# Scaling asymptotics of spectral Wigner functions

Boris Hanin, Steve Zelditch

###### Abstract.

We prove that smooth Wigner-Weyl spectral sums at an energy level $E$ exhibit
Airy scaling asymptotics across the classical energy surface $\Sigma_{E}$.
This was proved earlier by the authors for the isotropic harmonic oscillator
and the proof is extended in this article to all quantum Hamiltonians
$-\hbar^{2}\Delta+V$ where $V$ is a confining potential with at most quadratic
growth at infinity. The main tools are the Herman-Kluk initial value
parametrix for the propagator and the Chester-Friedman-Ursell normal form for
complex phases with a one-dimensional cubic degeneracy. This gives a rigorous
account of Airy scaling asymptotics of spectral Wigner distributions of M.V.
Berry, A. Ozorio de Almeida and other physicists.

S.Z. was partially supported by NSF grant DMS-1810747.

B.H. was funded by NSF CAREER grant DMS-2143754 as well as NSF grants
DMS-1855684, DMS-2133806 and an ONR MURI on Foundations of Deep Learning.

This article is thus concerned with spectral Wigner functions for a
Schrödinger operator

$\widehat{H}_{\hbar}=-\frac{\hbar^{2}}{2}\Delta_{{\mathbb{R}}^{d}}+V(x),$ (1)

where $V\in C^{\infty}({\mathbb{R}}^{d})$ is a real-valued potential that
satisfies $V(x)\to\infty$ as $||x||\to\infty$, and is at most of quadratic
growth at infinity,

$\forall\gamma\in\mathbb{N}^{d}\text{ s.t. }\left\lvert\gamma\right\rvert\geq
2~{}~{}~{}\exists
C_{\gamma}>0,\,\,\sup_{x}\left\lvert\partial_{x}^{\gamma}V(x)\right\rvert<C_{\gamma}.$
(2)

Let $\\{\psi_{j}(\hbar)\\}$ be a complete orthonormal basis for
$L^{2}({\mathbb{R}}^{d},dx)$ consisting of eigenfunctions of
$\widehat{H}_{\hbar}$:

$\widehat{H}_{\hbar}\psi_{j}(\hbar,x)=E_{j}(\hbar)\psi_{j}(\hbar,x),$

and let

$W_{\psi_{j},\psi_{j}}(x,\xi)=(2\pi\hbar)^{-d}\int_{{\mathbb{R}}^{d}}\psi_{j}(\hbar,x+\frac{v}{2})\psi_{j}(\hbar,x-\frac{v}{2})e^{-\frac{i}{\hbar}v\xi}\frac{dv}{(2\pi
h)^{d}}.$ (3)

be the Wigner functions of the individual eigenstates. The Weyl-Wigner
spectral functions for an energy interval $I_{\hbar}(E)=[E-a,E+b]$ centered at
$E$ are defined by,

$W_{\hbar,I_{\hbar}}(x,\xi):=\sum_{j:E_{j}(\hbar)\in
I_{\hbar}(E)}W_{\psi_{j},\psi_{j}}(x,\xi).$ (4)

In the recent articles [HZ20, HZb], the authors obtained scaling asymptotics
of Airy type for (4) around the energy surface

$\Sigma_{E}=\\{(x,\xi):H(x,\xi)=E\\},\;\;H(x,\xi)=||\xi||^{2}+V(x)$ (5)

in the special case of the isotropic Harmonic oscillator where
$V(x)=||x||^{2}$. The asymptotics involve two types of localization: (i)
spectral localization to the interval $I_{\hbar}(E)$ and (ii) phase space
localization, where $(x,\xi)$ is localized either near $\Sigma_{E}$ or at some
prescribed distance from it, either in the energy ball,

$B_{E}=\\{(x,\xi):H(x,\xi)\leq E\\}$

or outside the energy surface. The purpose of this article is to present one
generalization of the Airy scaling results to all Schrödinger operators (1)
for which $H$ is strictly convex, hence

$B_{E}\;\rm{is\;strictly\;convex}\;$ (6)

and to relate the scaling results to statements in M.V. Berry’s article
[Ber89] as well as to the related article [O98] of Ozorio de Almeida (see also
[TL]).

The result we generalize here is the following [HZ20, Theorem 1.4]. Let
$E_{N}(\hbar)=\hbar(N+\frac{d}{2})$ be the $N$th distinct eigenvalue of the
isotropic harmonic oscillator (see Section 5 for notation and background) and
define $\hbar=\hbar_{N}(E):=\frac{E}{N+\frac{d}{2}},$ so that
$E_{N}(\hbar)=E$. Let $\Pi_{\hbar,E_{N}(\hbar)}$ be the orthogonal projection
onto the corresponding eigenspace $V_{N}(\hbar)$ and let
$W_{\hbar,E_{N}(\hbar)}(x,\xi)$ be its Wigner distribution (in the notation of
(4) this means we choose $a=b=0$). Then, with $s=\frac{H(x,\xi)}{E}$, and
where for simplicity we assume that $s=\frac{H(x,\xi)}{E}\in(0,1]$, it follows
from [HZ20, (50)] that,

$\displaystyle W_{\hbar,E_{N}(\hbar)}(x,\xi)$
$\displaystyle=\frac{2}{(2\pi\hbar)^{d}}\left[\hbar_{E}^{1/3}\operatorname{Ai}\left(\hbar_{E}^{-2/3}B^{2}\left(s\right)\right)\left(1+O\left((s-1)^{2/3}\right)\right)+\varepsilon_{1}(h_{E},s)\right]$
(7)

where we’ve sest

$B(s)=i(3\beta(s)/2)^{1/3},\qquad\beta(s):={\textstyle\frac{1}{2}}[\cos^{-1}\sqrt{s}-\sqrt{s-s^{2}}].$
(8)

Note that in [HZ20, (50)] we use the large parameter $\nu=\frac{2}{\hbar_{E}}$
and that the argument of the Airy function there is
$\nu^{\frac{2}{3}}B^{2}(s)$; this introduces some additional universal
constants in the asymptotics. Note also that since we’ve restricted for
simplicity to $s\leq 1$, the formula (8) for $B(s)$ is pure imaginary, so that
its square is negative and the Airy function is evaluated in its oscillatory
region. In Section 5 we interpret the formulae geometrically and simplify (7)
when $(x,\xi)$ is very close to $\Sigma_{E}$.

The present article generalizes the scaling results (7) for the more general
Hamiltonians (1). There are both sharp and smooth versions of the scaling
results, depending on whether we sum over an energy interval as in (4) or
whether we use a smooth cutoff function $f$ to define general smoothed Weyl-
Wigner sums,

$\begin{array}[]{lll}W_{\hbar,f,\delta,E}(x,\xi):=\sum_{j=1}^{\infty}f\left(\hbar^{-\delta}(E-E_{j}(\hbar))\right)W_{\psi_{j},\psi_{j}}(x,\xi)\end{array}$
(9)

where $f\in\mathcal{S}({\mathbb{R}})$ is a Schwartz function with $\hat{f}\in
C_{0}^{\infty}({\mathbb{R}})$. In this article we only consider $\delta=1$ and
we concentrate on the behavior of (9) in a thin $\hbar^{1/3}$ neighborhood of
$\Sigma_{E}$. In Section 0.4 we review other scaling results in [HZ20, HZb]
where $\delta=0,\frac{2}{3}$, i.e. where we sum over larger spectral
intervals.

To state our first result about $W_{\hbar,f,1,E}(x,\xi)$, we need some
notation. Let us denote the Hamilton vector field of
$H:T^{*}{\mathbb{R}}^{d}\to{\mathbb{R}}$ by $\Xi_{H}$, and the Hamiltonian
flow of $\Xi_{H}$ by $\Phi^{t}$. The classical trajectory with initial data
$(q,p)$ is denoted by

$\Phi^{t}(q,p)=(q_{t},p_{t}),$ (10)

and we will write

$M_{t}(q,p):=D_{q,p}\Phi^{t}(q,p)$ (11)

for its Jacobian. It was pointed out in [Ber89, O98] that the asymptotics of
$W_{\hbar,f,1,E}(x,\xi)$ for $(x,\xi)\in B_{E}$ sufficiently close to
$\Sigma_{E}$ is governed in a sense that we will describe by the midpoint map,
$\Theta^{*}:{\mathbb{R}}\times\Sigma_{E}\rightarrow B_{E}$ given by

$\Theta^{*}(t,q,p)=\Theta^{t}(q,p):={\textstyle\frac{1}{2}}(I+\Phi^{t})(q,p)=\left(\frac{q+q_{t}}{2},\frac{p+p_{t}}{2}\right).$
(12)

The assumption (6) ensures that (12) takes its values in $B_{E}$. The
asymptotics of $W_{\hbar,f,1,E}(x,\xi)$ depend on the solutions $(t,q,p)$ of
the equations

$\Theta^{t}(q,p)=(x,\xi),\;\;(q,p)\in\Sigma_{E},\;\;(x,\xi)\in B_{E},$ (13)

with $(E,x,\xi)$ are fixed, which we will see in Proposition 3.1 below arise
as critical point equations for a certain phase function related to
$W_{\hbar,f,1,E}$. An important point is that there is an obvious symmetry in
the equation (13), namely

$\Theta^{t}(q,p)=\Theta^{-t}(\Phi^{t}(q,p)),$ (14)

which fixes the Hamiltonian arc from $(q,p)$ to $(q_{t},p_{t})$ but reverses
the endpoints. It is proved in Section 3.3 (see Lemma 3.4) that under
assumption 6, there exists $\varepsilon>0$ so that for $(x,\xi)$ in an
$\varepsilon$-tube around $\Sigma_{E}$ there exist unique $(t,q,p)$ up to the
symmetry (14) satisfying the midpoint equation,

$\Theta(t,q,p)=(x,\xi).$ (15)

Put another way, let us denote by $s\in[0,1]\to\gamma_{t,x,\xi}(s)$ the unique
Hamiltonian arc (up to the involution (14)) with $|t|$ sufficiently small on
$\Sigma_{E}$ such that

$(x,\xi)={\textstyle\frac{1}{2}}(\gamma_{t,x,\xi}(0)+\gamma_{t,x,\xi}(t)).$
(16)

In Lemma 4.4 it is proved that, there exists $\varepsilon_{0}>0$ so that if
$(x,\xi)$ is $\varepsilon_{0}$-close to $\Sigma_{E}$, then there exist exactly
two critical times $t=t_{\pm}(E,x,\xi)\in(-\varepsilon_{0},\varepsilon_{0})$
(see Proposition 2.1) of the form $t_{\pm}=\pm t$ such that
$\Theta^{t}(q,p)=(x,\xi),H(q,p)=E.$ For ease of future reference, we make the
following

######  Definition 0.1.

We denote by $t=t_{\pm}(E,x,\xi)$ with $t_{-}(E,x,\xi)=-t_{+}(E,x,\xi)$ the
critical times where $\Theta^{t}(q,p)=(x,\xi),H(q,p)=E.$

A direct computation provided in (56) shows that $\frac{dt}{dE}(E,x,\xi)$ is
well-defined for each $E>0$ and $(x,\xi)$ in a sufficiently small tubular
neighborhood of $\Sigma_{E}$. With this notation, we are now ready to present
our first result.

######  Theorem 0.2.

Let $\widehat{H}_{\hbar}$ be a Hamiltonian as in (1) satisfying the
assumptions (2) and (6). Further, fix $f\in\mathcal{S}({\mathbb{R}})$ such
that

$\hat{f}\in
C_{0}^{\infty},\qquad\mathrm{Supp}\hat{f}\subset(-a,a),\qquad\hat{f}(\tau)=1\,\,\forall\tau\in(-a/2,a/2)$

where $a<T_{\min}$ is less than the periodic of the minimal periodic orbit on
$\Sigma_{E}$. Then for $(x,\xi)\in B_{E}$ such that
$H(x,\xi)=E+O(\hbar^{2/3}),$ there exist $b,\sigma,u_{0,\nu},u_{1,\nu}\in
C^{\infty}(B_{E}),$ such that the smoothed spectral Weyl-Wigner functions
$W_{\hbar,f,1,E}$ defined in (9) with $\delta=1$ admit the pointwise semi-
classical asymptotic expansion

$\begin{array}[]{lll}W_{\hbar,f,1,E}(x,\xi)&\simeq&\hbar^{-d}\left(\hbar^{\frac{1}{3}}\mathrm{Ai}(-\hbar^{-\frac{2}{3}}\rho(x,\xi))\sum_{\nu=0}^{\infty}u_{0,\nu}(x,\xi)\hbar^{\nu}\right)+\\\
&&\\\
&+&\hbar^{-d}\left(\hbar^{\frac{2}{3}}\mathrm{Ai}^{\prime}(-\hbar^{-\frac{2}{3}}\rho(x,\xi))\sum_{\nu=0}^{\infty}u_{1,\nu}(x,\xi)\hbar^{\nu}\right),\end{array}$
(17)

where

$\begin{array}[]{l}\frac{4}{3}\rho^{3/2}={\textstyle\frac{1}{2}}\int_{t_{-}(E,x,\xi)}^{t_{+}(E,x,\xi)}(p-\xi)\cdot
dq=\int_{\beta_{t,x,\xi}}\omega,\end{array}$ (18)

where $\beta$ is defined in the next paragraph and where the leading
coefficient $u_{0,0}$ is given by,

$u_{0,0}=\sqrt{\pi}\rho(x,\xi)^{1/4}\left|\frac{dt_{+}}{dE}(E,x,\xi)\det(1+M_{t_{+}(E,x,\xi)}(x,\xi))\right|^{-{\textstyle\frac{1}{2}}},$
(19)

The integral in (18) is the integral over the oriented closed curve obtained
by connecting the endpoints of $\gamma_{t,x,\xi}$ by the chord

$\alpha_{t,x,\xi}(s)=(1-s)\gamma_{t,x,\xi}(0)+s\gamma_{t,x,\xi},\qquad
s\in[0,1].$

Since the closed curve bounds the two-dimensional surface $\beta_{t,x,\xi}$
consisting of line segments joining $(x,\xi)$ to points of the Hamilton orbit,
the integral (18) equals the oriented area $\int_{\beta}\omega$ where
$\omega=dp\wedge dq$ is the standard symplectic form of
$T^{*}{\mathbb{R}}^{d}$. In Section 5 we show that $\rho$ in (17) equals
$-B^{2}(s)$ up to universal constants in (7) in the case of the isotropic
harmonic oscillator.

The sign conventions in the argument of the Airy function is discussed below
Proposition 4.1. It is clearly consistent with (7), i.e. for $H(x,\xi)\leq E$
the argument of the Airy function is negative and therefore the asymptotics
are oscillatory.

Next, we remark that the result of Theorem 0.2 is stated and a proof is
sketched in [O98, TL]. Our motivation for presenting a rigorous proof is to
generalize our earlier result (7) and to relate these special asymptotics to
other types of asymptotics expansions for Wigner functions. The argument of
the Airy function and the leading coefficient and argument of the Airy
function agree with the calculations in [O98, TL]. To connect the notations,
in those articles, the leading coefficient is given by
$A_{0}(x,\xi,E)S_{0}^{1/6}$ where $A_{0}$ in [TL, (17)] is the coefficient
$|\frac{dt}{dE}\det(1+M)|^{-{\textstyle\frac{1}{2}}}$. Also, $S_{0}^{2/3}$ is
the argument of Airy in [TL]. In our notation, which follows that of [HoI],
the Airy argument is $\rho$ and
$\rho^{1/4}=(\rho^{3/2})^{\frac{1}{6}}=(S_{0})^{1/6}.$

Further, our proof of Theorem 0.2 relies on a stationary phase with fold
singularities (see [HoI, Page 236]) from which we see that when
$\rho(x,\xi)\approx 0$ Theorem 0.2 gives an asymptotic expansion in powers of
$\hbar^{1/3}$ that remains valid in any region of $T^{*}{\mathbb{R}}^{d}$ in
which $\hbar^{-\frac{2}{3}}\rho(x,\xi)$ stays bounded. However, due to the
exponential decay of the Airy functions on the positive axis, $\rho(x,\xi)$
becomes very small when $\hbar^{-\frac{2}{3}}\rho(x,\xi)$ is positive, and
stationary phase asymptotics given below in Theorem 0.5 then become valid.

