# Symplectic Normal Form and Growth of Sobolev Norm

Zhenguo Liang  School of Mathematical Sciences and Key Lab of Mathematics for
Nonlinear Science, Fudan University, Shanghai 200433, China
<EMAIL_ADDRESS>,  Jiawen Luo  School of Mathematical Sciences and Key
Lab of Mathematics for Nonlinear Science, Fudan University, Shanghai 200433,
China<EMAIL_ADDRESS>and  Zhiyan Zhao  Université Côte d’Azur,
CNRS, Laboratoire J. A. Dieudonné, 06108 Nice, France  zhiyan.zhao@univ-
cotedazur.fr

###### Abstract.

For a class of reducible Hamiltonian partial differential equations (PDEs)
with arbitrary spatial dimension, quantified by a quadratic polynomial with
time-dependent coefficients, we present a comprehensive classification of
long-term solution behaviors within Sobolev space. This classification is
achieved through the utilization of Metaplectic and Schrödinger
representations. Each pattern of Sobolev norm behavior corresponds to a
specific $n-$dimensional symplectic normal form, as detailed in Theorems 1.1
and 1.2.

When applied to periodically or quasi-periodically forced $n-$dimensional
quantum harmonic oscillators, we identify novel growth rates for the
$\mathcal{H}^{s}-$norm as $t$ tends to infinity, such as $t^{(n-1)s}e^{\lambda
st}$ (with $\lambda>0$) and $t^{(2n-1)s}+\iota t^{2ns}$ (with $\iota\geq 0$).
Notably, we demonstrate that stability in Sobolev space, defined as the
boundedness of the Sobolev norm, is essentially a unique characteristic of
one-dimensional scenarios, as outlined in Theorem 1.3.

As a byproduct, we discover that the growth rate of the Sobolev norm for the
quantum Hamiltonian can be directly described by that of the solution to the
classical Hamiltonian which exhibits the “fastest” growth, as articulated in
Theorem 1.4.

Keywords: Quadratic Hamiltonian; Reducibility; Symplectic normal form;
Metaplectic representation; Schrödinger representation; Classification of
growth of Sobolev norm

MSC 2020: 35Q40, 37C60, 35Q41, 47G30

Z. Liang was partially supported by National Natural Science Foundation of
China (Grants No. 12071083).

Z. Zhao was partially supported by the French government through the National
Research Angency (ANR) grant for the project KEN ANR-22-CE40-0016 and
partially supported by NSFC grant 12271091.

##  1\. Introduction of the Main Results

This paper focuses on the Sobolev norms of solutions to Hamiltonian partial
differential equations (PDEs), particularly aiming to introduce new behavioral
patterns and to provide a thorough classification of growth rates for certain
quantum Hamiltonians. This task is approached through a reducibility argument
and a complete classification of quantum normal forms.

In view of the significance of the quadratic quantum Hamiltonian in
illustrating wave propagation phenomena and the connection with the coherent
state, the study concentrates on Hamiltonian PDEs quantized by a quadratic
polynomial with time dependent coefficients. Specifically, the work delves
into the $n-$dimensional PDE

$\frac{1}{2\pi{\rm i}}\partial_{t}\psi(t,x)=H_{0}(t,D,X)\psi(t,x),\qquad
x=(x_{1},\cdots,x_{n})\in{\mathbb{R}}^{n},$ (1.1)

where $\displaystyle D=\begin{pmatrix}D_{1}\\\ \vdots\\\
D_{n}\end{pmatrix}:=\frac{1}{2\pi{\rm i}}\nabla$, $\displaystyle
X:=\begin{pmatrix}X_{1}\\\ \vdots\\\ X_{n}\end{pmatrix}$ with $X_{j}$ the
multiplication by $j-$th coordinate function, i.e., $(X_{j}f)(x)=x_{j}f(x)$,
and $H_{0}(t,D,X)$ is a time dependent quadratic polynomial in $(D,X)$, i.e.,

$\displaystyle H_{0}(t,D,X)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left\langle
D,A_{12}(t)D\right\rangle-\frac{1}{2}\left\langle
D,A_{11}(t)X\right\rangle-\frac{1}{2}\left\langle
X,A_{11}(t)^{*}D\right\rangle-\frac{1}{2}\left\langle
X,A_{21}(t)X\right\rangle$ $\displaystyle+\,\left\langle
l_{1}(t),D\right\rangle+\left\langle l_{2}(t),X\right\rangle+c(t),$

with the superscript “$*$” the transpose of matrix or vector, and $c(\cdot)\in
C_{b}^{0}({\mathbb{R}},{\mathbb{R}})$, $l_{j}(\cdot)\in
C_{b}^{0}({\mathbb{R}},{\mathbb{R}}^{n})$, $A_{ij}(\cdot)\in
C_{b}^{0}({\mathbb{R}},{\rm gl}(n,{\mathbb{R}}))$ satisfying
$A_{12}(t)=A_{12}(t)^{*}$, $A_{21}(t)=A_{21}(t)^{*}$. Here, “$C_{b}^{0}$”
means the continuous class with uniform boundedness w.r.t. $t\in{\mathbb{R}}$
(see Section 1.4–(5)).

Define the column of operators $Z$ by
$Z^{*}=(D^{*},X^{*})=(D_{1},\cdots,D_{n},X_{1},\cdots,X_{n})$. Then the
quadratic quantum Hamiltonian (1.1) is rewritten as

$\frac{1}{2\pi{\rm
i}}\partial_{t}\psi=({\mathcal{Q}}_{{\mathcal{A}}(t)}(Z)+{\mathcal{L}}_{\ell(t)}(Z)+c(t))\psi,$
(1.2)

with the linear operator $H_{0}(t,D,X)$ in Eq. (1.1) decomposed into

  * •

Homogeneous part ${\mathcal{Q}}_{{\mathcal{A}}(t)}(Z):=-\frac{1}{2}\langle
Z,\mathcal{A}(t){\mathbb{J}}_{n}Z\rangle$ with

${\mathcal{A}}(\cdot)=\left(\begin{array}[]{cc}A_{11}(\cdot)&A_{12}(\cdot)\\\
A_{21}(\cdot)&-A_{11}(\cdot)^{*}\end{array}\right)\in
C_{b}^{0}({\mathbb{R}},{\rm
sp}(n,{\mathbb{R}})),\qquad{\mathbb{J}}_{n}:=\left(\begin{array}[]{cc}0&I_{n}\\\
-I_{n}&0\end{array}\right),$

  * •

Linear part ${\mathcal{L}}_{\ell(t)}(Z):=\langle\ell(t),Z\rangle$ with
$\ell(\cdot)=(l_{1}(\cdot)^{*},l_{2}(\cdot)^{*})^{*}\in
C_{b}^{0}({\mathbb{R}},{\mathbb{R}}^{2n})$,

  * •

Scalar part $c(t)$.

It is quantized by the quadratic polynomial classical hamiltonian (usually
called the symbol of $H_{0}$)

$h_{0}(t,\xi,x)=-\frac{1}{2}\langle
z,\mathcal{A}(t){\mathbb{J}}_{n}z\rangle+\langle\ell(t),z\rangle+c(t),\qquad\xi,x\in{\mathbb{R}}^{n},$
(1.3)

where $z\in{\mathbb{R}}^{2n}$ such that $z^{*}=(\xi^{*},x^{*})$, and the three
terms of $h_{0}(t,\xi,x)$ are also called homogeneous part, linear part and
scalar part respectively in the classical hamiltonian sense.

As a typical example of the quantum Hamiltonian $H_{0}$ and a well-known
“equilibrium state”, the $n-$D quantum harmonic oscillator (QHO for short
through the paper),

${\mathcal{T}}=\sum_{j=1}^{n}{\mathcal{T}}_{j}:=2\pi\left(\langle
D,D\rangle+\langle
X,X\rangle\right)=-\frac{1}{2\pi}\Delta+2\pi|X|^{2},\qquad{\mathcal{T}}_{j}:=2\pi(D_{j}^{2}+X_{j}^{2}),$
(1.4)

i.e., $H_{0}$ with constant coefficients
$\mathcal{A}(\cdot)=4\pi{\mathbb{J}}_{n}$, $\ell(\cdot)=0$, $c(\cdot)=0$, as
well as their perturbations, are well investigated in many recent works and it
is much related to the present paper.

###  1.1. Reducibility and symplectic normal forms

For a specific quantum Hamiltonian $H_{0}(t,D,X)$ with constant coefficients,
a straightforward approach exists for analyzing the propagators of Eq. (1.2).
This includes, for example, Mehler’s formula for the $n-$D QHO ${\mathcal{T}}$
defined in (1.4). The properties of the solutions are explored through direct
computations. Consequently, to efficiently describe the behavior of solutions
to the time-dependent equation, it is practical to remove the time-dependence
from the original equation via an appropriate transformation. This process is
referred to as reducibility of the quantum Hamiltonian in this paper.

A considerable amount of prior research has focused on the reducibility of
time-dependent PDEs. For $1-$D QHOs, time periodic smooth perturbations were
initially examined [14, 17, 21, 31, 36]. In the context of time quasi-periodic
perturbations, both bounded [28, 40, 54, 55] and unbounded disturbances [2, 3,
9, 38, 42, 45] have been thoroughly investigated. The study of reducibility
issues has led to the comprehensive development of infinite-dimensional KAM
theory for $1-$D PDEs with unbounded perturbations, as seen in works by
Bambusi-Graffi [4], Kuksin [37], and Liu-Yuan [44].

Regarding higher-dimensional PDEs, the concept of reducibility was pioneered
by Eliasson-Kuksin [19, 20] for the quasi-periodic Schrödinger equation. For
higher-dimensional QHOs, references [29, 39, 41] discuss bounded potentials,
and [5] examines unbounded perturbations, including Hamiltonian PDEs quantized
by quadratic polynomials. Through the lens of reducibility, these results
offer insights into the long-term behavior of solutions (at least in terms of
upper bounds of Sobolev norms) and the spectral characteristics of the Floquet
operator.

As deduced from a quantization argument (see Lemma 4.3 and 4.4), the
reducibility of the quadratic quantum Hamiltonian (1.1) is closely linked to
the reducibility of the corresponding finite-dimensional system. To fully
comprehend the dynamics of solutions to Eq. (1.2), it is crucial to understand
the precise forms of the reduced constant equations, termed here as the
quantum normal forms (QNFs). In essence, a comprehensive classification of
quantum normal forms provides a complete depiction of the long-term behavior
of solutions.

Prior to presenting the main theorem regarding QNFs, it is essential to define
reducibility for both finite-dimensional and infinite-dimensional systems
precisely.

######  Definition 1.

(Reducibility in classical and quantum Hamiltonians) The affine system

$z^{\prime}={\mathcal{A}}(t)z+\ell(t),\qquad{\mathcal{A}}(\cdot)\in
C_{b}^{0}({\mathbb{R}},{\rm sp}(n,{\mathbb{R}})),\qquad\ell(\cdot)\in
C_{b}^{0}({\mathbb{R}},{\mathbb{R}}^{2n})$

is called reducible, if there exist ${\mathcal{B}}\in{\rm
sp}(n,{\mathbb{R}})$, $l\in{\mathbb{R}}^{2n}$ and $S(\cdot)\in
C_{b}^{1}({\mathbb{R}},{\rm Sp}(n,{\mathbb{R}}))$, $p(\cdot)\in
C_{b}^{1}({\mathbb{R}},{\mathbb{R}}^{2n})$
222$C_{b}^{1}({\mathbb{R}},\bullet)$, with $\bullet={\rm Sp}(n,{\mathbb{R}})$
or ${\mathbb{R}}^{2n}$, is the subspace of $C^{0}_{b}({\mathbb{R}},\bullet)$
whose entries are all $C^{1}$ with uniformly bounded derivatives on
${\mathbb{R}}$. See Section 1.4–(5)., such that it is conjugated to
$y^{\prime}={\mathcal{B}}y+l$ under the change of variable $z=S(t)y+p(t)$. The
quadratic quantum Hamiltonian with time dependent coefficients
$\frac{1}{2\pi{\rm i}}\partial_{t}\psi=H(t,Z)\psi$ is called reducible, if
there exist an $L^{2}-$unitary transformation
$\psi(t)={\mathcal{U}}(t)\varphi(t)$, and a quadratic quantum Hamiltonian
${\mathcal{R}}(Z)$ with constant coefficients, such that it is conjugated to
$\frac{1}{2\pi{\rm i}}\partial_{t}\varphi={\mathcal{R}}(Z)\varphi$.

######  Remark 1.

It is important to note that in the definition of reducibility, the regularity
required for the time-dependence in the variable transformation can differ
based on the regularity of the original system. For further details on this
variation, refer to Section 8.

######  Theorem 1.1.

(Reducibility and classification of QNFs) If the system
$z^{\prime}={\mathcal{A}}(t)z+\ell(t)$, with ${\mathcal{A}}(\cdot)\in
C_{b}^{0}({\mathbb{R}},{\rm sp}(n,{\mathbb{R}}))$ and $\ell(\cdot)\in
C_{b}^{0}({\mathbb{R}},{\mathbb{R}}^{2n})$ as in Eq. (1.2), is reducible, then
Eq. (1.2) is also reducible, via some $L^{2}-$unitary transformation
$\psi(t)={\mathcal{U}}(t)\varphi(t)$, to $\frac{1}{2\pi{\rm
i}}\partial_{t}\varphi(t,x)={\mathcal{R}}(Z)\varphi(t,x)$, where the operator
${\mathcal{R}}(Z)$ is a time-independent operator, and in one of following
forms.

  * QNF1

For $n\in{\mathbb{N}}^{*}$,
${\mathcal{R}}(Z)={\mathcal{R}}^{(n)}_{1}(Z):={\mathcal{Q}}_{{\mathcal{B}}^{(n)}_{1}(\lambda)}=-\frac{1}{2}\left\langle
Z,{\mathcal{B}}^{(n)}_{1}(\lambda){\mathbb{J}}_{n}Z\right\rangle$,
$\lambda>0$, with

${\mathcal{B}}^{(n)}_{1}(\lambda):=\left(\begin{array}[]{cccc:cccc}-\lambda&-1&&&&&&\\\
&\ddots&\ddots&&&&&\\\ &&-\lambda&-1&&&&\\\ &&&-\lambda&&&&\\\
\hdashline&&&&\lambda&&&\\\ &&&&1&\lambda&&\\\ &&&&&\ddots&\ddots&\\\
&&&&&&1&\lambda\end{array}\right)\in{\rm sp}(n,{\mathbb{R}}).$ (1.5)

In particular, ${\mathcal{B}}^{(1)}_{1}(\lambda)=\begin{pmatrix}-\lambda&0\\\
0&\lambda\end{pmatrix}$.

  * QNF2

For $n\in{\mathbb{N}}^{*}$ even,
${\mathcal{R}}(Z)={\mathcal{R}}^{(n)}_{2}(Z)={\mathcal{Q}}_{{\mathcal{B}}^{(n)}_{2}(\lambda_{1},\lambda_{2})}$,
$\lambda_{1},\lambda_{2}>0$, with

${\mathcal{B}}^{(n)}_{2}(\lambda_{1},\lambda_{2}):=\left(\begin{array}[]{cccc:cccc}A_{2}&&&&&&&\\\
-I_{2}&A_{2}&&&&&&\\\ &\ddots&\ddots&&&&&\\\ &&-I_{2}&A_{2}&&&&\\\
\hdashline&&&&-A_{2}^{*}&I_{2}&&\\\ &&&&&\ddots&\ddots&\\\
&&&&&&-A_{2}^{*}&I_{2}\\\ &&&&&&&-A_{2}^{*}\end{array}\right)\in{\rm
sp}(n,{\mathbb{R}}),$ (1.6)

where $A_{2}:=\begin{pmatrix}-\lambda_{1}&-\lambda_{2}\\\
\lambda_{2}&-\lambda_{1}\end{pmatrix}$. In particular,
${\mathcal{B}}^{(2)}_{2}(\lambda_{1},\lambda_{2})=\begin{pmatrix}A_{2}&0\\\
0&-A_{2}^{*}\end{pmatrix}$.

  * QNF3

For $n\in{\mathbb{N}}^{*}$, for
${\mathcal{R}}(Z)={\mathcal{R}}^{(n)}_{3}(Z)={\mathcal{Q}}_{{\mathcal{B}}^{(n)}_{3}(\mu,\gamma)}$,
$\gamma=\pm 1$, $\mu>0$, where

${\mathcal{B}}^{(n)}_{3}(\mu,\gamma):=\gamma\left(\begin{array}[]{cccc:cccc}&&&&&&-1&\mu\\\
&&&&&\begin{sideways}$\ddots$\end{sideways}&\begin{sideways}$\ddots$\end{sideways}&\\\
&&&&-1&\mu&&\\\ &&&&\mu&&&\\\ \hdashline&&&-\mu&&&&\\\ &&-\mu&1&&&&\\\
&\begin{sideways}$\ddots$\end{sideways}&\begin{sideways}$\ddots$\end{sideways}&&&&&\\\
-\mu&1&&&&&&\end{array}\right)\in{\rm sp}(n,{\mathbb{R}}).$ (1.7)

In particular,
${\mathcal{B}}^{(1)}_{3}(\mu,\gamma)=\gamma\begin{pmatrix}0&\mu\\\
-\mu&0\end{pmatrix}$.

  * QNF4

For $n\in{\mathbb{N}}^{*}$,
${\mathcal{R}}(Z)={\mathcal{R}}^{(n)}_{4}(Z)={\mathcal{Q}}_{{\mathcal{B}}^{(n)}_{4}(\gamma)}+{\mathcal{L}}_{l^{(n)}_{4}(\iota_{1})}$,
$\gamma=\pm 1$, $\iota_{1}\in{\mathbb{R}}$, with

${\mathcal{B}}^{(n)}_{4}(\gamma)=\gamma\left(\begin{array}[]{cccc:cccc}0&&&&&&&\\\
-1&0&&&&&&\\\ &\ddots&\ddots&&&&&\\\ &&-1&0&&&&\\\ \hdashline&&&&0&1&&\\\
&&&&&\ddots&\ddots&\\\ &&&&&&0&1\\\
&&&(-1)^{n+1}&&&&0\end{array}\right)\in{\rm sp}(n,{\mathbb{R}}),$ (1.8)

and $l^{(n)}_{4}(\iota_{1})=\iota_{1}{\bf e}_{1}\in{\mathbb{R}}^{2n}$, where
$\\{{\bf e}_{1},\cdots,{\bf e}_{2n}\\}$ is the canonical basis of column
vectors in ${\mathbb{R}}^{2n}$. In particular,
${\mathcal{B}}^{(1)}_{4}(\gamma)=\gamma\begin{pmatrix}0&0\\\
1&0\end{pmatrix}$.

  * QNF5

For $n\in{\mathbb{N}}^{*}$ with $n$ odd,
${\mathcal{R}}(Z)={\mathcal{R}}^{(n)}_{5}(Z)={\mathcal{Q}}_{{\mathcal{B}}^{(n)}_{5}}+{\mathcal{L}}_{l^{(n)}_{5}(\iota_{1},\iota_{2})}$,
$\iota_{1},\iota_{2}\in{\mathbb{R}}$, with

${{\mathcal{B}}}^{(n)}_{5}=\left(\begin{array}[]{cccc:cccc}0&&&&&&&\\\
-1&0&&&&&&\\\ &\ddots&\ddots&&&&&\\\ &&-1&0&&&&\\\ \hdashline&&&&0&1&&\\\
&&&&&\ddots&\ddots&\\\ &&&&&&0&1\\\ &&&&&&&0\end{array}\right)\in{\rm
sp}(n,{\mathbb{R}}),$ (1.9)

and $l^{(n)}_{5}(\iota_{1},\iota_{2})=\iota_{1}{\bf e}_{1}+\iota_{2}{\bf
e}_{2n}\in{\mathbb{R}}^{2n}$. In particular,
${\mathcal{B}}^{(1)}_{5}=\begin{pmatrix}0&0\\\ 0&0\end{pmatrix}$ and
$l^{(1)}_{5}(\iota_{1},\iota_{2})=\iota_{1}{\bf e}_{1}$.

  * QNF6

(Decomposable case) For $n\in{\mathbb{N}}^{*}$ with $n\geq 2$,
${\mathcal{R}}(Z)={\mathcal{R}}_{6}^{(n)}(Z)$ is composed as a direct sum of
operators from the previously mentioned five cases, each operating in a lower-
dimensional space. More precisely, there exists a mutually disjoint partition
$\bigcup_{j\in\Lambda}{{\mathcal{I}}_{j}}=\\{1,\cdots,n\\}$ with
$\\#{\mathcal{I}}_{j}=m_{j}<n$ and $k_{j}\in\\{1,\cdots,5\\}$ such that

${\mathcal{R}}_{6}^{(n)}(Z)=\sum_{j\in\Lambda}{\mathcal{R}}_{k_{j}}^{(m_{j})}(Z_{{\mathcal{I}}_{j}}),\qquad
Z_{{\mathcal{I}}_{j}}:=\begin{pmatrix}(D_{l})_{l\in{\mathcal{I}}_{j}}\\\
(X_{l})_{l\in{\mathcal{I}}_{j}}\end{pmatrix}.$ (1.10)

######  Remark 2.

The categorization of the homogeneous part in the quantum Hamiltonian is based
on the classification of the homogeneous Hamiltonian $h_{0}(t,\xi,x)$ (as in
(1.3), with $\ell$ and $c$ omitted), which effectively corresponds to
classifying the symplectic Lie algebra ${\rm sp}(n,{\mathbb{R}})$. This
classification is detailed by Hörmander (see Cases a) to g) in Theorem 3.1 of
[35]). In this paper, we reclassify the homogeneous part in an indecomposable
manner, since our objective is to analyze the growth of Sobolev norms, and
quantum quadratic Hamiltonians may contain linear terms. The differences
between our classification and the one in Theorem 3.1 of [35] are as follows:

  * •

Cases a) – c) in [35] correspond precisely to QNF1 – QNF3 in Theorem 1.1,
respectively.

  * •

Cases d) and e) are categorized into QNF5 and QNF4 for $n=1$, respectively.

  * •

Cases f) and g) are classified into QNF4 and QNF5 for $n>1$, respectively.

Since the QNF classification depends on the symplectic Lie algebra
classification, and each QNF is characterized by a Hamiltonian matrix in ${\rm
sp}(n,{\mathbb{R}})$, the normal forms in Theorem 1.1 are also referred to as
(quantum) symplectic normal forms, as indicated in the title of this paper.

######  Remark 3.

In this paper, the familiar classical-quantum correspondence, typically
established via Weyl quantization (refer to (1.15)), is slightly modified for
the sake of simplicity in expression. For the symbol $h_{0}(t,\xi,x)$ as
defined in (1.3), the corresponding system of equations of motion is given by
$z^{\prime}={\mathcal{A}}(t)^{*}z-{\mathbb{J}}_{n}\ell(t)$. It is important to
note that this alteration does not impact the validity or the formulation of
the theorems presented.

###  1.2. Classification of growth rates (GRs) of Sobolev norms

Proposed by Bourgain [11] and expanded upon in subsequent pioneering works
(e.g., [13, 24, 30, 32]), the behavior of solutions to Hamiltonian PDEs in
Sobolev space has been a significant topic in Mathematical Physics over the
past decades. Particularly, the existence of unbounded trajectories in Sobolev
space is intrinsically linked to weak turbulent effects and energy cascades.
In recent years, considerable advancements have been made in this area for a
variety of quantum Hamiltonian systems.

As discussed in the previous subsection, examining the growth of Sobolev norms
of solutions is intimately connected to the reducibility of the original
quantum Hamiltonian, especially in the case of linear Hamiltonians. Following
[26], Bambusi-Grébert-Maspero-Robert [5] analyzed a $1-$D QHO with a time-
periodic linear potential, demonstrating $t^{s}$ polynomial growth for
${{\mathcal{H}}}^{s}-$norms through reducibility to a transport equation. For
a $1-$D QHO with time quasi-periodic perturbations, which are quadratic
polynomials of $(D,X)$, various growth rates of ${{\mathcal{H}}}^{s}-$norms
have been observed depending on the normal forms of reducibility [42, 45]:
$t^{s}$ polynomial growth for parabolic normal form, exponential growth for
hyperbolic normal form, and $t^{2s}$ polynomial growth when reducible to the
Stark Hamiltonian.

