# The semiclassical limit of a quantum Zeno dynamics

Fabio Deelan Cunden  Dipartimento di Matematica, Univerisità degli Studi di
Bari, I-70125 Bari, Italy, and INFN, Sezione di Bari, I-70126 Bari, Italy
<EMAIL_ADDRESS>,  Paolo Facchi  Dipartimento di Fisica, Univerisità
degli Studi di Bari, I-70126 Bari, Italy, and INFN, Sezione di Bari, I-70126
Bari, Italy<EMAIL_ADDRESS>and  Marilena Ligabò  Dipartimento di
Matematica, Univerisità degli Studi di Bari, I-70125 Bari, Italy
<EMAIL_ADDRESS>
###### Abstract.

Motivated by a quantum Zeno dynamics in a cavity quantum electrodynamics
setting, we study the asymptotics of a family of symbols corresponding to a
truncated momentum operator, in the semiclassical limit of vanishing Planck
constant $\hbar\to 0$ and large quantum number $N\to\infty$, with $\hbar N$
kept fixed. In a suitable topology, the limit is the discontinuous symbol
$p\chi_{D}(x,p)$ where $\chi_{D}$ is the characteristic function of the
classically permitted region $D$ in phase space. A refined analysis shows that
the symbol is asymptotically close to the function $p\chi_{D}^{(N)}(x,p)$,
where $\chi_{D}^{(N)}$ is a smooth version of $\chi_{D}$ related to the
integrated Airy function. We also discuss the limit from a dynamical point of
view.

##  1\. Introduction

In the _quantum Zeno effect_ , frequent projective measurements can slow down
the evolution of a quantum system and eventually hinder any transition to
states different from the initial one. The situation is much richer when the
measurement does not confine the system in a single state, but rather in a
multidimensional subspace of its Hilbert space. This gives rise to a _quantum
Zeno dynamics_ (QZD): the system evolves in the projected subspace under the
action of its projected Hamiltonian. This phenomenon, first considered by
Beskow and Nilsson [4] in their study of the decay of unstable systems, was
dubbed quantum Zeno effect (QZE) by Misra and Sudarshan [38] who suggested a
parallelism with the paradox of the ‘flying arrow at rest’ by the philosopher
Zeno of Elea. Since then, QZE has received constant attention by physicists
and mathematicians, who explored different aspects of the phenomenon.

From the mathematical point of view, QZD is related to the limit of a product
formula obtained by intertwining the dynamical time evolution group with the
orthogonal projection associated with the measurements performed on the
system. It can be viewed as a generalization of Trotter-Kato product formulas
[8, 32, 51, 52] to more singular objects in which one semigroup is replaced by
a projection. The structure of the QZD product formula has been thoroughly
investigated and has been well characterized under quite general assumptions
[17, 18, 19, 20, 25, 26, 27, 36, 44, 45, 21].

QZE has been observed experimentally in a variety of systems, on experiments
involving photons, nuclear spins, ions, optical pumping, photons in a cavity,
ultracold atoms, and Bose-Einstein condensates, see [22] and references
therein. In all the abovementioned implementations, the quantum system is
forced to remain in its initial state through a measurement associated with a
one-dimensional projection. The present study is inspired by a proposal by
Raimond _et al._ [40, 41] for generating a multidimensional QZD in a cavity
quantum electrodynamics experiment. We briefly describe the proposal, skipping
most of the non-mathematical details.

The mode of the quantized electromagnetic field in a cavity can be
conveniently described in the Fock space representation. The Hamiltonian of
the quantized field is that of a harmonic oscillator (with angular frequency
$\omega=1$)

(1.1) $\displaystyle
H_{\operatorname{h.o.}}=\frac{1}{2}\left(-\hbar^{2}\frac{d^{2}}{dx^{2}}+\hat{x}^{2}\right),$

where $\hat{x}$ is the position operator and $\hat{p}=-i\hbar\frac{d}{dx}$ is
the momentum operator on $L^{2}({\mathbb{R}})$. The operators $\hat{x}$,
$\hat{p}$, and $H_{\operatorname{h.o.}}$ are essentially self-adjoint on the
common core $\SS({\mathbb{R}})$, the Schwartz space of rapidly decreasing
functions. The eigenfunction $\psi_{n}$ of $H_{\operatorname{h.o.}}$
represents a cavity state with $n$ photons ($n=0,1,2,\dots$) and energy
$\lambda_{n}=\hbar(n+1/2)$.

The cavity field undergoes a stroboscopic evolution alternating a short
continuous time evolution $e^{-i\frac{\tau}{\hbar}\hat{p}}$ given by a
_displacement operator_ , that without loss of generality is taken to be
generated by $\hat{p}$, and an instantaneous interaction

(1.2)
$P_{<N}=\sum_{k=0}^{N-1}|\psi_{k}\rangle\langle\psi_{k}|=\chi_{(-\infty,\hbar
N)}(H_{\operatorname{h.o.}}),$

with atoms injected into the cavity to ascertain whether in the cavity there
are less than $N$ photons ($N\geq 1$ is a chosen maximal photon number).

The quantum Zeno dynamics consists in performing a series of
$P_{<N}$-measurements in a fixed time interval $[0,t]$ at times $t_{j}=j\tau$,
$j=0,\dots,n$, with period $\tau=t/n$. The intertwining of the continuous time
evolutions and the projective measurements corresponds to the evolution
operator

$V_{n}(t)=\left(P_{<N}e^{-\frac{it}{n\hbar}\hat{p}}P_{<N}\right)^{n}.$

Observe that since $\operatorname{Ran}P_{<N}\subset
D(\hat{p})=H^{1}({\mathbb{R}})$, we have [21]

$\lim_{n\to\infty}V_{n}(t)=P_{<N}e^{-itH_{N}/\hbar},$

in the strong operator topology, uniformly for $t$ in compact subsets of
${\mathbb{R}}$, where the _Zeno Hamiltonian_ $H_{N}$ is a rank-$N$ truncation
of $\hat{p}$:

$\displaystyle H_{N}$ $\displaystyle=P_{<N}\hat{p}P_{<N}$ (1.3)
$\displaystyle=\chi_{(-\infty,\hbar
N)}(H_{\operatorname{h.o.}})\,\hat{p}\,\chi_{(-\infty,\hbar
N)}(H_{\operatorname{h.o.}}).$

Hence the QZD establishes a sort of ‘hard wall’ in the Hilbert space: the
state of the system evolves unitarily within the $N$-dimensional _Zeno
subspace_ spanned by states with at most $(N-1)$ photons,
$\psi_{0},\dots,\psi_{N-1}$. This hard wall prevents the state to escape from
the Zeno subspace and induces remarkable features in the quantum evolution
[40, 41].

The question addressed in this paper is: _What is the semiclassical limit of
the Zeno Hamiltonian $H_{N}$ and of its corresponding quantum dynamics?_

###  1.1. Semiclassical limit of the Zeno Hamiltonian

Semiclassical theory concerns the asymptotic analysis for vanishing Planck
constant ($\hbar\to 0$) of operators and vectors, with the ultimate goal of
understanding the quantum-to-classical transition. It is therefore convenient
to use a phase space description of quantum mechanics where operators are
represented by functions on the classical phase space (called Weyl symbols),
states are described by quasi-probability distributions (called Wigner
functions), and the noncommutative product of operators is mapped in a twisted
convolution product of symbols (called Moyal product), see e.g. [24, 42].

If we describe the QZD in the phase space, in the semiclassical limit
$N\to\infty$, $\hbar\to 0$ with the product $\hbar N=\mu$ kept fixed, we
expect the motion to be confined in the classically allowed region. The level
sets of the classical harmonic oscillator

(1.4)
$\mathfrak{h}_{\operatorname{h.o.}}(x,p)=\frac{1}{2}\left(p^{2}+x^{2}\right)$

are circles centered at the origin of the phase space
${\mathbb{R}}_{x}\times{\mathbb{R}}_{p}$ . In qualitative terms, the hard wall
can be viewed in the phase space as a circle with a radius $\propto\sqrt{\hbar
N}$. In the limit, the corresponding classically allowed region is the disk

$D:=\\{(x,p)\in{\mathbb{R}}^{2}\colon\mathfrak{h}_{\operatorname{h.o.}}(x,p)<\mu\\}=\\{p^{2}+x^{2}<2\mu\\},$

whose boundary

$\partial
D:=\\{(x,p)\in{\mathbb{R}}^{2}\colon\mathfrak{h}_{\operatorname{h.o.}}(x,p)=\mu\\}=\\{p^{2}+x^{2}=2\mu\\}$

is the circle of radius $\sqrt{2\mu}$. This is what Raimond _et al._ [40, 41]
called the ‘exclusion circle’: it separates $D$ from the classically forbidden
region where $\mathfrak{h}_{\operatorname{h.o.}}>\mu$.

Figure 1. Plot of the symbol ${\sigma}_{H_{N}}^{\hbar}$ in the phase plane
$(x,p)$. Here $N=17$ and $\mu=2$. Already for such a small value of $N$, the
graph of the symbol resembles a (rippled) tilted coin in the disk $D$ and zero
outside.

Let
$\chi_{D}(x,p)=\chi_{(-\infty,\mu)}\big{(}\mathfrak{h}_{\operatorname{h.o.}}(x,p)\big{)}$
be the characteristic function of the disk $D$. The first main result of the
paper is the identification of the limit of the Weyl symbols
${\sigma}_{P_{<N}}^{\hbar}(x,p)$ and ${\sigma}_{H_{N}}^{\hbar}(x,p)$ of the
projection operator $P_{<N}$ and the Zeno Hamiltonian $H_{N}$, respectively.
(The definition of the Weyl symbol of an operator is given in Definition 1.)

######  Theorem 1 (Weak convergence of the symbols).

Set $\mu>0$. Then,

(1.5) $\displaystyle\lim_{\begin{subarray}{c}N\to\infty,\hbar\to 0\\\ \hbar
N=\mu\end{subarray}}\int\limits_{{\mathbb{R}}_{x}\times{\mathbb{R}}_{p}}\left[{\sigma}_{P_{<N}}^{\hbar}(x,p)-\chi_{D}(x,p)\right]\varphi(x,p)dxdp=0,$
(1.6) $\displaystyle\lim_{\begin{subarray}{c}N\to\infty,\hbar\to 0\\\ \hbar
N=\mu\end{subarray}}\int\limits_{{\mathbb{R}}_{x}\times{\mathbb{R}}_{p}}\left[{\sigma}_{H_{N}}^{\hbar}(x,p)-p\chi_{D}(x,p)\right]\varphi(x,p)dxdp=0,$

for all $\varphi\in\mathcal{A}$.

######  Remark 1.

Here $\mathcal{A}$ is the space of test functions introduced by Lions and Paul
[33] as the completion of the smooth functions of compact support in the phase
space $C^{\infty}_{c}({\mathbb{R}}_{x}\times{\mathbb{R}}_{p})$ under the norm

(1.7)
$\|\varphi\|_{\mathcal{A}}:=\int_{{\mathbb{R}}}dy\sup_{x}\left|\mathcal{F}_{2}\varphi(x,y)\right|.$

In this paper $\mathcal{F}_{2}\varphi$ denotes the partial Fourier transform
of $\varphi$ in the second variable,

(1.8)
$\mathcal{F}_{2}\varphi(x,y):=\int_{{\mathbb{R}}}\varphi(x,p)e^{-ipy}dp.$

Quantum | Classical |
---|---|---
$N\in{\mathbb{N}}$ |  $\hbar\to 0$, $N\to\infty$ |
$\hbar>0$ | with $\hbar N=\mu$ |
$P_{<N}=\chi_{(-\infty,\hbar N)}(H_{\operatorname{h.o.}})$ | $\chi_{(-\infty,\mu)}(\mathfrak{h}_{\operatorname{h.o.}}(x,p))=\chi_{(-\infty,\sqrt{2\mu})}(\sqrt{x^{2}+p^{2}})$ |
$H_{N}=\chi_{(-\infty,\hbar N)}(H_{\operatorname{h.o.}})\,\hat{p}\,\chi_{(-\infty,\hbar N)}(H_{\operatorname{h.o.}})$ | $p\chi_{(-\infty,\mu)}(\mathfrak{h}_{\operatorname{h.o.}}(x,p))=p\chi_{(-\infty,\sqrt{2\mu})}(\sqrt{x^{2}+p^{2}})$ |
Table 1. Summary of the operators and their semiclassical limits.

######  Remark 2.

Theorem 1 makes precise the heuristic expectation that the symbol
${\sigma}_{P_{<N}}^{\hbar}(x,p)$ of the projection operator converges to the
characteristic function $\chi_{D}(x,p)$ of the classically allowed region, and
the symbol of the Zeno Hamiltonian ${\sigma}_{H_{N}}^{\hbar}(x,p)$ converges
to $p\chi_{D}(x,p)$. The content of Theorem 1 is schematically summarised in
Table 1.

######  Remark 3.

A plot of the Weyl symbol exhibits pronounced oscillations, also known as
_quantum ripples_ [6], in
$D=\\{\mathfrak{h}_{\operatorname{h.o.}}(x,p)<\mu\\}$. If the oscillations are
smoothed out, then the graph of ${\sigma}_{H_{N}}^{\hbar}$ is asymptotically
close to a ‘tilted coin’. See Fig. 1 and Fig. 2.

Figure 2. Comparison between the symbol of the Zeno Hamiltonian,
${\sigma}_{H_{N}}^{\hbar}$, and its semiclassical versions, $p\chi_{D}^{(N)}$
and $p\chi_{D}$, as functions of $p$ with $x=0$ fixed. The inset (a zoom of
the shaded area) show how $\chi_{D}^{(N)}$ better approximates the symbol
${\sigma}_{H_{N}}^{\hbar}$ near the edge. Here $N=57$, $\mu=2$.