The Airy scaling asymptotics of $W_{\hbar,f,1,E}(x,\xi)$ for $(x,\xi)$ in the
boundary layer around $\Sigma_{E}$ (i.e. $\hbar^{1/3}$ close to $\Sigma_{E}$)
are due to a fold singularity in the map (12). More precisely, in Section 3.3,
it is explained that the Airy asymptotics are due to the fact that the
relevant Lagrangian submanifold $\Lambda_{E}\subset
T^{*}(T^{*}{\mathbb{R}}^{d})$ in the second cotangent bundle (defined by (57))
has a fold singularity around $\Sigma_{E}$, i.e. $\Sigma_{E}$ is a caustic for
the Wigner function (see Section 3.3). The proof of Theorem 0.2 is based on
the use of the Herman-Kluk propagator and on the Airy asymptotics results of
Chester-Friedman-Ursell [CFU] as given in [HoI, Section 7] and [GSt]. Theorem
0.2 is a rigorous version of the result stated in [O98, (7.21)]. It does not
seem to appear in [Ber89].

Finally, we conjecture that when the smooth test function $f$ is replaced by
the indicator function of the spectral interval, then the sharp Wigner-Weyl
sum has the asymptotics,

$\sum_{j:|E_{j}(\hbar)-E|\leq
C\hbar}W_{\psi_{j},\psi_{j}}(x,\xi)=\hbar^{-d}\left(\hbar^{\frac{1}{3}}\rm{Ai}(-\hbar^{-\frac{2}{3}}\rho(x,\xi))\sum_{\nu=0}^{\infty}u_{0,\nu}(x,\xi)\hbar^{\nu}\right)+O(\hbar^{-d+\frac{2}{3}}).$

Classically such asymptotics for spectral intervals are obtained from the
smoothed results (17) by applying cosine Tauberian theorems. But Tauberian
theorems make the hypothesis that the terms of the sums are non-negative,
whereas it is a well-known and important phenomenon that Wigner functions are
almost never globally positive. This raises the question whether
$W_{\psi_{j},\psi_{j}}(x,\xi)\geq 0$ for $(x,\xi)$ in an $\hbar^{2/3}$ tube
around $\Sigma_{E}$ when the eigenvalue of $\psi_{j}$ satisfies
$|E_{j}(\hbar)-E|\leq C\hbar$. This is plausible, since asymptotically the
Airy function is only evaluated where it is positive; moreover, in a weak
sense, the Wigner functions tend to a delta function on $\Sigma_{E}$. But this
is far from sufficient to prove positivity. A possible source of counter-
examples where $W_{\psi_{j},\psi_{j}}(x,\xi)$ can be negative at some points
of $\Sigma_{E}$ is the Wigner distribution of product eigenstates of the
isotropic harmonic oscillator,

$\varphi_{\alpha,h}(x)=h^{-d/4}p_{\alpha}\left(x\cdot
h^{-1/2}\right)e^{-x^{2}/2h},$ (20)

where $\alpha=\left(\alpha_{1},\ldots,\alpha_{d}\right)\geq(0,\ldots,0)$ is a
$d-$dimensional multi-index and $p_{\alpha}(x)$ is the product
$\prod_{j=1}^{d}p_{\alpha_{j}}(x_{j})$ of the hermite polynomials $p_{k}$ (of
degree $k$) in one variable. There are similar products for generic
oscillators, where the multiplicity of each eigenvalue equals one. The Wigner
distribution is the product of those of the factors and are given by products
in the variables $(x_{1},\dots,x_{d})$ of one-dimensional Laguerre polynomials
of the variables $x_{j}^{2}$. The product is positive if and only if the
number of negative factors is even. It seems non-obvious whether one can
construct products with an odd number of negative factors when
$|E_{j}(\hbar)-E|\leq C\hbar$ and $\sum_{j=1}^{d}x_{j}^{2}=E$. Another source
of counter-examples could come from Wigner distributions of coherent states
along elliptic periodic orbits. In the case of the isotropic harmonic
oscillator, the Wigner distribution is positive near the periodic orbit on
$\Sigma_{E}$ but appears to be negative at some points on $\Sigma_{E}$ away
from the orbit [Lo]. As far as we know, the study of negativity sets of Wigner
distributions of eigenfunctions under the above constraints has never been
studied.

There are many further scaling asymptotics results for the smoothed and sharp
Weyl-Wigner sums (4)-(9). In Section 0.4 we review some model results for the
isotropic harmonic oscillator for various types of spectral intervals that
should admit generalizations to (1). We note that the asymptotics results for
the longer spectral intervals involve integrals of the Airy function rather
values of the Airy function as for intervals of order $O(\hbar)$.

###  0.1. Pointwise Semi-Classical Asymptotics of Wigner Transform of the
Propagator

As a interesting and useful warm-up to the proof of Theorem 0.2, we first give
a new proof of what is essentially a well-known result from the physics
literature (e.g. [Ber89]) giving pointwise semi-classical asymptotics for the
Wigner transformation of the Schrödinger operator $\widehat{H}_{\hbar}$.
Before stating the precise result in Proposition 0.3, we recall some notation.
First, the semi-classical Wigner transform is defined to be the unitary
operator

$\mathcal{W}_{\hbar}:L^{2}({\mathbb{R}}^{d}\times{\mathbb{R}}^{d})\to
L^{2}(T^{*}{\mathbb{R}}^{d}),$

taking Hilbert-Schmidt kernels $K_{\hbar}\in
L^{2}({\mathbb{R}}^{d}\times{\mathbb{R}}^{d})$ to their Wigner distributions.
The semi-classical Wigner transform extends to temperate (i.e. Schwartz)
distributions
$K_{\hbar}\in\mathcal{S}^{\prime}({\mathbb{R}}^{d}\times{\mathbb{R}}^{d})$ in
the dual space of Schwartz space by,

$\mathcal{W}_{\hbar}(K_{\hbar})(x,\xi):=\int_{{\mathbb{R}}^{d}}K_{\hbar}\left(x+\frac{v}{2},x-\frac{v}{2}\right)e^{-\frac{i}{\hbar}v\xi}\frac{dv}{(2\pi
h)^{d}}.$ (21)

In particular, each Wigner function $W_{\psi_{j},\psi_{j}}$ appearing in
Theorem 0.2 can be written as the Wigner transform

$W_{\psi_{j},\psi_{j}}(x,\xi)=\mathcal{W}_{\hbar}(\Pi_{j,\hbar})(x,\xi),$

of the kernel

$\Pi_{j,\hbar}(x,y):=\psi_{j}(\hbar,x)\psi_{j}(\hbar,y)$

of the rank one projection onto the state $\psi_{j}(\hbar,\cdot)$. One
advantage of viewing Wigner functions in this way is that, while Wigner
functions $W_{\psi_{j},\psi_{j}}$ are quadratic in $\psi_{j}$, the Wigner
transform $\mathcal{W}_{\hbar}(K_{\hbar})$ is linear in $K_{\hbar}$. We will
make use of this when proving Theorem 0.2. But first we consider the
‘propagator’ $U_{\hbar}(t)=e^{-{\frac{i}{h}}t\widehat{H}_{h}}$, i.e. the
solution operator of the Cauchy problem for the Schrödinger equation

$i\hbar\frac{\partial}{\partial t}u=\widehat{H}_{\hbar}u.$

More precisely, we will consider the Schwartz kernel

$U_{h}(t,x,y)=e^{-{\frac{i}{h}}t\widehat{H}_{h}}(x,y)=\sum_{j\geq
0}e^{-{\frac{i}{h}}tE_{j}(\hbar)}\Pi_{j,\hbar}(x,y)$ (22)

of $U_{\hbar}(t)$ and its Wigner transform

$\mathcal{U}_{\hbar}(t,x,\xi):=\mathcal{W}(U_{\hbar}(t,\cdot,\cdot))(x,\xi)=\int
e^{-{\frac{i}{h}}2\pi\xi\cdot
v}U_{\hbar}\left(t,x+\frac{v}{2},x-\frac{v}{2}\right)\frac{dv}{(2\pi\hbar)^{d}}.$
(23)

Following Berry [Ber89], our first task will be to obtain, for fixed
$(t,x,\xi)$, semi-classical pointwise asymptotics for
$\mathcal{U}_{\hbar}(t,x,\xi)$. Specifically, we prove the following formula
for pointwise semi-classical asymptotics of the Wigner function of the
propagator, stated in a heuristic way by M.V. Berry [Ber89, (21)] (see also
[TL] and [O98].

######  Proposition 0.3.

Let $\widehat{H}_{\hbar}$ satisfy (2). Let $(t,x,\xi)$ be fixed and consider
the set of points $(q_{j},p_{j})\in T^{*}{\mathbb{R}}^{d}$ and the Hamiltonian
arcs $\gamma_{j}=\gamma_{j;t,x,\xi}$ such that
$\gamma_{j;t,x,\xi}(0)=(q_{j},p_{j})$ and

$x=\frac{1}{2}\left(\gamma_{j;t,x,\xi}(0)+\gamma_{j;t,x,\xi}(t)\right),\qquad\xi=\frac{1}{2}\left(\dot{\gamma}_{j;t,x,\xi}(0)+\dot{\gamma}_{j;t,x,\xi}(t)\right).$
(24)

Consider the Jacobian

$M_{j}(t,x,\xi):=D_{q,p}\Phi^{t}(q,p)\big{|}_{(q,p)=(q_{j}(t,x,\xi),p_{j}(t,x,\xi))}$

of the Hamilton flow $\Phi^{t}$ of the endpoint
$\gamma_{j;t,x,\xi}(t)=\Phi^{t}(q_{j},p_{j})$ with respect to initial point
$(q,p)$, and assume that

$\det(I+M_{j}(x,\xi))\not=0,\qquad\forall j.$

Then, (23) admits the pointwise semi-classical asymptotics,

$\mathcal{U}_{\hbar}(t,x,\xi)=2^{d}\sum_{j}\frac{\exp\left(i\left[\frac{S_{j}}{\hbar}+\eta_{j}\right]\right)}{\det^{1/2}(I+M_{j}(x,\xi))}+O(\hbar),$

where $S_{j}$ is the classical action and $\eta_{j}$ is the Maslov index
associated to the path $\gamma_{j;t,x,\xi}$ (Section 1).

This Proposition is essentially well known in the physics literature (e.g.
[Ber89] and [O98]). We prove it in a new way using the Herman-Kluk parametrix
for the propagator (Section 2.1). The basic idea, explained in §2, of the
proof of Proposition 0.3 is to use the Herman-Kluk parameterix [R10] for the
kernel of the propagator $U_{\hbar}(t,x,y)$ to obtain a parametrix for its
Wigner distribution of the form

$\mathcal{U}_{\hbar}(t,x,\xi)\sim\int_{{\mathbb{R}}^{2d}}e^{{\frac{i}{h}}\Psi(q,p,v;t,x,\xi)}a_{\hbar}(t,q,p)dqdp,$
(25)

where $\Psi$ is an explicit complex-valued phase function depending on the
underlying the classical Hamiltonian flow (see (37)) and $a_{\hbar}$ is a
polyhomogeneous symbol (see (35)). This parametrix is valid for long times due
to the sub-quadratic assumption (6) on our Hamiltonians. A straight forward
stationary phase argument, detailed in §2, yields Proposition 0.3.

###  0.2. Outline of the Proof of Theorem 0.2

We will deduce Theorem 0.2 by analyzing the following relation between
$W_{\hbar,f,1,E}$ and the Wigner function of the propagator:

$W_{\hbar,f,1,E}(x,\xi)=\int_{{\mathbb{R}}}\widehat{f}(t)e^{itE/\hbar}\mathcal{U}_{\hbar}(t,x,\xi)\frac{dt}{2\pi},$
(26)

which is a consequence of Fourier inversion and the linearity of the Wigner
transform. As discussed in detail in [HZ20, HZb], $U_{\hbar}(t,x,y)$ is not
locally $L^{1}$ but is nonetheless well-defined as a tempered distribution in
the sense that integral of the form

$\mathcal{U}_{\hbar,f}(x,\xi):=\int_{{\mathbb{R}}}\hat{f}(t)\mathcal{U}_{\hbar}(t,x,\xi)dt$
(27)

are well-defined for $f\in\mathcal{S}({\mathbb{R}}).$ It therefore has a well-
defined distributional Wigner transform, making (26) a valid expression. To
analyze (26), just as in the derivation of Proposition 0.3, we start with the
Herman-Kluk [R10] parameterix to obtain a parametrix for the Wigner transform
of the propagator as in (25). In combination with (26) this yields an
oscillatory integral representation

$W_{\hbar,f,1,E}(x,\xi)\sim\int_{{\mathbb{R}}^{3d}}e^{{\frac{i}{h}}(\Psi(q,p,v;t,x,\xi)+tE)}a_{\hbar}(t,q,p)\frac{dqdpdv}{(2\pi\hbar)^{d}}\frac{dt}{2\pi}.$
(28)

To obtain the asymptotic expansion in Proposition 0.3 we apply stationary
phase. The key point is that the critical points with respect to $p,q,v$ are
non-degenerate (as they were in the proof of Proposition 0.3) but that the
critical points with respect to $t$ are degenerate with a fold singularity at
the only critical point in the support to $f$, namely at $t=0$.

After integrating out the $p,q,v$ variables, to obtain asymptotics for
$W_{\hbar,f,1,E}(x,\xi)$ we mst integrate in $t$. Theorem 0.2 follows by
applying the Chester-Friedman-Ursell (or Malgrange preparation theorem) to the
phase of the $dt$ integral resulting from Proposition 0.3. In Section 4, we
use the fold singularity at $t=0$ to complete the proof of Theorem 0.2.
Leveraging Theorem 0.5 and Proposition 0.3, we will see in Proposition 3.1
that $W_{\hbar,f,1,E}(x,\xi)$ is a semi-classical Fourier integral kernel with
a complex phase. The critical points in $(t,q,p,v)$ at which the imaginary
part of the phase vanishes (and hence the phase integrand is not exponentially
small in $\hbar$) are all solutions of

$\Theta^{t}(q,p)=(x,\xi),\qquad H(q,p)=E.$

The singularities of (28) are determined geometrically by the following
result, which is proved at the same time as Proposition 3.3 below.

######  Proposition 0.4.

The map $\Theta^{*}$ defined in (12) has a fold singularity along
$\\{0\\}\times\Sigma_{E}.$ That is, (12) fixes the diagonal when $t=0$ and the
kernel of its derivative is spanned by the vector field
$\frac{\partial}{\partial t}$.

Proposition 0.4 shows that the Lagrangian generated by the phase in (28) has a
fold singularity for critical points at $t=0$. To analyze it, we rely on
variants of the Malgrange preparation theorem for fold singularities, in the
stronger form given in [CFU, L61], to put the phase into cubic normal form.
See Proposition 4.1 and Proposition 4.2. We note that the latter statements
assume that the phase is real valued, while in [CFU, L61] the phase and
amplitude are assumed to be complex analytic. Neither assumption holds for the
phase of the spectral Wigner distribution in the generality of this article.
For this reason, we first integrate out all but the time $t$ variable in
Section 2.1 (see in particular Section 3.1), so that we may use Proposition
4.1.

###  0.3. Contributions to Smooth Wigner Functions Coming from Non-degenerate
Critical Times

Although we do not use it in the proof of Theorem 0.2, we state a second known
result on the asymptotics of (9) for which the stationary phase method
applies. We only use it to make comparisons with the Airy asymptotics as
$(x,\xi)$ moves away from the fold singularity. In the next Proposition we
retain the notation of Proposition 0.3.

