notation. Recall in Heisenberg coordinates, we have
$z=(z_{0},\dotsc,z_{m-1})$ in terms of coordinates in the ambient Grauert tube
with the property that if in addition $z\in\partial M_{\tau}$, then
$z=(\operatorname{Re}z_{0},z_{1},\dotsc,z_{m-1})$ in terms of coordinates on
the Grauert tube boundary. Here, we have a point
$(\frac{\theta}{\lambda},\frac{u}{\sqrt{\lambda}})$ in coordinates on
$\partial M_{\tau}$. In terms of the ambient coordinates on
$M_{\tau_{\mathrm{max}}}$, we denote the same point by
$(\Theta(\lambda),\frac{u}{\sqrt{\lambda}})$, where
$\operatorname{Re}\Theta(\lambda)=\frac{\theta}{\lambda}$ and
$\operatorname{Im}\Theta(\lambda)$ is determined by
$\varphi_{\tau}(\Theta(\lambda),\frac{u}{\sqrt{\lambda}})=0$.

######  Lemma .

We have

$\displaystyle\operatorname{Im}\Theta(\lambda)=\frac{1}{\lambda}\sum_{j=1}^{m-1}\lvert
u_{j}\rvert^{2}+O(\lambda^{-\frac{3}{2}}).$ (115)

In particular,
$\lvert\operatorname{Im}\Theta(\lambda)\rvert=O(\lambda^{-1})=\lvert\operatorname{Re}\Theta(\lambda)\rvert$.

###### Proof.

The implicit function theorem applied to
$\varphi_{\tau}(\Theta(\lambda),\frac{u}{\sqrt{\lambda}})=0$ shows
$\operatorname{Im}\Theta={O}(\frac{1}{\sqrt{\lambda}})$. Substituting $\lvert
z_{0}\rvert=\lvert\Theta(\lambda)\rvert={O}(\frac{1}{\sqrt{\lambda}})$ in the
remainder term of (97) shows $\operatorname{Im}\Theta={O}(\frac{1}{\lambda})$.
Since $\operatorname{Re}\Theta=O(\frac{1}{\lambda})$, we can in fact
substitute $\lvert
z_{0}\rvert=\lvert\Theta(\lambda)\rvert={O}(\frac{1}{\lambda})$ in the
remainder term of (97), proving the Lemma. ∎

We write the spatial variable $w$ of integration as

$\displaystyle w$ $\displaystyle=(w_{0},w^{\prime})$
$\displaystyle\quad\text{with $w^{\prime}=(w_{1},\dotsc,w_{m-1})$ in
coordinates on $M_{\tau}$},$ (116) $\displaystyle w$
$\displaystyle=(\operatorname{Re}w_{0},w^{\prime})$
$\displaystyle\quad\text{in coordinates on $\partial M_{\tau}$},$ (117)
$\displaystyle w(t)$ $\displaystyle=G^{t}_{\tau}(w).$ (118)

We are now ready to Taylor expand $i\psi_{\tau}=\varphi_{\tau}$ that appears
in the phase (107). By subsection 3.1, Taylor expansions in the ambient
coordinates take the form

$\displaystyle\begin{split}i\psi_{\tau}\bigg{(}\bigg{(}\Theta(\lambda),\frac{u}{\sqrt{\lambda}}\bigg{)},w\bigg{)}&=\frac{i}{2}\left(\Theta(\lambda)-\overline{w}_{0}\right)+\frac{1}{\sqrt{\lambda}}\sum_{j=1}^{m-1}u_{j}\overline{w}_{j}\\\
&\quad+R\bigg{(}\Theta(\lambda),\overline{w}_{0},\frac{u}{\sqrt{\lambda}},\overline{w}^{\prime}\bigg{)},\end{split}$
(119)
$\displaystyle\begin{split}i\psi_{\tau}\bigg{(}G^{t}_{\tau}(w),\bigg{(}\Phi(\lambda),\frac{v}{\sqrt{\lambda}}\bigg{)}\bigg{)}&=\frac{i}{2}(w_{0}(t)-\overline{\Phi}(\lambda))+\frac{1}{\sqrt{\lambda}}\sum_{j=1}^{m-1}w_{j}(t)\overline{v}_{j}\\\
&\quad+R\bigg{(}w_{0}(t),\overline{\Phi}(\lambda),w^{\prime}(t),\frac{\overline{v}}{\sqrt{\lambda}}\bigg{)}.\end{split}$
(120)

######  Remark .

The obvious notation $(\Phi(\lambda),\frac{v}{\sqrt{\lambda}})$ denotes the
ambient coordinates of $(\frac{\varphi}{\lambda},\frac{v}{\sqrt{\lambda}})$.
The form of the remainder $R$ is explicitly stated in subsection 3.1.