We see that at the boundary $\partial D$ the symbols
${\sigma}_{P_{<N}}^{\hbar}$ and ${\sigma}_{H_{N}}^{\hbar}$ develop a jump, for
large $N$. The second main result of the paper concerns a finer asymptotics of
${\sigma}_{P_{<N}}^{\hbar}$ and ${\sigma}_{H_{N}}^{\hbar}$ near $\partial D$.
By zooming in at the edge $\partial D$, one sees that the symbols have
nontrivial scaling limits related to the integrated Airy function

$\operatorname{Ai}_{1}(\xi):=\int_{\xi}^{+\infty}\operatorname{Ai}(u)\,du,\qquad\xi\in{\mathbb{R}},$

see (B.11). More precisely, set

(1.9)
$\chi_{D}^{(N)}(x,p):=\operatorname{Ai}_{1}\left(\frac{(2N)^{\frac{2}{3}}}{\mu}\left(\mathfrak{h}_{\operatorname{h.o.}}(x,p)-\mu\right)\right),\qquad
x,p\in{\mathbb{R}}.$

It follows from (B.12) that $\chi_{D}^{(N)}$ is a sequence of rotational
symmetric smooth functions on the phase space that approximate the
characteristic function,

$\lim_{N\to\infty}\chi_{D}^{(N)}(x,p)=\chi_{D}(x,p)$

for all $(x,p)\notin\partial D$. (On the boundary $\partial D$,
$\chi_{D}^{(N)}=1/3$, for all $N$.)

We can now state our second main result.

######  Theorem 2 (Weak asymptotics at the boundary).

Fix $\mu>0$. For all $g\in C^{\infty}_{c}({\mathbb{R}})$,

(1.10) $\lim_{\begin{subarray}{c}N\to\infty,\hbar\to 0\\\ \hbar
N=\mu\end{subarray}}\int\limits_{{\mathbb{R}}_{x}\times{\mathbb{R}}_{p}}\left[{\sigma}_{P_{<N}}^{\hbar}(x,p)-\chi_{D}^{(N)}(x,p)\right]\frac{1}{\hbar^{\frac{2}{3}}}g\left(\frac{x^{2}+p^{2}-2\mu}{\hbar^{\frac{2}{3}}}\right)dxdp=0.$

and

(1.11) $\lim_{\begin{subarray}{c}N\to\infty,\hbar\to 0\\\ \hbar
N=\mu\end{subarray}}\int\limits_{{\mathbb{R}}_{x}\times{\mathbb{R}}_{p}}\left[{\sigma}_{H_{N}}^{\hbar}(x,p)-p\chi_{D}^{(N)}(x,p)\right]\frac{1}{\hbar^{\frac{2}{3}}}g\left(\frac{x^{2}+p^{2}-2\mu}{\hbar^{\frac{2}{3}}}\right)dxdp=0.$

######  Remark 4.

In order to zoom at $\partial D$, we need to integrate the symbols
${\sigma}_{P_{<N}}^{\hbar}$ and ${\sigma}_{H_{N}}^{\hbar}$ against (sequences
of) compactly supported test functions that concentrate around $\partial D$.
Since $\partial D$ is invariant under rotations, without loss of generality we
consider test functions that are also rotational symmetric. The idea is to
consider, for $g\in C^{\infty}_{c}({\mathbb{R}})$, the rescaling
$\epsilon^{-2}g(\epsilon^{-2}(x^{2}+p^{2}-2\mu))$ that is nonzero in a region
of order $\mathrm{O}(\epsilon)$ within the boundary $\partial D$. The blow-up
scale that gives rise to a nontrivial limit is $\epsilon=\hbar^{\frac{1}{3}}$.
The reason for this choice will emerge in the following (see Section 3). Note
that the space of test functions $\mathcal{A}$ in Theorem 1 does not depend on
the details of the model. On the contrary, in Theorem 2 we integrate the
symbols ${\sigma}_{P_{<N}}^{\hbar}$ and ${\sigma}_{H_{N}}^{\hbar}$ against
test functions that concentrate around $\partial D$ in a suitable way.

######  Remark 5.

The limits in Theorems 1 and 2 do _not_ hold pointwise, in general. For
instance, it is easy to show (by using the parity of the harmonic oscillator
eigenfunctions) that

(1.12) $\displaystyle{\sigma}_{P_{<N}}^{\hbar}(0,0)=1+(-1)^{N+1}.$

The reader is invited to have a glance at Fig. 2. Inside the disk $D$, the
symbols oscillate with frequency of order $\operatorname{O}(N)$, while in the
classically forbidden region
$D^{c}=({\mathbb{R}}_{x}\times{\mathbb{R}}_{p})\setminus D$ the symbols are
exponentially suppressed. The monotonic behaviour outside the disk suggests
that for $(x,p)\in D^{c}$ the convergence to the limits may hold in a stronger
sense. In fact, a slight adaptation of the proof of Theorem 2 shows that
outside the disk, ${\sigma}_{P_{<N}}^{\hbar}$ and ${\sigma}_{H_{N}}^{\hbar}$
converge pointwise to the limit symbols.

######  Theorem 3 (Pointwise asymptotics in the classically forbidden region).

Fix $\mu>0$. For all $(x,p)\in D^{c}$,

(1.13) $\lim_{\begin{subarray}{c}N\to\infty,\hbar\to 0\\\ \hbar
N=\mu\end{subarray}}\left[{\sigma}_{P_{<N}}^{\hbar}(x,p)-\chi_{D}^{(N)}(x,p)\right]=0,$

and

(1.14) $\lim_{\begin{subarray}{c}N\to\infty,\hbar\to 0\\\ \hbar
N=\mu\end{subarray}}\left[{\sigma}_{H_{N}}^{\hbar}(x,p)-p\chi_{D}^{(N)}(x,p)\right]=0.$

###  1.2. Semiclassical limit of the quantum Zeno dynamics

The quantum dynamics in phase space is ruled by two elements: the Weyl symbol
of the Zeno Hamiltonian ${\sigma}_{H_{N}}^{\hbar}$ and the Moyal bracket (that
does depend on $\hbar$) [24, 42]. Hence, the semiclassical limit of the
dynamics should encompass a simultaneous $\hbar\to 0$ limit of the symbol (the
generator of the dynamics) _and_ the Moyal structure.

By Theorem 1, the symbol ${\sigma}_{H_{N}}^{\hbar}$ of the Zeno Hamiltonian
converges as $\hbar\to 0$, $N\to\infty$, with $\hbar N=\mu>0$, to
$p\chi_{D}(x,p)$. Moreover, the Moyal bracket has an asymptotic expansion in
powers of $\hbar$ whose leading term (zero-th order) is the classical Poisson
bracket. Hence, it is reasonable to expect that the limiting dynamics is well
described by the Hamiltonian evolution (i.e. Poisson) in phase space where the
Hamiltonian is the limit symbol $p\chi_{D}(x,p)$.

However, in this naïve approach we immediately face an obstruction: the symbol
$p\chi_{D}(x,p)$ is _not_ smooth, and hence it is not possible to write
Hamilton’s equations of motion! If we insist in writing, formally, Hamilton’s
equations, we get

(1.15) $\begin{cases}\displaystyle\dot{x}=\dfrac{\partial}{\partial
p}\left(p\chi_{D}(x,p)\right)=\chi_{[0,\sqrt{2\mu})}(r)+p\delta_{\sqrt{2\mu}}(r)\dfrac{p}{r},\\\
\displaystyle\dot{p}=-\dfrac{\partial}{\partial
x}\left(p\chi_{D}(x,p)\right)=-p\delta_{\sqrt{2\mu}}(r)\dfrac{x}{r},\end{cases}\qquad(\diamondsuit)$

where $r=\sqrt{x^{2}+p^{2}}$. The Dirac delta $\delta_{\sqrt{2\mu}}(r)$ arises
as distributional derivative of the step function. We stress again that the
above expressions are formal: the Hamiltonian is discontinuous at $\partial
D$, and its vector field in $(\diamondsuit)$ is singular.

We can now look at the corresponding phase portrait. First, the Hamiltonian
vector field is zero outside the closure of the disk $D$. Thus, all points
there are equilibrium points. If the particle is in $D$, then the equation of
motions are $\dot{x}=1,\dot{p}=0$, and the particle moves with constant
momentum

$x(t)=x_{0}+t,\qquad p(t)=p_{0}.$

It is thus proceeding at a constant velocity along the $x$-axis. When it hits
the boundary $\partial D$, the evolution is given by the singular
contributions, proportional to the delta functions:
$\dot{x}=p\delta_{\sqrt{2\mu}}(r)p/r,\dot{p}=-p\delta_{\sqrt{2\mu}}(r)x/r$.
Heuristically, these equations would correspond to a field tangential to the
boundary of $D$ that yields a motion along the circle $\partial D$ at
‘infinite’ speed. The particle reappears on the other side of the boundary
(with the same momentum $p=p_{0}$) and resumes its motion along the $x$-axis
at a constant velocity. The collision at the edge $\partial D$ thus realizes,
in this semiclassical picture, a reflection around the $p$-axis of the phase
space, transforming $\left(\sqrt{2\mu-p_{0}^{2}},p_{0}\right)$ into
$\left(-\sqrt{2\mu-p_{0}^{2}},p_{0}\right)$.

An interesting interpretation of the semiclassical limit of the Zeno dynamics
is as follows. In the limit dynamics, the points $(x,p)$, $(-x,p)$ on the
cirle $\partial D\subset{\mathbb{R}}_{x}\times{\mathbb{R}}_{p}$ are
identified. Hence, one can think of the $N\to\infty,\hbar\to 0$ limit, with
$\hbar N=\mu$, as yielding a _change of topology_ : the dynamics on the disk
becomes a motion on the sphere! We emphasise again that all this is formal,
although very close to what was observed in [40, 41], and called ‘phase
inversion mechanism’. The function $p\chi_{D}(x,p)$ is _not_ smooth and
therefore it is not the generator of a classical Hamiltonian dynamics.

(a) Trajectories generated by the Weyl symbol ${\sigma}_{H_{N}}^{\hbar}(x,p)$
with $N=7$.

(b) Trajectories generated by the smooth function $p\chi_{D}^{(N)}(x,p)$ with
$N=7$. A particle near $\partial D$ moves at speed $\propto N^{\frac{2}{3}}$.

(c) Trajectories generated by the discontinuous function $p\chi_{D}(x,p)$. The
motion on $\partial D$ is at ‘infinite’ speed.

Figure 3. Phase portraits for the Hamiltonian dynamics. The red solid line is
the boundary $\partial D$ of the disk. Here $\mu=2$.

We know, however, by Theorem 2, that the symbol
${\sigma}_{H_{N}}^{\hbar}(x,p)$ is asymptotically close to a smoothed version

(1.16)
$p\chi_{D}^{(N)}(x,p)=p\operatorname{Ai}_{1}\left(\frac{(2N)^{\frac{2}{3}}}{2\mu}\left(r^{2}-2\mu\right)\right).$

For each $N$, it makes sense to consider the Hamiltonian system generated by
$p\chi_{D}^{(N)}(x,p)$,

(1.17) $\begin{cases}\displaystyle\dot{x}=\dfrac{\partial}{\partial
p}\left(p\chi_{D}^{(N)}(x,p)\right)\\\
\displaystyle\dot{p}=-\dfrac{\partial}{\partial
x}\left(p\chi_{D}^{(N)}(x,p)\right)\end{cases}\qquad(\clubsuit).$

This is a family of well-defined Hamilton equations and we can expect, for
large $N$, the solutions of $(\clubsuit)$ to be ‘close’ to the sought
semiclassical limiting dynamics.

We give here a sketch of an argument showing that for large $N$, the solutions
of $(\clubsuit)$ behave as the formal solutions of the singular problem
$(\diamondsuit)$. The equations of motions from $(\clubsuit)$ are

(1.18)
$\displaystyle\displaystyle\dot{x}=\chi^{(N)}_{[0,\sqrt{2\mu})}(r)+p\delta^{(N)}_{\sqrt{2\mu}}(r)\dfrac{p}{r},\qquad\displaystyle\dot{p}=-p\delta^{(N)}_{\sqrt{2\mu}}(r)\dfrac{x}{r},$

where

(1.19)
$\chi^{(N)}_{[0,\sqrt{2\mu})}(r):=\operatorname{Ai}_{1}\left(\frac{(2N)^{\frac{2}{3}}}{2\mu}\left(r^{2}-2\mu\right)\right),\quad\delta^{(N)}_{\sqrt{2\mu}}(r):=-r\frac{(2N)^{\frac{2}{3}}}{\mu}\operatorname{Ai}\left(\frac{(2N)^{\frac{2}{3}}}{2\mu}\left(r^{2}-2\mu\right)\right).$

Observe that $\chi^{(N)}_{[0,\sqrt{2\mu})}(r)$ are uniformly bounded functions
that converge, as $N\to\infty$ to the characteristic function
$\chi_{[0,\sqrt{2\mu})}$, see (B.12). The corresponding component of the field
is of order $\mathrm{O}(1)$. The sequence of functions
$\delta^{(N)}_{\sqrt{2\mu}}(r)$ converges to $\delta_{\sqrt{2\mu}}(r)$ in a
distributional sense, as $N\to\infty$. This can be seen, in Fourier space,
from the identity
$\int_{{\mathbb{R}}}\operatorname{Ai}(x)e^{-ikx}dx=e^{ik^{3}/3}$.

