# Growth of Sobolev norms in linear Schrödinger equations
as a dispersive phenomenon

A. Maspero111 International School for Advanced Studies (SISSA), Via Bonomea
265, 34136, Trieste, Italy
Email<EMAIL_ADDRESS>
###### Abstract

In this paper we consider linear, time dependent Schrödinger equations of the
form ${\rm i}\partial_{t}\psi=K_{0}\psi+V(t)\psi$, where $K_{0}$ is a strictly
positive selfadjoint operator with discrete spectrum and constant spectral
gaps, and $V(t)$ a time periodic potential. We give sufficient conditions on
$V(t)$ ensuring that $K_{0}+V(t)$ generates unbounded orbits. The main
condition is that the resonant average of $V(t)$, namely the average with
respect to the flow of $K_{0}$, has a nonempty absolutely continuous spectrum
and fulfills a Mourre estimate. These conditions are stable under
perturbations. The proof combines pseudodifferential normal form with
dispersive estimates in the form of local energy decay.
We apply our abstract construction to the Harmonic oscillator on
${\mathbb{R}}$ and to the half-wave equation on ${\mathbb{T}}$; in each case,
we provide large classes of potentials which are transporters.

##  1 Introduction

We consider the abstract linear Schrödinger equation

${\rm i}\partial_{t}\psi=K_{0}\psi+V(t)\psi\ $ (1.1)

on a scale of Hilbert spaces ${\mathcal{H}}^{r}$; here $V(t)$ is a time
$2\pi$-periodic potential and $K_{0}$ a selfadjoint, strictly positive
operator with compact resolvent, pure point spectrum and constant spectral
gaps. We prove some abstract results ensuring, $\forall r>0$, the existence of
solutions $\psi(t)$ whose ${\mathcal{H}}^{r}$-norms grow polynomially fast,

$\|\psi(t)\|_{r}\geq C_{r}\,\left\langle t\right\rangle^{r},\quad\forall t\gg
1\ ,$

whereas their ${\mathcal{H}}^{0}$-norms are constant for all times,
$\|\psi(t)\|_{0}=\|\psi(0)\|_{0}$ $\forall t$. Here $\left\langle
t\right\rangle:=\sqrt{1+t^{2}}$.
These solutions therefore exhibit weak turbulent behavior in the form of
energy cascade towards high frequencies.
We apply our abstract result to two models: the Harmonic oscillator on
${\mathbb{R}}$ and the half-wave equation on ${\mathbb{T}}$. In both cases we
exhibit large classes of potentials $V(t)$, bounded, smooth and periodic in
time, so that the Hamiltonian $K_{0}+V(t)$ generates unbounded orbits.

The phenomenon is purely perturbative: for $V=0$ each norm of each solution is
constant for all times. So the central question is the existence of potentials
able to transport energy to high-frequencies; we formalize this notion in the
following definition:

######  Definition 1.1.

We shall say that $V(t)$ is a transporter if $\forall r>0$ there exists a
solution $\psi(t)\in{\mathcal{H}}^{r}$ of (1.1) with unbounded growth of norm,
i.e.

$\limsup_{t\to\infty}\|\psi(t)\|_{r}=\infty.$ (1.2)

If this happens for every nonzero solution, we shall say that $V(t)$ is a
universal transporter.

Starting with the pioneering work of Bourgain [9], in the last few years there
has been some efforts to construct both transporters [17, 22, 44] and
universal transporters [6, 34] for different types of Schrödinger equations.
All these papers provide explicit examples of potentials, constructed ad hoc
for the problem at hand.

The novelty of our result is that we identify sufficient, explicit and robust
(i.e. stable under perturbations) conditions ensuring $V(t)$ to be a
transporter. Precisely, its resonant average

$\displaystyle\left\langle
V\right\rangle:=\frac{1}{2\pi}\int_{0}^{2\pi}e^{{\rm i}sK_{0}}\,V(s)\,e^{-{\rm
i}sK_{0}}\,{\rm d}s\ $ (1.3)

must have nontrivial absolutely continuous spectrum in an interval, and over
this interval it has to fulfill a Mourre estimate – see (2.7) below (actually
we also require that both $K_{0}$ and $V(t)$ belong to some abstract graded
algebra of pseudodifferential operators, as in [5]).

The crucial point is that these conditions imply dispersive estimates for
$\left\langle V\right\rangle$ of the form

$\|K_{0}^{-k}e^{-{\rm i}t\left\langle
V\right\rangle}P_{c}\phi\|_{0}\lesssim\left\langle
t\right\rangle^{-k}\|K_{0}^{k}\phi\|_{0}\ ,\quad\forall t\in{\mathbb{R}},$
(1.4)

where $P_{c}$ is a projection on a subset of the absolutely continuous
spectral space of $\left\langle V\right\rangle$. A consequence of (1.4) is
that we obtain solutions of ${\rm i}\partial_{t}\phi=\left\langle
V\right\rangle\phi$ with decaying negative Sobolev norms and so, by duality,
growing positive Sobolev norms.
The fact that Mourre estimates imply dispersive estimates as above has origin
from the work of Sigal-Soffer in quantum scattering theory [41] and it has
been extended by many authors (see e.g. [42, 23, 31, 30, 24, 2]). See also the
recent results [13, 12, 20] where similar dispersive properties are studied
for pseudodifferential operators of order 0 on compact manifolds of dimension
greater equal $2$.

To explain the connection between the dynamics of (1.1) and the dispersive
properties of the flow of $\left\langle V\right\rangle$, let us briefly
describe the main ideas of the proof. The first step is to put system (1.1)
into its resonant pseudodifferential normal form. This is the resonant variant
of the normal form developed in [5] for non-resonant systems (and essentially
an abstract version of the normal form of Delort [17]); it allows, $\forall
N\in{\mathbb{N}}$, to conjugate equation (1.1) to

${\rm
i}\partial_{t}{\varphi}=\big{(}K_{0}+Z^{(N)}(t)+R^{(N)}(t)\big{)}{\varphi}$
(1.5)

where $Z^{(N)}(t)$ is a time dependent operator fulfilling

${\rm i}\partial_{t}Z^{(N)}(t)=[K_{0},Z^{(N)}(t)],\qquad
Z^{(N)}(0)=\left\langle V\right\rangle+\mbox{lower order terms}$ (1.6)

whereas $R^{(N)}(t)$ is an $N$-smoothing operator (it maps
${\mathcal{H}}^{r}\to{\mathcal{H}}^{r+N}$ continuously $\forall r$). The
difference with the non-resonant case of [5] is that, in that paper,
$Z^{(N)}(t)$ commutes with $K_{0}$. This is not anymore true in the resonant
case we deal with; however (1.6) implies that $e^{{\rm
i}tK_{0}}Z^{(N)}(t)e^{-{\rm i}tK_{0}}$ is time independent and thus coincides
with $Z^{(N)}(0)$. Thus, conjugating (1.5) with $e^{-{\rm i}tK_{0}}$, we
arrive at the equation

${\rm i}\partial_{t}\phi=\big{(}\left\langle V\right\rangle+T+R(t)\big{)}\phi$
(1.7)

where $T:=Z^{(N)}(0)-\left\langle V\right\rangle$ is a time independent
selfadjoint compact operator and $R(t)$ is $N$-smoothing.

Then we analyze the dynamics of the truncated equation

${\rm i}\partial_{t}\phi=\big{(}\left\langle V\right\rangle+T\big{)}\phi$
(1.8)

and prove that it has solutions with decaying negative Sobolev norms and so,
by duality, growing positive Sobolev norms. This is the core of the proof;
after this step, it is not difficult to construct a solution of the complete
equation (1.7) exhibiting energy cascade, exploiting that $R(t)$ is
regularizing. So let us concentrate on (1.8). The goal is to prove a
dispersive estimate of the form (1.4) with $\left\langle V\right\rangle$
replaced by $\left\langle V\right\rangle+T$. However the point is delicate
because the absolutely continuous spectrum of $\left\langle V\right\rangle$
(which exists by assumption) could be completely destroyed by adding $T$; a
celebrated theorem by Weyl-von Neumann ensures that any selfadjoint operator
(in a separable Hilbert space) can be perturbed by a (arbitrary small) compact
selfadjoint operator so that its spectrum becomes pure point (see e.g. [32,
pag. 525]). This is exactly the situation we want to avoid, as pure point
spectrum prevents dispersive estimates. To get around this, we exploit that
Mourre estimates are stable under pseudodifferential perturbations. This
allows us to prove that $\left\langle V\right\rangle+T$ fulfills Mourre
estimates and thus a dispersive estimate as (1.4).

We also stress that fulfilling a Mourre estimate seems to be a quite general
condition, and in the applications we exhibit large classes of operators which
are transporters. For example, for the half wave equation we prove that any
operator of the form $\cos(mt)v(x)$ with $v\in
C^{\infty}({\mathbb{T}},{\mathbb{R}})$ and $m\in{\mathbb{Z}}$ is a transporter
provided the $m$-th Fourier coefficient of $v(x)$ is not zero.

Finally, the conditions we identity to be transporters are robust: if a
potential $V(t)$ fulfills them, so does $V(t)+W(t)$ for any sufficiently small
pseudodifferential operator $W(t)$. This shows that weakly turbulent phenomena
induced by certain transporters are stable under perturbations. Up to our
knowledge, this “stability of instability” is new in the literature and we
consider it one of the main novelty of the paper.

We conclude the introduction by reviewing the known results about existence of
transporters for linear time dependent Schrödinger equations. As we already
mentioned, the first result is due to Bourgain [9], who constructed a
transporter for the Schrödinger equation on the torus; in this case $V(t,x)$
is a bounded real analytic function. Delort [17] constructs a transporter for
the harmonic oscillator on ${\mathbb{R}}$, which is a time $2\pi$-periodic
pseudodifferential operator of order zero. In [6] we proved that $ax\sin(t)$,
$a>0$, is a universal transporter for the harmonic oscillator on
${\mathbb{R}}$; in this case the potential is an unbounded operator. In [34]
we constructed universal transporters for the abstract equation (1.1), and
applied the result to the harmonic oscillator on ${\mathbb{R}}$, the half-wave
equation on ${\mathbb{T}}$ and on a Zoll manifold; in all cases the universal
transporters are time periodic pseudodifferential operators of order 0.
Finally recently Faou-Raphael [22] constructed a transporter for the harmonic
oscillator on ${\mathbb{R}}$ which is a time dependent function (and not a
pseudodifferential operator), and Thomann [44] has constructed a transporter
for the harmonic oscillator on the Bargman-Fock space. Finally we recall the
long-time growth result [25] for the semiclassical anharmonic oscillator on
${\mathbb{R}}^{d}$.

Acknowledgments: We thank Matteo Gallone for helpful discussions on spectral
theory and Dario Bambusi and Didier Robert for useful suggestions during the
preparation of this work.

##  2 The abstract result

We start with a Hilbert space ${\mathcal{H}}$, endowed with the scalar product
$\left\langle\cdot,\cdot\right\rangle$, and a reference operator $K_{0}$,
which we assume to be selfadjoint, positive, namely such that

$\langle\psi;K_{0}\psi\rangle\geq c_{K_{0}}\|\psi\|^{2}\ ,\quad\forall\psi\in
D(K_{0}^{1/2})\ ,\quad c_{K_{0}}>0\ ,$

and with compact resolvent.
We define as usual a scale of Hilbert spaces by
${\mathcal{H}}^{r}:=D(K_{0}^{r})$ (the domain of the operator $K_{0}^{r}$) if
$r\geq 0$, and ${\mathcal{H}}^{r}=({\mathcal{H}}^{-r})^{\prime}$ (the dual
space) if $r<0$. Finally we denote by
${\mathcal{H}}^{-\infty}=\bigcup_{r\in{\mathbb{R}}}{\mathcal{H}}^{r}$ and
${\mathcal{H}}^{+\infty}=\bigcap_{r\in{\mathbb{R}}}{\mathcal{H}}^{r}$. We
endow ${\mathcal{H}}^{r}$ with the natural norm
$\|\psi\|_{r}:=\|(K_{0})^{r}\psi\|_{0}$, where $\|\cdot\|_{0}$ is the norm of
${\mathcal{H}}^{0}\equiv{\mathcal{H}}$. Notice that for any
$m\in{\mathbb{R}}$, ${\mathcal{H}}^{+\infty}$ is a dense linear subspace of
${\mathcal{H}}^{m}$ (this is a consequence of the spectral decomposition of
$K_{0}$).

######  Remark 2.1.

By the very definition of ${\mathcal{H}}^{r}$, the unperturbed flow $e^{-{\rm
i}tK_{0}}$ preserves each norm, $\|e^{-{\rm i}tK_{0}}\psi\|_{r}=\|\psi\|_{r}$
$\,\forall t\in{\mathbb{R}}$. Consequently, every orbit of equation (1.1) with
$V(t)=0$ is bounded.

Following [5], we introduce now a graded algebra ${\mathcal{A}}$ of operators
which mimic some fundamental properties of different classes of
pseudodifferential operators. For $m\in{\mathbb{R}}$ let ${\mathcal{A}}_{m}$
be a linear subspace of
$\bigcap_{s\in{\mathbb{R}}}{\mathcal{L}}({\mathcal{H}}^{s},{\mathcal{H}}^{s-m})$
and define ${\mathcal{A}}:=\bigcup_{m\in{\mathbb{R}}}{\mathcal{A}}_{m}$. We
notice that the space
$\bigcap_{s\in{\mathbb{R}}}{\mathcal{L}}({\mathcal{H}}^{s},{\mathcal{H}}^{s-m})$
is a Fréchet space equipped with the semi-norms:
$\|A\|_{m,s}:=\|A\|_{{\mathcal{L}}({\mathcal{H}}^{s},{\mathcal{H}}^{s-m})}$.

We shall need to control the smoothing properties of the operators in the
scale $\\{{\mathcal{H}}^{r}\\}_{r\in{\mathbb{R}}}$. If $A\in{\mathcal{A}}_{m}$
then $A$ is more and more smoothing if $m\rightarrow-\infty$ and the opposite
as $m\rightarrow+\infty$. We will say that $A$ is of order $m$ if
$A\in{\mathcal{A}}_{m}$.