######  Proposition 0.5.

Let $H_{\hbar}$ satisfy (2). Fix $(E,x,\xi)$ with $(x,\xi)\in B_{E}$ and
consider all solutions $(t_{j},q_{j},p_{j})$ of (24) for which
$(q_{j},p_{j})\in\Sigma_{E}$ and $t_{j}\in\rm{supp}\;\hat{f}$. Assume that
$0\not\in\mathrm{supp}(\hat{f})$ and that
$\frac{dt_{j}}{dE}\det(1+M_{j}(x,t_{j}(E))\not=0$ for all $j$. Then the
smoothed spectral Wigner function (9) admits the pointwise semi-classical
asymptotics,

$W_{\hbar,f,1,E}(x,\xi):=\frac{2^{d+1}}{\sqrt{2\pi\hbar}}\sum_{j}\hat{f}(t_{j})\left|\frac{dt_{j}}{dE}\det(1+M_{j}(q_{j},p_{j},t_{j}(E))^{-1}\right|^{{\textstyle\frac{1}{2}}}\cos\left(\frac{S_{j}(x,\xi,E)}{\hbar}+m_{j}\right)+O(\hbar^{1/2}).$

Here, $S_{j}(x,\xi,E)$ is the action along the trajectory
$\gamma_{j;t,x,\xi}$.

Under the assumption (6), there are no solutions $(t_{j},q_{j},p_{j})$ unless
$(x,\xi)\in B_{E}$. Moreover, as explained in [Ber89], if the trajectory
through $(q,p)$ is periodic of period $T$, and if $(t,q,p)$ is a solution of
(24), then $(t+T,q,p)$ is another solution, but the non-degeneracy condition
is not satisfied.

###  0.4. Further scaling results

The results of M.V. Berry [Ber89] pertain mainly to the the contribution of
periodic orbits to the Wigner spectral asymptotics. It is evident from (13)
that if $(q,p)$ is a periodic point of period $T$ then one gets further
solutions by replacing $(q,p)$ by $\Phi^{T}(q,p)$. The fold singularity along
$\Sigma_{E}$ at $t=0$ also occurs at time $t=T$ at the periodic points of
$\Sigma_{E}$ of period $T$. The order of the asymptotics is higher than it
would be for non-degenerate critical points, and Berry therefore referred to
the caustic enhancement of such periodic orbits as ‘scars’. The methods of
this article extend to the periodic orbit case with few modifications but for
the sake of brevity we do not include them here.

In [HZb], further scaling asymptotics are proved in the case of the isotropic
harmonic oscillator. The essential difference to Theorem 0.2 is that much
larger spectral intervals are assumed. We do not generalize these results to
general Hamiltonians satisfying (2) or (6) in this article, but it is very
likely that generalizations do exist. We briefly review these additional
scaling results.

Instead of spectral intervals of width $\hbar$ we consider intervals of width
$\hbar^{2/3}$, e.g. $[E-\lambda_{-}\hbar^{2/3},E+\lambda_{+}\hbar^{2/3}$. In
the smoothing asymptotics we let $\delta=\frac{2}{3}$ and consider

$W_{\hbar,f,\frac{2}{3},E}(x,\xi):=\sum_{j=1}^{\infty}f\left(\hbar^{-\frac{2}{3}}(E-E_{j}(\hbar))\right)W_{\psi_{j},\psi_{j}}(x,\xi).$

We define the rescaled variable $u=u(x,\xi)$ centered at the energy surface
$\Sigma_{E}$ by

$H(x,\xi)=E+u\left(\hbar/2E\right)^{2/3}.$

We then prove that,

$W_{2/3,E,\lambda_{\pm}}(x,\xi)=(2\pi\hbar)^{-d}C_{E}\int_{-\lambda_{+}}^{-\lambda_{-}}\operatorname{Ai}\left(\frac{u}{E}+\lambda
C_{E}\right)d\lambda+O(\hbar^{-d+1/3-\delta}),\qquad C_{E}=(E/4)^{1/3}.$

Instead of getting the value of the Airy function at the scaled parameter $u$,
we get an integral over the values due to the larger spectral interval.

Furthermore, in the even larger interval $I_{\hbar}=[0,E]$, for any
$\varepsilon>0$, we obtained the bulk asymptotics,

$W_{\hbar,[0,E]}(x,\xi)=\left(2\pi\hbar\right)^{-d}\left[\int_{0}^{\infty}\operatorname{Ai}\left(\frac{u}{E}+\lambda\right)d\lambda+O(\hbar^{1/3-\varepsilon}\left\lvert
u\right\rvert^{1/2})+O(\left\lvert
u\right\rvert^{5/2}\hbar^{2/3-\varepsilon})\right],$ (29)

where the $O$-symbol depends only on $d,\varepsilon.$

The articles [Ber89, O98] use the special (Lorentzian) test function
$f_{\varepsilon}(E-E_{j}(\hbar))=\delta_{\varepsilon}(E-E_{j}(\hbar)):=-\frac{1}{\pi}{\operatorname{Im}\,}(E+i\varepsilon-x)^{-1}$,
with various choices of $\varepsilon$ stated in [Ber89, Pages 220-221] as
ranging from the mean level spacing of order $\hbar^{d}$ and the semi-
classical scaling $\frac{\hbar}{T_{\min}}$ where $T_{\min}$ is the length of
the shortest periodic orbit on $\Sigma_{E}$. The same test functions and
energy scales are used in [O98, Section 7]. The mathematical techniques of
this article (along with other spectral asymptotics articles in the
mathematics literature) are not valid below the semi-classical scale, and in
particular do not give scaling results on the length scale
$\delta=\hbar^{\nu}$ with $\nu>1$; they do apply on the scale
$\frac{\hbar}{T_{\min}}$, and we prove Theorem 0.2 by using a special case of
the Malgrange preparation theorem due to Chester-Friedman-Ursell [CFU] to put
the phase into normal form; the relevant theorems are explained in detail in
[HoI, Chapter 7] and in [GSt, Page 439, Page 444]. Such asymptotics are used
to determine the asymptotics of oscillatory integrals whose phase functions
exhibt fold singularities, such as occur in diffraction theory. In Section 3.3
we identify the relevant folding map.

Besides linking the somewhat heuristic asymptotics calculations of [Ber89,
O98] to the mathematical literature, in particular making more precise the
scale of the asymptotics, a novelty of our presentation is to use the Herman-
Kluk propagator as discussed by D. Robert [R10] to construct a ‘parametrix’
for (9).

###  0.5. Acknowledgements

Thanks to Mike Geis and Nick Lohr for many helpful comments that have improved
the exposition.

##  1\.  Background on Classical mechanics

In this section, we recall some basic results from classical mechanics and set
notation. With $\Phi^{t}$ the Hamiltonian flow defined in (10), we continue to
denote its derivative at $(q,p)$ by

$D_{q,p}\Phi^{t}:=M_{t}=\left(\begin{array}[]{cc}A_{t}&B_{t}\\\
C_{t}&D_{t}\end{array}\right),$ (30)

where

$A_{t}=\partial_{q}q_{t},\quad B_{t}=\partial_{p}q_{t},\quad
C_{t}=\partial_{q}p_{t},\quad D_{t}=\partial_{p}p_{t},$

with rows corresponding to components and columns to derivatives:

$\left(\partial_{q_{j}}q_{t}\right)_{k}=\left(A_{t}\right)_{k,j}.$

Since $\Phi^{t}$ is a Hamiltonian flow, we have

$\Omega=M_{t}^{T}\Omega M_{t}$

where $\Omega=\left(\begin{array}[]{cc}0&I\\\ -I&0\end{array}\right)$ is the
symplectic form [F] and we therefore have

$\displaystyle A_{t}C_{t}=C_{t}^{T}A_{t},\qquad
A_{t}^{T}D_{t}-C_{t}^{T}B_{t}=\operatorname{Id},\qquad
B_{t}^{T}D_{t}=D_{t}B_{t}.$ (31)

The action along the $\Phi^{t}$ orbit with initial value $(q,p)$ is defined
by,

$S(t,q,p):=\int_{0}^{t}(\dot{q}_{s}\cdot p_{s}-H(s,q_{s},p_{s}))ds$ (32)

We only deal with autonomous Hamiltonians, for which
$H(s,q_{s},p_{s})=H(q_{s},p_{s})$ and then $H(q_{s},p_{s})$ is constant along
Hamilton orbits and the second term is $tH(q,p)$. We will have occasion to use
the following elementary result.

######  Lemma 1.1.

We have

$\left\\{\begin{array}[]{l}\partial_{q}S(t,q,p)=\left(\partial_{q}q_{t}\right)\cdot
p_{t}-p,\\\ \\\ \partial_{p}S(t,q,p)=\left(\partial_{p}q_{t}\right)\cdot
p_{t},\\\ \\\ \partial_{t}S(t,q,p)=\dot{q}_{t}\cdot
p_{t}-H(q_{t},p_{t}).\end{array}\right.$

###### Proof.

Using the equations of motion $\partial_{p}H=\dot{q}_{s}$ and
$\partial_{q}H=-\dot{p}_{s}$, we get

$\displaystyle\partial_{q}S(t,q,p)$
$\displaystyle=\partial_{q}\left(\int_{0}^{t}\dot{q}_{s}\cdot
p_{s}-H(q_{s},p_{s})ds\right)$
$\displaystyle=\int_{0}^{t}\left(\partial_{q}\dot{q}_{s}\cdot
p_{s}+\dot{q}_{s}\cdot\partial_{q}p_{s}-\partial_{q}H(q_{s},p_{s})\partial_{q}q_{s}-\partial_{p}H(q_{s},p_{s})\partial_{q}p_{s}\right)ds$
$\displaystyle=\int_{0}^{t}\langle\partial_{q}\dot{q}_{s},p_{s}\rangle+\langle\dot{q}_{s},\partial_{q}p_{s}\rangle+\langle\dot{p}_{s},\partial_{q}q_{s}\rangle-\langle\dot{q}(s),\partial_{q}p_{s}\rangle
ds$
$\displaystyle=\int_{0}^{t}\langle\partial_{q}\dot{q}_{s},p_{s}\rangle+\langle\dot{p}_{s},\partial_{q}q_{s}\rangle
ds=\int_{0}^{t}\partial_{s}\langle\partial_{q}q_{s},p_{s}\rangle
ds=\langle\partial_{q}q_{s},p_{s}\rangle|_{0}^{t},$

proving the first statement. Similarly,

$\displaystyle\partial_{p}S(t,q,p)$
$\displaystyle=\partial_{p}\left(\int_{0}^{t}\dot{q}_{s}\cdot
p_{s}-H(q_{s},p_{s})ds\right)$
$\displaystyle=\int_{0}^{t}\left(\partial_{p}\dot{q}_{s}\cdot
p_{s}+\dot{q}_{s}\cdot\partial_{p}p_{s}-\partial_{q}H(q_{s},p_{s})\partial_{p}q_{s}-\partial_{p}H(q_{s},p_{s})\partial_{p}p_{s}\right)ds$
$\displaystyle=\int_{0}^{t}\left(\partial_{p}\dot{q}_{s}\cdot
p_{s}+\dot{q}_{s}\cdot\partial_{p}p_{s}+\langle\dot{p}_{s},\partial_{p}q_{s}\rangle-\langle\dot{q}_{s},\partial_{p}p_{s}\rangle\right)ds$
$\displaystyle=\int_{0}^{t}\left(\partial_{p}\dot{q}_{s}\cdot
p_{s}+\langle\dot{p}_{s},\partial_{p}q_{s}\rangle\right)ds=\int_{0}^{t}\frac{d}{ds}\langle
p_{s},\partial_{p}q_{s}\rangle ds=\langle
p_{s},\partial_{p}q_{s}\rangle|_{0}^{t},$

proving the second statement. The third statement is obvious.

∎

##  2\.  Herman-Kluk Parametrix and the Proof of Proposition 0.3

In this section, we prove Proposition 0.3. For this, we recall in §2.1 the
Herman-Kluk Parametrix for the propagator. Then we use this parameterix in
conjunction with stationary phase to complete the proof of Proposition 0.3 in
§2.2. We then provide in §2.3 and 2.4 two conceptual remarks on the proof that
will be useful for guiding our subsequent proof of Theorem 0.2.

###  2.1.  Herman-Kluk Parametrix for the Wigner Transform of the Propagator

As explained in [R10, RS] (see [R10, (1.7)]), for subquadratic Hamiltonians
satisfying 2, one can construct to construct a long-time parametrix

$U_{\hbar}(t,x,y)\sim\int_{{\mathbb{R}}^{2d}}e^{\frac{i}{\hbar}\Psi_{HK}(t,q,p;x,y)}a_{\hbar}(t,q,p)dqdp$
(33)

for the propagator $U_{\hbar}(t,x,y)$ (see (22)) with the complex Herman-Kluk
phase,

$\Psi_{HK}(t,q,p;x,y):=S(t,q,p)+p_{t}\left(x-q_{t}\right)-p(y-q)+\frac{i}{2}\left(\left\lvert
x-q_{t}\right\rvert^{2}+\left\lvert y-q\right\rvert^{2}\right).$ (34)

Here, $S(t,q,p)$ is the action given by (32). The phase $\Psi_{HK}$ is called
a positive complex phase since its imaginary part is positive. Moreover,
$a\sim\sum_{j}\hbar^{j}a_{j}(q,p)$ is a polyhomogeneous symbol with

$a_{0}=\left(\det\left(A_{t}+D_{t}+i(B_{t}-C_{t})\right)\right)^{1/2}\exp(-itH(q,p))$
(35)

where we’ve written as in (30) Section 1

$A_{t}=\frac{\partial q_{t}}{\partial q},\quad B_{t}=\frac{\partial
q_{t}}{\partial p},\quad C_{t}=\frac{\partial p_{t}}{\partial q},\quad
D_{t}=\frac{\partial p_{t}}{\partial p}.$

The parametrix is valid for all times $t<\log(1/\hbar)$ and the symbol
estimates are uniform over compact intervals in $t.$ The parameterix (33) for
the propagator $U_{\hbar}(t,x,y)$ reveals that its Wigner transform
$\mathcal{U}_{\hbar}(t,x,\xi)$ is an oscillatory integral with positive
complex phase. Namely,

$\mathcal{U}_{\hbar}(t,x,\xi)\sim(2\pi\hbar)^{-d}\int_{{\mathbb{R}}^{3d}}e^{\frac{i}{\hbar}\Psi(t,x,\xi;q,p,v)}a_{\hbar}(t,q,p)\frac{dqdpdv}{(2\pi\hbar)^{d}},$
(36)

where

$\displaystyle\Psi(t,x,\xi;q,p,v)$
$\displaystyle:=S_{t}+p_{t}\left(x+\frac{v}{2}-q_{t}\right)-p\left(x-\frac{v}{2}-q\right)-v\xi$
(37) $\displaystyle\qquad+\frac{i}{2}\left(\left\lvert
x+\frac{v}{2}-q_{t}\right\rvert^{2}+\left\lvert
x-\frac{v}{2}-q\right\rvert^{2}\right).$

The integral is of course exponentially small in $\hbar$ unless the imaginary
part vanishes, i.e.

${\operatorname{Im}\,}\Psi=\left\lvert
x+\frac{v}{2}-q_{t}\right\rvert^{2}+\left\lvert
x-\frac{v}{2}-q\right\rvert^{2}=0.$ (38)

###  2.2. Proof of Proposition 0.3

We are now ready to complete the proof of Proposition 0.3. In the notation of
Section 1, the main technical result is the following.

######  Proposition 2.1.

The Wigner function $\mathcal{U}_{\hbar}(t,x,\xi)$ of the propagator is a
semi-classical Fourier integral kernel with a positive complex phase. The
dominant critical points (in $(q,p,v)$) at which (38) holds are solutions of

$x=\frac{q+q_{t}}{2},\quad\xi=\frac{p+p_{t}}{2},\quad v=q_{t}-q.$

Moreover, there exist $\varepsilon,\varepsilon_{0}>0$ such that for $(x,\xi)$
in an $\varepsilon$-neighborhood of $\Sigma_{E}$ and
$t\in(-\varepsilon_{0},\varepsilon_{0})$, critical points are uniformly non-
degenerate. The critical value of the phase at a critical point $(q,p,v)$ is
given by:

$\Psi_{c}(t,x,\xi):=\int_{\gamma_{t,x,\xi}}(pdq-
Hds)-\langle(q_{t}-q),\xi\rangle,$ (39)

where $\gamma_{t,x,\xi}$ was defined in (16) as the unique Hamiltonian arc (up
to the involution (14)) with $|t|$ small on $\Sigma_{E}$ such that
$(x,\xi)={\textstyle\frac{1}{2}}(\gamma_{t,x,\xi}(0)+\gamma_{t,x,\xi}(t))$.