To apply stationary phase, the variables $t\in\mathbb{R}$ and $w\in\partial
M_{\tau}$ need to be rescaled:

$\displaystyle
t\mapsto\frac{t}{\sqrt{\lambda}}\quad\text{and}\quad\text{$(\operatorname{Re}w_{0},w^{\prime})\mapsto\bigg{(}\frac{\operatorname{Re}w_{0}}{\sqrt{\lambda}},\frac{w^{\prime}}{\sqrt{\lambda}}\bigg{)}$}.$
(121)

We write $w_{0}(\lambda)$ for the image of $w_{0}$ under this rescaling, so
that
$\operatorname{Re}w_{0}(\lambda)=\frac{\operatorname{Re}w_{0}}{\sqrt{\lambda}}$
and $\operatorname{Im}w(\lambda)$ is determined by
$\varphi_{\tau}(w_{0}(\lambda),\frac{w^{\prime}}{\sqrt{\lambda}})=0$.

######  Remark .

Since $w$ is only an integration variable, it is not necessary that rescaling
of $w$ respects Heisenberg-type order. Indeed, the uniform
$\sqrt{\lambda}$-rescaling (121) is needed for stationary phase.

Under the change-of-variables (121), the expansion (119) turns into

$\displaystyle
i\psi_{\tau}\bigg{(}\bigg{(}\Theta(\lambda),\frac{u}{\sqrt{\lambda}}\bigg{)},w(\lambda)\bigg{)}$
$\displaystyle=\frac{i}{2}(\Theta(\lambda)-\overline{w}_{0}(\lambda))+\frac{1}{{\lambda}}\sum_{j=1}^{m-1}u_{j}\overline{w}_{j}$
(122)
$\displaystyle\quad+R\bigg{(}\Theta(\lambda),\overline{w}_{0}(\lambda),\frac{u}{\sqrt{\lambda}},\frac{\overline{w}^{\prime}}{\sqrt{\lambda}}\bigg{)}.$
(123)

Multiplying through by $\lambda$ and using subsection 3.1, subsection 4.1 to
compute the remainder, we find

$\displaystyle
i\lambda\psi_{\tau}\bigg{(}\bigg{(}\Theta(\lambda),\frac{u}{\sqrt{\lambda}}\bigg{)},w(\lambda)\bigg{)}=-\sqrt{\lambda}\frac{i}{2}\operatorname{Re}w_{0}+\tilde{R}\bigg{(}\frac{\theta}{\lambda},\frac{\operatorname{Re}w_{0}}{\sqrt{\lambda}},\frac{u}{\sqrt{\lambda}},\frac{w^{\prime}}{\sqrt{\lambda}}\bigg{)},$
(124)

where

$\displaystyle\tilde{R}=\frac{i}{2}\theta-\frac{\lvert
u\rvert^{2}}{2}-\frac{\lvert w\rvert^{2}}{2}+u\cdot\overline{w}+\lambda
Q\bigg{(}\frac{\theta}{\lambda},\frac{u}{\sqrt{\lambda}},\frac{\operatorname{Re}w_{0}}{\sqrt{\lambda}},\frac{w^{\prime}}{\sqrt{\lambda}}\bigg{)}$
(125)

and $Q$ has the following asymptotic expansion

$\displaystyle
Q\bigg{(}\frac{\theta}{\lambda},\frac{u}{\sqrt{\lambda}},\frac{\operatorname{Re}w_{0}}{\sqrt{\lambda}},\frac{w^{\prime}}{\sqrt{\lambda}}\bigg{)}$
$\displaystyle\sim\sum_{j=1}\sum_{j+k+l=2}P_{j,k,l}\bigg{(}\frac{u}{\sqrt{\lambda}},\frac{\operatorname{Re}w_{0}}{\sqrt{\lambda}},\frac{w^{\prime}}{\sqrt{\lambda}}\bigg{)}$
(126) $\displaystyle\quad+\sum_{j+\frac{k+l+m}{2}\geq
3}P_{j,k,l,m}\bigg{(}\frac{\theta}{\lambda},\frac{u}{\sqrt{\lambda}},\frac{\operatorname{Re}w_{0}}{\sqrt{\lambda}},\frac{w^{\prime}}{\sqrt{\lambda}}\bigg{)}.$
(127)

Here, $P_{j,k,l}$ is a polynomial in three variables that is of degree $j,k,l$
in the first, second, and third slot, respectively. (The polynomial
$P_{j,k,l,m}$ is similarly defined.)