######  Definition 2.2.

We say that
$S\in{\mathcal{L}}({\mathcal{H}}^{+\infty},{\mathcal{H}}^{-\infty})$ is
$N$-smoothing if $\forall\kappa\in{\mathbb{R}}$, it can be extended to an
operator in ${\mathcal{L}}({\mathcal{H}}^{\kappa},{\mathcal{H}}^{\kappa+N})$.
When this is true for every $N\geq 0$, we say that $S$ is a smoothing
operator.

The first set of assumptions concerns the properties of ${\mathcal{A}}_{m}$:

Assumption I: Pseudodifferential algebra

  * (i)

For each $m\in{\mathbb{R}}$, $K_{0}^{m}\in{\mathcal{A}}_{m}$; in particular
$K_{0}$ is an operator of order one.

  * (ii)

For each $m\in{\mathbb{R}}$, ${\mathcal{A}}_{m}$ is a Fréchet space for a
family of filtering semi-norms $\\{\wp^{m}_{j}\\}_{j\geq 0}$ such that the
embedding
${\mathcal{A}}_{m}\hookrightarrow\bigcap_{s\in{\mathbb{R}}}{\mathcal{L}}({\mathcal{H}}^{s},{\mathcal{H}}^{s-m})$
is continuous222A family of seminorms $\\{\wp^{m}_{j}\\}_{j\geq 0}$ is called
filtering if for any $j_{1},j_{2}\geq 0$ there exist $k\geq 0$ and
$c_{1},c_{2}>0$ such that the two inequalities $\wp^{m}_{j_{1}}(A)\leq
c_{1}\wp^{m}_{k}(A)$ and $\wp^{m}_{j_{2}}(A)\leq c_{2}\wp^{m}_{k}(A)$ hold for
any $A\in{\mathcal{A}}_{m}$..
If $m^{\prime}\leq m$ then
${\mathcal{A}}_{m^{\prime}}\subseteq{\mathcal{A}}_{m}$ with a continuous
embedding.

  * (iii)

${\mathcal{A}}$ is a graded algebra, i.e. $\forall m,n\in{\mathbb{R}}$: if
$A\in{\mathcal{A}}_{m}$ and $B\in{\mathcal{A}}_{n}$ then
$AB\in{\mathcal{A}}_{m+n}$ and the map $(A,B)\mapsto AB$ is continuous from
${\mathcal{A}}_{m}\times{\mathcal{A}}_{n}$ into ${\mathcal{A}}_{m+n}$.

  * (iv)

${\mathcal{A}}$ is a graded Lie-algebra333This property will impose the choice
of the semi-norms $\\{\wp^{m}_{j}\\}_{j\geq 1}$. We will see in the examples
that the natural choice $(\|\cdot\|_{m,s})_{s\geq 0}$ has to be refined.  : if
$A\in{\mathcal{A}}_{m}$ and $B\in{\mathcal{A}}_{n}$ then the commutator
$[A,B]\in{\mathcal{A}}_{m+n-1}$ and the map $(A,B)\mapsto[A,B]$ is continuous
from ${\mathcal{A}}_{m}\times{\mathcal{A}}_{n}$ into ${\mathcal{A}}_{m+n-1}$.

  * (v)

${\mathcal{A}}$ is closed under perturbation by smoothing operators in the
following sense: let $A$ be a linear map:
${\mathcal{H}}^{+\infty}\rightarrow{\mathcal{H}}^{-\infty}$. If there exists
$m\in{\mathbb{R}}$ such that for every $N>0$ we have a decomposition
$A=A^{(N)}+S^{(N)}$, with $A^{(N)}\in{\mathcal{A}}_{m}$ and $S^{(N)}$ is
$N$-smoothing, then $A\in{\mathcal{A}}_{m}$.

  * (vi)

If $A\in{\mathcal{A}}_{m}$ then also the adjoint operator
$A^{*}\in{\mathcal{A}}_{m}$. The duality here is defined by the scalar product
$\langle\cdot,\cdot\rangle$ of ${\mathcal{H}}={\mathcal{H}}^{0}$. The adjoint
$A^{*}$ is defined by $\langle u,Av\rangle=\langle A^{*}u,v\rangle$ for
$u,v\in{\mathcal{H}}^{\infty}$ and extended by continuity.

It is well known that classes of pseudodifferential operators satisfy these
properties, provided one chooses for $K_{0}$ a suitable operator of the right
order (see e.g. [28]).

######  Remark 2.3.

One has that $\forall A\in{\mathcal{A}}_{m}$, $\forall B\in{\mathcal{A}}_{n}$

$\displaystyle\forall m,s\quad\exists N\ s.t.\ $ $\displaystyle\|A\|_{m,s}\leq
C_{1}\,\wp^{m}_{N}(A)\ ,$ (2.1) $\displaystyle\forall m,n,j\quad\exists N\
s.t.\ $ $\displaystyle\wp^{m+n}_{j}(AB)\leq
C_{2}\,\wp^{m}_{N}(A)\,\wp^{n}_{N}(B)\ ,$ (2.2) $\displaystyle\forall
m,n,j\quad\exists N\ s.t.\ $ $\displaystyle\wp^{m+n-1}_{j}([A,B])\leq
C_{3}\,\wp^{m}_{N}(A)\,\wp^{n}_{N}(B)\ ,$ (2.3)

for some positive constants $C_{1}(s,m)$, $C_{2}(m,n,j)$, $C_{3}(m,n,j)$.

######  Remark 2.4.

Any $A\in{\mathcal{A}}_{m}$ with $m<0$ is a compact operator on
${\mathcal{H}}$.
Indeed write $A=AK_{0}^{-m}\,K_{0}^{m}$. Then
$AK_{0}^{-m}\in{\mathcal{A}}_{0}$ is a bounded operator on ${\mathcal{H}}$
(Assumption I (i)–(iii)), whereas $K_{0}^{m}\equiv(K_{0}^{-1})^{-m}$ is
compact on ${\mathcal{H}}$, as $K_{0}^{-1}$ is a compact operator by
assumption.

For $\Omega\subseteq{\mathbb{R}}^{d}$ and ${\mathcal{F}}$ a Fréchet space, we
will denote by $C_{b}^{m}(\Omega,{\mathcal{F}})$ the space of $C^{m}$ maps
$f:\Omega\ni x\mapsto f(x)\in{\mathcal{F}}$ such that, for every seminorm
$\|\cdot\|_{j}$ of ${\mathcal{F}}$, one has

$\sup_{x\in\Omega}\|\partial_{x}^{\alpha}f(x)\|_{j}<+\infty\
,\quad\forall\alpha\in{\mathbb{N}}^{d}\ :\ \left|\alpha\right|\leq m\ .$ (2.4)

If (2.4) is true $\forall m$, we say $f\in
C^{\infty}_{b}(\Omega,{\mathcal{F}})$. Similarly we denote by
$C^{\infty}({\mathbb{T}},{\mathcal{F}})$ the space of smooth maps from the
torus ${\mathbb{T}}={\mathbb{R}}/(2\pi{\mathbb{Z}})$ to the Fréchet space
${\mathcal{F}}$.

The second set of assumptions concerns the operator $K_{0}$, its spectral
structure and an Egorov-like property, also well known for pseudo-differential
operators.

Assumption II: Properties of $K_{0}$

  * (i)

The operator $K_{0}$ has purely discrete spectrum fulfilling

${\rm spec}(K_{0})\subseteq{\mathbb{N}}+\lambda$ (2.5)

for some $\lambda\geq 0$.

  * (ii)

For any $m\in{\mathbb{R}}$ and $A\in{\mathcal{A}}_{m}$, the map defined on
${\mathbb{R}}$ by $\tau\mapsto A(\tau):={\rm e}^{{\rm i}\tau K_{0}}\,A\,{\rm
e}^{-{\rm i}\tau K_{0}}$ belongs to
$C^{\infty}_{b}({\mathbb{R}},{\mathcal{A}}_{m})$ and one has

$\forall j\quad\exists N\ s.t.\ \
\sup_{\tau\in{\mathbb{R}}}\wp^{m}_{j}(A(\tau))\leq C_{4}\,\wp^{m}_{N}(A)$
(2.6)

for some positive constant $C_{4}(m,j)$.

######  Remark 2.5.

Assumption II (i) guarantees that $e^{{\rm i}2\pi K_{0}}=e^{{\rm
i}2\pi\lambda}$. As a consequence, for any operator $V$, the map $\tau\mapsto
e^{{\rm i}\tau K_{0}}Ve^{-{\rm i}\tau K_{0}}$ is $2\pi$-periodic.

We denote by $C^{\infty}_{c}({\mathbb{R}}^{d},{\mathbb{R}}_{\geq 0})$ the set
of smooth functions with compact support from ${\mathbb{R}}^{d}$ to
${\mathbb{R}}_{\geq 0}$ (hence non-negative). Furthermore from now on, given
two operators $\mathsf{A},\mathsf{B}\in{\mathcal{L}}({\mathcal{H}})$, we write
$\mathsf{A}\leq\mathsf{B}$ with the meaning
$\left\langle\mathsf{A}{\varphi},{\varphi}\right\rangle\leq\left\langle\mathsf{B}{\varphi},{\varphi}\right\rangle$
$\,\forall{\varphi}\in{\mathcal{H}}$.

The last set of assumptions concerns the resonant average $\left\langle
V\right\rangle$ of the potential $V(t)$ (see (1.3)) and its spectrum
$\sigma(\left\langle V\right\rangle)$. Note that if $V(t)$ is selfadjoint
$\forall t$, so is $\left\langle V\right\rangle$.

Assumption III: Properties of the potential $V(t)$
The operator $V\in C^{\infty}({\mathbb{T}},{\mathcal{A}}_{0})$, $V(t)$
selfadjoint $\forall t$, and its resonant average $\left\langle
V\right\rangle$ fulfills:

  * (i)

There exists an interval $I_{0}\subset{\mathbb{R}}$ such that
$\left|\sigma(\left\langle V\right\rangle)\cap I_{0}\right|>0$; here
$\left|\cdot\right|$ denotes the Lebesgue measure.

  * (ii)

Mourre estimate over $I_{0}$: there exist a selfadjoint operator
$A\in{\mathcal{A}}_{1}$ and a function $g_{I_{0}}\in
C_{c}^{\infty}({\mathbb{R}},{\mathbb{R}}_{\geq 0})$ with $g_{I_{0}}\equiv 1$
on $I_{0}$ such that

$g_{I_{0}}(\left\langle V\right\rangle)\,{\rm i}[\left\langle
V\right\rangle,A]\,g_{I_{0}}(\left\langle
V\right\rangle)\geq\theta\,g_{I_{0}}(\left\langle V\right\rangle)^{2}+K$ (2.7)

for some $\theta>0$ and $K$ a selfadjoint compact operator.

The operator $g_{I_{0}}(\left\langle V\right\rangle)$ above is defined via
functional calculus, see Appendix B.
Following the literature, we shall say that $\left\langle V\right\rangle$ is
conjugated to $A$ over $I_{0}$.

######  Remark 2.6.

By Mourre theory [36] $\left\langle V\right\rangle$ has, in the interval
$I_{0}$, a nontrivial absolutely continuous spectrum with finitely many
eigenvalues of finite multiplicity and no singular continuous spectrum. In
general one cannot exclude the existence of embedded eigenvalues in the
absolutely continuous spectrum.444 For example consider
$H\in{\mathcal{L}}(L^{2}({\mathbb{T}}))$ given by
$(Hu)(x):=\cos(x)u(x)+\delta(1-\delta^{-1}\cos(x))\frac{1}{2\pi}\int_{\mathbb{T}}u(x)\big{(}1-\delta^{-1}\cos(x)\big{)}\,{\rm
d}x\ ,\quad\delta\in\left(-\frac{1}{2},\frac{1}{2}\right)\setminus\\{0\\}\ .$
$H$ is selfadjoint, a 1-rank perturbation of the multiplication operator by
$\cos(x)$, it has absolutely continuous spectrum in the interval $(-1,1)$, and
$\delta$ is an embedded eigenvalue with eigenvector $u(x)\equiv 1$. Moreover
$H$ is conjugated to $\sin(x)\frac{\partial_{x}}{{\rm
i}}+\frac{\partial_{x}}{{\rm i}}\sin(x)$ over $[-\frac{1}{2},\frac{1}{2}]$.

We are ready to state our main results. The first says that, under the set of
assumptions above, $V(t)$ is a transporter in the sense of Definition 1.1:

######  Theorem 2.7.

Assume that ${\mathcal{A}}$ is a graded algebra as in Assumption I, and that
$K_{0}$ and $V(t)\in C^{\infty}({\mathbb{T}},{\mathcal{A}}_{0})$ satisfy
Assumptions II and III. Then $V(t)$ is a transporter for the equation

${\rm i}\partial_{t}\psi=(K_{0}+V(t))\psi\ .$ (2.8)

More precisely, for any $r>0$ there exist a solution $\psi(t)$ of (2.8) in
${\mathcal{H}}^{r}$ and constants $C,T>0$ such that

$\|\psi(t)\|_{r}\geq C\left\langle t\right\rangle^{r},\quad\forall t\geq T\ .$
(2.9)

We also prove a stronger result: namely not only $V(t)$ is a transporter, but
also any operator sufficiently close to it (in the
${\mathcal{A}}_{0}$-topology). Here the precise statement:

######  Theorem 2.8.

With the same assumptions of Theorem 2.7, there exist $\epsilon_{0}>0$ and
${\mathtt{M}}\in{\mathbb{N}}$ such that for any $W\in
C^{\infty}({\mathbb{T}},{\mathcal{A}}_{0})$, $W(t)$ selfadjoint $\forall t$,
fulfilling

$\sup_{t\in{\mathbb{T}}}\,\wp^{0}_{\mathtt{M}}(W(t))\leq\epsilon_{0},$ (2.10)

then $V(t)+W(t)$ is a transporter for the equation

${\rm i}\partial_{t}\psi=\big{(}K_{0}+V(t)+W(t)\big{)}\psi\ .$ (2.11)

More precisely, for any $r>0$ there exist a solution $\psi(t)$ in
${\mathcal{H}}^{r}$ of (2.11) and constants $C,T>0$ such that

$\|\psi(t)\|_{r}\geq C\left\langle t\right\rangle^{r},\quad\forall t\geq T\ .$
(2.12)

Let us comment the above results.