It is important to note that almost reducibility (i.e., conjugation to a time-
independent system through a sequence of transformations without demonstrating
the convergence of this sequence) is also insightful for understanding the
long-term behaviors of solutions. For a $1-$D QHO with a time quasi-periodic
quadratic perturbation, it was shown in [43] that the growth of
${\mathcal{H}}^{s}$-norms can oscillate with an optimal $o(t^{s})$ upper bound
if the original equation is not reducible. Bambusi and his collaborators
employed a similar concept for time-dependent Schrödinger equations [6, 7, 8],
focusing on increasingly regularized remaining parts with each iteration,
instead of the usual assumption of asymptotic smallness. They achieved a
$t^{\epsilon}$ upper bound for any $\epsilon>0$ on the Sobolev norms of
solutions through this methodology.

Another method often used to achieve unbounded trajectories involves
constructing specific perturbations that induce infinite growth. For a $1-$D
QHO, Delort [15] (also see Maspero [47]) designed a time-periodic order-zero
pseudo-differential operator as the perturbation, leading to $t^{\frac{s}{2}}$
polynomial growth in ${{\mathcal{H}}}^{s}-$norms. In an abstract setting,
Maspero [48, 49] further explored time-periodic perturbations and provided
conditions under which certain solutions exhibit such polynomial growth. For a
$2-$D QHO, Faou-Raphaël [22] embedded Arnold diffusion into an infinite-
dimensional quantum system, constructing a time-decaying potential that
resulted in logarithmic growth of Sobolev norms over time. Similarly, Thomann
[53], based on studies of linear Lowest Landau Level equations with a time-
dependent potential [52], designed a perturbation that projects onto Bargmann-
Fock space, resulting in polynomial growth of Sobolev norms over time for some
traveling wave solutions. Bourgain [12] and Haus-Maspero [33] also
demonstrated logarithmic growth of Sobolev norms in various settings,
including for linear Schrödinger equations with quasi-periodic potentials and
semiclassical anharmonic oscillators with regular time-dependent potentials.

This paper aims to extend beyond the limitations of spatial dimension and
explore the long-term behavior of solutions to Eq. (1.2) in arbitrary
dimensions $n$. In the $n-$D quadratic Hamiltonian context, the Sobolev space
(also referred to as ${\mathcal{H}}^{s}-$space for a given $s\geq 0$) is
defined as

${\mathcal{H}}^{s}({\mathbb{R}}^{n}):=\left\\{u\in
L^{2}({\mathbb{R}}^{n}):{\mathcal{T}}^{\frac{s}{2}}u\in
L^{2}({\mathbb{R}}^{n})\right\\},$

with ${\mathcal{T}}$ being the $n-$D QHO defined as in (1.4), and equipped
with the Sobolev norm (also known as ${\mathcal{H}}^{s}-$norm)
$\|u\|_{{\mathcal{H}}^{s}({\mathbb{R}}^{n})}:=\left(\|{\mathcal{T}}^{\frac{s}{2}}u\|^{2}_{L^{2}}+\|u\|^{2}_{L^{2}}\right)^{\frac{1}{2}}$.
Several equivalent forms of the ${\mathcal{H}}^{s}-$norm will be presented in
Section 2.1.

Inspired by previous studies, the primary goal of this paper is to provide a
comprehensive classification of the growth rates of Sobolev norms for
solutions to Eq. (1.2) under the assumption of reducibility. All potential
growth rates of Sobolev norms for solutions to Eq. (1.2) will be precisely
delineated based on the classification of QNFs obtained in Theorem 1.1.

######  Theorem 1.2.

(Classification of GRs of Sobolev norms) Consider Eq. (1.2) with non-vanishing
initial condition $\psi(0)\in\mathcal{H}^{s}({\mathbb{R}}^{n})$, $s>0$. If the
system $z^{\prime}={\mathcal{A}}(t)z+\ell(t)$ is reducible through the
conjugation $z=U(t)y+v(t)$, $U\in C_{b}^{1}({\mathbb{R}},{\rm
Sp}(n,{\mathbb{R}}))$, $v\in C_{b}^{1}({\mathbb{R}},{\mathbb{R}}^{2n})$, then
there exists a constant $C>1$, depending on $s$, $\psi(0)$,
$\|U\|_{C_{b}^{0}}$ and $\|v\|_{C_{b}^{0}}$, and positive functions
$g^{(n)}_{k}(s,t)$, $1\leq k\leq 6$, such that, corresponding to QNF1 — QNF6
in Theorem 1.1,
$C^{-1}g^{(n)}_{k}(s,t)\leq\|\psi(t)\|_{{\mathcal{H}}^{s}({\mathbb{R}}^{n})}\leq
Cg^{(n)}_{k}(s,t)$ as $|t|\rightarrow\infty$. More precisely,

  * GR1

For ${\mathcal{R}}(Z)={\mathcal{R}}^{(n)}_{1}(Z)$,
$g^{(n)}_{1}(s,t)=|t|^{(n-1)s}e^{\lambda s|t|}$.

  * GR2

For $n$ even, ${\mathcal{R}}(Z)={\mathcal{R}}^{(n)}_{2}(Z)$,
$g^{(n)}_{2}(s,t)=|t|^{(\frac{n}{2}-1)s}e^{\lambda_{1}s|t|}$.

  * GR3

For ${\mathcal{R}}(Z)={\mathcal{R}}^{(n)}_{3}(Z)$,
$g^{(n)}_{3}(s,t)=|t|^{(n-1)s}$.

  * GR4

For ${\mathcal{R}}(Z)={\mathcal{R}}^{(n)}_{4}(Z)$,
$g^{(n)}_{4}(s,t)=|t|^{(2n-1)s}+|\iota_{1}|^{s}|t|^{2ns}$.

  * GR5

For $n$ odd, ${\mathcal{R}}(Z)={\mathcal{R}}^{(n)}_{5}(Z)$,
$g^{(n)}_{5}(s,t)=|t|^{(n-1)s}+(|\iota_{1}|+|\iota_{2}|)^{s}|t|^{ns}$.

  * GR6

(Decomposable case) For $n\geq 2$,
${\mathcal{R}}(Z)={\mathcal{R}}^{(n)}_{6}(Z)$,
$g^{(n)}_{6}(s,t)=\sum_{j\in\Lambda}g^{(m_{j})}_{k_{j}}(s,t)$.

######  Remark 4.

Among the growth rates mentioned, GR1 – GR5 are uniquely characteristic of the
$n-$D setting, in that they are not observed in any Hamiltonian PDE with
spatial dimension lower than $n$. To the best of our knowledge, this theorem
is the first to describe and comprehensively classify the growth of
${\mathcal{H}}^{s}({\mathbb{R}}^{n})-$norms across arbitrary spatial dimension
$n$. For greater clarity, and particularly to illustrate the decomposable case
more concretely, we provide detailed explanations of the quantum normal forms
and the corresponding growths of Sobolev norms for the $1-$D and $2-$D
scenarios in Appendix A.

Based on the classifications presented in Theorem 1.2, stability of solutions
to Eq. (1.2) within the Sobolev space, which is characterized by the
boundedness of the solution’s Sobolev norm, can be observed exclusively under
specific conditions. These conditions are either for QNF3 with $n=1$, or for
QNF5 with $n=1$ and both $\iota_{1}$ and $\iota_{2}$ equal to zero. This
corresponds to:

${\mathcal{R}}(Z)=\pm\frac{\mu}{2}\,(D_{1}^{2}+X_{1}^{2}),\quad\mu>0,\qquad{\rm
or}\qquad{\mathcal{R}}(Z)=0,$

or in the case of QNF6, which represents a direct sum of the aforementioned
$1-$D QNFs. Consequently, it appears that stability in the Sobolev space
manifests primarily as a $1-$D phenomenon.

######  Theorem 1.3.

Assume that the affine system $z^{\prime}={\mathcal{A}}(t)z+\ell(t)$ is
reducible with ${\mathcal{A}}(\cdot)\in C_{b}^{0}({\mathbb{R}},{\rm
sp}(n,{\mathbb{R}}))$ and $\ell(\cdot)\in
C_{b}^{0}({\mathbb{R}},{\mathbb{R}}^{2n})$ as in Eq. (1.2). Given $s>0$, if
the solution to Eq. (1.2) satisfies
$\sup_{t\in{\mathbb{R}}}\|\psi(t)\|_{{\mathcal{H}}^{s}({\mathbb{R}}^{n})}<\infty$,
then Eq. (1.2) is reduced to the constant Hamiltonian

${\mathcal{R}}(Z)=\sum_{j=1}^{n}c_{j}(D_{j}^{2}+X_{j}^{2}),\qquad
c_{j}\in{\mathbb{R}}.$

Additionally, there exists a notable correlation between classical and quantum
Hamiltonians regarding the long-term behaviors of their solutions. This
connection facilitates the direct observation of Sobolev norm growth through
computations performed on classical systems, thereby eliminating the need to
specify which symplectic normal form is responsible for this growth.

######  Theorem 1.4.

Assume that the affine system

$z^{\prime}={\mathcal{A}}(t)z+\ell(t),\qquad z\in{\mathbb{R}}^{2n},$ (1.11)

is reducible. Let $z^{\rm f}(t)$ be a solution to (1.11) that exhibits the
fastest growth as $|t|\to\infty$ in the sense that, for any solution
$z_{0}(t)$ to (1.11) we have

$\sup_{t\in{\mathbb{R}}}\frac{\|z_{0}(t)\|}{1+\|z^{\rm f}(t)\|}<\infty.$
(1.12)

Consequently, for the solution $\psi(t)$ to Eq. (1.2) with
$\psi(0)\in{\mathcal{H}}^{s}({\mathbb{R}}^{n})\setminus\\{0\\}$, $s>0$, there
exists a constant $C>1$, depending on $\psi(0)$ and $z^{\rm f}(0)$, such that

$C^{-1}\|z^{\rm f}(t)\|^{s}\leq\|\psi(t)\|_{s}\leq C\|z^{\rm f}(t)\|^{s}.$
(1.13)

######  Remark 5.

The asymptotic estimate presented in (1.13) is anticipated to be valid in more
expansive contexts, extending beyond the limits of the reducibility assumption
and the quadratic polynomial form of the Hamiltonian. Let us consider two
illustrative examples in the $1-$D framework.

In the case of a $1-$D QHO perturbed by a time quasi-periodic homogeneous
quadratic form of $(X,D)$, it has been demonstrated in [43] that the
${\mathcal{H}}^{s}-$norm of the solution exhibits oscillatory growth with an
optimal $o(t^{s})-$upper bound under the condition of a non-reducible quantum
Hamiltonian. Correspondingly, for the analogous classical system, a non-
reducible quasi-periodic ${\rm sl}(2,{\mathbb{R}})-$linear system, Eliasson
has established the existence of oscillatory unbounded solutions with an
$o(t)-$upper bound (refer to Theorem A2 and B2 in [18]).

In the case of a $1-$D QHO perturbed by a specific time-periodic order$-0$
pseudo-differential operator:

$\frac{1}{2\pi{\rm
i}}\partial_{t}\psi=\left(\frac{1}{4\pi}{\mathcal{T}}+h^{W}(t,D,X)\right)\psi,\qquad
h(t,\xi,x)=\epsilon\cos(2t)\eta(\xi,x)\frac{x\xi}{x^{2}+\xi^{2}},$

where $\eta\in C^{\infty}({\mathbb{R}}^{2},{\mathbb{R}}_{+})$ is defined such
that $\eta(\xi,x)$ equals $0$ when $x^{2}+\xi^{2}\leq\frac{1}{4}$ and $1$ when
$x^{2}+\xi^{2}\geq\frac{1}{2}$, and $h^{W}$ is the Weyl quantization of the
classical Hamiltonian $h$ on ${\mathbb{R}}^{2}$ (refer to (1.15)). Maspero
[49] has demonstrated that if $\epsilon>0$ is sufficiently small, then $h^{W}$
acts as a transporter, implying the existence of a solution
$\psi(t)\in{\mathcal{H}}^{s}$ for $s>0$, which satisfies $\|\psi(t)\|_{s}\sim
t^{\frac{s}{2}}$ as $t\to\infty$, within the context of this paper.
Conversely, the corresponding classical system

$x^{\prime}=\xi+\partial_{\xi}h(t,\xi,x),\qquad\xi^{\prime}=-x-\partial_{x}h(t,\xi,x),\qquad
x,\xi\in{\mathbb{R}},$ (1.14)

permits a solution for $t\geq\epsilon^{-1}$ of the form

$x(t)=\sqrt{\epsilon}\cos(t)\sqrt{t+\frac{\sin(4t)}{4}},\quad\xi(t)=-\sqrt{\epsilon}\sin(t)\sqrt{t+\frac{\sin(4t)}{4}}.$

Therefore, the quantity $\sqrt{x^{2}(t)+\xi^{2}(t)}$ scales approximately as
$(\epsilon t)^{\frac{1}{2}}$ when $t\to\infty$, representing the optimal
growth rate for solutions to the system (1.14).

###  1.3. Strategy of proof

The classical-quantum correspondence facilitated by Weyl quantization is well-
established. More specifically, for any given symbol $h=h(\xi,x)$, where
$\xi,x\in{\mathbb{R}}^{n}$ and $n\geq 1$, the Weyl operator $h^{W}$ associated
with $h$ is defined as follows:

$\left(h^{W}u\right)(x)=\int_{\xi,y\in{\mathbb{R}}^{n}}e^{2\pi{\rm
i}\langle\xi,x-y\rangle}\,h\left(\xi,\frac{x+y}{2}\right)u(y)\,dy\,d\xi,\qquad\forall\
u\in L^{2}({\mathbb{R}}^{n}).$ (1.15)

Notably, if $h$ is a quadratic polynomial in $(\xi,x)$, then the corresponding
Weyl operator $h^{W}$ becomes a quadratic polynomial in $(D,X)$ after
symmetrization. Therefore, to comprehensively understand Eq. (1.2), it is
essential to examine the associated finite-dimensional Hamiltonian system.
This approach was utilized in [5] to achieve reducibility and the almost
preservation of Sobolev norms for the time quasi-periodic perturbed QHO, by
implementing a classical KAM scheme for “most” frequency vectors. Similarly,
in the $1-$D context, the (almost) reducibility and diverse behaviors of
Sobolev norms have been explored in [42, 43, 45]. This was achieved by
adapting Eliasson’s KAM scheme for time quasi-periodic ${\rm
sl}(2,{\mathbb{R}})-$linear systems [18], where the reducibility
transformation is not necessarily close to the identity, to the quantum
Hamiltonian.

The reducibility assumption of the affine system

$z^{\prime}={\mathcal{A}}(t)z+\ell(t),\qquad{\mathcal{A}}(\cdot)\in
C_{b}^{0}({\mathbb{R}},{\rm sp}(n,{\mathbb{R}})),\qquad\ell(\cdot)\in
C_{b}^{0}({\mathbb{R}},{\mathbb{R}}^{2n}),$ (1.16)

means that there exist $U(\cdot)\in C_{b}^{1}({\mathbb{R}},{\rm
Sp}(n,{\mathbb{R}}))$ and $v(\cdot)\in
C_{b}^{1}({\mathbb{R}},{\mathbb{R}}^{2n})$ such that, via the transformation
$z=U(t)w+v(t)$, the system (1.16) is conjugated to the affine system with
constant coefficients $w^{\prime}={\mathcal{B}}w+l$. Similar to the strategy
of reducibility in [45], the reducibility of the quantum Hamiltonian (1.2) is
achieved in two steps.

  1. 1)

Through the transformation $z=U(t)y$, the system (1.16) is conjugated to the
affine system

$y^{\prime}={\mathcal{B}}y+\tilde{\ell}(t),\qquad{\mathcal{B}}\in{\rm
sp}(n,{\mathbb{R}}),\qquad\tilde{\ell}(\cdot)\in
C_{b}^{0}({\mathbb{R}},{\mathbb{R}}^{2n}),\ \ {\rm with}\ \
\tilde{\ell}(t)=U(t)^{-1}\ell(t).$ (1.17)

Simultaneously, Eq. (1.2) is conjugated to the quantum Hamiltonian

$\frac{1}{2\pi{\rm
i}}\partial_{t}\phi=({\mathcal{Q}}_{{\mathcal{B}}}+{\mathcal{L}}_{\tilde{\ell}(t)})\phi,$
(1.18)

via the $L^{2}-$unitary transformation $\psi={\mathcal{M}}(U(t))\phi$, where
${\mathcal{M}}(U(\cdot))$ denotes the Metaplectic representation of $U(\cdot)$
(refer to Section 2.3 for details).

  2. 2)

By applying the translation $y=w+U(t)^{-1}v(t)$, the system (1.17) is
transformed into

$w^{\prime}={\mathcal{B}}w+l,\qquad{\mathcal{B}}\in{\rm
sp}(n,{\mathbb{R}}),\qquad l\in{\mathbb{R}}^{2n},$

where all coefficients are time-independent. In parallel, the Hamiltonian PDE
(1.18) is conjugated to $\frac{1}{2\pi{\rm
i}}\partial_{t}\varphi=({\mathcal{Q}}_{{\mathcal{B}}}+{\mathcal{L}}_{l})\varphi$
(up to a scalar term, typically omitted, see Remark 11), through the
$L^{2}-$unitary transformation $\phi=\rho(U(t)^{-1}v(t))\varphi$. Here,
$\rho(U(\cdot)^{-1}v(\cdot))$ represents the Schrödinger representation of
$U(\cdot)^{-1}v(\cdot)$ (for details, see Section 2.2).

The strategy for proving reducibility is summarized in the following diagram.

$\begin{array}[]{rcccl}&{\rm Affine\ system}&&{\rm Hamiltonian\ PDE}&\\\
&&&&\\\ &z^{\prime}={\mathcal{A}}(t)z+\ell(t)&\stackrel{{\scriptstyle\rm
Quantized}}{{\longrightarrow}}&\frac{1}{2\pi{\rm
i}}\partial_{t}\psi=({\mathcal{Q}}_{{\mathcal{A}}(t)}+{\mathcal{L}}_{\ell(t)})\psi&\\\
&&&&\\\
z=U(t)y&\big{\downarrow}&&\big{\downarrow}&\psi={\mathcal{M}}(U(t))\phi\\\
&&&&\\\ &y^{\prime}={\mathcal{B}}y+\tilde{\ell}(t)&\stackrel{{\scriptstyle\rm
Quantized}}{{\longrightarrow}}&\frac{1}{2\pi{\rm
i}}\partial_{t}\phi=({\mathcal{Q}}_{{\mathcal{B}}}+{\mathcal{L}}_{\tilde{\ell}(t)})\phi&\\\
&&&&\\\
y=w+U(t)^{-1}v(t)&\big{\downarrow}&&\big{\downarrow}&\phi=\rho(U(t)^{-1}v(t))\varphi\\\
&&&&\\\ &w^{\prime}={\mathcal{B}}w+l&\stackrel{{\scriptstyle\rm
Quantized}}{{\longrightarrow}}&\frac{1}{2\pi{\rm
i}}\partial_{t}\varphi=({\mathcal{Q}}_{{\mathcal{B}}}+{\mathcal{L}}_{l})\varphi&\end{array}$

The classical-quantum correspondence in this context is achieved through the
Metaplectic and Schrödinger representations. Section 3, the cornerstone of
this article, presents detailed estimates of the Sobolev norms for both
representations (see Proposition 3.2 and 3.7). The provided upper and lower
bounds demonstrate the equivalence of the Sobolev norm under these
representations, which allows us to concentrate on the constant QNFs
established in Theorem 1.1 and to undertake concrete computations once the
quantum Hamiltonian flow is explicitly defined. Moreover, we establish a link
between the growth patterns of homogeneous classical and quantum Hamiltonian
flows (refer to Corollary 3.3 \- (ii)). Consequently, for QNF1 to QNF3 in
Theorem 1.1, which are all homogeneous, only the Metaplectic representation is
required and the behavior of solutions in Sobolev space can be directly
inferred from the corresponding classical linear system. For the remaining
cases, by employing an inductive approach based on the Baker-Campbell-
Hausdorff formula, the quantum Hamiltonian flow is still explicitly derivable
(as discussed in Section 6).

###  1.4. Index of notations

  1. (1)

With the dimension $n$ fixed, the notations “$\lesssim$” and “$\gtrsim$” are
used to denote upper and lower bounds, respectively, by a positive constant
that depends solely on $s\in{\mathbb{R}}$ and is independent of any other
variables. Specifically, for non-negative values $a$ and $b$, the inequality
$a\lesssim b$ (or $a\gtrsim b$) indicates the existence of a constant $c_{s}$,
which is only a function of $s$, such that $a\leq c_{s}b$ (or $a\geq c_{s}b$,
respectively). The expression $a\simeq b$ signifies that both $a\lesssim b$
and $a\gtrsim b$ hold true. Furthermore, when considering the dependence on
other factors, for instance,
$u\in{\mathcal{H}}^{s}({\mathbb{R}}^{n})\setminus\\{0\\}$, the notation
$a\gtrsim_{u}b$ implies that there is a specific constant $c_{s,u}$, dependent
only on $s$ and $u$, ensuring that $a\geq c_{s,u}b$ (refer to (3.3) in
Proposition 3.2 for an example).

  2. (2)

For $s,s^{\prime}\in{\mathbb{R}}$ with $s\geq s^{\prime}$, let
$B({{\mathcal{H}}}^{s},{{\mathcal{H}}}^{s^{\prime}})$ be the space of bounded
operators from ${{\mathcal{H}}}^{s}$ to ${{\mathcal{H}}}^{s^{\prime}}$ with
$\|\cdot\|_{B({{\mathcal{H}}}^{s},{{\mathcal{H}}}^{s^{\prime}})}$ the operator
norm, simplified to $B({{\mathcal{H}}}^{s})$ and
$\|\cdot\|_{B({{\mathcal{H}}}^{s})}$ if $s^{\prime}=s$.