We conclude that the Hamiltonian vector field generated by
$p\chi_{D}^{(N)}(x,p)$ converges to the singular vector field generated by
$p\chi_{D}(x,p)$. The component of the field containing
$\delta_{\sqrt{2\mu}}^{(N)}(r)$ is of order $\mathrm{O}(N^{\frac{2}{3}})$ and
generates a motion at speed $\propto N^{\frac{2}{3}}$, which becomes
‘infinite’ in the singular limit. Fig. 3 shows a comparison of the phase
portraits of the Hamiltonian dynamics generated by the Weyl symbol
${\sigma}_{H_{N}}^{\hbar}(x,p)$, the smooth Hamiltonian $p\chi_{D}^{(N)}(x,p)$
and the discontinuous function $p\chi_{D}(x,p)$. Note the effective change of
topology in the limit singular case that results from the instantaneous motion
along the circle $\partial D$.

###  1.3. Spectral analysis of the Zeno Hamiltonian $H_{N}$

The matrix representation of $H_{N}$ (in the Hermite basis
$\left\\{\psi_{k}^{\hbar}\right\\}_{k\in{\mathbb{N}}}$, see Appendix A) is the
$N\times N$ complex Hermitian matrix

$H_{N}=i\sqrt{\frac{\hbar}{2}}\begin{bmatrix}0&1&0&\cdots&0\\\
-1&0&\sqrt{2}&&\vdots\\\ 0&-\sqrt{2}&\ddots&&\\\ \vdots&&&0&\sqrt{N-1}\\\
0&\cdots&&-\sqrt{N-1}&0\end{bmatrix}.$

This is a _Jacobi matrix_ about which we have very precise spectral
information (characteristic polynomial, eigenvalues and their counting
measure) for all $N$.

######  Proposition 1.

For all $N\geq 1$,

(1.20)
$\det\left(yI_{N}-H_{N}\right)=\left(\frac{\sqrt{\hbar}}{2}\right)^{N}h_{N}\left(\sqrt{\hbar}y\right),$

where $h_{N}$ is the _Hermite polynomial_ of degree $N$, see Appendix A. In
particular, the eigenvalues of $H_{N}$ are the $N$ (simple and real) zeros of
the Hermite function $\psi^{\hbar}_{N}(z)$.

###### Proof.

$H_{N}$ is unitarily equivalent to $P_{<N}\hat{x}P_{<N}$ (see equations
(A.7)-(A.8)), and so the two operators have equal characteristic polynomial.
The claim now follows from a result for general orthogonal polynomials on the
real line due to Simon [47, Prop. 2.2]. ∎

If $y^{(j)}_{N}$ are the zeros of $\psi^{\hbar}_{N}(y)$, we define the
eigenvalues counting measure $\nu_{N}$ of the Zeno Hamiltonian $H_{N}$ to be
the nonnegative measure that puts weight $1/N$ on each eigenvalue of $H_{N}$
(the $y^{(j)}_{N}$’s). From well-known results on Hermite polynomials [14] it
follows that the measure $\nu_{N}$ weakly converges to the semicircular
density in the simultaneous limit $\hbar\to 0$, $N\to\infty$ with $\hbar N$
asymptotically fixed. See Figure 4.

######  Proposition 2.

For all continuous bounded functions $f$,

(1.21)
$\int_{{\mathbb{R}}}f(y)d\nu_{N}(y)\to\int_{{\mathbb{R}}}f(y)\rho_{\mu}(y)dy,$

as $N\to\infty$, $\hbar\to 0$, with the product $\hbar N$ converging to
$\mu>0$.

Figure 4. Illustration of Proposition 2. The histogram of the eigenvalues of
the Zeno Hamiltonian $H_{N}$ for $N=2000$, and $\hbar N=\mu=2$ is compared
with the semicircular density
$\rho_{\mu}(y)=\frac{1}{\pi\mu}\sqrt{(2\mu-y^{2})_{+}}$ of Eq. (B.1).

######  Remark 6.

The semicircular spectral distribution can be obtained formally from the
semiclassical limit of Theorem 1. Indeed, in the limit symbol $p\chi_{D}(x,p)$
of the Zeno Hamiltonian is concentrated on the disk $D$ of radius
$\sqrt{2\mu}$. Semiclassically, the density of the eigenvalues is the fraction
of the phase space volume with energy between $y$ and $y+dy$:

$\frac{\mathrm{Area}\left(\\{y\leq p\leq y+dy\\}\cap
D\right)}{\mathrm{Area}\left(D\right)}=\frac{2\sqrt{2\mu-y^{2}}dy}{\pi(2\mu)}=\rho_{\mu}(y).$

###  1.4. Proof strategy and relations to other works

When $N$ is large, the symbols ${\sigma}_{P_{<N}}^{\hbar}$ and
${\sigma}_{H_{N}}^{\hbar}$ are highly oscillating smooth functions. As
discussed in Remark 5, looking for a global semiclassical limit in a pointwise
sense is hopeless. It turns out that the sought convergence of the symbols
holds in a weak sense if the set of test functions is chosen to be
$\mathcal{A}$.

The proofs presented in this paper are based on the following observations:

  1. (1)

The asymptotics of integrals of the Weyl symbols
${\sigma}_{P_{<N}}^{\hbar}(x,p)$ and ${\sigma}_{H_{N}}^{\hbar}(x,p)$ against
functions on the phase space is related to the pointwise asymptotics of the
Fourier transforms $\mathcal{F}_{2}{\sigma}_{P_{<N}}^{\hbar}(x,y)$ and
$\mathcal{F}_{2}{\sigma}_{H_{N}}^{\hbar}(x,y)$;

  2. (2)

The function $\mathcal{F}_{2}{\sigma}_{P_{<N}}^{\hbar}(x,y)$ is a sum of $N$
terms (cross products of Hermite functions), see Eq. (2.13). However, thanks
to the Christoffel-Darboux formula (Lemma 3) this sum can be expressed in
terms of the $N$-th and $(N-1)$-th Hermite functions only. Hence, studying the
large $N$ asymptotics with $\hbar N=\mu$ amounts to study the large degree
asymptotics of Hermite functions. This is a well-studied topic in the theory
of orthogonal polynomials from which we can freely borrow explicit asymptotic
formulae. So, to prove the convergence of the symbols we will show the
convergence of the Christoffel-Darboux kernel along with its derivatives to
the _sine_ and _Airy kernels_ (in the formulation presented in the book of
Anderson, Guionnet and Zeitouni [2])

  3. (3)

The symbol ${\sigma}_{H_{N}}^{\hbar}(x,p)$ is ‘asymptotically close’ to
$p{\sigma}_{P_{<N}}^{\hbar}(x,p)$ in the dual space $\mathcal{A}^{\prime}$
(Proposition 7). This is suggested by the heuristic observation that, in the
limit $\hbar\to 0$, the algebra of observables should become commutative. What
we gain is that, once we know the asymptotics of
${\sigma}_{P_{<N}}^{\hbar}(x,p)$ we can directly deduce the asymptotics of
${\sigma}_{H_{N}}^{\hbar}(x,p)$.

The seminal paper by Lions and Paul [33] on the semiclassical limit of Wigner
measures, and the more recent developments [1, 3, 11, 23] were instrumental in
our study.

We mention that the symbol ${\sigma}_{P_{<N}}^{\hbar}(x,p)$ of the orthogonal
projection $P_{<N}$ studied in the present paper has close connection to the
fuzzy approximation of two-dimensional disk proposed by Lizzi, Vitale and
Zampini [34, 35]. A _fuzzy space_ is an approximation of an abelian algebra of
functions on an ordinary space with a sequence of finite-rank matrix algebras,
which preserve the symmetries of the original space, at the price of non-
commutativity. Eq. (1.5) of Theorem 1 is the precise mathematical statement
behind the numerical results of [34, 35]. To our knowledge, the finer
asymptotics of Theorem 2 is a new result, that has not been observed
numerically neither.

The convergence of symbols of projection operators to the characteristic
function of the classically allowed region is folklore in theoretical physics.
In recent years, there has been an explosion of results on the asymptotics of
the Christoffel-Darboux kernel for orthogonal polynomials on the real line,
especially in connections to eigenvalue statistics of random matrices and
integrable probability models [14, 37, 30, 43]. The interest to these
asymptotics in the theoretical and mathematical physics community has been
mostly motivated by applications to the number statistics of non-interacting
fermions. The asymptotics at the ‘edge’ has been also investigated at various
levels of rigour. See, e.g. [5, 6, 7, 9, 10, 12, 13, 15, 16, 49].

The semiclassical structure of quite general _cut quantum observables_ $\Pi
Q\Pi$ (with $\Pi$ a spectral projection and $Q$ a pseudodifferential operator)
was studied by Hernandez-Duenas and Uribe [29]. Their results is consistent
with Theorem 1 of the present paper. Those authors also studied the unitary
dynamics generated by the cut quantum observables (the analogue of the Zeno
Hamiltonian $H_{N}$ of the present paper), and numerically found fascinating
phenomena of splitting of the wave-packets and infinite propagation speed near
the boundary of the classically allowed region. This is again consistent with
our findings.

Given the universality results on the asymptotics of orthogonal polynomials
and random matrices [14], we expect that Theorem 1 is valid for a rather large
class of symbols associated to finite-rank orthogonal projections. The recent
paper by Deleporte and Lambert [15] suggests that Theorem 2 would be valid as
long as the gradient of the confining potential does not vanish at the points
of classical inversion of motion. In any case the statement of analogues of
Theorem 2 should depend on the geometry of the level sets of the corresponding
classical Hamiltonian function. Further study is in progress.

###  1.5. Outline of the paper

The structure of the paper is as follows. In the next section we recall some
preliminary background material, introduce a precise presentation of the model
and provide the calculation of the symbols. In Section 3 we discuss the
different scaling limits in Theorems 1 and 2. Section 4 is entirely devoted to
the proofs of the main technical results, and of Theorems 1, 2 and 3. The
paper includes two appendices. In Appendix A we collect known formulae on the
Hermite functions. Appendix B contains the definition and a few properties of
the sine and the Airy kernel.

##  2\. Notation, preliminaries, Weyl symbols and kernels

We first introduce some notation and preliminary notions that we use
throughout this work. For a linear operator $L$ on $L^{2}({\mathbb{R}})$ we
write $L\doteq L(u,v)$ to indicate that $L$ has kernel $L(u,v)\in
L^{2}({\mathbb{R}}\times{\mathbb{R}})$. In this paper all kernels are
continuous. For $A,B$ linear operators, we use the notation $[A,B]:=AB-BA$ for
the commutator. Let $D$ be the linear operator defined, for $f\in
H^{1}(\mathbb{R})$, by the formula $(Df)(u)=\frac{d}{du}f(u)$. We have

(2.1) $[D,L]\doteq\left(\frac{\partial}{\partial u}+\frac{\partial}{\partial
v}\right)L(u,v).$

For $x\in{\mathbb{R}}$ and $\gamma>0$, let

$\displaystyle V_{x,\gamma}\colon L^{2}({\mathbb{R}})$
$\displaystyle\longrightarrow$ $\displaystyle L^{2}({\mathbb{R}})$ (2.2)
$\displaystyle f(u)$ $\displaystyle\longmapsto$
$\displaystyle\left(V_{x,\gamma}f\right)(u):=\sqrt{\gamma}f\left(x+\gamma
u\right).$

Of course
$\left(V^{-1}_{x,\gamma}f\right)(u)=\sqrt{1/\gamma}f\left(\gamma^{-1}(u-x)\right)$,
and $V_{x,\gamma}$ is unitary. If we conjugate the operator $L$ by the scaling
unitary $V_{x,\gamma}$, its kernel gets changed into

(2.3) $\displaystyle V_{x,\gamma}LV_{x,\gamma}^{-1}\doteq\gamma
L\left(x+\gamma u,x+\gamma v\right).$

We shall consider the following space of test functions introduced by Lions
and Paul [33],

(2.4) $\mathcal{A}=\left\\{f\in
C_{0}({\mathbb{R}}_{x}\times{\mathbb{R}}_{p})\colon\|f\|_{\mathcal{A}}:=\int_{{\mathbb{R}}}dy\sup_{x}\left|\mathcal{F}_{2}f(x,y)\right|<\infty\right\\},$

where $C_{0}({\mathbb{R}}_{x}\times{\mathbb{R}}_{p})$ is the usual space of
continuous functions tending to zero at infinity.

$\mathcal{A}$ is a Banach algebra with the following properties (see [33]):

  * -

$\mathcal{S}({\mathbb{R}}_{x}\times{\mathbb{R}}_{p})$,
$C^{\infty}_{c}({\mathbb{R}}_{x}\times{\mathbb{R}}_{p})$, and
$\mathcal{B}=\\{f\in\mathcal{A}\colon\mathcal{F}_{2}f\in
C_{c}({\mathbb{R}}_{x}\times{\mathbb{R}}_{y})\\}$ are dense subspaces in
$\mathcal{A}$.

  * -

$\sup_{x,p}|f(x,p)|\leq(1/2\pi)\|f\|_{\mathcal{A}}$; hence $\mathcal{A}$ is
contained in the space of bounded continuous functions in the phase space
$C_{b}({\mathbb{R}}_{x}\times{\mathbb{R}}_{p})$.

Let $\mathcal{A}^{\prime}$ be the dual of $\mathcal{A}$. From the Parseval
identity it follows that

(2.5)
$\|h\|_{\mathcal{A}^{\prime}}=\frac{1}{2\pi}\sup_{y}\int\left|\mathcal{F}_{2}h(x,y)\right|dx.$

A basic property is
$\|h\|_{\mathcal{A}^{\prime}}\leq\frac{1}{2\pi}\|h\|_{L^{1}}$ (hence
$L^{1}({\mathbb{R}}_{x}\times{\mathbb{R}}_{p})\subset\mathcal{A}^{\prime}$).