Similarly, using in addition subsection 3.1, under the change-of-variables
(121) the expansion (120) turns into

$\displaystyle
i\lambda\psi_{\tau}\bigg{(}G^{\frac{t}{\sqrt{\lambda}}}_{\tau}(w(\lambda)),\bigg{(}\Phi(\lambda),\frac{v}{\sqrt{\lambda}}\bigg{)}\bigg{)}$
$\displaystyle=\sqrt{\lambda}\frac{i}{2}\left(\operatorname{Re}w_{0}+2\tau
t\right)$ (128)
$\displaystyle\quad+\tilde{S}\bigg{(}\frac{\phi}{\lambda},\frac{\operatorname{Re}w_{0}}{\sqrt{\lambda}},\frac{t}{\sqrt{\lambda}},\frac{v}{\sqrt{\lambda}},\frac{w^{\prime}}{\sqrt{\lambda}}\bigg{)},$

where

$\displaystyle\tilde{S}=-\frac{i}{2}\phi-\frac{\lvert
v\rvert^{2}}{2}-\frac{\lvert w\rvert^{2}}{2}+\overline{v}\cdot w+\lambda
T\bigg{(}\frac{\phi}{\lambda},\frac{v}{\sqrt{\lambda}},\frac{t}{\sqrt{\lambda}},\frac{\operatorname{Re}w_{0}}{\sqrt{\lambda}},\frac{w^{\prime}}{\sqrt{\lambda}}\bigg{)}$
(129)

and $T$ has the following asymptotic expansion

$\displaystyle
T\bigg{(}\frac{\phi}{\lambda},\frac{v}{\sqrt{\lambda}},\frac{t}{\sqrt{\lambda}},\frac{\operatorname{Re}w_{0}}{\sqrt{\lambda}},\frac{w^{\prime}}{\sqrt{\lambda}}\bigg{)}$
$\displaystyle\sim\sum_{i+j>0}\sum_{i+j+k+l=2}P_{i,j,k,l}\bigg{(}\frac{t}{\sqrt{\lambda}},\frac{\operatorname{Re}w_{0}}{\sqrt{\lambda}},\frac{v}{\sqrt{\lambda}},\frac{w^{\prime}}{\sqrt{\lambda}}\bigg{)}$
(130)
$\displaystyle\quad+\sum_{i+\frac{j+k+l+m}{2}}P_{i,j,k,l,m}\bigg{(}\frac{\phi}{\lambda},\frac{v}{\sqrt{\lambda}},\frac{t}{\sqrt{\lambda}},\frac{\operatorname{Re}w_{0}}{\sqrt{\lambda}},\frac{w^{\prime}}{\sqrt{\lambda}}\bigg{)}.$
(131)

It follows from (128) and (124) that under the change-of-variables (121), the
integral (114) becomes an oscillatory integral with parameter
$\sqrt{\lambda}$:

$\Pi_{\chi,\lambda}\bigg{(}\frac{\theta}{\lambda},\frac{u}{\sqrt{\lambda}},\frac{\phi}{\lambda},\frac{v}{\sqrt{\lambda}}\bigg{)}\sim\lambda^{m}\int
e^{i\sqrt{\lambda}\tilde{\raisebox{0.0pt}[0.9pt]{$\tilde{\Psi}$}}}\tilde{\raisebox{0.0pt}[0.9pt]{$\tilde{A}$}}\,d\sigma_{1}d\sigma_{2}d\mu_{\tau}(w)dt,$
(132)

where

$\displaystyle\tilde{\raisebox{0.0pt}[0.9pt]{$\tilde{\Psi}$}}$
$\displaystyle=-t-\frac{\sigma_{2}}{2}\operatorname{Re}w_{0}+\frac{\sigma_{1}}{2}(\operatorname{Re}w_{0}+2\tau
t),$ (133) $\displaystyle\tilde{\raisebox{0.0pt}[0.9pt]{$\tilde{A}$}}$
$\displaystyle=e^{\sigma_{2}\tilde{R}+\sigma_{1}\tilde{S}}\varrho_{\lambda}\tilde{A}\bigg{(}\frac{\theta}{\lambda},\frac{u}{\sqrt{\lambda}},\frac{\phi}{\lambda},\frac{v}{\sqrt{\lambda}},\frac{\operatorname{Re}w_{0}}{\sqrt{\lambda}},\frac{w^{\prime}}{\sqrt{\lambda}},\frac{t}{\sqrt{\lambda}},\sigma_{1},\sigma_{2}\bigg{)}J(w(\lambda)),$

with $J(\eta)$ the volume density in Heisenberg coordinates.