  3. (3)

Let ${\rm Sp}(n,{\mathbb{R}})$ (resp. ${\rm sp}(n,{\mathbb{R}})$) be the
symplectic group (resp. symplectic Lie algebra) 444See Section 2.3 for the
precise definitions. of $2n\times 2n$ real matrices with the matrix norm
defined by

$\|{\mathbb{A}}\|=\sum_{i,j=1}^{2n}|a_{ij}|,\quad{\mathbb{A}}=(a_{ij})_{i,j=1}^{2n}\in{\rm
Sp}(n,{\mathbb{R}})\ \ ({\rm or}\ \ {\rm sp}(n,{\mathbb{R}})).$ (1.19)

Moreover, we define $\\{{\bf e}_{1},\cdots,{\bf e}_{2n}\\}$ the canonical
basis of column vectors in ${\mathbb{R}}^{2n}$, the $n-$D identity matrix
$I_{n}$ and the $2n-$D standard symplectic matrix

${\mathbb{J}}_{n}:=\left(\begin{array}[]{cc}0&I_{n}\\\
-I_{n}&0\end{array}\right)=(-{\bf e}_{n+1},\cdots,\underset{\underset{n^{\rm
th}}{\uparrow}}{-{\bf e}_{2n}},{\bf e}_{1},\cdots,{\bf e}_{n})\in{\rm
Sp}(n,{\mathbb{R}}),$

as well as the $2n-$D anti-diagonal symplectic matrix

${\mathbb{J}}^{\rm a}_{n}:=(-{\bf e}_{2n},\cdots,-\underset{\underset{n^{\rm
th}}{\uparrow}}{{\bf e}_{n+1}},{\bf e}_{n},\cdots,{\bf e}_{1})\in{\rm
Sp}(n,{\mathbb{R}}).$ (1.20)

  4. (4)

Given ${\mathbb{A}}_{j}=\left(\begin{array}[]{cc}A_{j}&B_{j}\\\
C_{j}&F_{j}\end{array}\right)\in{\rm Sp}(n_{j},{\mathbb{R}})$(resp. ${\rm
sp}(n_{j},{\mathbb{R}})$), $j=1,2$, we define the “direct sum” of the two
matrices by

${\mathbb{A}}={\mathbb{A}}_{1}\bigoplus{\mathbb{A}}_{2}:=\left(\begin{array}[]{cccc}A_{1}&0&B_{1}&0\\\
0&A_{2}&0&B_{2}\\\ C_{1}&0&F_{1}&0\\\ 0&C_{2}&0&F_{2}\end{array}\right)\in{\rm
Sp}(n_{1}+n_{2},{\mathbb{R}})\ \ ({\rm or}\ \ {\rm
sp}(n_{1}+n_{2},{\mathbb{R}})).$

  5. (5)

Let $C^{0}_{b}({\mathbb{R}},\bullet)$, $\bullet={\rm Sp}(n,{\mathbb{R}})$,
${\rm sp}(n,{\mathbb{R}})$ or ${\mathbb{R}}^{d}$, be the space of
$\bullet-$valued functions whose entries are all $C^{0}$ and uniformly bounded
functions on ${\mathbb{R}}$, equipped with the norm

$\|U\|_{C^{0}_{b}}:=\sup_{t\in{\mathbb{R}}}\|U(t)\|,\qquad U\in
C^{0}_{b}({\mathbb{R}},\bullet).$

Let $C^{1}_{b}({\mathbb{R}},\bullet)$ be the subspace of
$C^{0}_{b}({\mathbb{R}},\bullet)$ whose entries are all $C^{1}$ with uniformly
bounded derivatives on ${\mathbb{R}}$, equipped with the norm

$\|U\|_{C^{1}_{b}}:=\sup_{t\in{\mathbb{R}}}(\|U(t)\|+\|U^{\prime}(t)\|),\qquad
U\in C^{1}_{b}({\mathbb{R}},\bullet).$

  6. (6)

As in Eq. (1.2), we define the columns of operators

$D=\begin{pmatrix}D_{1}\\\ \vdots\\\ D_{n}\end{pmatrix}:=\frac{1}{2\pi{\rm
i}}\nabla_{x},\quad X:=\begin{pmatrix}X_{1}\\\ \vdots\\\
X_{n}\end{pmatrix},\quad Z_{j}:=\begin{pmatrix}D_{j}\\\
X_{j}\end{pmatrix},\quad Z:=\begin{pmatrix}D\\\ X\end{pmatrix},$ (1.21)

where $X_{j}$ represents the multiplication by the $j-$th coordinate function,
i.e., $(X_{j}f)(x)=x_{j}f(x)$. It is important to note that the derivation in
$D_{j}$ always refers to the $j-$th spatial coordinate if other temporal
variable is involved. Naturally, for $p\in{\mathbb{R}}^{n}$ and
$v\in{\mathbb{R}}^{2n}$, we define

$\langle p,D\rangle:=\sum_{j=1}^{n}p_{j}D_{j},\quad\langle
p,X\rangle:=\sum_{j=1}^{n}p_{j}X_{j},\quad{\mathcal{L}}_{v}(Z)=\left\langle
v,Z\right\rangle:=\sum_{j=1}^{n}(v_{j}D_{j}+v_{n+j}X_{j}),$

and for ${\mathcal{A}}=\left(\begin{array}[]{cc}A&B\\\
C&-A^{*}\end{array}\right)\in{\rm sp}(n,{\mathbb{R}})$,

$\langle Z,{\mathcal{A}}{\mathbb{J}}_{n}Z\rangle:=-\left\langle
D,BD\right\rangle+\left\langle D,AX\right\rangle+\left\langle
X,A^{*}D\right\rangle+\left\langle X,CX\right\rangle,$

where the term $\left\langle
D,AX\right\rangle:=\sum_{i,j=1}^{n}A_{ij}D_{i}X_{j}$ and similar for the other
terms. Let ${\mathcal{Q}}_{{\mathcal{A}}}(Z):=-\frac{1}{2}\langle
Z,{\mathcal{A}}{\mathbb{J}}_{n}Z\rangle$, whose symbol is the homogeneous
quadratic polynomial

${\mathcal{Q}}_{{\mathcal{A}}}(\xi,x)=\frac{1}{2}\left(\langle\xi,B\xi\rangle-2\langle\xi,Ax\rangle-\langle
x,Cx\rangle\right),\qquad\xi,x\in{\mathbb{R}}^{n}.$ (1.22)

  7. (7)

For $x\in{\mathbb{R}}^{n}$, let $\langle x\rangle:=\sqrt{1+\langle
x,x\rangle}$. Define the operators $\langle X\rangle$ and $|X|$ by

$\left(\langle X\rangle u\right)(x):=\langle x\rangle
u(x),\qquad\left(|X|u\right)(x):=\sqrt{\langle x,x\rangle}u(x),$

and, with ${{\mathcal{F}}}^{-1}$ the inverse of Fourier transform, define
$\langle D\rangle$ and $|D|$ by

$\langle D\rangle u:={{\mathcal{F}}}^{-1}\left(\langle
X\rangle\hat{u}\right),\quad|D|u:={{\mathcal{F}}}^{-1}\left(|X|\hat{u}\right).$

Moreover, given $\alpha\in{\mathbb{N}}^{n}$, we define
$D^{\alpha}:=D_{1}^{\alpha_{1}}\cdots D_{n}^{\alpha_{n}}$ and
$X^{\alpha}:=X_{1}^{\alpha_{1}}\cdots X_{n}^{\alpha_{n}}$.

###  1.5. Description of the remaining of paper

In the preliminary Section 2, we explore the equivalent forms of Sobolev
norms, along with an introduction to the Metaplectic and Schrödinger
representations, highlighting their fundamental properties.

Section 3 presents the upper and lower bounds of Sobolev norms within these
two representations. This leads to a pivotal conclusion of the paper: all
$L^{2}-$unitary transformations, derived from these representations and
conjugating time-dependent equations, maintain the growth rate of the Sobolev
norms of solutions. This is articulated in Propositions 3.2 and 3.7. A
comprehensive proof of Proposition 3.2 is included in this section, while the
proof for Proposition 3.7 is deferred to Appendix B.

Section 4 delves into a classical-quantum correspondence for the quadratic
Hamiltonians, which facilitates the translation of conjugation in classical
Hamiltonian systems to quantum Hamiltonians.

In Section 5, we discuss the reducibility of the affine system (1.16) and
provide a complete classification of symplectic normal forms (see Proposition
5.2). From these findings, Theorem 1.1 is derived using the classical-quantum
correspondence.

The examination of Sobolev norm growth for each QNF, or each variant of
reduced constant quantum Hamiltonian, is conducted in Section 6. This analysis
culminates in the proof of Theorems 1.2 and 1.4.

Finally, the application of these theorems in time-periodic and time quasi-
periodic quadratic quantum Hamiltonians is explored in Sections 7 and 8.

##  2\. Preliminaries

Adhering to the conventions established by Folland [23], this section revisits
the definition and characteristics of Sobolev norms. Additionally, it offers a
review of the Metaplectic and Schrödinger representations. Key formulas,
essential for subsequent discussions, are introduced here without their
proofs.

###  2.1. Sobolev space and Sobolev norm

In the $n-$D setting, $n\geq 2$, with the columns of operators $D$ and $X$
defined as in (1.21), the $n-$D QHO in this paper,

${\mathcal{T}}=2\pi\left(\langle D,D\rangle+\langle
X,X\rangle\right)=-\frac{1}{2\pi}\Delta+2\pi|X|^{2},$

differs from the conventional harmonic operator “$-\Delta+|X|^{2}$.” This
distinction leads to variations in the eigenfunctions of the QHO, specifically
when normalized. Precisely, the $1-$D Hermite functions

$h_{j}(x)=\frac{2^{\frac{1}{4}}}{\sqrt{j!}}\left(\frac{-1}{2\sqrt{\pi}}\right)^{j}e^{\pi
x^{2}}\frac{d^{j}}{dx^{j}}\left(e^{-2\pi x^{2}}\right),\quad
x\in{\mathbb{R}},\quad j\in{\mathbb{N}},$

which adhere to the recurrence formula

$h_{j}(x)=\sqrt{\frac{\pi}{j}}(X-{\rm
i}D)h_{j-1}=\sqrt{\frac{\pi^{j}}{j!}}(X-{\rm i}D)^{j}h_{0},$ (2.1)

yield the eigenfunctions of ${\mathcal{T}}$ corresponding to the eigenvalue
$2|\alpha|+n$ for $\alpha\in{\mathbb{N}}^{n}$, given by

$h_{\alpha}(x)=h_{\alpha_{1}}(x_{1})\cdots h_{\alpha_{n}}(x_{n}).$

The Sobolev space (also referred to as the ${\mathcal{H}}^{s}$-space for a
given $s\geq 0$) is defined as

${\mathcal{H}}^{s}({\mathbb{R}}^{n}):=\left\\{f\in
L^{2}({\mathbb{R}}^{n}):{\mathcal{T}}^{\frac{s}{2}}f\in
L^{2}({\mathbb{R}}^{n})\right\\},\quad s\geq 0,$

equipped with the graph norm, known as the Sobolev norm or
${\mathcal{H}}^{s}$-norm:

$\|f\|_{{\mathcal{H}}^{s}({\mathbb{R}}^{n})}:=\left(\|{\mathcal{T}}^{\frac{s}{2}}f\|^{2}_{L^{2}({\mathbb{R}}^{n})}+\|f\|^{2}_{L^{2}({\mathbb{R}}^{n})}\right)^{\frac{1}{2}},$

and for $s<0$,
${\mathcal{H}}^{s}({\mathbb{R}}^{n})=({\mathcal{H}}^{-s}({\mathbb{R}}^{n}))^{\prime}$
by duality. For convenience, we sometimes denote $L^{2}$ and
${\mathcal{H}}^{s}$ instead of $L^{2}({\mathbb{R}}^{n})$ and
${\mathcal{H}}^{s}({\mathbb{R}}^{n})$ respectively, when the dimension of the
base space is explicitly stated, and we abbreviate
$\|\cdot\|_{{\mathcal{H}}^{s}}$ as $\|\cdot\|_{s}$.

Define the Fourier transform ${{\mathcal{F}}}$ on $L^{2}({\mathbb{R}}^{n})$
and its inverse ${{\mathcal{F}}}^{-1}$ as follows

$\displaystyle({{\mathcal{F}}}f)(\xi)\,=\,\hat{f}(\xi)$ $\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{n}}e^{-2\pi{\rm i}\langle
x,\xi\rangle}f(x)\,dx,\qquad f\in L^{2}({\mathbb{R}}^{n}),$
$\displaystyle({{\mathcal{F}}}^{-1}\hat{f})(x)\,=\,f(x)$ $\displaystyle=$
$\displaystyle\int_{{\mathbb{R}}^{n}}e^{2\pi{\rm i}\langle
x,\xi\rangle}\hat{f}(\xi)\,d\xi,\qquad\hat{f}\in L^{2}({\mathbb{R}}^{n}).$

Moreover, the partial Fourier transform ${{\mathcal{F}}}_{k}$, $k=1,\cdots,n$,
on $L^{2}({\mathbb{R}}^{n})$, and its inverse are defined as

$\displaystyle({{\mathcal{F}}}_{k}f)(x_{1},\cdots,x_{k-1},\xi_{k},x_{k+1},\cdots,x_{n})$
$\displaystyle=$ $\displaystyle\int_{\mathbb{R}}e^{-2\pi{\rm
i}x_{k}\xi_{k}}f(x)\,dx_{k},\qquad f\in L^{2}({\mathbb{R}}^{n}),$
$\displaystyle({{\mathcal{F}}}_{k}^{-1}\hat{f})(\xi_{1},\cdots,\xi_{k-1},x_{k},\xi_{k+1},\cdots,\xi_{n})$
$\displaystyle=$ $\displaystyle\int_{{\mathbb{R}}}e^{2\pi{\rm
i}x_{k}\xi_{k}}\hat{f}(\xi)\,d\xi_{k},\qquad\hat{f}\in
L^{2}({\mathbb{R}}^{n}).$

For given $s\geq 0$, it is well known that
${\mathcal{F}},{\mathcal{F}}_{k}:{\mathcal{H}}^{s}\to{\mathcal{H}}^{s}$, and
for $u\in{\mathcal{H}}^{s}$,

$\|u\|_{s}=\|{\mathcal{F}}u\|_{s}=\|{\mathcal{F}}_{k}u\|_{s},\quad\|X_{k}^{s}u\|_{L^{2}}=\|D_{k}^{s}{\mathcal{F}}_{k}u\|_{L^{2}},\quad\|X_{j}^{s}u\|_{L^{2}}=\|X_{j}^{s}{\mathcal{F}}_{k}u\|_{L^{2}},\quad
j\neq k.$ (2.2)

According to [22], for $u\in{\mathcal{H}}^{s}$, $s\geq 0$, the equivalent
forms of the ${\mathcal{H}}^{s}-$norm can be expressed as follows:

$\|u\|_{s}\simeq\|\langle D\rangle^{s}u\|_{L^{2}}+\|\langle
X\rangle^{s}u\|_{L^{2}}=\|\langle X\rangle^{s}\hat{u}\|_{L^{2}}+\|\langle
X\rangle^{s}u\|_{L^{2}}\simeq\|u\|_{L^{2}}+\||X|^{s}\hat{u}\|_{L^{2}}+\||X|^{s}u\|_{L^{2}},$

and, in accordance with [29], if $s\in{\mathbb{N}}^{*}$, then

$\displaystyle\|u\|_{s}\simeq\sup_{\alpha,\beta\in{\mathbb{N}}^{n}\atop{|\alpha|+|\beta|\leq
s}}\|X^{\alpha}D^{\beta}u\|_{L^{2}}.$

Furthermore, for $s>0$, $u\in{\mathcal{H}}^{s}\setminus\\{0\\}$ is equivalent
to one of the following arguments:

  * (i)

$\|u\|_{s}>0$,

  * (ii)

$\|\langle X\rangle^{s}u\|_{L^{2}}>0$ and $\|\langle
X\rangle^{s}{\mathcal{F}}u\|_{L^{2}}>0$,

  * (iii)

$\|X_{j}^{s}u\|_{L^{2}}>0$ and $\|X_{j}^{s}{\mathcal{F}}u\|_{L^{2}}>0$ for
$j=1,\cdots,n$.

###  2.2. Schrödinger representation

Let ${\bf H_{n}}$ represent the real $(2n+1)-$dimensional Heisenberg group,
which is ${\mathbb{R}}^{2n+1}$ equipped with the group law

$(p,q,t)(p^{\prime},q^{\prime},t^{\prime})=\left(p+p^{\prime},q+q^{\prime},t+t^{\prime}+\frac{1}{2}(\langle
p,q^{\prime}\rangle-\langle q,p^{\prime}\rangle)\right),\quad
p,q,p^{\prime},q^{\prime}\in{\mathbb{R}}^{n},\quad
t,t^{\prime}\in{\mathbb{R}}.$

The map $\rho$, from ${\bf H_{n}}$ to the group of unitary operators on
$L^{2}({\mathbb{R}}^{n})$, is defined as

$\rho(p,q,t)=e^{2\pi{\rm i}(\langle p,D\rangle+\langle q,X\rangle+t{\rm
Id})}=e^{2\pi{\rm i}t}e^{2\pi{\rm i}(\langle p,D\rangle+\langle
q,X\rangle)},\quad(p,q,t)\in{\bf H_{n}},$

meaning, for $f\in L^{2}({\mathbb{R}}^{n})$,

$\displaystyle\left(\rho(p,q,t)f\right)(x)=e^{2\pi{\rm i}t+2\pi{\rm i}\langle
q,x\rangle+\pi{\rm i}\langle p,q\rangle}f(x+p),$ (2.3)

$\rho$ then becomes an irreducible unitary representation of ${\bf H_{n}}$ on
the Hilbert space $L^{2}({\mathbb{R}}^{n})$, known as the Schrödinger
representation. Since the variable $t$ always acts in a straightforward
manner, it is often convenient to omit it entirely. Consequently, we redefine
$\rho$ from ${\mathbb{R}}^{2n}$ to $B({\mathcal{H}}^{s})$, $s\geq 0$, as:

$\rho\begin{pmatrix}p\\\ q\end{pmatrix}=\rho(p,q,0)=e^{2\pi{\rm i}(\langle
p,D\rangle+\langle q,X\rangle)},\qquad\forall\ \begin{pmatrix}p\\\
q\end{pmatrix}\in{\mathbb{R}}^{2n},$

and (2.3) is reformulated as

$\left(\rho\begin{pmatrix}p\\\ q\end{pmatrix}f\right)(x)=e^{2\pi{\rm i}\langle
q,x\rangle+\pi{\rm i}\langle p,q\rangle}f(x+p),\qquad f\in
L^{2}({\mathbb{R}}^{n}).$ (2.4)

A straightforward discussion shows that

$\displaystyle\rho\begin{pmatrix}p_{1}\\\
q_{1}\end{pmatrix}\rho\begin{pmatrix}p_{2}\\\ q_{2}\end{pmatrix}=e^{\pi{\rm
i}(\langle p_{1},q_{2}\rangle-\langle
q_{1},p_{2}\rangle)}\rho\begin{pmatrix}p_{1}+p_{2}\\\
q_{1}+q_{2}\end{pmatrix},\qquad p_{1},q_{1},p_{2},q_{2}\in{\mathbb{R}}^{n},$
(2.5)

and it can be easily verified that

$\frac{1}{2\pi{\rm
i}}\frac{d}{dt}\rho(t\ell)=\left\langle\ell,Z\right\rangle\rho(t\ell),\qquad\forall\
\ell\in{\mathbb{R}}^{2n}.$ (2.6)

######  Proposition 2.1.

For ${\mathcal{B}}\in{\rm sp}(n,{\mathbb{R}})$ and $w(\cdot)\in
C_{b}^{0}({\mathbb{R}},{\mathbb{R}}^{2n})$,

$\left.\partial_{\tau}\rho\left(e^{\tau{\mathcal{B}}}w(t)\right)\right|_{\tau=0}=\left.\partial_{\tau}\rho\left(w(t)+\tau{\mathcal{B}}w(t)\right)\right|_{\tau=0},\qquad\forall\
t\in{\mathbb{R}}.$ (2.7)

###### Proof.

For simplicity, let us express
$e^{\tau{\mathcal{B}}}w(t)=w(t)+\tau{\mathcal{B}}w(t)+g_{\mathcal{B}}(\tau)w(t)$,
where $|g_{\mathcal{B}}(\tau)|=O(\tau^{2})$ as $\tau\to 0$. Based on (2.5), we
obtain

$\rho\left(e^{\tau{\mathcal{B}}}w(t)\right)=e^{-\pi{\rm i}\langle
w(t)+\tau{\mathcal{B}}w(t),{\mathbb{J}}_{n}g_{\mathcal{B}}(\tau)w(t)\rangle}\rho\left(w(t)+\tau{\mathcal{B}}w(t)\right)\rho(g_{\mathcal{B}}(\tau)w(t)).$

Consequently,

$\displaystyle\left.\partial_{\tau}\rho\left(e^{\tau{\mathcal{B}}}w(t)\right)\right|_{\tau=0}$
$\displaystyle=$ $\displaystyle\left.\partial_{\tau}\left(e^{-\pi{\rm
i}\langle
w(t)+s{\mathcal{B}}w(t),{\mathbb{J}}_{n}g_{\mathcal{B}}(\tau)w(t)\rangle}\right)\right|_{\tau=0}\rho(w(t))$
$\displaystyle+\,\left.e^{-\pi{\rm i}\langle
w(t)+\tau{\mathcal{B}}w(t),{\mathbb{J}}_{n}g_{\mathcal{B}}(\tau)w(t)\rangle}\right|_{\tau=0}\left.\partial_{\tau}\rho\left(w(t)+\tau{\mathcal{B}}w(t)\right)\right|_{\tau=0}$
$\displaystyle+\,\left.e^{-\pi{\rm i}\langle
w(t)+\tau{\mathcal{B}}w(t),{\mathbb{J}}_{n}g_{\mathcal{B}}(\tau)w(t)\rangle}\right|_{\tau=0}\rho(w(t))\left.\partial_{\tau}\rho\left(g_{\mathcal{B}}(\tau)w(t)\right)\right|_{\tau=0}$
$\displaystyle=:$ $\displaystyle P_{1}+P_{2}+P_{3},$

where $g_{\mathcal{B}}(0)=g_{\mathcal{B}}^{\prime}(0)=0$. As a result, (2.7)
is established since $P_{1}=P_{3}=0$,
$P_{2}=\left.\partial_{\tau}\rho\left(w(t)+\tau{\mathcal{B}}w(t)\right)\right|_{\tau=0}$.∎

The following proposition can be established through a similar proof.

######  Proposition 2.2.

For $w(\cdot)\in C_{b}^{1}({\mathbb{R}},{\mathbb{R}}^{2n})$,
$\left.\partial_{\tau}\rho(w(t+\tau))\right|_{\tau=0}=\left.\partial_{\tau}\rho\left(w(t)+\tau
w^{\prime}(t)\right)\right|_{\tau=0}$, $t\in{\mathbb{R}}$.

Consider $w\in{\mathbb{R}}^{2n}$. The operator $\rho(w)$ is well-defined on
$L^{2}({\mathbb{R}}^{n})$, and consequently, on
${\mathcal{H}}^{s}({\mathbb{R}}^{n})$, $s\geq 0$, as delineated in (2.4). For
$s>0$, we extend the definition of $\rho(w)$ to
${\mathcal{H}}^{-s}({\mathbb{R}}^{n})$ by employing duality:

$\langle\rho(w)u,v\rangle=\langle u,\rho(-w)v\rangle,\qquad
u\in{\mathcal{H}}^{-s}({\mathbb{R}}^{n}),\quad
v\in{\mathcal{H}}^{s}({\mathbb{R}}^{n}).$ (2.8)

######  Proposition 2.3.

For $w(\cdot)\in C_{b}^{1}(\mathbb{R},{\mathbb{R}}^{2n})$, it follows that
$\partial_{t}\rho(w(t))\in B({\mathcal{H}}^{s},{\mathcal{H}}^{s-1})$ for any
$s\in{\mathbb{R}}$.

The proof of this proposition is provided in Appendix B.

###  2.3. Metaplectic representation

The symplectic group, denoted by ${\rm Sp}(n,{\mathbb{R}})$, comprises
$2n\times 2n$ real matrices that preserve the symplectic form. Specifically, a
matrix ${\mathbb{A}}=\begin{pmatrix}A&B\\\ C&F\end{pmatrix}$ belongs to ${\rm
Sp}(n,{\mathbb{R}})$, with $A$, $B$, $C$, and $F$ being its four $n\times n$
blocks, if and only if the conditions $A^{*}C=C^{*}A$, $B^{*}F=F^{*}B$ and
$A^{*}F-C^{*}B=I_{n}$ are satisfied. Such a matrix ${\mathbb{A}}\in{\rm
Sp}(n,{\mathbb{R}})$ is referred to as a symplectic matrix.

The symplectic Lie algebra, ${\rm sp}(n,{\mathbb{R}})$, consists of all
matrices ${\mathcal{A}}\in{\rm gl}(2n,{\mathbb{R}})$ such that
$e^{t{\mathcal{A}}}\in{\rm Sp}(n,{\mathbb{R}})$ for every $t\in{\mathbb{R}}$.
In other words, ${\mathcal{A}}=\begin{pmatrix}A&B\\\ C&F\end{pmatrix}$ is in
${\rm sp}(n,{\mathbb{R}})$, with $A$, $B$, $C$, and $F$ as the four $n\times
n$ blocks of ${\mathcal{A}}$, if and only if the following hold: $F=-A^{*}$,
$B=B^{*}$, and $C=C^{*}$. A matrix ${\mathcal{A}}\in{\rm sp}(n,{\mathbb{R}})$
is known as a Hamiltonian matrix.

Let ${\bf Q}$ represent the space of real homogeneous quadratic polynomials on
${\mathbb{R}}^{2n}$, equipped with the Poisson bracket. Specifically, for
smooth functions $Q_{1}$ and $Q_{2}$ on ${\mathbb{R}}^{2n}$, the Poisson
bracket is defined as

$\left\\{Q_{1}(\xi,x),Q_{2}(\xi,x)\right\\}:=\sum\limits_{j=1}^{n}\left(\frac{\partial
Q_{1}}{\partial\xi_{j}}\frac{\partial Q_{2}}{\partial x_{j}}-\frac{\partial
Q_{1}}{\partial x_{j}}\frac{\partial Q_{2}}{\partial\xi_{j}}\right).$

Recall the definition of the homogeneous quadratic polynomial
${\mathcal{Q}}_{{\mathcal{A}}}(\xi,x)$ given in (1.22).

######  Proposition 2.4.

(Proposition 4.42 in [23]) The mapping
${\mathcal{A}}\mapsto{\mathcal{Q}}_{{\mathcal{A}}}(\xi,x)$ constitutes a Lie
algebra isomorphism from ${\rm sp}(n,{\mathbb{R}})$ to ${\bf Q}$.

