# The Spectrum of an Almost Maximally Open Quantized Cat Map

Yonah Borns-Weil<EMAIL_ADDRESS>Department of Mathematics,
University of California, Berkeley, CA 94720

###### Abstract.

We consider eigenvalues of a quantized cat map (i.e. hyperbolic symplectic
integer matrix), cut off in phase space to include a fixed point as its only
periodic orbit on the torus. We prove a simple formula for the eigenvalues on
both the quantized real line and the quantized torus in the semiclassical
limit as $h\to 0$. We then consider the case with no fixed points, and prove a
superpolynomial decay bound on the eigenvalues. The results are illustrated
with numerical calculations.

##  1\. Introduction

The study of the classical-quantum correspondence dynamics consists of
comparing quantum evolution to classical dynamical systems. One example of
this correspondence is the relationship between continuous-time classical
dynamics and the theory ofscattering resonances – see the textbook by Dyatlov
and Zworski [8] for an introduction. Recently, there has been interest in the
discrete-time analogue given by _open quantum maps_. These maps correspond to
quantizations of symplectomorphisms on a given symplectic manifold, which are
“opened” by designating some of the points to “escape to infinity”. Of
particular interest are maps corresponding to _hyperbolic_ symplectomorphisms,
where one may study the relationship between quantum and classical dynamics
near a hyperbolic trapped set. See the paper by Nonnenmacher [15] for a
general overview of the literature.

Perhaps the most fundamental example in classical hyperbolic dynamics is the
generalized _Arnol’d cat map_ given by a matrix in
$\operatorname{SL}(2,\mathbb{Z})=\mathrm{Sp}(2,\mathbb{Z})$, which act on both
$\mathbb{R}^{2}$ and $\mathbb{T}^{2}$ and have a single hyperbolic fixed point
at the origin. The natural quantum-dynamical analogue is a class of
metaplectic operators, known as _quantized cat maps_. On the quantized torus,
such maps were treated by Bouzouina and De Biévre [2], who proved a version of
quantum ergodicity in this setting (see also Dyatlov–Jézéquel [7] and
Faure–Nonnenmacher–De Biévre [9]). These quantum cat maps can then be “opened”
as part of the general framework of open quantum maps, which corresponds to
sending some points “to infinity” to mimic scattering. Once adjusted in this
manner, such maps enjoy many similar results to continuous scattering
problems, and are a subject of current research. In the physics literature,
open quantum maps are used as a toy computational model for quantum chaos, see
Saraceno–Vallejos [19] and Nonnenmacher–Zworski [18] and references given
there. In addition, in Nonnenmacher, Sjöstrand, and Zworski [16] reduce the
study of some continuous-time systems that op open quantum maps. Schenck [20]
considers the similar case of partially open quantum maps, which gives a
discrete-time analogue to a damped wave equation.

In this paper, we consider a distinct but related problem, of determining the
quantum behavior of a cat map very near its hyperbolic fixed point.
Specifically, suppose $M$ is a cat map and $\widehat{M}$ is its quantization,
acting either on the quantized real line or the quantized torus. We may
consider $\operatorname{Op}_{h}\chi\widehat{M}$ or
$\operatorname{Op}_{N}\chi\widehat{M}_{N}$ respectively, where $\chi$ is a
bump function equal to $1$ near the origin and $\operatorname{Op}$ refers to
semiclassical quantization, see Sections 2.1 and 3.1. This has the effect of
isolating the behavior near the hyperbolic fixed point, and producing a
compact operator whose spectrum can be studied.

The case of such an “almost maximally open” map has been studied before in the
case of Schrödinger operators on $\mathbb{R}^{n}$. The first results in the
complicated setting of obstacle problems were provided by
Bardos–Guillot–Ralston [1], and the problem was settled by Ikawa [13] with
improvements by Gerard [11]. The semiclassical case was then handled by
Gerard–Sjöstrand [12] and Sjöstrand [21], see also Louati and Rouleux [14].
Surprisingly, to the author’s knowledge, the analogous situation for an open
quantum map has never been studied before. We shall show that the situations
on the torus and on the real line are similar, which matches our intuition
that everything away from a small local region had “escaped”. When localized
by a small enough microlocal cutoff operator and for small enough $h$, the
spectrum is discrete away from the origin, and the eigenvalues are given up to
$O(h^{\infty})$ error by $\lambda^{-\frac{2k+1}{2}}$, where $\lambda$ is the
positive eigenvalue of the underlying symplectic transformation.

In addition, we consider the “maximally open” case of
$\operatorname{Op}_{N}\nu\widehat{M}_{N}$, where $\nu$ is a bump function away
localized in a small annulus about the hyperbolic fixed point. This map
corresponds to a discrete-time analogue of the “nontrapping estimates” seen
for scattering resonances in Section 6.4 of [8]. In this case, the eigenvalues
decay to zero, and we shall show in Section 4 that they in fact decay to zero
like $O(h^{\infty})$.

###  1.1. The Classical Cat Map

The original Arnol’d cat map was defined as the matrix

$M=\begin{pmatrix}2&1\\\ 1&1\end{pmatrix}$

acting on $\mathbb{R}^{2}$, and more interestingly also acting on the two-
dimensional torus $\mathbb{T}^{2}$. As $M\in\mathrm{SL}(2,\mathbb{Z})$, one
sees that $M$ is invertible and measure-preserving with respect to the
Lebesgue measure. When considered as a measure-preserving map on
$\mathbb{T}^{2}$, $M$ is mixing, which in particular implies that it is
ergodic. In addition, $M$ is topologically transitive and has a dense set of
periodic points, which gives it to be _chaotic_ as defined by e.g. Teschl
[23].

These above properties hold for any $M\in\mathrm{SL}(2,\mathbb{Z})$ with
eigenvalues $\lambda,\frac{1}{\lambda}\not\in S^{1}$, so we call such matrices
_generalized cat maps_ , or for convenience just _cat maps_. We discuss the
set of fixed and periodic points in further detail. By an argument using the
pigeonhole principle, we may see that points on $\mathbb{T}^{2}$ are periodic
points of $M$ if and only if they have rational coordinates. We now provide an
elementary proposition about periodic orbits that will motivate our main
result.

######  Proposition 1.1.

Let $M=Q^{-1}DQ$ be a generalized cat map with $D$ diagonal and $\det Q=1$.
Then the only periodic orbit of $M$ on $\mathbb{T}^{2}$ contained in
$B_{\frac{1}{4\lambda\|Q\|^{2}}}(0)$ is the fixed point at $0$.

###### Proof.

We begin by proving this for symmetric $M$, which is equivalent to $\|Q\|=1$.
Write $v\in B_{\frac{1}{4\lambda}}(0)$ as $v=a_{1}v_{1}+a_{2}v_{2}$, where
$v_{1},v_{2}$ are the normalized eigenvectors of $M$ acting on
$\mathbb{R}^{2}$ corresponding to $\lambda,\lambda^{-1}$, so

$M^{n}v=\lambda^{n}a_{1}v_{1}+\lambda^{-n}a_{2}v_{2}.$ (1.1)

If $v$ is a periodic point, we can easily see that if one of $a_{1}$ is $0$
then $a_{2}$ must also be, so without loss of generality assume
$a_{1},a_{2}\neq 0$, and note that $|a_{1}|,|a_{2}|\leq\frac{1}{4\lambda}$.
Let $n=\left\lfloor\log_{\lambda}\frac{1}{4|a_{1}|}\right\rfloor$, so

$\frac{1}{4\lambda}<\lambda^{n}|a_{1}|\leq\frac{1}{4}$ (1.2)

and also

$|\lambda^{-n}a_{2}|\leq|a_{2}|\leq\frac{1}{4\lambda}.$ (1.3)

Combining (1.2) and (1.3) with (1.1) gives that
$\frac{1}{4\lambda}<\|M^{n}v\|\leq\frac{1}{2}$, which proves the result for
this case.

In the general case, we let $M=Q^{-1}DQ$. Let $v\in Q^{-1}B_{\varepsilon}(0)$.
Then after scaling, the proof for the symmetric case says that there is an
$n\in\mathbb{N}$ such that $D^{n}Qv\in B_{2\lambda\varepsilon}(0)\setminus
B_{\varepsilon}(0)$, or equivalently that $M^{n}v\in
Q^{-1}B_{2\lambda\varepsilon}(0)\setminus Q^{-1}B_{\varepsilon}(0)$. Then this
gives that any point int $B_{\frac{\varepsilon}{\|Q\|}}(0)$ will be taken to
$Q^{-1}B_{2\lambda\varepsilon}(0)\setminus Q^{-1}B_{\varepsilon}(0)$ which is
itself contained in $B_{2\lambda\varepsilon\|Q\|}(0)$. Letting
$\varepsilon=\frac{1}{4\lambda\|Q\|}$ gives the desired result. ∎

###  1.2. Overview of Theorems and Proofs

We work in the setting of operators on a quantum “state space,” which is a
Hilbert space on which “quantized” observables act. In the case of
$T^{*}\mathbb{R}$, the state space is simply $L^{2}(\mathbb{R})$, while in the
case of the torus $\mathbb{T}^{2}$, it is a space $H_{N}$ which is
$N$-dimensional and spanned by the Dirac comb distributions
$\frac{1}{\sqrt{N}}\sum_{k\in\mathbb{Z}}\delta_{x=k+\frac{n}{N}}$. Observables
are given by semiclassical pseudodifferential operators
$\operatorname{Op}_{h}\chi$ (on the real line) and $\operatorname{Op}_{N}\chi$
(on the torus). From that framework we can then define metaplectic operators
$\widehat{M}$, $\widehat{M_{N}}$ that “quantize” linear symplectic
transformations, which on $\mathbb{T}^{2}$ are precisely the cat maps
discussed in Section 1.1.

Our result on the real line is the following:

######  Theorem 1.

Let $\widehat{M}$ be a metaplectic operator with eigenvalues
$\lambda,\lambda^{-1}$ of $M$ where $\lambda>1$, and let $\chi\in
C^{\infty}_{0}(\mathbb{R}^{2})$ equal to $1$ near the origin. Then for all
$\delta>0$ there is an $h_{0}>0$ such that if $0<h<h_{0},$

$\operatorname{Spec}(\operatorname{Op}_{h}\chi\widehat{M})\cap\left\\{|z|\geq\delta\right\\}=\\{\mu_{0},\mu_{1},\dots\\}\cap\left\\{|z|\geq\delta\right\\},\quad\frac{1}{\mu_{k}}=\lambda^{\frac{2k+1}{2}}+O(h^{\infty}).$

On the quantized torus, the analogous result is:

######  Theorem 2.

Let $\widehat{M}_{N}$ be a metaplectic operator on $H_{N}$ with eigenvalues
$\lambda,\lambda^{-1}$ of $M$ where $\lambda>1$, and let $\chi\in
C^{\infty}_{0}(\mathbb{T}^{2})$ equal to $1$ near the origin with support on a
small enough neighborhood of $0$. Then for all $\delta>0$, there is an
$h_{0}>0$ such that if $0<h=\frac{1}{2\pi N}<h_{0}$,

$\operatorname{Spec}(\operatorname{Op}_{N}\chi\widehat{M}_{N})\cap\left\\{|z|\geq\delta\right\\}=\\{\mu_{0},\mu_{1},\dots\\}\cap\left\\{|z|\geq\delta\right\\},\quad\frac{1}{\mu_{k}}=\lambda^{\frac{2k+1}{2}}+O(h^{\infty}).$

For exact definitions of the terms in Theorems 1 and 2, see Sections 2.1 and
3.1 respectively.

We give an overview of the proofs of Theorems 1 and 2. There is a strong
similarity to the methods used in [11], and Theorem 1 could be proven using
his methods, though we elect to use different underlying function spaces. To
find the spectrum of $\operatorname{Op}_{h}\chi\widehat{M}$, it is enough to
consider the special case where $M=F$ is a diagonal matrix. Our strategy will
be to build a Grushin problem acting on
$L^{2}(\mathbb{R})\oplus\mathbb{C}^{K}$ given by

$\mathcal{P}(z):=\begin{pmatrix}I-z\chi(x,hD)\widehat{F}&\tilde{R}_{-}\\\
\tilde{R}_{+}&0\end{pmatrix}.$

Provided $\mathcal{P}(z)$ is invertible with inverse

$\mathcal{E}(z):=\begin{pmatrix}E(z)&\tilde{E}_{+}(z)\\\
\tilde{E}_{-}(z)&E{-+}(z)\end{pmatrix},$

then $I-z\chi(x,hD)\widehat{F}$ is invertible if and only if $E_{-+}$ is,
reducing the study of the spectrum to the invertibility of the matrix
$E_{-+}$. But $E_{-+}$ is (up to $O(h^{\infty})$ errors) conjugate to a simple
diagonal matrix, which gives the desired spectrum. To construct the operators
$\tilde{R}_{-},\tilde{R}_{+}$, we use the now standard approach as outlined by
Sjöstrand and Zworski in Section 3.4 of [22]. The main difficulty is showing
well-posedness; in other words, showing that $\mathcal{P}(z)$ is invertible.
This is achieved through careful bounding of $\chi(x,hD)\hat{F}$ on the range
of $I-\tilde{R}_{-}\tilde{R}_{+}$ with respect to an unusual norm, which is
different from but fulfills an analogous role to the one seen in [11]. The
necessary inequalities are collected and proven in Section 2.2.

To solve the corresponding problem on the quantized torus, we make use of
“sampling operators” $S_{\psi}:L^{2}(\mathbb{R})\to H_{N}$ defined in Section
3.2. The power of such operators comes from their interplay with others; in
particular, $S_{\psi}$ and $S_{\psi}S_{\psi}^{*}$ act as pseudodifferential
operators on $H_{N}$ as a subset of $\mathcal{S}^{\prime}(\mathbb{R})$, and
hence obey the rules of pseudodifferential calculus. They can then be
microlocally cut off to form “spectral operators” $T:L^{2}(\mathbb{R})\to
H_{N}$ and $T:H_{N}\to L^{2}(\mathbb{R})$ which map our Grushin problem on
$L^{2}(\mathbb{R})$ to one on $H_{N}$. The study of the spectrum can then be
carried out exactly as it was for Theorem 1.

For completeness, we also provide an analysis of the “maximally open” or
“nontrapping” case, corresponding to the classical situation of no periodic
orbits, such that every path escapes to infinity. It is not surprising that in
this case the eigenvalues will decay to $0$ as $h\to 0$. In fact, we may
strengthen that result by providing a superpolynomial rate of decay. Though an
analogous result should hold on $L^{2}(\mathbb{R})$, we only prove this fact
on the quantized torus.

######  Theorem 3.

Let $\widehat{M}_{N}$ be a metaplectic operator on $H_{N}$ with eigenvalues
$\lambda,\lambda^{-1}$ of $M$ where $\lambda>1$, and let $\nu\in
C^{\infty}_{0}(\mathbb{T}^{2})$ equal to $0$ near the origin with support on a
small enough neighborhood of $0$. Then

$\sup_{\lambda\in\operatorname{Spec}\operatorname{Op}_{N}\nu\widehat{M}_{N}}|\lambda|=O(h^{\infty}).$

A discussion and proof of Theorem 3 is given in Section 4.

Finally, we discuss how small the support of $\chi$ must be in Theorem 2. We
shall show that there is a constant $c$ such the results holds for
$\operatorname{supp}\chi,\operatorname{supp}\nu$ respectively contained in a
$\frac{c}{\lambda\|Q\|^{2}}$-neighborhood of the origin, where $Q$ is the
matrix of eigenvectors of $M$, normalized to have determinant $1$ (this
doesn’t quite fix $Q$ but bounds its norm below). The reciprocal dependence on
$\lambda$ and $\|Q\|^{2}$ matches the intuition that we are isolating a single
fixed point in a region with no other periodic orbits, as discussed in Section
1.1.

###  1.3. Numerical Experiments

An additional motivation for working on the finite-dimensional space $H_{N}$
is the ease by which one may perform numerics. Let $M$ be the original Arnol’d
cat map

$M=\begin{pmatrix}2&1\\\ 1&1\end{pmatrix}$

and for ease of computation, let
$\operatorname{Op}_{h}\chi=\operatorname{Op}_{h}^{L}(\rho(x)\rho(\xi))$ for
$\rho\in C^{\infty}(\mathbb{R})$ and $\operatorname{Op}_{h}^{L}$ the left
quantization as defined in Sections 2.1 and 3.1. By (2.1),
$\operatorname{Op}_{h}\chi$ is the quanitzation of a compactly supported bump
function up to $O(h^{\infty})$ errors. Restricting to $H_{N}$, we see in
Figure 1 the largest four eigenvalue real parts of
$\operatorname{Op}_{N}\chi\widehat{M}_{N}$, as functions of $N$. The imaginary
parts rapidly decay to $0$ and are omitted from the plot.

Figure 1. Real parts of four largest eigenvalues of
$\operatorname{Op}_{N}\chi\widehat{M}_{N}$ for original Arnol’d cat map $M$
and $\chi=1$ near the origin. The limits predicted by Theorem 2 are given as
dotted lines.