######  Remark .

In (121), the spatial rescaling introduces a factor of $\lambda^{-m+1/2}$,
while the time rescaling introduces a factor of $\lambda^{-1/2}$. Together
with the overall prefactor of $\lambda^{2m}$ in (114), we obtain the
$\lambda^{m}$ prefactor in (132). The explicit descriptions of $\tilde{R}$ and
$\tilde{S}$ imply that $e^{\sigma_{2}\tilde{R}+\sigma_{1}\tilde{S}}$ may be
absorbed into the amplitude.

We make one final localization argument. Let $\\{\eta,1-\eta\\}$ be a
partition of unity subordinate to the cover

$\left\\{(\sigma_{1},\sigma_{2}):0<\sigma_{1},\sigma_{2}<\frac{2}{\tau}\right\\}\quad\text{and}\quad\left\\{(\sigma_{1},\sigma_{2}):\sigma_{1},\sigma_{2}>\frac{3}{2\tau}\right\\}.$
(134)

Decompose (132) into two integrals:

$\displaystyle\Pi_{\chi,\lambda}\bigg{(}\frac{\theta}{\lambda},\frac{u}{\sqrt{\lambda}},\frac{\phi}{\lambda},\frac{v}{\sqrt{\lambda}}\bigg{)}$
$\displaystyle\sim I_{1}^{\prime}+I_{2}^{\prime},$ (135) $\displaystyle
I_{1}^{\prime}$ $\displaystyle=\lambda^{m}\int
e^{i\lambda\tilde{\raisebox{0.0pt}[0.9pt]{$\tilde{\Psi}$}}}\eta(\sigma_{1},\sigma_{2})\tilde{\raisebox{0.0pt}[0.9pt]{$\tilde{A}$}}\,d\sigma_{1}d\sigma_{2}d\mu_{\tau}(w)dt,$
(136) $\displaystyle I_{2}^{\prime}$ $\displaystyle=\lambda^{m}\int
e^{i\lambda\tilde{\raisebox{0.0pt}[0.9pt]{$\tilde{\Psi}$}}}(1-\eta(\sigma_{1},\sigma_{2}))\tilde{\raisebox{0.0pt}[0.9pt]{$\tilde{A}$}}\,d\sigma_{1}d\sigma_{2}d\mu_{\tau}(w)dt,$
(137)

with $\tilde{\raisebox{0.0pt}[0.9pt]{$\tilde{A}$}}$ and
$\tilde{\raisebox{0.0pt}[0.9pt]{$\tilde{\Psi}$}}$ as in (133).

######  Lemma .

We have $I_{2}^{\prime}=O(\lambda^{-\infty})$.

###### Proof.

Notice that

$\displaystyle\left\lvert\nabla_{\operatorname{Re}w_{0},t}\tilde{\raisebox{0.0pt}[0.9pt]{$\tilde{\Psi}$}}\right\rvert^{2}\geq\bigg{(}\frac{\sigma_{1}}{2}-\frac{\sigma_{2}}{2}\bigg{)}^{2}+(\tau\sigma_{1}-1)^{2}\geq\frac{1}{4}$
(138)

on the support of $1-\eta$. Thus, the Lemma follows from repeated integration
by parts in $(\operatorname{Re}w_{0},t)$. ∎

We have finally reduced the spectral localization kernel to the oscillatory
integral