The Metaplectic representation ${\mathcal{M}}$ of ${\rm Sp}(n,{\mathbb{R}})$
is defined as follows. Given $\mathbb{A}\in{\rm Sp}(n,{\mathbb{R}})$, it
induces an automorphism $T_{\mathbb{A}}$ of the Heisenberg group ${\bf H_{n}}$
by

$T_{\mathbb{A}}(p,q,t)=\left(\mathbb{A}\begin{pmatrix}p\\\
q\end{pmatrix},t\right),\qquad p,q\in{\mathbb{R}}^{n},\qquad
t\in{\mathbb{R}}.$

Since the Schrödinger representation $\rho$, defined in (2.3), is an
irreducible unitary representation of ${\bf H_{n}}$ on
$L^{2}({\mathbb{R}}^{n})$, the composite $\rho\circ T_{\mathbb{A}}$ is also an
irreducible unitary representation of ${\bf H_{n}}$, satisfying

$(\rho\circ T_{\mathbb{A}})(0,0,t)=e^{2\pi{\rm i}t},\qquad t\in{\mathbb{R}}.$

By the Stone-von Neumann theorem, the two irreducible unitary representations
$\rho$ and $\rho\circ T_{\mathbb{A}}$ are equivalent, meaning there exists a
unitary operator ${\mathcal{M}}(\mathbb{A})$ on $L^{2}({\mathbb{R}}^{n})$ such
that

$\rho\circ
T_{\mathbb{A}}({\mathcal{X}})={\mathcal{M}}(\mathbb{A})\rho({\mathcal{X}}){\mathcal{M}}(\mathbb{A})^{-1},\qquad{\mathcal{X}}\in{\bf
H_{n}}.$ (2.9)

As demonstrated in [23], ${\mathcal{M}}(\mathbb{A})$ can be uniquely chosen,
up to factors $\pm 1$, so that ${\mathcal{M}}$ becomes a double-valued unitary
representation of ${\rm Sp}(n,{\mathbb{R}})$. Hence,

${\mathcal{M}}(\mathbb{A}\mathbb{B})=\pm\,{\mathcal{M}}(\mathbb{A}){\mathcal{M}}(\mathbb{B}),\qquad\mathbb{A},\mathbb{B}\in{\rm
Sp}(n,{\mathbb{R}}),$ (2.10)

and, specifically,
${\mathcal{M}}(\mathbb{A}){\mathcal{M}}\left(\mathbb{A}^{-1}\right)=\pm\,{\rm
Id}$. Therefore, ${\mathcal{M}}$ maps ${\rm Sp}(n,{\mathbb{R}})$ into the
group of unitary operators on $L^{2}({\mathbb{R}}^{n})$ modulo ${\pm\,{\rm
Id}}$ and is referred to as the Metaplectic representation of ${\rm
Sp}(n,{\mathbb{R}})$. In explicit formulations, the ambiguity of $\pm 1$
typically manifests as the ambiguity in the sign of a square root. For
simplicity, we often omit the sign $\pm 1$.

Given $\mathbb{A}\in{\rm Sp}(n,{\mathbb{R}})$, the operator
${\mathcal{M}}(\mathbb{A})$ is well-defined on $L^{2}({\mathbb{R}}^{n})$, and
consequently, on ${\mathcal{H}}^{s}({\mathbb{R}}^{n})$ for $s\geq 0$, as
specified in (2.9). For $s>0$, we extend the definition of
${\mathcal{M}}(\mathbb{A})$ to ${\mathcal{H}}^{-s}({\mathbb{R}}^{n})$ using
duality:

$\langle{\mathcal{M}}(\mathbb{A})u,v\rangle=\langle
u,{\mathcal{M}}(\mathbb{A}^{-1})v\rangle,\qquad
u\in{\mathcal{H}}^{-s}({\mathbb{R}}^{n}),\quad
v\in{\mathcal{H}}^{s}({\mathbb{R}}^{n}).$ (2.11)

Define the infinitesimal version $d{\mathcal{M}}$ of the Metaplectic
representation as

$d{\mathcal{M}}({\mathcal{A}}):=\left.\frac{d}{dt}{\mathcal{M}}\left(e^{t{\mathcal{A}}}\right)\right|_{t=0},\qquad{\mathcal{A}}\in{\rm
sp}(n,{\mathbb{R}}).$ (2.12)

In this context and subsequently, the sign of
${\mathcal{M}}(e^{t{\mathcal{A}}})$ is chosen to ensure continuity in $t$ and
equality to $\rm Id$ at $t=0$. Under this framework, the following theorem is
established:

######  Theorem 2.5.

(Theorem 4.45 in [23]) For any $u$ in the Schwartz space
${\mathcal{S}}({\mathbb{R}}^{n})$ and ${\mathcal{A}}\in{\rm
sp}(n,{\mathbb{R}})$,

$d{\mathcal{M}}({\mathcal{A}})u=2\pi{\rm i}{\mathcal{Q}}_{{\mathcal{A}}}(Z)u.$

######  Remark 6.

The proof of the above theorem shows that for $s\in{\mathbb{R}}$,
$d{\mathcal{M}}({\mathcal{A}})=2\pi{\rm i}{\mathcal{Q}}_{{\mathcal{A}}}(Z)\in
B({\mathcal{H}}^{s},{\mathcal{H}}^{s-2})$.

Consider ${\mathcal{A}}\in{\rm sp}(n,{\mathbb{R}})$ and
$u\in{\mathcal{H}}^{s}$. Let us denote
$\psi(t,x)=({\mathcal{M}}(e^{t{\mathcal{A}}})u)(x)$. Given that
${\mathcal{M}}(e^{(t+\tau){\mathcal{A}}})={\mathcal{M}}(e^{\tau{\mathcal{A}}}){\mathcal{M}}(e^{t{\mathcal{A}}})$
for any $t,\tau\in{\mathbb{R}}$, a straightforward computation, in conjunction
with Theorem 2.5, leads us to the following observation:

$\partial_{t}\psi(t,x)=\lim\limits_{\tau\to
0}\frac{1}{\tau}\left(\psi(t+\tau,x)-\psi(t,x)\right)=\lim\limits_{\tau\rightarrow
0}\frac{1}{\tau}\left({\mathcal{M}}(e^{\tau{\mathcal{A}}})-{\rm
Id}\right)\psi(t,x)=2\pi{\rm i}{\mathcal{Q}}_{{\mathcal{A}}}(Z)\psi(t,x),$

which lies in ${\mathcal{H}}^{s-2}$. The aforementioned equality implies that
for ${\mathcal{A}}\in{\rm sp}(n,{\mathbb{R}})$,

$e^{2\pi{\rm
i}\tau{\mathcal{Q}}_{{\mathcal{A}}}(Z)}={\mathcal{M}}(e^{\tau{\mathcal{A}}}),\qquad\forall\
\tau\in{\mathbb{R}},$ (2.13)

thereby establishing a connection between the linear system

$z^{\prime}(t)={\mathcal{A}}z(t),\qquad z(0)=z_{0}\in{\mathbb{R}}^{2n},$
(2.14)

and its corresponding Hamiltonian PDE

$\frac{1}{2\pi{\rm
i}}\partial_{t}\psi={\mathcal{Q}}_{{\mathcal{A}}}(Z)\psi(t,x),\qquad\psi(0)=u\in{\mathcal{H}}^{s}({\mathbb{R}}^{n}).$
(2.15)

It is evident that $z(t)=e^{t{\mathcal{A}}}z_{0}$ solves the system (2.14),
while $\psi(t)={\mathcal{M}}(e^{t{\mathcal{A}}})u$ addresses Eq. (2.15).
Moreover, in light of Theorem 1.2 in [46], $\psi(t)$ is globally well-posed in
${\mathcal{H}}^{s}({\mathbb{R}}^{n})$ and exhibits an upper bound for the
${\mathcal{H}}^{s}$-norm that grows exponentially with $t$.

For specific matrices $\mathbb{A}\in{\rm Sp}(n,{\mathbb{R}})$, explicit
formulas of the Metaplectic representation can be provided:

######  Theorem 2.6.

(Theorem 4.51 & 4.53 in [23]) Let $\mathbb{A}=\begin{pmatrix}A&B\\\
C&F\end{pmatrix}\in{\rm Sp}(n,{\mathbb{R}})$ and $u\in{\mathcal{H}}^{s}$,
$s\in{\mathbb{R}}$.

  * (i)

If $\mathrm{det}\ A\neq 0$, then
$\displaystyle({\mathcal{M}}(\mathbb{A})u)(x)=(\mathrm{det}\
A)^{-\frac{1}{2}}\int_{{\mathbb{R}}^{n}}e^{2\pi{\rm
i}S(\xi,x)}\hat{u}(\xi)\,d\xi$, where

$S(\xi,x):=-\frac{1}{2}\langle
x,CA^{-1}x\rangle+\langle\xi,A^{-1}x\rangle+\frac{1}{2}\langle\xi,A^{-1}B\xi\rangle.$

  * (ii)

If $\mathrm{det}\ B\neq 0$, then
$\displaystyle({\mathcal{M}}(\mathbb{A})u)(x)={\rm
i}^{\frac{n}{2}}(\mathrm{det}\
B)^{-\frac{1}{2}}\int_{{\mathbb{R}}^{n}}e^{2\pi{\rm
i}S^{\prime}(x,y)}u(y)\,dy$, where

$S^{\prime}(x,y)=-\frac{1}{2}\langle x,FB^{-1}x\rangle+\langle
y,B^{-1}x\rangle-\frac{1}{2}\langle y,B^{-1}Ay\rangle.$

######  Remark 7.

According to [23], the aforementioned formulas are valid when $u\in
L^{2}({\mathbb{R}}^{n})$. Consequently, they also apply to
${\mathcal{H}}^{s}({\mathbb{R}}^{n})$ for $s\geq 0$. Furthermore, by utilizing
the definition by duality as presented in (2.11), it can be demonstrated that
these formulas remain applicable for
$u\in{\mathcal{H}}^{-s}({\mathbb{R}}^{n})$, with $s\geq 0$.

The oscillatory integrals presented in Theorem 2.6 do not provide an explicit
description for arbitrary $\mathbb{A}\in{\rm Sp}(n,{\mathbb{R}})$. This is due
to the prevalence of $2n\times 2n$ symplectic matrices whose $n\times n$
blocks are all singular. This issue will be further explored in Section 3.

##  3\. Bounds of Sobolev norms under the representations

It is established that Eq. (2.15) is globally well-posed in
${\mathcal{H}}^{s}({\mathbb{R}}^{n})$ for $s\geq 0$. To achieve a more
quantitative description of the ${\mathcal{H}}^{s}$-norm, which is the primary
objective of this paper, we provide both upper and lower bounds of Sobolev
norms. These bounds are determined under the application of the Metaplectic
representation for any $\mathbb{A}\in{\rm Sp}(n,{\mathbb{R}})$ and the
Schrödinger representation for any $w\in{\mathbb{R}}^{2n}$.

###  3.1. Bounds of Sobolev norms under Metaplectic representation

Recall from (2.13) that for ${\mathcal{B}}(\cdot)\in
C_{b}^{0}({\mathbb{R}},{\rm sp}(n,{\mathbb{R}}))$ and $|\tau|\leq 1$,
$e^{\tau{\mathcal{B}}(\cdot)}\in C_{b}^{0}({\mathbb{R}},{\rm
Sp}(n,{\mathbb{R}}))$ and satisfies
${\mathcal{M}}(e^{\tau{\mathcal{B}}(t)})=e^{2\pi{\rm
i}\tau{\mathcal{Q}}_{{\mathcal{B}}(t)}}$ for every $t\in{\mathbb{R}}$.
According to Lemma 2.8 in [5], there exist constants $c_{1}$ and $c_{2}>0$,
dependent on $s\geq 0$ and the uniform bounds of ${\mathcal{B}}(\cdot)$,
ensuring that

$c_{1}\|u\|_{s}\leq\|e^{2\pi{\rm
i}\tau{\mathcal{Q}}_{{\mathcal{B}}(t)}}u\|_{s}\leq c_{2}\|u\|_{s},\qquad
u\in{\mathcal{H}}^{s},\qquad t\in{\mathbb{R}}.$ (3.1)

This estimate is particularly effective when the $L^{2}-$unitary
transformation is of the exponential form, especially for near-identity
transformations. However, as noted by Djoković [16], not all elements in
$C_{b}^{0}({\mathbb{R}},{\rm Sp}(n,{\mathbb{R}}))$ are encompassed in (3.1).
Specifically, for a connected Lie group $G$, its Lie algebra $g$, and the
exponential map $\exp:g\rightarrow G$ of $G$, the index of an element $a\in
G$, denoted by ${\rm ind}(a)$, is the smallest positive integer $m$ such that
$a^{m}$ lies in the image of $\exp$.

######  Lemma 3.1.

(Djoković [16]) The image of ${\rm ind}$ from ${\rm Sp}(n,{\mathbb{R}})$ to
${\mathbb{Z}}$ is $\\{1,2\\}$. In other words, for any ${\mathbb{B}}\in{\rm
Sp}(n,{\mathbb{R}})$, there exists $\mathcal{B}\in{\rm sp}(n,{\mathbb{R}})$
such that ${\mathbb{B}}^{2}=e^{\mathcal{B}}$.

Considering Lemma 3.1, for $\mathbb{A}(\cdot)\in C_{b}^{0}({\mathbb{R}},{\rm
Sp}(n,{\mathbb{R}}))$, we cannot assure the existence of
${\mathcal{B}}(\cdot)\in C_{b}^{0}({\mathbb{R}},{\rm sp}(n,{\mathbb{R}}))$
such that $\mathbb{A}(\cdot)=e^{{\mathcal{B}}(\cdot)}$. To extend (3.1), we
propose the following proposition, which reveals that only uniform boundedness
is required, and even continuity is not a necessity.

######  Proposition 3.2.

For ${\mathbb{A}}\in{\rm Sp}(n,{\mathbb{R}})$, for any $s\in{\mathbb{R}}$, we
have

$\|{\mathbb{A}}\|^{-|s|}\|u\|_{s}\lesssim\|{\mathcal{M}}({\mathbb{A}})u\|_{s}\lesssim\|{\mathbb{A}}\|^{|s|}\|u\|_{s},\qquad\forall\
u\in{\mathcal{H}}^{s}.$ (3.2)

Furthermore, for $s>0$ and $u\in{\mathcal{H}}^{s}\setminus\\{0\\}$, it holds
that

$\|{\mathcal{M}}({\mathbb{A}})u\|_{s}\gtrsim_{u}\|{\mathbb{A}}\|^{s}.$ (3.3)

The estimate (3.2) is crucial for the ${\mathcal{H}}^{s}-$estimates of the
$L^{2}-$unitary transformations in the reducibility argument, demonstrating
the ${\mathcal{H}}^{s}-$equivalence between the original time-dependent
equation and the reduced equation with a constant homogeneous part.
Conversely, the lower bound (3.3) is applied to the time propagator of the
homogeneous quadratic Hamiltonian PDE via (2.13), essentially contributing to
the growth of the Sobolev norm for the solution to the reduced Hamiltonian
PDE. More specifically, we arrive at the following corollary.

######  Corollary 3.3.

Given $U\in C_{b}^{0}({\mathbb{R}},{\rm Sp}(n,{\mathbb{R}}))$,
${\mathcal{A}}\in{\rm sp}(n,{\mathbb{R}})$, we have, for $t\in{\mathbb{R}}$,

  * (i)

for $u\in{\mathcal{H}}^{s}$, $s\in{\mathbb{R}}$,
$\|U\|_{C_{b}^{0}}^{-|s|}\|u\|_{s}\lesssim\|{\mathcal{M}}(U(t))u\|_{s},\
\|{\mathcal{M}}(U(t)^{-1})u\|_{s}\lesssim\|U\|_{C_{b}^{0}}^{|s|}\|u\|_{s}$,

  * (ii)

for $u\in{\mathcal{H}}^{s}\setminus\\{0\\}$, $s>0$,
$\|e^{t{\mathcal{A}}}\|^{s}\lesssim_{u}\|{\mathcal{M}}(e^{t{\mathcal{A}}})u\|_{s}\lesssim\|e^{t{\mathcal{A}}}\|^{s}\|u\|_{s}$.

The remainder of this subsection is dedicated to the proof of Proposition 3.2.

####  3.1.1. Bounds for symplectic matrix with diagonal block form

######  Lemma 3.4.

For ${\mathbb{A}}=\begin{pmatrix}A&0\\\ 0&A^{*-1}\end{pmatrix}\in{\rm
Sp}(n,{\mathbb{R}})$, we have, for any $s>0$,

$\|{\mathbb{A}}\|^{-s}\|u\|_{s}\lesssim\|{\mathcal{M}}(\mathbb{A})u\|_{s}\lesssim\|{\mathbb{A}}\|^{s}\|u\|_{s},\quad\forall\
u\in{\mathcal{H}}^{s}.$ (3.4)

Moreover, if $A={\rm diag}\\{\zeta_{0},\cdots,\zeta_{n-1}\\}$, then, for
$u\in{\mathcal{H}}^{s}\setminus\\{0\\}$,

$\|{\mathcal{M}}(\mathbb{A})u\|_{s}\gtrsim\sum_{j=1}^{n}\left(|\zeta_{j-1}|^{s}\|X_{j}^{s}u\|_{L^{2}}+|\zeta_{j-1}|^{-s}\|X_{j}^{s}\hat{u}\|_{L^{2}}\right).$
(3.5)

###### Proof.

Since ${\mathbb{A}}=\begin{pmatrix}A&0\\\ 0&A^{*-1}\end{pmatrix}\in{\rm
Sp}(n,{\mathbb{R}})$ implies that $A$ is non-singular, by Theorem 2.6-(i), we
have

$\left({\mathcal{M}}\left({\mathbb{A}}\right)u\right)(x)=(\mathrm{det}\
A)^{-\frac{1}{2}}\int_{{\mathbb{R}}^{n}}e^{2\pi{\rm
i}\langle\xi,A^{-1}x\rangle}\hat{u}(\xi)d\xi=(\mathrm{det}\
A)^{-\frac{1}{2}}u\left(A^{-1}x\right),\qquad u\in{\mathcal{H}}^{s},\quad
s>0.$

A straightforward computation on the expression of solution implies that

$\displaystyle\||X|^{s}{\mathcal{M}}\left({\mathbb{A}}\right)u\|_{L^{2}}$
$\displaystyle=$ $\displaystyle|\mathrm{det}\
A|^{-\frac{1}{2}}\left(\int_{{\mathbb{R}}^{n}}|x|^{2s}\left|u\left(A^{-1}x\right)\right|^{2}dx\right)^{\frac{1}{2}}$
(3.6) $\displaystyle=$ $\displaystyle|\mathrm{det}\
A|^{-\frac{1}{2}}\left(\int_{{\mathbb{R}}^{n}}|Ay|^{2s}|u(y)|^{2}\,d\left(Ay\right)\right)^{\frac{1}{2}}\leq\|A\|^{s}\left(\int_{{\mathbb{R}}^{n}}|y|^{2s}|u(y)|^{2}dy\right)^{\frac{1}{2}},$

and, if $A={\rm diag}\\{\zeta_{0},\cdots,\zeta_{n-1}\\}$, then

$\||X|^{s}{\mathcal{M}}\left({\mathbb{A}}\right)u\|_{L^{2}}=\left(\int_{{\mathbb{R}}^{n}}\left(\sum_{j=1}^{n}\zeta^{2}_{j-1}y^{2}_{j}\right)^{s}|u(y)|^{2}\,dy\right)^{\frac{1}{2}}\gtrsim\sum_{j=1}^{n}|\zeta_{j-1}|^{s}\|X_{j}^{s}u\|_{L^{2}}.$
(3.7)

On the other hand, since

$\left({\mathcal{F}}{\mathcal{M}}\left({\mathbb{A}}\right)u\right)(\xi)=(\mathrm{det}\
A)^{-\frac{1}{2}}\int_{{\mathbb{R}}^{n}}e^{-2\pi{\rm i}\left\langle
A^{*}\xi,A^{-1}x\right\rangle}u\left(A^{-1}x\right)dx=|\mathrm{det}\
A|^{\frac{1}{2}}\hat{u}\left(A^{*}\xi\right),$

it follows that

$\left\||X|^{s}{\mathcal{F}}{\mathcal{M}}\left({\mathbb{A}}\right)u\right\|_{L^{2}}=\left(\int_{{\mathbb{R}}^{n}}\left|A^{*-1}\eta\right|^{2s}|\hat{u}(\eta)|^{2}d\eta\right)^{\frac{1}{2}}\leq\|A^{-1}\|^{s}\left(\int_{{\mathbb{R}}^{n}}\left|\eta\right|^{2s}|\hat{u}(\eta)|^{2}d\eta\right)^{\frac{1}{2}},$
(3.8)

and, if $A={\rm diag}\\{\zeta_{0},\cdots,\zeta_{n-1}\\}$, then

$\left\||X|^{s}{\mathcal{F}}{\mathcal{M}}\left({\mathbb{A}}\right)u\right\|_{L^{2}}=\left(\int_{{\mathbb{R}}^{n}}\left(\sum_{j=1}^{n}\zeta^{-2}_{j-1}\eta^{2}_{j}\right)^{s}|\hat{u}(\eta)|^{2}\,d\eta\right)^{\frac{1}{2}}\gtrsim\sum_{j=1}^{n}|\zeta_{j-1}|^{-s}\|X_{j}^{s}\hat{u}\|_{L^{2}}.$
(3.9)

Combining (3.6) and (3.8), we obtain

$\left\|{\mathcal{M}}\left({\mathbb{A}}\right)u\right\|_{s}\lesssim\left(\|A\|^{s}+\|A^{-1}\|^{s}\right)\|u\|_{s}\lesssim\|{\mathbb{A}}\|^{s}\|u\|_{s}.$

The property (2.10) implies that
${\mathcal{M}}\left({\mathbb{A}}^{-1}\right){\mathcal{M}}\left({\mathbb{A}}\right)={\rm
Id}$. Then, noting that $\|{\mathbb{A}}^{-1}\|=\|{\mathbb{A}}\|$, we have

$\|u\|_{s}=\left\|{\mathcal{M}}\left({\mathbb{A}}^{-1}\right){\mathcal{M}}\left({\mathbb{A}}\right)u\right\|_{s}\lesssim\|{\mathbb{A}}\|^{s}\left\|{\mathcal{M}}\left({\mathbb{A}}\right)u\right\|_{s},$
(3.10)

which implies that
$\left\|{\mathcal{M}}\left({\mathbb{A}}\right)u\right\|_{s}\gtrsim\|{\mathbb{A}}\|^{-s}\|u\|_{s}$.
Moreover, if $A={\rm diag}\\{\zeta_{0},\cdots,\zeta_{n-1}\\}$, then, combining
(3.7) and (3.9), the lower bound (3.5) is established.∎

####  3.1.2. Elementary symplectic (ES) matrices

To extend the application of Lemma 3.4 to any symplectic matrix, we introduce
the concept of elementary symplectic matrices within ${\rm
Sp}(n,{\mathbb{R}})$.

A matrix ${\mathbb{A}}\in{\rm Sp}(n,{\mathbb{R}})$ is termed elementary
symplectic, abbreviated as ES for brevity, if it falls into one of six
specific categories. These categories are collectively denoted by
${\mathbb{A}}\in{\bf ES}:=\bigcup_{{\bf k}={\bf 1}}^{\bf 6}{\bf ES(k)}$.

  * ES(1).

${\mathbb{A}}={\mathcal{J}}_{k}$ or ${\mathcal{J}}_{k}^{-1}$, $k=1,\cdots,n$,
where

${\mathcal{J}}_{k}:=({\bf e}_{1},\cdots,{\bf
e}_{k-1},\underset{\underset{k^{\rm th}}{\uparrow}}{-{\bf e}_{n+k}},{\bf
e}_{k+1},\cdots,{\bf e}_{n},{\bf e}_{n+1},\cdots,{\bf
e}_{n+k-1},\underset{\underset{(n+k)^{\rm th}}{\uparrow}}{{\bf e}_{k}},{\bf
e}_{n+k+1},\cdots,{\bf e}_{2n}).$

Note that, for $S\in{\rm Sp}(n,{\mathbb{R}})$ expressed by column vectors,
i.e., $S=(\xi_{1},\cdots,\xi_{n},\eta_{1},\cdots,\eta_{n})$,

$S{\mathcal{J}}_{k}=(\xi_{1},\cdots,\xi_{k-1},\underset{\underset{k^{\rm
th}}{\uparrow}}{-\eta_{k}},\xi_{k+1},\cdots,\xi_{n},\eta_{1},\cdots,\eta_{k-1},\underset{\underset{(n+k)^{\rm
th}}{\uparrow}}{\xi_{k}},\eta_{k+1},\cdots,\eta_{n}).$

while for $S\in{\rm Sp}(n,{\mathbb{R}})$ expressed by row vectors, i.e.,
$S=(\alpha_{1},\cdots,\alpha_{n},\beta_{1},\cdots,\beta_{n})^{*}$,

${\mathcal{J}}_{k}S=(\alpha_{1},\cdots,\alpha_{k-1},\underset{\underset{k^{\rm
th}}{\uparrow}}{\beta_{k}},\alpha_{k+1},\cdots,\alpha_{n},\beta_{1},\cdots,\beta_{k-1},\underset{\underset{(n+k)^{\rm
th}}{\uparrow}}{-\alpha_{k}},\beta_{k+1},\cdots,\beta_{n})^{*}.$

  * ES(2).