  1. 1.

The growth of Sobolev norms of Theorem 2.7 is truly an energy cascade
phenomenon; indeed the ${\mathcal{H}}^{0}$-norm of any solution of (2.8) is
preserved for all times, $\|\psi(t)\|_{0}=\|\psi(0)\|_{0}$, $\,\forall
t\in{\mathbb{R}}$. This is due to the selfadjointness of $K_{0}+V(t)$ (the
same happens to solutions of (2.11)).

  2. 2.

Estimates (2.9), (2.12) provide optimal lower bounds for the speed of growth
of the Sobolev norms. Indeed we proved [33] that, under the assumptions
above555 in particular the fact that $[K_{0},V(t)]$ and $[K_{0},V(t)+W(t)]$
are uniformly (in $t$) bounded operators on the scale ${\mathcal{H}}^{r}$, any
solution of (2.8) or (2.11) fulfills the upper bounds

$\forall r>0\ \ \
\exists\,{\widetilde{C}}_{r}>0\colon\quad\|\psi(t)\|_{r}\leq{\widetilde{C}}_{r}\left\langle
t\right\rangle^{r}\,\|\psi(0)\|_{r}.$

Thus, Theorems 2.7, 2.8 construct unbounded solutions with optimal growth.

  3. 3.

Theorem 2.8 proves robustness of certain type of transporters under small
pseudodifferential perturbations. This shows a sort of “stability of
instability”, which, up to our knowledge, is new in this context.

  4. 4.

Actually there are infinitely many distinct solutions undergoing growth of
Sobolev norms. Their initial data are constructed in a unique way starting
from functions belonging to the absolutely continuous spectral subspace of the
operator $\left\langle V\right\rangle$. We describe such initial data in
Corollary 3.16.

  5. 5.

Energy cascade is a resonant phenomenon; here it happens because $V(t)$
oscillates at frequency $\omega=1$ which resonates with the spectral gaps of
$K_{0}$. In [5] we proved that if $V(t)\equiv\mathsf{V}(\omega t)$ is
quasiperiodic in time with a frequency vector $\omega\in{\mathbb{R}}^{n}$
fulfilling the non-resonant condition

$\exists\gamma,\tau>0\colon\quad\left|\ell+\omega\cdot
k\right|\geq\frac{\gamma}{\left\langle
k\right\rangle^{\tau}}\quad\forall\ell,k\in{\mathbb{Z}}\times{\mathbb{Z}}^{n}\setminus\\{0\\}$

(which is violated if $V(t)$ is $2\pi$-periodic) then the Sobolev norms grow
at most as $\left\langle t\right\rangle^{\epsilon}$ $\forall\epsilon>0$. The
$\left\langle t\right\rangle^{\epsilon}$-speed of growth is also known for
systems with increasing [37, 33, 5] or shrinking [21, 35] spectral gaps and
for Schrödinger equation on ${\mathbb{T}}^{d}$ with bounded [10, 16, 8] and
even unbounded [7] potentials.

  6. 6.

In concrete models one can typically prove that if $V(t)$ is sufficiently
small in size and oscillates in time with a strongly non resonant frequency
$\omega$ (typically belonging to some Cantor set of large measure), then all
solutions have uniformly in time bounded Sobolev norms. Therefore the
stability/instability of the system depends only on the resonance property of
the frequency $\omega$. We mention just the recent results [4, 6] which deal
with the harmonic oscillator (as we consider it in the applications) and refer
to those papers for a complete bibliography.

  7. 7.

The most delicate assumption to verify is (2.7). In the applications, one can
try to construct an escape function for the principal symbol $\left\langle
v\right\rangle$ of $\left\langle V\right\rangle$. This means to find a symbol
$a(x,\xi)$ of order 1 such that the Poisson bracket $\\{\left\langle
v\right\rangle,a\\}$ is strictly positive in some energy levels:

$\exists c>0\colon\quad\\{\left\langle v\right\rangle,a\\}\geq c\qquad\mbox{
in }\\{(x,\xi):\ \ \left|\left\langle
v\right\rangle(x,\xi)-\lambda\right|\leq\delta\\}\ .$

Then symbolic calculus and sharp Gårding inequality imply that (2.7) holds in
the interval $I=(\lambda-\delta,\lambda+\delta)$; see [13] Section 6.2 for
details.

We finally note that the second theorem is stronger than the first one and
implies it in the special case $W(t)\equiv 0$. However we think that the
statement of Theorem 2.7 is clear and useful in the applications (see e.g.
Section 4), so we decided to state it on its own. Having said so, in the
sequel we shall only prove Theorem 2.8.

##  3 Proof of the abstract result

As already mentioned, we shall only prove Theorem 2.8. The proof is divided in
three steps; in the first one we put system (2.11) in its resonant
pseudodifferential normal form. In the second one we analyze the dynamics of
the effective Hamiltonian and prove the existence of solutions with decaying
negative Sobolev norms. The final step is to construct a solution of the
complete equation exhibiting growth of Sobolev norms.

###  3.1 Resonant pseudodifferential normal form

The goal of this section is to put system (2.11) into its resonant
pseudodifferential normal form up to an arbitrary $N$-smoothing operator. In
this first step we shall only require Assumptions I and II. It is slightly
more convenient to deal with the equation

${\rm i}\partial_{t}\psi=\big{(}K_{0}+\mathsf{V}(t)\big{)}\psi$ (3.1)

and then to specify the result for $\mathsf{V}(t)=V(t)+W(t)$ as in (2.11).
Given $\mathsf{V}\in C^{\infty}({\mathbb{T}},{\mathcal{A}}_{m})$,
$m\in{\mathbb{R}}$, we define the averaged operator

$\displaystyle{\widehat{\mathsf{V}}}(t):=\frac{1}{2\pi}\int_{0}^{2\pi}e^{{\rm
i}sK_{0}}\,\mathsf{V}(t+s)\,e^{-{\rm i}sK_{0}}\,{\rm d}s\ .$ (3.2)

We shall prove below that ${\widehat{\mathsf{V}}}(t)\in
C^{\infty}({\mathbb{T}},{\mathcal{A}}_{m})$ (see Lemma 3.2).

######  Proposition 3.1 (Resonant pseudodifferential normal form).

Consider equation (3.1) with $\mathsf{V}\in
C^{\infty}({\mathbb{T}},{\mathcal{A}}_{0})$, $\mathsf{V}(t)$ selfadjoint
$\forall t$. There exists a sequence $\\{X_{j}(t)\\}_{j\geq 1}$ of selfadjoint
(time-dependent) operators in ${\mathcal{H}}$ with $X_{j}\in
C^{\infty}({\mathbb{T}},{\mathcal{A}}_{1-j})$ and fulfilling

$\forall r\in{\mathbb{R}},\ \exists c_{r,j},C_{r,j}>0\colon\qquad
c_{r,j}\|{\varphi}\|_{r}\leq\|e^{\pm{\rm i}X_{j}(t)}{\varphi}\|_{r}\leq
C_{r,j}\|{\varphi}\|_{r},\qquad\forall t\in{\mathbb{R}},$ (3.3)

such that the following holds true. For any $N\geq 1$, the change of variables

$\psi=e^{-{\rm i}X_{1}(t)}\cdots e^{-{\rm i}X_{N}(t)}{\varphi}$ (3.4)

transforms (3.1) into the equation

${\rm
i}\partial_{t}{\varphi}=\big{(}K_{0}+Z^{(N)}(t)+\mathsf{V}^{(N)}(t)\big{)}{\varphi}\
;$ (3.5)

here $\mathsf{V}^{(N)}\in C^{\infty}({\mathbb{T}},{\mathcal{A}}_{-N})$ whereas
$Z^{(N)}\in C^{\infty}({\mathbb{T}},{\mathcal{A}}_{0})$, it is selfadjoint
$\forall t$, it fulfills

${\rm i}\partial_{t}Z^{(N)}(t)=[K_{0},Z^{(N)}(t)]$ (3.6)

and it has the expansion

$Z^{(N)}(t)={\widehat{\mathsf{V}}}(t)+T^{(N)}(t),\qquad T^{(N)}\in
C^{\infty}({\mathbb{T}},{\mathcal{A}}_{-1})\ .$ (3.7)

Here ${\widehat{\mathsf{V}}}(t)$ is the averaged operator defined in (3.2).

In order to prove the proposition we start with some preliminary results. The
first regards the properties of ${\widehat{\mathsf{V}}}(t)$.

######  Lemma 3.2.

Let $\mathsf{V}\in C^{\infty}({\mathbb{T}},{\mathcal{A}}_{m})$,
$m\in{\mathbb{R}}$, $\mathsf{V}(t)$ selfadjoint $\forall t$. Then the
following holds true.

  * (i)

${\widehat{\mathsf{V}}}\in C^{\infty}({\mathbb{T}},{\mathcal{A}}_{m})$, it is
selfadjoint $\forall t$, it commutes with ${\rm i}\partial_{t}-K_{0}$, i.e.
${\rm
i}\partial_{t}{\widehat{\mathsf{V}}}(t)=[K_{0},{\widehat{\mathsf{V}}}(t)]$ and

$\forall j,\ell\geq 0\quad\exists\,M\in{\mathbb{N}},\ C>0\ \ {\rm s.t.}\ \
\sup_{t\in{\mathbb{T}}}\wp^{m}_{j}(\partial_{t}^{\ell}{\widehat{\mathsf{V}}}(t))\leq
C\,\sup_{t\in{\mathbb{T}}}\ \wp^{m}_{M}(\mathsf{V}(t))\ .$ (3.8)

  * (ii)

The resonant averaged operator $\left\langle\mathsf{V}\right\rangle$, defined
in (1.3), belongs to ${\mathcal{A}}_{m}$, it is selfadjoint and

$\forall j\geq 0\quad\exists\,M\in{\mathbb{N}},\ C>0\ \ {\rm s.t.}\ \
\wp^{m}_{j}(\left\langle\mathsf{V}\right\rangle)\leq
C\,\sup_{t\in{\mathbb{T}}}\ \wp^{m}_{M}(\mathsf{V}(t))\ .$ (3.9)

  * (iii)

One has the chain of identities

${\widehat{\mathsf{V}}}(0)=\left\langle\mathsf{V}\right\rangle=e^{{\rm
i}tK_{0}}\,{\widehat{\mathsf{V}}}(t)\,e^{-{\rm
i}tK_{0}}=\langle\,{\widehat{\mathsf{V}}}\,\rangle,\qquad\forall
t\in{\mathbb{R}}\ .$ (3.10)

###### Proof.

$(i)$ The properties ${\widehat{\mathsf{V}}}\in
C^{\infty}({\mathbb{T}},{\mathcal{A}}_{m})$ and ${\widehat{\mathsf{V}}}(t)$
selfadjoint $\forall t$ follow from Assumption II and the fact that
$\mathsf{V}(t)$ is $2\pi$-periodic in $t$ and selfadjoint $\forall t$. Let us
prove it commutes with ${\rm i}\partial_{t}-K_{0}$. Using

$\partial_{s}\left(e^{{\rm i}sK_{0}}\,\mathsf{V}(t+s)\,e^{-{\rm
i}sK_{0}}\right)=e^{{\rm i}sK_{0}}\big{(}{\rm
i}[K_{0},\mathsf{V}(t+s)]+\partial_{s}\mathsf{V}(t+s)\big{)}e^{-{\rm
i}sK_{0}}$

we get

$\displaystyle\partial_{t}{\widehat{\mathsf{V}}}(t)$
$\displaystyle=\frac{1}{2\pi}\int_{0}^{2\pi}e^{{\rm
i}sK_{0}}\,\partial_{t}\mathsf{V}(t+s)\,e^{-{\rm i}sK_{0}}\,{\rm
d}s=\frac{1}{2\pi}\int_{0}^{2\pi}e^{{\rm
i}sK_{0}}\,\partial_{s}\mathsf{V}(t+s)\,e^{-{\rm i}sK_{0}}\,{\rm d}s$
$\displaystyle=\frac{1}{2\pi{\rm i}}\int_{0}^{2\pi}e^{{\rm
i}sK_{0}}\,[K_{0},\mathsf{V}(t+s)]\,e^{-{\rm i}sK_{0}}\,{\rm d}s={\rm
i}^{-1}\,[K_{0},{\widehat{\mathsf{V}}}(t)]$

where in the second line we used the periodicity of $s\mapsto e^{{\rm
i}sK_{0}}\,\mathsf{V}(t+s)\,e^{-{\rm i}sK_{0}}$ (see Remark 2.5) to remove the
boundary terms. Estimate (3.8) for $\ell=0$ follows from Assumption II. For
$\ell\geq 1$ we use induction: assume (3.8) is true up to a certain $\ell$;
using $\partial_{t}^{\ell+1}{\widehat{\mathsf{V}}}(t)=-{\rm
i}\partial_{t}^{\ell}[K_{0},{\widehat{\mathsf{V}}}(t)]=-{\rm
i}[K_{0},\partial_{t}^{\ell}{\widehat{\mathsf{V}}}(t)]$, we get $\forall
j\in{\mathbb{N}}$

$\wp_{j}^{m}(\partial_{t}^{\ell+1}{\widehat{\mathsf{V}}}(t))\leq\
\wp_{j}^{m}([K_{0},\partial_{t}^{\ell}{\widehat{\mathsf{V}}}(t)])\leq
C\wp_{j_{1}}^{m}(\partial_{t}^{\ell}{\widehat{\mathsf{V}}}(t))\leq
C\wp_{j_{2}}^{m}(\mathsf{V}(t))$

using also the inductive assumption. This proves (3.8).
$(ii)$ It is clear that $\left\langle\mathsf{V}\right\rangle$ is time
independent, selfadjoint and in ${\mathcal{A}}_{m}$ by Assumption II. Estimate
(3.9) follows from Assumption II.
$(iii)$ Clearly
${\widehat{\mathsf{V}}}(0)=\left\langle\mathsf{V}\right\rangle$. Then, as the
map $\tau\mapsto e^{{\rm i}\tau K_{0}}\,\mathsf{V}(\tau)\,e^{-{\rm i}\tau
K_{0}}$ is $2\pi$-periodic, one has $\forall t\in{\mathbb{R}}$

$e^{{\rm i}tK_{0}}\,{\widehat{\mathsf{V}}}(t)\,e^{-{\rm
i}tK_{0}}=\frac{1}{2\pi}\int_{0}^{2\pi}e^{{\rm
i}(t+s)K_{0}}\,\mathsf{V}(t+s)\,e^{-{\rm i}(s+t)K_{0}}\,{\rm
d}s=\left\langle\mathsf{V}\right\rangle\ .$

Finally, exploiting this last identity, one has

$\langle\,{\widehat{\mathsf{V}}}\,\rangle=\frac{1}{2\pi}\int_{0}^{2\pi}e^{{\rm
i}tK_{0}}\,{\widehat{\mathsf{V}}}(t)\,e^{-{\rm i}tK_{0}}{\rm
d}t=\frac{1}{2\pi}\int_{0}^{2\pi}\left\langle\mathsf{V}\right\rangle{\rm
d}t=\left\langle\mathsf{V}\right\rangle$

which completes the proof of (3.10). ∎

The second preliminary result regards how to solve the homological equations
which appear during the normal form procedure. More precisely we look for a
time periodic operator $X(t)$ solving the homological equation

$\partial_{t}X(t)+{\rm
i}[K_{0},X(t)]=\mathsf{V}(t)-{\widehat{\mathsf{V}}}(t),$ (3.11)

where ${\widehat{\mathsf{V}}}(t)$ is the averaged operator defined in (3.2).
This is done in the next lemma.