$\Pi_{\chi,\lambda}\bigg{(}\frac{\theta}{\lambda},\frac{u}{\sqrt{\lambda}},\frac{\phi}{\lambda},\frac{v}{\sqrt{\lambda}}\bigg{)}\sim\lambda^{m}\int_{\mathbb{R}\times\mathbb{R}^{+}\times\mathbb{R}^{+}\times\mathbb{R}\times\mathbb{C}^{m-1}}e^{i\sqrt{\lambda}\tilde{\raisebox{0.0pt}[0.9pt]{$\tilde{\Psi}$}}}B\,dw^{\prime}d(\operatorname{Re}w_{0})d\sigma_{1}d\sigma_{2}dt,$
(139)

with phase and amplitude

$\displaystyle\tilde{\raisebox{0.0pt}[0.9pt]{$\tilde{\Psi}$}}$
$\displaystyle=-t-\frac{\sigma_{2}}{2}\operatorname{Re}w_{0}+\frac{\sigma_{1}}{2}(\operatorname{Re}w_{0}+2\tau
t),$ (140) $\displaystyle\tilde{\raisebox{0.0pt}[0.9pt]{$\tilde{A}$}}$
$\displaystyle=e^{\sigma_{2}\tilde{R}+\sigma_{1}\tilde{S}}\eta\varrho_{\lambda}\hat{\chi}\tilde{A}J.$
(141)

######  Remark .

The cutoff functions $\varrho_{\lambda},\eta,\hat{\chi}$ localize $B$ to a
compact region. The $\lambda^{m-1}$ factor in $B$ comes from comparing the
amplitude $\tilde{\raisebox{0.0pt}[0.9pt]{$\tilde{A}$}}$ in the integral (132)
(which already has an overall factor of $\lambda^{m}$ from the change-of-
variables $\sigma_{j}\mapsto\lambda\sigma_{j}$ and (121)) with the amplitude
$A$ in (104).

We are now in the position to integrate with respect to
$\operatorname{Re}w_{0},\sigma_{1},\sigma_{2},t$ using the method of
stationary phase, thereby reducing (139) to a Gaussian integral over
$\mathbb{C}^{m-1}$. We note the following derivatives:

$\displaystyle\partial_{\sigma_{2}}\tilde{\raisebox{0.0pt}[0.9pt]{$\tilde{\Psi}$}}$
$\displaystyle=-\frac{1}{2}\operatorname{Re}w_{0},$
$\displaystyle\partial_{\sigma_{1}}\tilde{\raisebox{0.0pt}[0.9pt]{$\tilde{\Psi}$}}$
$\displaystyle=\frac{1}{2}(\operatorname{Re}w_{0}+2\tau t),$ (142)
$\displaystyle\partial_{t}\tilde{\raisebox{0.0pt}[0.9pt]{$\tilde{\Psi}$}}$
$\displaystyle=-1+\tau{\sigma_{1}},$
$\displaystyle\partial_{\operatorname{Re}w_{0}}\tilde{\raisebox{0.0pt}[0.9pt]{$\tilde{\Psi}$}}$
$\displaystyle=-\frac{\sigma_{2}}{2}+\frac{\sigma_{1}}{2}.$ (143)

The critical set of the phase is the point
$C=\\{\operatorname{Re}w_{0}=0,t=0,\sigma_{1}=\sigma_{2}=\frac{1}{\tau}\\}$.
The Hessian matrix and its inverse at the critical point are

$\displaystyle\tilde{\raisebox{0.0pt}[0.9pt]{$\tilde{\Psi}$}}^{\prime\prime}_{C}$
$\displaystyle=\left(\begin{array}[]{c|cccc}&t&\sigma_{1}&\sigma_{2}&\operatorname{Re}w_{0}\\\
\hline\cr t&0&\tau&0&0\\\ \sigma_{1}&\tau&0&0&\frac{1}{2}\\\
\sigma_{2}&0&0&0&-\frac{1}{2}\\\
\operatorname{Re}w_{0}&0&\frac{1}{2}&-\frac{1}{2}&0\end{array}\right),\quad(\tilde{\raisebox{0.0pt}[0.9pt]{$\tilde{\Psi}$}}^{\prime\prime}_{C})^{-1}=\begin{pmatrix}0&\frac{1}{\tau}&\frac{1}{\tau}&0\\\
\frac{1}{\tau}&0&0&0\\\ \frac{1}{\tau}&0&0&-2\\\ 0&0&-2&0\end{pmatrix}.$ (149)

Set

$\displaystyle
L_{\tilde{\raisebox{0.0pt}[0.9pt]{$\tilde{\Psi}$}}}=\left\langle(\tilde{\raisebox{0.0pt}[0.9pt]{$\tilde{\Psi}$}}^{\prime\prime}_{C})^{-1}D,D\right\rangle=\frac{2}{\tau}\partial_{\sigma_{1}}\partial_{t}+\frac{2}{\tau}\partial_{\sigma_{2}}\partial_{t}-4\partial_{\sigma_{2}}\partial_{\operatorname{Re}w_{0}}$
(150)