For $n\geq 2$, ${\mathbb{A}}={\mathbb{B}}_{jk}$ for $1\leq j<k\leq n$, where

${\mathbb{B}}_{jk}:=({\bf e}_{1},\cdots,\underset{\underset{j^{\rm
th}}{\uparrow}}{{\bf e}_{k}},\cdots,\underset{\underset{k^{\rm
th}}{\uparrow}}{{\bf e}_{j}},\cdots,\underset{\underset{(n+j)^{\rm
th}}{\uparrow}}{{\bf e}_{n+k}},\cdots,\underset{\underset{(n+k)^{\rm
th}}{\uparrow}}{{\bf e}_{n+j}}\cdots,{\bf e}_{2n}).$

Note that, for $S\in{\rm Sp}(n,{\mathbb{R}})$, $S{\mathbb{B}}_{jk}$ means the
exchanges between $j^{\rm th}$, $k^{\rm th}$ columns and between $(n+j)^{\rm
th}$, $(n+k)^{\rm th}$ columns, i.e., for
$S=(\xi_{1},\cdots,\xi_{n},\eta_{1},\cdots,\eta_{n})$,

$S{\mathbb{B}}_{jk}=(\xi_{1},\cdots,\underset{\underset{j^{\rm
th}}{\uparrow}}{\xi_{k}},\cdots,\underset{\underset{k^{\rm
th}}{\uparrow}}{\xi_{j}},\cdots,\xi_{n},\eta_{1},\cdots,\underset{\underset{(n+j)^{\rm
th}}{\uparrow}}{\eta_{k}},\cdots,\underset{\underset{(n+k)^{\rm
th}}{\uparrow}}{\eta_{j}},\cdots,\eta_{n}),$

while ${\mathbb{B}}_{jk}S$ means the exchanges between $j^{\rm th}$, $k^{\rm
th}$ rows and between $(n+j)^{\rm th}-$row and $(n+k)^{\rm th}-$rows, i.e.,
for $S=(\alpha_{1},\cdots,\alpha_{n},\beta_{1},\cdots,\beta_{n})^{*}$,

${\mathbb{B}}_{jk}S=:(\alpha_{1},\cdots,\underset{\underset{j^{\rm
th}}{\uparrow}}{\alpha_{k}},\cdots,\underset{\underset{k^{\rm
th}}{\uparrow}}{\alpha_{j}},\cdots,\alpha_{n},\beta_{1},\cdots,\underset{\underset{(n+j)^{\rm
th}}{\uparrow}}{\beta_{k}},\cdots,\underset{\underset{(n+k)^{\rm
th}}{\uparrow}}{\beta_{j}},\cdots,\beta_{n})^{*}.$

  * ES(3).

For $n\geq 2$, ${\mathbb{A}}=\mathbb{E}_{q,j}$ for $2\leq j\leq n$ and
$|q|\leq 1$, where

$\mathbb{E}_{q,j}:=({\bf e}_{1},{\bf e}_{2},\cdots,\underset{\underset{j^{\rm
th}}{\uparrow}}{{\bf e}_{j}+q{\bf e}_{1}},\cdots,{\bf
e}_{n},\underset{\underset{(n+1)^{\rm th}}{\uparrow}}{{\bf e}_{n+1}-q{\bf
e}_{n+j}},\cdots,{\bf e}_{2n}).$

Note that, for $S=(\xi_{1},\cdots,\xi_{n},\eta_{1},\cdots,\eta_{n})\in{\rm
Sp}(n,{\mathbb{R}})$, we have

$S\mathbb{E}_{q,j}=(\xi_{1},\cdots,\underset{\underset{j^{\rm
th}}{\uparrow}}{\xi_{j}+q\xi_{1}},\cdots,\xi_{n},\underset{\underset{(n+1)^{\rm
th}}{\uparrow}}{\eta_{1}-q\eta_{j}},\cdots,\eta_{j},\cdots,\eta_{n}).$

  * ES(4).

For $n\geq 2$, ${\mathbb{A}}=\mathbb{F}_{q,j}$ for $2\leq j\leq n$ and
$|q|\leq 1$, where

$\mathbb{F}_{q,j}:=({\bf e}^{*}_{1},\cdots,\underset{\underset{j^{\rm
th}}{\uparrow}}{{\bf e}^{*}_{j}+q{\bf e}^{*}_{1}},\cdots,{\bf
e}^{*}_{n},\underset{\underset{(n+1)^{\rm th}}{\uparrow}}{{\bf
e}^{*}_{n+1}-q{\bf e}^{*}_{n+j}},\cdots,{\bf e}^{*}_{n+j},\cdots,{\bf
e}^{*}_{2n})^{*}.$

Note that, for
$S=(\alpha^{*}_{1},\cdots,\alpha^{*}_{n},\beta^{*}_{1},\cdots,\beta^{*}_{n})^{*}$,
we have

$\mathbb{F}_{q,j}S=(\alpha^{*}_{1},\cdots,\underset{\underset{j^{\rm
th}}{\uparrow}}{\alpha^{*}_{j}+q\alpha^{*}_{1}},\cdots,\alpha^{*}_{n},\underset{\underset{(n+1)^{\rm
th}}{\uparrow}}{\beta^{*}_{1}-q\beta^{*}_{j}},\cdots,\beta^{*}_{j},\dots,\beta^{*}_{n})^{*}.$

  * ES(5).

For ${\mathbb{A}}=\mathbb{T}_{\ell}$ for $1\leq j\leq n$ and
$\ell\in[-n,n]\times[-1,1]^{n-1}$ (or $[-1,1]$ for $n=1$), where

$\mathbb{T}_{\ell}:=({\bf e}_{1},\dots,{\bf e}_{n},{\bf
e}_{n+1}+\sum\limits_{j=1}^{n}\ell_{j}{\bf e}_{j},{\bf e}_{n+2}+\ell_{2}{\bf
e}_{1},\cdots,{\bf e}_{2n}+\ell_{n}{\bf e}_{1}).$

Note that, for $S=(\xi_{1},\cdots,\xi_{n},\eta_{1},\cdots,\eta_{n})\in{\rm
Sp}(n,{\mathbb{R}})$, we have

$S\mathbb{T}_{\ell}=(\xi_{1},\cdots,\xi_{n},\eta_{1}+\sum\limits_{j=1}^{n}\ell_{j}\xi_{j},\eta_{2}+\ell_{2}\xi_{1},\cdots,\eta_{n}+\ell_{n}\xi_{1}).$

  * ES(6).

${\mathbb{A}}=\mathbb{S}_{\ell}$ for $\ell\in[-n,n]\times[-1,1]^{n-1}$, where

$\mathbb{S}_{\ell}:=({\bf e}^{*}_{1},\cdots,{\bf e}^{*}_{n},{\bf
e}^{*}_{n+1}+\sum\limits_{j=1}^{n}\ell_{j}{\bf e}^{*}_{j},{\bf
e}^{*}_{n+2}+\ell_{2}{\bf e}^{*}_{1},\cdots,{\bf e}^{*}_{2n}+\ell_{n}{\bf
e}^{*}_{1})^{*}.$

Note that, for
$S=(\alpha^{*}_{1},\cdots,\alpha^{*}_{n},\beta^{*}_{1},\cdots,\beta^{*}_{n})^{*}$,
we have

$\mathbb{S}_{\ell}S=(\alpha^{*}_{1},\cdots,\alpha^{*}_{n},\beta^{*}_{1}+\sum\limits_{j=1}^{n}\ell_{j}\alpha^{*}_{j},\beta^{*}_{2}+\ell_{2}\alpha^{*}_{1},\cdots,\beta^{*}_{n}+\ell_{n}\alpha^{*}_{1})^{*}.$

For ${\bf k}={\bf 1},\cdots,{\bf 6}$, it is easy to verify that
${\mathbb{A}}^{-1}\in{\bf ES(k)}$ if ${\mathbb{A}}\in{\bf ES(k)}$.

######  Lemma 3.5.

For ${\mathbb{A}}\in{\bf ES(k)}$, ${\bf k}={\bf 1},\cdots,{\bf 6}$, for
$u\in{\mathcal{H}}^{s}\setminus\\{0\\}$, $s>0$, we have

$\left\|{\mathcal{M}}({\mathbb{A}})u\right\|_{s}\simeq\left\|u\right\|_{s},\qquad\|X^{s}_{j}{\mathcal{M}}({\mathbb{A}})u\|_{L^{2}},\
\|X^{s}_{j}{\mathcal{F}}{\mathcal{M}}({\mathbb{A}})u\|_{L^{2}}\gtrsim_{u}1,\quad
j=1,\cdots,n.$ (3.11)

###### Proof.

Let us show (3.11) for every type of ES matrix.

For ${\mathbb{A}}={\mathcal{J}}_{k}$ or ${\mathcal{J}}_{k}^{-1}\in{\bf
ES(1)}$, in view of (2.2), it is sufficient to show
${\mathcal{M}}({\mathcal{J}}_{k})={\rm i}^{\frac{1}{2}}\mathcal{F}_{k}^{-1}$.
Without loss of generality, let us suppose $k=1$. For
$J_{1}:=\left(\begin{array}[]{cccc:cccc}&&&&1&&&\\\ &&&&&0&&\\\
&&&&&&\ddots&\\\ &&&&&&&0\\\ \hdashline-1&&&&&&&\\\ &0&&&&&&\\\
&&\ddots&&&&&\\\ &&&0&&&&\end{array}\right)\in{\rm sp}(n,{\mathbb{R}})$,
noting that

$e^{tJ_{1}}=\left(\begin{array}[]{cccc:cccc}\cos(t)&&&&\sin(t)&&&\\\
&1&&&&0&&\\\ &&\ddots&&&&\ddots&\\\ &&&1&&&&0\\\
\hdashline-\sin(t)&&&&\cos(t)&&&\\\ &0&&&&1&&\\\ &&\ddots&&&&\ddots&\\\
&&&0&&&&1\end{array}\right)\in{\rm Sp}(n,{\mathbb{R}}),$

and especially, $e^{\frac{\pi}{2}J_{1}}={\mathcal{J}}_{1}$, we have, by
Theorem 2.5, that

$d{\mathcal{M}}(J_{1})u=2\pi{\rm i}{\mathcal{Q}}_{J_{1}}(Z)u=\pi{\rm
i}(D_{1}^{2}+X_{1}^{2})u,\qquad u\in\mathcal{S}({\mathbb{R}}^{n}).$

Then, for $h_{\alpha}(x)=h_{\alpha_{1}}(x_{1})\cdots h_{\alpha_{n}}(x_{n})$,
with $\alpha=(\alpha_{1},...,\alpha_{n})\in{\mathbb{N}}^{n}$ and $h_{k}$ the
$k-$th Hermite function,

$d{\mathcal{M}}(J_{1})h_{\alpha}=\pi{\rm
i}(D_{1}^{2}+X_{1}^{2})h_{\alpha_{1}}(x_{1})...h_{\alpha_{n}}(x_{n})=\frac{\rm
i}{2}(2\alpha_{1}+1)h_{\alpha}.$

Since
$v_{\alpha}(t,\cdot):=\left({\mathcal{M}}(e^{tJ_{1}}\right)h_{\alpha})(\cdot)$
solves the equation $\partial_{t}v(t,\cdot)=d{\mathcal{M}}(J_{1})v(t,\cdot)$
with $v(0,\cdot)=h_{\alpha}(\cdot)$, we have that
${\mathcal{M}}(e^{tJ_{1}})h_{\alpha}=e^{td{\mathcal{M}}(J_{1})}h_{\alpha}=e^{\frac{{\rm
i}t}{2}(2\alpha_{1}+1)}h_{\alpha}$, and hence, for $t=\frac{\pi}{2}$, for
$\alpha\in{\mathbb{N}}^{n}$,
${\mathcal{M}}\left({\mathcal{J}}_{1}\right)h_{\alpha}={\rm
i}^{\alpha_{1}+\frac{1}{2}}h_{\alpha}$. On the other hand, with
$\mathcal{F}h_{0}=h_{0}$ and by the recurrence formula (2.1),

$\mathcal{F}h_{k}=\sqrt{\frac{\pi^{k}}{k!}}\mathcal{F}(X-{\rm
i}D)^{k}h_{0}=\sqrt{\frac{\pi^{k}}{k!}}(-D-{\rm
i}X)^{k}\mathcal{F}h_{0}=\sqrt{\frac{\pi^{k}}{k!}}(-{\rm i})^{k}(-{\rm
i}D+X)^{k}h_{0}=(-{\rm i})^{k}h_{k}.$

Thus, for every $\alpha\in{\mathbb{N}}^{n}$,
$\mathcal{F}_{1}^{-1}h_{\alpha}={\rm i}^{\alpha_{1}}h_{\alpha}={\rm
i}^{-\frac{1}{2}}{\mathcal{M}}({\mathcal{J}}_{1})h_{\alpha}$. Since
$\\{h_{\alpha}\\}$ is a complete orthonormal set, this establishes
${\mathcal{M}}({\mathcal{J}}_{1})={\rm i}^{-\frac{1}{2}}\mathcal{F}_{1}^{-1}$.

For ${\mathbb{A}}={\mathbb{B}}_{jk}\in{\bf ES(2)}$, $1\leq j<k\leq n$, we have
that

$({\mathcal{M}}({\mathbb{A}})u)(x)=u(x_{1},\cdots,\underset{\underset{j^{\rm
th}}{\uparrow}}{x_{k}},\cdots,\underset{\underset{k^{\rm
th}}{\uparrow}}{x_{j}},\cdots,x_{n}),\quad({\mathcal{F}}{\mathcal{M}}({\mathbb{A}})u)(\xi)=\hat{u}(\xi_{1},\cdots,\underset{\underset{j^{\rm
th}}{\uparrow}}{\xi_{k}},\cdots,\underset{\underset{k^{\rm
th}}{\uparrow}}{\xi_{j}},\cdots,\xi_{n}).$

Hence,
$\|X^{s}_{l}{\mathcal{M}}({\mathbb{A}})u\|_{L^{2}}=\|X^{s}_{l}u\|_{L^{2}}$,
$\|X^{s}_{l}{\mathcal{F}}{\mathcal{M}}({\mathbb{A}})u\|_{L^{2}}=\|X^{s}_{l}{\mathcal{F}}u\|_{L^{2}}$
for $l\neq k,j$, and

$\|X^{s}_{j}{\mathcal{M}}({\mathbb{A}})u\|_{L^{2}}=\|X^{s}_{k}u\|_{L^{2}},\quad\|X^{s}_{k}{\mathcal{M}}({\mathbb{A}})u\|_{L^{2}}=\|X^{s}_{j}u\|_{L^{2}},$
$\|X^{s}_{j}{\mathcal{F}}{\mathcal{M}}({\mathbb{A}})u\|_{L^{2}}=\|X^{s}_{k}{\mathcal{F}}u\|_{L^{2}},\quad\|X^{s}_{k}{\mathcal{F}}{\mathcal{M}}({\mathbb{A}})u\|_{L^{2}}=\|X^{s}_{j}{\mathcal{F}}u\|_{L^{2}},$

which implies (3.11).

For ${\mathbb{A}}=\mathbb{E}_{q,j}\in{\bf ES(3)}$, we see that
$\|{\mathbb{A}}\|\simeq 1$ since $-1\leq q\leq 1$. Noting that ${\mathbb{A}}$
is of diagonal block form, in view of Lemma 3.4, we have
$\left\|{\mathcal{M}}({\mathbb{A}})u\right\|_{s}\simeq\left\|u\right\|_{s}$
for $u\in{\mathcal{H}}^{s}$. A direct computation shows that

$({\mathcal{M}}({\mathbb{A}})u)(x)=u(x_{1}-qx_{j},x_{2},\cdots,x_{n}),\quad({\mathcal{F}}{\mathcal{M}}({\mathbb{A}})u)(\xi)=\hat{u}(\xi_{1},\cdots,\underset{\underset{j^{\rm
th}}{\uparrow}}{\xi_{j}-q\xi_{1}},\cdots,\xi_{n}),\quad
u\in{\mathcal{H}}^{s}.$

Hence,
$\|X^{s}_{l}{\mathcal{M}}({\mathbb{A}})u\|_{L^{2}}=\|X^{s}_{l}u\|_{L^{2}}$ for
$l\neq 1$,
$\|X^{s}_{l}{\mathcal{F}}{\mathcal{M}}({\mathbb{A}})u\|_{L^{2}}=\|X^{s}_{l}{\mathcal{F}}u\|_{L^{2}}$
for $l\neq j$. Moreover,

$\|X^{s}_{1}{\mathcal{M}}({\mathbb{A}})u\|_{L^{2}}=\left(\int_{{\mathbb{R}}^{n}}|x_{1}|^{2s}|u(x_{1}-qx_{j},\cdots,x_{n})|^{2}dx\right)^{\frac{1}{2}}=\left(\int_{{\mathbb{R}}^{n}}|y_{1}+qy_{j}|^{2s}|u(y)|^{2}dy\right)^{\frac{1}{2}}=:F_{u}(q).$

Since the function $F_{u}$ is non-vanishing and continuous on $[-1,1]$
provided that $u\in{\mathcal{H}}^{s}\setminus\\{0\\}$, we see that
$F_{u}(q)\gtrsim_{u}1$. Similarly,
$\|X^{s}_{j}{\mathcal{F}}{\mathcal{M}}({\mathbb{A}})u\|_{L^{2}}\gtrsim_{u}1$,
which completes the proof of (3.11) for ${\mathbb{A}}\in{\bf ES(3)}$. The
proof is similar for ${\mathbb{A}}\in{\bf ES(4)}$.

For ${\mathbb{A}}=\mathbb{T}_{\ell}\in{\bf ES(5)}$, according to Theorem
2.6-(i), we have

$({\mathcal{M}}({\mathbb{A}})u)(x)=\int_{{\mathbb{R}}^{n}}e^{2\pi{\rm
i}(\langle\xi,x\rangle+\frac{1}{2}\langle\xi,B_{\ell}\xi\rangle)}\hat{u}(\xi)\,d\xi,\quad
B_{\ell}:=\begin{pmatrix}\ell_{1}&\ell_{2}&\cdots&\ell_{n}\\\ \ell_{2}&0&&\\\
\vdots&&\ddots&\\\ \ell_{n}&&&0\end{pmatrix}.$

Then $({\mathcal{F}}{\mathcal{M}}({\mathbb{A}})u)(\xi)=e^{\pi{\rm
i}\langle\xi,B_{\ell}\xi\rangle}\hat{u}(\xi)$. For $j=1,\cdots,n$, we have
$\|X_{j}^{s}{\mathcal{F}}{\mathcal{M}}({\mathbb{A}})u\|_{L^{2}}=\|X_{j}^{s}{\mathcal{F}}u\|_{L^{2}}$,

$\|X_{j}^{s}{\mathcal{M}}({\mathbb{A}})u\|_{L^{2}}=\left(\int_{{\mathbb{R}}^{n}}|x_{j}|^{2s}\left|\int_{{\mathbb{R}}^{n}}e^{2\pi{\rm
i}(\langle\xi,x\rangle+\frac{1}{2}\langle\xi,B_{\ell}\xi\rangle)}\hat{u}(\xi)\,d\xi\right|^{2}dx\right)^{\frac{1}{2}}=:G_{j,u}(\ell).$

Since the function $G_{j,u}$ is non-vanishing and continuous on
$[-n,n]\times[-1,1]^{n-1}$ provided that
$u\in{\mathcal{H}}^{s}\setminus\\{0\\}$, we see that
$G_{j,u}(\ell)\gtrsim_{u}1$. Moreover, for $s\in{\mathbb{N}}^{*}$, we have

$\|X_{j}^{s}{\mathcal{M}}({\mathbb{A}})u\|_{L^{2}}=\|D_{j}^{s}{\mathcal{F}}{\mathcal{M}}({\mathbb{A}})u\|_{L^{2}}\lesssim\sum_{k=0}^{s}\sup_{|\beta|=s-k}\|\langle
X\rangle^{k}D^{\beta}\hat{u}\|_{L^{2}}\lesssim\|u\|_{s}.$

By interpolation and (3.10), we obtain (3.11) for ${\mathbb{A}}\in{\bf
ES(5)}$. The proof is similar for ${\bf ES(6)}$.∎

####  3.1.3. Proof of (3.2) for $s>0$.

The subsequent proposition demonstrates that Lemma 3.4 can always be applied,
up to a finite number of ES transformations. By combining this with Lemma 3.5,
we can establish the estimates (3.2) and (3.3) presented in Proposition 3.2
for the case where $s>0$.

######  Proposition 3.6.

(ES-digonalization of symplectic matrix) For given ${\mathbb{A}}\in{\rm
Sp}(n,{\mathbb{R}})$, there exist
$\zeta_{0},\zeta_{1},\cdots,\zeta_{n-1}\in{\mathbb{R}}$ satisfying

$|\zeta_{0}|\simeq\|{\mathbb{A}}\|,\qquad 1\leq|\zeta_{j}|\leq
4^{j}\left(\frac{n!}{(n-j)!}\right)^{2}\|{\mathbb{A}}\|,\quad j=0,\cdots,n-1,$
(3.12)

such that, through at most $(2n^{2}+6n)$ ES transformations, $L_{1},\cdots
L_{k_{L}},R_{1},\cdots R_{k_{R}}\in{\bf ES}$,

$\left(\prod_{l=k_{L}}^{1}L_{l}\right){\mathbb{A}}\left(\prod_{j=1}^{k_{R}}R_{j}\right)={\bf\Lambda}:={\rm
diag}\left\\{\zeta_{0},\cdots,\zeta_{n-1},\zeta^{-1}_{0},\cdots,\zeta^{-1}_{n-1}\right\\}.$
(3.13)

###### Proof.

Let us denote the quantities in (3.12) as

$M_{j}:=4^{j}\left(\frac{n!}{(n-j)!}\right)^{2}\|{\mathbb{A}}\|,\qquad
j=0,\cdots,n.$

We shall “diagonalize” ${\mathbb{A}}$ through ES transformations, utilizing an
induction argument based on the ES block-diagonalization process. Throughout
the proof, the superscript “$j$” (without parentheses), for $j=0,1,\cdots,n$,
corresponds to a specific stage in the induction process (but is not an index
of power). Furthermore, to streamline the notations, we define the ES
transformations as follows:

  * •

$\mathbb{A}\xrightarrow{R}\mathbb{B}$ indicates that
${\mathbb{A}}R={\mathbb{B}}$ for some $R\in{\bf ES}$,

  * •

${\mathbb{A}}\xrightarrow[L]{}{\mathbb{B}}$ signifies that
$L{\mathbb{A}}={\mathbb{B}}$ for some $L\in{\bf ES}$.

$\bullet$ First ES block-diagonalisation for ${\mathbb{A}}$.

Initially, let us “move” the largest entry of ${\mathbb{A}}$ to the
$(1,1)$-entry through exchanges of columns and rows. For
${\mathbb{A}}={\mathbb{A}}^{0}=\left(a^{0}_{ij}\right)_{i,j\in\\{1,\cdots,2n\\}}\in{\rm
Sp}(n,{\mathbb{R}})$, let $\chi_{0}$ be the matrix entry such that
$|\chi_{0}|=\max_{i,j}\left|a^{0}_{ij}\right|$. In light of the matrix norm
definition (1.19), it follows that
$\frac{|{\mathbb{A}}|}{4n^{2}}\leq|\chi_{0}|\leq|{\mathbb{A}}|$. By
multiplying $\mathbb{J}_{n}$ or $-\mathbb{J}_{n}$ (consisting of the product
of $n$ matrices in ES(1)) to each side of ${\mathbb{A}}^{0}$ at most once,
which entails exchanges among the four $n\times n$ blocks of
${\mathbb{A}}^{0}$ with a possible change of sign, we can obtain a new
symplectic matrix, still denoted by ${\mathbb{A}}^{0}$, with the block form
$\mathbb{A}^{0}=\begin{pmatrix}A^{0}_{11}&A^{0}_{12}\\\
A^{0}_{21}&A^{0}_{22}\end{pmatrix}$, ensuring that $\chi_{0}$ is one of the
entries in the upper-left block $A^{0}_{11}$. Assuming that $\chi_{0}$ is the
$(i_{0},j_{0})$-entry of ${\mathbb{A}}^{0}$ for some
$i_{0},j_{0}\in\\{1,\cdots,n\\}$, we apply the ES transformations
${\mathbb{A}}^{0}\xrightarrow{{\mathbb{B}}_{1j_{0}}}\
\xrightarrow[\mathbb{B}_{1i_{0}}]{}{\mathbb{A}}_{1}^{0}$, resulting in
${\mathbb{A}}_{1}^{0}\in{\rm Sp}(n,{\mathbb{R}})$ with $\chi_{0}$ as its
$(1,1)$-entry.