Define the action,

$S(t,x,\xi)=\int_{\gamma_{t,x,\xi}}(pdq-Hds)$ (40)

along the path $\gamma_{t,x,\xi}$.

###### Proof of Proposition 2.1.

As mentioned above, the imaginary part of the phase is strictly positive
unless

$x+\frac{v}{2}-q_{t}=x-\frac{v}{2}-q=0.$ (41)

Hence, any critical points where these conditions are satisfied will provide
the dominant contribution to the Wigner function. It is immediate from (41)
that at the dominant critical points,

$x=\frac{q+q_{t}}{2},\;\;\qquad v=q_{t}-q.$ (42)

In addition, the equation $d_{v}\Psi=0$ implies that

$\xi=\frac{p+p_{t}}{2}.$

∎

To complete the proof of Proposition 0.3, we fix $t>0$, consider the full
$dqdpdv$ integral (36), and apply the method of stationary phase for positive
complex phase functions (cf. [HoI, Theorem 7.7.5]). For this, we need to
compute the Hessian of (37) at the dominant critical points. We use the
notation (30), which we repeat for the reader’s convenience,

$A_{t}(q,p)=\partial_{q}q_{t}(q,p),\quad
B_{t}(q,p)=\partial_{p}q_{t}(q,p),\quad
C_{t}(q,p)=\partial_{q}p_{t}(q,p),\quad D_{t}(q,p)=\partial_{p}p_{t}(q,p).$

The key result is the following computation.

######  Lemma 2.2.

Fix $t,x,\xi$ and suppose $(q_{c},p_{c},v_{c})$ is a dominant critical point
of $\Psi(q,p,v;t,x,\xi)$ (37) in the variables $(q,p,v)$ for which
$\partial_{p}q_{t}\neq 0.$ Then
$\det\left(\text{Hess}(\Psi)\right)(q_{c},p_{c},v_{c})$ is given by,

$(-1)^{d}\cdot\det(1+M_{t}(q_{c},p_{c}))\det\left(A_{t}(q_{c},p_{c})+D_{t}(q_{c},p_{c})+i(B_{t}(q_{c},p_{c})-C_{t}(q_{c},p_{c}))\right).$

###### Proof.

We claim that at the critical point,

$\displaystyle\partial_{pp}\Psi$
$\displaystyle=B_{t}^{T}\left(-D_{t}+iB_{t}\right),\qquad$
$\displaystyle\partial_{pq}\Psi=B_{t}^{T}\left(-C_{t}+iA_{t}\right)$
$\displaystyle\partial_{pv}\Psi$
$\displaystyle=\frac{1}{2}\left(\operatorname{Id}+D_{t}-iB_{t}\right)^{T},\qquad$
$\displaystyle\partial_{vv}\Psi=\frac{i}{2}\operatorname{Id}$
$\displaystyle\partial_{qq}\Psi$
$\displaystyle=-A_{t}^{T}C_{t}+i\left(A_{t}^{T}A_{t}+\operatorname{Id}\right),\qquad$
$\displaystyle\partial_{qv}\Psi=\frac{1}{2}\left(C_{t}+i\left(-A_{t}+\operatorname{Id}\right)\right).$

We summarize the key points of the calculations. Apriori, the Hessian involves
second derivatives in $(q,p)$ of $(q_{t},p_{t},\dot{q}_{t},\dot{p}_{t})$ but
in fact these cancel at a dominant critical point. First,

$\partial_{p_{j}}\Psi=\partial_{p_{j}}p_{t}\left(x+\frac{v}{2}-q_{t}\right)-\left(x_{j}-\frac{v_{j}}{2}-q_{j}\right)+i\left(-\partial_{p_{j}}q_{t}\left(x+\frac{v}{2}-q_{t}\right)\right),$
(43)

and at a critical point,

$\partial_{p_{k},p_{j}}\Psi=-\partial_{p_{j}}p_{t}\partial_{p_{k}}q_{t}+i\left(\partial_{p_{j}}q_{t}\partial_{p_{k}}q_{t}\right)=(-\partial_{p_{j}}(p_{t}+i\partial_{p_{j}}q_{t})\partial_{p_{k}}q_{t}=\left((-D_{t}+iB_{t})^{T}B_{t}\right)_{k,j}$

Also,

$\partial_{q_{j}}\Psi=\partial_{q_{j}}p_{t}\left(x+\frac{v}{2}-q_{t}\right)-i\left(\partial_{q_{j}}q_{t}\left(x+\frac{v}{2}-q_{t}\right)+\left(x_{j}-\frac{v_{j}}{2}-q_{j}\right)\right),$
(44)

and at a critical point,

$\partial_{p_{k},q_{j}}\Psi=-\partial_{q_{j}}p_{t}\partial_{p_{k}}q_{t}+i\left(\partial_{q_{j}}q_{t}\partial_{p_{k}}q_{t}\right).$

Starting from (43), we have

$\partial_{v_{k}p_{j}}\Psi=\frac{1}{2}\left((\partial_{p_{j}}p_{t})_{k}+\delta_{j,k}-i(\partial_{p_{j}}q_{t})_{k}\right)=\frac{1}{2}\left(\operatorname{Id}+D_{t}-iB_{t}\right)_{k,j}.$

Again starting from (43), we have

$\partial_{tp_{j}}\Psi=\left(-\partial_{p_{j}}p_{t}+i\partial_{p_{j}}q_{t}\right)\dot{q}_{t}=\left[\dot{q}_{t}(-D_{t}+iB_{t})\right]_{j}.$

Starting from (44), we have

$\partial_{q_{k}q_{j}}\Psi=\partial_{q_{j}}p_{t}\partial_{q_{k}}q_{t}+i\left(\partial_{q_{j}}q_{t}\partial_{q_{k}}q_{t}+\delta_{j,k}\right)=\left[A_{t}^{T}C_{t}+i\left(A_{t}^{T}A_{t}+\operatorname{Id}\right)\right]_{k,j}.$

Finally, from (44) we obtain

$\partial_{v_{k}q_{j}}\Psi=\frac{1}{2}\left((\partial_{q_{j}}p_{t})_{k}+i\left(-(\partial_{q_{j}}q_{t})_{k}+\delta_{k,j}\right)\right).$

The Hessian of $\Psi$ at $(q_{c},p_{c},v_{c})$ is therefore

$\left(\begin{array}[]{ccc}\frac{i}{2}\operatorname{Id}&\frac{1}{2}\left(C_{t}+i(-A_{t}+\operatorname{Id})\right)&\frac{1}{2}\left(D_{t}+\operatorname{Id}-iB_{t}\right)\\\
\frac{1}{2}\left(C_{t}^{T}+i(-A_{t}+\operatorname{Id})\right)&-A_{t}C_{t}+i(A_{t}^{2}+\operatorname{Id})&-C_{t}^{T}B_{t}+iA_{t}B_{t}\\\
\frac{1}{2}\left(D_{t}+\operatorname{Id}-iB_{t}^{T}\right)&-B_{t}^{T}C_{t}+iB_{t}^{T}A_{t}&B_{t}^{T}(-D_{t}+iB_{t})\end{array}\right),$

as claimed. We now calculate the determinant of the Hessian. Multiplying the
first row on the left by $A_{t}$ and adding the result to the second row and
then multiplying the first row by $B_{t}^{T}$ on the left and adding the
result to the third row shows that the determinant of
$\text{Hess}(\Psi)(q_{c},p_{c},v_{c})$ is the same as the determinant of

$\left(\begin{array}[]{ccc}\frac{i}{2}\operatorname{Id}&\frac{1}{2}\left(C_{t}+i(-A_{t}+\operatorname{Id})\right)&\frac{1}{2}\left(\widetilde{D}_{t}-iB_{t}\right)\\\
\frac{1}{2}\left(C_{t}^{T}+i\widetilde{A}_{t}\right)&-i\widetilde{A}_{t}&\widetilde{A}_{t}\\\
\frac{1}{2}\left(\widetilde{D}_{t}+iB_{t}^{T}\right)&iB_{t}^{T}&B_{t}^{T}\end{array}\right),$

where for any matrix $K,$ we write $\widetilde{K}=\operatorname{Id}+K.$ Next,
multiplying the first column by $2$ and subtracting it from the second column,
we find that the determinant of $\text{Hess}(\Psi)(q_{c},p_{c},v_{c})$ is the
same as the determinant of

$\left(\begin{array}[]{ccc}\frac{i}{2}\operatorname{Id}&\frac{1}{2}\left(C_{t}-i\widetilde{A}_{t}\right)&\frac{1}{2}\left(\widetilde{D}_{t}-iB_{t}\right)\\\
\frac{1}{2}\left(C_{t}^{T}+i\widetilde{A}_{t}\right)&-C_{t}^{T}&\widetilde{A}_{t}\\\
\frac{1}{2}\left(\widetilde{D}_{t}+iB_{t}^{T}\right)&-\widetilde{D}_{t}&B_{t}^{T}\end{array}\right).$

Next, by dividing the first row and column by $\sqrt{2/i}$, we find that the
determinant of $\text{Hess}(\Psi)(q_{c},p_{c},v_{c})$ is the same as the
determinant of

$\left(\begin{array}[]{ccc}\operatorname{Id}&\frac{1}{\sqrt{2i}}\left(C_{t}-i\widetilde{A}_{t}\right)&\frac{1}{\sqrt{2i}}\left(\widetilde{D}_{t}-iB_{t}\right)\\\
\frac{1}{\sqrt{2i}}\left(C_{t}^{T}+i\widetilde{A}_{t}\right)&-C_{t}^{T}&\widetilde{A}_{t}\\\
\frac{1}{\sqrt{2i}}\left(\widetilde{D}_{t}+iB_{t}^{T}\right)&-\widetilde{D}_{t}&B_{t}^{T}\end{array}\right)$

times $(i/2)^{d}$. Next, writing
$\Omega=\left(\begin{array}[]{cc}0&\operatorname{Id}\\\
-\operatorname{Id}&0\end{array}\right),$ note that

$\left(\begin{array}[]{cc}-C_{t}^{T}&\widetilde{A}_{t}\\\
-\widetilde{D}_{t}&B_{t}^{T}\end{array}\right)=\widetilde{M}_{t}^{T}\Omega.$

Further, observe that

$\left(C_{t}-i\widetilde{A}_{t},\quad\widetilde{D}_{t}-iB_{t}\right)=(-i\operatorname{Id}\quad\operatorname{Id})\widetilde{M}$

and that

$\left(\begin{array}[]{c}C_{t}^{T}+i\widetilde{A}_{t}\\\
\widetilde{D}_{t}+iB_{t}^{T}\end{array}\right)=\widetilde{M}^{T}\left(\begin{array}[]{c}i\operatorname{Id}\\\
\operatorname{Id}\end{array}\right).$

Hence, the determinant of $\text{Hess}(\Psi)(q_{c},p_{c},v_{c})$ is the same
as $(i/2)^{d}$ times the determinant of

$\left(\begin{array}[]{cc}\operatorname{Id}&0\\\
0&\widetilde{M}^{T}\end{array}\right)\left(\begin{array}[]{cc}\operatorname{Id}&\frac{1}{\sqrt{2i}}\left(-i\operatorname{Id}~{}~{}~{}\operatorname{Id}\right)\widetilde{M}\\\
\frac{1}{\sqrt{2i}}\left(\begin{array}[]{c}i\operatorname{Id}\\\
\operatorname{Id}\end{array}\right)&\Omega\par\end{array}\right).$

Therefore, the determinant of $\text{Hess}(\Psi)(q_{c},p_{c},v_{c})$ equals
$(2/i)^{d}$ times $\det(\widetilde{M})$ times the determinant of

$\left(\begin{array}[]{cc}\operatorname{Id}&\frac{1}{\sqrt{2i}}\left(-i\operatorname{Id}~{}~{}~{}\operatorname{Id}\right)\widetilde{M}\\\
\frac{1}{\sqrt{2i}}\left(\begin{array}[]{c}i\operatorname{Id}\\\
\operatorname{Id}\end{array}\right)&\Omega\par\end{array}\right).$

Take the transpose of this matrix and using the Schur complement formula, we
find that the determinant of this matrix is the same as the determinant of

$\operatorname{Id}-\frac{1}{2i}\left[\left(i\operatorname{Id}~{}~{}~{}\operatorname{Id}\right)\Omega\widetilde{M}^{T}\left(-i\operatorname{Id}~{}~{}~{}\operatorname{Id}\right)^{T}\right]=\frac{1}{2i}\left(-\operatorname{Id}~{}~{}~{}i\operatorname{Id}\right)\Omega
M^{T}\left(-i\operatorname{Id}~{}~{}~{}\operatorname{Id}\right)^{T},$

where we have used that

$\frac{1}{2i}\left[\left(i\operatorname{Id}~{}~{}~{}\operatorname{Id}\right)\Omega\left(-i\operatorname{Id}~{}~{}~{}\operatorname{Id}\right)^{T}\right]=2i\operatorname{Id}.$

Finally, since

$\left(-\operatorname{Id}~{}~{}~{}i\operatorname{Id}\right)\Omega
M^{T}\left(-i\operatorname{Id}~{}~{}~{}\operatorname{Id}\right)^{T}=A_{t}+D_{t}+i(B_{t}^{T}-C_{t}^{T})$

and the determinant is invariant under transposing the matrix, we find that

$\det\left(\text{Hess}(\Psi)(q_{c},p_{c},v_{c})\right)=2^{2d}\det\left(\operatorname{Id}+M_{t}\right)\cdot\det(A_{t}+D_{t}+i(B_{t}-C_{t})),$
(45)

as claimed.