In the nontrapping case, we let
$\operatorname{Op}_{N}\nu=\operatorname{Op}_{N}^{L}((\rho(x)-\rho(2x))(\rho(\xi)-\rho(2\xi)))$
as an example of the cutoff operator considered in Theorem 3. The first plot
in Figure 2 shows rapid decay to zero of the largest eigenvalue of
$\operatorname{Op}_{N}\nu\widehat{M}_{N}$ on $H_{N}$, exactly as was predicted
by Theorem 3.

To further show the rate of decay of the spectrum, the second plot in Figure 2
shows the norms of largest eigenvalues of
$\operatorname{Op}_{N}\nu\widehat{M}_{N}$ plotted on a log-log plot.

Figure 2. Left: Norm of largest eigenvalue of
$\operatorname{Op}_{N}\nu\widehat{M}_{N}$, with $\nu=0$ near the origin.
Right: Log-log plot of norm of the largest eigenvalues of
$\operatorname{Op}_{N}\nu\widehat{M}_{N}$, which illustrates Theorem 3.

We see a decrease to $-\infty$ in the slope of the curve, which points to an
$O(h^{\infty})$ decay rate of $|\lambda_{0}|$ consistent with Theorem 3.

##  2\. The Operator on the Real Line

###  2.1. Review of Pseudodifferential and Metaplectic Operators

We recall briefly the semiclassical pseduodifferential and metaplectic calculi
on $\mathbb{R}^{n}$, which will be used repeatedly throughout the paper. The
reader may refer to the book by Zworski [24] for a complete introduction.

A function $m:T^{*}\mathbb{R}^{n}\to(0,\infty)$ is called an _order function_
if $m(x,\xi)\leq C\langle|x-y,\xi-\eta|\rangle^{N}m(y,\eta)$ for some $C,N$.
The corresponding _symbol class_ of $m$ is defined as

$S(m):=\\{a\in
C^{\infty}(T^{*}\mathbb{R}^{n}):\forall\alpha,\beta\in\mathbb{Z}^{n},\,|\partial_{x}^{\alpha}\partial_{\xi}^{\beta}a(x,\xi)|\leq
C_{\alpha\beta}m(x,\xi)\\}.$

Note that elements of $S(m)$ (called _symbols_) may depend on $h$, and in such
a case the constant $C_{\alpha\beta}$ in the definition must be uniform in
$h$.

The _Weyl quantization_ of a symbol is an operator
$\operatorname{Op}_{h}a:\mathcal{S}(\mathbb{R}^{n})\to\mathcal{S}^{\prime}(\mathbb{R}^{n})$
(sometimes written $a(x,hD)$) given by

$(\operatorname{Op}_{h}(a)u)(x):=\frac{1}{(2\pi h)^{n}}\iint
e^{\frac{i}{h}\langle
x-y,\xi\rangle}a\left(\frac{x+y}{2},\xi\right)u(y)\,dy\,d\xi.$

These operators are called _semiclassical pseudodifferential operators_.

A classical result of Calderon and Vaillancourt (see [24]) gives that for
$a\in S(1)$, $\operatorname{Op}_{h}a:L^{2}(\mathbb{R}^{n})\to
L^{2}(\mathbb{R}^{n})$ is bounded with norm independent of $h$. In addition,
if $a$ is real, $\operatorname{Op}_{h}a$ is a (possibly unbounded) symmetric
on $L^{2}$. In general, the formal adjoint of $\operatorname{Op}_{h}a$ is
$(\operatorname{Op}_{h}a)^{*}$.

We may also consider the _left quantization_ (also called the _standard
quantization_), given by

$(\operatorname{Op}^{L}_{h}(a)u)(x):=\frac{1}{(2\pi h)^{n}}\iint
e^{\frac{i}{h}(x-y)\xi}a\left(x,\xi\right)u(y)\,dy\,s\xi.$

We can go from one quantization to the other via the _change of quantization
formula_

$\operatorname{Op}_{h}a=\operatorname{Op}^{L}_{h}a_{L}$

with

$a_{L}=e^{\frac{ih}{2}\langle D_{x},D_{\xi}\rangle}a$ (2.1)

Let $a\in S(m_{1}),b\in S(m_{2})$. The composition of Weyl quantizations is
given by the following formula:

$\operatorname{Op}_{h}a\operatorname{Op}_{h}b=\operatorname{Op}_{h}{a\\#b}$

where $a\\#b\in S(m_{1}m_{2})$ is the _Moyal product_ given by

$a\\#b(x,\xi):=e^{ih\sigma(D_{x},D_{\xi},D_{y},D_{\eta})}(a(x,\xi)b(y,\eta))|_{y=x,\eta=\xi}$

which gives the asymptotic formula

$a\\#b(x,\xi)\sim\sum_{k=0}^{\infty}\frac{i^{k}h^{k}}{k!}\sigma(D_{x},D_{\xi},D_{y},D_{\eta})^{k}(a(x,\xi)b(y,\eta))|_{y=x,\eta=\xi}$

in $S(m_{1}m_{2})$. In particular, if $a,b\in S(1)$ have disjoint supports,
then

$\|\operatorname{Op}_{h}(a)\operatorname{Op}_{h}(b)\|_{L^{2}(\mathbb{R}^{n})\to
L^{2}(\mathbb{R}^{n})}=O(h^{\infty})$

It also follows that if $a,b\in S(1)$, the commutator of quantizations is
given by

$[\operatorname{Op}_{h}a,\operatorname{Op}_{h}b]=-ih\operatorname{Op}_{h}(\\{a,b\\})+O(h^{3})_{L^{2}(\mathbb{R}^{n})\to
L^{2}(\mathbb{R}^{n})},$

where $\\{\cdot,\cdot\\}$ is the Poisson bracket.

Furthermore, if $a\in S$ is bounded below by $c_{0}>0$, the _Gårding
inequality_ gives that for every $\varepsilon>0$ there is a $C>0$ such that
for $h$ small enough,

$\langle\operatorname{Op}_{h}(a)u,u\rangle\geq
C(c_{0}-\varepsilon)\|u\|_{L^{2}(\mathbb{R}^{n})}.$ (2.2)

We remark that (2.2) may be strengthened to the _Sharp Gårding inequality_ and
furthermore to the _Fefferman-Phong inequality_ , which give negative
$h$-dependent lower bounds when $\varepsilon=0$ (see Theorem 4.32 in [24]),
though we will not need such bounds for this paper.

The exponential of a pseudodifferential operator is also a pseudodifferential
operator. If $g=\log m$ for order function $m$, then
$e^{t\operatorname{Op}_{h}g}$ is the semiclassical quantization of some
$b_{t}\in S(m^{t})$. In particular, if $g\in
C^{\infty}_{0}(T^{*}\mathbb{R}^{n})$, then there is a $C$ such that

$\|e^{t\operatorname{Op}_{h}g}\|_{L^{2}(\mathbb{R}^{n})\to
L^{2}(\mathbb{R}^{n})}\leq e^{C|t|}.$

Finally, we discuss the slightly less standard theory of functional calculus
applied to Weyl pseudodifferential operators.

######  Lemma 2.1.

If $a\in S(1)$ or $a\in S(m)$ is elliptic in the sense that
$a\geq\frac{1}{C}m-C$, and $\phi\in C^{\infty}_{0}(T^{*}\mathbb{R}^{n})$, then
$\phi(\operatorname{Op}_{h}a)$ is a Weyl pseudodiffernetial operator with
symbol in $S(m^{-k})$ for every $k$. Suppose $b\in S^{1}$ such that
$\operatorname{supp}b\cap a^{-1}(\operatorname{supp}\phi)=\emptyset$. Then
$\phi(\operatorname{Op}_{h}a)\operatorname{Op}_{h}b=O(h^{\infty})_{L^{2}(\mathbb{R}^{n})\to
L^{2}(\mathbb{R}^{n})}$.

The proof of Lemma 2.1 follows from the Helffer-Sjöstrand formula ((8.13) in
[6]), together (8.15–8.16) in [6] which provide the asymptotic expansion for
the symbol of $\phi(\operatorname{Op}_{h}a)$ as $\sum_{j=0}^{\infty}p_{j}$
with

$p_{j}(x,\xi)=\frac{1}{(2j)!}\partial_{t}^{2j}\left(q_{j}(x,\xi,t)\phi(t)\right)|_{t=a_{0}(x,\xi)},$

where $a_{0}=\lim_{h\to 0}a$ and $q_{j}$ are symbols coming from terms in the
semiclassical expansion of the resolvent $(z-\operatorname{Op}_{h}a)^{-1}$.

We now discuss the calculus of metaplectic operators. Let
$M=\begin{pmatrix}A&B\\\ C&D\end{pmatrix}\in Sp(2n,\mathbb{R})$ be a real
symplectic matrix. Then the _semiclassical metaplectic operators associated to
$M$_ are $\widehat{M}:L^{2}(\mathbb{R}^{n})\to L^{2}(\mathbb{R}^{n})$ given in
[24][7]. In the special case when $\det A\neq 0$, we have (up to a global
phase) that

$\widehat{M}u(y)=\frac{|\det A|^{-\frac{1}{2}}}{(2\pi h)^{n}}\iint
e^{\frac{i}{h}\left(\frac{1}{2}\langle CA^{-1}x,x\rangle+\langle
A^{-1}x,\eta\rangle-\frac{1}{2}\langle A^{-1}B\eta,\eta\rangle-\langle
y,\eta\rangle\right)}u(y)\,dy\,d\eta.$

These operators are unitary, and form a representation of the _metaplectic
group_ , which is a double cover of the symplectic group. The importance of
metaplectic operators is that they “quantize” linear changes of phase
variables, in the sense given by the _exact Egorov theorem_ [24][7]

$\operatorname{Op}_{h}(a)\widehat{M}=\widehat{M}\operatorname{Op}_{h}(a\circ
M).$

###  2.2. Preliminary Inequalities

We begin by studying the operator $\chi(x,hD)\widehat{F}$ on $\mathbb{R}$,
where
$\widehat{F}u(x)=\frac{1}{\sqrt{\lambda}}u\left(\frac{x}{\lambda}\right)$ and
$\chi\in C^{\infty}(T^{*}\mathbb{R})$. A similar operator on different spaces
was considered by Gerard [11] and on yet different spaces by Faure and Tsujii
[10].

For $m\in\mathbb{N}$, we define the following weighted Sobolev-type norm:

$\|u\|_{Y}^{2}:=\sum_{k=0}^{m}\|\langle
x\rangle^{-m}u^{(k)}\|_{L^{2}(\mathbb{R}^{n})}^{2}.$

Denote by $Y$ the closure of $C^{\infty}_{0}(\mathbb{R})$ in the $Y$-norm.

Fix $\Lambda>0$, and define the “cut-off” version of the $Y$-norm as follows:

$\|u\|_{B}^{2}:=\|U_{h}\mathbbm{1}_{P_{h}\leq\Lambda}u\|_{Y}^{2}+\|\mathbbm{1}_{P_{h}\geq\Lambda}u\|_{L^{2}(\mathbb{R})}^{2}.$

Here $P_{h}=P(x,hD)$ is the semiclassical harmonic oscillator
$h^{2}\Delta+x^{2}$, and $U_{h}u(x)=h^{\frac{1}{4}}u(h^{\frac{1}{2}}x)$ is a
unitary semiclassical rescaling. We remark that the $B$-norm depends on
$h,\Gamma$, and $m$, though we generally suppress these constants in writing.

In fact, the $B$-norm is just a dressed up $L^{2}$ norm, as the following
proposition shows.

######  Proposition 2.2.

There is a $C=C(m,\Lambda)$ such that

$C^{-1}h^{\frac{m}{2}}\|u\|_{L^{2}(\mathbb{R})}\leq\|u\|_{B}\leq
Ch^{-\frac{m}{2}}\|u\|_{L^{2}(\mathbb{R})}.$

The purpose of Proposition 2.2 is to be able to write
$O(h^{\infty})\|u\|_{L^{2}(\mathbb{R})}$ and $O(h^{\infty})\|u\|_{B}$
indistinguishably, which we do throughout the remainder of the paper.

###### Proof.

We first prove the lower bound. By subtracting off
$\|\mathbbm{1}_{P_{h}\geq\Lambda}u\|_{L^{2}(\mathbb{R})}^{2}$ from both sides,
we must only show that for some $c>0$,

$\|U_{h}w\|_{Y}^{2}\geq ch^{m}\|w\|_{L^{2}(\mathbb{R})}^{2}$ (2.3)

for $w=\mathbbm{1}_{P_{h}\leq\Lambda}u$. But

$\displaystyle\|U_{h}w\|_{Y}^{2}$ $\displaystyle=\sum_{k=0}^{m}\|\langle
x\rangle^{-m}\partial^{k}(U_{h}w)\|_{L^{2}(\mathbb{R})}^{2}\geq\|\langle
x\rangle^{-m}U_{h}w\|_{L^{2}(\mathbb{R})}^{2}=h^{\frac{1}{2}}\int\langle
x\rangle^{-2m}\left|w\left(h^{\frac{1}{2}}x\right)\right|^{2}\,dx$
$\displaystyle=\int\left\langle
h^{-\frac{1}{2}}x\right\rangle^{-2m}|w(x)|^{2}\,dx=\left\|\left\langle
h^{-\frac{1}{2}}x\right\rangle^{-m}w\right\|_{L^{2}(\mathbb{R})}^{2}\geq
h^{m}\|\langle x\rangle^{-m}w\|_{L^{2}(\mathbb{R})}^{2}.$

Now let $\phi,\tilde{\phi}\in C^{\infty}_{0}(\\{|x|<2\Lambda\\})$ such that
$\tilde{\phi}=1$ on $\\{|x|\leq\Lambda\\}$, $\phi=1$ on
$\operatorname{supp}\tilde{\phi}$. Then $w=\tilde{\phi}(P_{h})w$, and we have

$\left\|\langle x\rangle^{-m}\right\|_{L^{2}(\mathbb{R})}^{2}=\left\|\langle
x\rangle^{-m}\tilde{\phi}(P_{h})w\right\|_{L^{2}(\mathbb{R})}^{2}=\left\|\left(\langle
x\rangle^{-m}\phi(P_{h})+I-\phi(P_{h})\right)\tilde{\phi}(P_{h})w\right\|_{L^{2}(\mathbb{R})}^{2}.$

Consider $A=\langle x\rangle^{-m}\phi(P_{h})+I-\phi(P_{h})$ as a semiclassical
pseudodifferential operator. We have by the Helffer-Sjöstrand construction
(see [6] Chapter 8) that $\phi(P_{h})$ is a smoothing operator with principal
symbol $\phi(x^{2}+\xi^{2})$, so in particular is in $S(1)$, and hence $A$ has
$S(1)$ symbol equal to $\langle
x\rangle^{-m}\phi(x^{2}+\xi^{2})+1-\phi(x^{2}+\xi^{2})+O_{L^{2}\to L^{2}}(h)$.
This symbol is elliptic, so we get for $h$ small enough that:

$\left\|\left(\langle
x\rangle^{-m}\phi(P_{h})+I-\phi(P_{h})\right)\tilde{\phi}(P_{h})w\right\|_{L^{2}(\mathbb{R})}\geq
C\|\tilde{\phi}(P_{h})w\|_{L^{2}(\mathbb{R})}=\|w\|_{L^{2}(\mathbb{R})}.$

Putting the inequalities together gives (2.3).

We now prove the upper bound. As before, by subtracting off
$\|\mathbbm{1}_{P_{h}\geq\Lambda}u\|_{L^{2}(\mathbb{R})}^{2}$ from both sides,
we must only show that

$\|U_{h}w\|_{Y}^{2}\leq Ch^{-m}\|w\|_{L^{2}(\mathbb{R})}^{2}$ (2.4)

for $w=\mathbbm{1}_{P_{h}\leq\Lambda}u$. But

$\displaystyle\|U_{h}w\|_{Y}^{2}=\sum_{k=0}^{m}\left\|\langle
x\rangle^{-m}\partial^{k}(U_{h}w)\right\|_{L^{2}(\mathbb{R})}^{2}$
$\displaystyle\leq\sum_{k=0}^{m}\left\|\partial^{k}(U_{h}w)\right\|_{L^{2}(\mathbb{R})}^{2}=\sum_{k=0}^{m}\left\|U_{h}h^{\frac{k}{2}}\partial^{k}w\right\|_{L^{2}(\mathbb{R})}^{2}$
$\displaystyle=\sum_{k=0}^{m}h^{-k}\left\|(hD)^{k}w\right\|_{L^{2}(\mathbb{R})}^{2}.$

Let $\phi$ be as before, so $(hD)^{k}w=(hD)^{k}\phi(P_{h})w$. As $\phi(P_{h})$
has symbol $\phi(x^{2}+\xi^{2})\in S(\langle\xi\rangle^{-k})$,
$(hD)^{k}\phi(P_{h})$ has symbol in $S(1)$ and hence is bounded on
$L^{2}(\mathbb{R})$. This gives that

$\|U_{h}w\|_{Y}^{2}\leq\sum_{k=0}^{m}h^{-k}\left\|w\right\|_{L^{2}(\mathbb{R})}^{2}\leq
Ch^{-m}\left\|w\right\|_{L^{2}(\mathbb{R})}^{2}$

which is (2.4). ∎

The key property of the $Y$ norm is that we can find sharp bounds for
$\widehat{F}$ on certain closed subspaces. Remark that by Sobolev embedding,
functions in $Y$ have $C^{K-1}$ smoothness for $K\leq m$, so we may define the
subspace

$V_{K}:=\\{u\in Y:u^{(k)}(0)=0\quad\forall k<K\\}.$ (2.5)

The following inequality is inspired by Proposition 3.5 in [11]. Although
seemingly elementary, is in some sense the crux of the entire argument to
follow.