Let us now focus on the first row and first column of ${\mathbb{A}}_{1}^{0}$
and symbolically represent ${\mathbb{A}}_{1}^{0}$ as follows:

$\mathbb{A}_{1}^{0}:=\left(\begin{array}[]{cccc:cccc}\chi_{0}&b_{2}&\cdots&b_{n}&g_{1}&g_{2}&\cdots&g_{n}\\\
a_{2}&&&&&&&\\\ \vdots&&*&&&&*&\\\ a_{n}&&&&&&&\\\ \hdashline
a_{n+1}&&&&&&&\\\ \vdots&&*&&&&*&\\\ a_{2n}&&&&&&&\end{array}\right),$

with “$*$” denoting other entries which are temporarily irrelevant. We then
apply $n-1$ column transformations to $\mathbb{A}_{1}^{0}$:

$\mathbb{A}_{1}^{0}\xrightarrow{\mathbb{E}_{-b_{2}/\chi_{0},2}}\cdots\xrightarrow{\mathbb{E}_{-b_{n}/\chi_{0},n}}\left(\begin{array}[]{cccc:cccc}\chi_{0}&0&\cdots&0&g_{1}+\chi_{0}^{-1}\sum\limits_{j=2}^{n}{b_{j}g_{j}}&g_{2}&\cdots&g_{n}\\\
a_{2}&&&&&&&\\\ \vdots&&*&&&*&&\\\ a_{n}&&&&&&&\\\ \hdashline
a_{n+1}&&&&&&&\\\ \vdots&&*&&&*&&\\\
a_{2n}&&&&&&&\end{array}\right)=:{\mathbb{A}}_{2}^{0},$

resulting in ${\mathbb{A}}_{2}^{0}\in{\rm Sp}(n,{\mathbb{R}})$ with the
$(1,n+1)$-matrix entry
$g^{\prime}:=g_{1}+\chi_{0}^{-1}\sum\limits_{j=2}^{n}{b_{j}g_{j}}$ satisfying
$|g^{\prime}|\leq n\|{\mathbb{A}}\|$. A subsequent column transformation:

${\mathbb{A}}_{2}^{0}\xrightarrow{\mathbb{T}_{\ell}^{(n)}}{\mathbb{A}}_{3}^{0}:=\left(\begin{array}[]{cccc:cccc}\chi_{0}&0&\cdots&0&0&0&\cdots&0\\\
a_{2}&&&&&&&\\\ \vdots&&*&&&&*&\\\ a_{n}&&&&&&&\\\ \hdashline
a_{n+1}&&&&&&&\\\ \vdots&&*&&&&*&\\\ a_{2n}&&&&&&&\end{array}\right)\quad{\rm
with}\quad\ell:=-\chi_{0}^{-1}\left(g^{\prime},g_{2},\cdots,g_{n}\right)$

yields ${\mathbb{A}}_{3}^{0}$, where every entry falls within
$\left[-2n\|{\mathbb{A}}\|,2n\|{\mathbb{A}}\|\right]$. In the first column of
${\mathbb{A}}_{3}^{0}$, we have:

$|\chi_{0}|,\ |a_{2}|,\ \cdots,\ |a_{2n}|\leq\|{\mathbb{A}}\|.$

Then, using $n-1$ row transformations on ${\mathbb{A}}^{0}_{3}$, we obtain
${\mathbb{A}}^{0}_{4}$:

${\mathbb{A}}^{0}_{3}\xrightarrow[\mathbb{F}_{-a_{2}/\chi_{0},2}]{}\cdots\xrightarrow[\mathbb{F}_{-a_{n}/\chi_{0},n}]{}\left(\begin{array}[]{cccc:cccc}\chi_{0}&0&\cdots&0&0&0&\cdots&0\\\
0&&&&&&&\\\ \vdots&&*&&&&*&\\\ 0&&&&&&&\\\ \hdashline a^{\prime}&&&&&&&\\\
a_{n+2}&&&&&&&\\\ \vdots&&*&&&&*&\\\
a_{2n}&&&&&&&\end{array}\right)=:{\mathbb{A}}^{0}_{4}$

with $a^{\prime}:=a_{n+1}+\chi_{0}^{-1}\sum\limits_{j=2}^{n}a_{j}a_{n+j}$. All
entries of ${\mathbb{A}}^{0}_{4}$ are within
$\left[-4n^{2}\|{\mathbb{A}}\|,4n^{2}\|{\mathbb{A}}\|\right]$. A further row
transformation:

${\mathbb{A}}^{0}_{4}\xrightarrow[\mathbb{S}_{\tau}]{}\left(\begin{array}[]{cccc:cccc}\chi_{0}&0&\cdots&0&0&0&\cdots&0\\\
0&&&&&&&\\\ \vdots&&*&&&&*&\\\ 0&&&&&&&\\\ \hdashline 0&&&&&&&\\\ 0&&&&&&&\\\
\vdots&&*&&&&*&\\\ 0&&&&&&&\end{array}\right)=:{\mathbb{A}}_{5}^{0}\quad{\rm
with}\quad\tau:=-\chi_{0}^{-1}\left(a^{\prime},a_{n+2},\cdots,a_{2n}\right),$

leads to ${\mathbb{A}}_{5}^{0}$, where $\chi_{0}^{-1}$ is the sole non-
vanishing entry in both the $(n+1)^{\rm th}$ row and column, i.e.,

${\mathbb{A}}_{5}^{0}=\left(\begin{array}[]{cccc:cccc}\chi_{0}&0&\cdots&0&0&0&\cdots&0\\\
0&&&&0&&&\\\ \vdots&&*&&\vdots&&*&\\\ 0&&&&0&&&\\\ \hdashline
0&0&\cdots&0&\chi_{0}^{-1}&0&\cdots&0\\\ 0&&&&0&&&\\\ \vdots&&*&&\vdots&&*&\\\
0&&&&0&&&\end{array}\right).$ (3.14)

Indeed, by writing ${\mathbb{A}}_{5}^{0}\in{\rm Sp}(n,{\mathbb{R}})$ in the
forms of column vectors and row vectors, i.e.,

${\mathbb{A}}_{5}^{0}=:(\xi_{1},\cdots,\xi_{n},\eta_{1},\cdots,\eta_{n})=:(\alpha_{1},\cdots,\alpha_{n},\beta_{1},\cdots,\beta_{n})^{*},$

by the facts that $\langle\xi_{1},\mathbb{J}\xi_{j}\rangle=0$,
$\langle\xi_{1},\mathbb{J}\eta_{j}\rangle=\delta_{ij}$, and
$({\mathbb{A}}_{5}^{0})^{*}\in{\rm Sp}(n,{\mathbb{R}})$, one obtains
$\beta_{1}=\chi_{0}^{-1}{\bf e}_{n+1}$ and $\eta_{1}=\chi_{0}^{-1}{\bf
e}_{n+1}$.

Now, from (3.14), we observe that ${\mathbb{A}}_{5}^{0}$ is already in a
block-diagonal form. If $|\chi_{0}|<1$, then by multiplying
${\mathcal{J}}_{1}$ and $-{\mathcal{J}}_{1}$ to each side of
${\mathbb{A}}_{5}^{0}$ respectively, we can facilitate the exchanges between
$\chi_{0}$ and $\chi_{0}^{-1}$, thereby positioning $\chi_{0}^{-1}$ as the
$(1,1)$-entry. Consequently, it can be seen that, through a maximum of
$(4n+4)$ ES transformations, ${\mathbb{A}}={\mathbb{A}}^{0}$ is converted into
the first block-diagonal form:

${\mathbb{A}}^{0}_{6}={\mathbb{A}}_{1}\bigoplus{\mathbb{A}}^{1}\quad{\rm
with}\quad{\mathbb{A}}_{1}=\left(\begin{array}[]{cc}\zeta_{0}&0\\\
0&\zeta_{0}^{-1}\end{array}\right),\quad{\mathbb{A}}^{1}\in{\rm
Sp}(n-1,{\mathbb{R}}),$

where $\zeta_{0}=\chi_{0}$ or $\chi_{0}^{-1}$ and satisfies
$\max\left\\{\frac{\|{\mathbb{A}}\|}{4n^{2}},1\right\\}\leq|\zeta_{0}|\leq
4n^{2}\|{\mathbb{A}}\|=M_{1}$. Furthermore, every entry of ${\mathbb{A}}^{1}$,
corresponding to the implicit entries denoted by “$*$” in (3.14), falls within
$\left[-4n^{2}\|{\mathbb{A}}\|,4n^{2}\|{\mathbb{A}}\|\right]=[-M_{1},M_{1}]$.

$\bullet$ Successive ES block-diagonalisations.

In pursuit of the ES diagonalization (3.13), we repeatedly apply the ES block-
diagonalization process to reduce the dimension of the non-diagonal block.

Suppose that ${\mathbb{A}}$ has been transformed into
${\mathbb{A}}_{n-m}\bigoplus{\mathbb{A}}^{n-m}$, for $1\leq m\leq n-1$, where

${\mathbb{A}}_{n-m}:={\rm
diag}\left\\{\zeta_{0},\cdots,\zeta_{n-m-1},\zeta_{0}^{-1},\cdots,\zeta_{n-m-1}^{-1}\right\\},$

with $\zeta_{j}$ satisfying $1\leq|\zeta_{j}|\leq M_{j+1}$ for
$j=0,\cdots,n-m-1$. Here, ${\mathbb{A}}^{n-m}\in{\rm Sp}(m,{\mathbb{R}})$ has
entries in the range of $[-M_{n-m},M_{n-m}]$. Following the previously
outlined ES block-diagonalization process, through at most $(4m+4)$ ES
transformations, ${\mathbb{A}}^{n-m}$ is converted to
$\left(\begin{array}[]{cc}\zeta_{n-m}&0\\\
0&\zeta_{n-m}^{-1}\end{array}\right)\bigoplus{\mathbb{A}}^{n-m+1}$, where
$\zeta_{n-m}$ is either the largest matrix entry of ${\mathbb{A}}^{n-m}$ or
its inverse, such that $1\leq|\zeta_{n-m}|\leq 4m^{2}M_{n-m}=M_{n-m+1}$. The
matrix ${\mathbb{A}}^{n-m+1}\in{\rm Sp}(m-1,{\mathbb{R}})$ has every entry
within $\left[-M_{n-m+1},M_{n-m+1}\right]$. Subsequently, ${\mathbb{A}}$ is
transformed into

${\mathbb{A}}_{n-m+1}\bigoplus{\mathbb{A}}^{n-m+1}\quad{\rm
with}\quad{\mathbb{A}}_{n-m+1}:={\rm
diag}\left\\{\zeta_{0},\cdots,\zeta_{n-m},\zeta_{0}^{-1},\cdots,\zeta_{n-m}^{-1}\right\\}.$

Notably, in the final state when $m=1$, ${\mathbb{A}}$ is transformed into
${\bf\Lambda}$ as specified in (3.13). Furthermore, there are at most
$\sum_{m=1}^{n}(4m+4)=(2n^{2}+6n)$ ES transformations between ${\mathbb{A}}$
and ${\bf\Lambda}$.∎

By synthesizing Proposition 3.6, Lemma 3.4 (specifically, (3.4)), and Lemma
3.5, we establish the bounds (3.2) in Proposition 3.2 for the case where
$s>0$.

Moreover, (3.13) in Proposition 3.6 suggests that

${\mathbb{A}}=\left(\prod^{k_{L}}_{l=1}L_{l}^{-1}\right){\bf\Lambda}\left(\prod^{1}_{j=k_{R}}R_{j}^{-1}\right).$
(3.15)

Applying (2.10) and Lemma 3.5, we deduce that

$\|{\mathcal{M}}({\mathbb{A}})u\|_{s}=\left\|\prod^{1}_{l=k_{L}}{\mathcal{M}}\left(L_{l}^{-1}\right){\mathcal{M}}({\bf\Lambda})\prod^{1}_{j=k_{R}}{\mathcal{M}}\left(R_{j}^{-1}\right)u\right\|_{s}\simeq\left\|{\mathcal{M}}({\bf\Lambda})\prod^{1}_{j=k_{R}}{\mathcal{M}}\left(R_{j}^{-1}\right)u\right\|_{s}.\
$

Let $\tilde{u}:=\prod_{j}{\mathcal{M}}\left(R_{j}^{-1}\right)u$. For
$u\in{\mathcal{H}}^{s}\setminus\\{0\\}$, Lemma 3.5 ensures that
$\tilde{u}\in{\mathcal{H}}^{s}\setminus\\{0\\}$ and
$\|X_{1}^{s}\tilde{u}\|_{L^{2}}\gtrsim_{u}1$. Therefore, in conjunction with
(3.5) in Lemma 3.4, it follows that

$\|{\mathcal{M}}({\mathbb{A}})u\|_{s}\simeq\|{\mathcal{M}}({\bf\Lambda})\tilde{u}\|_{s}\gtrsim|\zeta_{0}|^{s}\|X_{1}^{s}\tilde{u}\|_{L^{2}}\gtrsim_{u}\|{\mathbb{A}}\|^{s},$

which leads to the derivation of the lower bound (3.3).

####  3.1.4. Proof of (3.2) for $s<0$.

Fix $s<0$. Given $\mathbb{A}\in{\rm Sp}(n,{\mathbb{R}})$, as per (2.11), we
have:

$\left\langle{\mathcal{M}}(\mathbb{A})u,v\right\rangle=\left\langle
u,{\mathcal{M}}\left(\mathbb{A}^{-1}\right)v\right\rangle,\qquad\forall\
u\in{\mathcal{H}}^{s},\ v\in{\mathcal{H}}^{-s}.$ (3.16)

Considering that (3.2) is valid for ${\mathbb{A}}^{-1}\in{\rm
Sp}(n,{\mathbb{R}})$ and $-s>0$, it follows that:

$\|{\mathcal{M}}(\mathbb{A})u\|_{s}=\sup_{\|v\|_{-s}\leq
1}|\langle{\mathcal{M}}(\mathbb{A})u,v\rangle|\leq\sup_{\|v\|_{-s}\leq
1}\|u\|_{s}\left\|{\mathcal{M}}\left(\mathbb{A}^{-1}\right)v\right\|_{-s}\lesssim\|{\mathbb{A}}\|^{-s}\|u\|_{s}.$
(3.17)

Furthermore, for $u\in{\mathcal{H}}^{s}$ and $v\in{\mathcal{H}}^{-s}$,
equality (3.16) implies:

$\left\langle{\mathcal{M}}\left(\mathbb{A}^{-1}\right){\mathcal{M}}(\mathbb{A})u,v\right\rangle=\left\langle{\mathcal{M}}(\mathbb{A})u,{\mathcal{M}}(\mathbb{A})v\right\rangle=\left\langle
u,{\mathcal{M}}\left(\mathbb{A}^{-1}\right){\mathcal{M}}(\mathbb{A})v\right\rangle,$

which equals to $\langle u,v\rangle$, considering the fact that
${\mathcal{M}}\left(\mathbb{A}^{-1}\right){\mathcal{M}}(\mathbb{A})={\rm Id}$
on ${\mathcal{H}}^{-s}$ for $-s>0$. Thus, for $u\in{\mathcal{H}}^{s}$ with
$s<0$, we have
$\left\|{\mathcal{M}}\left(\mathbb{A}^{-1}\right){\mathcal{M}}(\mathbb{A})u\right\|_{s}=\|u\|_{s}$.
As demonstrated in (3.17),

$\|u\|_{s}=\left\|{\mathcal{M}}\left(\mathbb{A}^{-1}\right){\mathcal{M}}(\mathbb{A})u\right\|_{s}\lesssim\|{\mathbb{A}}\|^{-s}\|{\mathcal{M}}(\mathbb{A})u\|_{s}.$
(3.18)

By combining (3.17) and (3.18), we establish (3.2) for $s<0$.

###  3.2. Bounds of Sobolev norms under Schrödinger representation

######  Proposition 3.7.

For $w\in{\mathbb{R}}^{2n}$, for $u\in{\mathcal{H}}^{s}$, $s\in{\mathbb{R}}$,
$\|w\|^{-|s|}\|u\|_{s}\lesssim\|\rho(w)u\|_{s}\lesssim\|w\|^{|s|}\|u\|_{s}$.

The proof of this proposition is more straightforward than that of Proposition
3.2. We defer it to Appendix B for brevity.

######  Corollary 3.8.

Given $w(\cdot)\in C_{b}^{0}({\mathbb{R}},{\mathbb{R}}^{2n})$, for
$s\in{\mathbb{R}}$ and $t\in{\mathbb{R}}$, we have

$\|w\|_{C_{b}^{0}}^{-|s|}\|u\|_{s}\lesssim\|\rho(w(t))u\|_{s}\lesssim\|w\|_{C_{b}^{0}}^{|s|}\|u\|_{s},\qquad\forall\
u\in{\mathcal{H}}^{s}.$

##  4\. Classical-quantum correspondence for quadratic Hamiltonians

In this section, we will elucidate the relationship between reducibility in
classical quadratic Hamiltonian systems and the corresponding Hamiltonian
PDEs.

Initially, we introduce the following preparatory lemmas.

######  Lemma 4.1.

Given $a>0$, for $\mathbb{A}(\cdot)\in C^{1}(]-a,a[,{\rm Sp}(n,{\mathbb{R}}))$
satisfying

$\mathbb{A}(0)=I_{2n},\qquad\left.\frac{d}{d\tau}\mathbb{A}(\tau)\right|_{\tau=0}=0,$
(4.1)

we have that, for $s\in{\mathbb{R}}$,

$\displaystyle\left.\partial_{\tau}{\mathcal{M}}(\mathbb{A}(\tau))u\right|_{\tau=0}=0,\qquad\forall\
u\in{\mathcal{H}}^{s}.$ (4.2)

###### Proof.

If $\mathbb{A}(\cdot)=\begin{pmatrix}A(\cdot)&0\\\
0&A(\cdot)^{*-1}\end{pmatrix}$ with $A(\cdot)\in C^{1}(]-a,a[,{\rm
GL}(n,{\mathbb{R}}))$ satisfying

$A(0)=A(0)^{-1}=I_{n},\qquad\left.\frac{d}{d\tau}A(\tau)\right|_{\tau=0}=\left.\frac{d}{d\tau}A(\tau)^{-1}\right|_{\tau=0}=0,$

then, for $|\tau|$ sufficiently small, we have
$\left({\mathcal{M}}({\mathbb{A}}(\tau))u\right)(x)=(\mathrm{det}\
A(\tau))^{-\frac{1}{2}}u\left(A(\tau)^{-1}x\right)$, and

$\displaystyle\left.\partial_{\tau}\left((\mathrm{det}\
A(\tau))^{-\frac{1}{2}}u\left(A(\tau)^{-1}x\right)\right)\right|_{\tau=0}$
$\displaystyle=$ $\displaystyle\left.\frac{d}{d\tau}(\mathrm{det}\
A(\tau))^{-\frac{1}{2}}\right|_{\tau=0}u(x)+\left.\partial_{\tau}u\left(A(\tau)^{-1}x\right)\right|_{\tau=0}$
$\displaystyle=$ $\displaystyle 2\pi{\rm
i}\left\langle\left.\frac{d}{d\tau}A(\tau)^{-1}\right|_{\tau=0}x,(Du)(x)\right\rangle\
=\ 0.$

If $\mathbb{A}(\cdot)=\begin{pmatrix}I_{n}&0\\\
C(\cdot)&I_{n}\end{pmatrix}\in{\rm Sp}(n,{\mathbb{R}})$ with
$C(\cdot)=C(\cdot)^{*}\in C^{1}((-a,a),{\rm gl}(n,{\mathbb{R}}))$ satisfying

$C(0)=0,\qquad\left.\frac{d}{d\tau}C(\tau)\right|_{\tau=0}=0,$

then, for $|\tau|$ sufficiently small, we have
$\left({\mathcal{M}}({\mathbb{A}}(\tau))u\right)(x)=e^{-\pi{\rm i}\langle
x,C(\tau)x\rangle}u(x)$, we obtain (4.2) since

$\left.\partial_{\tau}\left(e^{-\pi{\rm i}\langle
x,C(\tau)x\rangle}u(x)\right)\right|_{\tau=0}=-\pi{\rm i}\left\langle
x,\left.\frac{d}{d\tau}C(\tau)\right|_{\tau=0}x\right\rangle u(x)=0.$

For the general $\mathbb{A}(\cdot)=\begin{pmatrix}A(\cdot)&B(\cdot)\\\
C(\cdot)&F(\cdot)\end{pmatrix}\in C^{1}(]-a,a[,{\rm Sp}(n,{\mathbb{R}}))$
satisfying (4.1), it follows that

$\displaystyle A(0)=F(0)=I_{n},\qquad B(0)=C(0)=0,$ (4.3)
$\displaystyle\left.\frac{d}{d\tau}A(\tau)\right|_{\tau=0}=\left.\frac{d}{d\tau}B(\tau)\right|_{\tau=0}=\left.\frac{d}{d\tau}C(\tau)\right|_{\tau=0}=\left.\frac{d}{d\tau}F(\tau)\right|_{\tau=0}=0.$
(4.4)

Considering $A(0)=I_{n}$, $\mathbb{A}(\tau)$ can be decomposed for
sufficiently small $|\tau|$ as

$\mathbb{A}(\tau)=\begin{pmatrix}I&0\\\
C(\tau)A(\tau)^{-1}&I\end{pmatrix}\begin{pmatrix}A(\tau)&0\\\
0&A(\tau)^{*-1}\end{pmatrix}{\mathbb{J}}_{n}\begin{pmatrix}I&0\\\
-A(\tau)^{-1}B(\tau)&I\end{pmatrix}{\mathbb{J}}_{n}^{-1},$

and from the above two elementary cases, it follows from the conditions (4.3)
and (4.4) that

$\displaystyle\left.\partial_{\tau}{\mathcal{M}}(\mathbb{A}(\tau))u\right|_{\tau=0}$
$\displaystyle=$
$\displaystyle\left.\partial_{\tau}{\mathcal{M}}\begin{pmatrix}I&0\\\
C(\tau)A^{-1}(\tau)&I\end{pmatrix}u\right|_{\tau=0}+\left.\partial_{\tau}{\mathcal{M}}\begin{pmatrix}A(\tau)&0\\\
0&A(\tau)^{*-1}\end{pmatrix}u\right|_{\tau=0}$
$\displaystyle+\,\left.{\mathcal{M}}({\mathbb{J}}_{n})\partial_{\tau}{\mathcal{M}}\begin{pmatrix}I&0\\\
-A(\tau)^{-1}B(\tau)&I\end{pmatrix}{\mathcal{M}}({\mathbb{J}}_{n}^{-1})u\right|_{\tau=0}\
=\ 0.\qed$

######  Lemma 4.2.

Given $Y(\cdot)\in C^{1}({\mathbb{R}},{\rm Sp}(n,{\mathbb{R}}))$ 666It is easy
to verify that $Y^{-1}(\cdot)Y^{\prime}(\cdot)\in C^{0}({\mathbb{R}},{\rm
sp}(n,{\mathbb{R}}))$ for $Y(\cdot)\in C^{1}({\mathbb{R}},{\rm
Sp}(n,{\mathbb{R}}))$. and $u\in{\mathcal{H}}^{s}$, $s\in{\mathbb{R}}$, we
have

$\displaystyle\frac{1}{2\pi{\rm
i}}\partial_{t}{\mathcal{M}}(Y(t))u={\mathcal{M}}(Y(t)){\mathcal{Q}}_{Y^{-1}(t)Y^{\prime}(t)}(Z)u\in{\mathcal{H}}^{s-2},\quad\forall\
t\in{\mathbb{R}}.$ (4.5)

######  Remark 8.

Lemma 4.2 is perceived as an extension of Theorem 2.5, wherein the time-
parameterized infinitesimal variant of the Metaplectic representation on
${\mathcal{H}}^{s}({\mathbb{R}}^{n})$ is considered, applicable for any
$s\in{\mathbb{R}}$.