∎

To complete the proof of Proposition 0.3, we note that when $(x,\xi)$ and
$t=0$, we have $M_{t}=I$ and obviously the modulus of (45) is a positive
constant. Hence the same is true for $(x,\xi)$ in a tubular neighborhood of
some positive radius around $\Sigma_{E}$. Thus, Proposition 0.3 follows by
stationary phase for positive complex phases [HoI]. $\square$

###  2.3.  Remark: Integrating Out $dv$ First

We remark that a somewhat simpler analysis is possible for the proof of
Proposition 0.3 by first integrating in $dv$, to get a reduced oscillatory
integral in $dqdp$. This is natural since the $dv$ integral is essentially a
Fourier transform of a Gaussian, slightly distorted by the non-constant
amplitude. In particular the Hessian $d_{v}^{2}\Psi$ at the critical point is
non-degenerate. Eliminating $dv$ we obtain the reduced integral,

$\mathcal{U}_{\hbar}(t,x,\xi)=\frac{1}{(2\pi\hbar)^{d}}\int_{{\mathbb{R}}^{d}}\int_{{\mathbb{R}}^{d}}e^{\frac{i}{\hbar}\Psi_{c}(t,q,p;x,\xi)}A(\hbar,q,p,t)dqdp$
(46)

with reduced phase,

$\begin{array}[]{l}\Psi_{c}(t,x,\xi;q,p):=S_{t}(q,p)+(p_{t}-p)\left(x-\frac{q_{t}+q}{2}\right)-\langle
q_{t}-q,\xi\rangle+i\left(\left\lvert
x-(\frac{q_{t}+q}{2})\right\rvert^{2}\right)\par\end{array}$ (47)

Here, we did not use the equation $\left(x-\frac{q_{t}+q}{2}\right)=0$. The
amplitude is that of the original $dqdpdv$ integral evaluated at $v=q_{t}-q$
and multiplied by the constant Hessian determinant in $v$. We need to further
apply stationary phase for complex phase functions in $dqdp$ to obtain this as
a critical point equation. If we do that, we obtain the critical point
equation for the real part of (47) simplify to,

$\left\\{\begin{array}[]{l}x=\frac{q_{t}+q}{2},\\\ \\\
d_{q}S_{t}-{\textstyle\frac{1}{2}}(p_{t}-p)(I-d_{q}q_{t})-\langle
d_{q}q_{t},\xi\rangle+\xi=0,\\\ \\\
d_{p}S_{t}-{\textstyle\frac{1}{2}}(p_{t}-p)d_{p}q_{t}-\langle
d_{p}q_{t},\xi\rangle=0.\end{array}\right.,$

and by Lemma 1.1 reduce further to,

$\left\\{\begin{array}[]{l}\left({\textstyle\frac{1}{2}}(p_{t}+p)-\xi\right)[-I+d_{q}q_{t}]=0,\\\
\\\
=\left({\textstyle\frac{1}{2}}(p_{t}+p)-\xi)\right)d_{p}q_{t}=0,\end{array}\right.$

which (again) imply ${\textstyle\frac{1}{2}}(p_{t}+p)=\xi$. At a dominant
critical point, the real part of the Hessian is given by,

${\operatorname{Re}\,}\rm{Hess}_{q,p}\Psi_{c,crit}=\begin{pmatrix}&\\\
A_{t}B_{t}-{\textstyle\frac{1}{2}}C_{t}(I+A_{t})&[A_{t}D_{t}-{\textstyle\frac{1}{2}}B_{t}C_{t}]^{T}\\\
&&\\\
A_{t}D_{t}-{\textstyle\frac{1}{2}}B_{t}C_{t}&B_{t}D_{t}-{\textstyle\frac{1}{2}}D_{t}C_{t}+{\textstyle\frac{1}{2}}C_{t}\end{pmatrix}|_{\Theta^{t}(q,p)=(x,\xi)},$

and the imaginary part of the Hessian is given by,

${\operatorname{Im}\,}\rm{Hess}=\begin{pmatrix}\frac{1}{4}[2I+2\langle\partial_{q}q_{t},\partial_{q}q_{t}\rangle+2\partial_{q}q_{t}]&2\langle\partial_{p}q_{t},\partial_{q}q_{t}\rangle+2\partial_{p}q_{t}\\\
&\\\
2\langle\partial_{p}q_{t},\partial_{q}q_{t}\rangle+2\partial_{p}q_{t}&{\textstyle\frac{1}{2}}\langle\partial_{p}q_{t},\partial_{p}q_{t}\rangle\end{pmatrix}$

We leave this approach at this point since we have already calculated the full
Hessian determinant.

###  2.4. Remark: Interpretation of Proof of Proposition 0.3 by Lagrangian
Submanifolds

For fixed $(t,x,\xi)$, the critical set of the phase (37) in the sense of
[HoIII] is therefore

$C_{W_{\hbar,t}}=\\{(x,\xi,q,p,v):x=\frac{q+q_{t}}{2},\quad\xi=\frac{p+p_{t}}{2},\quad
v=q_{t}-q\\}.$

Using Lemma 1.1, we find that the associated Lagrangian submanifold of
$T^{*}({\mathbb{R}}\times T^{*}{\mathbb{R}}^{d})$ is given by,

$\begin{array}[]{lll}\Lambda_{t}&=&\\{(x,d_{x}\Psi,\xi,d_{\xi}\Psi:(x,\xi,q,p,v)\in
C_{W_{\hbar,t}(x,\xi))}\\}\\\ &&\\\
&=&\\{(x,(p_{t}-p),\xi,q_{t}-q),x=\frac{q+q_{t}}{2},\quad\xi=\frac{p+p_{t}}{2}.\\}\\}\end{array}$
(48)

We also consider the space-time Lagrangian

$\begin{array}[]{lll}\Gamma&=&\\{t,d_{t}\Psi,x,d_{x}\Psi,\xi,d_{\xi}\Psi:(t,x,\xi,q,p,v)\in
C_{W_{\hbar,t}(x,\xi))}\\}\subset T^{*}({\mathbb{R}}\times
T^{*}{\mathbb{R}}^{d}))\\\ &&\\\
&=&\\{(t,-H(q,p),x,(p_{t}-p),\xi,q_{t}-q),x=\frac{q+q_{t}}{2},\quad\xi=\frac{p+p_{t}}{2}.\\}\end{array}$
(49)

We have the natural projection,

$\pi:\Gamma\to{\mathbb{R}}\times T^{*}{\mathbb{R}}^{d},$ (50)

given by

$(t,-H(q,p)+\dot{p}_{t}(x+\frac{q_{t}-q}{2}-q_{t}),x,(p_{t}-p),\xi,q_{t}-q),(q,p,v)\mapsto(t,x,\xi).$

Since $v=q_{t}-q$ is uniquely determined once $(q,p)$ are determined, the
fiber of (50) is the set of solutions of

$\pi^{-1}(t,x,\xi)=\\{(q,p,v):x=\frac{q+q_{t}}{2},\quad\xi=\frac{p+p_{t}}{2},\quad
v=q_{t}-q\\}\leftrightarrow\\{(t,\tau,q,p):\Theta^{t}(q,p)=(x,\xi)\\}.$

as defined in (12). By the inverse function theorem, the solution is locally
unique if $D_{(q,p)}\Theta^{t}$ does not have $-1$ as an eigenvalue. As an
example where it is non-unique, we could let $(x,\xi)=(0,0)$ and consider
‘anti-podal’ times when $\Theta^{t}(q,p)=-(q,p)$. For instance, in the case of
the isotropic oscillator such a time is $t=\pi$. An oscillator on
${\mathbb{R}}^{d}$ with $k\leq d$ equal frequencies provides an example where
the fiber is not discrete but has dimension $d-k$.

##  3\. Proof of Theorem 0.2: the $t$ Integral and Energy Asymptotics

In this section we extend the analysis from the previous section, which
involved computing the integral (36) in $v,q,p$ to computing pointwise
asymptotics for the Wigner transform $\mathcal{U}_{\hbar}(t,x,\xi)$ of the
propagator, to include also the $t$ integral (see eg (9))

$W_{\hbar,f,1,E}(x,\xi)\sim\int_{{\mathbb{R}}^{3d}}e^{{\frac{i}{h}}(\Psi(q,p,v;t,x,\xi)+tE)}a_{\hbar}(t,q,p)\frac{dqdpdv}{(2\pi\hbar)^{d}}\frac{dt}{2\pi}$
(51)

to determine the asymptotics of the Wigner distribution of the smoothed
spectral projector.

As in [O98, TL], it is the $t$-integral which introduces a degeneracy in the
phase and a fold singularity in the associated Lagrangian submanifold (see
Section 3.2 and Section 3.3 for the geometric analysis of folds). In the next
Section 4, we put the phase into Malgrange normal form, or more correctly a
stronger version of it for cubic phases due to Chester-Friedman-Ursell [CFU]
and Levinson [L61].

We consider two approaches to evaluating (51): (i) integrate in $(t,q,p,v)$ at
once; (ii) integrate in $(q,p,v)$ first, reduce to a one dimensional integral
in $dt$ and then determine the asymptotics of this integral. The advantage of
(i), which we carry out in §3.0.1, is that the critical point equations are
considerably simpler to solve. However, the phase is complex-valued and the
standard normal forms results need to be modified to apply to it. The
advantage of (ii), which we carry out in §3.1, is that we can directly apply
the asymptotics results from the literature [CFU, HoI, GSt]. But the critical
point equations become more complicated.

####  3.0.1. The $dtdqdpdv$ integral

We extend Proposition 2.1 and Proposition 2.2 by including the additional
integration in $t$.

######  Proposition 3.1.

Let $f\in\mathcal{S}({\mathbb{R}})$. The Wigner function

$W_{\hbar,E,f}(x,\xi)=\int_{{\mathbb{R}}^{3d}}\int_{{\mathbb{R}}}\hat{\rho}(t)W_{\hbar,t}(x,\xi)e^{\frac{itE}{\hbar}}dqdpdvdt$

of the smoothed spectral function is a semi-classical Fourier integral kernel
with a complex phase. The critical points in $(t,q,p,v)$ at which the
imaginary part of the phase

$\widetilde{\Psi}_{E}(t,q,p,v;x,\xi):=\Psi(t,q,p,v;x,\xi)+tE$

vanishes are all solutions of

$x=\frac{q+q_{t}}{2},\quad\xi=\frac{p+p_{t}}{2},\quad v=q_{t}-q,\quad
H(q,p)=E.$

We postpone the proof of the last statement to Section 3.1.2, where we first
integrate out the $(q,p,v)$ variables and reduce to a one dimensional
integral.

###### Proof.

The additional integration leads to the additional critical point equation,

$d_{t}\left(S(t,q,p)+p_{t}(x+\frac{v}{2}-q_{t})+tE+\frac{i}{2}\left(\left\lvert
x+\frac{v}{2}-q_{t}\right\rvert^{2}+\left\lvert
x-\frac{v}{2}-q\right\rvert^{2}\right))\right)=0,$

whose real part is the equation,

$\begin{array}[]{l}d_{t}\left(S(t,q,p)+p_{t}(x+\frac{v}{2}-q_{t})+tE\right)=0\\\
\\\
\iff\dot{q}(t)p(t)-H(q,p)+\dot{p}(x+\frac{v}{2}-q_{t})-p_{t}\dot{q_{t}}+E=0.\end{array}$

Due to the vanishing of the imaginary part of the equation, we also get
$\dot{p}(x+\frac{v}{2}-q_{t})-p_{t}\dot{q_{t}}=0$ at the critical point, and
the additional critical point equation simplifies to

$\dot{q}(t)p(t)-H(q,p)-p_{t}\dot{q_{t}}+E=0\iff H(q,p)=E.$

This proves all but the final statement, deferred to Section 3.1.2.

∎

###  3.1.  Reduction to a one-dimensional integral

We now re-do the proof of Proposition 3.1 by first eliminating the variables
$(q,p,v)$ by stationary phase and reducing to a single $dt$ integral. To state
the result, we need to introduce some notation which will be clarified in the
course of the proof. Recall the Hamiltonian arc with
${\textstyle\frac{1}{2}}(\gamma_{t,x,\xi}(0)+\gamma_{t,x,\xi}(t))=(x,\xi)$ of
(16),

$\gamma_{t,x,\xi}(s)=\Phi^{s}(q(t,x,\xi),p(t,x,\xi)),\;\;\;s\in[0,t]$

whose endpoints are given by,

$\gamma_{t,x,\xi}(0)=(q(t,x,\xi),p(t,x,\xi)),\;\gamma_{t,x,\xi}(t)=\Phi^{t}(q(t,x,\xi),p(t,x,\xi)),$

where $(q(t,x,\xi),p(t,x,\xi))$ are local solutions of

$(q(t,x,\xi),p(t,x,\xi))=(\Theta^{t})^{-1}(x,\xi)\iff{\textstyle\frac{1}{2}}(I+\Phi^{t})(q(t,x,\xi),p(t,x,\xi))=(x,\xi).$

Note that $(\Theta^{t})^{-1}(x,\xi)$ is multi-valued but has well-defined
branches near $\Sigma_{E}$ (see Lemma 3.4).

We now rewrite (39) in an advantageous form,

$\Psi_{c}(t,x,\xi):=\int_{\hat{\gamma}_{t,x,\xi}}(pdq)-\int_{\gamma_{t,x,\xi}}Hds=\int_{\beta_{t,x,\xi}}\omega-\int_{\gamma_{t,x,\xi}}Hds,$
(52)

where $\hat{\gamma}_{t,x,\xi}$ is the completed contour obtained by adding in
the chordal link to obtain the oriented closed curve connecting the endpoints
of $\gamma_{t,x,\xi}$ by the chord
$\alpha_{t,x,\xi}(s)=(1-s)\gamma_{t,x,\xi}(0)+s\gamma_{t,x,\xi}(t)$ (see (18)
). Note that

$\begin{array}[]{l}\int_{\alpha_{t,x,\xi}}pdq=\int_{0}^{1}((1-s)p_{t,x,\xi}(0)+sp_{t,x,\xi}(t))d(1-s)q_{t,x,\xi}(0)+sq_{t,x,\xi}(t)\\\
\\\
=\left(\int_{0}^{1}((1-s)p_{t,x,\xi}(0)+sp_{t,x,\xi}(t))ds\right)\cdot\left(q_{t,x,\xi}(t)-q_{t,x,\xi}(0)\right)\\\
\\\ =\langle\xi,q_{t,x,\xi}(t)-q_{t,x,\xi}(0)\rangle.\end{array}$

As mentioned below (18), $\hat{\gamma}_{t,x,\xi}$ bounds the two-dimensional
surface $\beta_{t,x,\xi}$ consisting of line segments joining $(x,\xi)$ to
points of the Hamilton orbit, the integral (18) equals the oriented area
$\int_{\beta_{t,x,\xi}}\omega$ where $\omega=dp\wedge dq$ is the standard
symplectic form of $T^{*}{\mathbb{R}}^{d}$.

######  Proposition 3.2.

The Wigner function
$W_{\hbar,E,\rho}(x,\xi)=\int_{{\mathbb{R}}}\hat{\rho}(t)W_{\hbar,t}(x,\xi)e^{\frac{itE}{\hbar}}dt$
of the smoothed spectral function is a semi-classical Fourier integral kernel
with real phase,

$\Psi_{E}(t,x,\xi):=\int_{\beta_{t,x,\xi}}\omega-tH(\gamma_{t,x,\xi}(0))+tE,$
(53)

and with amplitude
$\left|\det(1+M_{j}(q_{j},p_{j},t_{j}(E))\right|^{-{\textstyle\frac{1}{2}}}$,
where $q_{j},p_{j},t_{j}(E)$ are as in Proposition 0.3 and Proposition 0.5.
For $(x,\xi)$ near $\Sigma_{E}$, the critical point in $t$ of the phase occur
at times $t$ such that there exist $(q,p)\in\Sigma_{E}$ for which
$(x,\xi)=\Theta^{t}(q,p)$. If $t_{\pm}(E,x,\xi)$ are the times $t$ such that
$\Theta^{t}(q,p)=(x,\xi)$ has a solution, then at a critical point,
$\frac{\partial^{2}}{\partial
t^{2}}\Psi_{E}(t)=-dH(\gamma_{t,x,\xi}(0))(J_{t,x,\xi}(0))=(\frac{dt}{dE})^{-1}$.

###### Proof.

The equation (53) is the same as (52). We start with a general principle: Let
$f:{\mathbb{R}}_{x}^{n}\times{\mathbb{R}}_{y}^{m}\to{\mathbb{R}}$ be a smooth
function, and consider the critical point equation $d_{x,y}f(x,y)=0$. Let
$(a,b)$ be a critical point such that the partial Hessian
$(\frac{\partial^{2}}{\partial y_{j}\partial y_{k}}f)(a,b)$ is non degenerate.
Then by the implicit function theorem, there exists an open set
$U\subset{\mathbb{R}}^{n}$ containing $a$ and a unique differentiable function
$g:U\to{\mathbb{R}}^{m}$ such that $g(a)=b$ and $d_{y}f(x,g(x))=0$ identically
in $U$. Moreover, the critical point equations $d_{x}f(x,g(x))=0$ and
$d_{y}f(x,g(x))=0$ are equivalent in $U$ to $d_{x,y}f(x,y)=0$. Indeed,
$\frac{\partial}{\partial x}f(x,g(x))=\frac{\partial f}{\partial
x}(x,g(x))+\frac{\partial f}{\partial y}(x,g(x))\frac{\partial g(x)}{\partial
x}.$ The second term is zero in $U$ by definition of $g(x)$, and then the
first term must be zero when the left side is zero.

We apply this to the case where $n=1,m=3d$ and $f(t,q,p,v)$ of the phase (37).
It follows from Proposition 2.1 that the non-degneracy condition is satisfied
as long as $D\Theta^{t}$ is non-singular, i.e. $D\Theta^{t}$ does not have
$-1$ as an eigenvalue. We then get locally defined branches
$g:{\mathbb{R}}\to{\mathbb{R}}^{3d}$, $g(t)=(q_{t},p_{t},v_{t})$ such that
$d_{q,p,v}\Psi(t,(q_{t},p_{t},v_{t}))=0$. The phase of the additional $dt$
integral is $\Psi(t,(q_{t},p_{t},v_{t}))+tE$. By the general principal, its
critical points are the same as for those of $d_{t,q,p,v}\Psi=0$ and by
Proposition 3.1 these are given by the equations stated there.