######  Definition 1.

Given a number $\hbar>0$, the Weyl symbol of the operator $L\doteq L(u,v)$ is
defined as

(2.6) $\sigma_{L}^{\hbar}(x,p):=\int_{{\mathbb{R}}_{y}}\hbar
L\left(x-\frac{\hbar y}{2},x+\frac{\hbar y}{2}\right)e^{ipy}dy.$

Equivalently, $\sigma_{L}^{\hbar}$ is defined by the identity

(2.7)
$\left(\mathcal{F}_{2}\sigma_{L}^{\hbar}\right)(x,y)=\left(2\pi\hbar\right)L\left(x-\frac{\hbar
y}{2},x+\frac{\hbar y}{2}\right)$

and, by Plancherel’s theorem,

(2.8)
$\int\limits_{{\mathbb{R}}_{x}\times{\mathbb{R}}_{p}}\overline{\sigma_{L}^{\hbar}(x,p)}\varphi(x,p)dxdp=\int\limits_{{\mathbb{R}}_{x}\times{\mathbb{R}}_{y}}\overline{\hbar
L\left(x-\frac{\hbar y}{2},x+\frac{\hbar
y}{2}\right)}\mathcal{F}_{2}\varphi(x,y)dxdy.$

We will often use the shorthand

(2.9)
$\langle\sigma,f\rangle:=\int_{{\mathbb{R}}_{x}\times{\mathbb{R}}_{p}}\overline{\sigma(x,p)}f(x,p)dxdp.$

We recall that the Weyl symbol of the product of two operators $A$, $B$ is
_not_ the ordinary product of the symbols
$\sigma^{\hbar}_{AB}\neq\sigma^{\hbar}_{A}\sigma^{\hbar}_{B}$, unless $A$ and
$B$ commute. The noncommutative Moyal product $\sharp$ is defined as the
composition law that does the job:
$\sigma^{\hbar}_{AB}=\sigma^{\hbar}_{A}\,\sharp\,\sigma^{\hbar}_{B}$ [24].

######  Definition 2.

Given two linear operators $A$ and $B$ on $L^{2}({\mathbb{R}})$ with Weyl
symbols $\sigma_{A}^{\hbar}$ and $\sigma_{B}^{\hbar}$ respectively, the Moyal
product is defined as follows:

$\sigma^{\hbar}_{A}\,\sharp\,\sigma^{\hbar}_{B}(x,p)=\int_{{\mathbb{R}}^{4}}\sigma^{\hbar}_{A}(x_{1},p_{1})\sigma^{\hbar}_{B}(x_{2},p_{2})e^{\frac{2i}{\hbar}[(x-x_{1})(p-p_{2})-(x-x_{2})(p-p_{1})]}\frac{dx_{1}dp_{1}dx_{2}dp_{2}}{(\pi\hbar)^{2}}.$

Recall that the normalised eigenfunctions of the harmonic oscillator operator
$H_{\operatorname{h.o.}}$ in (1.1) are the Hermite functions

(2.10)
$\psi_{k}^{\hbar}(x)=\sqrt{\frac{\alpha}{\sqrt{\pi}2^{k}k!}}\exp\left(-\frac{1}{2}\alpha^{2}x^{2}\right)h_{k}(\alpha
x),\qquad k=0,1,2,\ldots$

where $\alpha^{2}=1/\hbar$ and

(2.11) $h_{k}(y)=(-1)^{k}e^{y^{2}}\frac{d^{k}}{dy^{k}}e^{-y^{2}}$

is the $k$-th _Hermite polynomials_ , see Appendix A. Consider the _orthogonal
projection_

$P_{<N}=\chi_{(-\infty,\hbar N)}(H_{\operatorname{h.o.}})$

onto the span of the first $N$ Hermite eigenfunctions in (1.2). The _Zeno
Hamiltonian_ in (1.3) is the _truncated momentum operator_

(2.12) $H_{N}=P_{<N}\hat{p}P_{<N}=\hat{p}P_{<N}-[\hat{p},P_{<N}]P_{<N}.$

######  Proposition 3 (Integral kernels).

(2.13) $P_{<N}\doteq
K_{N}(u,v)=\sum_{k=0}^{N-1}\psi_{k}^{\hbar}(u)\psi_{k}^{\hbar}(v)$
$\displaystyle P_{<N}\hat{p}P_{<N}\doteq Q_{N}(u,v)$
$\displaystyle=\int_{{\mathbb{R}}}K_{N}(u,w)\left(-i\hbar\frac{\partial}{\partial
w}\right)K_{N}(w,v)dw$ (2.14)
$\displaystyle=i\sqrt{\frac{\hbar}{2}}\sum_{j=0}^{N-2}\sqrt{j+1}\left[\psi^{\hbar}_{j+1}\left(u\right)\psi^{\hbar}_{j}\left(v\right)-\psi^{\hbar}_{j}\left(u\right)\psi^{\hbar}_{j+1}\left(v\right)\right],$
(2.15) $[\hat{p},P_{<N}]P_{<N}\doteq R_{N}(u,v)=i\sqrt{\frac{\hbar
N}{2}}\psi^{\hbar}_{N}\left(u\right)\psi^{\hbar}_{N-1}\left(v\right).$

###### Proof.

Formula (2.13) follows directly by the definition of the Hermite functions.
Formula (3) is obtained by a direct calculation using the three-term
recurrence (A.8), while (2.15) follows by applying the identity (2.1) to
$P_{<N}$, and using the orthonormality of the eigenfunctions
$\\{\psi^{\hbar}_{k}\\}_{k\in{\mathbb{N}}}$. ∎

######  Remark 7.

The kernels $K_{N}$, $Q_{N}$ and $R_{N}$ are rapidly decreasing functions in
$\SS({\mathbb{R}}_{u}\times{\mathbb{R}}_{v})$.

The Weyl symbols of $P_{<N}$ and $H_{N}$

(2.16) $\displaystyle{\sigma}_{P_{<N}}^{\hbar}(x,p)=\int_{{\mathbb{R}}}\hbar
K_{N}\left(x-\frac{\hbar y}{2},x+\frac{\hbar y}{2}\right)e^{ipy}dy,$ (2.17)
$\displaystyle{\sigma}_{H_{N}}^{\hbar}(x,p)=\int_{{\mathbb{R}}}\hbar
Q_{N}\left(x-\frac{\hbar y}{2},x+\frac{\hbar y}{2}\right)e^{ipy}dy,$

have explicit representations in terms of associated Laguerre polynomials
(this is a manifestation of the so-called ‘Laguerre connection’ [24, §1.9]).

######  Proposition 4 (Weyl symbols).

For all $x,p\in{\mathbb{R}}$:

(2.18) $\displaystyle{\sigma}_{P_{<N}}^{\hbar}(x,p)$
$\displaystyle=2e^{-(p^{2}+x^{2})/\hbar}\sum_{j=0}^{N-1}(-1)^{j}L_{j}\left(2(p^{2}+x^{2})/\hbar\right),$
(2.19) $\displaystyle{\sigma}_{H_{N}}^{\hbar}(x,p)$
$\displaystyle=4pe^{-(p^{2}+x^{2})/\hbar}\sum_{j=0}^{N-2}(-1)^{j}L_{j}^{(1)}\left(2(p^{2}+x^{2})/\hbar\right),$

where

$L_{k}^{(j)}(y)=\sum_{m=0}^{k}\frac{(k+j)!}{(k-m)!(j+m)!m!}(-y)^{m}$

are the _associated Laguerre polynomials_.

###### Proof.

A consequence of the following formula by Groenewold [28] valid for all $j\leq
k$ (we write the formula as in [11, Eq. (30)]),

(2.20)
$\int\psi_{j}^{\hbar}\left(x-\frac{y}{2}\right)\psi_{k}^{\hbar}\left(x+\frac{y}{2}\right)e^{ipy}dy\\\
=2\sqrt{\left(\frac{2}{\hbar}\right)^{k-j}\frac{j!}{k!}}\left(x+ip\right)^{k-j}e^{-(p^{2}+x^{2})/\hbar}(-1)^{j}L_{j}^{(k-j)}\left(2(p^{2}+x^{2})/\hbar\right).$

∎

######  Remark 8.

The symbols ${\sigma}_{P_{<N}}^{\hbar}$ and ${\sigma}_{H_{N}}^{\hbar}$ are
rapidly decreasing functions in $\SS({\mathbb{R}}_{x}\times{\mathbb{R}}_{p})$.
Notice that ${\sigma}_{P_{<N}}^{\hbar}$ is rotational symmetric. It may be
convenient in the following to consider ${\sigma}_{P_{<N}}^{\hbar}$ and
${\sigma}_{H_{N}}^{\hbar}$ as complex-valued functions defined on the
complexification ${\mathbb{C}}_{x}\times{\mathbb{C}}_{p}$ of the real phase
space. They are entire functions in both variables $x$ and $p$.

##  3\. Scaling limits

In this section we provide an heuristic explanation of the different scaling
limits in Theorems 1 and 2. The following discussion is somewhat breezy. For a
more careful exposition of similar ideas, see [7, 9, 15].

Recall that $P_{<N}\doteq K_{N}(u,v)$ and

(3.1) $\mathcal{F}_{2}{\sigma}_{P_{<N}}^{\hbar}(x,y)=2\pi\hbar
K_{N}\left(x-\frac{\hbar y}{2},x+\frac{\hbar y}{2}\right).$

At scale $\hbar$, the kernel has an asymptotic limit that can be identified as
follows. We start by writing the rescaled kernel in terms of the conjugation
of a unitary transformation on the operator. If we conjugate the projection
$P_{<N}$ by the scaling unitary $V_{x,\hbar}$ in (2.2), we get that the kernel
of the rescaled projection is the rescaled kernel:

(3.2) $\displaystyle V_{x,\hbar}P_{<N}V_{x,\hbar}^{-1}=\chi_{(-\infty,\hbar
N)}\left(V_{x,\hbar}H_{\operatorname{h.o.}}V_{x,\hbar}^{-1}\right)\doteq\hbar
K_{N}\left(x+\hbar u,x+\hbar v\right).$

The action of the rescaled harmonic oscillator operator on a function $f$ in
its domain is

$\displaystyle\left(V_{x,\hbar}H_{\operatorname{h.o.}}V_{x,\hbar}^{-1}f\right)(u)=\frac{1}{2}\left[-\frac{d^{2}}{du^{2}}+\hbar^{2}u^{2}+2\hbar
ux+x^{2}\right]f(u).$

So we expect that

(3.3) $\chi_{(-\infty,\hbar
N)}\left(V_{x,\hbar}H_{\operatorname{h.o.}}V_{x,\hbar}^{-1}\right)\simeq\chi_{(-\infty,2\mu-x^{2})}\left(-\frac{d^{2}}{du^{2}}\right),\quad\text{
for $\hbar\to 0$, $N\to\infty$, with $\hbar N=\mu$}.$

We recall the following result adapted from [7, Lemma A.5].

######  Lemma 1.

The operator $-\frac{d^{2}}{du^{2}}$ is essentially self-adjoint on
$C^{\infty}_{c}({\mathbb{R}})$, and its unique self-adjoint extension has only
absolutely continuous spectrum
$\sigma\left(-\frac{d^{2}}{du^{2}}\right)=\sigma_{\mathrm{ac}}\left(-\frac{d^{2}}{du^{2}}\right)=[0,\infty)$.
Moreover,

(3.4)
$\chi_{(-\infty,2\mu-x^{2})}\left(-\frac{d^{2}}{du^{2}}\right)\doteq\mu\rho_{\mu}(x)K_{\operatorname{sine}}\left(\mu\rho_{\mu}(x)u,\mu\rho_{\mu}(x)v\right),$

where $K_{\operatorname{sine}}$ is the sine kernel (B.3).

From (3.1), we see that a rescaling $\hbar$ in the Hilbert space
$L^{2}({\mathbb{R}})$ corresponds to zooming at scale $\hbar^{0}$ in the phase
space. The precise statement of (3.3) is Proposition 5 in Section 4.

To explain how a different asymptotics arises at the boundary $\partial D$, we
need to study the rescaled harmonic oscillator operator in a neighbourhood of
the classical turning points $x=\pm\sqrt{2\mu}$. Let us zoom at scale
$\hbar^{\alpha}$, with $\alpha>0$ an exponent to be determined:

$\displaystyle\left(V_{\sqrt{2\mu},\hbar^{\alpha}}H_{\operatorname{h.o.}}V_{\sqrt{2\mu},\hbar^{\alpha}}^{-1}f\right)(u)=\frac{1}{2}\left[-\hbar^{2(1-\alpha)}\frac{d^{2}}{du^{2}}+\hbar^{2\alpha}u^{2}+2^{\frac{3}{2}}\mu^{\frac{1}{2}}\hbar^{\alpha}u+2\mu\right]f(u).$

If we choose $\alpha=\frac{2}{3}$ we then expect that

(3.5) $\chi_{(-\infty,\hbar
N)}\left(V_{\sqrt{2\mu},\hbar^{\frac{2}{3}}}H_{\operatorname{h.o.}}V_{\sqrt{2\mu},\hbar^{\frac{2}{3}}}^{-1}\right)\simeq\chi_{(-\infty,0)}\left(-\frac{d^{2}}{du^{2}}+c_{\mu}^{3}\hat{u}\right),$

for $\hbar\to 0$, $N\to\infty$, with $\hbar N=\mu$, where $\hat{u}$ is the
position operator and $c_{\mu}=2^{\frac{1}{2}}\mu^{\frac{1}{6}}$ is a constant
given in (B.2). Thus, the limit at the edge is related to the Airy
differential operator for which we have the following spectral result, adapted
from [7, Lemma A.7].