###### Proof.

Recalling (2.12) and Remark 6, we have for $t\in{\mathbb{R}}$,

${\mathcal{Q}}_{Y^{-1}(t)Y^{\prime}(t)}(Z)u=\frac{1}{2\pi{\rm
i}}\left.\partial_{\tau}{\mathcal{M}}\left(e^{\tau
Y^{-1}(t)Y^{\prime}(t)}\right)\right|_{\tau=0}u\in{\mathcal{H}}^{s-2},\qquad
u\in{\mathcal{H}}^{s},$

then (4.5) is equivalent to

$\left.\partial_{\tau}{\mathcal{M}}(Y^{-1}(t)Y(t+\tau))\right|_{\tau=0}u=\left.\partial_{\tau}{\mathcal{M}}\left(e^{\tau
Y^{-1}(t)Y^{\prime}(t)}\right)\right|_{\tau=0}u.$ (4.6)

Noting that, for $t,\tau\in{\mathbb{R}}$,

$\displaystyle{\mathcal{M}}\left(Y^{-1}(t)Y(t+\tau)\right)-{\mathcal{M}}\left(e^{\tau
Y^{-1}(t)Y^{\prime}(t)}\right)$ $\displaystyle=$
$\displaystyle\left({\mathcal{M}}\left(Y^{-1}(t)Y(t+\tau)e^{-\tau
Y^{-1}(t)Y^{\prime}(t)}\right)-{\rm Id}\right){\mathcal{M}}\left(e^{\tau
Y^{-1}(t)Y^{\prime}(t)}\right),$

for obtaining (4.6), it is sufficient to show

$\left.\partial_{\tau}{\mathcal{M}}\left(Y^{-1}(t)Y(t+\tau)e^{-\tau
Y^{-1}(t)Y^{\prime}(t)}\right)\right|_{\tau=0}u=0,\qquad\forall\
u\in{\mathcal{H}}^{s}.$ (4.7)

With any fixed $t\in{\mathbb{R}}$, define
$\mathbb{A}_{t}(\tau):=Y^{-1}(t)Y(t+\tau)e^{-\tau Y^{-1}(t)Y^{\prime}(t)}$,
which satisfies

${\mathbb{A}}_{t}(\cdot)\in C^{1}({\mathbb{R}},{\rm
Sp}(n,{\mathbb{R}})),\qquad{\mathbb{A}}_{t}(0)=I_{2n},\qquad\left.\frac{d}{d\tau}\mathbb{A}_{t}(\tau)\right|_{\tau=0}=0.$

Applying Lemma 4.1 to ${\mathbb{A}}_{t}(\cdot)$, we obtain (4.7).∎

The subsequent proposition can be viewed as analogous to Proposition 2.7 in
[5].

######  Lemma 4.3.

Given two Hamiltonian systems

$z_{j}^{\prime}={\mathcal{A}}_{j}(t)z_{j}+\ell_{j}(t),\quad{\mathcal{A}}_{j}(\cdot)\in
C_{b}^{0}(\mathbb{R},{\rm sp}(n,{\mathbb{R}})),\quad\ell_{j}(\cdot)\in
C_{b}^{0}({\mathbb{R}},{\mathbb{R}}^{2n}),\qquad j=1,2,$ (4.8)

if there is $S(\cdot)\in C_{b}^{1}({\mathbb{R}},{\rm Sp}(n,{\mathbb{R}}))$
such that $z_{1}(t)=S(t)z_{2}(t)$, then, for the two Hamiltonian PDEs

$\frac{1}{2\pi{\rm
i}}\partial_{t}\psi_{j}=\left({\mathcal{Q}}_{{\mathcal{A}}_{j}(t)}(Z)+{\mathcal{L}}_{\ell_{j}(t)}(Z)\right)\psi_{j},\qquad
j=1,2,$ (4.9)

we have the conjugation under the $L^{2}-$unitary transformation
$\psi_{1}(t)={\mathcal{M}}(S(t))\psi_{2}(t)$.

######  Remark 9.

Regarding the two Hamiltonian PDEs in (4.9), if the initial conditions
$\psi_{1}(0)$ and $\psi_{2}(0)$ are both in ${\mathcal{H}}^{s}$, for $s\geq
0$, then the solutions are globally well-posed in ${\mathcal{H}}^{s}$.
Furthermore, as $S(\cdot)\in C_{b}^{0}({\mathbb{R}},{\rm
Sp}(n,{\mathbb{R}}))$, in accordance with Corollary 3.3 \- (i), the norms
$\|\psi_{1}(t)\|_{s}$ and $\|\psi_{2}(t)\|_{s}$ demonstrate parallel growth
rates with respect to time $t$ as $|t|\to\infty$. This is evident from the
relation:

$\|S\|_{C_{b}^{0}}^{-s}\|\psi_{2}(t)\|_{s}\lesssim\|\psi_{1}(t)\|_{s}\lesssim\|S\|_{C_{b}^{0}}^{s}\|\psi_{2}(t)\|_{s}.$

###### Proof.

The conjugation $z_{1}(t)=S(t)z_{2}(t)$ between two Hamiltonian systems in
(4.8) means that

$S^{\prime}(t)={\mathcal{A}}_{1}(t)S(t)-S(t){\mathcal{A}}_{2}(t),\qquad\ell_{1}(t)=S(t)\ell_{2}(t).$
(4.10)

Since $S(\cdot)\in C_{b}^{1}({\mathbb{R}},{\rm Sp}(n,{\mathbb{R}}))$, from
Proposition 3.2 and Lemma 4.2, we have

${\mathcal{M}}(S(t))\in
B({\mathcal{H}}^{s}),\qquad\frac{d}{dt}{\mathcal{M}}(S(t))\in
B({\mathcal{H}}^{s},{\mathcal{H}}^{s-2}),\qquad t\in{\mathbb{R}}.$

To prove the conjugation $\psi_{1}(t)={\mathcal{M}}(S(t))\psi_{2}(t)$ between
two equations in (4.9), it is sufficient to prove, in
$B({\mathcal{H}}^{s},{\mathcal{H}}^{s-2})$, that

$\displaystyle\frac{1}{2\pi{\rm
i}}{\mathcal{M}}\left(S(t)\right)^{-1}\frac{d}{dt}{\mathcal{M}}(S(t))$
$\displaystyle=$
$\displaystyle{\mathcal{M}}\left(S(t)\right)^{-1}{\mathcal{Q}}_{{\mathcal{A}}_{1}(t)}(Z){\mathcal{M}}(S(t))-{\mathcal{Q}}_{{\mathcal{A}}_{2}(t)}(Z),$
(4.11) $\displaystyle{\mathcal{L}}_{\ell_{2}(t)}(Z)$ $\displaystyle=$
$\displaystyle{\mathcal{M}}(S(t))^{-1}{\mathcal{L}}_{\ell_{1}(t)}(Z){\mathcal{M}}(S(t)).$
(4.12)

According to Theorem 2.5 and (4.10), we have that

$\displaystyle{\mathcal{M}}\left(S(t)\right)^{-1}{\mathcal{Q}}_{{\mathcal{A}}_{1}(t)}(Z){\mathcal{M}}(S(t))$
$\displaystyle=$ $\displaystyle\frac{1}{2\pi{\rm
i}}{\mathcal{M}}\left(S(t)^{-1}\right)\left.\partial_{\tau}{\mathcal{M}}\left(e^{\tau{\mathcal{A}}_{1}(t)}\right)\right|_{\tau=0}{\mathcal{M}}\left(S(t)\right)$
$\displaystyle=$ $\displaystyle\frac{1}{2\pi{\rm
i}}\left.\partial_{\tau}{\mathcal{M}}\left(e^{\tau
S(t)^{-1}{\mathcal{A}}_{1}(t)S(t)}\right)\right|_{\tau=0}$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi{\rm
i}}\left.\partial_{\tau}{\mathcal{M}}\left(e^{\tau\left(S(t)^{-1}S^{\prime}(t)+{\mathcal{A}}_{2}(t)\right)}\right)\right|_{\tau=0}$
$\displaystyle=$
$\displaystyle{\mathcal{Q}}_{S(t)^{-1}S^{\prime}(t)+{\mathcal{A}}_{2}(t)}(Z)\
=\
{\mathcal{Q}}_{S(t)^{-1}S^{\prime}(t)}(Z)+{\mathcal{Q}}_{{\mathcal{A}}_{2}(t)}(Z),$

which means that the right hand side of (4.11) equals to
${\mathcal{Q}}_{S(t)^{-1}S^{\prime}(t)}(Z)$. Combining with Lemma 4.2, we
obtain (4.11). As for (4.12), notice that the property (2.9) of the
Schrödinger representation, as well as the second equality in (4.10), we have

$\displaystyle{\mathcal{M}}(S(t))^{-1}{\mathcal{L}}_{\ell_{1}(t)}(Z){\mathcal{M}}(S(t))$
$\displaystyle=$ $\displaystyle\frac{1}{2\pi{\rm
i}}{\mathcal{M}}\left(S(t)^{-1}\right)\left.\partial_{\tau}\rho\left(\tau\ell_{1}(t)\right)\right|_{\tau=0}{\mathcal{M}}(S(t))$
$\displaystyle=$ $\displaystyle\frac{1}{2\pi{\rm
i}}\left.\partial_{\tau}\left({\mathcal{M}}\left(S(t)^{-1}\right)\rho\left(\tau\ell_{1}(t)\right){\mathcal{M}}(S(t))\right)\right|_{\tau=0}$
$\displaystyle=$ $\displaystyle\frac{1}{2\pi{\rm
i}}\left.\partial_{\tau}\rho\left(\tau
S(t)^{-1}\ell_{1}(t)\right)\right|_{\tau=0}\ =\
{\mathcal{L}}_{\ell_{2}(t)}(Z).\qed$

######  Lemma 4.4.

Given two Hamiltonian systems

$z_{j}^{\prime}={\mathcal{A}}z_{j}+\ell_{j}(t),\quad{\mathcal{A}}\in{\rm
sp}(n,{\mathbb{R}}),\quad\ell_{j}(\cdot)\in
C_{b}^{0}({\mathbb{R}},{\mathbb{R}}^{2n}),\qquad j=1,2,$ (4.13)

if there is $w(\cdot)\in C_{b}^{1}(\mathbb{R},{\mathbb{R}}^{2n})$ such that
$z_{1}(t)=z_{2}(t)+w(t)$, then the Hamiltonian PDE

$\frac{1}{2\pi{\rm
i}}\partial_{t}\psi_{1}=\left({\mathcal{Q}}_{{\mathcal{A}}}(Z)+{\mathcal{L}}_{\ell_{1}(t)}(Z)\right)\psi_{1}$
(4.14)

is conjugated, via the $L^{2}-$unitary transformation
$\psi_{1}=\rho(w(t))\psi_{2}$, to

$\frac{1}{2\pi{\rm
i}}\partial_{t}\psi_{2}=\left({\mathcal{Q}}_{{\mathcal{A}}}(Z)+{\mathcal{L}}_{\ell_{2}(t)}(Z)+{\mathcal{C}}(t)\right)\psi_{2},\qquad{\mathcal{C}}(t):=-\frac{1}{2}\langle
w(t),{\mathbb{J}}_{n}(\ell_{1}(t)+\ell_{2}(t))\rangle.$ (4.15)

######  Remark 10.

In parallel with Remark 9, the condition that $w(\cdot)\in
C_{b}^{0}(\mathbb{R},{\mathbb{R}}^{2n})$ enables us to employ Corollary 3.8.
From this, we can infer that the growth rates of the solutions to Eq. (4.14)
and Eq. (4.15) are comparable.

######  Remark 11.

Eq. (4.15) is transformable into the equation

$\frac{1}{2\pi{\rm
i}}\partial_{t}\tilde{\psi}_{2}=\left({\mathcal{Q}}_{{\mathcal{A}}}(Z)+{\mathcal{L}}_{\ell_{2}(t)}(Z)\right)\tilde{\psi}_{2}$

via the scalar transformation $\psi_{2}=e^{2\pi{\rm
i}\int_{0}^{t}{\mathcal{C}}(\tau)d\tau}\tilde{\psi}_{2}$. Consequently, the
scalar part is often omitted when considering the norm of the solution to Eq.
(4.15).

###### Proof.

The conjugation $z_{1}(t)=z_{2}(t)+w(t)$ between the two systems in (4.13)
implies that

$\ell_{2}(t)={\mathcal{B}}w(t)-w^{\prime}(t)+\ell_{1}(t),\qquad\forall\
t\in{\mathbb{R}}.$ (4.16)

For $w(\cdot)\in C_{b}^{1}({\mathbb{R}},{\mathbb{R}}^{2n})$, Propositions 2.3
and 3.7 lead to

$\rho(w(t))\in B({\mathcal{H}}^{s}),\quad\partial_{t}\rho(w(t))\in
B({\mathcal{H}}^{s},{\mathcal{H}}^{s-1}),\qquad t\in{\mathbb{R}}.$

To establish the conjugation $\psi_{1}(t)=\rho(w(t))\psi_{2}(t)$ between Eq.
(4.14) and Eq. (4.15), it suffices to demonstrate, within
$B({\mathcal{H}}^{s},{\mathcal{H}}^{s-2})$, that

$\rho(w(t))^{-1}\left({\mathcal{Q}}_{{\mathcal{B}}}(Z)\rho(w(t))-\frac{1}{2\pi{\rm
i}}\frac{d}{dt}\rho(w(t))+{\mathcal{L}}_{\ell_{1}(t)}(Z)\rho(w(t))\right)={\mathcal{Q}}_{{\mathcal{B}}}(Z)+{\mathcal{L}}_{\ell_{2}(t)}(Z)+{\mathcal{C}}(t).$

Since, through (2.9), we have

$\displaystyle\rho(w(t))^{-1}{\mathcal{Q}}_{{\mathcal{B}}}(Z)\rho(w(t))$
$\displaystyle=$ $\displaystyle\frac{1}{2\pi{\rm
i}}\left.\partial_{\tau}\left(\rho(w(t))^{-1}{\mathcal{M}}\left(e^{\tau{\mathcal{B}}}\right)\rho(w(t)){\mathcal{M}}\left(e^{-\tau{\mathcal{B}}}\right){\mathcal{M}}\left(e^{\tau{\mathcal{B}}}\right)\right)\right|_{\tau=0}$
$\displaystyle=$ $\displaystyle\frac{1}{2\pi{\rm
i}}\left.\partial_{\tau}\left(\rho(w(t))^{-1}\rho\left(e^{\tau{\mathcal{B}}}w(t)\right){\mathcal{M}}\left(e^{\tau{\mathcal{B}}}\right)\right)\right|_{\tau=0}$
$\displaystyle=$ $\displaystyle\frac{1}{2\pi{\rm
i}}\left.\left(\rho(w(t))^{-1}\rho\left(e^{\tau{\mathcal{B}}}w(t)\right)\right)\right|_{\tau=0}+\frac{1}{2\pi{\rm
i}}\left.\frac{d}{d\tau}{\mathcal{M}}\left(e^{\tau{\mathcal{B}}}\right)\right|_{\tau=0}$
$\displaystyle=$ $\displaystyle\frac{1}{2\pi{\rm
i}}\left.\rho(w(t))^{-1}\partial_{\tau}\rho(e^{\tau{\mathcal{B}}}w(t))\right|_{\tau=0}+{\mathcal{Q}}_{\mathcal{B}}(Z),$

it remains to show that

$\rho(w(t))\left({\mathcal{L}}_{\ell_{2}(t)}(Z)+{\mathcal{C}}(t)\right)=\frac{1}{2\pi{\rm
i}}\left.\partial_{\tau}\rho\left(e^{\tau{\mathcal{B}}}w(t)\right)\right|_{\tau=0}-\frac{1}{2\pi{\rm
i}}\frac{d}{dt}\rho(w(t))+{\mathcal{L}}_{\ell_{1}(t)}(Z)\rho(w(t)).$ (4.17)

Indeed, following Proposition 2.1 and 2.2, the terms in the right hand side of
(4.17) satisfy that

$\displaystyle{\mathcal{L}}_{\ell_{1}(t)}(Z)\rho(w(t))$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi{\rm
i}}\left.\partial_{\tau}\left(\rho(\tau\ell_{1}(t))\rho(w(t))\right)\right|_{\tau=0},$
$\displaystyle\frac{d}{dt}\rho(w(t))\ =\
\left.\partial_{\tau}\rho(w(t+\tau))\right|_{\tau=0}$ $\displaystyle=$
$\displaystyle\left.\partial_{\tau}\rho\left(w(t)+\tau
w^{\prime}(t)\right)\right|_{\tau=0},$
$\displaystyle\left.\partial_{\tau}\rho\left(e^{\tau{\mathcal{B}}}w(t)\right)\right|_{\tau=0}$
$\displaystyle=$
$\displaystyle\left.\partial_{\tau}\rho\left(w(t)+\tau{\mathcal{B}}w(t)\right)\right|_{\tau=0},$

then, through (2.5) and (4.16), the right hand side of (4.17) equals to

$\displaystyle\frac{1}{2\pi{\rm
i}}\left(\left.\partial_{\tau}\left(e^{2\pi{\rm i}\tau\langle
w(t),{\mathbb{J}}_{n}\ell_{1}(t)\rangle}\rho(w(t))\rho(\tau\ell_{1}(t))\right)\right|_{\tau=0}-\left.\partial_{\tau}\left(e^{-\pi{\rm
i}\tau\langle
w(t),{\mathbb{J}}_{n}w^{\prime}(t)\rangle}\rho(w(t))\rho\left(\tau
w^{\prime}(t)\right)\right)\right|_{\tau=0}\right.$ $\displaystyle\ \ \ \ \ \
\ \ \ \ +\left.\left.\partial_{\tau}\left(e^{-\pi{\rm i}\tau\langle
w(t),{\mathbb{J}}_{n}{\mathcal{B}}w(t)\rangle}\rho(w(t))\rho\left(\tau{\mathcal{B}}w(t)\right)\right)\right|_{\tau=0}\right)$
$\displaystyle=$ $\displaystyle\frac{1}{2}\left(2\langle
w(t),{\mathbb{J}}_{n}\ell_{1}(t)\rangle+\langle
w(t),{\mathbb{J}}_{n}w^{\prime}(t)\rangle-\langle
w(t),{\mathbb{J}}_{n}{\mathcal{B}}w(t)\rangle\right)\rho(w(t))$
$\displaystyle+\,\frac{1}{2\pi{\rm
i}}\rho(w(t))\left.\partial_{\tau}\left(\rho\left(\tau{\mathcal{B}}w(t)\right)-\rho\left(\tau
w^{\prime}(t)\right)+\rho(\tau\ell_{1}(t))\right)\right|_{\tau=0}$
$\displaystyle=$ $\displaystyle-\frac{1}{2}\left\langle
w,{\mathbb{J}}_{n}({\mathcal{B}}w-w^{\prime}(t)+2\ell_{1}(t))\right\rangle\rho(w(t))+\rho(w(t))\left\langle{\mathcal{B}}w(t)-w^{\prime}(t)+\ell_{1}(t),Z\right\rangle$
$\displaystyle=$ $\displaystyle-\frac{1}{2}\left\langle
w(t),{\mathbb{J}}_{n}(\ell_{1}(t)+\ell_{2}(t))\right\rangle\rho(w(t))+\rho(w(t))\left\langle\ell_{2}(t),Z\right\rangle$
$\displaystyle=$
$\displaystyle\rho(w(t))\left({\mathcal{L}}_{\ell_{2}(t)}(Z)-\frac{1}{2}\left\langle
w(t),{\mathbb{J}}_{n}(\ell_{1}(t)+\ell_{2}(t))\right\rangle\right).$

Hence Lemma 4.4 is shown. ∎

Combining Lemma 4.3, Lemma 4.4 and Remark 11, we have

######  Proposition 4.5.

If the affine system $z^{\prime}={\mathcal{A}}(t)z+\ell(t)$ with
${\mathcal{A}}(\cdot)\in C_{b}^{0}(\mathbb{R},{\rm sp}(n,{\mathbb{R}}))$ and
$\ell(\cdot)\in C_{b}^{0}({\mathbb{R}},{\mathbb{R}}^{2n})$ is reducible, i.e.,
it is conjugated to $y^{\prime}={\mathcal{B}}y+l$, ${\mathcal{B}}\in{\rm
sp}(n,{\mathbb{R}})$, $l\in{\mathbb{R}}^{2n}$, through some transformation

$z(t)=U(t)y(t)+v(t),\qquad U(\cdot)\in C_{b}^{1}({\mathbb{R}},{\rm
Sp}(n,{\mathbb{R}})),\qquad v(\cdot)\in
C_{b}^{1}(\mathbb{R},{\mathbb{R}}^{2n}),$

then, for the two Hamiltonian PDEs

$\frac{1}{2\pi{\rm
i}}\partial_{t}\psi_{1}=\left({\mathcal{Q}}_{{\mathcal{A}}(t)}(Z)+{\mathcal{L}}_{\ell(t)}(Z)\right)\psi_{1},\qquad\frac{1}{2\pi{\rm
i}}\partial_{t}\psi_{2}=\left({\mathcal{Q}}_{{\mathcal{B}}}(Z)+{\mathcal{L}}_{l}(Z)\right)\psi_{2},$

we have $\psi_{1}(t)=e^{2\pi{\rm
i}\int_{0}^{t}{\mathcal{C}}(\tau)d\tau}\rho(v(t)){\mathcal{M}}(U(t))\psi_{2}(t)$
with for some ${\mathcal{C}}(\cdot)\in C_{b}^{0}({\mathbb{R}},{\mathbb{R}})$.

######  Remark 12.

In light of the diagram presented in Section 1.3, the aforementioned
$L^{2}$-unitary transformation can be expressed, modulo a scalar coefficient,
as $\psi_{1}(t)={\mathcal{M}}(U(t))\rho(U(t)^{-1}v(t))\psi_{2}(t)$. In fact,
according to (2.9), it follows that
${\mathcal{M}}(U(t))\rho(U(t)^{-1}v(t))=\rho(v(t)){\mathcal{M}}(U(t))$.

##  5\. Quantum symplectic normal forms – Proof of Theorem 1.1

Let us return to the time-dependent quadratic quantum Hamiltonian as presented
in (1.2). Considering Remark 11, we set aside the scalar component and
concentrate on the equation

$\frac{1}{2\pi{\rm
i}}\partial_{t}\psi=({\mathcal{Q}}_{\mathcal{A}(t)}(Z)+{\mathcal{L}}_{\ell(t)}(Z))\psi=\left(-\frac{1}{2}\langle
Z,\mathcal{A}(t)\mathbb{J}Z\rangle+\langle\ell(t),Z\rangle\right)\psi,\quad
x\in{\mathbb{R}}^{n},$ (5.1)

where $\mathcal{A}(\cdot)\in C_{b}^{0}({\mathbb{R}},{\rm sp}(n,{\mathbb{R}}))$
and $\ell(\cdot)\in C_{b}^{0}({\mathbb{R}},{\mathbb{R}}^{2n})$. Our objective
is to establish the reducibility framework for Eq. (5.1) through the
classical-quantum correspondence delineated in Section 4, thereby providing a
comprehensive categorization of quantum symplectic normal forms. Under the
assumption of reducibility for the linear system

$z^{\prime}=\mathcal{A}(t)z+\ell(t),$ (5.2)

we identify $\tilde{\mathcal{B}}\in{\rm sp}(n,{\mathbb{R}})$,
$\tilde{l}\in{\mathbb{R}}^{2n}$, and transformations $S(\cdot)\in
C_{b}^{1}({\mathbb{R}},{\rm Sp}(n,{\mathbb{R}}))$, $r(\cdot)\in
C_{b}^{1}({\mathbb{R}},{\mathbb{R}}^{2n})$, such that the system (5.2) is
conjugated to $y^{\prime}=\tilde{\mathcal{B}}y+\tilde{l}$ under the change of
variable $z=S(t)y+r(t)$. In light of Proposition 4.5, our focus is directed
towards the classification of the reduced classical system
$y^{\prime}=\tilde{\mathcal{B}}y+\tilde{l}$.

It was shown by Hörmander [35] that any Hamiltonian matrix can be conjugated
to the direct sum of the normal forms as delineated in (1.5) – (1.9). We will
now present this theorem in the context of Hamiltonian matrices.

######  Lemma 5.1.

[35] For any $\tilde{\mathcal{B}}\in{\rm sp}(n,{\mathbb{R}})$, there exist
$T\in{\rm Sp}(n,{\mathbb{R}})$ and a mutually disjoint partition
$\bigcup_{j\in\Lambda}{{\mathcal{I}}_{j}}=\\{1,\cdots,n\\}$ with
$\\#{\mathcal{I}}_{j}=m_{j}>0$ and $k_{j}\in\\{1,\cdots,5\\}$, such that

$T^{-1}\tilde{\mathcal{B}}T=\bigoplus_{j\in\Lambda}{\mathcal{B}}_{k_{j}}^{(m_{j})}.$
(5.3)

If $\\#\Lambda=1$, then $T^{-1}\tilde{\mathcal{B}}T={\mathcal{B}}_{k}^{(n)}$
with ${\mathcal{B}}_{k}^{(n)}$, $k=1,\cdots,5$, introduced in (1.5) – (1.9).
Specifically,

  * Case 1.