######  Lemma 2.3.

There is a $C_{0}$ such that for $K\leq m$, $u\in V_{K}$,

$\|\widehat{F}u\|_{Y}\leq C_{0}\lambda^{-\frac{M}{4}}\|u\|_{Y}.$ (2.6)

###### Proof.

We first write:

$\displaystyle\left\|\langle
x\rangle^{-m}\partial_{x}^{k}\widehat{F}u\right\|_{L^{2}}^{2}$
$\displaystyle=\lambda^{-2k}\left\|\langle
x\rangle^{-m}\widehat{F}\partial_{x}^{k}u\right\|_{L^{2}}^{2}=\lambda^{-2k-1}\int\langle
x\rangle^{-2m}\left|u^{(k)}\left(\frac{x}{\lambda}\right)\right|^{2}\,dx$
$\displaystyle=\lambda^{-2k}\int\langle\lambda
x\rangle^{-2m}|u^{(k)}(x)|^{2}\,dx.$

When $k\geq K$, we then directly see that

$\left\|\langle
x\rangle^{-m}\partial_{x}^{k}\widehat{F}u\right\|_{L^{2}}^{2}\leq\lambda^{-2K}\|\langle
x\rangle^{-m}u^{(k)}\|_{L^{2}}^{2}.$

When $k<K$, we require an additional argument.

######  Lemma 2.4.

If $r<1$, $u\in V_{K}$, $0\leq k<K$,

$\|u^{(k)}\|_{L^{2}([-r,r])}\leq\frac{2\sqrt{2}r^{K-k+\frac{1}{2}}}{(K-k)!}\left(\|u^{(K)}\|_{L^{2}([-1,1])}+\|u^{(K+1)}\|_{L^{2}([-1,1])}\right)$

###### Proof of Lemma 2.4.

We easily see that

$\|u^{(k)}\|_{L^{2}([-r,r])}\leq\sqrt{2r}\|u^{(k)}\|_{L^{\infty}([-r,r])}.$
(2.7)

But

$\displaystyle u^{(k)}(x)$
$\displaystyle=\frac{1}{(K-k-1)!}\int_{0}^{x}u^{(K)}(t)(x-t)^{K-k-1}\,dt$
$\displaystyle=\frac{x^{K-k}}{(K-k-1)!}\int_{0}^{1}u^{(K)}(sx)(1-s)^{K-k-1}\,ds.$

This gives that

$\|u^{(k)}\|_{L^{\infty}([-r,r])}\leq\frac{r^{K-k}}{(K-k)!}\|u^{(K)}\|_{L^{\infty}([-r,r])}.$
(2.8)

But by the Sobolev Inequality,

$\|u^{(K)}\|_{L^{\infty}([-r,r])}\leq
2\left(\|u^{(K)}\|_{L^{2}([-1,1])}+\|u^{(K+1)}\|_{L^{2}([-1,1])}\right).$
(2.9)

Combining (2.7), (2.8), and (2.9) gives the inequality. ∎

Fix $s=s(\lambda)\in\left(1,\sqrt{\frac{\lambda^{2}+1}{2}}\right)$ to be
chosen later, and let $r=\sqrt{\frac{s^{2}-1}{\lambda-s^{2}}}<1$, which is
chosen such that $x>r\implies\langle\lambda x\rangle>s\langle x\rangle$. Then

$\displaystyle\left\|\langle
x\rangle^{-m}\partial_{x}\widehat{F}u\right\|_{L^{2}}^{2}$
$\displaystyle=\lambda^{-2k}\int_{|x|\leq r}\langle\lambda
x\rangle^{-2m}|u^{(k)}(x)|^{2}\,dx+\lambda^{-2k}\int_{|x|>r}\langle\lambda
x\rangle^{-2m}|u^{(k)}(x)|^{2}\,dx.$
$\displaystyle\leq\lambda^{-2k}\int_{|x|\leq r}\langle\lambda
x\rangle^{-2m}|u^{(k)}(x)|^{2}\,dx+\lambda^{-2k}s^{-2m}\int_{|x|>r}\langle
x\rangle^{-2m}|u^{(k)}(x)|^{2}\,dx.$

But by Lemma 2.4 the first term is

$\displaystyle\lambda^{-2k}\int_{|x|\leq r}\langle\lambda x\rangle^{-2m}$
$\displaystyle|u^{(k)}(x)|^{2}\,dx\leq\lambda^{-2k}\int_{|x|\leq
r}|u^{(k)}(x)|^{2}\,dx$
$\displaystyle\leq\lambda^{-2k}\left(2\sqrt{2}\frac{r^{K-k+\frac{1}{2}}}{(K-k)!}\left(\|u^{(K)}\|_{L^{2}([-1,1])}+\|u^{(K+1)}\|_{L^{2}([-1,1])}\right)\right)^{2}$
$\displaystyle\leq
16\lambda^{-2k}\frac{r^{2K-2k+1}}{(K-k)!^{2}}\left(\|u^{(K)}\|_{L^{2}([-1,1])}^{2}+\|u^{(K+1)}\|_{L^{2}([-1,1])}^{2}\right).$

We now wish to optimize $r$ and $s$ to get the best possible bound. Set
$r=s^{-1}$ and get $s=\lambda^{\frac{1}{4}}$, $r=\lambda^{-\frac{1}{4}}$.
Summing everything together gives the bound (2.6). ∎

We also need a bound for a smoothing operator in the $Y$-norm.

######  Lemma 2.5.

Let $\chi\in C^{\infty}_{0}(T^{*}\mathbb{R})$, and define operator
$\widetilde{\chi}$ by $\widetilde{\chi}=\chi(\sqrt{h}x,\sqrt{h}D)$. Then there
are $C>0$, $f:\mathbb{N}\to\mathbb{R}^{+}$ such that

$\|\widetilde{\chi}u\|_{Y}\leq(C+f(m)\sqrt{h})\|u\|_{Y}.$ (2.10)

###### Proof.

Let $\tilde{h}=\sqrt{h}$. We compute the commutator of $\widetilde{\chi}$ with
a derivative:

$\displaystyle[D_{x},\widetilde{\chi}]u(x)$ $\displaystyle=D_{x}\iint
e^{i(x-y)\xi}\chi(\tilde{h}\frac{x+y}{2},\tilde{h}\xi)u(y)\,dy\,d\xi$
$\displaystyle\quad-\iint
e^{i(x-y)\xi}\chi(\tilde{h}\frac{x+y}{2},\tilde{h}\xi)D_{y}u(y)\,dy\,d\xi$
$\displaystyle=\iint
D_{x}\left(e^{i(x-y)\xi}\chi(\tilde{h}\frac{x+y}{2},\tilde{h}\xi)\right)u(y)\,dy\,d\xi$
$\displaystyle\quad+\iint
D_{y}\left(e^{i(x-y)\xi}\chi(\tilde{h}\frac{x+y}{2},\tilde{h}\xi)\right)u(y)\,dy\,d\xi$
$\displaystyle=\tilde{h}\iint\left(e^{i(x-y)\xi}\chi^{\prime}(\tilde{h}\frac{x+y}{2},\tilde{h}\xi)\right)u(y)\,dy\,d\xi.$

By an easy induction, we see that

$[D_{x}^{k},\widetilde{\chi}]=\tilde{h}\chi_{k}(\tilde{h}x,\tilde{h}D)$ (2.11)

where $\chi_{k}$ is a compactly supported smooth function independent of $h$.

We then have:

$\displaystyle\|\widetilde{\chi}u\|_{Y}^{2}$
$\displaystyle=\sum_{k=0}^{m}\|\langle
x\rangle^{-m}D^{k}\widetilde{\chi}u\|_{L^{2}(\mathbb{R})}^{2}$
$\displaystyle=\sum_{k=0}^{m}\|\langle
x\rangle^{-m}(\widetilde{\chi}D^{k}+[D^{k},\widetilde{\chi}])u\|_{L^{2}(\mathbb{R})}^{2}$
$\displaystyle\leq\sum_{k=0}^{m}\left(\|\langle
x\rangle^{-m}\widetilde{\chi}D^{k}u\|_{L^{2}(\mathbb{R})}+\|\langle
x\rangle^{-m}[D^{k},\widetilde{\chi}]u\|_{L^{2}(\mathbb{R})}\right)^{2}$
$\displaystyle\leq 2\sum_{k=0}^{m}\|\langle
x\rangle^{-m}\widetilde{\chi}D^{k}u\|_{L^{2}(\mathbb{R})}^{2}+2\tilde{h}^{2}\sum_{k=0}^{m}\|\langle
x\rangle^{-m}\chi_{k}(\tilde{h}x,\tilde{h}D)u\|_{L^{2}(\mathbb{R})}^{2}.$
(2.12)

Define function $\chi_{s}(x,\xi)=\chi(\tilde{h}x,\xi)$, so
$\widetilde{\chi}=\operatorname{Op}_{\tilde{h}}\chi_{s}$. Note that while
$\chi_{s}$ is $\tilde{h}$-dependent, this dependence only contributes to
further decay of its derivatives, so it is still in all the same symbol
classes as $\chi$. Note also that $\langle x\rangle^{-m}\in S(\langle
x\rangle^{-m})$, so we have (with $\cdot$ referring to the $x$ variable):

$\displaystyle\langle x\rangle^{-m}\chi_{s}(x,\tilde{h}D)$
$\displaystyle\langle x\rangle^{m}=\chi_{s}(x,\tilde{h}D)+[\langle
x\rangle^{-m},\chi_{s}(x,\tilde{h}D)]\langle x\rangle^{m}$
$\displaystyle=\chi_{s}(x,\tilde{h}D)+\left(\frac{\tilde{h}}{i}\\{\langle\cdot\rangle^{-m},\chi_{s}\\}(x,\tilde{h}D)+O_{S(\langle
x\rangle^{-m})}(\tilde{h}^{3})\right)\langle x\rangle^{m}$
$\displaystyle=\chi_{s}(x,\tilde{h}D)+\left(\frac{\tilde{h}}{i}(m\langle\cdot\rangle^{-m-2}(\cdot)\partial_{\xi}\chi_{s})^{w}(x,\tilde{h}D)+O_{S(\langle
x\rangle^{-m})}(\tilde{h}^{3})\right)\langle x\rangle^{m}$
$\displaystyle=\chi_{s}(x,\tilde{h}D)+\left(O_{S(\langle
x\rangle^{-m})}(\tilde{h})\right)\langle x\rangle^{m}.$

Then $\langle x\rangle^{-m}\chi_{s}(x,\tilde{h}D)\langle x\rangle^{m}$ is
bounded by $C+O(\tilde{h})$ from $L^{2}(\mathbb{R})$ to $L^{2}(\mathbb{R})$
where the big-O depends on $m$ but the $C$ does not. This proves that:

$\|\langle
x\rangle^{-m}\widetilde{\chi}D^{k}u\|_{L^{2}(\mathbb{R})}\leq(C+O(\tilde{h}))\|\langle
x\rangle^{-m}D^{k}u\|_{L^{2}(\mathbb{R})}.$ (2.13)

Similarly we see that

$\langle x\rangle^{-m}\chi_{k,r}(x,\tilde{h}D)\langle
x\rangle^{m}\leq\chi_{k,r}(x,\tilde{h}D)+\left(O_{S(\langle
x\rangle^{-m})}(\tilde{h})\right)\langle x\rangle^{m}$

where $\chi_{k,r}(x,\xi)=\chi_{k}(\tilde{h}x,\xi)$, which gives

$\|\langle
x\rangle^{-m}\widetilde{\chi}_{k}(\tilde{h}x,\tilde{h}\xi)u\|_{L^{2}(\mathbb{R})}\leq(C+O(\tilde{h}))\|\langle
x\rangle^{-m}u\|_{L^{2}(\mathbb{R})}.$ (2.14)

Combining (2.12), (2.13), and (2.14) gives (2.10). ∎

We put the pieces together to prove an estimate for $\chi(x,hD)\widehat{F}$ in
the $B$-norm.

######  Proposition 2.6.

Suppose that
$\operatorname{supp}\chi\subseteq\\{|x|^{2}+|\xi|^{2}<\Lambda/(2\lambda^{2})\\}$.
Then there are $C>0,f:\mathbb{N}\to\mathbb{R}^{+}$ such that if $u\in Y\cap
V_{K}$, then

$\|\chi(x,hD)\widehat{F}u\|_{B}\leq\left(C\lambda^{-\frac{K}{4}}+f(m)\sqrt{h}\right)\|u\|_{B}.$
(2.15)

We remark that the independence of $C$ from $m$ in Proposition 2.6 (and hence
also in Proposition 2.5) is crucial, as we will eventually take $K$ very large
to make the left hand side of (2.15) small, but require that $m\geq K$ for
$V_{K}$ to even be defined.

###### Proof.

Recall that

$\|\chi(x,hD)\widehat{F}u\|_{B}^{2}=\|U_{h}\mathbbm{1}_{P_{h}\leq\Lambda}\chi(x,hD)\widehat{F}u\|_{Y}^{2}+\|\mathbbm{1}_{P_{h}\geq\Lambda}\chi(x,hD)\widehat{F}u\|_{L^{2}(\mathbb{R})}^{2}.$

By Theorem 6.6 in [24] (or an argument similar to that of Proposition 2.2),

$\|\mathbbm{1}_{P_{h}\geq\Lambda}\chi(x,hD)\widehat{F}u\|_{L^{2}(\mathbb{R})}\leq
O(h^{\infty})\|u\|_{L^{2}}$

so by Proposition 2.2 we must only bound
$\|U_{h}\mathbbm{1}_{P_{h}\leq\Lambda}\chi(x,hD)\widehat{F}u\|_{Y}$.

But

$\left\|U_{h}\mathbbm{1}_{P_{h}\leq\Lambda}\chi(x,hD)\widehat{F}u\right\|_{Y}\leq\left\|U_{h}\mathbbm{1}_{P_{h}\geq\Lambda}\chi(x,hD)\widehat{F}u\right\|_{Y}+\left\|U_{h}\chi(x,hD)\widehat{F}u\right\|_{Y}.$
(2.16)

We bound the first term in (2.16) first. We have:

$\displaystyle\left\|U_{h}\mathbbm{1}_{P_{h}\geq\Lambda}\chi(x,hD)\widehat{F}u\right\|_{Y}^{2}$
$\displaystyle=\sum_{k=0}^{m}\left\|\langle
x\rangle^{-m}\partial^{k}(U_{h}\mathbbm{1}_{P_{h}\geq\Lambda}\chi(x,hD)\widehat{F}u)\right\|_{L^{2}(\mathbb{R})}^{2}$
$\displaystyle\leq\sum_{k=0}^{m}\left\|\partial^{k}(U_{h}\mathbbm{1}_{P_{h}\geq\Lambda}\chi(x,hD)\widehat{F}u)\right\|_{L^{2}(\mathbb{R})}^{2}$
$\displaystyle=\sum_{k=0}^{m}h^{-k}\left\|(hD)^{k}(\mathbbm{1}_{P_{h}\geq\Lambda}\chi(x,hD)\widehat{F}u)\right\|_{L^{2}(\mathbb{R})}^{2}$

Let $\phi\in C^{\infty}_{0}(\\{|x|<\Lambda\\})$ have $\phi=1$ on
$\operatorname{supp}\chi$, and let $\tilde{\phi}\in
C^{\infty}_{0}(\\{|x|<2\Lambda\\})$ have $\tilde{\phi}=1$ on
$\\{|x|<\Lambda\\}$.