######  Lemma 2.

The operator $-\frac{d^{2}}{du^{2}}+c_{\mu}^{3}\hat{u}$ is essentially self-
adjoint on $C^{\infty}_{c}({\mathbb{R}})$, and its self-adjoint extension has
only absolutely continuous spectrum
$\sigma\left(-\frac{d^{2}}{du^{2}}+c_{\mu}^{3}\hat{u}\right)=\sigma_{\mathrm{ac}}\left(-\frac{d^{2}}{du^{2}}+c_{\mu}^{3}\hat{u}\right)=(-\infty,\infty)$.
Moreover,

(3.6)
$\chi_{(-\infty,0)}\left(-\frac{d^{2}}{du^{2}}+c_{\mu}^{3}\hat{u}\right)\doteq
c_{\mu}K_{\operatorname{Ai}}(c_{\mu}u,c_{\mu}v).$

where $K_{\operatorname{Ai}}$ is the Airy kernel (B.4).

The precise statement of (3.5) is Proposition 6 in Section 4.

From (3.1), we see that a rescaling $\hbar^{\frac{2}{3}}$ at the edge in the
Hilbert space $L^{2}({\mathbb{R}})$ corresponds to zooming at scale
$\hbar^{\frac{2}{3}-1}=\hbar^{-\frac{1}{3}}$ around the boundary $\partial D$
in the phase space. This explains the rescaling in Theorem 2.

##  4\. Proofs

The proofs presented in this section are based on the following three
observations:

  1. (1)

The asymptotics of the Weyl symbols ${\sigma}_{P_{<N}}^{\hbar}$,
${\sigma}_{H_{N}}^{\hbar}$ is related (by Fourier transform in the second
variable $\mathcal{F}_{2}$) to the asymptotics of the integral kernels
${K_{N}}(x,y)$ and ${Q_{N}}(x,y)$.

  2. (2)

The kernel ${K_{N}}(x,y)$ is a sum of $N$ terms (cross products of Hermite
functions), see Eq. (2.13). However, thanks to Christoffel-Darboux formula
this sum can be expressed in terms of the $N$-th and $(N-1)$-th Hermite
functions only. Hence, studying the large $N$ asymptotics with $\hbar
N\sim\mu$ amounts to study the large degree asymptotics of the Hermite
functions.

  3. (3)

${\sigma}_{H_{N}}^{\hbar}(x,p)$ is ‘asymptotically close’ to
$p{\sigma}_{P_{<N}}^{\hbar}(x,p)$ for $N\to\infty$, $\hbar\to 0$ with $\hbar
N=\mu>0$, see Proposition 7, therefore once we know the asymptotics of $K_{N}$
(and hence of ${\sigma}_{P_{<N}}^{\hbar}$) we can directly deduce the
asymptotics of ${\sigma}_{H_{N}}^{\hbar}$.

###  4.1. Asymptotics of the kernels

By telescoping the sum in (2.13) and using the three-term relation (A.7), we
get the celebrated _Christoffel-Darboux formula_ [46].

######  Lemma 3.

For all $u,v\in\mathbb{R}$,

(4.1) $K_{N}(u,v)=\begin{cases}\displaystyle\sqrt{\frac{\hbar
N}{2}}\frac{\psi_{N}^{\hbar}(u)\psi_{N-1}^{\hbar}(v)-\psi_{N-1}^{\hbar}(u)\psi_{N}^{\hbar}(v)}{u-v}&\text{if
$u\neq v$}\\\ \\\ \displaystyle\sqrt{\frac{\hbar
N}{2}}({\psi_{N}^{\hbar}}^{\prime}(u)\psi_{N-1}^{\hbar}(u)-\psi_{N}^{\hbar}(u){\psi_{N-1}^{\hbar}}^{\prime}(u))&\text{if
$u=v$}\end{cases}.$

Thus, the large-$N$ asymptotics of the _Christoffel-Darboux kernel_
$K_{N}(x,y)$ boils down to the classical subject of large degree asymptotics
of orthogonal polynomials. A consequence of the Plancherel-Rotach asymptotics
for $\psi_{N}^{\hbar}(x)$ (Equations (A.10)-(A.12)) are the following
asymptotic behaviours of the kernel $K_{N}(x,y)$.

######  Proposition 5 (Bulk asymptotics of the Christoffel-Darboux kernel).

Suppose that $\hbar=\hbar_{N}$ is the sequence defined by the condition $\hbar
N=\mu$. Then, for any compact sets $U\Subset{\mathbb{R}}$ and
$V\Subset{\mathbb{R}}^{2}$, and for any $\alpha,\beta\in\\{0,1\\}$, there
exists a constant $C>0$ such that

(4.2) $\sup_{x\in U}\sup_{(t,s)\in
V}\left|\partial^{\alpha}_{t}\partial^{\beta}_{s}\left\\{\hbar
K_{N}\left(x+\hbar t,x+\hbar
s\right)-\mu\rho_{\mu}(x)K_{\operatorname{sine}}\left(\mu\rho_{\mu}(x)t,\mu\rho_{\mu}(x)s\right)\right\\}\right|\leq
C\hbar,$

where $\rho_{\mu}(x)$ is the semicircular density (B.1), and
$K_{\operatorname{sine}}$ is the sine kernel (B.3).

######  Proposition 6 (Edge asymptotics of the Christoffel-Darboux kernel).

Suppose that $\hbar=\hbar_{N}$ is the sequence defined by the condition $\hbar
N=\mu$. For any compact set $W\Subset{\mathbb{C}}^{2}$, and for any
$\alpha,\beta\in\\{0,1\\}$, there exists a constant $C>0$ such that

(4.3) $\sup_{(t,s)\in
W}\left|\partial^{\alpha}_{t}\partial^{\beta}_{s}\left\\{\hbar^{\frac{2}{3}}K_{N}\left(\sqrt{2\mu}+\hbar^{\frac{2}{3}}t,\sqrt{2\mu}+\hbar^{\frac{2}{3}}s\right)-c_{\mu}K_{\operatorname{Ai}}\left(c_{\mu}t,c_{\mu}s\right)\right\\}\right|\leq
C\hbar^{\frac{1}{3}},$

where $c_{\mu}$ is given in (B.2), and $K_{\operatorname{Ai}}$ is the Airy
kernel (B.4).

The scaling limits of the Christoffel-Darboux kernel to the sine and Airy
kernel are well-known results. It is perhaps less known that the local uniform
convergence can be promoted to their derivatives as well. We outline here a
proof, adapting the presentation of the book by Anderson, Guionnet and
Zeitouni [2, Chap. 3].

######  Notation.

From now on, $(\hbar_{N})_{N\geq 1}$ is the positive sequence such that
product $\hbar_{N}N=\mu$, where $\mu$ is a fixed positive number. We will
write $\hbar$ instead of $\hbar_{N}$ for short, when no confusion arises. We
will also use the following shorthand

(4.4) $\displaystyle K_{N,x_{0},\gamma}(t,s):=\gamma K_{N}(x_{0}+\gamma
t,x_{0}+\gamma s).$

###### Proof of Propositions 5 and 6.

Consider first the case $\alpha=\beta=0$:

(4.5) $\sup_{x\in U}\sup_{(t,s)\in
V}\left|K_{N,x,\hbar}\left(t,s\right)-\mu\rho_{\mu}(x)K_{\operatorname{sine}}\left(\mu\rho_{\mu}(x)t,\mu\rho_{\mu}(x)s\right)\right|\leq
C\hbar,$

and

(4.6) $\sup_{(t,s)\in
W}\left|K_{N,\sqrt{2\mu},\hbar^{\frac{2}{3}}}\left(t,s\right)-c_{\mu}K_{\operatorname{Ai}}\left(c_{\mu}t,c_{\mu}s\right)\right|\leq
C\hbar^{\frac{1}{3}}.$

It is useful to get rid of the removable singularity $t=s$ in $K_{N,x,\hbar}$.
Toward this end, noting that for any differentiable functions $f,g$ on
${\mathbb{R}}$,

$\frac{f(t)g(s)-f(s)g(t)}{t-s}=g(s)\int_{0}^{1}f^{\prime}(\lambda
t+(1-\lambda)s)d\lambda-f(s)\int_{0}^{1}g^{\prime}(\lambda
t+(1-\lambda)s)d\lambda,$

we deduce that

$\displaystyle K_{N,x,\hbar}(t,s)$ $\displaystyle=\sqrt{\frac{\hbar
N}{2}}\psi_{N-1}^{\hbar}(x+\hbar
s)\int_{0}^{1}{\psi_{N}^{\hbar}}^{\prime}(\lambda(x+\hbar
t)+(1-\lambda)(x+\hbar s))d\lambda$ $\displaystyle-\sqrt{\frac{\hbar
N}{2}}\psi_{N}^{\hbar}(x+\hbar
s)\int_{0}^{1}{{\psi}_{N-1}^{\hbar}}^{\prime}(\lambda(x+\hbar
t)+(1-\lambda)(x+\hbar s))d\lambda$ $\displaystyle=\sqrt{\frac{\hbar
N}{2}}\psi_{N-1}^{\hbar}(x+\hbar
s)\int_{0}^{1}\left(\sqrt{\frac{2N}{\hbar}}\psi_{N-1}^{\hbar}(z)-\frac{z}{\hbar}\psi_{N}^{\hbar}(z)\right)_{z=x+\hbar[\lambda
t+(1-\lambda)s]}d\lambda$ $\displaystyle-\sqrt{\frac{\hbar
N}{2}}\psi_{N}^{\hbar}(x+\hbar
s)\int_{0}^{1}\left(\sqrt{\frac{2N-2}{\hbar}}\psi_{N-2}^{\hbar}(z)-\frac{z}{\hbar}\psi_{N-1}^{\hbar}(z)\right)_{z=x+\hbar[\lambda
t+(1-\lambda)s]}d\lambda$

where we used relation (A.9) in the last equality.

We can now insert the uniform Plancherel-Rotach asymptotics (A.10)-(A.11),
perform the integrals and use elementary trigonometric identities to conclude
the proof of (4.5).

To prove the $C^{1}$-local uniform convergence, we start by taking the
derivative(s) of the Christoffel-Darboux kernel
$\partial^{\alpha}_{t}\partial^{\beta}_{s}K_{N,x,\hbar}\left(t,s\right)$. This
entails computing the derivatives of Hermite functions. Now the trick is to
write the derivative ${\psi_{n}^{\hbar}}^{\prime}$ as a combination of Hermite
functions (not differentiated) using again formula (A.9). Hence, the local
uniform asymptotics of ${\psi_{n}^{\hbar}}^{\prime}$ can be read off from the
Plancherel-Rotach asymptotics (A.10)-(A.11) of $\psi_{n}^{\hbar}$. The proof
of the $C^{1}$-convergence is therefore a simple modification of the proof of
(4.5).

To prove (4.6), we use again (A.9) to write the kernel as

$\displaystyle K_{N}(x,y)$
$\displaystyle={\frac{\hbar}{2}}\frac{\psi_{N}^{\hbar}(x){\psi_{N}^{\hbar}}^{\prime}(y)-\psi_{N}^{\hbar}(y){\psi_{N}^{\hbar}}^{\prime}(x)}{x-y}-\frac{1}{2}\psi_{N}^{\hbar}(x)\psi_{N}^{\hbar}(y).$

If we set

(4.7)
$\Psi_{N}^{\hbar}(t):=\hbar^{-\frac{1}{6}}\left(V_{\sqrt{2\mu},\hbar^{\frac{2}{3}}}\psi_{N}^{\hbar}\right)(t)=\hbar^{\frac{1}{6}}\psi_{N}^{\hbar}(\sqrt{2\mu}+\hbar^{\frac{2}{3}}t),$

then,

(4.8)
$K_{N,\sqrt{2\mu},\hbar^{\frac{2}{3}}}\left(t,s\right)={\frac{1}{2}}\frac{\Psi_{N}^{\hbar}(t){\Psi_{N}^{\hbar}}^{\prime}(s)-\Psi_{N}^{\hbar}(s){\Psi_{N}^{\hbar}}^{\prime}(t)}{t-s}-\frac{\hbar^{\frac{1}{3}}}{2}\Psi_{N}^{\hbar}(t)\Psi_{N}^{\hbar}(s).$

By the Plancherel-Rotach asymptotics (A.12), for any compact set
$J\Subset{\mathbb{C}}$,

(4.9) $\lim_{N\to\infty}\sup_{t\in
J}|\Psi_{N}^{\hbar}(t)-2^{\frac{1}{2}}c_{\mu}^{-\frac{1}{2}}\operatorname{Ai}(c_{\mu}t)|=0.$

Since the functions $\Psi_{N}^{\hbar}$ are entire, the above locally uniform
convergence entails the uniform convergence of ${\Psi_{N}^{\hbar}}^{\prime}$
to $\operatorname{Ai}^{\prime}$ on compact subsets of ${\mathbb{C}}$ (a
standard application of Cauchy’s integral formula).