####  3.1.1. Calculation of $\frac{d}{dt}\Psi_{E}(t_{c})$: Reynolds formula

A direct proof without using implicit functions can be obtained from the
Reynolds transport formula for the first term of (52),

$\frac{d}{dt}\int_{\beta_{t,x,\xi}}\omega=\int_{\partial\beta_{t,x,\xi}}\iota_{J_{t,x,\xi}}\omega=\int_{\hat{\gamma}}\iota_{J_{t,x,\xi}}\omega.$

Here, $J_{t,x,\xi}$ is the variational (Jacobi) vector field along
$\partial\beta_{t,x,\xi}=\hat{\gamma}_{t,x,\xi}$ obtained by varying
$\gamma_{t,x,\xi}$ and the chord $\alpha_{t,x,\xi}$. $J_{t,x,\xi}$ is
determined by $\gamma_{t,x,\xi}(0)$, since
$\Phi^{s}\gamma_{t,x,\xi}(0)=\gamma_{t,x,\xi}(s).$ Hence,

$J_{t,x,\xi}(s):=\frac{\partial}{\partial
t}\Phi^{s}\gamma_{t,x,\xi}(0)=D_{\gamma_{t,x,\xi}(0)}\Phi^{s}\left(\frac{d}{dt}\gamma_{t,x,\xi}(0)\right).$

$J_{t,x,\xi}(s)$ is a Jacobi field along $\gamma_{t,x,\xi}(s)$, i.e. the
variational vector field obtained by varying a curve of of Hamilton orbits. It
is not a Jacobi field on $\alpha_{t,x,\xi}$ since these are not extremals, but
for brevity we continue to call it a Jacobi field. We compute the integral by
writing $\hat{\gamma}_{t,x,\xi}=\gamma_{t,x,\xi}\cup\alpha_{t,x,\xi}$ and
studying each integral separately. Next, recall that $\Xi_{H}$ is the Hamilton
vector field of $H$. Since
$\gamma_{t,x,\xi}^{\prime}(s)=\Xi_{H}(\gamma_{t,x,\xi}(s))$, we find

$\begin{array}[]{l}\int_{\gamma_{t,x,\xi}}\iota_{J_{t,x,\xi}}\omega=\int_{0}^{t}\omega(J_{t,x,\xi}(s),\Xi)ds\\\
\\\
=\int_{0}^{t}dH(J_{t,x,\xi}(s))=\int_{0}^{t}(\Phi^{s})^{*}dH(J_{t,x,\xi}(0))ds=tdH(J_{t,x,\xi}(0)).\end{array}$
(54)

On the other hand, since
$\frac{d}{ds}\alpha_{t,x,\xi}(s)=\frac{d}{ds}((1-s)\gamma_{t,x,\xi}(0)+s\gamma_{t,x,\xi}(t))=\gamma_{t,x,\xi}(t)-\gamma_{t,x,\xi}(0),$
and since along $\alpha_{t,x,\xi}$,
$J_{t,x,\xi}=(1-s)J_{t,x,\xi}(0)+sJ_{t,x,\xi}(t),$ we have

$\begin{array}[]{lll}\int_{\alpha_{t,x,\xi}}\iota_{J_{t,x,\xi}}\omega=\int_{0}^{1}\omega(J_{t,x,\xi}(s),\alpha_{t,x,\xi}^{\prime}(s))ds&=&\omega({\textstyle\frac{1}{2}}(J_{t,x,\xi}(0)+J_{t,x,\xi}(t)),\gamma_{t,x,\xi}(t)-\gamma_{t,x,\xi}(0))=0,\end{array}$

since

${\textstyle\frac{1}{2}}(J_{t,x,\xi}(0)+J_{t,x,\xi}(t))={\textstyle\frac{1}{2}}\frac{d}{dt}(\gamma_{t,x,\xi}(0)+\gamma_{t,x,\xi}(t))=\frac{d}{dt}(x,\xi)=0.$

That leaves the second term of (52),

$\frac{d}{dt}\int_{\gamma_{t,x,\xi}}Hds=\frac{d}{dt}tH(\gamma_{t,x,\xi}(0))=H(\gamma_{t,x,\xi}(0))+tdH(J_{t,x,\xi}(0)).$
(55)

It follows from (54) - (55) that

$\frac{\partial}{\partial t}\Psi_{E}=-H(\gamma_{t,x,\xi}(0))+E,$

proving Proposition 3.2.

####  3.1.2.  Calculation of $\Psi_{E}^{\prime\prime}(t_{c})$.

We differentiate $\frac{\partial}{\partial
t}\Psi_{E}=-H(\gamma_{t,x,\xi}(0))+E$ once more to get

$\frac{\partial^{2}}{\partial
t^{2}}\Psi_{E}=-\frac{d}{dt}H(\gamma_{t,x,\xi}(0))=-dH(\gamma_{t,x,\xi}(0))(J_{t,x,\xi}(0)).$

We claim that if $t(E,x,\xi)$ is implicitly one of the solutions of
$\Theta^{t}(q,p)=(x,\xi)$ with $(q,p)=\gamma_{t,x,\xi}(0)\in\Sigma_{E}$ then

$\left(\frac{dt(E,x,\xi)}{dE}\right)^{-1}=-dH(\gamma_{t,x,\xi}(0))(J_{t,x,\xi}(0))|_{t=t(E,x,\xi)}.$
(56)

This follows because $t_{\pm}(E,x,\xi)$ may be defined as the locally unique
pair of solutions near $t=0$ of the equation $H((\Theta^{t})^{-1}(x,\xi))=E$.
Thus,

$\frac{d}{dE}H((\Theta^{t})^{-1}(x,\xi))=-dH_{(\Theta^{t})^{-1}(x,\xi)}\frac{d}{dt}((\Theta{t})^{-1}(x,\xi)\cdot\frac{dt}{dE}=1.$

Here, as above, $(\Theta^{t})^{-1}$ is multi-valued, so that
$\Theta^{t})^{-1}(x,\xi)$ is a set but near $\Sigma_{E}$ it is double-valued
and we mean that the equations hold for either branch. One such branch is
given by $(\Theta^{t})^{-1}(x,\xi)=\gamma_{t,x,\xi}(0)$ (the other being
$g_{t,x,\xi}(t)$ and $\frac{d}{dt}((\Theta^{t})^{-1}(x,\xi)=J_{t,x,\xi}(0)$
(resp. $J_{t,x,\xi}(t)$. This proves (56). Regarding the amplitude, after
eliminating $(q,p,v)$ in Proposition 2.2, the amplitude

$a_{0}=\left(\det\left(A_{t}+D_{t}+i(B_{t}-C_{t})\right)\right)^{1/2}\exp(-itH(q,p))$

gets multiplied by
$\left(\det\left(\text{Hess}(\Psi)\right)(q_{c},p_{c},v_{c})\right)^{-{\textstyle\frac{1}{2}}}$
where the Hessian is,

$(-1)^{d}\cdot\det(1+M_{t}(q_{c},p_{c}))\det\left(A_{t}(q_{c},p_{c})+D_{t}(q_{c},p_{c})+i(B_{t}(q_{c},p_{c})-C_{t}(q_{c},p_{c}))\right).$

This cancels all factors of the amplitude except for $\det(I+M_{t})$. This
concludes the proof of Proposition 3.2.

∎

###  3.2. Background on folds

In the next Section 3.3 we discuss fold singularities of Lagrangian
submanifolds associated to the Wigner spectral function. In preparation, we
review the definitions and properties of folding maps $f:X\to Y$ for general
manifolds. References include [G, HoIII].

We follow the exposition in [G, Appendix 1, Page 109]. Let $f:X\to Y$ be a
smooth map of n-dimensional manifolds and let $S\subset X$ be a hypersurface
(codimension one submanifold) of $X$. Let $dV_{Y}$ be a volume form on $Y$.

Then $f:X\to Y$ is a folding map with $S$ as the fold locus if the following
are satisfied,

  * •

$S$ is set of critical points of $f$, i.e. $D_{s}f:T_{s}X\to T_{f(s)}Y$ fails
to be surjective ;

  * •

At every $s\in S,$ the kernel $\ker D_{s}f$ is one-dimensional and is
transversal to $T_{s}S$;

  * •

$f^{*}dV_{Y}$ vanishes to first order at each $s\in S$, i.e. $\det D_{s}f$
vanishes to first order on $S$.

In this case, one can find coordinates $x_{1},\dots,x_{n}$ on $X$ and
coordinates $y_{1},\dots,y_{n}$ on $Y$ so that $S=\\{x_{n}=0\\}$ and
$f(S)=\\{y_{n}=0\\}$ and so that $f^{*}y_{j}=x_{j}$, for $j\leq n-1$ and
$f^{*}y_{n}=x_{n}^{2}$. Associated to the folding map is a canonical
involution $\sigma:V\to V$ in a neighborhood $V$ of $S$ in $X$, fixing $S$ and
having the local form
$\sigma(x_{1},\dots,x_{n-1},x_{n})=(x_{1}\dots,x_{n-1},-x_{n})$.

###  3.3.  Analysis of the fold singularity as a singularity of the projection
of the Lagrangian

The time integral is the semi-classical Fourier transform of the $dqdpdv$
integral, and the associated Lagrangian submanifold generated by the phase is
the classical analogue of the Fourier transform, $(t,\tau)\to(-\tau,t)$
applied to (48) with $\tau=E$, namely

$\begin{array}[]{lll}\Lambda_{E}&=&\\{(x,\xi,d_{x,\xi}\Psi_{E}):d_{q,p,v,t}\widetilde{\Psi}_{E}=0\\}\subset
T^{*}(T^{*}{\mathbb{R}}^{d}))\\\ &&\\\
&=&\\{(x,\xi,p_{t}-p,q-q_{t}):x=\frac{q+q_{t}}{2},\quad\xi=\frac{p+p_{t}}{2},H(q,p)=E\\}\\\
&&\\\
&=&\\{(x,\xi,-2(x-q),2(\xi-p)):x=\frac{q+q_{t}}{2},\quad\xi=\frac{p+p_{t}}{2},H(q,p)=E\\}.\end{array}$
(57)

Note that the change of the Lagrangian submanifold $\Lambda_{t}\to\Lambda_{E}$
of (48) under the $dt$ integral corresponds to the symplectic map
$(t,\tau)\to(-\tau,t)$ in (49). The phase $\widetilde{\Psi}_{E}$ induces the
Lagrangian immersion,

$\begin{array}[]{lll}\iota_{\widetilde{\Psi}_{E}}:(t,q,p,x,\xi)\in
C_{\Psi_{E}}&\to&(\frac{q+q_{t}}{2},\frac{p+p_{t}}{2},q-q_{t},p_{t}-p)\\\
&&\\\ &=&(x,\xi,-2(x-q),2(\xi-p)),\end{array}$ (58)

which is closely related to the map (12).

The set of $\lambda\in\Lambda_{E,x,\xi}$ on which the natural projection

$\pi:\Lambda_{E}\to T^{*}{\mathbb{R}}^{d}$ (59)

is a submersion is called the regular set and its complement where
$D_{\lambda}\pi$ is singular is called the (Maslov) singular cycle. The image
of the Maslov singular cycle under $\pi$ is called the ‘caustic’. When
$\Sigma_{E}$ is strictly convex, the image is $B_{E}$ and there is an apparent
fold over its boundary $\Sigma_{E}$. We prove this folding statement along
with the closely related Proposition 0.4.

######  Proposition 3.3.

  * •

(i) If $\Sigma_{E}$ is convex, the map (59) $\pi:\Lambda_{E}\to B_{E}$ has a
fold singularity along the zero section $\Sigma_{E}\times\\{0\\}\subset
T^{*}(T^{*}{\mathbb{R}}^{d}))$.

  * •

(ii) The map $\widetilde{\Theta}:{\mathbb{R}}\times\Sigma_{E}\to B_{E}$ with
$\widetilde{\Theta}(t,q,p)=\Theta^{t}(q,p)$ (12) fixes the diagonal when $t=0$
and the kernel of its derivative is $2\frac{\partial}{\partial t}-\Xi_{H},$
i.e. $D\widetilde{\Theta}\left(2\frac{\partial}{\partial
t}-\Xi_{H}\right)|_{t=0}=0$.

This is the folding result relevant to the asymptotics of the smoothed Weyl-
Wigner asymptotics of Theorem 0.2. First we prove Proposition 0.4.

###### Proof.

We first relate (59) and (12) through the diagram,

$\begin{array}[]{llll}\iota_{\widetilde{\Psi}_{E}}:C_{\widetilde{\Psi}_{E}}&\rightarrow&\Lambda_{E}\\\
&&\\\ \pi\downarrow&&\downarrow\pi\\\ &&\\\
\Theta:{\mathbb{R}}\times\Sigma_{E}&\rightarrow&B_{E},\end{array}$

where in each case $\pi$ denotes the natural projection to the base. This
shows that $C_{\Psi_{E}}$ is the fiber-product
$\Lambda_{E}\times_{B_{E}}({\mathbb{R}}\times\Sigma_{E})$ with respect to
$B_{E}$, i.e. pairs of points in the product space which get mapped to the
same point in $B_{E}$. This description guides the analysis of the fibers of
the maps in the diagram.

It is evident that the projection
$\pi:C_{\widetilde{\Psi}_{E}}\to({\mathbb{R}}\times\Sigma_{E})$ is a
diffeomorphism, since $(t,q,p;x,\xi)\in C_{\widetilde{\Psi}_{E}}$ occurs only
when $(x,\xi)=\Theta^{t}(q,p)$, so $(x,\xi)$ is redundant. From the formula in
(58) we see that $(q,p)$ are uniquely determined by the point
$(x,\xi,-2(x-q),2(\xi-p))\in\Lambda_{E}$; since $t$ is defined by
$q_{t}=2x-q,p_{t}=2\xi-p$ , the point $(t,q,p,x,\xi)\in
C_{\widetilde{\Psi}_{E}}$ is uniquely determined. Hence, also
$\iota_{\widetilde{\Psi}_{E}}$ is a diffeomorphism. It follows that the fold
singularities of $\pi:\Lambda_{E}\to B_{E}$ and
$\Psi:{\mathbb{R}}\times\Sigma_{E}\to B_{E}$ are equivalent.

The fiber of the projection (59) is,

$\begin{array}[]{lll}\pi^{-1}(x,\xi)&=&\\{(x,\xi,q-q_{t},p_{t}-p):H(q,p)=E,(x,\xi)=(\frac{q+q_{t}}{2},\frac{p+p_{t}}{2})\\}\\\
&&\\\
&=&\\{((x,\xi),q-q_{t},p_{t}-p)),\;(q,p)\in(\Theta^{t})^{-1}(x,\xi)\cap\Sigma_{E})\\}.\end{array}$

It is evident that the inverse image of the boundary under (59) is the zero
section $0_{T^{*}(T^{*}{\mathbb{R}}^{d})}$ over $\Sigma_{E}$. It is the fold
locus of the the map $\Theta:{\mathbb{R}}\times\Sigma_{E}$ at $t=0$ and
therefore of the natural projection, $\pi:\Lambda_{E}\to
T^{*}{\mathbb{R}}^{d}.$ It is the fixed point set of the involution
$\sigma:(t,q,p)\to(-t,\Phi^{t}(q,p))$. reversing the endpoints of the chord
whose midpoint is $(x,\xi)$.