######  Lemma 3.3.

Let $\mathsf{V}\in C^{\infty}({\mathbb{T}},{\mathcal{A}}_{m})$,
$m\in{\mathbb{R}}$, $\mathsf{V}(t)$ selfadjoint $\forall t$. The homological
equation (3.11) has a solution $X\in
C^{\infty}({\mathbb{T}},{\mathcal{A}}_{m})$ and $X(t)$ is selfadjoint $\forall
t$.

###### Proof.

We look for a solution of (3.11) using the method of variation of constants.
In particular we take $X(t)=e^{-{\rm i}tK_{0}}\,Y(t)\,e^{{\rm i}tK_{0}}$ for
some $Y\in C^{\infty}({\mathbb{R}},{\mathcal{A}}_{m})$ with $Y(0)=0$ to be
determined. Then $X$ solves (3.11) provided $\partial_{t}Y(t)=e^{{\rm
i}tK_{0}}\,(\mathsf{V}(t)-{\widehat{\mathsf{V}}}(t))\,e^{-{\rm i}tK_{0}}$,
giving

$Y(t)=\int_{0}^{t}e^{{\rm
i}sK_{0}}\big{(}\mathsf{V}(s)-{\widehat{\mathsf{V}}}(s)\big{)}\,e^{-{\rm
i}sK_{0}}\,{\rm d}s.$

By Lemma 3.2 and Assumption II, $Y\in
C^{\infty}({\mathbb{R}},{\mathcal{A}}_{m})$ and it is selfadjoint $\forall t$.
Therefore one gets

$X(t)=\int_{0}^{t}e^{{\rm
i}(s-t)K_{0}}\big{(}\mathsf{V}(s)-{\widehat{\mathsf{V}}}(s)\big{)}\,e^{-{\rm
i}(s-t)K_{0}}\,{\rm d}s.$

Again $X\in C^{\infty}({\mathbb{R}},{\mathcal{A}}_{m})$ and it is selfadjoint
$\forall t$. Finally (recall Remark 2.5)

$\displaystyle X(t+2\pi)-X(t)$ $\displaystyle=\int_{t}^{t+2\pi}e^{{\rm
i}(s-t)K_{0}}\,\big{(}\mathsf{V}(s)-{\widehat{\mathsf{V}}}(s)\big{)}\,e^{-{\rm
i}(s-t)K_{0}}\,{\rm d}s$ $\displaystyle=e^{-{\rm
i}tK_{0}}\int_{0}^{2\pi}e^{{\rm
i}sK_{0}}\,\big{(}\mathsf{V}(s)-{\widehat{\mathsf{V}}}(s)\big{)}\,e^{-{\rm
i}sK_{0}}\,{\rm d}s\ e^{{\rm i}tK_{0}}$ $\displaystyle=2\pi e^{-{\rm
i}tK_{0}}\big{(}\left\langle\mathsf{V}\right\rangle-\langle\,{\widehat{\mathsf{V}}}\,\rangle\big{)}e^{{\rm
i}tK_{0}}\stackrel{{\scriptstyle\eqref{av.W}}}{{=}}0$

which proves the periodicity of $t\mapsto X(t)$. ∎

We are ready to prove Proposition 3.1. During the proof we shall use some
results proved in [5] about the flow generated by pseudodifferential
operators; we collect them, for the reader’s convenience, in Appendix A.

###### Proof of Proposition 3.1.

The proof is inductive on $N$. Let us start with $N=1$. We look for a change
of variables of the form $\psi=e^{-{\rm i}X_{1}(t)}{\varphi}$ where
$X_{1}(t)\in C^{\infty}({\mathbb{T}},{\mathcal{A}}_{0})$ is selfadjoint
$\forall t$, to be determined. By Lemma A.1, ${\varphi}$ fulfills the
Schrödinger equation ${\rm i}\partial_{t}{\varphi}=H^{+}(t){\varphi}$ with

$\displaystyle H^{+}(t)$ $\displaystyle:=e^{{\rm
i}X_{1}(t)}\,\big{(}K_{0}+\mathsf{V}(t)\big{)}\,e^{-{\rm
i}X_{1}(t)}-\int_{0}^{1}e^{{\rm i}sX_{1}(t)}\,(\partial_{t}X_{1}(t))\,e^{-{\rm
i}sX_{1}(t)}\ {\rm d}s\ .$

Then a commutator expansion, see Lemma A.2, gives

$\displaystyle H^{+}(t)$ $\displaystyle=K_{0}+{\rm
i}[X_{1}(t),K_{0}]+\mathsf{V}(t)-\partial_{t}X_{1}+\mathsf{V}^{(1)}(t)$

with $\mathsf{V}^{(1)}\in C^{\infty}({\mathbb{T}},{\mathcal{A}}_{-1})$,
selfadjoint $\forall t$. By Lemma 3.3, we choose $X_{1}\in
C^{\infty}({\mathbb{T}},{\mathcal{A}}_{0})$, selfadjoint $\forall t$, s.t.

${\rm
i}[K_{0},X_{1}(t)]+\partial_{t}X_{1}(t)=\mathsf{V}(t)-{\widehat{\mathsf{V}}}(t)\
,$ (3.12)

where ${\widehat{\mathsf{V}}}(t)$ is the averaged operator (see (3.2)). With
this choice we have

$\displaystyle H^{+}(t)$ $\displaystyle=K_{0}+Z^{(1)}(t)+\mathsf{V}^{(1)}(t)\
,\quad Z^{(1)}(t):={\widehat{\mathsf{V}}}(t)\ .$ (3.13)

By Lemma 3.2, $Z^{(1)}\in C^{\infty}({\mathbb{T}},{\mathcal{A}}_{0})$, it is
selfadjoint $\forall t$, it commutes with ${\rm i}\partial_{t}-K_{0}$. The map
$e^{-{\rm i}X_{1}(t)}$ fulfills (3.3) thanks to Lemma A.3. This concludes the
first step.

The iterative step $N\to N+1$ is proved following the same lines, just adding
the remark that $e^{{\rm i}X_{N+1}}Z^{(N)}e^{-{\rm i}X_{N+1}}-Z^{(N)}\in
C^{\infty}({\mathbb{T}},{\mathcal{A}}_{-N-1})$, and solving the homological
equation

${\rm
i}[K_{0},X_{N+1}(t)]+\partial_{t}X_{N+1}(t)=\mathsf{V}^{(N)}(t)-{\widehat{\mathsf{V}^{(N)}}}(t)\
.$ (3.14)

So one puts $Z^{(N+1)}:=Z^{(N)}+{\widehat{\mathsf{V}^{(N)}}}$. Note that
${\widehat{\mathsf{V}^{(N)}}}\in C^{\infty}({\mathbb{T}},{\mathcal{A}}_{-N})$,
so $Z^{(N)}$ has an expansion in operators of decreasing order.

∎

It turns out that property (3.6) implies that $e^{{\rm
i}tK_{0}}\,Z^{(N)}(t)\,e^{-{\rm i}tK_{0}}$ is time independent. A consequence
of this fact is the following corollary.

######  Corollary 3.4.

Consider equation (3.1) with $\mathsf{V}\in
C^{\infty}({\mathbb{T}},{\mathcal{A}}_{0})$, $\mathsf{V}(t)$ selfadjoint
$\forall t$. Fix $N\in{\mathbb{N}}$ arbitrary. There exists a change of
coordinates ${\mathcal{U}}_{N}(t)$ unitary in ${\mathcal{H}}$ and fulfilling

$\forall r\geq 0\quad\exists c_{r},C_{r}>0\colon\qquad
c_{r}\|{\varphi}\|_{r}\leq\|{\mathcal{U}}_{N}(t)^{\pm}{\varphi}\|_{r}\leq
C_{r}\|{\varphi}\|_{r},\qquad\forall t\in{\mathbb{R}},$ (3.15)

such that $\psi(t)$ is a solution of (3.1) if and only if
$\phi(t):={\mathcal{U}}_{N}(t)\psi(t)$ solves

${\rm
i}\partial_{t}\phi=\big{(}\left\langle\mathsf{V}\right\rangle+T_{N}+R_{N}(t)\big{)}\phi\
;$ (3.16)

here $\left\langle\mathsf{V}\right\rangle$ is the resonant average of
$\mathsf{V}$ (see (1.3)), $T_{N}\in{\mathcal{A}}_{-1}$ is time independent and
selfadjoint and $R_{N}\in C^{\infty}({\mathbb{T}},{\mathcal{A}}_{-N})$.

###### Proof.

Fix $N\in{\mathbb{N}}$ and apply Proposition 3.1 to conjugate equation (3.1)
to the form (3.5) via the change of variables (3.4). Then we gauge away
$K_{0}$ by the change of coordinates ${\varphi}=e^{-{\rm i}tK_{0}}\phi$,
getting

${\rm i}\partial_{t}\phi=e^{{\rm
i}tK_{0}}\,\big{(}Z^{(N)}(t)+\mathsf{V}^{(N)}(t)\big{)}\,e^{-{\rm
i}tK_{0}}\,\phi.$

Define

$\mathsf{H}_{N}:=e^{{\rm i}tK_{0}}\,Z^{(N)}(t)\,e^{-{\rm i}tK_{0}},\qquad
R_{N}(t):=e^{{\rm i}tK_{0}}\,\mathsf{V}^{(N)}(t)\,e^{-{\rm i}tK_{0}}.$

The operator $R_{N}\in C^{\infty}({\mathbb{T}},{\mathcal{A}}_{-N})$ by
Assumption II since $\mathsf{V}^{(N)}\in
C^{\infty}({\mathbb{T}},{\mathcal{A}}_{-N})$.
Let us now prove that $\mathsf{H}_{N}$ is time independent. We know by Lemma
3.1 that $Z^{(N)}(t)$ commutes with ${\rm i}\partial_{t}-K_{0}$; therefore

$\partial_{t}\big{(}e^{{\rm i}tK_{0}}\,Z^{(N)}(t)\,e^{-{\rm
i}tK_{0}}\big{)}=e^{{\rm i}tK_{0}}\,\big{(}{\rm
i}[K_{0},Z^{(N)}(t)]+\partial_{t}Z^{(N)}(t)\big{)}\,e^{-{\rm i}tK_{0}}=0$

and we get

$\mathsf{H}_{N}=e^{{\rm i}tK_{0}}\,Z^{(N)}(t)\,e^{-{\rm
i}tK_{0}}|_{t=0}=Z^{(N)}(0)\stackrel{{\scriptstyle\eqref{Z.exp}}}{{=}}{\widehat{\mathsf{V}}}(0)+T^{(N)}(0)\stackrel{{\scriptstyle\eqref{av.W}}}{{=}}\left\langle\mathsf{V}\right\rangle+T^{(N)}(0).$

So we put $T_{N}:=T^{(N)}(0)$; clearly it belongs to ${\mathcal{A}}_{-1}$, it
is selfadjoint and time independent.

Finally we put ${\mathcal{U}}_{N}(t):=e^{{\rm i}tK_{0}}\,e^{{\rm
i}tX_{N}(t)}\,\cdots e^{{\rm i}tX_{1}(t)}$; estimate (3.15) follows from (3.3)
and Remark 2.1. ∎

Coming back to the original equation (2.11), we apply Corollary 3.4 with
$\mathsf{V}=V+W\in C^{\infty}({\mathbb{T}},{\mathcal{A}}_{0})$, getting the
following result:

######  Corollary 3.5.

With the same assumptions of Theorem 2.8, the following holds true. Fix
$N\in{\mathbb{N}}$ arbitrary. There exists a change of coordinates
${\mathcal{U}}_{N}(t)$, unitary in ${\mathcal{H}}$ and fulfilling (3.15) such
that $\psi(t)$ is a solution of (2.11) if and only if
$\phi(t):={\mathcal{U}}_{N}(t)\psi(t)$ solves

${\rm i}\partial_{t}\phi=\big{(}\left\langle V\right\rangle+\left\langle
W\right\rangle+T_{N}+R_{N}(t)\big{)}\phi$ (3.17)

where $T_{N}\in{\mathcal{A}}_{-1}$ is selfadjoint and time independent whereas
$R_{N}\in C^{\infty}({\mathbb{T}},{\mathcal{A}}_{-N})$.