Then
$\mathbbm{1}_{P_{h}\geq\Lambda}=\mathbbm{1}_{P_{h}\geq\Lambda}(I-\phi(P_{h}))$,
so

$\displaystyle\left\|(hD)^{k}\mathbbm{1}_{P_{h}\geq\Lambda}\chi(x,hD)\widehat{F}u\right\|_{L^{2}(\mathbb{R})}=\left\|(hD)^{k}\mathbbm{1}_{P_{h}\geq\Lambda}(I-\phi(P_{h}))\chi(x,hD)\widehat{F}u\right\|_{L^{2}(\mathbb{R})}$
$\displaystyle\leq\left\|(hD)^{k}((I-\phi(P_{h}))\chi(x,hD)\widehat{F}u\right\|_{L^{2}(\mathbb{R})}+\left\|(hD)^{k}\mathbbm{1}_{P_{h}\leq\Lambda}(I-\phi(P_{h}))\chi(x,hD)\widehat{F}u\right\|_{L^{2}(\mathbb{R})}$
$\displaystyle=\left\|(hD)^{k}(I-\phi(P_{h}))\chi(x,hD)\widehat{F}u\right\|_{L^{2}(\mathbb{R})}+\left\|(hD)^{k}\tilde{\phi}(P_{h})(I-\phi(P_{h}))\chi(x,hD)\widehat{F}u\right\|_{L^{2}(\mathbb{R})}.$

But by pseudodifferential calculus,
$\|(hD)^{k}(I-\phi(P_{h}))\chi(x,hD)\|_{L^{2}\to L^{2}}$ and
$\|(hD)^{k}\tilde{\phi}(P_{h})\|_{L^{2}\to L^{2}}$ are both $O(h^{\infty})$,
and all the other operators are uniformly bounded, so we have for each $k$,

$\left\|(hD)^{k}\mathbbm{1}_{P_{h}\geq\Lambda}\chi(x,hD)\widehat{F}u\right\|_{L^{2}(\mathbb{R})}=O(h^{\infty})\|u\|_{L^{2}}$

and hence by Lemma 2.2:

$\left\|U_{h}\mathbbm{1}_{P_{h}\geq\Lambda}\chi(x,hD)\widehat{F}u\right\|_{Y}=O(h^{\infty})\|u\|_{B}$

To handle the second term in (2.16), we commute terms and apply Lemmas 2.5 and
2.3:

$\displaystyle\left\|U_{h}\chi(x,hD)\widehat{F}u\right\|_{Y}$
$\displaystyle=\left\|U_{h}\chi(x,hD)\mathbbm{1}_{\frac{x^{2}}{\lambda^{2}}+\lambda^{2}(hD)^{2}\leq\Lambda}\widehat{F}u\right\|_{Y}+O(h^{\infty})\|u\|_{L^{2}(\mathbb{R})}$
$\displaystyle=\left\|\widetilde{\chi}U_{h}\mathbbm{1}_{\frac{x^{2}}{\lambda^{2}}+\lambda^{2}(hD)^{2}\leq\Lambda}\widehat{F}u\right\|_{Y}+O(h^{\infty})\|u\|_{L^{2}(\mathbb{R})}$
$\displaystyle\leq(C+f(m)\sqrt{h})\|U_{h}\mathbbm{1}_{\frac{x^{2}}{\lambda^{2}}+\lambda^{2}(hD)^{2}\leq\Lambda}\widehat{F}u\|_{Y}+O(h^{\infty})\|u\|_{L^{2}(\mathbb{R})}$
$\displaystyle=(C+f(m)\sqrt{h})\|\widehat{F}U_{h}\mathbbm{1}_{P_{h}\leq\Lambda}u\|_{Y}+O(h^{\infty})\|u\|_{L^{2}(\mathbb{R})}$
$\displaystyle\leq(C\lambda^{-\frac{K}{4}}+f(m)\sqrt{h})\|U_{h}\mathbbm{1}_{P_{h}\leq\Lambda}u\|_{Y}+O(h^{\infty})\|u\|_{L^{2}(\mathbb{R})}.$

This gives (2.15). ∎

Finally, we will need pointwise control of $\widetilde{\chi}u$ near $x=0$. We
assume from now on that $m\geq K+2$.

######  Lemma 2.7.

There is an $f:\mathbb{N}\times\mathbb{R}^{+}\to\mathbb{R}^{+}$ such that for
$k<m-1$,

$\left|\partial_{x}^{k}\widetilde{\chi}u(0)-\partial_{x}^{k}u(0)\right|\leq
f(m,\varepsilon)h^{\frac{m-k-1}{2}-\varepsilon}\|u\|_{Y}$ (2.17)

where $\widetilde{\chi}=\chi(\sqrt{h}x,\sqrt{h}D)$ as in Lemma 2.5.

###### Proof.

We recall the notation form Lemma 2.5 and write $\tilde{h}:=\sqrt{h}$ and
$\chi_{s}(x,\xi):=\chi(\tilde{h}x,\xi)$ so
$\widetilde{\chi}=\chi_{s}(x,\tilde{h}D)$.

We first prove the result for $k=0$. Write
$\chi_{s}(x,\tilde{h}D)=\operatorname{Op}^{L}_{\tilde{h}}(\chi_{L})+O(\tilde{h}^{\infty})_{L^{2}(\mathbb{R})\to
L^{2}(\mathbb{R})}$, where $\operatorname{Op}^{L}$ is the left quantization,
and let $\psi(x)=\check{\chi}_{L}(0,x)$, where the inverse Fourier transform
is in the second variable. We then compute:

$\widetilde{\chi}u(0)=\int\chi_{L}(0,\tilde{h}\xi)\hat{u}(\xi)\,d\xi+O(\tilde{h}^{\infty})\|u\|_{Y}.$

By Parseval’s Identity and Taylor’s theorem:

$\displaystyle\int$
$\displaystyle\chi_{L}(0,\tilde{h}\xi)\hat{u}(\xi)\,d\xi=\tilde{h}^{-1}\int\psi(\tilde{h}^{-1}x)u(x)\,dx$
$\displaystyle=\tilde{h}^{-1}\int\psi(\tilde{h}^{-1}x)\left(u(0)+\sum_{j=1}^{p-1}\frac{x^{j}}{j!}u^{(j)}(0)+\frac{x^{p}}{(p-1)!}\int_{0}^{1}(1-t)^{p-1}u^{(p)}(tx)\,dt\right)\,dx$
$\displaystyle=u(0)+\frac{\tilde{h}^{-1}}{(p-1)!}\int\psi(\tilde{h}^{-1}x)x^{p}\int_{0}^{1}(1-t)^{p-1}u^{(p)}(tx)\,dt\,dx$
(2.18)

where $p<m$.

We split the integral error term in (2.18) as follows. Let:

$I_{1}:=\tilde{h}^{-1}\int_{|x|<r}\psi(\tilde{h}^{-1}x)x^{p}\int_{0}^{1}(1-t)^{p-1}u^{(p)}(tx)\,dt\,dx$
$I_{2}:=\tilde{h}^{-1}\int_{|x|\geq
r}\psi(\tilde{h}^{-1}x)x^{p}\int_{0}^{\frac{r}{x}}(1-t)^{p-1}u^{(p)}(tx)\,dt\,dx$
$I_{3}:=\tilde{h}^{-1}\int_{|x|\geq
r}\psi(\tilde{h}^{-1}x)x^{p}\int_{\frac{r}{x}}^{1}(1-t)^{p-1}u^{(p)}(tx)\,dt\,dx$

where $0<r<1$ is to be chosen later, so by (2.18),

$\widetilde{\chi}u(0)-u(0)=\frac{1}{(p-1)!}(I_{1}+I_{2}+I_{3})+O(h^{\infty})\|u\|_{Y}$
(2.19)

We must bound each $I_{j}$ in turn. Note that as the result is the same for
$\psi(-\cdot),u(-\cdot)$, we may only bound the integrals where $x\geq 0$. We
then have:

$\displaystyle\begin{split}|I_{1}|&\leq\tilde{h}^{-1}\int_{|x|<r}|\psi(\tilde{h}^{-1}x)||x|^{p}\int_{0}^{1}|(1-t)^{p-1}u^{(p)}(tx)|\,dt\,dx\\\
&\leq
r^{p}\tilde{h}^{-1}\int_{|x|<r}|\psi(\tilde{h}^{-1}x)|\,dx\|u^{(p)}\|_{L^{\infty}([-r,r])}\\\
&\leq
Cr^{p}\left(\|u^{(p)}\|_{L^{2}([-1,1])}+\|u^{(p+1)}\|_{L^{2}([-1,1])}\right)\\\
&\leq C(m)r^{p}\left(\|\langle
x\rangle^{-m}u^{(p)}\|_{L^{2}(\mathbb{R})}+\|\langle
x\rangle^{-m}u^{(p+1)}\|_{L^{2}(\mathbb{R})}\right).\end{split}$ (2.20)

Also

$\displaystyle\begin{split}|I_{2}|&\leq\tilde{h}^{-1}\int_{|x|\geq
r}|\psi(\tilde{h}^{-1}x)|\int_{0}^{r}|x-t|^{p-1}|u^{(p)}(t)|\,dt\,dx\\\ &\leq
C(p)\tilde{h}^{-1}\int_{|x|\geq r}|\psi(\tilde{h}^{-1}x)|\langle
x\rangle^{p}\,dx\|u^{(p)}\|_{L^{\infty}([-r,r])}\\\ &\leq
C(p)\tilde{h}^{-1}\int_{|x|\geq
r}|\psi(\tilde{h}^{-1}x)|\langle\tilde{h}^{-1}x\rangle^{p}\,dx\left(\|\langle
x\rangle^{-m}u^{(p)}\|_{L^{2}(\mathbb{R})}+\|\langle
x\rangle^{-m}u^{(p+1)}\|_{L^{2}(\mathbb{R})}\right)\\\
&=C(p)\int_{|y|\geq\tilde{h}^{-1}r}|\psi(y)|\langle
y\rangle\,dy\left(\|\langle
x\rangle^{-m}u^{(p)}\|_{L^{2}(\mathbb{R})}+\|\langle
x\rangle^{-m}u^{(p+1)}\|_{L^{2}(\mathbb{R})}\right)\\\ &\leq
C(m)O(\tilde{h}^{\infty})\left(\|\langle
x\rangle^{-m}u^{(p)}\|_{L^{2}(\mathbb{R})}+\|\langle
x\rangle^{-m}u^{(p+1)}\|_{L^{2}(\mathbb{R})}\right)\end{split}$ (2.21)

provided $r>\tilde{h}^{1-\varepsilon}$.

Finally we get

$\displaystyle|I_{3}|$ $\displaystyle\leq\tilde{h}^{-1}\int_{|x|\geq
r}|\psi(\tilde{h}^{-1}x)|\int_{r}^{x}|x-t|^{p-1}|u^{(p)}(t)|\,dt\,dx$
$\displaystyle=\int_{t\geq
r}\tilde{h}^{-1}\int_{t}^{\infty}|\psi(\tilde{h}^{-1}x)||x-t|^{p-1}\,dx|u^{(p)}(t)|\,dt.$

But

$\displaystyle\tilde{h}^{-1}\int_{t}^{\infty}|\psi(\tilde{h}^{-1}x)||x-t|^{p-1}\,dx$
$\displaystyle\leq\tilde{h}^{-1}\int_{t}^{\infty}|\psi(\tilde{h}^{-1}x)||\tilde{h}^{-1}(x-t)|^{p-1}\,dx$
$\displaystyle=\int_{\tilde{h}^{-1}t}^{\infty}|\psi(y)||y-\tilde{h}^{-1}t|^{p-1}\,dy$
$\displaystyle=\Psi(\tilde{h}^{-1}t)$

for $\Psi\in\mathcal{S}(\mathbb{R})$, so

$\displaystyle\begin{split}|I_{3}|&\leq\int_{t\geq
r}\Psi(\tilde{h}^{-1}t)|u^{(p)}(t)|\,dt\\\ &\leq\int_{t\geq
r}\Psi(\tilde{h}^{-1}t)\langle\tilde{h}^{-1}t\rangle^{m}\langle
t\rangle^{-m}|u^{(p)}(t)|\,dt\\\
&\leq\left(\int_{t>r}\left|\Psi(\tilde{h}^{-1}t)\right|^{2}\langle\tilde{h}^{-1}t\rangle^{2m}\right)^{\frac{1}{2}}\left(\int_{t>r}\langle
t\rangle^{-2m}\left|u^{(p)}(t)\right|^{2}\right)^{\frac{1}{2}}\\\
&=C(m)O(\tilde{h}^{\infty})\|\langle
x\rangle^{-m}u^{(p)}\|_{L^{2}(\mathbb{R})}.\end{split}$ (2.22)

The bounds (2.20), (2.21), (2.22) together with (2.19) give (2.17) for the
case $k=0$, by choosing $p$ as large as possible and
$r=\tilde{h}^{1-\varepsilon}$.

For larger powers, we use the fact from the proof of Lemma 2.5, specifically
(2.11), which says that
$[D_{x}^{k},\widetilde{\chi}]=\tilde{h}\chi_{k}(\tilde{h}x,\tilde{h}D)$ where
$\chi_{k}(x,\xi)=0$ near $x,\xi=0$. Then

$\left|\partial_{x}^{k}\widetilde{\chi}u(0)-\partial_{x}^{k}u(0)\right|\leq\left|\widetilde{\chi}\partial_{x}^{k}u(0)-\partial_{x}^{k}u(0)\right|+|\tilde{h}\chi_{k}(\tilde{h}x,\tilde{h}D)u(0)|,$
(2.23)

and we may simply apply the result for $k=0$ on the first term. On the second
term, we write $\chi_{k}=\chi_{k}+\phi-\phi$ where $\phi=1$ near $0$ and
$\operatorname{supp}\phi\subseteq\chi_{k}^{-1}(0)$. Then by (2.23) we get

$\displaystyle|\tilde{h}\chi_{k}(\tilde{h}x,\tilde{h}D)$ $\displaystyle
u(0)|=\tilde{h}|\chi_{k}(\tilde{h}x,\tilde{h}D)u(0)+\phi(\tilde{h}x,\tilde{h}D)u(0)-\phi(\tilde{h}x,\tilde{h}D)u(0)|$
$\displaystyle\leq\tilde{h}\left|\left(\chi_{k}(\tilde{h}x,\tilde{h}D)+\phi(\tilde{h}x,\tilde{h}D)\right)u(0)-u(0)\right|+\left|\phi(\tilde{h}x,\tilde{h}D)u(0)-u(0)\right|$

on which we can again apply the result for $k=0$. This proves (2.17) in
general. ∎

###  2.3. Grushin Problems

To determine the spectrum of $\operatorname{Op}_{h}\chi\widehat{F}$, we use
the technique of building a Grushin problem, which relies on the well-known
Schur complement formula. We briefly review these techniques here, and refer
to Sjöstrand and Zworski [22] for a more extensive exposition. We also remark
that the particular Grushin problem in this section is similar to those in
[11].

The _Schur complement formula_ in elementary linear algebra says that for
block matrices

$\mathcal{P}:=\begin{pmatrix}P&R_{-}\\\
R_{+}&0\end{pmatrix},\quad\mathcal{P}^{-1}:=\mathcal{E}:=\begin{pmatrix}E&E_{+}\\\
E_{-}&E_{-+}\end{pmatrix},$ (2.24)

$P$ is invertible if and only if $E_{-+}$ is, and in fact we have the formulas

$P^{-1}=E-E_{+}E_{-+}^{-1}E_{-},\quad E_{-+}^{-1}=-R_{+}P^{-1}R_{-}.$

More generally, the above formula applies even if the operators act on
infinite-dimensional spaces. Specifically, let $H_{1},H_{2},H_{+},H_{-}$ be
Hilbert spaces, and let $P:H_{1}\to H_{2}$ be an operator. Let $R_{-}:H_{-}\to
H$, $R_{+}:H\to H_{+}$, and define $\mathcal{P}:H_{1}\oplus H_{-}\to
H_{2}\oplus H_{+}$ by (2.24). If $\mathcal{P}$ is invertible, we say that the
Grushin problem is _well-posed_. In practice, $H_{\pm}$ are often finite-
dimensional or otherwise simpler spaces, and we refer to $E_{-+}$ as the
_effective Hamiltonian_ of $P$. We remark in particular that if $P$ is a
Fredholm operator, it is always possible to construct a Grushin problem with
finite-dimensional $H_{\pm}$, which matches the intuition of Fredholm
operators being “effectively finite-dimensional.”

The starting point for our computation are the following simple Taylor
coefficient operators:

$R_{-}:\mathbb{C}^{K+1}\to C^{\infty}(\mathbb{R}),\quad
R_{+}:C^{\infty}(\mathbb{R})\to\mathbb{C}^{K+1}$
$R_{-}(a_{0},\dots,a_{K}):=\sum_{j=0}^{K}a_{j}\frac{x^{j}}{j!},\quad
R_{+}u:=\left(u(0),u^{\prime}(0),\dots,u^{(K)}(0)\right)$

Note that $R_{+}R_{-}=I_{\mathbb{C}^{K+1}}$ and $R_{-}R_{+}$ returns the $K$th
order Taylor polynomial at $0$. In particular, this means that
$\mathrm{Ran}(I-R_{-}R_{+})\subseteq V_{K+1}$ as defined in (2.5).