If $\tilde{\mathcal{B}}$ has two real eigenvalues $\pm\lambda$, $\lambda>0$,
each with multiplicity $n$, then
$T^{-1}\tilde{\mathcal{B}}T={\mathcal{B}}_{1}^{(n)}={\mathcal{B}}_{1}^{(n)}(\lambda)$.

  * Case 2.

For $n$ even, if $\tilde{\mathcal{B}}$ has four complex eigenvalues
$\pm\lambda_{1}\pm{\rm i}\lambda_{2}$, $\lambda_{1},\lambda_{2}>0$, each with
multiplicity $\frac{n}{2}$, then
$T^{-1}\tilde{\mathcal{B}}T={\mathcal{B}}_{2}^{(n)}={\mathcal{B}}_{2}^{(n)}(\lambda_{1},\lambda_{2})$.

  * Case 3.

If $\tilde{\mathcal{B}}$ has two purely imaginary eigenvalues $\pm{\rm i}\mu$,
$\mu>0$, each with multiplicity $n$, then
$T^{-1}\tilde{\mathcal{B}}T={\mathcal{B}}_{3}^{(n)}={\mathcal{B}}_{3}^{(n)}(\mu,\gamma)$.

  * Case 4.

If $\tilde{\mathcal{B}}^{2n}=0$ and $\tilde{\mathcal{B}}^{2n-1}\neq 0$, then
$T^{-1}\tilde{\mathcal{B}}T={\mathcal{B}}_{4}^{(n)}={\mathcal{B}}_{4}^{(n)}(\gamma)$.

  * Case 5.

For $n$ odd, if $\tilde{\mathcal{B}}^{n}=0$ and, for $n\geq 3$, ${\rm
Rank}(\tilde{\mathcal{B}}^{n-1})=2$, then
$T^{-1}\tilde{\mathcal{B}}T={\mathcal{B}}^{(n)}_{5}$.

######  Remark 13.

One may naturally question why ${\mathcal{B}}^{(n)}_{5}$, which remains
definable as in (1.9), is not included in the classification for the case when
$n$ is even. Indeed, as elucidated in [35], such a matrix can be conjugated to
${\mathcal{B}}^{(\frac{n}{2})}_{4}(1)\bigoplus{\mathcal{B}}^{(\frac{n}{2})}_{4}(-1)$.
This result categorizes it into the decomposable case.

Given that the full dimension $n$ is arbitrarily specified, our attention
turns to the simplified system

$y^{\prime}={\mathcal{B}}y+\tilde{l},\qquad{\mathcal{B}}={\mathcal{B}}_{k}^{(n)},\quad
k=1,\cdots,5.$ (5.4)

For $k=1,2,3$, where ${\mathcal{B}}={\mathcal{B}}^{(n)}_{k}$ is invertible,
the system (5.4) can be conjugated to $w^{\prime}={\mathcal{B}}w$ via the
translation $y=w-{\mathcal{B}}^{-1}\tilde{l}$. For $k=4,5$, with
${\mathcal{B}}={\mathcal{B}}^{(n)}_{k}$ being nilpotent, the systems

${\mathcal{B}}^{(n)}_{4}f_{4}=\tilde{l}-\langle\tilde{l},{\bf
e}_{1}\rangle{\bf
e}_{1},\qquad{\mathcal{B}}^{(n)}_{5}f_{5}=\tilde{l}-\langle\tilde{l},{\bf
e}_{1}\rangle{\bf e}_{1}-\langle\tilde{l},{\bf e}_{2n}\rangle{\bf e}_{2n},$

admit solutions $f_{4}$ and $f_{5}$, respectively. Therefore, for $k=4,5$, the
system (5.4) is conjugated to $w^{\prime}={\mathcal{B}}^{(n)}_{k}+l^{(n)}_{k}$
using the translation $y=w-f_{k}$. Here, $l^{(n)}_{k}$ is defined as
$l^{(n)}_{k}:=\left\\{\begin{array}[]{ll}\langle\tilde{l},{\bf
e}_{1}\rangle{\bf e}_{1},&k=4\\\ \langle\tilde{l},{\bf e}_{1}\rangle{\bf
e}_{1}+\langle\tilde{l},{\bf e}_{2n}\rangle{\bf
e}_{2n},&k=5\end{array}\right.$. Moreover, for the special case of $n=1$ and
$k=5$, where ${\mathcal{B}}^{(1)}_{5}=\begin{pmatrix}0&0\\\ 0&0\end{pmatrix}$,
the reduced system $w^{\prime}=l^{(1)}_{5}=\begin{pmatrix}\iota_{1}\\\
\iota_{2}\end{pmatrix}$, given that $(\iota_{1},\iota_{2})\neq(0,0)$, can be
further conjugated to
$\tilde{w}^{\prime}=\begin{pmatrix}\sqrt{\iota_{1}^{2}+\iota_{2}^{2}}\\\
0\end{pmatrix}$ under the rotation
$w=\frac{1}{\sqrt{\iota_{1}^{2}+\iota_{2}^{2}}}\begin{pmatrix}\iota_{1}&-\iota_{2}\\\
\iota_{2}&\iota_{1}\end{pmatrix}\tilde{w}$.

To summarize, the following proposition elucidates Theorem 1.1 in light of
Proposition 4.5.

######  Proposition 5.2.

(Classical normal forms) If the system (5.2) is reducible, then there exists
$U(\cdot)\in C_{b}^{1}({\mathbb{R}},{\rm Sp}(n,{\mathbb{R}}))$ and
$v(\cdot)\in C_{b}^{1}({\mathbb{R}},{\mathbb{R}}^{2n})$ such that, via the
transformation $z(t)=U(t)w(t)+v(t)$, the system (5.2) is conjugated to

$w^{\prime}={\mathcal{B}}^{(n)}_{k}w+l^{(n)}_{k},\qquad
l^{(n)}_{k}=\left\\{\begin{array}[]{ll}0,&k=1,2,3\\\\[5.69054pt] \iota_{1}{\bf
e}_{1},&k=4\\\\[5.69054pt] \iota_{1}{\bf e}_{1}+\iota_{2}{\bf
e}_{2n},&k=5\end{array}\right.,\qquad\iota_{1},\iota_{2}\in{\mathbb{R}},$
(5.5)

where, for $n=1$, $l^{(1)}_{5}=\iota_{1}{\bf e}_{1}$, or to
$w^{\prime}={\mathcal{B}}^{(n)}_{6}w+l^{(n)}_{6}$, which is a direct product
of systems in the above $5$ cases with lower dimensions, i.e., there exists a
mutually disjoint partition
$\bigcup_{j\in\Lambda}{{\mathcal{I}}_{j}}=\\{1,\cdots,n\\}$ with
$\\#{\mathcal{I}}_{j}=m_{j}<n$ and $k_{j}\in\\{1,\cdots,5\\}$ such that
${\mathcal{B}}_{6}^{(n)}=\bigoplus_{j\in\Lambda}{\mathcal{B}}_{k_{j}}^{(m_{j})}$,
$l_{6}^{(n)}=\bigoplus_{j\in\Lambda}l_{k_{j}}^{(m_{j})}$, and (5.2) is
conjugated to the direct product of

$w^{\prime}_{{\mathcal{I}}_{j}}={\mathcal{B}}_{k_{j}}^{(m_{j})}w_{{\mathcal{I}}_{j}}+l_{k_{j}}^{(m_{j})},\qquad
w_{{\mathcal{I}}_{j}}:=\begin{pmatrix}(w_{d})_{d\in{\mathcal{I}}_{j}}\\\
(w_{n+d})_{d\in{\mathcal{I}}_{j}}\end{pmatrix}\in{\mathbb{R}}^{2m_{j}}.$

Proof of Theorem 1.1. Proposition 5.2 demonstrates that the non-homogeneous
system (5.2) can be reduced to symplectic normal forms in the indecomposable
cases. Specifically, the system (5.2) is conjugated to the system (5.5) via
the transformation $z(t)=U(t)w(t)+v(t)$. Following Proposition 4.5, the
Hamiltonian PDE (1.2) is then conjugated to the quantum normal form

$\frac{1}{2\pi{\rm
i}}\partial_{t}\varphi={\mathcal{R}}(Z)\varphi=({\mathcal{Q}}_{{\mathcal{B}}}(Z)+{\mathcal{L}}_{l}(Z))\varphi$
(5.6)

through an $L^{2}-$unitary transformation, up to a scalar multiplication that
does not alter the norm,

$\psi(t)={\mathcal{U}}(t)\varphi(t),\qquad{\mathcal{U}}(t):=\rho(v(t)){\mathcal{M}}(U(t)).$
(5.7)

The five specific forms of ${\mathcal{B}}$ and $l$ in (5.5) yield the quantum
Hamiltonian ${\mathcal{R}}(Z)$, as delineated in QNF1 – QNF5 of Theorem 1.1.

In the decomposable case of ${\mathcal{B}}$, symplectic normal forms of
$\\#\Lambda$ lower-dimensional independent systems can be deduced using
Proposition 5.2. According to Proposition 4.5, Eq. (1.2) is also conjugated to
Eq. (5.6), which effectively becomes the direct sum of $\\#\Lambda$
independent quantum Hamiltonians,

${\mathcal{R}}(Z)=\sum_{j\in\Lambda}{\mathcal{R}}_{k_{j}}^{(m_{j})}(Z_{{\mathcal{I}}_{j}})=\sum_{j\in\Lambda}\left({\mathcal{Q}}_{{\mathcal{B}}_{k_{j}}^{(m_{j})}}(Z_{{\mathcal{I}}_{j}})+{\mathcal{L}}_{l_{k_{j}}^{(m_{j})}}(Z_{{\mathcal{I}}_{j}})\right),\quad
Z_{{\mathcal{I}}_{j}}:=\begin{pmatrix}(D_{d})_{d\in{\mathcal{I}}_{j}}\\\
(X_{d})_{d\in{\mathcal{I}}_{j}}\end{pmatrix}.$

This is consistent with QNF6 in Theorem 1.1.

##  6\. Growth rate of Sobolev norms – Proof of Theorem 1.2 & 1.4

Considering Remark 9 and 10, all $L^{2}-$unitary transformations derived
through the Metaplectic or Schrödinger representation, as expressed in (5.7),
preserve the growth rate of Sobolev norms. It is therefore essential to
concentrate on the QNFs ${\mathcal{R}}_{k}^{(n)}(Z)$, $k=1,\cdots,6$, and
evaluate $\|e^{2\pi{\rm i}t{\mathcal{R}}^{(n)}_{k}}u\|_{s}$ for
$u\in{\mathcal{H}}^{s}({\mathbb{R}}^{n})\setminus\\{0\\}$.

For $k=1,2,3$, the QNF is homogeneous, meaning
${\mathcal{R}}^{(n)}_{k}={\mathcal{Q}}_{{\mathcal{B}}^{(n)}_{k}}$. Through
straightforward calculations, we find that for
${\mathcal{B}}^{(n)}_{1}={\mathcal{B}}^{(n)}_{1}(\lambda)$ with $\lambda>0$,

$e^{t{\mathcal{B}}^{(n)}_{1}(\lambda)}=\left(\begin{array}[]{cccc:cccc}e^{-t\lambda}&(-t)e^{-t\lambda}&\dots&\frac{(-t)^{n-1}}{(n-1)!}e^{-t\lambda}&&&&\\\
&\ddots&\ddots&\vdots&&&&\\\ &&e^{-t\lambda}&(-t)e^{-t\lambda}&&&&\\\
&&&e^{-t\lambda}&&&&\\\ \hdashline&&&&e^{t\lambda}&&&\\\
&&&&te^{t\lambda}&e^{t\lambda}&&\\\ &&&&\vdots&\ddots&\ddots&\\\
&&&&\frac{t^{n-1}}{(n-1)!}e^{t\lambda}&\dots&te^{t\lambda}&e^{t\lambda}\end{array}\right),$
(6.1)

for ${\mathcal{B}}^{(n)}_{2}={\mathcal{B}}^{(n)}_{2}(\lambda_{1},\lambda_{2})$
with $\lambda_{1},\lambda_{2}>0$,

$e^{t{\mathcal{B}}^{(n)}_{2}(\lambda_{1},\lambda_{2})}=\left(\begin{array}[]{cccc:cccc}e^{tA_{2}}&&&&&&&\\\
-te^{tA_{2}}&e^{tA_{2}}&&&&&&\\\ \vdots&\ddots&\ddots&&&&&\\\
\frac{(-t)^{\frac{n}{2}-1}}{(\frac{n}{2}-1)!}e^{tA_{2}}&\dots&-te^{tA_{2}}&e^{tA_{2}}&&&&\\\\[5.69054pt]
\hdashline&&&&e^{-tA_{2}^{*}}&te^{-tA_{2}^{*}}&\dots&\frac{t^{\frac{n}{2}-1}}{(\frac{n}{2}-1)!}e^{-tA_{2}^{*}}\\\
&&&&&\ddots&\ddots&\vdots\\\ &&&&&&e^{-tA_{2}^{*}}&te^{-tA_{2}^{*}}\\\
&&&&&&&e^{-tA_{2}^{*}}\end{array}\right),$ (6.2)

where
$e^{tA_{2}}=\begin{pmatrix}e^{-\lambda_{1}t}\cos(\lambda_{2}t)&-e^{-\lambda_{1}t}\sin(\lambda_{2}t)\\\
e^{-\lambda_{1}t}\sin(\lambda_{2}t)&e^{-\lambda_{1}t}\cos(\lambda_{2}t)\end{pmatrix}$,
and for ${\mathcal{B}}_{3}^{(n)}={\mathcal{B}}_{3}^{(n)}(\mu,\gamma)$,
$\gamma=\pm 1$, $\mu>0$,

$e^{t{\mathcal{B}}_{3}^{(n)}(\mu,\gamma)}=\left(\begin{array}[]{cccc:cccc}a_{0}(t)&a_{1}(t)&\cdots&a_{n-1}(t)&b_{n-1}(t)&\cdots&b_{1}(t)&b_{0}(t)\\\
&\ddots&\ddots&\vdots&\vdots&\begin{sideways}$\ddots$\end{sideways}&\begin{sideways}$\ddots$\end{sideways}&\\\
&&a_{0}(t)&a_{1}(t)&b_{1}(t)&b_{0}(t)&&\\\ &&&a_{0}(t)&b_{0}(t)&&&\\\
\hdashline&&&-b_{0}(t)&a_{0}(t)&&&\\\
&&-b_{0}(t)&-b_{1}(t)&a_{1}(t)&a_{0}(t)&&\\\
&\begin{sideways}$\ddots$\end{sideways}&\begin{sideways}$\ddots$\end{sideways}&\vdots&\vdots&\ddots&\ddots&\\\
-b_{0}(t)&-b_{1}(t)&\cdots&-b_{n-1}(t)&a_{n-1}(t)&\cdots&a_{1}(t)&a_{0}(t)\end{array}\right),$
(6.3)

where $a_{j}(t):=\frac{(\gamma t)^{j}}{j!}\cos(\gamma\mu t+\frac{j\pi}{2})$,
$b_{j}(t):=\frac{(\gamma t)^{j}}{j!}\sin(\gamma\mu t-\frac{j\pi}{2})$. Hence,
as $|t|\to\infty$,

$\|e^{t{\mathcal{B}}_{1}^{(n)}(\lambda)}\|\simeq|t|^{n-1}e^{\lambda|t|},\qquad\|e^{t{\mathcal{B}}_{2}^{(n)}(\lambda_{1},\lambda_{2})}\|\simeq|t|^{\frac{n}{2}-1}e^{\lambda_{1}|t|},\qquad\|e^{t{\mathcal{B}}_{3}^{(n)}(\mu,\gamma)}\|\simeq|t|^{n-1}.$

Leveraging Corollary 3.3 \- (ii), for $s>0$, the norm $\|e^{2\pi{\rm
i}t{\mathcal{R}}^{(n)}_{k}}u\|_{s}$ aligns with the growth of
$\|e^{t{\mathcal{B}}_{k}^{(n)}}\|^{s}=g^{(n)}_{k}(s,t)$, $k=1,2,3$, thus
substantiating GR1 – GR3 in Theorem 1.2.

######  Proposition 6.1.

Given $u\in{\mathcal{H}}^{s}\setminus\\{0\\}$, $s>0$, we have $\|e^{2\pi{\rm
i}t{\mathcal{R}}^{(n)}_{k}}u\|_{s}\simeq_{u}g^{(n)}_{k}(s,t)$ for $k=4,5$.

###### Proof.

For QNF4
${\mathcal{R}}^{(n)}_{4}={\mathcal{Q}}_{{\mathcal{B}}^{(n)}_{4}}+{\mathcal{L}}_{l_{4}^{(n)}}$,
let us focus on $\gamma=1$. A direct computation yields that

$\displaystyle e^{t{\mathcal{B}}^{(n)}_{4}(1)}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccc:cccc}1&&&&&&&\\\ -t&1&&&&&&\\\
\vdots&\ddots&\ddots&&&&&\\\ \frac{(-t)^{n-1}}{(n-1)!}&\dots&-t&1&&&&\\\
\hdashline\frac{t^{2n-1}}{(2n-1)!}&\cdots&(-1)^{n}\frac{t^{n+1}}{(n+1)!}&(-1)^{n+1}\frac{t^{n}}{n!}&1&t&\dots&\frac{t^{n-1}}{(n-1)!}\\\
\vdots&&\vdots&\vdots&&\ddots&\ddots&\vdots\\\
\frac{t^{n+1}}{(n+1)!}&\cdots&(-1)^{n}\frac{t^{3}}{3!}&(-1)^{n+1}\frac{t^{2}}{2!}&&&1&t\\\
\frac{t^{n}}{n!}&\cdots&(-1)^{n}\frac{t^{2}}{2!}&(-1)^{n+1}t&&&&1\end{array}\right)$
(6.12) $\displaystyle=:$ $\displaystyle\begin{pmatrix}A(t)&0\\\
C(t)&A(t)^{*-1}\end{pmatrix}\in{\rm Sp}(n,{\mathbb{R}}).$ (6.13)

For $t\in{\mathbb{R}}$, define $b_{4}(t)\in{\mathbb{R}}^{2n}$ and
$c_{4}(t)\in{\mathbb{R}}$ by

$b_{4}(t)^{*}=\iota_{1}\left(t,\cdots,\frac{t^{n}}{n!},-\frac{(-t)^{2n}}{(2n)!},\cdots,-\frac{(-t)^{n+1}}{(n+1)!}\right)=:(p_{4}(t)^{*},q_{4}(t)^{*}),\qquad
c_{4}(t):=-\frac{\iota_{1}^{2}}{2}\cdot\frac{t^{2n+1}}{(2n+1)!}.$

Then, a straightforward calculation verifies that

$\displaystyle\left(e^{2\pi{\rm i}t{\mathcal{R}}^{(n)}_{4}}u\right)(x)$
$\displaystyle=$ $\displaystyle e^{2\pi{\rm
i}c_{4}(t)}\left({\mathcal{M}}\left(e^{t{\mathcal{B}}^{(n)}_{4}(1)}\right)\rho(b_{4}(t))\,u\right)(x)$
(6.14) $\displaystyle=$ $\displaystyle e^{2\pi{\rm i}c_{4}(t)}e^{-\pi{\rm
i}\langle x,C(t)A(t)^{-1}x\rangle}e^{2\pi{\rm i}\langle
q_{4}(t),A(t)^{-1}x\rangle+\pi{\rm i}\langle
p_{4}(t),q_{4}(t)\rangle}u(A(t)^{-1}x+p_{4}(t)).\ $

For $1\leq j\leq n$, we have

$\displaystyle\left\|X_{j}^{s}e^{2\pi{\rm
i}t{\mathcal{R}}^{(n)}_{4}}u\right\|_{L^{2}}$ $\displaystyle=$
$\displaystyle\left(\int_{{\mathbb{R}}^{n}}|x_{j}|^{2s}|u(A(t)^{-1}x+p_{4}(t))|^{2}dx\right)^{\frac{1}{2}}$
(6.15) $\displaystyle=$
$\displaystyle\left(\int_{{\mathbb{R}}^{n}}|(A(t)y-A(t)p_{4}(t))_{j}|^{2s}|u(y)|^{2}dy\right)^{\frac{1}{2}}.$

Then, by noting that $\|A(t)\|\simeq|t|^{n-1}$ and
$(A(t)p_{4}(t))^{*}=\iota_{1}\left(t,-\frac{t^{2}}{2},\cdots,\frac{(-1)^{n+1}}{n!}t^{n}\right)$,
we obtain

$\sum_{j=1}^{n}\left\|X_{j}^{s}e^{2\pi{\rm
i}t{\mathcal{R}}^{(n)}_{4}}u\right\|_{L^{2}}\simeq|t|^{(n-1)s}\|X_{1}^{s}u\|_{L^{2}}+|\iota_{1}|^{s}|t|^{ns}\|u\|_{L^{2}},\qquad|t|\to\infty.$
(6.16)

On the other hand, for $1\leq j\leq n$ and $s\in{\mathbb{N}}^{*}$,

$\displaystyle\left|\left(D_{j}^{s}e^{2\pi{\rm
i}t{\mathcal{R}}^{(n)}_{4}}u\right)(x)\right|$ $\displaystyle=$
$\displaystyle\left|\frac{1}{(2\pi{\rm
i})^{s}}\partial^{s}_{x_{j}}\left(e^{-\pi{\rm i}\langle
x,C(t)A(t)^{-1}x\rangle}e^{2\pi{\rm i}\langle
q_{4}(t),A(t)^{-1}x\rangle}u(A(t)^{-1}x+p_{4}(t))\right)\right|$
$\displaystyle=$
$\displaystyle\frac{1}{(2\pi)^{s}}\left|\sum_{k=0}^{s}C_{s}^{k}\partial^{s-k}_{x_{j}}\left(e^{-2\pi{\rm
i}\left\langle\frac{1}{2}C(t)A(t)^{-1}x-A(t)^{*-1}q_{4}(t),x\right\rangle}\right)\partial^{k}_{x_{j}}\left(u(A(t)^{-1}x+p_{4}(t))\right)\right|.$

By computations, we have, as $|t|\to\infty$, $\|A(t)^{-1}\|\simeq|t|^{n-1}$,
$\|C(t)A(t)^{-1}\|\simeq|t|^{2n-1}$ and

$(A(t)^{*-1}q_{4}(t))_{1}=-\iota_{1}t^{2n}\frac{(-1)^{n+1}}{2(n!)^{2}},\qquad|(A(t)^{*-1}q_{4}(t))_{j}|\simeq|\iota_{1}||t|^{2n+1-j},\quad
2\leq j\leq n,$
$(C(t)p_{4}(t))_{1}=\iota_{1}t^{2n}\left(\frac{(-1)^{n+1}}{2(n!)^{2}}+\frac{1}{(2n)!}\right),\qquad|(C(t)p_{4}(t))_{j}|\simeq|\iota_{1}||t|^{2n+1-j},\quad
2\leq j\leq n.$

Hence, as $|t|\to\infty$,

$\displaystyle\left\|D_{j}^{s}e^{2\pi{\rm
i}t{\mathcal{R}}^{(n)}_{4}}u\right\|_{L^{2}}$ $\displaystyle\lesssim$
$\displaystyle\left(\int_{{\mathbb{R}}^{n}}|(C(t)A(t)^{-1}x-A(t)^{*-1}q_{4}(t))_{j}|^{2s}|u(A(t)^{-1}x+p_{4}(t))|^{2}dx\right)^{\frac{1}{2}}$
(6.17) $\displaystyle=$
$\displaystyle\left(\int_{{\mathbb{R}}^{n}}|(C(t)y-C(t)p_{4}(t)-A(t)^{*-1}q_{4}(t))_{j}|^{2s}|u(y)|^{2}dy\right)^{\frac{1}{2}}$
$\displaystyle\lesssim$
$\displaystyle|t|^{(2n-2)s\|u\|_{s}}+|\iota_{1}|^{s}|t|^{(2n-1)s}\|u\|_{L^{2}},\quad
2\leq j\leq n,$ $\displaystyle\left\|D_{1}^{s}e^{2\pi{\rm
i}t{\mathcal{R}}^{(n)}_{4}}u\right\|_{L^{2}}$ $\displaystyle\simeq$
$\displaystyle\left(\int_{{\mathbb{R}}^{n}}|(C(t)A(t)^{-1}x-A(t)^{*-1}q_{4}(t))_{1}|^{2s}|u(A(t)^{-1}x+p_{4}(t))|^{2}dx\right)^{\frac{1}{2}}$
(6.18) $\displaystyle=$