###  3.2 Local energy decay estimates

From now on we are going to assume also Assumption III. In the previous
section we have conjugated the original equation (2.11) to the resonant
equation (3.17). In this section we consider the effective equation obtained
removing $R_{N}(t)$ from (3.17), namely

${\rm i}\partial_{t}{\varphi}=H_{N}{\varphi},\qquad H_{N}:=\left\langle
V\right\rangle+\left\langle W\right\rangle+T_{N},$ (3.18)

with $T_{N}\in{\mathcal{A}}_{-1}$ of Corollary 3.5. Note that $H_{N}$ is
selfadjoint by Lemma 3.2 and Corollary 3.5. The goal is to construct a
solution of (3.18) with polynomially in time growing Sobolev norms. Actually
we will prove the following slightly stronger result, namely the existence of
a solution with decaying negative Sobolev norms:

######  Proposition 3.6 (Decay of negative Sobolev norms).

With the same assumptions of Theorem 2.8, consider the operator $H_{N}$ in
(3.18). For any $k\in{\mathbb{N}}$, there exist a nontrivial solution
${\varphi}(t)\in{\mathcal{H}}^{k}$ of (3.18) and $\forall r\in[0,k]$ a
constant $C_{r}>0$ such that

$\|{\varphi}(t)\|_{{-r}}\leq C_{r}\left\langle
t\right\rangle^{-r}\,\|{\varphi}(0)\|_{r}\ ,\qquad\forall t\in{\mathbb{R}}\ .$
(3.19)

######  Remark 3.7.

As $H_{N}$ is selfadjoint, the conservation of the ${\mathcal{H}}^{0}$-norm
and Cauchy-Schwartz inequality give

$\|{\varphi}(0)\|_{0}^{2}=\|{\varphi}(t)\|_{0}^{2}\leq\|{\varphi}(t)\|_{r}\
\|{\varphi}(t)\|_{{-r}}\ ,\qquad\forall t\in{\mathbb{R}}\ ,$

so that (3.19) implies the growth of positive Sobolev norms:

$\|{\varphi}(t)\|_{r}\geq\frac{1}{C_{r}}\frac{\|{\varphi}(0)\|_{0}^{2}}{\|{\varphi}(0)\|_{r}}\,\left\langle
t\right\rangle^{r}\ ,\quad\forall t\in{\mathbb{R}}\ .$

The rest of the section is devoted to the proof of Proposition 3.6. As we
shall see, it follows from a local energy decay estimate for the operator
$H_{N}$, namely a dispersive estimate of the form

$\|\left\langle A\right\rangle^{-k}\,e^{-{\rm
i}H_{N}t}\,g_{J}(H_{N})\,{\varphi}\|_{0}\leq C_{k}\left\langle
t\right\rangle^{-k}\|\left\langle
A\right\rangle^{k}g_{J}(H_{N}){\varphi}\|_{0}\ ,\qquad\forall
t\in{\mathbb{R}}$ (3.20)

where $A\in{\mathcal{A}}_{1}$, $J\subset I_{0}$ is an interval and $g_{J}\in
C^{\infty}_{c}({\mathbb{R}},{\mathbb{R}}_{\geq 0})$ with $g_{J}\equiv 1$ on
$J$.

######  Remark 3.8.

Actually estimate (3.20) show the existence of infinitely many solutions of
(3.18) with decaying negative Sobolev norms. In particular this happens to any
solution whose initial datum ${\varphi}(0)$ belongs to the (infinite
dimensional) set ${\rm Ran}\,E_{J}(H_{N})$, where $E_{J}(H_{N})$ is the
spectral projection of $H_{N}$ corresponding to the interval $J$.

A possible approach (which we will follow here) to obtain such estimate is via
Sigal-Soffer minimal velocity estimates [42, 23, 31, 30, 24, 2]. These
estimates are based on Mourre theory, let us recall this last one.

#### Mourre theory.

Let $\mathsf{H}$ be a selfadjoint operator on the Hilbert space
${\mathcal{H}}$, and denote by $\sigma(\mathsf{H})$ its spectrum. We further
denote by $\sigma_{d}(\mathsf{H})$ its discrete spectrum,
$\sigma_{ess}(\mathsf{H})$ its essential spectrum, $\sigma_{pp}(\mathsf{H})$
its pure point spectrum, $\sigma_{ac}(\mathsf{H})$ its absolutely continuous
spectrum and $\sigma_{sc}(\mathsf{H})$ its singular spectrum; see e.g. [38]
pag. 236 and 231 for their definitions. Furthermore we denote by
$E_{\Omega}(\mathsf{H})$ the spectral projection of $\mathsf{H}$ corresponding
to the Borel set $\Omega$ and by $m_{\varphi}(\Omega):=\left\langle
E_{\Omega}(\mathsf{H}){\varphi},{\varphi}\right\rangle$ the spectral measure
associated to ${\varphi}\in{\mathcal{H}}$.

Assume a selfadjoint operator $\mathsf{A}$ can be found such that
$D(\mathsf{A})\cap{\mathcal{H}}$ is dense in ${\mathcal{H}}$. We put

${\rm ad}^{0}_{\mathsf{A}}(\mathsf{H}):=\mathsf{H},\qquad{\rm
ad}_{\mathsf{A}}(\mathsf{H}):=[\mathsf{H},\mathsf{A}],\qquad{\rm
ad}^{n}_{\mathsf{A}}(\mathsf{H}):=[{\rm
ad}^{n-1}_{\mathsf{A}}(\mathsf{H}),\mathsf{A}],\quad\forall n\geq 2\ .$ (3.21)

Consider the following properties:

  * (M1)

For some ${\mathtt{N}}\geq 1$, the operators ${\rm
ad}^{n}_{\mathsf{A}}(\mathsf{H})$ with $n=1,\ldots,{\mathtt{N}}$, can all be
extended to bounded operators on ${\mathcal{H}}$.

  * (M2)

Mourre estimate: there exists an open interval $I\subset{\mathbb{R}}$ with
compact closure and a function $g_{I}\in
C^{\infty}_{c}({\mathbb{R}},{\mathbb{R}}_{\geq 0})$ with $g_{I}\equiv 1$ on
$I$ such that

$g_{I}(\mathsf{H})\,{\rm
i}[\mathsf{H},\mathsf{A}]\,g_{I}(\mathsf{H})\geq\theta
g_{I}(\mathsf{H})^{2}+\mathsf{K}$ (3.22)

for some $\theta>0$ and $\mathsf{K}$ a selfadjoint compact operator on
${\mathcal{H}}$.

If the estimate (3.22) holds true with $\mathsf{K}=0$, we shall say that
$\mathsf{H}$ fulfills a strict Mourre estimate.
Mourre theorem [36] says the following:

######  Theorem 3.9 (Mourre).

Assume conditions (M1) – (M2) with ${\mathtt{N}}=2$. In the interval $I$, the
operator $\mathsf{H}$ can have only absolutely continuous spectrum and
finitely many eigenvalues of finite multiplicity. If $\mathsf{K}=0$, there are
no eigenvalues in the interval $I$, i.e. $\sigma(\mathsf{H})\cap
I=\sigma_{ac}(\mathsf{H})\cap I$.

######  Remark 3.10.

The version stated here of Mourre theorem is taken from [3, Lemma 5.6] and
[14, Theorem 4.7 – 4.9], and it has slightly weaker assumptions compared to
[36].

######  Remark 3.11.

Mourre theorem guarantees that $\sigma_{sc}(\mathsf{H})\cap I=\emptyset$ and,
in case $\mathsf{K}=0$, $\sigma_{pp}(\mathsf{H})\cap I=\emptyset$. However it
does not guarantee that $\sigma(\mathsf{H})\cap I\neq\emptyset$; in our case
we shall verify this property explicitly.

The key point is that if $H_{N}$ fulfills a strict Mourre estimate (namely
with $\mathsf{K}=0$) then one can prove a local energy decay estimate like
(3.20) for the Schrödinger flow of $H_{N}$. This is a quite general fact which
follows exploiting minimal velocity estimates [30] and we prove it for
completeness in Appendix C.

So the next goal is to prove that $H_{N}$ satisfies a strict Mourre estimate
over a certain interval $J\subset I_{0}$. During the proof we will use some
standard results from functional calculus; we recall them in Appendix B. We
shall also use the following lemma:

######  Lemma 3.12.

Let $\mathsf{H}\in{\mathcal{L}}({\mathcal{H}})$ be selfadjoint. If
$\lambda\in\sigma_{ac}(\mathsf{H})$, then $\forall\delta>0$ one has

$\left|[\lambda-\delta,\lambda+\delta]\cap\sigma(\mathsf{H})\right|>0\ .$

###### Proof.

By contradiction, assume that $\exists\delta_{0}>0$ such that
$\left|[\lambda-\delta_{0},\lambda+\delta_{0}]\cap\sigma(\mathsf{H})\right|=0$.
As $\lambda\in\sigma_{ac}(\mathsf{H})$, there exists $f\in{\mathcal{H}}$ such
that $E_{[\lambda-\delta_{0},\lambda+\delta_{0}]}(\mathsf{H})f\neq 0$ and the
spectral measure $m_{f}=\left\langle E(\mathsf{H})f,f\right\rangle$ is
absolutely continuous. Then

$0=m_{f}([\lambda-\delta_{0},\lambda+\delta_{0}])=\left\langle
E_{[\lambda-\delta_{0},\lambda+\delta_{0}]}(\mathsf{H})f,f\right\rangle=\|E_{[\lambda-\delta_{0},\lambda+\delta_{0}]}(\mathsf{H})f\|_{0}^{2}>0$

giving a contradiction. ∎

######  Lemma 3.13.

There exist $\epsilon_{0},{\mathtt{M}}>0$ such that, provided $W$ fulfills
(2.10), the following holds true:

  * (i)

There exists an interval $I\subset I_{0}$ such that
$\left|I\cap\sigma(H_{N})\right|>0$.

  * (ii)

$H_{N}$ fulfills a strict Mourre estimate over $I$: there exists a function
$g_{I}\in C^{\infty}_{c}({\mathbb{R}},{\mathbb{R}}_{\geq 0})$ with ${\rm
supp}\,g_{I}\subset I_{0}$, $g_{I}\equiv 1$ on $I$, and $\theta^{\prime}>0$
such that

$g_{I}(H_{N})\,{\rm
i}[H_{N},A]\,g_{I}(H_{N})\geq\theta^{\prime}g_{I}(H_{N})^{2}\ .$ (3.23)

Here $I_{0}$ is the interval and $A$ is the operator of Assumption III.

###### Proof.

During the proof we shall often use that for
$\mathsf{A},\mathsf{B},\mathsf{C}\in{\mathcal{L}}({\mathcal{H}})$ and
selfadjoints

$\displaystyle\mathsf{A}\leq\mathsf{B}\ \ \Rightarrow\ \
\mathsf{C}\mathsf{A}\mathsf{C}\leq\mathsf{C}\mathsf{B}\mathsf{C},\qquad\|\mathsf{A}\|_{{\mathcal{L}}({\mathcal{H}})}\leq
a\ \ \Rightarrow\ \ -a\leq\mathsf{A}\leq a\ .$ (3.24)

To shorten notation, throughout the proof we shall put

$H_{0}:=\left\langle V\right\rangle\ .$

We split the proof in several steps.

Step 1: By Assumption III, $H_{0}$ fulfills a Mourre estimate over the
interval $I_{0}$. The first step of the proof is to exhibit a subinterval
$I_{1}\subset I_{0}$ containing only absolutely continuous spectrum of
$H_{0}$, namely

$\sigma(H_{0})\cap I_{1}=\sigma_{ac}(H_{0})\cap I_{1}\
,\qquad\left|\sigma(H_{0})\cap I_{1}\right|>0\ ,$ (3.25)

and over which $H_{0}$ fulfills a strict Mourre estimate: $\exists
g_{I_{1}}\in C^{\infty}_{c}({\mathbb{R}},{\mathbb{R}}_{\geq 0})$,
$g_{I_{1}}\equiv 1$ on $I_{1}$, ${\rm supp}\,g_{I_{1}}\subset I_{0}$, such
that

$g_{I_{1}}(H_{0})\,{\rm
i}[H_{0},A]\,g_{I_{1}}(H_{0})\geq\frac{\theta}{2}g_{I_{1}}(H_{0})^{2}\ .$
(3.26)

To prove this claim, first apply Mourre theorem to $H_{0}$ (note that (M1) and
(M2) are verified $\forall{\mathtt{N}}\in{\mathbb{N}}$ by symbolic calculus
and Assumption III), getting that $\sigma(H_{0})\cap I_{0}$ contains only
finitely many eigenvalues with finite multiplicity and absolutely continuous
spectrum. In particular $|\overline{\sigma_{pp}(H_{0})}\cap I_{0}|=0$ and by
Assumption III $(i)$ it follows that $|\sigma_{ac}(H_{0})\cap
I_{0}|=|\sigma(H_{0})\cap I_{0}|>0$.
So we take $\lambda_{0}\in
I_{0}\cap(\sigma_{ac}(H_{0})\setminus\sigma_{pp}(H_{0}))$ and a sufficiently
small interval
$I_{1}(\overline{\delta}):=(\lambda_{0}-\overline{\delta},\lambda_{0}+\overline{\delta})\subset
I_{0}$, $\overline{\delta}>0$, which does not contain eigenvalues of $H_{0}$;
this is possible as the eigenvalues of $H_{0}$ in $I_{0}$ are finite. Moreover
by Lemma 3.12, $\left|\sigma(H_{0})\cap I_{1}(\delta)\right|>0$ for any
$\delta>0$. Now take $\delta\in(0,\overline{\delta})$ and a function
$g_{\delta}\in C^{\infty}_{c}({\mathbb{R}},{\mathbb{R}}_{\geq 0})$ with ${\rm
supp}\,g_{\delta}\subset I_{1}(\delta)$ and $g_{\delta}=1$ on
$I_{1}(\frac{\delta}{2})$. We claim that provided
$\delta\in(0,\overline{\delta})$ is sufficiently small