We now define the “Gaussian” versions of $R_{\pm}$ as follows:

$\hat{R}_{-}(a_{0},\dots,a_{K}):=e^{-\frac{x^{2}}{2}}R_{-}(a_{0},\dots,a_{K}),\quad\hat{R}_{+}u:=R_{+}\left(e^{\frac{x^{2}}{2}}u\right).$

The property

$\hat{R}_{+}\hat{R}_{-}=I_{\mathbb{C}^{K+1}}$ (2.25)

still holds, but $\hat{R}_{-}\hat{R}_{+}$ is more complicated. We nevertheless
still have that $\mathrm{Ran}(I-\hat{R}_{-}\hat{R}_{+})\subseteq V_{K+1}$, as
for any $u\in C^{\infty}$,

$(I-\hat{R}_{-}\hat{R}_{+})u=u-e^{-\frac{x^{2}}{2}}R_{-}R_{+}(e^{\frac{x^{2}}{2}}u)=e^{-\frac{x^{2}}{2}}(I-R_{-}R_{+})(e^{\frac{x^{2}}{2}}u)\in
V_{K+1}.$ (2.26)

We also note the relation $\hat{R}_{+}=B_{K}R_{+}$, where $B_{K}$ is an
explicitly computable lower-triangular matrix with all $1$’s on the diagonal.

We are now ready to define the “localized” $R_{\pm}$ operators. Define

$\tilde{R}_{-}:\mathbb{C}^{K+1}\to
L^{2}(\mathbb{R}),\quad\tilde{R}_{+}:L^{2}(\mathbb{R})\to\mathbb{C}^{K+1}$

given by

$\tilde{R}_{-}:=\mathbbm{1}_{P_{h}\leq\Lambda}U_{h}^{*}\hat{R}_{-},\quad\tilde{R}_{+}:=\hat{R}_{+}U_{h}\mathbbm{1}_{P_{h}\leq\Lambda},$

where we note that for $h\leq\frac{\Lambda}{2(K+1)}$ the
$\mathbbm{1}_{P_{h}\leq\Lambda}$ in the definition of $\tilde{R}_{-}$ is
actually superfluous. We see that by the Sobolev inequality

$\displaystyle(R_{+}U_{h}\mathbbm{1}_{P_{h}\leq\Lambda}u)_{k}$
$\displaystyle=\partial^{k}U_{h}\mathbbm{1}_{P_{h}\leq\Lambda}u(x)|_{x=0}$
$\displaystyle\lesssim\left(\|\partial^{k}U_{h}\mathbbm{1}_{P_{h}\leq\Lambda}u(x)\|_{L^{2}([-1,1])}+\|\partial^{k+1}U_{h}\mathbbm{1}_{P_{h}\leq\Lambda}u(x)\|_{L^{2}([-1,1])}\right)$
$\displaystyle\lesssim\|u\|_{B}$

with constants independent of $h$, so $\tilde{R}_{+}:B\to\mathbb{C}^{K+1}$ is
bounded independent of $h$. Similarly

$\|\mathbbm{1}_{P_{h}\leq\Lambda}U_{h}^{*}\hat{R}_{-}v\|_{B}=\|U_{h}\mathbbm{1}_{P_{h}\leq\Lambda}U_{h}^{*}\hat{R}_{-}v\|_{Y}=\|\mathbbm{1}_{P\leq\frac{\Lambda}{h}}\hat{R}_{-}v\|_{Y}=\|\hat{R}_{-}v\|_{Y}$

for $h$ small enough, which gives in that case that
$\tilde{R}_{-}:\mathbb{C}^{K+1}\to B$ is also bounded independent of $h$.

The analogues of $(\ref{RmRp})$ and $(\ref{RpRm})$ for $\tilde{R}_{\pm}$ are
given in the following proposition.

######  Proposition 2.8.

If $h\leq\frac{\Lambda}{2(K+1)}$, then:

$\tilde{R}_{+}\tilde{R}_{-}=I_{\mathbb{C}^{K+1}}$

and $\mathbbm{1}_{P_{h}\leq\Lambda}-\tilde{R}_{-}\tilde{R}_{+}$ has range
$V_{K+1}$.

###### Proof.

We have that for $h$ small enough:

$\displaystyle\tilde{R}_{+}\tilde{R}_{-}=\hat{R}_{+}\mathbbm{1}_{P\leq\frac{\Lambda}{h}}\hat{R}_{-}=I_{\mathbb{C}^{K+1}}$

and also

$\displaystyle\tilde{R}_{-}\tilde{R}_{+}=\mathbbm{1}_{P_{h}\leq\Lambda}U_{h}^{*}\hat{R}_{-}\hat{R}_{+}U_{h}\mathbbm{1}_{P_{h}\leq\Lambda}=U_{h}^{*}\mathbbm{1}_{P\leq\frac{\Lambda}{h}}\hat{R}_{-}\hat{R}_{+}\mathbbm{1}_{P\leq\frac{\Lambda}{h}}U_{h}$

hence

$\displaystyle\mathbbm{1}_{P_{h}\leq\Lambda}-\tilde{R}_{-}\tilde{R}_{+}$
$\displaystyle=U_{h}^{*}\mathbbm{1}_{P\leq\frac{\Lambda}{h}}^{2}U_{h}-U_{h}^{*}\mathbbm{1}_{P\leq\frac{\Lambda}{h}}\hat{R}_{-}\hat{R}_{+}\mathbbm{1}_{P\leq\frac{\Lambda}{h}}U_{h}$
$\displaystyle=U_{h}^{*}\mathbbm{1}_{P\leq\frac{\Lambda}{h}}(I-\hat{R}_{-}\hat{R}_{+})\mathbbm{1}_{P\leq\frac{\Lambda}{h}}U_{h}$
$\displaystyle=U_{h}^{*}(I-\hat{R}_{-}\hat{R}_{+})\mathbbm{1}_{P\leq\frac{\Lambda}{h}}U_{h}$

which has range $V_{K+1}$. ∎

In the argument to come it will be necessary to apply $\tilde{R}_{+}$ to
$\chi(x,hD)\widehat{F}u.$ The analysis is given by the following lemma, which
makes use of Lemma 2.7 in a fundamental way.

######  Lemma 2.9.

For any $u\in L^{2}(\mathbb{R})$ we have

$\tilde{R}_{+}\chi(x,hD)\widehat{F}u=L_{K}\tilde{R}_{+}u+Ru$ (2.27)

where $L_{K}$ is a lower triangular matrix independent of $u$ and $|Ru|\leq
C_{\varepsilon}h^{\frac{m-K-1}{2}-\varepsilon}\|u\|_{B}$.

###### Proof.

We compute the following (using analogous arguments to those of Lemma 2.2):

$\displaystyle\begin{split}\tilde{R}_{+}\chi(x,hD)\widehat{F}u&=\hat{R}_{+}U_{h}\mathbbm{1}_{P_{h}\leq\Lambda}\chi(x,hD)\widehat{F}u\\\
&=\hat{R}_{+}U_{h}\chi(x,hD)\widehat{F}u+O(h^{\infty})\|u\|_{L^{2}(\mathbb{R})}\\\
&=\hat{R}_{+}\widehat{F}U_{h}\chi_{\lambda}(x,hD)u+O(h^{\infty})\|u\|_{L^{2}(\mathbb{R})}\\\
&=\hat{R}_{+}\widehat{F}U_{h}\chi_{\lambda}(x,hD)\mathbbm{1}_{P_{h}\leq\Lambda}u+O(h^{\infty})\|u\|_{L^{2}(\mathbb{R})}\\\
&=\hat{R}_{+}\widehat{F}\widetilde{\chi_{\lambda}}U_{h}\mathbbm{1}_{P_{h}\leq\Lambda}u+O(h^{\infty})\|u\|_{L^{2}(\mathbb{R})}\\\
&=B_{K}D_{K}R_{+}\widetilde{\chi_{\lambda}}U_{h}\mathbbm{1}_{P_{h}\leq\Lambda}u+O(h^{\infty})\|u\|_{L^{2}(\mathbb{R})}\end{split}$
(2.28)

where
$D_{K}=\frac{1}{\sqrt{\lambda}}\mathrm{diag}(1,\frac{1}{\lambda},\dots,\frac{1}{\lambda^{K}})$.

We must now commute $R_{+}$ and $\widetilde{\chi_{\lambda}}$. We have by Lemma
2.7:

$\displaystyle\partial_{x}^{k}\widetilde{\chi_{\lambda}}U_{h}\mathbbm{1}_{P_{h}\leq\Lambda}u(x)|_{x=0}$
$\displaystyle=\partial_{x}^{k}U_{h}\mathbbm{1}_{P_{h}\leq\Lambda}u(x)|_{x=0}+O_{m}(h^{\frac{m-k-1}{2}-\varepsilon})\|U_{h}\mathbbm{1}_{P_{h}\leq\Lambda}u\|_{Y}.$

Then

$B_{K}D_{K}R_{+}\widetilde{\chi_{\lambda}}U_{h}\mathbbm{1}_{P_{h}\leq\Lambda}u=B_{K}D_{K}B_{K}^{-1}\tilde{R}_{+}u+O_{m}(h^{\frac{m-K-1}{2}-\varepsilon})\|u\|_{B}.$
(2.29)

Combining (2.28) and (2.29) gives (2.27). ∎

###  2.4. Inverting the Grushin Problem

Let

$\mathcal{P}(z):=\begin{pmatrix}I-z\chi(x,hD)\widehat{F}&\tilde{R}_{-}\\\
\tilde{R}_{+}&0\end{pmatrix}.$

Split $u\in L^{2}(\mathbb{R})$ as $u=u_{1}+u_{2}$ with
$u_{1}=\mathbbm{1}_{P_{h}\leq\Lambda}u$ and
$u_{2}=\mathbbm{1}_{P_{h}\geq\Lambda}u$, and similarly for $v=v_{1}+v_{2}$.
Consider $\mathcal{P}$ as an operator acting on
$\mathbbm{1}_{P_{h}\leq\Lambda}L^{2}(\mathbb{R})\oplus\mathbbm{1}_{P_{h}\geq\Lambda}L^{2}(\mathbb{R})\oplus\mathbb{C}^{K+1}$.
Let

$\mathcal{P}(z)\begin{pmatrix}u\\\ u_{-}\end{pmatrix}=\begin{pmatrix}v\\\
v_{+}\end{pmatrix}$

so

$(I-z\chi\widehat{F})(u_{1}+u_{2})+\tilde{R}_{-}u_{-}=v_{1}+v_{2}$
$\tilde{R}_{+}(u_{1}+u_{2})=v_{+}.$

The equations decouple (up to $O(h^{\infty})$ errors) as

$(I-z\chi\widehat{F}+R_{0})u_{1}+R_{1}u_{2}+\tilde{R}_{-}u_{-}=v_{1}$
$(I+R_{3})u_{2}+R_{2}u_{1}=v_{2}$ $\tilde{R}_{+}u_{1}=v_{+}.$

where $R_{0},R_{1},R_{2},R_{3}=O(h^{\infty})_{L^{2}\to L^{2}}$.

Define the following operator on
$L^{2}(\mathbb{R})\oplus\mathbb{C}^{K+1}=\mathbbm{1}_{P_{h}\leq\Lambda}L^{2}(\mathbb{R})\oplus\mathbbm{1}_{P_{h}\geq\Lambda}L^{2}(\mathbb{R})\oplus\mathbb{C}^{K+1}$:

$\tilde{\mathcal{E}}(z):=\begin{pmatrix}\tilde{E}&0&\tilde{E}_{+}\\\ 0&I&0\\\
\tilde{E}_{-}&0&\tilde{E}_{-+}\end{pmatrix}$ (2.30)

and define the nontrivial entries as follows:

$\tilde{E}:=\sum_{j=0}^{\infty}z^{j}(\chi(x,hD)\widehat{F})^{j}(I-\tilde{R}_{-}\tilde{R}_{+})$
$\tilde{E}_{+}:=\tilde{R}_{-}-\sum_{j=0}^{\infty}z^{j}(\chi(x,hD)\widehat{F})^{j}(I-\tilde{R}_{-}\tilde{R}_{+})(I-z\mathbbm{1}_{P_{h}\leq\Lambda}\chi(x,hD)\widehat{F})\tilde{R}_{-}$
$\tilde{E}_{-}:=\tilde{R}_{+}$
$\tilde{E}_{-+}:=-(I_{\mathbb{C}^{K+1}}-zL_{K}).$

Recall that by Proposition 2.8, $I-\tilde{R}_{-}\tilde{R}_{+}$ has range
$V_{K+1}$, so by Proposition 2.6, there is a $c>0$ independent of $m$ such
that for $h$ small enough

$\tilde{E},\tilde{E}_{+}\text{ defined and bounded for }|z|\leq
c\lambda^{\frac{K+1}{4}}.$ (2.31)

######  Lemma 2.10.

For all $m\in\mathbb{N},K\leq m-2$, there is an $r_{K}$ with $r_{K}\to\infty$
such that for $|z|\leq r_{K}$, $h$ small enough, the operator
$\tilde{\mathcal{E}}(z)$ is well-defined and satisfies

$\mathcal{P}(z)\tilde{\mathcal{E}}(z)=I+\mathcal{T}(z)$

with

$\|\mathcal{T}(z)\|_{B\oplus\mathbb{C}^{K+1}\to
B\oplus\mathbb{C}^{K+1}}=O(h^{\frac{m-K-1}{4}-\varepsilon}).$ (2.32)

###### Proof.

Let $r_{K}=c\lambda^{\frac{K}{8}}$, where $c$ is given in (2.31).

Let

$\mathcal{T}(z)=\begin{pmatrix}T_{11}&T_{12}&T_{1+}\\\ T_{21}&T_{22}&T_{2+}\\\
T_{+1}&T_{+2}&T_{++}\end{pmatrix}$

on
$\mathbbm{1}_{P_{h}\leq\Lambda}L^{2}(\mathbb{R})\oplus\mathbbm{1}_{P_{h}\geq\Lambda}L^{2}(\mathbb{R})\oplus\mathbb{C}^{K+1}$,
so expanded out we have:

$\begin{pmatrix}I+T_{11}&T_{12}&T_{1+}\\\ T_{21}&I+T_{22}&T_{2+}\\\
T_{+1}&T_{+2}&I+T_{++}\end{pmatrix}=\\\
\begin{pmatrix}I-z\chi(x,hD)\widehat{F}+R_{0}&R_{1}&\tilde{R}_{-}\\\
R_{2}&I+R_{3}&0\\\
\tilde{R}_{+}&0&0\end{pmatrix}\begin{pmatrix}\tilde{E}&0&\tilde{E}_{+}\\\
0&I&0\\\ \tilde{E}_{-}&0&\tilde{E}_{-+}\end{pmatrix}$ (2.33)

It remains to compute the product entries in (2.33). We have by telescoping
that:

$\displaystyle\begin{split}I+T_{11}&=(I-z\chi(x,hD)\widehat{F})\left(\sum_{j=0}^{\infty}z^{j}(\chi(x,hD)\widehat{F})^{j}(I-\tilde{R}_{-}\tilde{R}_{+})\right)+\tilde{R}_{-}\tilde{R}_{+}+R_{0}\tilde{E}\\\
&=I-\tilde{R}_{-}\tilde{R}_{+}+\tilde{R}_{-}\tilde{R}_{+}+R_{0}\tilde{E}\\\
&=I+O(h^{\infty}).\end{split}$

The equation

$T_{12}=R_{1}=O(h^{\infty})$

is trivial, and for $T_{1+}$ we compute by Lemma 2.9:

$\displaystyle\begin{split}T_{1+}&=(I-z\chi(x,hD)\widehat{F})\tilde{R}_{-}-(I-\tilde{R}_{-}\tilde{R}_{+})(I-z\mathbbm{1}_{P_{h}\leq\Lambda}\chi(x,hD)\widehat{F})\tilde{R}_{-}\\\
&\quad+\tilde{R}_{-}(-I_{\mathbb{C}^{K+1}}+zL_{K})+R_{0}\tilde{E}_{+}\\\
&=(I-z\chi(x,hD)\widehat{F})\tilde{R}_{-}-(I-\tilde{R}_{-}\tilde{R}_{+})(I-z\chi(x,hD)\widehat{F})\tilde{R}_{-}+\tilde{R}_{-}(-I_{\mathbb{C}^{K+1}}+zL_{K})\\\
&\quad+(I-\tilde{R}_{-}\tilde{R}_{+})(I-z\mathbbm{1}_{P_{h}\geq\Lambda}\chi(x,hD)\widehat{F})\tilde{R}_{-}+R_{0}\tilde{E}_{+}\\\
&=\tilde{R}_{-}\tilde{R}_{+}(I-z\chi(x,hD)\widehat{F})\tilde{R}_{-}+\tilde{R}_{-}(-I_{\mathbb{C}^{K+1}}+zL_{K})\\\
&\quad-(I-\tilde{R}_{-}\tilde{R}_{+})(z\mathbbm{1}_{P_{h}\geq\Lambda}\chi(x,hD)\widehat{F})\tilde{R}_{-}+R_{0}\tilde{E}_{+}\\\
&=\tilde{R}_{-}R\tilde{R}_{-}-(I-\tilde{R}_{-}\tilde{R}_{+})(z\mathbbm{1}_{P_{h}\geq\Lambda}\chi(x,hD)\widehat{F})\tilde{R}_{-}+R_{0}\tilde{E}_{+}\\\
&=O(h^{\infty}).\end{split}$

The next three

$T_{21}=R_{2}\tilde{E},\quad I+T_{22}=I+R_{3},\quad T_{2+}=R_{2}\tilde{E}_{+}$

are trivially all equal to $O(h^{\infty})$, and $T_{+2}=0$.