By the very same argument, each finite-$N$ kernel
$K_{N,\sqrt{2\mu},\hbar^{\frac{2}{3}}}$ is analytic, and hence their
derivatives converge to the derivatives of the Airy kernel. The proof is
complete. ∎

######  Remark 9.

Since
${\sigma}_{P_{<N}}^{\hbar}\in\SS({\mathbb{R}}_{x}\times{\mathbb{R}}_{p})$, we
have that the Fourier transform are rapidly decreasing functions too,
$\mathcal{F}_{2}{\sigma}_{P_{<N}}^{\hbar}\in\SS({\mathbb{R}}_{x}\times{\mathbb{R}}_{y})$.
On the contrary,

$\mathcal{F}_{2}\chi_{D}(x,y)=\mu\rho_{\mu}(x)K_{\operatorname{sine}}\left(-\mu\rho_{\mu}(x)y/2,\mu\rho_{\mu}(x)y/2\right)=\frac{\sin\left[\sqrt{(2\mu-x^{2})_{+}}y\right]}{\pi
y}$

is not integrable in ${\mathbb{R}}_{x}\times{\mathbb{R}}_{y}$ and this tells
that we cannot get in (4.2) a convergence stronger than uniform on compact
subsets.

In order to prove Theorem 2 we will also need to show that the Airy kernel on
the antidiagonal is dominated by and integrable function.

######  Lemma 4.

There exist positive constants $C$, $c$ such that, for all $N\in\mathbb{N}$,
with $\hbar N$ fixed,

$\left|K_{N,\sqrt{2\mu},\hbar^{\frac{2}{3}}}\left(-y,y\right)\right|\leq
Ce^{-c\left|y\right|^{\frac{3}{2}}},\quad\text{for all $y\in{\mathbb{R}}$}.$

###### Proof.

We start from the formula (recall the notation (2.2)):

(4.10) $K_{N,\sqrt{2\hbar
N},\hbar^{\frac{1}{2}}\mu^{\frac{1}{6}}N^{-\frac{1}{6}}}(s,t)=\\\
\sqrt{2}\mu^{\frac{1}{6}}N^{\frac{1}{3}}\int_{0}^{\infty}\left(V_{\sqrt{2\hbar
N},\hbar^{\frac{1}{2}}\mu^{\frac{1}{6}}N^{-\frac{1}{6}}}\psi_{N}^{\hbar}\right)\left(u+s\right)\left(V_{\sqrt{2\hbar
N},\hbar^{\frac{1}{2}}\mu^{\frac{1}{6}}N^{-\frac{1}{6}}}\psi_{N}^{\hbar}\right)\left(u+t\right)\,du\\\
+\frac{\mu^{\frac{1}{3}}}{2N^{\frac{1}{3}}}\int_{0}^{\infty}(s+t+2u)\left(V_{\sqrt{2\hbar
N},\hbar^{\frac{1}{2}}\mu^{\frac{1}{6}}N^{-\frac{1}{6}}}\psi_{N}^{\hbar}\right)\left(u+s\right)\left(V_{\sqrt{2\hbar
N},\hbar^{\frac{1}{2}}\mu^{\frac{1}{6}}N^{-\frac{1}{6}}}\psi_{N}^{\hbar}\right)\left(u+t\right)\,du\\\
+\frac{1}{2}\int_{0}^{\infty}\left(V_{\sqrt{2\hbar
N},\hbar^{\frac{1}{2}}\mu^{\frac{1}{6}}N^{-\frac{1}{6}}}\psi_{N}^{\hbar}\right)^{\prime}\left(u+s\right)\left(V_{\sqrt{2\hbar
N},\hbar^{\frac{1}{2}}\mu^{\frac{1}{6}}N^{-\frac{1}{6}}}\psi_{N}^{\hbar}\right)\left(u+t\right)+\\\
\bigl{[}\left(V_{\sqrt{2\hbar
N},\hbar^{\frac{1}{2}}\mu^{\frac{1}{6}}N^{-\frac{1}{6}}}\psi_{N}^{\hbar}\right)\left(u+s\right)\left(V_{\sqrt{2\hbar
N},\hbar^{\frac{1}{2}}\mu^{\frac{1}{6}}N^{-\frac{1}{6}}}\psi_{N}^{\hbar}\right)^{\prime}\left(u+t\right)\bigr{]}\,du.$

This is an identity true for all $\hbar$, $N$, and $\mu$. It can be proved
from the representation (4.8) using the differential equation for the Hermite
functions.

If $\hbar N=\mu$, on the antidiagonal $-s=t=y$ we thus get

(4.11)
$K_{N,\sqrt{2\mu},\hbar^{\frac{2}{3}}}(-y,y)=\underbrace{\sqrt{2}\mu^{\frac{2}{3}}\int_{0}^{\infty}\Psi_{N}^{\hbar}(u-y)\Psi_{N}^{\hbar}(u+y)\,du}_{I_{1}}\\\
+\underbrace{\hbar^{\frac{2}{3}}\int_{0}^{\infty}u\,\Psi_{N}^{\hbar}(u-y)\Psi_{N}^{\hbar}(u+y)\,du}_{I_{2}}\\\
+\underbrace{\frac{\hbar^{\frac{1}{3}}}{2}\int_{0}^{\infty}\left[{\Psi_{N}^{\hbar}}^{\prime}(u-y)\Psi_{N}^{\hbar}(u+y)+\Psi_{N}^{\hbar}(u-y){\Psi_{N}^{\hbar}}^{\prime}(u+y)\right]\,du}_{I_{3}},$

where we used the notation (4.7). To estimate $I_{1}$, $I_{2}$, and $I_{3}$,
we need some explicit bounds on the rescaled wavefunctions $\Psi_{N}^{\hbar}$.
Note that $K_{N,\sqrt{2\mu},\hbar^{\frac{2}{3}}}(-y,y)$ is even, and so it
suffices to study the case $y>0$. A useful bound is

$\left|\psi_{N}^{\hbar}(y)\right|\leq\frac{C^{\prime}}{N^{\frac{1}{12}}\hbar^{\frac{1}{4}}},$

for all $y$, see [31]. Hence the rescaled wavefunctions are uniformly bounded
by a constant

(4.12)
$\left|\Psi_{N}^{\hbar}(y)\right|\leq\frac{C^{\prime}}{N^{\frac{1}{12}}\hbar^{\frac{1}{12}}}\leq
C^{\prime\prime}.$

To get an integrable estimate on $\Psi_{N}^{\hbar}(y)$ for $y>0$ we employ a
theorem by Sonin and Polya [48, Theorem 7.31.1] giving quantitative growth
information on the solutions of Sturm-Liouville equation. We observe that
$\Psi_{N}^{\hbar}(y)$ satisfies the differential equation

${\Psi_{N}^{\hbar}}^{\prime\prime}=V\Psi_{N}^{\hbar},\quad\text{where}\quad
V(y)=2\sqrt{2\mu}y+\hbar^{\frac{2}{3}}y^{2}-\hbar^{\frac{1}{3}},$

for all $y\in{\mathbb{R}}$. If $b$ denotes the positive zero of $V$, then we
have

  1. (1)

$\Psi_{N}^{\hbar}>0$ on $[b,+\infty)$;

  2. (2)

$\lim_{y\to\infty}\left(\log\Psi_{N}^{\hbar}(y)\right)^{\prime}=-\infty$.

  3. (3)

$V>0$ and $V^{\prime}>0$ on $[b,+\infty)$;

(For the first we use known bounds [31] on the largest zero of the Hermite
polynomial of degree $N$; the second is true because $\Psi_{N}^{\hbar}$ is a
polynomial times a Gaussian.) The above mentioned theorem of Sonin and Polya
(see the formulation in [2, Lemma 3.9.31]) allows to conclude that

$\left(\log\Psi_{N}^{\hbar}(y)\right)^{\prime}\leq-\sqrt{V}\quad\text{on
$[b,+\infty)$}.$

Hence,

$\displaystyle\Psi_{N}^{\hbar}(y)$
$\displaystyle\leq\Psi_{N}^{\hbar}(b)\exp\left(-\int_{b}^{y}\sqrt{V(y^{\prime})}dy^{\prime}\right)$
$\displaystyle\leq\Psi_{N}^{\hbar}(b)\exp\left(-\int_{0}^{y}\sqrt{\left(2\sqrt{2\mu}y+\hbar^{\frac{2}{3}}y^{2}-\hbar^{\frac{1}{3}}\right)_{+}}dy^{\prime}\right)$
$\displaystyle\leq\Psi_{N}^{\hbar}(b)\exp\left(-\int_{0}^{y}\sqrt{\left(2\sqrt{2\mu}y-\hbar^{\frac{1}{3}}\right)_{+}}dy^{\prime}\right)$
$\displaystyle\leq
c^{\prime}\exp\left(-\frac{2}{3}c^{\prime}\left(y-c^{\prime}\hbar^{\frac{1}{3}}\right)^{\frac{3}{2}}\right),$

for all $y\geq b$. A short calculation shows that
$0<b<\frac{1}{2}\frac{\hbar^{\frac{1}{3}}}{(2\mu)^{\frac{1}{2}}}+\frac{1}{8}\frac{\hbar^{\frac{4}{3}}}{(2\mu)^{\frac{3}{2}}}$.
Since $\Psi_{N}^{\hbar}(y)\to\operatorname{Ai}(y)$ pointwise, with different
constants we have

(4.13) $\Psi_{N}^{\hbar}(y)\leq
c^{\prime}\exp\left(-c^{\prime}y^{\frac{3}{2}}\right),\quad\text{for $y\geq
0$}.$

We can now estimate, for $y>0$,

$\displaystyle I_{1}$ $\displaystyle\leq
C\int_{0}^{\infty}\left|\Psi_{N}^{\hbar}(u-y)\right|\left|\Psi_{N}^{\hbar}(u+y)\right|\,du$
$\displaystyle\leq
C\int_{0}^{+\infty}\left|\Psi_{N}^{\hbar}(u-y)\right|\exp\left(-c\left(u+y\right)^{\frac{3}{2}}\right)du$
$\displaystyle\leq
C\exp\left(-cy^{\frac{3}{2}}\right)\int_{0}^{+\infty}\left|\Psi_{N}^{\hbar}(u-y)\right|du$
$\displaystyle\leq
C\exp\left(-cy^{\frac{3}{2}}\right)\left(\int_{0}^{y}\left|\Psi_{N}^{\hbar}(u-y)\right|du+\int_{y}^{\infty}\left|\Psi_{N}^{\hbar}(u-y)\right|du\right)$
$\displaystyle\leq
C\exp\left(-cy^{\frac{3}{2}}\right)\left(\int_{-y}^{0}\Psi_{N}^{\hbar}(u)du+\int_{0}^{\infty}\Psi_{N}^{\hbar}(u)du\right)$
$\displaystyle\leq C\exp\left(-cy^{\frac{3}{2}}\right)\left(Cy+C\right)$
$\displaystyle\leq C\exp\left(-cy^{\frac{3}{2}}\right).$

where $C,c$ denote different constants in each line. In the second to last
step we used the uniform bound (4.12) on ${\mathbb{R}}$ and the integrable
bound (4.13) on $[0,\infty)$.

The analysis of $I_{2}$ and $I_{3}$ as functions of $y$ proceeds almost
verbatim. Moreover, they are $o(1)$ as $N\to\infty$, so that their
contribution is negligible. ∎

######  Remark 10.

Since $K_{N,\sqrt{2\mu},\hbar^{\frac{2}{3}}}(-y,y)\to
K_{\operatorname{Ai}}(-y,y)$ pointwise, it follows that
$K_{\operatorname{Ai}}(-y,y)$ is also dominated by $Ce^{-c|y|^{\frac{3}{2}}}$.
For an illustration of the kernels see Fig. 5.

Figure 5. The kernel at the edge $K_{N,\sqrt{2\mu},\hbar^{\frac{2}{3}}}(-y,y)$
compared to its limit $c_{\mu}K_{\operatorname{Ai}}(-c_{\mu}y,c_{\mu}y)$. Here
$\mu=2$. Note the integrable tails (see Lemma 4).

###  4.2. Asymptotics of the symbols

Note that ${\sigma}_{H_{N}}^{\hbar}(x,p)\neq p{\sigma}_{P_{<N}}^{\hbar}(x,p)$.
This is _not_ surprising since the operators $\hat{p}$ and $P_{<N}$ do not
commute.

######  Lemma 5.

For all $x,p\in{\mathbb{R}}$:

(4.14) ${\sigma}_{H_{N}}^{\hbar}(x,p)=\ p{\sigma}_{P_{<N}}^{\hbar}(x,p)\\\
+\frac{i}{2}\sqrt{\frac{\hbar
N}{2}}\int_{{\mathbb{R}}_{y}}\hbar\left[\psi^{\hbar}_{N-1}\left(x-\frac{\hbar
y}{2}\right)\psi^{\hbar}_{N}\left(x+\frac{\hbar
y}{2}\right)-\psi^{\hbar}_{N}\left(x-\frac{\hbar
y}{2}\right)\psi^{\hbar}_{N-1}\left(x+\frac{\hbar
y}{2}\right)\right]e^{ipy}dy.$

###### Proof.

An application of the three-term recurrence of the Hermite functions (A.9). ∎

######  Proposition 7.

The families $\\{{\sigma}_{P_{<N}}^{\hbar}\\}_{N\geq 1}$ and
$\\{{\sigma}_{H_{N}}^{\hbar}\\}_{N\geq 1}$ are bounded in
$\mathcal{A}^{\prime}$. Moreover,

(4.15)
$\|{\sigma}_{H_{N}}^{\hbar}-p{\sigma}_{P_{<N}}^{\hbar}\|_{\mathcal{A}^{\prime}}\leq\hbar\sqrt{\frac{\mu}{2}},$

thus the distance between ${\sigma}_{H_{N}}^{\hbar}(x,p)$ and
$p{\sigma}_{P_{<N}}^{\hbar}(x,p)$ is asymptotically small in
$\mathcal{A}^{\prime}$, as $N\to\infty$, $\hbar\to 0$, with $\hbar N\to\mu>0$.