$\|g_{\delta}(H_{0})Kg_{\delta}(H_{0})\|_{{\mathcal{L}}({\mathcal{H}})}\leq\frac{\theta}{2}\
,$ (3.27)

where $\theta>0$ is the one of Assumption III. Indeed in
$I_{1}(\overline{\delta})$ the spectrum of $H_{0}$ is absolutely continuous;
this means that $\forall{\varphi}\in{\mathcal{H}}$, the vector
${\varphi}^{\prime}:=E_{I_{1}(\overline{\delta})}(H_{0}){\varphi}$ belongs to
the absolutely continuous subspace of $H_{0}$, namely its spectral measure
$m_{{\varphi}^{\prime}}$ is absolutely continuous w.r.t. the Lebesgue measure.
Now, since for any ${\varphi}\in{\mathcal{H}}$ one has by functional calculus
$g_{\delta}(H_{0})=g_{\delta}(H_{0})E_{I_{1}(\overline{\delta})}(H_{0})$, one
has that

$\|g_{\delta}(H_{0}){\varphi}\|_{0}^{2}=\|g_{\delta}(H_{0})E_{I_{1}(\overline{\delta})}(H_{0}){\varphi}\|_{0}^{2}=\int_{\mathbb{R}}g_{\delta}(\lambda)^{2}\,{\rm
d}m_{{\varphi}^{\prime}}(\lambda)\to 0\ \ \ \mbox{ as }\delta\to 0$

by Lebesgue dominated convergence theorem. In particular $g_{\delta}(H_{0})\to
0$ strongly as $\delta\to 0$ and then, being $K$ compact,
$g_{\delta}(H_{0})K\to 0$ uniformly as $\delta\to 0$ (see e.g. [1]). Therefore
for $\delta\in(0,\overline{\delta})$ sufficiently small (3.27) holds true.
Using the assumption (2.7), (3.27) and (3.24) we deduce that

$\displaystyle g_{\delta}(H_{0})\,g_{I_{0}}(H_{0})\,{\rm
i}[H_{0},A]\,g_{I_{0}}(H_{0})\,g_{\delta}(H_{0})$ $\displaystyle\geq\theta
g_{\delta}(H_{0})\,g_{I_{0}}(H_{0})^{2}\,g_{\delta}(H_{0})-\frac{\theta}{2}\
;$

next apply $g_{\frac{\delta}{2}}(H_{0})$ to the right and left of the previous
inequality, use again (3.24) and the identity
$g_{I_{0}}(H_{0})\,g_{\delta}(H_{0})\,g_{\frac{\delta}{2}}(H_{0})=g_{\frac{\delta}{2}}(H_{0})$
(which follows from
$g_{I_{0}}\,g_{\delta}\,g_{\frac{\delta}{2}}=g_{\frac{\delta}{2}}$), to get
the strict Mourre estimate

$g_{I_{1}}(H_{0})\,{\rm
i}[H_{0},A]\,g_{I_{1}}(H_{0})\geq\frac{\theta}{2}g_{I_{1}}(H_{0})^{2}$ (3.28)

where $I_{1}:=I_{1}(\frac{\delta}{4})$ and $g_{I_{1}}:=g_{\frac{\delta}{2}}$
fulfills $g_{I_{1}}\equiv 1$ on $I_{1}$, ${\rm supp}\,g_{I_{1}}\subset
I_{1}(\frac{\delta}{2})$. Clearly $I_{1}$ fulfills (3.25).

Step 2: We shall prove that the selfadjoint operator

$H_{\left\langle W\right\rangle}:=H_{0}+\left\langle W\right\rangle$

has a nontrivial spectrum in a subinterval $I_{2}\subseteq I_{1}$, and over
this interval it fulfills the strict Mourre estimate

$g_{I_{2}}\big{(}H_{\left\langle W\right\rangle}\big{)}\,{\rm
i}[H_{\left\langle W\right\rangle},A]\,g_{I_{2}}\big{(}H_{\left\langle
W\right\rangle}\big{)}\geq\frac{\theta}{4}g_{I_{2}}\big{(}H_{\left\langle
W\right\rangle}\big{)}^{2}\ $ (3.29)

for any $g_{I_{2}}\in C^{\infty}_{c}({\mathbb{R}},{\mathbb{R}}_{\geq 0})$ with
${\rm supp}\,g_{I_{2}}\subset I_{1}$, $g_{I_{2}}\equiv 1$ on $I_{2}$. To prove
this, we exploit that $\left\langle W\right\rangle\in{\mathcal{A}}_{0}$ is a
small bounded perturbation of $H_{0}$, fulfilling, by (2.1), (3.9)

$\exists M_{0}\in{\mathbb{N}},C_{0}>0\colon\qquad\|\left\langle
W\right\rangle\|_{{\mathcal{L}}({\mathcal{H}})}{\leq}\,C_{0}[W]_{M_{0}},$
(3.30)

where we denoted

$[W]_{M}:=\sup_{t\in{\mathbb{T}}}\,\wp^{0}_{M}(W(t))\ .$

First let us prove that $\sigma(H_{\left\langle W\right\rangle})\cap
I_{1}\neq\emptyset$. Take again the same $\lambda_{0}\in\sigma(H_{0})\cap
I_{1}$ as in the previous step. We claim that

${\rm dist}\big{(}\lambda_{0},\sigma(H_{\left\langle
W\right\rangle})\big{)}\leq C_{0}\,[W]_{M_{0}}.$ (3.31)

If $\lambda_{0}\in\sigma(H_{\left\langle W\right\rangle})$ this is trivial. So
assume that $\lambda_{0}$ belongs to the resolvent set of $H_{\left\langle
W\right\rangle}$. As $\lambda_{0}\in\sigma(H_{0})$, by Weyl criterion
$\exists(f_{n})_{n\geq 1}\in{\mathcal{H}}$ with $\|f_{n}\|_{0}=1$ such that
$\|(H_{0}-\lambda_{0})f_{n}\|_{0}\to 0$ as $n\to\infty$. Then $\forall n\geq
1$

$\displaystyle 1$ $\displaystyle=\|f_{n}\|_{0}=\|(H_{\left\langle
W\right\rangle}-\lambda_{0})^{-1}\,(H_{\left\langle
W\right\rangle}-\lambda_{0})f_{n}\|_{0}\leq\frac{1}{{\rm
dist}\big{(}\lambda_{0},\sigma(H_{\left\langle
W\right\rangle})\big{)}}\|(H_{\left\langle
W\right\rangle}-\lambda_{0})f_{n}\|_{0}$
$\displaystyle\stackrel{{\scriptstyle\eqref{pm.s20}}}{{\leq}}\frac{1}{{\rm
dist}\big{(}\lambda_{0},\sigma(H_{\left\langle
W\right\rangle})\big{)}}\Big{(}\|(H_{0}-\lambda_{0})f_{n}\|_{0}+C_{0}[W]_{M_{0}}\Big{)}$

which proves (3.31) passing to the limit $n\to\infty$. Then, provided
$[W]_{M_{0}}$ is sufficiently small, (3.31) implies that ${\rm
dist}\big{(}\lambda_{0},\sigma(H_{\left\langle
W\right\rangle})\big{)}<\delta/8$. From this we learn that (recall
$I_{1}=(\lambda_{0}-\frac{\delta}{4},\lambda_{0}+\frac{\delta}{4})$)

$\sigma(H_{\left\langle W\right\rangle})\cap I_{1}\neq\emptyset\ .$ (3.32)

Next we prove the Mourre estimate (3.29); we shall work perturbatively from
(3.26). First

$\displaystyle g_{I_{1}}(H_{0})\,{\rm i}[H_{\left\langle
W\right\rangle},A]\,g_{I_{1}}(H_{0})=g_{I_{1}}(H_{0})\,{\rm
i}[H_{0},A]\,g_{I_{1}}(H_{0})+g_{I_{1}}(H_{0})\,{\rm i}[\left\langle
W\right\rangle,A]\,g_{I_{1}}(H_{0});$

we bound the first term in the right hand side above from below using (3.26).
Concerning the second term, we use

$\displaystyle\exists\,M_{1}\in{\mathbb{N}},\,C_{1}>0\colon\quad\|{\rm
i}[\left\langle W\right\rangle,A]\|_{{\mathcal{L}}({\mathcal{H}})}\leq
C_{1}[W]_{M_{1}}\ $ (3.33)

(by (2.1), (2.3), (3.9)) and the inequalities (3.24) to bound it from above
getting

$g_{I_{1}}(H_{0})\,{\rm i}[\left\langle
W\right\rangle,A]\,g_{I_{1}}(H_{0})\geq-
C_{1}\,[W]_{M_{1}}\,g_{I_{1}}(H_{0})^{2}\ .$

Therefore we find

$\displaystyle g_{I_{1}}(H_{0})\,{\rm i}[H_{\left\langle
W\right\rangle},A]\,g_{I_{1}}(H_{0})\geq\left(\frac{\theta}{2}-C_{1}[W]_{M_{1}}\right)\,g_{I_{1}}(H_{0})^{2}\
.$ (3.34)

Take now an open interval $I_{2}\subset I_{1}$ such that
$\sigma(H_{\left\langle W\right\rangle})\cap{I}_{2}\neq\emptyset$ (it is
possible by (3.32)); take also $g_{I_{2}}\in
C^{\infty}_{c}({\mathbb{R}},{\mathbb{R}}_{\geq 0})$ with ${\rm
supp}\,g_{I_{2}}\subseteq I_{1}$ and $g_{I_{2}}\equiv 1$ on $I_{2}$; remark
that $g_{I_{1}}g_{I_{2}}=g_{I_{2}}$. Now we wish to replace $g_{I_{1}}(H_{0})$
by $g_{I_{2}}(H_{\left\langle W\right\rangle})$ in (3.34), thus getting the
claimed estimate (3.29). So write

$\displaystyle g_{I_{2}}(H_{\left\langle W\right\rangle})\,{\rm
i}[H_{\left\langle W\right\rangle},A]$
$\displaystyle\,g_{I_{2}}(H_{\left\langle
W\right\rangle})=g_{I_{2}}(H_{\left\langle
W\right\rangle})\,g_{I_{1}}(H_{\left\langle W\right\rangle})\,{\rm
i}[H_{\left\langle W\right\rangle},A]\,g_{I_{1}}(H_{\left\langle
W\right\rangle})\,g_{I_{2}}(H_{\left\langle W\right\rangle})$
$\displaystyle=g_{I_{2}}(H_{\left\langle
W\right\rangle})\,g_{I_{1}}(H_{0})\,{\rm i}[H_{\left\langle
W\right\rangle},A]\,g_{I_{1}}(H_{0})\,g_{I_{2}}(H_{\left\langle
W\right\rangle})$ (3.35) $\displaystyle\ +g_{I_{2}}(H_{\left\langle
W\right\rangle})\,\Big{(}\big{(}g_{I_{1}}(H_{\left\langle
W\right\rangle})-g_{I_{1}}(H_{0})\big{)}\,{\rm i}[H_{\left\langle
W\right\rangle},A]\,g_{I_{1}}(H_{0})$ (3.36) $\displaystyle\
+g_{I_{1}}(H_{\left\langle W\right\rangle})\,{\rm i}[H_{\left\langle
W\right\rangle},A]\,\big{(}g_{I_{1}}(H_{\left\langle
W\right\rangle})-g_{I_{1}}(H_{0})\big{)}\Big{)}\,g_{I_{2}}(H_{\left\langle
W\right\rangle})$ (3.37)

Again we estimate (3.35) from below and the other lines from above. First

$\displaystyle\eqref{pm6}$
$\displaystyle\stackrel{{\scriptstyle\eqref{pm5}}}{{\geq}}\left(\frac{\theta}{2}-C_{1}[W]_{M_{1}}\right)\,g_{I_{2}}(H_{\left\langle
W\right\rangle})\,g_{I_{1}}(H_{0})^{2}\,g_{I_{2}}(H_{\left\langle
W\right\rangle})\ .$ (3.38)

We still have to bound from below $g_{I_{2}}(H_{\left\langle
W\right\rangle})\,g_{I_{1}}(H_{0})^{2}\,g_{I_{2}}(H_{\left\langle
W\right\rangle})$. To proceed we use that $g_{I_{1}}(H_{\left\langle
W\right\rangle})-g_{I_{1}}(H_{0})$ is small in size, being bounded, via Lemma
B.6 and (3.30), by

$\displaystyle\|g_{I_{1}}(H_{\left\langle
W\right\rangle})-g_{I_{1}}(H_{0})\|_{{\mathcal{L}}({\mathcal{H}})}\leq
C\,[W]_{M_{0}}\,.$ (3.39)

So write

$\displaystyle g_{I_{2}}(H_{\left\langle
W\right\rangle})\,g_{I_{1}}(H_{0})^{2}\,g_{I_{2}}(H_{\left\langle
W\right\rangle})=g_{I_{2}}(H_{\left\langle
W\right\rangle})\,g_{I_{1}}(H_{\left\langle
W\right\rangle})^{2}\,g_{I_{2}}(H_{\left\langle W\right\rangle})$ (3.40)
$\displaystyle\ \ +g_{I_{2}}(H_{\left\langle
W\right\rangle})\,\Big{(}g_{I_{1}}(H_{\left\langle
W\right\rangle})\,(g_{I_{1}}(H_{0})-g_{I_{1}}(H_{\left\langle
W\right\rangle}))\,+(g_{I_{1}}(H_{0})-g_{I_{1}}(H_{\left\langle
W\right\rangle}))\,g_{I_{1}}(H_{0})\Big{)}\,g_{I_{2}}(H_{\left\langle
W\right\rangle}).$

Therefore, using $g_{I_{1}}g_{I_{2}}=g_{I_{2}}$, estimates (3.39) and (3.24),
we deduce