The bounds on $T_{+1}$ and $T_{+3}$ require a more delicate argument. Fix $J$
to determined later. Then we have:

$\displaystyle T_{+1}$
$\displaystyle=\tilde{R}_{+}\sum_{j=0}^{\infty}z^{j}(\chi(x,hD)\widehat{F})^{j}(I-\tilde{R}_{-}\tilde{R}_{+})$
$\displaystyle=\sum_{j=0}^{J}z^{j}\tilde{R}_{+}(\chi(x,hD)\widehat{F})^{j}(I-\tilde{R}_{-}\tilde{R}_{+})+\tilde{R}_{+}\sum_{j=J+1}^{\infty}z^{j}(\chi(x,hD)\widehat{F})^{j}(I-\tilde{R}_{-}\tilde{R}_{+}).$

By Lemma 2.6, for $|z|\leq r_{k}$

$\displaystyle\begin{split}&\left\|\tilde{R}_{+}\sum_{j=J+1}^{\infty}z^{j}(\chi(x,hD)\widehat{F})^{j}(I-\tilde{R}_{-}\tilde{R}_{+})\right\|_{B\to\mathbb{C}^{K+1}}\\\
&\leq
C\sum_{j=J+1}^{\infty}z^{j}\|(\chi(x,hD)\widehat{F})^{j}(I-\tilde{R}_{-}\tilde{R}_{+})\|_{B\to
B}\leq C\sum_{j=J+1}^{\infty}\lambda^{-\frac{K}{8}}\leq
C\lambda^{-\frac{JK}{8}}.\end{split}$ (2.34)

Meanwhile repeated application of Lemma 2.9 gives

$R_{+}(\chi(x,hD)\widehat{F})^{j}=L^{j}\tilde{R}_{+}+\sum_{i=0}^{j-1}L^{i}R(\chi(x,hD)\hat{F})^{j-1-i}$

with

$\left\|\sum_{i=0}^{j-1}L^{i}R(\chi(x,hD)\hat{F})^{j-1-i}\right\|_{B\to\mathbb{C}^{K+1}}\leq
CJh^{s}$

for $s=\frac{m-K-1}{2}-\varepsilon$ and $C=C(\varepsilon)$. Then

$\left\|\sum_{j=0}^{J}z^{j}\tilde{R}_{+}(\chi(x,hD)\widehat{F})^{j}(I-\tilde{R}_{-}\tilde{R}_{+})\right\|_{B\to\mathbb{C}^{K}}\leq
CJr_{K}^{J}h^{s}=CJ\lambda^{\frac{JK}{8}}h^{s}.$ (2.35)

Combining (LABEL:LateTerms1) and (2.35) gives:

$\|T_{+1}\|_{B\to\mathbb{C}^{K}}\leq
C\left(\lambda^{-\frac{JK}{8}}+J\lambda^{\frac{JK}{8}}h^{s}\right).$

To optimize, set $J=\frac{4s}{K\log\lambda}\log\frac{1}{h}$, which gives the
bound

$\|T_{+1}\|_{B\to\mathbb{C}^{K+1}}\leq Ch^{\frac{s}{2}}\log\frac{1}{h}.$
(2.36)

Finally to bound $T_{++}$ we use (2.36) to get:

$\displaystyle\begin{split}I+T_{++}&=\tilde{R}_{+}\tilde{R}_{-}-\tilde{R}_{+}\sum_{j=0}^{\infty}z^{j}(\chi(x,hD)\widehat{F})^{j}(I-\tilde{R}_{-}\tilde{R}_{+})(I-z\mathbbm{1}_{P_{h}\leq\Lambda}\chi(x,hD)\widehat{F})\tilde{R}_{-}\\\
&=I-T_{+1}\tilde{R}_{-}\\\
&=I+O\left(h^{\frac{s}{2}}\log\frac{1}{h}\right)_{\mathbb{C}^{K+1}\to\mathbb{C}^{K+1}}.\end{split}$

By choosing a new $\varepsilon$, this proves the bound (2.32) for $h$ small
enough, and completes the proof of the lemma. ∎

######  Corollary 2.11.

For all $m\in\mathbb{N},K\leq m-2$, there is an $r_{K}$ with $r_{K}\to\infty$,
such that for $h$ small enough there is a
$\mathcal{E}(z):L^{2}(\mathbb{R})\oplus\mathbb{C}^{K+1}\to
L^{2}(\mathbb{R})\oplus\mathbb{C}^{K+1}$ defined on $\\{|z|\leq R_{K}\\}$ such
that

$\mathcal{P}(z)\mathcal{E}(z)=\mathcal{E}(z)\mathcal{P}(z)=I.$

###### Proof.

We first find a right inverse via Neumann series. For $h$ small enough we may
define

$\mathcal{E}(z):=\tilde{\mathcal{E}}(z)\left(\sum_{j=0}^{\infty}(-1)^{j}\mathcal{T}(z)^{j}\right)$
(2.37)

which gives that $\mathcal{P}(z)\mathcal{E}(z)=I.$

We now claim that $\mathcal{P}(z)$ is an analytic family of Fredholm operators
of index $0$. For indeed for $z=0$ we have

$\mathcal{P}(0)=\begin{pmatrix}I&\tilde{R}_{-}\\\
\tilde{R}_{+}&0\end{pmatrix}$

and one may easily check that

$\mathcal{P}(0)^{-1}=\begin{pmatrix}I-\tilde{R}_{-}\tilde{R}_{+}&\tilde{R}_{-}\\\
\tilde{R}_{+}&-I\end{pmatrix}$

is a two-sided inverse. Then by continuity $\mathcal{P}(z)$ has index zero for
all $z$.

This shows that the right inverse $\mathcal{E}(z)$ defined in (2.37) is a two-
sided inverse, as desired. ∎

We may now estimate the spectrum of $\chi\widehat{F}$ using the Schur
complement formula. We have:

$\left(\mathcal{T}^{j}\right)_{+-}=O(h^{pj})$

for $p=\frac{m-K-1}{4}-\varepsilon$. This gives that

$E_{-+}=-I+zL_{K}+O(h^{p})=B_{K}(-I+zD_{K}+O(h^{p}))B_{K}^{-1},$

which shows that (using boundedness of all operators involved to bound
$\frac{1}{z}$ below) $\frac{1}{z}\in\sigma(\chi\widehat{F})$ if and only if
$D_{K}-\frac{1}{z}I=O(h^{p})$. This implies
$\frac{1}{z}\in\sigma_{h^{p}}(D_{K})$ (where $\sigma_{\varepsilon}$ denotes
the $\varepsilon$-pseudospectrum), meaning by self-adjointness that
$\frac{1}{z}=\lambda^{\frac{2k+1}{2}}+O(h^{p})$.

###### Proof of Theorem 1.

We generalize the above to metaplectic operators on $L^{2}(\mathbb{R})$. Let
$M\in Sp(2,\mathbb{R})$ have eigenvalues $\lambda,\lambda^{-1}$ with
$\lambda>1$. Then $M=Q^{-1}FQ$, where $F=\begin{pmatrix}\lambda&0\\\
0&\lambda^{-1}\end{pmatrix}$ is as before, so

$\operatorname{Op}_{h}\chi\widehat{M}=\operatorname{Op}_{h}\chi\widehat{Q^{-1}}\widehat{F}\widehat{Q}=\widehat{Q^{-1}}\operatorname{Op}_{h}(\chi\circ
Q^{-1})\widehat{F}\widehat{Q}.$ (2.38)

and hence has the same spectrum as $\operatorname{Op}_{h}(\chi\circ
Q^{-1})\widehat{F}$.

Thus by picking $K$ such that $\frac{1}{r_{K}}<\delta$ and $m\gg K$ as large
as we want, we have shown Theorem 1. ∎

In Section 3 of the paper, we will need an actual Grushin problem for the
general metaplectic operator. But replacing $\mathcal{P}(z)$ with
$\mathcal{P}^{Q}(z)=\mathcal{Q}^{-1}\mathcal{P}(z)\mathcal{Q}$ where
$\mathcal{Q}=\begin{pmatrix}\widehat{Q}&0\\\ 0&I\end{pmatrix}$, we have our
new block matrix

$\mathcal{P}^{Q}(z)=\begin{pmatrix}I-z\operatorname{Op}_{h}(\chi_{Q})\widehat{M}&\widehat{Q}^{-1}\tilde{R}_{-}\\\
\tilde{R}_{+}\widehat{Q}&0\end{pmatrix}$ (2.39)

where $\chi_{Q}=\chi\circ Q^{-1}$. Then replacing $\mathcal{E}(z)$ with
$\mathcal{Q}^{-1}\mathcal{E}(z)\mathcal{Q}$ gives the desired well-posedness.

##  3\. The operator on the torus

###  3.1. Quantization of the Torus

We describe the space of quantum states on $\mathbb{T}^{2}$, as seen in [4].
Let

$H_{N}:=\mathrm{span}\left\\{\frac{1}{\sqrt{N}}\sum_{k\in\mathbb{Z}}\delta_{x=k+\frac{n}{N}}:n\in\\{0,1,\dots,N-1\\}\right\\}$

with $h=\frac{1}{2\pi N}$. These are precisely the elements of $\mathcal{S}$
which are periodic in physical and phase space under the semiclassical Fourier
transform $\mathcal{F}_{h}$. For simplicity, define
$Q_{n}=\frac{1}{\sqrt{N}}\sum_{k\in\mathbb{Z}}\delta_{x=k+\frac{n}{N}}$, which
we decide by definition to be an orthonormal basis of $H_{N}$. One may remark
that this is an elementary example of the general geometric Toeplitz
quantization studied in [5].

We review the pseudodifferential calculus on $H_{N}$, which also may be found
in [3], [7], [20]. Let $a\in C^{\infty}(\mathbb{T}^{2})$. Then $a$ lifts to a
doubly-periodic function $\tilde{a}$ on $T^{*}\mathbb{R}$, which is seen to be
in the symbol class $S(1)$. Define

$\operatorname{Op}_{N}a=\operatorname{Op}_{h}a|_{H_{N}}$

which is a map from $H_{N}$ to itself that is also given in coordinates by

$\operatorname{Op}_{N}(a)Q_{j}=\sum_{m=0}^{N-1}A_{mj}Q_{m}$

with

$A_{mj}=\sum_{k,l\in\mathbb{Z}}\hat{a}(k,j-m-lN)(-1)^{kl}e^{\pi
i\frac{(j+m)k}{N}}.$

Most results from pseudodifferential calculus carry over to $H_{N}$, see [3],
[7] for details. In particular, there is an analogue of Lemma 2.1, given as
follows.

######  Lemma 3.1.

If $a\in S(1)$ on $\mathbb{T}^{2}$, and $\phi\in
C^{\infty}_{0}(\mathbb{T}^{2})$, then $\phi(\operatorname{Op}_{N}a)$ is a Weyl
pseudodiffernetial operator on $H_{N}$. Suppose $b\in S(1)$ on
$\mathbb{T}^{2}$ such that $\operatorname{supp}b\cap
a^{-1}(\operatorname{supp}\phi)=\emptyset$. Then
$\phi(\operatorname{Op}_{N}a)\operatorname{Op}_{N}b=O(h^{\infty})_{H_{N}\to
H_{N}}$.

The proof of Lemma 3.1 essentially consists of showing that the action of
$\phi(\operatorname{Op}_{N}a)$ on $H_{N}$ is the same as the action of the
matrix $\phi$ applied to the matrix with entries
$\langle\phi(\operatorname{Op}_{N}a)Q_{j},Q_{k}\rangle$. The details are given
(in a slightly more general situation) as Lemma 2.8 in [3].

In addition, a particular class of metaplectic operators can be defined on
$H_{N}$. Let $M\in Sp(2n,\mathbb{Z})$ be an integer symplectic matrix, and let
$N$ be even. Define $\widehat{M}_{N}=\widehat{M}|_{H_{N}}$, which is a unitary
map on $H_{N}$. Consequently, we have an analogous exact Egorov’s theorem:

$\operatorname{Op}_{N}(a)\widehat{M}_{N}=\widehat{M}_{N}\operatorname{Op}_{N}(a\circ
M).$

###  3.2. Transplantation Operators

We consider the following “sampling” maps between the torus and the real line.
Let $\psi\in\mathbb{C}_{0}^{\infty}(\mathbb{T}^{*}\mathbb{R})$ be a bump
function at $(0,0)$. The sampling operator $S_{\psi}:L^{2}(\mathbb{R})\to
H_{N}$ is defined by

$S_{\psi}u(x)=N^{-\frac{1}{2}}\sum_{n=0}^{N-1}a_{n}Q_{n}$

where

$a_{n}=\sum_{k\in\mathbb{Z}}(\operatorname{Op}_{h}(\psi)u)(k+\frac{n}{N})$

and as usual $h=\frac{1}{2\pi N}$. We may also write
$S_{\psi}=\Pi_{N}\operatorname{Op}_{h}\psi$, where
$\Pi_{N}:\mathcal{S}(\mathbb{R})\to H_{N}$ given by

$\Pi_{N}u=N^{-\frac{1}{2}}\sum_{n=0}^{N-1}\sum_{k\in\mathbb{Z}}u(k+\frac{n}{N})Q_{n}$

are the projection operators considered in [7]. As
$\operatorname{Op}_{h}(\psi):L^{2}(\mathbb{R})\to\mathcal{S}(\mathbb{R})$, we
easily see that $S_{\psi}$ is well defined and bounded. In fact, we shall soon
show that it is bounded independent of $h$.

The benefit of working with $S_{\psi}$ comes from the particularly nice form
of $S_{\psi}^{*}$ and $S_{\psi}S_{\psi}^{*}$.

######  Proposition 3.2.

The adjoint of $S_{\psi}$ is $\operatorname{Op}_{h}\overline{\psi}$, acting on
$H_{N}$ as a subset of $\mathcal{S}^{\prime}(\mathbb{R})$.

###### Proof.

Let $K(x,y)$ be the Schwartz kernel of $\operatorname{Op}_{h}\psi$. Then for
$u\in L^{2}(\mathbb{R})$, $v=\sum_{n=0}^{N-1}Q_{n}$, we have

$\langle
S_{\psi}u,v\rangle_{H_{N}}=N^{-\frac{1}{2}}\sum_{n=0}^{N-1}\overline{v_{n}}\sum_{k\in\mathbb{Z}}(\operatorname{Op}_{h}(\psi)\,u)(k+\frac{n}{N})N^{-\frac{1}{2}}\int\sum_{n=0}^{N-1}\overline{v_{n}}\sum_{k\in\mathbb{Z}}K(k+\frac{n}{N},y)u(y)\,dy$

which gives

$S_{\psi}^{*}v(y)=N^{-\frac{1}{2}}\sum_{n=0}^{N-1}\sum_{k\in\mathbb{Z}}\overline{K(k+\frac{n}{N},y)}v_{n}\\\
=(\operatorname{Op}_{h}(\overline{\psi})v)(y)$

as desired. ∎

######  Proposition 3.3.

If $\psi\in C^{\infty}_{0}(\mathbb{T}^{2})\subseteq
C^{\infty}_{0}(\mathbb{R}^{2})$, then

$\Pi_{N}\operatorname{Op}_{h}\psi|_{H_{N}}=\operatorname{Op}_{N}\tilde{\psi},$
(3.1)

where $\tilde{\psi}(x,\xi)=\sum_{k,l\in\mathbb{Z}}\psi(x+k,\xi+l)$ is the
periodization of $\psi$.

###### Proof.

Let $u\in H_{N}$. By periodicity of $u$, we have formally:

$\displaystyle\Pi_{N}\operatorname{Op}_{h}\psi u$
$\displaystyle=N^{-\frac{1}{2}}\sum_{n=0}^{N-1}\sum_{k\in\mathbb{Z}}\frac{1}{2\pi
h}\iint
e^{i(k+\frac{n}{N}-y)\xi}\psi\left(\frac{k+\frac{n}{N}+y}{2},\xi\right)u(y)\,dy\,d\xi
Q_{n}$
$\displaystyle=N^{-\frac{1}{2}}\sum_{n=0}^{N-1}\sum_{k\in\mathbb{Z}}\frac{1}{2\pi
h}\iint
e^{i(\frac{n}{N}-y)\xi}\psi\left(\frac{\frac{n}{N}+y}{2}+k,\xi\right)u(y+k)\,dy\,d\xi
Q_{n}$ $\displaystyle=N^{-\frac{1}{2}}\sum_{n=0}^{N-1}\frac{1}{2\pi h}\iint
e^{i(\frac{n}{N}-y)\xi}\sum_{k\in\mathbb{Z}}\psi\left(\frac{\frac{n}{N}+y}{2}+k,\xi\right)u(y)\,dy\,d\xi
Q_{n}$
$\displaystyle=N^{-\frac{1}{2}}\sum_{n=0}^{N-1}(\operatorname{Op}_{h}(\psi_{p})u)Q_{n}$

where $\psi_{p}(x,\xi)=\sum_{k\in\mathbb{Z}}\psi(x+k,\xi)$.