###### Proof of Proposition 7.

Let $f\in\mathcal{A}$. From Plancherel’s theorem

$\displaystyle\langle{\sigma}_{P_{<N}}^{\hbar},f\rangle$
$\displaystyle=\int_{{\mathbb{R}}_{x}\times{\mathbb{R}}_{y}}\overline{\hbar
K_{N}\left(x-\frac{\hbar y}{2},x+\frac{\hbar
y}{2}\right)}\mathcal{F}_{2}f(x,y)dydx,$
$\displaystyle\langle{\sigma}_{H_{N}}^{\hbar},f\rangle$
$\displaystyle=\int_{{\mathbb{R}}_{x}\times{\mathbb{R}}_{y}}\overline{\hbar
Q_{N}\left(x-\frac{\hbar y}{2},x+\frac{\hbar
y}{2}\right)}\mathcal{F}_{2}f(x,y)dydx.$

We can estimate

$\displaystyle\left|\langle{\sigma}_{P_{<N}}^{\hbar},f\rangle\right|$
$\displaystyle\leq\hbar\left(\int\sup_{x}\left|\mathcal{F}_{2}f(x,y)\right|dy\right)\left(\sup_{y}\int\left|K_{N}\left(x-\frac{\hbar
y}{2},x+\frac{\hbar y}{2}\right)\right|dx\right)$
$\displaystyle\leq\hbar\|f\|_{\mathcal{A}}\sup_{y}\sum_{j=0}^{N-1}\left(\int\left|\psi_{k}^{\hbar}\left(x-\hbar
y/2\right)\psi_{k}^{\hbar}\left(x+\hbar y/2\right)\right|dx\right)$
$\displaystyle\leq\hbar\|f\|_{\mathcal{A}}\sup_{y}\sum_{j=0}^{N-1}\left(\int\left|\psi_{k}^{\hbar}\left(x-\hbar
y/2\right)\right|^{2}dx\,\int\left|\psi_{k}^{\hbar}\left(x+\hbar
y/2\right)\right|^{2}dx\right)^{1/2}$ $\displaystyle\leq\hbar
N\|f\|_{\mathcal{A}}.$

Similarly,

$\displaystyle\left|\langle\sigma^{\hbar}_{H_{N}},f\rangle\right|\leq\hbar\|f\|_{\mathcal{A}}\sqrt{\frac{\hbar}{2}}\sup_{y}\sum_{j=0}^{N-2}2\sqrt{j+1}\leq\sqrt{2}(\hbar
N)^{3/2}\|f\|_{\mathcal{A}}.$

The convergent sequence $\hbar N$ is bounded from above. The proof of the
uniform boundedness of the symbols is complete.

With the help of Lemma 5, similar calculations are used in the proof of
(4.15),

$\displaystyle\|{\sigma}_{H_{N}}^{\hbar}-p{\sigma}_{P_{<N}}^{\hbar}\|_{\mathcal{A}^{\prime}}$
$\displaystyle\leq\sup_{y}\frac{\hbar}{2}\sqrt{\frac{\mu}{2}}\int_{{\mathbb{R}}}\left|\psi^{\hbar}_{N}\left(x-\frac{\hbar
y}{2}\right)\psi^{\hbar}_{N-1}\left(x+\frac{\hbar
y}{2}\right)-\psi^{\hbar}_{N-1}\left(x-\frac{\hbar
y}{2}\right)\psi^{\hbar}_{N}\left(x+\frac{\hbar y}{2}\right)\right|dx$
$\displaystyle\leq\hbar\sqrt{\frac{\mu}{2}}\|\psi_{N}^{\hbar}\|_{2}\|\psi_{N-1}^{\hbar}\|_{2}.$

∎

###  4.3. Proofs of Theorems 1, 2 and 3

###### Proof of Theorem 1.

Notice that, by Proposition 5, for any compact sets $U,V\Subset{\mathbb{R}}$,
there is a constant $C=C(U,V,\mu)>0$, such that

(4.16) $\displaystyle\sup_{x\in U}\sup_{y\in
V}\left|\left(\mathcal{F}_{2}{\sigma}_{P_{<N}}^{\hbar}-\mathcal{F}_{2}\chi_{D}\right)\left(x,y\right)\right|\leq\frac{C}{N},$
(4.17) $\displaystyle\sup_{x\in U}\sup_{y\in
V}\left|\left(\mathcal{F}_{2}p{\sigma}_{P_{<N}}^{\hbar}-\mathcal{F}_{2}p\chi_{D}\right)\left(x,y\right)\right|\leq\frac{C}{N},$

for all $N\geq 1$, where

$\mathcal{F}_{2}\chi_{D}(x,y)=\mu\rho_{\mu}(x)K_{\operatorname{sine}}\left(-\mu\rho_{\mu}(x)y/2,\mu\rho_{\mu}(x)y/2\right).$

It is enough to show the two claims (1.5)-(1.6) for all $f\in\mathcal{B}$. By
the density of $\mathcal{B}$ in $\mathcal{A}$ the thesis will follow. Let
$f\in\mathcal{B}$ (so that $\mathcal{F}_{2}f$ has compact support
$J\Subset{\mathbb{R}}_{x}\times{\mathbb{R}}_{y}$, see Section 2). Then,

$\displaystyle\left|\int\overline{\left[{\sigma}_{P_{<N}}^{\hbar}\left(x,p\right)-\chi_{D}(x,p)\right]}f(x,p)dxdp\right|$
$\displaystyle=\left|\int\overline{\left[\mathcal{F}_{2}{\sigma}_{P_{<N}}^{\hbar}(x,y)-\mathcal{F}_{2}\chi_{D}(x,y)\right]}\mathcal{F}_{2}f(x,y)dxdy\right|$
$\displaystyle\leq\|\mathcal{F}_{2}f\|_{\infty}\int_{J}\left|\mathcal{F}_{2}{\sigma}_{P_{<N}}^{\hbar}(x,y)-\mathcal{F}_{2}\chi_{D}(x,y)\right|dxdy\leq
C\|\mathcal{F}_{2}f\|_{\infty}N^{-1}.$

for some constant $C$ (dependent on $J$). Hence, for all $f\in\mathcal{B}$,

$\lim_{N\to\infty}\langle{\sigma}_{P_{<N}}^{\hbar}-\chi_{D},f\rangle=0.$

Similarly, for $f\in\mathcal{B}$,

$\left|\langle{\sigma}_{H_{N}}^{\hbar}-p\chi_{D},f\rangle\right|\leq\left|\langle{\sigma}_{H_{N}}^{\hbar}-p{\sigma}_{P_{<N}}^{\hbar},f\rangle\right|+\left|\langle
p{\sigma}_{P_{<N}}^{\hbar}-p\chi_{D},f\rangle\right|.$

The first term is of order $O(N^{-1})$ by Proposition 7; the second term is
also $O(N^{-1})$ by Eq. (4.17). Hence, for all $f\in\mathcal{B}$,

$\lim_{N\to\infty}\langle{\sigma}_{H_{N}}^{\hbar}-p\chi_{D},f\rangle=0.$

∎

Before proving Theorem 2, we need some notation. Consider the change of
coordinates

$\displaystyle T\colon{\mathbb{R}}_{x}\times{\mathbb{R}}_{p}$
$\displaystyle\to{\mathbb{C}}^{2}$ (4.18) $\displaystyle(x,p)$
$\displaystyle\mapsto(\theta,\zeta)$

where $\zeta$ is solution of

(4.19) $\displaystyle x^{2}+p^{2}=2\mu+\zeta^{2},$

and $\theta\in[0,2\pi)$ is given by

(4.20) $\theta=\begin{cases}\arctan\frac{p}{x}&\text{if $x\neq 0$}\\\
\frac{\pi}{2}&\text{if $x=0$, $p>0$}\\\ \frac{3\pi}{2}&\text{if $x=0$,
$p<0$}\end{cases}.$

If $(x,p)\notin D$ then $\zeta\in(0,\infty)$; if t$(x,p)\in D$, then
$\zeta\in[0,i\sqrt{2\mu}]$. $T$ is a bijection from
${\mathbb{R}}_{x}\times{\mathbb{R}}_{p}$ to
$T({\mathbb{R}}^{2}\setminus(0,0))=[0,2\pi)\times({\mathbb{R}}_{+}\cup[0,i\sqrt{2\mu}])$
with Jacobian determinant

(4.21) $|J|:=\left|\det\frac{\partial
T(x,p)}{\partial(\zeta,\theta)}\right|=|\zeta|.$

We can now prove Theorem 2.

###### Proof of Theorem 2.

Let $g\in C^{\infty}_{c}(\mathbb{R})$. We have

$\displaystyle\int_{{\mathbb{R}}_{x}\times{\mathbb{R}}_{p}}{\sigma}_{P_{<N}}^{\hbar}(x,p)\frac{1}{\hbar^{\frac{2}{3}}}g\left(\frac{x^{2}+p^{2}-2\mu}{\hbar^{\frac{2}{3}}}\right)dxdp$
$\displaystyle=\int_{D^{c}}{\sigma}_{P_{<N}}^{\hbar}(x,p)\frac{1}{\hbar^{\frac{2}{3}}}g\left(\frac{x^{2}+p^{2}-2\mu}{\hbar^{\frac{2}{3}}}\right)dxdp+\int_{D}{\sigma}_{P_{<N}}^{\hbar}(x,p)\frac{1}{\hbar^{\frac{2}{3}}}g\left(\frac{x^{2}+p^{2}-2\mu}{\hbar^{\frac{2}{3}}}\right)dxdp$
$\displaystyle=2\pi\int_{0}^{+\infty}{\sigma}_{P_{<N}}^{\hbar}(\sqrt{2\mu},z\hbar^{\frac{1}{3}})zg(z^{2})dz+2\pi\int_{0}^{+\infty}{\sigma}_{P_{<N}}^{\hbar}(\sqrt{2\mu},iz\hbar^{\frac{1}{3}})zg(-z^{2})\chi_{(0,\frac{2\mu}{\hbar^{2/3}})}(z)dz$
$\displaystyle=2\pi\int_{{\mathbb{R}}_{y}}\int_{0}^{+\infty}K_{N,\sqrt{2\mu},\hbar^{\frac{2}{3}}}\left(-y/2,y/2\right)e^{-izy}zg(z^{2})dzdy$
$\displaystyle+2\pi\int_{{\mathbb{R}}_{y}}\int_{0}^{+\infty}K_{N,\sqrt{2\mu},\hbar^{\frac{2}{3}}}\left(-y/2,y/2\right)e^{zy}zg(-z^{2})\chi_{(0,\frac{2\mu}{\hbar^{2/3}})}(z)dzdy,$

where in the third line we used the rotational symmetry of the symbol
${\sigma}_{P_{<N}}^{\hbar}$ and we considered the symbol
${\sigma}_{P_{<N}}^{\hbar}$ as a function on
${\mathbb{C}}_{x}\times{\mathbb{C}}_{p}$ (see Remark 8). Similarly,

$\displaystyle\int_{{\mathbb{R}}_{x}\times{\mathbb{R}}_{p}}\chi_{D}^{(N)}\left(x,p\right)\frac{1}{\hbar^{\frac{2}{3}}}g\left(\frac{x^{2}+p^{2}-2\mu}{\hbar^{\frac{2}{3}}}\right)dxdp$
$\displaystyle=2\pi\int_{0}^{+\infty}\operatorname{Ai}_{1}\left(\frac{1}{2^{\frac{1}{3}}\mu^{\frac{1}{3}}}\left(\frac{x^{2}+p^{2}-2\mu}{\hbar^{\frac{2}{3}}}\right)\right)\frac{1}{\hbar^{\frac{2}{3}}}g\left(\frac{x^{2}+p^{2}-2\mu}{\hbar^{\frac{2}{3}}}\right)dxdp$
$\displaystyle=2\pi\int_{{\mathbb{R}}_{y}}\int_{0}^{+\infty}c_{\mu}K_{\operatorname{Ai}}(-c_{\mu}y/2,c_{\mu}y/2)e^{-izy}zg(z^{2})dzdy$
$\displaystyle+2\pi\int_{{\mathbb{R}}_{y}}\int_{0}^{+\infty}c_{\mu}K_{\operatorname{Ai}}(-c_{\mu}y/2,c_{\mu}y/2)e^{zy}zg(-z^{2})\chi_{(0,\frac{2\mu}{\hbar^{2/3}})}(z)dzdy.$

Hence,

$\displaystyle\left|\int_{{\mathbb{R}}_{x}\times{\mathbb{R}}_{p}}\left[{\sigma}_{P_{<N}}^{\hbar}(x,p)-\chi_{D}^{(N)}\left(x,p\right)\right]\frac{1}{\hbar^{\frac{2}{3}}}g\left(\frac{x^{2}+p^{2}-2\mu}{\hbar^{\frac{2}{3}}}\right)dxdp\right|\leq
I_{N}+J_{N},$

where

$\displaystyle
I_{N}=2\pi\int_{0}^{+\infty}\left(\int_{{\mathbb{R}}_{y}}f_{N}(y)dy\right)\left|zg(z^{2})\right|dz,\quad
J_{N}=2\pi\int_{0}^{+\infty}\left(\int_{{\mathbb{R}}_{y}}f_{N}(y)e^{zy}dy\right)\left|zg(z^{2})\right|dz,$

and
$f_{N}(y):=\left|K_{N,\sqrt{2\mu},\hbar^{\frac{2}{3}}}\left(-y/2,y/2\right)-c_{\mu}K_{\operatorname{Ai}}(-c_{\mu}y/2,c_{\mu}y/2)\right|$.
By Proposition 6 $f_{N}(y)$ tends to zero uniformly on compact sets (and hence
pointwise). The tail estimate of Lemma 4 implies that the sequences $f_{N}(y)$
and $f_{N}(y)e^{zy}$ are dominated by an integrable function, and so by the
dominated convergence theorem both $\int f_{N}(y)dy$ and $\int
f_{N}(y)e^{zy}dy$ tend to zero (the latter for any $z$). Since
$\operatorname{supp}g\Subset\mathbb{R}_{z}$, we conclude that both $I_{N}$ and
$J_{N}$ go to zero as $N\to\infty$. This proves (1.10) from which (1.11)
follows by recalling Lemma 5. ∎

###### Proof of Theorem 3.

It is again enough to prove (1.13). The second claim (1.14) will follow by
Lemma 5. Fix $\epsilon>0$. By rotational symmetry we can assume
$x=\sqrt{2\mu}$ and $p=z\in{\mathbb{R}}$.