By the method of stationary phase ([20, Theorem 7.75]), we have

$\int_{\mathbb{R}\times\mathbb{R}^{+}\times\mathbb{R}^{+}\times\mathbb{R}}e^{i\sqrt{\lambda}\tilde{\raisebox{0.0pt}[0.9pt]{$\tilde{\Psi}$}}}\tilde{\raisebox{0.0pt}[0.9pt]{$\tilde{A}$}}\,dw^{\prime}d\sigma_{1}d\sigma_{2}dt\\\
=\gamma\lambda^{m}\sum_{j=0}^{N-1}\lambda^{-\frac{j}{2}}\sum_{\nu-\mu=j}\sum_{2\nu\geq
3\mu}\frac{1}{i^{j}2^{\nu}}L_{\tilde{\raisebox{0.0pt}[0.9pt]{$\tilde{\Psi}$}}}^{\nu}\left[e^{\mu(\sigma_{2}\tilde{R}+\sigma_{1}\tilde{S})}\eta\varrho_{\lambda}\hat{\chi}\tilde{A}J\right]_{C}+\hat{R}_{N}.$
(151)

The leading coefficient $\gamma$ is given by

$\displaystyle\gamma=e^{i\sqrt{\lambda}\tilde{\raisebox{0.0pt}[0.9pt]{$\tilde{\Psi}$}}|_{C}}\left(\det\left(\sqrt{\lambda}\tilde{\raisebox{0.0pt}[0.9pt]{$\tilde{\Psi}$}}^{\prime\prime}_{C}/2\pi
i\right)\right)^{-\frac{1}{2}}=\frac{8\pi^{2}}{\lambda\tau}.$ (152)

The stationary phase remainder estimate implies

$\displaystyle\int_{\mathbb{C}^{m-1}}$
$\displaystyle\lvert\hat{R}_{N}\rvert\,dw^{\prime}\leq\int_{\mathbb{C}^{m-1}}\lambda^{-\frac{N}{2}}\sum_{\lvert\alpha\rvert\leq
2N}\sup\lvert
D^{\alpha}(\eta\rho_{\lambda}\hat{\chi}\tilde{A}J)\rvert\,dw^{\prime}\leq
C_{N}\lambda^{-\frac{N}{2}}$ (153)

because $\tilde{A}$ is a symbol of order zero by subsection 4.1. (Here, the
supremum and the derivative $D^{\alpha}$ are taken over
$t,\sigma_{1},\sigma_{2},\operatorname{Re}w_{0}$ and the integral is with
respect to the remaining variable $w^{\prime}$.)

It remains to integrate the asymptotic expansion (151) in $w^{\prime}$. The
left-hand side is (139); the right-hand side can be integrated term-by-term
thanks to the estimate above. After substituting expressions (125) and (129)
and performing a Taylor expansion near the
$p=(0,0)\in\mathbb{R}\times\mathbb{C}^{m-1}$, we find an asymptotic expansion
for the kernel in increasing half-powers of $\lambda$:

$\displaystyle\Pi_{\chi,\lambda}\bigg{(}\frac{\theta}{\lambda},\frac{u}{\sqrt{\lambda}},\frac{\phi}{\lambda},\frac{v}{\sqrt{\lambda}}\bigg{)}$
$\displaystyle\sim\frac{8\pi^{2}\lambda^{m-1}}{\tau}\int_{\mathbb{C}^{m-1}}\bigg{[}e^{\frac{1}{\tau}\big{(}\frac{i}{2}(\theta-\phi)-\frac{|u|^{2}}{2}-\frac{|v|^{2}}{2}-|w^{\prime}|^{2}+u\cdot\overline{w}^{\prime}+w^{\prime}\cdot\overline{v}\big{)}}$
(154) $\displaystyle\quad\times\sum_{j\geq
0}C_{j}\lambda^{-\frac{j}{2}}P_{j}(\theta,u,\phi,v,w^{\prime})\bigg{]}\,dw^{\prime}$
(155)
$\displaystyle=\frac{8\pi^{2}\lambda^{m-1}}{\tau}e^{\frac{i(\theta-\phi)}{2\tau}-\frac{|u|^{2}}{2\tau}-\frac{|v|^{2}}{2\tau}+\frac{v\cdot\overline{u}}{\tau}}\int_{\mathbb{C}^{m-1}}\bigg{[}e^{\frac{1}{\tau}\big{(}-|z|^{2}+z\cdot(\overline{v}-\overline{u})\big{)}}$
(156) $\displaystyle\quad\times\sum_{j\geq
0}C_{j}{\lambda}^{-\frac{j}{2}}{P_{j}}(\theta,u,\phi,v,w^{\prime})\bigg{]}\,dz.$
(157)