Now let $\psi_{L}=e^{\frac{i}{2}h\langle D_{x},D_{\xi}\rangle}\psi_{p},$ and
remark that $\psi_{L}$ is also periodic in the $x$ variable. Then we get the
left quantization

$\displaystyle
N^{-\frac{1}{2}}\sum_{n=0}^{N-1}(\operatorname{Op}_{h}^{L}\psi_{L}u)Q_{n}$
$\displaystyle=N^{-1}\sum_{k\in\mathbb{Z}}\sum_{n=0}^{N-1}\delta_{x=k+\frac{n}{N}}\frac{1}{2\pi
h}\int e^{\frac{i}{h}x\xi}\chi_{L}(x,\xi)\hat{u}(\xi)\,d\xi$
$\displaystyle=\sum_{m\in\mathbb{Z}}\delta_{x=\frac{m}{n}}\int
e^{\frac{i}{h}x\xi}\chi_{L}(x,\xi)\hat{u}(\xi)\,d\xi$
$\displaystyle=N\sum_{m\in\mathbb{Z}}e^{2\pi iNmx}\int
e^{\frac{i}{h}x\xi}\chi_{L}(x,\xi)\hat{u}(\xi)\,d\xi$
$\displaystyle=\frac{1}{2\pi
h}\int\sum_{m\in\mathbb{Z}}e^{\frac{i}{h}x(\xi+m)}\chi_{L}(x,\xi)\hat{u}(\xi)\,d\xi$
$\displaystyle=\frac{1}{2\pi
h}\int\sum_{m\in\mathbb{Z}}e^{\frac{i}{h}x\xi}\sum_{m\in\mathbb{Z}}\chi_{L}(x,\xi)\hat{u}(\xi+m)\,d\xi,$

where the third line follows from Poisson summation formula and the final line
from periodicity of $\hat{u}$. But as $e^{\frac{i}{2}h\langle
D_{x},D_{\xi}\rangle}$ commutes with periodization, we have this is just
$\operatorname{Op}_{h}(\tilde{\chi})u$, as desired. ∎

######  Corollary 3.4.

$S_{\psi}S_{\psi^{*}}=\operatorname{Op}_{N}\left(\widetilde{\psi\\#\overline{\psi}}\right),$
(3.2)

and consequently $S_{\psi}:L^{2}(\mathbb{R})\to H_{N}$ is bounded
independently of $h$.

###### Proof.

We simply compute:

$S_{\psi}S_{\psi}^{*}=\Pi_{N}\operatorname{Op}_{h}\psi\operatorname{Op}_{h}\overline{\psi}|_{H_{N}}=\Pi_{N}\operatorname{Op}_{h}\left(\widetilde{\psi\\#\overline{\psi}}\right)|_{H_{N}}=\operatorname{Op}_{N}\left(\widetilde{\psi\\#\overline{\psi}}\right).$

which gives (3.2). ∎

We now construct “spectral cutoffs” which will allow us to move our result
about $L^{2}$ onto $H_{N}$. Let $f\in C^{\infty}(\mathbb{T}^{2})$ have
$f(x,\xi)=(x^{2}+\xi^{2})$ on a neighborhood of $(0,0)$ that will be specified
later, and $f(x,\xi)=1$ outside a slightly larger neighborhood. In addition,
we assume $f$ to be nondecreasing with $|x|^{2}+|\xi|^{2}$. Let $v_{1},\dots
v_{B}\in H_{N}$ be eigenfunctions of $\operatorname{Op}_{N}F$ corresponding to
the smallest $B$ eigenvalues $\lambda_{1},\dots,\lambda_{B}$, such that
$\lambda_{B}<\Lambda^{\prime},\lambda_{B+1}>\Lambda^{\prime}+\Omega(h^{k})$
for some $k$, and for $\Lambda^{\prime}>\Lambda$ to be determined later. We do
suppose that $\psi=1$ on $\\{|x|^{2}+|\xi|^{2}\leq\Lambda^{\prime}\\}$. (By
Weyl’s law, such an $\Lambda^{\prime}$, $B$ exist in any $O(h)$-neighborhood.)
Let $H_{\Lambda^{\prime}}=\mathrm{span}\\{v_{1},\dots v_{B}\\}$. These
eigenfunctions serve intuitively as “$H_{N}$-analogues” to the familiar
Hermite functions, and we shall use this analogy in constructing our local
Grushin problem. Refer to Figure 3 for a diagram of the functions $\chi,\psi$,
and $f$ in relation to the balls of radii
$\sqrt{\Lambda},\sqrt{\Lambda^{\prime}}$.

Let
$T_{0}=S_{\psi}^{*}\mathbbm{1}_{F\leq\Lambda^{\prime}}=S_{\psi}^{*}\Pi_{H_{\Lambda^{\prime}}}$.
A key fact is that $T_{0}$ is almost unitary on $H_{\Lambda^{\prime}}$.

######  Lemma 3.5.

Let $f=x^{2}+\xi^{2}$ on $\operatorname{supp}\psi$. Then

$T_{0}^{*}T_{0}=\Pi_{H_{\Lambda^{\prime}}}(I+R_{\Lambda^{\prime}})$

where $R_{\Lambda^{\prime}}|_{H_{\Lambda^{\prime}}^{\perp}}=0$ and
$\|R_{\Lambda^{\prime}}\|_{H_{N}\to H_{N}}=O(h^{\infty})$.

###### Proof.

We trivially see that for $w\in H_{\Lambda^{\prime}}^{\perp}$, $T_{0}w=0$. Let
$v\in H_{\Lambda^{\prime}}$. Then for $\psi\in C^{\infty}(\mathbb{T}^{2})$
with $\phi=1$ on $|x|^{2}+|\xi|^{2}\leq E$ and $\tilde{\psi}=1$ on
$\operatorname{supp}\phi$, we get by Lemma 2.1 that

$\displaystyle T_{0}^{*}T_{0}v$
$\displaystyle=\Pi_{H_{\Lambda^{\prime}}}S_{\psi}S_{\psi}^{*}\Pi_{H_{\Lambda^{\prime}}}v=\Pi_{H_{\Lambda^{\prime}}}\operatorname{Op}_{N}(\psi\sharp\overline{\psi})\mathbbm{1}_{F\leq\Lambda^{\prime}}v=\Pi_{H_{\Lambda^{\prime}}}\operatorname{Op}_{N}(\psi\sharp\overline{\psi})\phi(F)\mathbbm{1}_{F\leq\Lambda^{\prime}}v$
$\displaystyle=\Pi_{H_{\Lambda^{\prime}}}\phi(F)\mathbbm{1}_{F\leq\Lambda^{\prime}}v+\Pi_{H_{\Lambda^{\prime}}}O(h^{\infty})v=\Pi_{H_{\Lambda^{\prime}}}(I+R_{\Lambda^{\prime}})v$

as desired. ∎

We now show that for any eigenvector $v$ of $F$ with eigenvalue at most
$\Lambda^{\prime}$, $T_{0}v$ is very close to being an eigenvector of $P_{h}$
with the same eigenvalue.

######  Lemma 3.6.

If $Fv=\lambda v$ with $\lambda\leq\Lambda^{\prime}$, then
$P_{h}T_{0}v=\lambda v+w$ with $\|w\|_{L^{2}(\mathbb{R})}=O(h^{\infty}).$

###### Proof.

If $Fv=\lambda v$, then letting $\phi\in C^{\infty}(\mathbb{T})$ be as in
Lemma 3.5 gives

$\displaystyle P_{h}S_{\psi}^{*}\mathbbm{1}_{F\leq\Lambda^{\prime}}v$
$\displaystyle=P_{h}S_{\psi}^{*}\phi(F)v=S_{\psi}^{*}\phi(F)Fv+O(h^{\infty})=\lambda
S_{\psi}^{*}\mathbbm{1}_{F\leq\Lambda^{\prime}}v+O(h^{\infty}).$

from which the result follows. ∎

We are now ready to construct our spectral cutoffs. Recall from Lemma 3.5 that
$T_{0}^{*}T_{0}=\Pi_{H_{\Lambda^{\prime}}}(I+R_{\Lambda^{\prime}})$. By the
$O(h^{\infty})$ bound on $R_{\Lambda^{\prime}}$, for $h$ small enough
$(I+R_{\Lambda^{\prime}})^{-\frac{1}{2}}$ is a well-defined bounded self-
adjoint operator, so let $T=T_{0}(I+R_{\Lambda^{\prime}})^{-\frac{1}{2}}$.
Then $T$ is a unitary transformation on $H_{\Lambda^{\prime}}$, and by Lemma
3.6 and eigenvalue separation near $\Lambda^{\prime}$, one has
$T:H_{\Lambda^{\prime}}\to\mathbbm{1}_{P_{h}\leq\Lambda}L^{2}(\mathbb{R})$.

The other key property of $T$ is that it plays well with our cutoff
metaplectic operator.

######  Proposition 3.7.

Given metaplectic $\widehat{M}$ with $M\in Sp(2,\mathbb{Z})$,

$T^{*}(I-z\chi(x,hD)\widehat{M})T=\Pi_{\mathrm{Ran}T^{*}}-z\operatorname{Op}_{N}\chi\widehat{M}_{N}+O(h^{\infty})_{H_{N}\to
H_{N}}.$

###### Proof.

We must only compute $T^{*}\chi(x,hD)\widehat{M}T$. But this is

$\displaystyle\mathbbm{1}_{F\leq\Lambda^{\prime}}S_{\psi}\chi(x,hD)\widehat{M}S_{\psi}^{*}\mathbbm{1}_{F\leq\Lambda^{\prime}}$
$\displaystyle=\mathbbm{1}_{F\leq\Lambda^{\prime}}S_{\psi}\chi(x,hD)\operatorname{Op}_{N}(\psi\circ
M^{-1})\widehat{M}_{N}\mathbbm{1}_{F\leq\Lambda^{\prime}}+O(h^{\infty})$
$\displaystyle=\mathbbm{1}_{F\leq\Lambda^{\prime}}\operatorname{Op}_{N}\chi\widehat{M}_{N}\mathbbm{1}_{F\leq\Lambda^{\prime}}+O(h^{\infty}).$

Let $\tilde{\phi}$ have
$\operatorname{supp}\tilde{\phi}\subseteq\\{|x|^{2}+|\xi|^{2}<\Lambda^{\prime}\\}$,
with $\tilde{\phi}=1$ on
$\operatorname{supp}\chi\cup\operatorname{supp}\chi_{\lambda}$. Then

$\displaystyle\mathbbm{1}_{F\leq\Lambda^{\prime}}\operatorname{Op}_{N}\chi\widehat{M}_{N}\mathbbm{1}_{F\leq\Lambda^{\prime}}$
$\displaystyle=\mathbbm{1}_{F\leq\Lambda^{\prime}}\tilde{\phi}(F)\operatorname{Op}_{N}\chi\widehat{M}_{N}\tilde{\phi}(F)\mathbbm{1}_{F\leq\Lambda^{\prime}}+O(h^{\infty})$
$\displaystyle=\tilde{\phi}(F)\operatorname{Op}_{N}\chi\widehat{M}_{N}\tilde{\phi}(F)+O(h^{\infty})$
$\displaystyle=\operatorname{Op}_{N}\chi\widehat{M}_{N}+O(h^{\infty})$

which completes the proof. ∎

###  3.3. Grushin Problem on the Torus

Let

$\mathcal{P}_{N}(z):=\begin{pmatrix}I-z\operatorname{Op}_{N}\chi_{Q}\widehat{M}_{N}&T^{*}\widehat{Q}^{-1}\tilde{R}_{-}\\\
\tilde{R}_{+}\widehat{Q}T&0\end{pmatrix}$ (3.3)

as a map from $H_{N}\oplus\mathbb{C}^{K+1}\to H_{N}\oplus\mathbb{C}^{K+1}$. We
need one more lemma about different sized spectral cutoffs on
$L^{2}(\mathbb{R}).$

######  Lemma 3.8.

If $\Lambda^{\prime}>\|Q\|^{2}\Lambda$, then

$\mathbbm{1}_{P_{h}\leq\Lambda}\widehat{Q}^{-1}\mathbbm{1}_{P_{h}\leq\Lambda^{\prime}}=\mathbbm{1}_{P_{h}\leq\Lambda}\widehat{Q}^{-1}+O(h^{\infty})_{L^{2}(\mathbb{R})\to
L^{2}(\mathbb{R})}.$

###### Proof.

Let $\phi_{1},\phi_{2}\in C^{\infty}(T^{*}\mathbb{R})$ with the following
properties:

  * •

$\operatorname{supp}\phi_{2}\subseteq|x|^{2}+|\xi|^{2}<\Lambda^{\prime}$.

  * •

$\phi_{2}(x,\xi)=1$ whenever
$(\|Q\|^{-1}x,\|Q\|^{-1}\xi)\in\operatorname{supp}\phi_{1}$.

  * •

$\phi_{1}(x,\xi)=1$ whenever $|x|^{2}+|\xi|^{2}\leq\Lambda$.

Let $\phi_{1}(P_{h})=\operatorname{Op}_{h}\phi_{P1}$,
$\phi_{2}(P_{h})=\operatorname{Op}_{h}\phi_{P2}$. Then by repeated
applications of Lemma 2.1 and exact Egorov theorem:

$\displaystyle\mathbbm{1}_{P_{h}\leq\Lambda}\widehat{Q}^{-1}\mathbbm{1}_{P_{h}\leq\Lambda^{\prime}}$
$\displaystyle=\mathbbm{1}_{P_{h}\leq\Lambda}\phi_{1}(P_{h})\widehat{Q}^{-1}\mathbbm{1}_{P_{h}\leq\Lambda^{\prime}}$
$\displaystyle=\mathbbm{1}_{P_{h}\leq\Lambda}\widehat{Q}^{-1}\operatorname{Op}_{h}(\phi_{P1}\circ
Q)\mathbbm{1}_{P_{h}\leq\Lambda^{\prime}}$
$\displaystyle=\mathbbm{1}_{P_{h}\leq\Lambda}\widehat{Q}^{-1}\operatorname{Op}_{h}(\phi_{P1}\circ
Q)\operatorname{Op}_{h}\phi_{P2}\mathbbm{1}_{P_{h}\leq\Lambda^{\prime}}+O(h^{\infty})_{L^{2}(\mathbb{R})\to
L^{2}(\mathbb{R})}$
$\displaystyle=\mathbbm{1}_{P_{h}\leq\Lambda}\widehat{Q}^{-1}\operatorname{Op}_{h}(\phi_{P1}\circ
Q)\phi_{2}(P_{h})\mathbbm{1}_{P_{h}\leq\Lambda^{\prime}}+O(h^{\infty})_{L^{2}(\mathbb{R})\to
L^{2}(\mathbb{R})}$
$\displaystyle=\mathbbm{1}_{P_{h}\leq\Lambda}\widehat{Q}^{-1}\operatorname{Op}_{h}(\phi_{P1}\circ
Q)\operatorname{Op}_{h}\phi_{P2}+O(h^{\infty})_{L^{2}(\mathbb{R})\to
L^{2}(\mathbb{R})}$
$\displaystyle=\mathbbm{1}_{P_{h}\leq\Lambda}\operatorname{Op}_{h}(\phi_{P1})\widehat{Q}^{-1}+O(h^{\infty})_{L^{2}(\mathbb{R})\to
L^{2}(\mathbb{R})}$
$\displaystyle=\mathbbm{1}_{P_{h}\leq\Lambda}\widehat{Q}^{-1}+O(h^{\infty})_{L^{2}(\mathbb{R})\to
L^{2}(\mathbb{R})}$

which completes the proof. ∎

Split $u\in H_{N}$ as $u=u_{1}+u_{2}$ with
$u_{1}=\mathbbm{1}_{F\leq\Lambda^{\prime}}u$ and
$u_{2}=\mathbbm{1}_{F\geq\Lambda^{\prime}}u$, and similarly for
$v=v_{1}+v_{2}$.