$\displaystyle\left|{\sigma}_{P_{<N}}^{\hbar}(\sqrt{2\mu},z)-\chi_{D}^{(N)}(\sqrt{2\mu},z)\right|$
$\displaystyle\leq\int_{{\mathbb{R}}_{y}}\left|K_{N,\sqrt{2\mu},\hbar^{\frac{2}{3}}}\left(y/2,-y/2\right)-c_{\mu}K_{\operatorname{Ai}}(-c_{\mu}y/2,c_{\mu}y/2)\right|dy$
$\displaystyle\leq\int_{-L}^{L}\left|K_{N,\sqrt{2\mu},\hbar^{\frac{2}{3}}}\left(-y/2,y/2\right)-c_{\mu}K_{\operatorname{Ai}}(-c_{\mu}y/2,c_{\mu}y/2)\right|dy$
$\displaystyle+2\int_{L}^{+\infty}\left|K_{N,\sqrt{2\mu},\hbar^{\frac{2}{3}}}\left(-y/2,y/2\right)\right|dy$
$\displaystyle+2\int_{L}^{+\infty}\left|c_{\mu}K_{\operatorname{Ai}}(-c_{\mu}y/2,c_{\mu}y/2)\right|dy:=I_{1}+I_{2}+I_{3}$

for every $L>0$. Choose $L_{0}$ such that the $I_{2}$ and $I_{3}$ are each
bounded by $\epsilon/3$. The first integral $I_{1}$ is bounded by
$C\hbar^{\frac{1}{3}}$, with $C=C(L_{0})$. Take
$\hbar=\left(\epsilon/(3C)\right)^{3}$ and conclude the proof. ∎

## Acknowledgements

FDC thanks Nick Simm for helpful correspondence. The authors would also like
to thank Gerardo Hernandez-Duenas and Alejandro Uribe for pointing out the
paper [29]. We acknowledge the support by the Italian National Group of
Mathematical Physics (GNFM-INdAM), by PNRR MUR projects CN00000013-‘Italian
National Centre on HPC, Big Data and Quantum Computing’ and PE0000023-NQSTI,
by Regione Puglia through the project ‘Research for Innovation’ - UNIBA024,
and by Istituto Nazionale di Fisica Nucleare (INFN) through the project
‘QUANTUM’.

## Declarations

### Funding and/or Conflicts of interests/Competing interests

The authors have no competing interests to declare that are relevant to the
content of this article.

### Data availability

Data sharing not applicable to this article as no datasets were generated or
analysed during the current study.

##  Appendix A Quantum harmonic oscillator and Hermite polynomials

The classical harmonic oscillator Hamiltonian function is

(A.1)
$\mathfrak{h}_{\operatorname{h.o.}}(x,p)=\frac{1}{2}\left(p^{2}+x^{2}\right).$

The _harmonic oscillator Schrödinger operator_ is

(A.2)
$H_{\operatorname{h.o.}}=\frac{1}{2}\left(-\hbar^{2}\partial^{2}_{x}+\hat{x}^{2}\right).$

The _Hermite functions_ $(\alpha^{2}=1/\hbar)$ are

(A.3)
$\psi_{k}^{\hbar}(x)=\sqrt{\frac{\alpha}{\sqrt{\pi}2^{k}k!}}\exp\left(-\frac{1}{2}\alpha^{2}x^{2}\right)h_{k}(\alpha
x),\qquad k=0,1,2,\ldots$

where

(A.4) $h_{k}(y)=(-1)^{k}e^{y^{2}}\frac{d^{k}}{dy^{k}}e^{-y^{2}}$

is the $k$-th _Hermite polynomials_. The Hermite functions are eigenfunctions
of the harmonic oscillator

(A.5)
$H_{\operatorname{h.o.}}\psi_{k}^{\hbar}(x)=\lambda_{k}\psi_{k}^{\hbar}(x),\quad
k=0,1,2,\ldots,$

with eigenvalues $\lambda_{k}=\hbar\left(k+\frac{1}{2}\right)$, and form an
orthonormal basis in $L^{2}({\mathbb{R}})$

(A.6)
$\int_{{\mathbb{R}}}\psi_{k}^{\hbar}(x)\psi_{\ell}^{\hbar}(x)dx=\delta_{k,\ell}.$

Useful formulae are the following three-term relations written in terms of
position operator $\hat{x}$ and momentum operator
$\hat{p}=-i\hbar\frac{d}{dx}$:

(A.7) $\displaystyle(\hat{x}\psi_{k}^{\hbar})(x)$
$\displaystyle=\sqrt{\frac{\hbar}{2}}\left[\sqrt{k+1}\psi_{k+1}^{\hbar}(x)+\sqrt{k}\psi_{k-1}^{\hbar}(x)\right]$
(A.8) $\displaystyle(\hat{p}\psi_{k}^{\hbar})(x)$
$\displaystyle=i\sqrt{\frac{\hbar}{2}}\left[\sqrt{k+1}\psi_{k+1}^{\hbar}(x)-\sqrt{k}\psi_{k-1}^{\hbar}(x)\right].$

When combined they give the useful relation

(A.9)
$\frac{d}{dz}{\psi_{k}^{\hbar}}(z)=\sqrt{\frac{2k}{\hbar}}\psi_{k-1}^{\hbar}(z)-\frac{z}{\hbar}\psi_{k}^{\hbar}(z).$

We have the following Plancherel-Rotach asymptotics formulae (see [48, Theorem
8.22.9]). Let $\epsilon<\epsilon^{\prime}$ be fixed positive numbers, and
$n\in{\mathbb{Z}}$ fixed. Let $\hbar=\hbar_{N}$ so that $\hbar_{N}N=\mu$ ,
where $\mu>0$ is a fixed number. The following asymptotics hold true:

  1. (i)

If $x=\sqrt{(2+1/N)\mu}\cos\phi$, $\epsilon\leq\phi\leq\pi-\epsilon$, then

(A.10)
$\psi_{N+n}^{\hbar}\left(x\right)=\left(\frac{2}{\mu}\right)^{\frac{1}{4}}\left(\frac{1}{\pi\sin\phi}\right)^{\frac{1}{2}}\left\\{\sin\left[\left(\frac{N}{2}+\frac{1}{4}\right)(\sin(2\phi)-2\phi)+\frac{3\pi}{4}-n\phi\right]+O(N^{-1})\right\\};$

  2. (ii)

If $x=\sqrt{(2+1/N)\mu}\cosh\phi$, $\epsilon\leq\phi\leq\epsilon^{\prime}$,
then

(A.11)
$\psi_{N+n}^{\hbar}\left(x\right)=\left(\frac{1}{8\mu}\right)^{\frac{1}{4}}\left(\frac{1}{\pi\sinh\phi}\right)^{\frac{1}{2}}\exp\left(-\left(\frac{N}{2}+\frac{1}{4}\right)(\sinh(2\phi)-2\phi)+n\phi\right)\left(1+O(N^{-1})\right);$

  3. (iii)

If $x=\sqrt{(2+1/N)\mu}+\sqrt{\mu/2}{N^{-\frac{2}{3}}}t$, with $t$ complex and
bounded, then,

(A.12)
$\psi_{N}^{\hbar}\left(x\right)=\left(\sqrt{2/\mu}N^{1/3}\right)^{\frac{1}{2}}\operatorname{Ai}(t)+O(N^{-1/2}).$

In all these formulae, the $O$-terms hold uniformly. Note that the choice
$\hbar N=\mu$ is the right scaling of a vanishing Planck constant that gives
rise to a nontrivial asymptotics.

##  Appendix B Sine kernel, Airy kernel and their Fourier transforms

We denote the normalised _semicircular density_ of radius $\sqrt{2\mu}>0$,

(B.1) $\rho_{\mu}(x)=\frac{1}{\pi\mu}\sqrt{(2\mu-x^{2})_{+}},$

and

(B.2) $c_{\mu}=2^{\frac{1}{2}}\mu^{\frac{1}{6}}.$

Here is why these factors pop out in all formulae. If the energy of a
(classical) harmonic oscillator is $\mathfrak{h}_{\mathrm{h.o.}}(x,p)=\mu$,
then the momentum as a function of position is $p(x)=\sqrt{(2\mu-x^{2})_{+}}$.
At the points of inversion of motion $x=\pm\sqrt{2\mu}$, the momentum is zero,
and its one-side derivative is $|p^{\prime}(\pm\sqrt{2\mu})|=c_{\mu}^{3}$.

The sine and the Airy kernels are

(B.3) $\displaystyle K_{\operatorname{sine}}(u,v)$
$\displaystyle=\frac{\sin\pi\left(u-v\right)}{\pi(u-v)}\quad$
$\displaystyle\text{(\emph{sine kernel})},$ (B.4) $\displaystyle
K_{\operatorname{Ai}}(u,v)$
$\displaystyle=\frac{\operatorname{Ai}(u)\operatorname{Ai}^{\prime}(v)-\operatorname{Ai}^{\prime}(u)\operatorname{Ai}(v)}{u-v}\quad$
$\displaystyle\text{(\emph{Airy kernel})},$

where the _Airy function_ is defined by the formula

(B.5) $\operatorname{Ai}(x):=\frac{1}{2\pi
i}\int_{C}e^{\zeta^{3}/3-x\zeta}d\zeta,$

with $C$ a contour in the complex $\zeta$-plane consisting of the ray joining
$e^{-i\pi/3}{\infty}$ to the origin plus the ray joining the origin to
$e^{i\pi/3}{\infty}$.

The kernels (B.3)-(B.4) are defined for $u=v$ in the unique way making them
continuous (and in fact $C^{\infty}$). The sine kernel can be viewed as the
‘square’ of another symmetric kernel,

(B.6) $K_{\operatorname{sine}}(u,v)=\int_{-1}^{1}e^{\pi
iu\lambda}\overline{e^{\pi iv\lambda}}d\lambda.$

A similar identity holds for the Airy kernel,

(B.7)
$K_{\operatorname{Ai}}(u,v)=\int_{0}^{+\infty}\operatorname{Ai}(u+\lambda)\operatorname{Ai}(v+\lambda)d\lambda.$

(Use (2.1) and the Airy differential equation, see [50].) Using a trick one
gets the following useful representation:

(B.8)
$K_{\operatorname{Ai}}(u,v)=\int_{{\mathbb{R}}}e^{iq(t-s)}\left(\int_{0}^{+\infty}\operatorname{Ai}\left(\lambda+2^{2/3}q^{2}+(u+v)/2^{1/3}\right)d\lambda\right)\frac{dq}{2\pi}.$

Note that $K_{\operatorname{sine}}(u,-u)$ is locally integrable (but not in
$L_{1}({\mathbb{R}})$), while
$\int_{{\mathbb{R}}}\left|K_{\operatorname{Ai}}(u,-u)\right|du<\infty$.

$K_{\operatorname{sine}}$ is the Fourier transform of the characteristic
function of the disk:

(B.9)
$\int_{{\mathbb{R}}}\mu\rho_{\mu}(x)K_{\operatorname{sine}}(-\mu\rho_{\mu}(x)y/2,\mu\rho_{\mu}(x)y/2)e^{ipy}dy=\chi_{D}(x,p).$

From (B.8) we also get an explicit formula for the Fourier transform of
$K_{\operatorname{Ai}}$:

(B.10)
$\int_{{\mathbb{R}}}c_{\mu}K_{\operatorname{Ai}}(-c_{\mu}y/2,c_{\mu}y/2)e^{izy}dy=\operatorname{Ai}_{1}\left(\frac{z^{2}}{\left(2\mu\right)^{\frac{1}{3}}}\right),$

where

(B.11)
$\operatorname{Ai}_{1}(\xi):=\int_{\xi}^{+\infty}\operatorname{Ai}\left(u\right)du$

is the _integrated Airy function_. We have
$\operatorname{Ai}_{1}(-\infty)=\int_{{\mathbb{R}}}\operatorname{Ai}\left(u\right)du=1$.
The function $\operatorname{Ai}_{1}(\xi)$ has the following large $|\xi|$
asymptotics [39, Eq. (9.10.4)-(9.10.6)]:

(B.12)
$\operatorname{Ai}_{1}(\xi)\sim\begin{cases}\dfrac{1}{2\pi^{1/2}|\xi|^{3/4}}e^{-\frac{2}{3}|\xi|^{2/3}},&\text{for
$\xi\to+\infty$},\\\ \\\
1-\dfrac{1}{\pi^{1/2}|\xi|^{3/4}}\cos\left({\frac{2}{3}|\xi|^{2/3}+\frac{\pi}{4}}\right),&\text{for
$\xi\to-\infty$}.\end{cases}$

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