To verify that $\Theta$ has a fold singularity along
$\\{0\\}\times\Sigma_{E}$, i.e. satisfies the conditions for a fold in Section
3.2 Consider the curves,

$\alpha(s)=\alpha_{q,p}(s):=(2s,\Phi^{-s}(q,p)):{\mathbb{R}}\to{\mathbb{R}}\times\Sigma_{E}.$

Then $\alpha(0)=(0,q,p)$, $\alpha^{\prime}(0)=2\frac{\partial}{\partial
t}-\Xi_{H}$ and (referring to (12))

$\widetilde{\Theta}(\alpha(s))={\textstyle\frac{1}{2}}(\Phi^{-s}(q,p)+\Phi^{2s}(\Phi^{-s}(q,p))={\textstyle\frac{1}{2}}(\Phi^{-s}(q,p)+\Phi^{s}(q,p)),$

hence

$(D_{(0,q,p)}\widetilde{\Theta})\alpha^{\prime}(0)=\frac{d}{ds}|_{s=0}\widetilde{\Theta}(\alpha(s))=-\Xi_{H}(q,p)+\Xi_{H}(q,p)=0.$

we see that $\widetilde{\Theta}$ has a fold along the points
$\\{0\\}\times\Sigma_{E}$.

∎

######  Corollary 3.4.

If $\Sigma_{E}$ is convex, there exists $\varepsilon>0$ so that the map
$\Theta$ is 2-1 on $(-\varepsilon,0)\cup(0,\varepsilon)\times\Sigma_{E}$, i.e.
if $|t|\leq\varepsilon$, $(x,\xi)\in B_{E}$ and
$d((x,\xi),\Sigma_{E})<\varepsilon$ then there is a unique Hamiltonian arc of
time $t$ (up to orientation reversal) for which $(x,\xi)$ is the chordal
midpoint.

The statement follows from the local description of fold singularities in
Section 3.2. As noted above, the canonical involution is orientation reversal,
$\sigma(t,q,p)=(-t,\Theta^{t}(q,p))$, i.e.
$\Theta^{t}(q,p)={\textstyle\frac{1}{2}}((q,p)+\Phi^{t}(q,p))=\Theta^{-t}(\Phi^{t}(q,p))={\textstyle\frac{1}{2}}(\Phi^{t}(q,p)+\Phi^{-t}\Phi^{t}(q,p))$

On the other hand, if $\Sigma_{E}$ is convex, then there always exists at
least one closed orbit of the Hamiltonian flow on $\Sigma_{E}$ if $H$ is
convex [Rab, W]. If $(q,p)$ is a periodic point of period $T$,then
$\iota_{\Psi}(0,q,p)=\iota_{\Psi}(T,q,p)$. In this case,
$\iota_{\Psi}^{-1}(x,\xi)=\\{(kT,qp)\\}$ has a discrete inverse image. But it
is also possible to have an inverse image of positive dimension, for instance
if $V(x)=||x||^{2}$ and $(x,\xi)=(0,0)$ for instance if periodic orbits come
in positive dimensional families. Another such fiber occurs at a non-period of
the isotropic oscillator: at time $T=\pi$, $\Phi^{T}(q,p)=-(q,p)$ and so
$\iota_{\Psi}^{-1}(0,0)=\\{T\\}\times\Sigma_{E}.$

Remark: The phase function $\Psi_{E}$ of (53) is equivalent to the phase
function $\widetilde{\Psi}_{E}$ in the sense of [HoIII, GSt], namely they both
parametrize the same Lagrangian.

##  4\.  Proof of Theorem 0.2

The proof of Theorem 0.2 is based on a special case of the Malgrange
preparation theorem which was proved earlier by Chester-Friedmann-Ursell
[CFU]. The Malgrange preparation theorem asserts that if $f(t,x)$ is a smooth
function of $(t,x)\in{\mathbb{R}}\times{\mathbb{R}}^{n}$, and if $k$ is the
positive integer so that

$f(0,0)=0,\frac{\partial}{\partial
t}f(0,0)=0,\;\;\cdots,\frac{\partial^{k-1}}{\partial
t^{k-1}}f(0,0)=0,\frac{\partial^{k}}{\partial t^{k}}f(0,0)\not=0,$ (60)

then there exists a smooth function $c(t,x)$ which is non-vanishing near
$(0,0)$ and smooth functions $a_{j}(x),j=1,\dots,k-1$ so that

$f(t,x)=c(t,x)\left(t^{k}+a_{k-1}(x)t^{k-1}+\cdots a_{0}(x)\right).$

We only apply this theorem when $k=3$ and use a stronger form from [CFU, L61]
and also proved in detail in [HoI, Theorem 7.5.13] (see also [GSt, Page 444]).
With no loss of generality assume that $\frac{\partial^{3}f}{\partial
t^{3}}(0,0)>0$. Then, there exists a $C^{\infty}$ function $T=T(t,x)$ near
$(0,0)$ with $T(0,0)=0,\frac{\partial T}{\partial t}(0,0)>0$ and $C^{\infty}$
functions $\rho,\mu$ near $(0,0)$ such that

$f(t,x)=\frac{T^{3}}{3}+a(x)T+\mu(x),$ (61)

with $a(0)=0,\mu(0)=f(0,0)$.

The detailed statement of the result is given in [HoI, Theorem 7.7.19].

######  Proposition 4.1.

Let $f(t,x)$ be real valued on ${\mathbb{R}}^{n+1}$ satisfying (60) with
$k=3$. Then there there exist $C^{\infty}$ functions $u_{0,\nu},u_{1,\nu}$
such that, in the notation above,

$\begin{array}[]{lll}\int u(t,x)e^{i\tau f(t,x)}dt&\sim&e^{i\tau
b(x)}\tau^{-\frac{n}{2}}\left(\tau^{-\frac{1}{3}}\rm{Ai}(\tau^{\frac{2}{3}}a(x))\sum_{\nu=0}^{\infty}u_{0,\nu}(x)\tau^{-\nu}\right)+\\\
&&\\\ &+&e^{i\tau
b(x)}\tau^{-\frac{n}{2}}\left(\tau^{-\frac{2}{3}}\rm{Ai}^{\prime}(\tau^{\frac{2}{3}}a(x))\sum_{\nu=0}^{\infty}u_{1,\nu}(x)\tau^{-\nu}\right).\end{array}$

Remark: Note that $T^{2}+a(x)=0$, so that $a(x)<0$. In [GSt, Page 444]), the
normal form is stated in the form, $\frac{T^{3}}{3}-\rho(x)T+\mu(x).$ Hence,
in what follows, $a(x)=-\rho(x)$.

We apply the result to the oscillatory integral
$W_{\hbar,E,\rho}(x,\xi)=\int_{{\mathbb{R}}^{3d}}\int_{{\mathbb{R}}}\hat{\rho}(t)W_{\hbar,t}(x,\xi)e^{\frac{itE}{\hbar}}dqdpdvdt$
and to the one-dimensional reduction in Section 3.1.

For the full ${\mathbb{R}}^{3d+1}$ dimensional integral, we use

######  Proposition 4.2.

Let $f(y,x)$ be complex valued function with positive imaginary part on
${\mathbb{R}}^{n+m}$. Assume that ${\operatorname{Im}\,}f(0,0)=0$, that
$\rm{Hess}({\operatorname{Im}\,}f)(0,0)$ is non-degenerate, and that

  * •

(i) ${\operatorname{Re}\,}f_{x}^{\prime}(0,0)=0.$

  * •

(ii) $\rm{Rank}{\operatorname{Re}\,}f_{xx}^{\prime\prime}(0,0)=n-1.$

  * •

(iii) $\langle X,\partial/\partial
x\rangle^{3}{\operatorname{Re}\,}f(0,0)\not=0$ if
$0\not=X\in\ker{\operatorname{Re}\,}f_{xx}^{\prime\prime}(0,0)$.

  * •

(iv) $f(x_{1},0)$ is real valued.

Then there exist $C^{\infty}$ functions $\rho(x),\mu(x)$ near $(0,0)$ such
that $\rho(0)=0,\mu(0)=f(0)$ and such that, for any $u\in C_{0}^{\infty}$ with
$\rm{Supp}u$ sufficiently close to $(0,0)$, there exist $C^{\infty}$ fuctions
$u_{0,\nu},u_{1,\nu}$ such that,

$\begin{array}[]{lll}\int u(y,x)e^{i\tau
f(y,x)}dx&\sim&e^{i\tau\mu(y)}\tau^{-\frac{n}{2}}\left(\tau^{-\frac{1}{3}}\rm{Ai}(-\tau^{\frac{2}{3}}\rho(y))\sum_{\nu=0}^{\infty}u_{0,\nu}(y)\tau^{-\nu}\right)+\\\
&&\\\
&+&e^{i\tau\mu(y)}\tau^{-\frac{n}{2}}\left(\tau^{-\frac{2}{3}}\rm{Ai}^{\prime}(-\tau^{\frac{2}{3}}\rho(y))\sum_{\nu=0}^{\infty}u_{1,\nu}(y)\tau^{-\nu}\right).\end{array}$

The proof is similar to that of [HoI, Theorem 7.7.18] and also in the
reduction to one integral. One integrates first in $(x_{2},\dots,x_{n})$ where
stationary phase applies, then applies Proposition 4.1 to the remaining one
dimensional integral. For the sake of completeness, we provide more details as
in [HoI, Theorem 7.7. 19], although it explicitly assumed there that the phase
is real-valued. We label the coordinates so that $x_{1}=t$ and
$(q,p,v)=(x_{2},\dots,x_{n})$ with $n=3d+1$. Let
$f={\operatorname{Re}\,}\Psi$. Define $X_{j}(x_{1},y)$ for $j=2.\dots,n$ so
that the equations $\frac{\partial f(x,y)}{\partial x_{j}}=0$ determine
$x_{j}$ as functions $x_{j}=X_{j}(x_{1},y)$. Change variables in the
$dx_{2}\cdots dx_{n}$ integral to $x_{j}^{\prime}=x_{j}-X_{j}$. Then the
critical points of the $dx_{2}\cdots dx_{n}$ integral become
$x_{j}=0,j=2,\dots,n$. By applying stationary phase with a positive complex
phase to this integral, we obtain an asymptotic of the integrand with fixed
$x_{1}$ of the form,

$\tau^{-\frac{n-1}{2}}e^{i\tau f(x_{1},0)}U(x_{1},y,0).$

Here we use that the imaginary part ${\operatorname{Im}\,}\Psi$ of the phase
vanishes at the critical set.

To apply the results to our integrals, it is necessary to show that the
hypotheses of Proposition 4.1 hold and then to calculate the coefficients
$\rho,\mu$ in (61) and the principal coefficients $u_{0,0},u_{1,0}$ in
Proposition 4.1. The following general calculation is due to [CFU, Section 3];
see also [GSt, (6.7), Page 440]. By (61), and the convention that $\rho=-a$,
we may express $\varphi(t,x)=\frac{T^{3}}{3}-\rho T+\mu.$ For each critical
point $d_{t}\varphi(t,x)$ there exist two roots $t_{\pm}(x)$. By
$\varphi_{c}(t_{\pm},x)$ we denote the critical value of the phase at each
critical point. We then have,

######  Lemma 4.3.

In the notation above,

$\left\\{\begin{array}[]{l}\rho^{3/2}(x)=\frac{3}{4}(\varphi_{c}(t_{-},x)-\varphi_{c}(t_{+},x))\\\
\\\ \mu(x)=\frac{1}{2}(\varphi_{c}(t_{-},x)+\varphi_{c}(t_{+},x))\
.\end{array}\right.$

We review the proof, since we need to do the calculations.

###### Proof.

We assume that $\varphi(t,x)=\frac{T^{3}}{3}-\rho(x)T+\mu(x)$ is a smooth
expression (here we write $a=-\rho$). The critical point equation in $t$
(viewing $x$ as a parameter) is

$\frac{\partial}{\partial t}\varphi(t,x)=\frac{\partial}{\partial
T}\varphi(T,x)\frac{\partial T}{\partial t}=\frac{\partial T}{\partial
t}\left(T^{2}-\rho(x)\right)=0.$

Hence, since $\frac{\partial T}{\partial t}\neq 0$ for $t,x$ sufficiently
close to $0$, we have $T^{2}=\rho(x)$ on the critical point set, and there
exist two roots $T_{\pm}=\pm\sqrt{\rho(x)}$ when $T\not=0$. It follows that
$\frac{\partial t}{\partial T}\not=0$ when $T\not=0$. The critical values of
the phase at the two critical points are given by,

$\varphi_{c}(t_{\pm},x)=\frac{T_{\pm}^{3}}{3}-\rho(x)T_{\pm}+\mu(x)=\frac{\pm\rho^{\frac{3}{2}}}{3}-\rho(x)(\pm\rho^{{\textstyle\frac{1}{2}}})+\mu(x)=\mp\frac{2}{3}\rho^{\frac{3}{2}}+\mu(x).$

Thus,

$\varphi_{c}(t_{+},x)=-\frac{2}{3}\rho^{\frac{3}{2}}(x)+\mu(x),\;\;\varphi_{c}(t_{-},x)=+\frac{2}{3}\rho^{\frac{3}{2}}(x)+\mu(x).$

∎

###  4.1. Completion of the proof of Theorem 0.2

####  4.1.1. Calculation of $\Psi_{E}^{\prime\prime}(t_{c})$ and of
$\Psi_{E}^{\prime\prime\prime}(t_{c})$.

###### Proof.

When $t=0$ and $(x,\xi)\in\Sigma_{E}$, we have

$\frac{\partial^{2}}{\partial
t^{2}}\Psi_{E}=-\frac{d}{dt}H(\gamma_{t,x,\xi}(0))=-dH(\gamma_{t,x,\xi}(0))(J_{t,x,\xi}(0))|_{t=0,(x,\xi)\in\Sigma_{E}}$

In computing the time derivatives we may assume $(x,\xi)\in\Sigma_{E}$. But
$H(\gamma_{t,x,\xi}(0))$ has a local minimum when $t=0$ in that case. Indeed,
for any $t$ such that
$(x,\xi)={\textstyle\frac{1}{2}}(\gamma_{t,x,\xi}(0)+\gamma_{t,x,\xi}(t))$, it
is necessary that $H(\gamma_{t,x,\xi}(0))\geq H(x,\xi)=E.$ It follows also
that $\frac{d}{dt}(\Theta^{t})^{-1}(x,\xi)|_{t=0}\in T\Sigma_{E}$, and that

$dH{(\gamma_{t,x,\xi}(0))}|_{t=0}=0,\;\;(x,\xi)\in\Sigma_{E}.$ (62)

We now show that

$\frac{\partial^{3}}{\partial
t^{3}}\Psi_{E}|_{t=0}=-\frac{d^{2}}{dt^{2}}H(\gamma_{t,x,\xi}(0))|_{t=0}\not=0.$

We have $\Theta^{t}(\gamma_{t,x,\xi}(0))=(x,\xi)$ and
$\frac{d}{dt}\Theta^{t}={\textstyle\frac{1}{2}}\frac{d}{dt}\Phi^{t}={\textstyle\frac{1}{2}}\Xi_{H}$.
Then,
$0=\frac{d}{dt}\Theta^{t}(\gamma_{t,x,\xi}(0))={\textstyle\frac{1}{2}}\Xi_{H}(x,\xi)+D\Theta^{t}\frac{d}{dt}\gamma_{t,x,\xi}|_{t=0}$.
If the second term is zero, we would get $\Xi_{H}(x,\xi)=0$, a contradiction.
In fact, $D\Theta^{t}\frac{d}{dt}|_{t=0}=Id$ and we get
$J_{0,x,\xi}(0)=-{\textstyle\frac{1}{2}}\Xi_{H}(x,\xi)$. But then, by strict
convexity of $H$,

$\frac{d^{2}}{dt^{2}}H(\gamma_{t,x,\xi}(0))|_{t=0}=\rm{Hess}_{x,\xi}H(J_{0,x,\xi}(0),J_{0,x,\xi}(0))>0.$

Note that the other term
$dH_{(\gamma_{t,x,\xi}(0))}\frac{d^{2}}{dt^{2}}\gamma_{t,x,\xi}(0)=0$ by (62).
This completes the proof that the conditions (60) are satisfied.

We may therefore apply Proposition 4.1 to obtain an asymptotic expansion of
the form stated in Theorem 0.2. The remaining step is to use Lemma 4.3 to
calculate the parameters $\rho,\mu,u_{0,0},u_{1,0}$.