$g_{I_{2}}(H_{\left\langle
W\right\rangle})\,g_{I_{1}}(H_{0})^{2}\,g_{I_{2}}(H_{\left\langle
W\right\rangle})\geq\big{(}1-C[W]_{M_{0}}\big{)}\,g_{I_{2}}(H_{\left\langle
W\right\rangle})^{2}\ .$

Thus we can finally estimate line (3.35) from below using (3.38) and the
previous estimate, concluding

$\eqref{pm6}\geq\left(\frac{\theta}{2}-C_{1}[W]_{M_{1}}\right)\big{(}1-C[W]_{M_{0}}\big{)}\,g_{I_{2}}(H_{\left\langle
W\right\rangle})^{2}.$ (3.41)

Next consider lines (3.36), (3.37). We use the bound (see (3.33))

$\|[H_{\left\langle
W\right\rangle},A]\|_{{\mathcal{L}}({\mathcal{H}}^{0})}\leq
C\big{(}1+[W]_{M_{1}}\big{)}\ ,$

and (3.39) to get

$\eqref{pm7}+\eqref{pm8}\geq-C\,[W]_{M_{0}}\,(1+[W]_{M_{1}})\,g_{I_{2}}(H_{\left\langle
W\right\rangle})^{2}.$ (3.42)

Putting together (3.41) and (3.42) we finally find

$g_{I_{2}}(H_{\left\langle W\right\rangle})\,{\rm i}[H_{\left\langle
W\right\rangle},A]\,g_{I_{2}}(H_{\left\langle
W\right\rangle})\geq\left(\frac{\theta}{2}-C([W]_{M_{1}}+[W]_{M_{0}}+[W]_{M_{0}}\,[W]_{M_{1}})\right)\,g_{I_{2}}(H_{\left\langle
W\right\rangle})^{2}.$

Thus, provided (2.10) holds true for ${\mathtt{M}}$ sufficiently large and
$\epsilon_{0}$ sufficiently small, the strict Mourre estimate (3.29) follows.
Mourre theorem implies that the spectrum of $H_{\left\langle W\right\rangle}$
in $I_{2}$ is absolutely continuous and by (3.32) it is also nonempty;
summarizing (use also Lemma 3.12)

$\sigma(H_{\left\langle W\right\rangle})\cap I_{2}=\sigma_{ac}(H_{\left\langle
W\right\rangle})\cap I_{2}\qquad\mbox{and}\qquad\left|\sigma(H_{\left\langle
W\right\rangle})\cap I_{2}\right|>0\ .$ (3.43)

Step 3: The last step is to consider the operator $H_{N}=H_{0}+\left\langle
W\right\rangle+T_{N}=H_{\left\langle W\right\rangle}+T_{N}$, which, for the
remaining part of the proof, we shall denote just by $H$. We shall constantly
use that any pseudodifferential operator of strictly negative order is a
compact operator on ${\mathcal{H}}$ (see Remark 2.4); in particular
$T_{N}\in{\mathcal{A}}_{-1}$ is compact. We begin by proving that

$\left|\sigma(H)\cap I_{2}\right|>0\ .$ (3.44)

Indeed by Weyl theorem $\sigma_{ess}(H)=\sigma_{ess}(H_{\left\langle
W\right\rangle})$ and therefore

$\displaystyle\sigma(H)\cap I_{2}$ $\displaystyle\supset\sigma_{ess}(H)\cap
I_{2}=\sigma_{ess}(H_{\left\langle W\right\rangle})\cap
I_{2}=\sigma(H_{\left\langle W\right\rangle})\cap I_{2}\ ,$

since $\sigma_{d}(H_{\left\langle W\right\rangle})\cap I_{2}=\emptyset$ having
$H_{\left\langle W\right\rangle}$ no eigenvalues in $I_{2}$. Then (3.44)
follows by (3.43).

Next we prove that $H$ fulfills a Mourre estimate over $I_{2}$, i.e.

$g_{I_{2}}\big{(}H\big{)}\,{\rm
i}[H,A]\,g_{I_{2}}\big{(}H\big{)}\geq\frac{\theta}{4}g_{I_{2}}\big{(}H\big{)}^{2}+K$
(3.45)

with $K$ a compact operator. We work perturbatively from (3.29). Again first
we compute

$g_{I_{2}}\big{(}H_{\left\langle W\right\rangle}\big{)}\,{\rm
i}[H,A]\,g_{I_{2}}\big{(}H_{\left\langle
W\right\rangle}\big{)}=g_{I_{2}}\big{(}H_{\left\langle
W\right\rangle}\big{)}\,{\rm i}[H_{\left\langle
W\right\rangle},A]\,g_{I_{2}}\big{(}H_{\left\langle
W\right\rangle}\big{)}+g_{I_{2}}\big{(}H_{\left\langle
W\right\rangle}\big{)}\,{\rm i}[T_{N},A]\,g_{I_{2}}\big{(}H_{\left\langle
W\right\rangle}\big{)}\ ;$

we estimate the first term in the r.h.s. above by (3.29), whereas the second
term is a compact operator since $[T_{N},A]\in{\mathcal{A}}_{-1}$. We obtain

$g_{I_{2}}\big{(}H_{\left\langle W\right\rangle}\big{)}\,{\rm
i}[H,A]\,g_{I_{2}}\big{(}H_{\left\langle
W\right\rangle}\big{)}\geq\frac{\theta}{4}g_{I_{2}}\big{(}H_{\left\langle
W\right\rangle}\big{)}^{2}+K_{1}$ (3.46)

with $K_{1}$ a compact operator. Now we must replace
$g_{I_{2}}\big{(}H_{\left\langle W\right\rangle}\big{)}$ with $g_{I_{2}}(H)$.
We write

$\displaystyle g_{I_{2}}(H)\,{\rm
i}[H,A]\,g_{I_{2}}(H)=g_{I_{2}}(H_{\left\langle W\right\rangle})\,{\rm
i}[H,A]\,g_{I_{2}}(H_{\left\langle W\right\rangle})$ (3.47)
$\displaystyle\quad+\big{(}g_{I_{2}}(H)-g_{I_{2}}(H_{\left\langle
W\right\rangle})\big{)}\,{\rm i}[H,A]\,g_{I_{2}}(H_{\left\langle
W\right\rangle})+g_{I_{2}}(H)\,{\rm
i}[H,A]\,\big{(}g_{I_{2}}(H)-g_{I_{2}}(H_{\left\langle
W\right\rangle})\big{)}$ (3.48)

This time we use that $g_{I_{2}}(H)-g_{I_{2}}(H_{\left\langle
W\right\rangle})$ is a compact operator, see Lemma B.6. Thus

$\displaystyle\eqref{pm600}$
$\displaystyle\stackrel{{\scriptstyle\eqref{pm.12}}}{{\geq}}\frac{\theta}{4}\,g_{I_{2}}\big{(}H_{\left\langle
W\right\rangle}\big{)}^{2}+K_{1}=\frac{\theta}{4}g_{I_{2}}(H)^{2}+K_{2}$

where $K_{1}$, $K_{2}$ are compact operators. Similarly, using that ${\rm
i}[H,A]\in{\mathcal{A}}_{0}$ is a bounded operator, we deduce that (3.48) is a
compact operator. Estimate (3.45) follows.
In particular $H$ is conjugated to $A$ over the interval $I_{2}$ fulfilling
(3.43). Proceeding as in Step 1, we produce a subinterval $I\subset I_{2}$
such that

$\left|I\cap\sigma(H)\right|>0\ ,\qquad I\cap\sigma(H)=I\cap\sigma_{ac}(H)$

and over which $H$ fulfills the strict Mourre estimate (3.23). ∎

The previous result has proved the existence of an interval $I$ over which
$H_{N}$ fulfills a strict Mourre estimate. This implies that $H_{N}$ fulfills
dispersive estimates in the form of local energy decay. In the literature
there are various variants of this result, thus in Appendix C we state and
prove the one we apply here.

######  Corollary 3.14.

Fix $k\in{\mathbb{N}}$. For any interval $J\subset I$, any function $g_{J}\in
C^{\infty}_{c}({\mathbb{R}},{\mathbb{R}}_{\geq 0})$ with ${\rm
supp}\,g_{J}\subset I$, $g_{J}\equiv 1$ on $J$, there exists a constant
$C_{k}>0$ such that

$\|\left\langle A\right\rangle^{-k}\,e^{-{\rm
i}H_{N}t}\,g_{J}(H_{N})\,{\varphi}\|_{0}\leq C_{k}\left\langle
t\right\rangle^{-k}\|\left\langle
A\right\rangle^{k}g_{J}(H_{N}){\varphi}\|_{0}\ ,\quad\forall t\in{\mathbb{R}}\
,\ \ \ \forall{\varphi}\in{\mathcal{H}}^{k}\ .$ (3.49)

Moreover $J$ can be chosen so that $\left|J\cap\sigma(H_{N})\right|>0$ and
$\sigma(H_{N})\cap J=\sigma_{ac}(H_{N})\cap J$.

###### Proof.

Apply Theorem C.1, noting that condition (M1) at page (M1) is trivially
satisfied $\forall n\in{\mathbb{N}}$ as ${\rm
ad}^{n}_{A}(H_{N})\in{\mathcal{A}}_{0}\subset{\mathcal{L}}({\mathcal{H}})$,
whereas the whole point of Lemma 3.13 was to verify (M2). This gives estimate
(3.49). The right hand side is finite for ${\varphi}\in{\mathcal{H}}^{k}$ by
Lemma 3.15 below, which ensures that
$g_{J}(H_{N}){\varphi}\in{\mathcal{H}}^{k}$. Finally note that, since
$\left|I\cap\sigma(H_{N})\right|>0$, it is certainly possible to choose
$J\subset I$ so that $\left|J\cap\sigma(H_{N})\right|>0$; as $H_{N}$ fulfills
a strict Mourre estimate over $I$, its spectrum in this interval is absolutely
continuous, so the same is true in $J$.

∎

######  Lemma 3.15.

For any $k\in{\mathbb{N}}$, $g_{J}(H_{N})$ extends to a bounded operator
${\mathcal{H}}^{k}\to{\mathcal{H}}^{k}$.

###### Proof.

As
$K_{0}^{k}\,g_{J}(H_{N})\,K_{0}^{-k}=g_{J}(H_{N})-[g_{J}(H_{N}),K_{0}^{k}]K_{0}^{-k},$
it is clearly sufficient to show that $[g_{J}(H_{N}),K_{0}^{k}]K_{0}^{-k}$ is
bounded on ${\mathcal{H}}$. The adjoint formula (B.6) gives

$[g_{J}(H_{N}),K_{0}^{k}]K_{0}^{-k}=\sum_{j=1}^{k}c_{k,j}\,{\rm
ad}^{j}_{K_{0}}(g_{J}(H_{N}))\,K_{0}^{-j}\ ;$

then it is enough to show that ${\rm
ad}^{j}_{K_{0}}(g_{J}(H_{N}))\in{\mathcal{L}}({\mathcal{H}})$. As ${\rm
ad}^{j}_{K_{0}}(H_{N})$ is a bounded operator $\forall j$ (symbolic calculus),
the result is an immediate application of Lemma B.5. ∎

We finally prove Proposition 3.6.

###### Proof of Proposition 3.6.

First we show that for any $k\in{\mathbb{N}}$, there exists $C_{2k}>0$ such
that

$\|e^{-{\rm i}tH_{N}}g_{J}(H_{N}){\varphi}\|_{-2k}\leq C_{2k}\,\left\langle
t\right\rangle^{-2k}\,\|g_{J}(H_{N}){\varphi}\|_{2k},\qquad\forall
t\in{\mathbb{R}},\ \ \ \forall{\varphi}\in{\mathcal{H}}^{2k}\ .$ (3.50)

This follows from Corollary 3.14 with $k\leadsto 2k$. Indeed, as
$A\in{\mathcal{A}}_{1}$, the operator $\left\langle
A\right\rangle^{2k}=(1+A^{2})^{k}\in{\mathcal{A}}_{2k}$ and therefore, by
symbolic calculus, $K_{0}^{-2k}\left\langle A\right\rangle^{2k}$ and
$\left\langle A\right\rangle^{2k}K_{0}^{-2k}$ belong to
${\mathcal{A}}_{0}\subset{\mathcal{L}}({\mathcal{H}})$. Then

$\displaystyle\|e^{-{\rm i}tH_{N}}g_{J}(H_{N}){\varphi}\|_{-2k}$
$\displaystyle\leq\|K_{0}^{-2k}\,\left\langle
A\right\rangle^{2k}\|_{{\mathcal{L}}({\mathcal{H}})}\,\|\left\langle
A\right\rangle^{-2k}e^{-{\rm i}tH_{N}}g_{J}(H_{N}){\varphi}\|_{0}$
$\displaystyle\leq C_{2k}\left\langle t\right\rangle^{-2k}\|\left\langle
A\right\rangle^{2k}g_{J}(H_{N}){\varphi}\|_{0}$ $\displaystyle\leq
C_{2k}\left\langle t\right\rangle^{-2k}\|\left\langle
A\right\rangle^{2k}K_{0}^{-2k}\|_{{\mathcal{L}}({\mathcal{H}})}\|g_{J}(H_{N}){\varphi}\|_{2k}$

proving (3.50). Then linear interpolation with the equality $\|e^{-{\rm
i}tH_{N}}{\varphi}_{0}\|_{0}=\|{\varphi}_{0}\|_{0}$ $\,\forall t$ gives
$\forall r\in[0,2k]$

$\|e^{-{\rm i}tH_{N}}g_{J}(H_{N}){\varphi}\|_{-r}\leq C_{r}\left\langle
t\right\rangle^{-r}\|g_{J}(H_{N}){\varphi}\|_{r}\ ,\quad\forall
t\in{\mathbb{R}}\ ,\ \ \ \forall{\varphi}\in{\mathcal{H}}^{r}\ .$