The equality follows from the change-of-variables $w=z+u$. The principal term
in the series is given by

$P_{0}=\tau^{-2(m-1)}s_{0}(p,p)^{2}\sigma_{0,\tau}(p)J(p).$ (158)

Since

$\displaystyle\int_{\mathbb{C}^{m-1}}e^{\frac{1}{\tau}\big{(}-|z|^{2}+z\cdot(\overline{v}-\overline{u})\big{)}}\,dz$
$\displaystyle=(\tau\pi)^{m-1},$ (159)

we conclude

$\displaystyle\Pi_{\chi,\lambda}\bigg{(}\frac{\theta}{\lambda},\frac{u}{\sqrt{\lambda}},\frac{\phi}{\lambda},\frac{v}{\sqrt{\lambda}}\bigg{)}$
$\displaystyle\sim
C_{m}\frac{\lambda^{m-1}}{\tau^{m-2}}e^{\frac{i(\theta-\phi)}{2\tau}-\frac{|u|^{2}}{2\tau}-\frac{|v|^{2}}{2\tau}+\frac{v\cdot\overline{u}}{\tau}}$
(160)
$\displaystyle\quad\times\left(1+\sum_{j=1}^{N}\lambda^{-\frac{j}{2}}P_{j}(u,v,\theta,\phi)\right)$
(161)
$\displaystyle\quad+\lambda^{{}^{m-1-\frac{N+1}{2}}}R(\theta,u,\phi,v,\lambda).$
(162)

###  4.2. Proof of Theorem 1.2

We look at the near-diagonal scaling asymptotics for the tempered spectral
projection kernel (77) under Heisenberg-type scaling. Similar to the proof of
Theorem 1.1, we write out the kernel using subsection 2.4 and (65):

$\displaystyle
P_{\chi,\lambda}\bigg{(}\frac{\theta}{\lambda},\frac{u}{\sqrt{\lambda}},\frac{\varphi}{\lambda},\frac{v}{\sqrt{\lambda}}\bigg{)}\sim\int_{\mathbb{R}\times\partial
M_{\tau}\times\mathbb{R}^{+}\times\mathbb{R}^{+}}e^{i\lambda\Psi}B\,d\sigma_{1}d\sigma_{2}d\mu_{\tau}(w)dt,$
(163)

in which the phase $\Psi$ and the amplitude $B$ are given by

$\displaystyle\Psi$
$\displaystyle=-t+\frac{1}{\lambda}\sigma_{2}\psi_{\tau}\bigg{(}\bigg{(}\frac{\theta}{\lambda},\frac{u}{\sqrt{\lambda}}\bigg{)},w\bigg{)}+\frac{1}{\lambda}\sigma_{1}\psi_{\tau}\bigg{(}G^{t}_{\tau}(w),\bigg{(}\frac{\phi}{\lambda},\frac{v}{\sqrt{\lambda}}\bigg{)}\bigg{)},$
(164) $\displaystyle B$
$\displaystyle=\hat{\chi}(t)s\bigg{(}\frac{\theta}{\lambda},\frac{u}{\sqrt{\lambda}},w,\sigma_{2}\bigg{)}s\bigg{(}G^{t}_{\tau}(w),\frac{\phi}{\lambda},\frac{v}{\sqrt{\lambda}},\sigma_{1}\bigg{)}\sigma_{t,\tau}(w,\sigma_{1}).$

As we point out in subsection 2.5, despite identical notation, the
unitarization symbol $\sigma_{t,\tau}$ in the expression for $B$ is of order
$-(m-1)/2$, whereas $\sigma_{t,\tau}$ in the expression for $A$ in (105) is of
order zero. Hence, the oscillatory integral expression (163) is exactly
$\lambda^{-(m-1)/2}$ times the expression (104). The rest of the computations
proceed in the same manner.

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