We therefore fix $(x,\xi)\in B_{E}$ sufficiently close to $\Sigma_{E}$ and
define $(t,q,p)\in(-\varepsilon_{0},\varepsilon_{0})\times\Sigma_{E}$ as above
by $\Psi^{t}(q,p)=(x,\xi)$. Further, we denote by $t_{\pm}(E,x,\xi)$ the two
roots of the critical point equation for the phase $\Psi_{E}(t;x,\xi)$ (see
Definition 0.1).

######  Lemma 4.4.

If $(x,\xi)$ is $\varepsilon_{0}$-close to $\Sigma_{E}$, then there exist
exactly two critical times
$t=t_{\pm}(E,x,\xi)\in(-\varepsilon_{0},\varepsilon_{0})$ (see Proposition
2.1) and they are of the form $t_{\pm}=\pm t$ where
$\Theta^{t}(q,p)=(x,\xi),H(q,p)=E.$ Moreover, the action changes sign under
the reversal of $\gamma_{t,x,\xi}$.

Indeed, for such $(x,\xi)$ there is a unique Hamiltonian arc of time $t$ with
$|t|\leq\varepsilon$ (up to the symmetry (14)), for which $(x,\xi)$ is the
midpoint of the chord with endpoints $(q,p),(q_{t},p_{t})$. The two critical
times points $(t,q,p)$ correspond to $\gamma_{t,x,\xi}$ and its orientation
reversal. Reversing the orientation means to change
$\gamma_{t,x,\xi}(s)\to\gamma^{*}_{t,x,\xi}(s):=\gamma_{t,x,\xi}(t-s)$, so
that
$\frac{d}{ds}\gamma^{*}_{t,x,\xi}(s):=-(\frac{d}{ds}\gamma_{t,x,\xi})(t-s)$.
We write $\gamma_{t,x,\xi}(s)=(q_{t,x,\xi}(s),p_{t,x,\xi}(s))$ and
$\gamma^{*}_{t,x,\xi}(s)=(q^{*}_{t,x,\xi}(s),p^{*}_{t,x,\xi}(s))=(q_{t,x,\xi}(t-s),p_{t,x,\xi}(t-s))$
and show that

$\int_{0}^{t}p_{t,x,\xi}(s))\frac{d}{ds}q_{t,x,\xi}(s))ds=-\int_{0}^{t}p^{*}_{t,x,\xi}(s))\frac{d}{ds}q^{*}_{t,x,\xi}(s))ds.$

Indeed, the right side equals

$\begin{array}[]{l}\int_{0}^{t}p_{t,x,\xi}(t-s))\frac{d}{ds}q_{t,x,\xi}(t-s))ds=-\int_{0}^{t}p_{t,x,\xi}(t-s))\dot{q}_{t,x,\xi}(t-s))ds\\\
\\\
=\int_{t}^{0}p_{t,x,\xi}(s))\dot{q}_{t,x,\xi}(s))ds=-\int_{0}^{t}p_{t,x,\xi}(s))\dot{q}_{t,x,\xi}(s))ds.\end{array}$

####  4.1.2. Calculation of $\rho$

By the first equation of Lemma 4.3, we need to calculate the “odd” part of the
phase, defined by

$\begin{array}[]{l}{\textstyle\frac{1}{2}}(\Psi_{E}(t_{+}(E,x,\xi);x,\xi)-\Psi_{E}(t_{-}(E,x,\xi);x,\xi))\\\
\\\
={\textstyle\frac{1}{2}}(S_{t_{+}(E,x,\xi)}-S_{t_{-}(E,x,\xi)})-\langle{\textstyle\frac{1}{2}}(q_{t_{+}(E,x,\xi)}-q_{t_{-}(E,x,\xi)}),\xi\rangle+(t_{+}(E,x,\xi)-t_{-}(E,x,\xi))E,\end{array}$

where $S_{t}(q,p)=\int_{\gamma_{t,q,p}}(pdq-Hds)$ is the action integral along
the phase space trajectory $\gamma_{t,q,p}$ with initial value $(q,p)$. The“
odd” part of the action integral is

${\textstyle\frac{1}{2}}(\int_{0}^{t_{+}(E,x,\xi)}pdq-\int_{0}^{t_{-}(E,x,\xi)}pdq)={\textstyle\frac{1}{2}}\int_{t_{-}(E,x,\xi)}^{t_{+}(E,x,\xi)}pdq,$

where the integral is taken along the Hamilton path with endpoints
$(q_{t_{-}},p_{t_{-}}),(q_{t_{+}},p_{t_{+}})$ where
$t_{\pm}=t_{\pm}(E,x,\xi)$. Since $H(q_{s},p_{s})=H(q,p)$, the second term of
$S_{t}$ combines with $tE$ to produce
$(t_{+}(E,x,\xi)-t_{-}(E,x,\xi))(E-H(q,p))=0$ at the critical time. One also
has,

$\langle{\textstyle\frac{1}{2}}(q_{t_{+}(E,x,\xi)}-q_{t_{-}(E,x,\xi)}),\xi\rangle={\textstyle\frac{1}{2}}\int_{t_{-}(E,x,\xi)}^{t_{+}(E,x,\xi)}\xi\cdot
dq.$

Hence,

$\begin{array}[]{l}\frac{4}{3}\rho^{3/2}={\textstyle\frac{1}{2}}\int_{t_{-}(E,x,\xi)}^{t_{+}(E,x,\xi)}(p-\xi)\cdot
dq=\int_{\beta_{t,x,\xi}}\omega,\end{array}$ (63)

where in the second inequality we use the observation noted in (18) and in the
proof of Proposition 3.2 that this action integral is the oriented area of the
surface bounded by the oriented closed curve $\hat{\gamma}_{t,x,\xi}$
consisting of the Hamilton arc followed by the chord.

####  4.1.3. Calculation that $\mu=0$

We claim that $\mu=0$. This follows from Lemma 4.4, since the two critical
times correspond to the unique Hamilton orbit and its orientation reversal.
The same is true of $\langle q_{t}-q,\xi\rangle$, which is clearly odd under
orientation reversal.

####  4.1.4.  Calculation of $u_{0,0}$

We claim that

$u_{0,0}=\sqrt{\pi}\rho^{1/4}\left|\frac{dt_{j}}{dE}\det(1+M_{j}(x,t_{j}(E))\right|^{-{\textstyle\frac{1}{2}}},$
(64)

This follows from a standard matching result, using the stationary phase
asymptotics of the Airy function, and is given in detail in [GSt, Page 459].
We wish to match the Airy asymptotic expansion [GSt, Page 442, (6.9)],

$e^{i\tau\sigma}\\{\frac{g_{0}}{\tau^{1/3}}\rm{Ai}(-\tau^{2/3}\rho)+\frac{g_{1}}{i\tau^{2/3}}\rm{Ai}^{\prime}(-\tau^{2/3}\rho)\\}$

to the stationary phase of the same integral for for $\tau^{2/3}\rho>>0$,
given by

$\frac{\tau^{-1/3}e^{i\tau\sigma}}{\sqrt{\pi}[\tau^{2/3}\rho]^{1/4}}\left(g_{0}\cos(\frac{2\tau\rho^{2/3}}{3}-\pi/4))-g_{1}\rho^{1/2}\sin(\frac{2\tau\rho^{2/3}}{3}-\pi/4)\right).$

See [GSt, Page 443, (6.15)]. The coefficients of the expansions match if and
only if (in an obvious notation)

$\rm{Airy}\;\;g_{0}\iff\rm{StPh}\;\;\frac{1}{\sqrt{\pi}}\rho^{-1/4}g_{0}.$

Note that
$\tau^{-1/3}[\tau^{-2/3}]^{1/4}=\tau^{-1/3-1/6}=\tau^{-{\textstyle\frac{1}{2}}}.$
The stationary phase calculation uses that,

$\rm{Ai}(t)\simeq\frac{1}{\sqrt{\pi}t^{1/4}}\cos(\frac{2}{3}t^{3/2}-\frac{\pi}{4}).$

For stationary phase points very close to $\Sigma_{E}$ one gets by Proposition
0.5 the amplitude

$\frac{2^{d+1}}{\sqrt{2\pi\hbar}}\sum_{j}e^{-\varepsilon
t_{j}/\hbar}A_{j}(x,E)\cos\left(\frac{S_{j}(x,E)}{\hbar}+m_{j}\right),$

and applying this to the critical points $(t,q,p)$ corresponding to $(x,\xi)$
close to $\Sigma_{E}$ gives that

$g_{0}=\left|\frac{dt_{j}}{dE}\det(1+M_{j}(x,t_{j}(E))^{-1}\right|^{{\textstyle\frac{1}{2}}},$

proving (64) and concluding the proof of Theorem 0.2.

∎

##  5\.  Example: Isotropic harmonic oscillator.

In this section, we evaluate all of the objects above in the simplest case of
the isotropic harmonic oscillator,
$H(q,p)={\textstyle\frac{1}{2}}(||p||^{2}+||q||^{2})$ and check the
consistency of (7) and (17).

The first simplifying feature is that the Hamilton flow $\Phi^{t}$, resp. the
midpoint map $\Theta^{t}={\textstyle\frac{1}{2}}(I+\Phi^{t})$ are linear; they
are given respectively by,

$M_{t}=\Phi^{t}=\bigoplus\begin{pmatrix}\cos t&\sin t\\\ &\\\ -\sin t&\cos
t\end{pmatrix},\;\;\Theta^{-t}=\bigoplus\frac{2}{1+\cos
t}\begin{pmatrix}(1+\cos t)&-\sin t\\\ &\\\ \sin t&(1+\cos t),\end{pmatrix}.$
(65)

i.e. $(q_{t},p_{t})=(\cos tq+\sin tp,-\sin tq+\cos tp)$ and on the critical
point set of the $(q,p,v)$ integral,

$\begin{pmatrix}q\\\ \\\ p\end{pmatrix}=\Theta^{-t}(x,\xi)=\frac{2}{1+\cos
t}\begin{pmatrix}(1+\cos t)x-\sin t\xi\\\ \\\ \sin tx+(1+\cos
t)\xi\end{pmatrix}.$ (66)

Given $(x,\xi)$ we can solve for the critical time $t=t(x,\xi,E)$ at which
there exists $(q,p)\in\Sigma_{E}$ for which $(x,\xi)$ is the midpoint of the
Hamilton orbit of time $t$ starting at $(q,p)$. Namely,

$\cos\frac{t}{2}=\frac{\sqrt{H(x,\xi)}}{\sqrt{H(q,p)}}=\sqrt{s},\;\;(H(q,p)=E)$

where in the second equality we use the notation
$s=\frac{H(x,\xi)}{E}\in(0,1]$ in (8). There are two solutions
$t=t_{\pm}(E,x,\xi)$ with $t_{-}=-t_{+}$.

####  5.0.1. Simple geometric formulae for spheres

Let $C_{R}$ be the circle of radius $R$ around the origin. Since the
Hamiltonian orbits of an isotropic oscillator are great circles of the sphere
of radius $\sqrt{2E}$, we may regard them as circular arcs on $C_{\sqrt{2E}}$.
The travel time $t$ of the Hamiltonian flow $\Phi^{t}$ on this arc has length
$Rt=\sqrt{2E}\;t$.

The chord between endpoints of the Hamilton arc of time $t$ has length
${\textstyle\frac{1}{2}}\sqrt{R^{2}-r^{2}}$ where $r$ is the distance from the
origin to the midpoint of the chord. Since $(x,\xi)$ is the midpoint of the
chord, $r=\sqrt{2H(x,\xi)}$ and the length of the chord is
${\textstyle\frac{1}{2}}\sqrt{2E-2H(x,\xi)}$. The area of the circular sector
bounded by the Hamilton arc and the chord is given by,

$A=R^{2}\cos^{-1}(\frac{r}{R})-rR\sqrt{1-\frac{r^{2}}{R^{2}}}=(2E)\cos^{-1}\sqrt{\frac{H(x,\xi)}{E}}-2\sqrt{H(x,\xi)E}\sqrt{1-\frac{H(x,\xi)}{E}}.$

In the In the notation of (8) with $\sqrt{s}=\frac{r}{R}$, and comparing with
(63),

$A=R^{2}\left(\cos^{-1}\sqrt{s}-\sqrt{s}\sqrt{1-s}\right)=\frac{4}{3}\rho^{3/2}.$

On the other hand,
$\beta(s):={\textstyle\frac{1}{2}}[\cos^{-1}\sqrt{s}-\sqrt{s-s^{2}}],$ so
$A=2R^{2}\beta$ and $B^{2}=-[(2R^{2})^{-1}A]^{\frac{2}{3}}$. We recall that
$\hbar=\hbar_{N}(E):=\frac{E}{N+\frac{d}{2}}$ and that the argument of the
Airy function in (7) is $\hbar_{E}^{-2/3}B^{2}\left(s\right)$. We cannot and
do not use the same convention for a general Hamiltonian, so that the factor
$[(2R^{2})^{-1}]^{\frac{2}{3}}=E^{-\frac{2}{3}}$ accounts for the change in
the Airy argument between $\hbar_{E}^{-2/3}B^{2}\left(s\right)$ in (7) and
$\hbar^{-\frac{2}{3}}\rho(x,\xi))$ (17).

###  5.1. Simplified asymptotics

For $(x,\xi)\in T^{*}{\mathbb{R}}^{d}$ define the rescaled variable
$u=u(x,\xi)$ centered at the energy surface $\Sigma_{E}$ by

$\left(\left\lVert
x\right\rVert^{2}+\left\lVert\xi\right\rVert^{2}\right)/2=E+u\left(\hbar/2E\right)^{2/3}.$
(67)

It is proved in [HZ20] that, for $\left\lvert u\right\rvert<\hbar^{-1/3}$,

$W_{\hbar,E}(x,\xi)=\begin{cases}\frac{2}{(2\pi\hbar)^{d}}\left(\frac{\hbar}{2E}\right)^{1/3}\left(\operatorname{Ai}(u/E)+O\left((1+\left\lvert
u\right\rvert)^{1/4}u^{2}\hbar^{2/3}\right)\right),&\qquad u<0\\\
\frac{2}{(2\pi\hbar)^{d}}\left(\frac{\hbar}{2E}\right)^{1/3}\operatorname{Ai}(u/E)\left(1+O\left((1+\left\lvert
u\right\rvert)^{3/2}u\hbar^{2/3}\right)\right),&\qquad u>0\end{cases}$ (68)

The simplified asymptotics follow by Taylor expansion of $\beta$ in (8) and of
the amplitude $u_{0,0}$ denoted in [HZ20] by

$\alpha_{0}(s)=\begin{cases}s^{(1-\alpha)/2}\frac{\sqrt{2|B(s)|}}{(1-s)^{1/4}s^{3/4}},&\quad
0<s<1\\\ s^{(1-\alpha)/2}\frac{\sqrt{2B(s)}}{(s-1)^{1/4}s^{3/4}},&\quad s\geq
1\end{cases}.$ (69)

Namely, the Taylor expansions are,

$B^{2}(1+t)=2^{-2/3}t(1+O(t)),\qquad\alpha_{0}(1+t)=2^{1/3}(1+O(t)).$ (70)

These estimates yield

$\nu^{2/3}B^{2}\left(s\right)=\left(\frac{2}{\hbar_{E}}\right)^{2/3}B^{2}(1+(u/E)\hbar_{E}^{2/3})=\frac{u}{E}+O\left(\left\lvert
u\right\rvert^{2}\hbar_{E}^{2/3}\right),$ (71)

which leads in [HZ20] to,

$\nu^{-1/3}\operatorname{Ai}(\nu^{2/3}B^{2}(s))\alpha_{0}(s)=\hbar_{E}^{1/3}\left[\operatorname{Ai}(u/E)+O((1+\left\lvert
u\right\rvert)^{1/4}u^{2}\hbar_{E}^{2/3})\right]$

when $u<0$ and

$\nu^{-1/3}\operatorname{Ai}(\nu^{2/3}B^{2}(s))\alpha_{0}(s)=\hbar_{E}^{1/3}\operatorname{Ai}(u/E)\left[1+O((1+\left\lvert
u\right\rvert)^{3/2}u\hbar_{E}^{2/3})\right]$

when $u>0.$

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