Consider $\mathcal{P}_{N}$ as an operator acting on
$H_{\Lambda^{\prime}}\oplus
H_{\Lambda^{\prime}}^{\perp}\oplus\mathbb{C}^{K+1}$. Let

$\mathcal{P}_{N}(z)\begin{pmatrix}u\\\ u_{-}\end{pmatrix}=\begin{pmatrix}v\\\
v_{+}\end{pmatrix}$

so

$(I-z\operatorname{Op}_{N}\chi_{Q}\widehat{M}_{N})(u_{1}+u_{2})+T^{*}\widehat{Q}^{-1}\tilde{R}_{-}u_{-}=v_{1}+v_{2}$
$\tilde{R}_{+}\widehat{Q}T(u_{1}+u_{2})=v_{+}.$

As before, the equations decouple (up to $O(h^{\infty})$ errors) as

$(I-z\operatorname{Op}_{n}\chi_{Q}\widehat{M}_{N}+S_{0})u_{1}+S_{1}u_{2}+T^{*}\widehat{Q}^{-1}\tilde{R}_{-}u_{-}=v_{1}$
$(I+S_{3})u_{2}+S_{2}u_{1}=v_{2}$ $\tilde{R}_{+}\widehat{Q}Tu_{1}=v_{+}.$

where $R_{1},R_{2}=O(h^{\infty})_{L^{2}\to L^{2}}$. Then we can write

$\mathcal{P}_{N}(z):=\begin{pmatrix}I-z\operatorname{Op}_{N}\chi_{Q}\widehat{M}_{N}+S_{0}&S_{1}&T^{*}\widehat{Q}^{-1}\tilde{R}_{-}\\\
S_{2}&I+S_{3}&0\\\ \tilde{R}_{+}\widehat{Q}T&0&0\end{pmatrix}$ (3.4)

as an operator on
$\mathbbm{1}_{F\leq\Lambda^{\prime}}H_{N}\oplus\mathbbm{1}_{Q>\Lambda^{\prime}}H_{N}\oplus\mathbb{C}^{N+1}$,
with $S_{0},S_{1},S_{2},S_{3}=O(h^{\infty})_{H_{N}\to H_{N}}$.

$x$$\xi$$\frac{1}{2}$$\operatorname{supp}\psi$$\operatorname{supp}\chi$$\operatorname{supp}\chi$$\sqrt{\Lambda}$$\sqrt{\Lambda^{\prime}}$$f(x,\xi)=x^{2}+\xi^{2}$$f(x,\xi)=(x-1)^{2}+\xi^{2}$
Figure 3. Nested cutoff functions used in the construction of
$\tilde{\mathcal{E}}_{N}$. The functions $\chi$ and $f$ are defined on the
torus, so two fundamental domains are shown. The inner dotted ellipse is the
support of $\chi_{Q}$, which must be contained in the disc
$\left\\{x^{2}+\xi^{2}\leq\frac{\Lambda}{2\lambda^{2}}\right\\}$ (see (2.39)
and Proposition 2.6).

Define

$\tilde{\mathcal{E}}_{N}(z):=\begin{pmatrix}T^{*}{E}_{Q}T&0&T^{*}{E}_{Q,+}\\\
0&I&0\\\ {E}_{Q,-}T&0&{E}_{Q,-+}\end{pmatrix}$ (3.5)

and where

$\mathcal{Q}^{-1}\mathcal{E}(z)\mathcal{Q}:=\mathcal{E}_{Q}(z)=\begin{pmatrix}E_{Q}&E_{Q,+}\\\
E_{Q,-}&E_{Q,-+}\end{pmatrix}$

is the inverse from Corollary 2.11 conjugated by $\mathcal{Q}$ acting on
$\mathbbm{1}_{P_{h}\leq\Lambda}L^{2}(\mathbb{R})\oplus\mathbb{C}^{K+1}$ where
we have by Lemma 2.10, Lemma 2.2, and Lemma 3.8 that all terms acting on
$\mathbbm{1}_{P_{h}>\Lambda}L^{2}(\mathbb{R})$ are of norm $O(h^{\infty})$.
Then we multiply (3.4) and (3.5) to get:

$\displaystyle\begin{split}&\mathcal{P}_{N}(z)\tilde{\mathcal{E}}_{N}(z)\\\
&=\begin{pmatrix}(I-z\operatorname{Op}_{N}\chi_{Q}\widehat{M}_{N})T^{*}E_{Q}T+T^{*}\tilde{R}_{-}E_{Q,-}T&S_{1}&(I-z\operatorname{Op}_{N}\chi_{Q}\widehat{M}_{N})T^{*}E_{Q,+}+T^{*}\tilde{R}_{-}E_{Q,-+}\\\
S_{2}T^{*}E_{Q}T&I+S_{3}&S_{2}T^{*}E_{Q,+}\\\
\tilde{R}_{+}TT^{*}E_{Q}T&0&R_{+}TT^{*}E_{Q,-+}\end{pmatrix}\\\
&=\begin{pmatrix}T^{*}\left((I-z\operatorname{Op}_{h}\chi_{Q}\widehat{M})E_{Q}+\tilde{R}_{-}E_{Q,-}\right)T&0&T^{*}\left((I-z\operatorname{Op}_{h}\chi_{Q}\widehat{M})E_{Q,+}+\tilde{R}_{-}E_{Q,-+}\right)\\\
0&I&0\\\ \tilde{R}_{+}E_{Q}T&0&R_{+}E_{Q,-+}\end{pmatrix}+O(h^{\infty})\\\
&:=I+\mathcal{S}(z)\end{split}$ (3.6)

where $\|\mathcal{S}(z)\|_{H_{N}\oplus\mathbb{C}^{K}\to
H_{N}\oplus\mathbb{C}^{K}}=O(h^{\infty}).$

We now resolve the Grushin problem in a similar manner to Corollary 2.11.

######  Corollary 3.9.

For all $K\in\mathbb{N}$, there is an $r_{K}$ with $r_{K}\to\infty$, such that
for $h$ small enough there is a
$\mathcal{E}_{N}(z):H_{N}\oplus\mathbb{C}^{K+1}\to
H_{N}\oplus\mathbb{C}^{K+1}$ defined on $\\{|z|\leq r_{K}\\}$ such that

$\mathcal{P}_{N}(z)\mathcal{E}_{N}(z)=\mathcal{E}_{N}(z)\mathcal{P}_{N}(z)=I.$

###### Proof.

The proof is analogous to that of Corollary 2.11, but simpler as everything is
finite-dimensional. We write:

$\mathcal{E}_{N}(z):=\tilde{\mathcal{E}}_{N}(z)\left(\sum_{j=0}^{\infty}(-1)^{j}\mathcal{S}(z)^{j}\right)$
(3.7)

which converges for $h$ small enough by (LABEL:AppInvT) and gives that
$\mathcal{E}_{N}(z)$ is a right inverse. As $H_{N}$ is a finite-dimensional
space, it must also be a left inverse. ∎

As in Theorem 1, the Schur complement formula gives:

###### Proof of Theorem 2.

We have:

$\left(\mathcal{S}^{j}\right)_{+-}=O(h^{\infty}).$

This gives that for $p=\frac{m-K-1}{4}-\varepsilon$,

$E_{Q,-+}=-I+zL_{K}+O(h^{p})=B_{K}(-I+zD_{K}+O(h^{p}))B_{K}^{-1},$

from which the result follows via the same argument as Theorem 1. ∎

Finally, we discuss the constraints on the support of $\chi$ for Theorem 2. We
require
$\operatorname{supp}\chi,\operatorname{supp}\nu\subseteq\\{|x|^{2}+|\xi|^{2}\leq\Lambda\\}$
to use Proposition 2.6, and in (2.39) we replaced $\chi$ with the pulled back
cutoff $\chi_{Q}$. For the theorems on the quantized real line, $\Lambda$ may
be fixed arbitrarily large, but on $H_{N}$ there are conditions imposed by
periodicity. Specifically, Lemma 3.8 requires
$\Lambda^{\prime}>\|Q\|^{2}\Lambda$, the construction of the spectral cutoffs
$T_{0}$ require that $\psi=1$ on $\\{|x|^{2}+|\xi|^{2}\leq\Lambda^{\prime}\\}$
and $f(x,\xi)=x^{2}+\xi^{2}$ on $\operatorname{supp}\psi$ for Lemmas 3.5 and
3.6 to apply, and $f$ must in particular be a smooth function, so may not
equal $x^{2}+\xi^{2}$ on a ball about the origin of radius larger than
$\frac{1}{2}$. These requirements are satisfied for certain $\psi$, $f$,
$\Lambda$, $\Lambda^{\prime}$ provided
$\operatorname{supp}\chi,\operatorname{supp}\nu$ is contained in a ball of
radius $\frac{c}{\lambda\|Q\|^{2}}$ about the origin, as claimed in the
discussion in Section 1.2.

##  4\. Rates of Decay for Nontrapping Eigenvalues

We now move on to the proof of Theorem 3. We use the technique of a microlocal
weight function, similar to (but simpler than) that used by Nonnenmacher,
Sjöstrand, and Zworski in [17] and also to Section 6.4 of [8].

###### Proof of Theorem 3.

Let $g\in C^{\infty}_{0}(\mathbb{T}^{2};\mathbb{R})$ have
$g(x,\xi)=x^{2}-\xi^{2}$ on a neighborhood of $\operatorname{supp}\nu$. Let

$M_{tg}:=e^{-t\operatorname{Op}_{N}g}\operatorname{Op}_{N}\nu\widehat{M}_{N}e^{t\operatorname{Op}_{N}g}=e^{-t\operatorname{Op}_{N}g}\widehat{M}_{N}\operatorname{Op}_{N}(M^{*}\nu)e^{t\operatorname{Op}_{N}g}$

for some $t$ that may depend on $N$. We shall bound the norm of $M_{tg}$ and
optimize by taking $t=K^{\prime}\log\left(\frac{1}{h}\right)$ at the end.
Recall that by results in Section 2.1 adapted to the calculus on $H_{N}$,
$e^{\pm t\operatorname{Op}_{N}g}$ are semiclassical pseudodifferential
operators with norm at most $O\left(e^{C|t|}\right)$. We compute:

$\displaystyle
M_{tg}=e^{-t\operatorname{Op}_{N}g}\widehat{M}_{N}e^{t\operatorname{Op}_{N}g}e^{-t\operatorname{Op}_{N}g}\operatorname{Op}_{N}(M^{*}\nu)e^{t\operatorname{Op}_{N}g}$
$\displaystyle=e^{-t\operatorname{Op}_{N}g}e^{t\operatorname{Op}_{N}\left(\left(M^{-1}\right)^{*}g\right)}\widehat{M}_{N}\operatorname{Op}_{N}\nu_{tg}$
$\displaystyle=e^{-t\operatorname{Op}_{N}g}e^{t\operatorname{Op}_{N}\left(\left(M^{-1}\right)^{*}g\right)}e^{t\left(\operatorname{Op}_{N}g-\operatorname{Op}_{N}\left(\left(M^{-1}\right)^{*}g\right)\right)}e^{-t\left(\operatorname{Op}_{N}g-\operatorname{Op}_{N}\left(\left(M^{-1}\right)^{*}g\right)\right)}\widehat{M}_{N}\operatorname{Op}_{N}\nu_{tg}.$

where
$\operatorname{Op}_{N}\nu_{tg}=e^{-t\operatorname{Op}_{N}g}\operatorname{Op}_{N}(M^{*}\nu)e^{t\operatorname{Op}_{N}g}$
is a pseudodifferential operator and the second equality comes from the exact
Egorov’s theorem applied to the functional calculus expression
$e^{t\operatorname{Op}_{N}g}$. Note also that if $t=O(\log\frac{1}{h})$, then
$\|e^{\pm t\operatorname{Op}_{N}g}\|_{H_{N}\to H_{N}}\leq O(h^{M})$ for some
$M$, so the microlocal support of $\operatorname{Op}_{N}\nu$ is preserved.

We claim that
$e^{-t\operatorname{Op}_{N}g}e^{t\operatorname{Op}_{N}\left(\left(M^{-1}\right)^{*}g\right)}e^{t\left(\operatorname{Op}_{N}g-\operatorname{Op}_{N}\left(\left(M^{-1}\right)^{*}g\right)\right)}$
is bounded independent of $t$ and $N$. For convenience let
$G_{1}=\operatorname{Op}_{N}g$,
$G_{2}=\operatorname{Op}_{N}\left(\left(M^{-1}\right)^{*}g\right)$, so we must
bound $W(t):=e^{-tG_{1}}e^{tG_{2}}e^{t(G_{1}-G_{2})}$. We have that

$\displaystyle\begin{split}\partial_{t}W(t)&=-G_{1}e^{-tG_{1}}e^{tG_{2}}e^{t(G_{1}-G_{2})}+e^{-tG_{1}}e^{tG_{2}}G_{1}e^{t(G_{1}-G_{2})}\\\
&=\left(-G_{1}+e^{-tG_{1}}e^{tG_{2}}G_{1}e^{-tG_{2}}e^{tG_{1}}\right)W(t)\\\
&:=Z(t)W(t)\end{split}$ (4.1)

where $Z$ is a pseudodifferential operator. To establish bounds on $Z(t)$, we
argue as in Theorem 6.21 in [8]. We see that

$\displaystyle Z(t)$
$\displaystyle=-G_{1}+e^{tG_{2}}G_{1}e^{-tG_{2}}-e^{tG_{2}}G_{1}e^{-tG_{2}}+e^{-tG_{1}}e^{tG_{2}}G_{1}e^{-tG_{2}}e^{tG_{1}}$
$\displaystyle=(-I+e^{\operatorname{ad}_{tG_{2}}})G_{1}+(-I+e^{\operatorname{ad}_{-tG_{1}}})\left(e^{tG_{2}}G_{1}e^{-tG_{2}}\right).$
(4.2)

We bound the first term of (4.2), and the second will follow similarly. Fix
$K$ large, and expand using Taylor’s formula:

$(-I+e^{\operatorname{ad}_{tG_{2}}})G_{1}=\sum_{k=1}^{K}\frac{t^{k}}{k!}\operatorname{ad}_{G_{2}}^{k}G_{1}+\frac{t^{K+1}}{K!}\int_{0}^{1}(1-s)^{K}e^{-tsG_{2}}\operatorname{ad}_{G_{2}}^{K+1}G_{1}e^{tsG_{2}}\,ds.$
(4.3)

Note that by pseudodifferential calculus
$\|\operatorname{ad}_{G_{2}}^{k}G_{1}\|_{H_{N}\to H_{N}}=O(h^{k})$. Plugging
into (4.3) gives for some $C>0$

$\|(-I+e^{\operatorname{ad}_{tG_{2}}})G_{1}\|_{H_{N}\to
H_{N}}=O(th+(th)^{K}e^{Ct})$

and an analogous argument for
$(-I+e^{\operatorname{ad}_{-tG_{1}}})\left(e^{tG_{2}}G_{1}e^{-tG_{2}}\right)$
gives that

$\|Z(t)\|_{H_{N}\to H_{N}}=O(th+(th)^{K}e^{Ct}).$

Now choose $t=K^{\prime}\log\frac{1}{h}$ for fixed $K^{\prime}$. Making $K$
large enough, we have $\|Z(s)\|_{H_{N}\to H_{N}}=O(h\log\frac{1}{h})$
uniformly in $s\leq t$. By (4.1) and Gronwall’s inequality, we see that

$\|W(t)\|_{H_{N}\to H_{N}}\leq
e^{tCh\log\frac{1}{h}}=e^{C_{0}h\left(\log\frac{1}{h}\right)^{2}}\leq C_{1}.$

We now show the norm of
$e^{-t\left(\operatorname{Op}_{N}g-\operatorname{Op}_{N}\left(\left(M^{-1}\right)^{*}g\right)\right)}\widehat{M}_{N}\operatorname{Op}_{N}\nu_{tg}$
to be small as $t$ grows large. Note that
$g-\left(\left(M^{-1}\right)^{*}g\right)\geq c_{0}$ on a neigborhood away from
the origin, which we may take as $\operatorname{supp}\nu_{tg}$ up to
$O(h^{\infty})$ errors, so by the same method as Lemma 3.6 in [17] we may
replace it with
$\left(g-\left(\left(M^{-1}\right)^{*}g\right)\right)\nu_{1}+(1-\nu_{1})$ with
$\nu_{1}=1$ on $\operatorname{supp}\nu$ and $g=x^{2}-\xi^{2}$ on
$\operatorname{supp}\nu_{1}$. Then by the Gårding inequality,
$\operatorname{Op}_{N}(g-(M^{-1})^{*}g)\nu_{1}+(1-\nu_{1})$ has spectrum
bounded below by $\frac{c_{0}}{2}$ for $h$ small enough. This gives

$\left\|e^{-t\left(\operatorname{Op}_{N}g-\operatorname{Op}_{N}\left(\left(M^{-1}\right)^{*}g\right)\right)}\right\|\leq
e^{-\frac{t}{2}c_{0}}=h^{K^{\prime}\frac{c_{0}}{2}}$

where $K^{\prime}$ can be chosen arbitrarily large, which completes the proof.
∎

##  5\. Acknowledgements

We would like to thank Maciej Zworski for suggesting the problem and giving
guidance along the way. We also acknowledge the partial support under the NSF
grant DMS-1952939.

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