Finally we must show that this estimate is not trivial, namely that
$\exists{\varphi}\in{\mathcal{H}}^{k}$ so that $g_{J}(H_{N}){\varphi}\neq 0$.
So take $J\subset I$ with $\left|J\cap\sigma(H_{N})\right|>0$ and
$\sigma(H_{N})\cap J=\sigma_{ac}(H_{N})\cap J$, which is possible by Corollary
3.14. As $g_{J}(H_{N}){\mathcal{H}}\neq\\{0\\}$ and ${\mathcal{H}}^{k}$ is
dense in ${\mathcal{H}}$, we have that
$g_{J}(H_{N}){\mathcal{H}}^{k}\neq\\{0\\}$. Then it is enough to take
$f\in{\mathcal{H}}^{k}$ so that $g_{J}(H_{N})f\neq 0$, and put
${\varphi}_{0}:=g_{J}(H_{N})f$ which, by Lemma 3.15, belongs to
${\mathcal{H}}^{k}$. Such initial datum fulfills the claim of Proposition 3.6.
∎

###  3.3 Proof of Theorem 2.8

We are finally in position of proving Theorem 2.8. Recall that in Corollary
3.5 we have conjugated equation (2.11) to (3.17) with a change of variables
bounded ${\mathcal{H}}^{r}\to{\mathcal{H}}^{r}$ uniformly in time, whereas in
Proposition 3.6 we have constructed a solution of the effective equation ${\rm
i}\partial_{t}\psi=H_{N}\psi$ with decaying negative Sobolev norms, therefore
with growing positive Sobolev norms. The last step is to construct a solution
of the full equation (3.17) with growing Sobolev norms. To achieve this, we
exploit that the perturbation $R_{N}(t)$ is $N$-smoothing (Definition 2.2).
So to proceed we fix the parameters. First fix $r>0$, then choose
$N,k\in{\mathbb{N}}$ such that

$N\geq 2r+2,\quad k\geq N-r.$ (3.51)

Apply Corollary 3.5 with such $N$, producing the operators $T_{N}$, $R_{N}(t)$
and conjugating (2.11) to (3.17). By Proposition 3.6,
$\exists\,{\varphi}_{0}\in{\mathcal{H}}^{k}$ such that ${\varphi}(t):=e^{-{\rm
i}tH_{N}}{\varphi}_{0}$ fulfills $\forall{\tt r}\in[0,k]$:

$\|{\varphi}(t)\|_{-{\tt r}}\leq C_{{\tt r},N}\left\langle
t\right\rangle^{-{\tt r}}\,\|{\varphi}_{0}\|_{{\tt r}},\qquad\forall
t\in{\mathbb{R}}\ .$ (3.52)

We look for an exact solution $\phi(t)$ of (3.17) of the form
$\phi(t)={\varphi}(t)+u(t)$, i.e. $u(t)$ has to satisfy

${\rm i}\partial_{t}u=\big{(}H_{N}+R_{N}(t)\big{)}u+R_{N}(t){\varphi}(t).$
(3.53)

Denoting by $U_{N}(t,s)$ the linear propagator of $H_{N}+R_{N}(t)$, we choose

$u(t):={\rm
i}\int\limits_{t}^{+\infty}U_{N}(t,s)\,R_{N}(s)\,{\varphi}(s)\,{\rm d}s.$
(3.54)

We estimate the ${\mathcal{H}}^{r}$ norm of $u(t)$. As

$\sup_{t}\|[H_{N}+R_{N}(t),\,K_{0}]\|_{{\mathcal{L}}({\mathcal{H}}^{m})}<C_{m}<\infty\
,\qquad\forall m\in{\mathbb{R}},$

Theorem 1.5 of [33] guarantees that the propagator $U_{N}(t,s)$ extends to a
bounded operator ${\mathcal{H}}^{r}\to{\mathcal{H}}^{r}$ fulfilling666apply
the theorem with $\tau=0$ and note that in that paper we defined
$\|\psi\|_{r}\equiv\|K_{0}^{r/2}\psi\|_{0}$, therefore the estimate in that
paper reads explicitly $\|K_{0}^{r/2}U_{N}(t,s)\psi\|_{0}\leq
C_{r}\,\left\langle t-s\right\rangle^{r/2}\|K_{0}^{r/2}\psi\|_{0}$

$\forall r>0\ \ \
\exists\,C_{r}>0\colon\qquad\|U_{N}(t,s)\|_{{\mathcal{L}}({\mathcal{H}}^{r})}\leq
C_{r}\,\left\langle t-s\right\rangle^{r},\quad\forall t,s\in{\mathbb{R}}\ .$

This estimate, the smoothing property
$R_{N}(t)\colon{\mathcal{H}}^{r-N}\to{\mathcal{H}}^{r}$ and (3.52) with ${\tt
r}:=N-r\in[0,k]$ give

$\displaystyle\|u(t)\|_{r}$ $\displaystyle\leq
C_{r}\int\limits_{t}^{+\infty}\left\langle
t-s\right\rangle^{r}\|R_{N}(s)\,{\varphi}(s)\|_{r}\,{\rm d}s\leq
C_{r}\int\limits_{t}^{+\infty}\left\langle
t-s\right\rangle^{r}\,\|{\varphi}(s)\|_{-(N-r)}\,{\rm d}s$ $\displaystyle\leq
C_{r,N}\,\|{\varphi}_{0}\|_{N-r}\int\limits_{t}^{+\infty}\left\langle
t-s\right\rangle^{r}\,\frac{1}{\left\langle s\right\rangle^{N-r}}\,\,{\rm
d}s\leq C_{r,N}\,\|{\varphi}_{0}\|_{k}\left\langle t\right\rangle^{-1}\ .$

In particular the ${\mathcal{H}}^{r}$ norm of $u(t)$ decreases to 0 as
$t\to\infty$. Then $\phi(t)={\varphi}(t)+u(t)$ fulfills

$\|\phi(t)\|_{r}\geq\|{\varphi}(t)\|_{r}-\|u(t)\|_{r}\geq
c_{r}\frac{\|{\varphi}_{0}\|_{0}^{2}}{\|{\varphi}_{0}\|_{r}}\left\langle
t\right\rangle^{r}-C_{r,N}\|{\varphi}_{0}\|_{k}\left\langle
t\right\rangle^{-1}\geq C\left\langle t\right\rangle^{r}\ ,\quad\forall|t|\geq
T\,,$ (3.55)

where we used (3.52) with ${\tt r}=r$ and Remark 3.7.
Finally we get a solution of the original equation (2.11) putting
$\psi(t)={\mathcal{U}}_{N}(t)^{-1}\phi(t)$, recall Proposition 3.4. The
operator ${\mathcal{U}}_{N}(t)$ fulfills (3.15), thus $\psi(t)$ has
polynomially growing Sobolev norms as (2.12), concluding the proof of Theorem
2.8.

We can also prove the existence of infinitely many solutions undergoing growth
of Sobolev norms.

######  Corollary 3.16.

There are infinitely many distinct solutions of equation (2.11) with growing
Sobolev norms.

###### Proof.

We fix $r>0$ and choose $N,k$ as in (3.51). From the previous proof, it
follows that any initial data of the form

$\psi(0):=({\rm Id}+{\mathcal{K}}_{0}){\varphi}\
,\qquad{\mathcal{K}}_{t}{\varphi}:={\rm
i}\int_{t}^{+\infty}U_{N}(t,s)R_{N}(s)e^{-{\rm i}sH_{N}}{\varphi}\,{\rm
d}s,\,\qquad t\geq 0,$

with ${\varphi}\in{\rm Ran}\,g_{J}(H_{N})\cap{\mathcal{H}}^{k}$, gives rise to
a solution with growing Sobolev norms (see also Remark 3.8). Here $J$ is the
interval of Corollary 3.14. In particular, as
$\left|J\cap\sigma(H_{N})\right|>0$ and $\sigma(H_{N})\cap
J=\sigma_{ac}(H_{N})\cap J$, the set ${\rm Ran}\,g_{J}(H_{N})$ has infinite
dimension. Let us prove that ${\rm Id}+{\mathcal{K}}_{0}$ is injective. Assume
there are ${\varphi}_{1}\neq{\varphi}_{2}\in{\rm
Ran}\,g_{J}(H_{N})\cap{\mathcal{H}}^{k}$ with $({\rm
Id}+{\mathcal{K}}_{0}){\varphi}_{1}=({\rm
Id}+{\mathcal{K}}_{0}){\varphi}_{2}$. Put
$u_{j}(t):={\mathcal{K}}_{t}{\varphi}_{j}$, $j=1,2$; arguing as in the
previous proof one has $\|u_{j}(t)\|_{r}\to 0$ as $t\to\infty$.
Then ${\mathcal{U}}_{N}(t)^{-1}(e^{-{\rm i}tH_{N}}{\varphi}_{j}+u_{j}(t))$,
$j=1,2$, both solve (2.11) and have the same initial datum, so they are the
same solution $\psi(t)$ of equation (2.11). Then

$\displaystyle\|{\varphi}_{1}-{\varphi}_{2}\|_{0}$ $\displaystyle=\|e^{-{\rm
i}tH_{N}}({\varphi}_{1}-{\varphi}_{2})\|_{0}$ $\displaystyle\leq
C_{r}\|{\mathcal{U}}_{N}^{-1}(t)e^{-{\rm
i}tH_{N}}({\varphi}_{1}-{\varphi}_{2})\|_{r}\leq
C_{r}\big{(}\|u_{1}(t)\|_{r}+\|u_{2}(t)\|_{r}\big{)}\to 0$

as $t\to\infty$. Hence ${\varphi}_{1}={\varphi}_{2}$. ∎

##  4 Applications

In the following section we apply Theorem 2.8 to the harmonic oscillator on
${\mathbb{R}}$ and the half-wave equation on ${\mathbb{T}}$. In both cases we
construct transporters which are stable under small, time periodic,
pseudodifferential perturbations.

###  4.1 Harmonic oscillator on ${\mathbb{R}}$

Consider the quantum harmonic oscillator

$\displaystyle{\rm
i}\partial_{t}\psi=\frac{1}{2}(-\partial_{x}^{2}+x^{2})\psi+V(t,x,D)\psi,\quad
x\in{\mathbb{R}}.$ (4.1)

Here $K_{0}:=\frac{1}{2}\left(-\partial_{x}^{2}+x^{2}\right)$ is the quantum
Harmonic oscillator, the scale of Hilbert spaces is defined as usual by
${\mathcal{H}}^{r}={\rm Dom}\left(K_{0}^{r}\right)$, and the base space
$({\mathcal{H}}^{0},\left\langle\cdot,\cdot\right\rangle)$ is
$L^{2}({\mathbb{R}},{\mathbb{C}})$ with its standard scalar product. The
perturbation $V$ is chosen as the Weyl quantization of a symbol belonging to
the following class:

######  Definition 4.1.

A function $f$ is a symbol of order  $\rho\in{\mathbb{R}}$ if $f\in
C^{\infty}({\mathbb{R}}_{x}\times{\mathbb{R}}_{\xi},{\mathbb{C}})$ and
$\forall\alpha,\beta\in{\mathbb{N}}_{0}$, there exists $C_{\alpha,\beta}>0$
such that

$|\partial_{x}^{\alpha}\,\partial_{\xi}^{\beta}f(x,\xi)|\leq C_{\alpha,\beta}\
(1+|x|^{2}+|\xi|^{2})^{\rho-\frac{\beta+\alpha}{2}}\ .$

We will write $f\in S^{\rho}_{{\rm har}}$.

We endow $S^{\rho}_{\rm har}$ with the family of seminorms

$\wp^{\rho}_{j}(f):=\sum_{|\alpha|+|\beta|\leq j}\ \
\sup_{(x,\xi)\in{\mathbb{R}}^{2}}\frac{\left|\partial_{x}^{\alpha}\,\partial_{\xi}^{\beta}f(x,\xi)\right|}{\left(1+|x|^{2}+|\xi|^{2}\right)^{\rho-\frac{\beta+\alpha}{2}}}\
,\qquad j\in{\mathbb{N}}\cup\\{0\\}\ .$

Such seminorms turn $S^{\rho}_{\rm har}$ into a Fréchet space. If a symbol $f$
depends on additional parameters (e.g. it is time dependent), we ask that all
the seminorms are uniform w.r.t. such parameters.
To a symbol $f\in S^{\rho}_{\rm har}$ we associate the operator $f(x,D)$ by
standard Weyl quantization

$\Big{(}f(x,D)\psi\Big{)}(x):=\frac{1}{2\pi}\iint_{y,\xi\in{\mathbb{R}}}{\rm
e}^{{\rm i}(x-y)\xi}\,f\left(\frac{x+y}{2},\xi\right)\,\psi(y)\,{\rm d}y{\rm
d}\xi\ .$

######  Definition 4.2.

We say that $F\in{\mathcal{A}}_{\rho}$ if it is a pseudodifferential operator
with symbol of class $S^{\rho}_{{\rm har}}$, i.e., if there exists $f\in
S^{\rho}_{{\rm har}}$ and $S$ smoothing (in the sense of Definition 2.2) such
that $F=f(x,D_{x})+S$.

######  Remark 4.3.

With our numerology, the symbol of the harmonic oscillator $K_{0}$ is of order
1, $\frac{1}{2}({x^{2}+\xi^{2}})\in S^{1}_{{\rm har}}$, and not of order 2 as
typically in the literature.

As an application of the abstract theorems, we describe a class of operators
which are transporters. This class, which we call smooth Töplitz operators, is
easily described in terms of their matrix elements, which we now introduce. We
denote by $\\{{\bf e}_{n}\\}_{n\in{\mathbb{N}}}$ the Hermite basis, formed by
the (orthonormal) eigenvectors of the Harmonic oscillator $K_{0}$:

$K_{0}{\bf e}_{n}=\left(n-\frac{1}{2}\right){\bf e}_{n},\quad\|{\bf
e}_{n}\|_{0}=1,\quad n\in{\mathbb{N}}\ .$ (4.2)

To each operator $\mathsf{H}\in{\mathcal{L}}({\mathcal{H}})$ we associate its
matrix $(\mathsf{H}_{mn})_{m,n\in{\mathbb{N}}}$ with respect to the Hermite
basis, whose elements are given by

$\mathsf{H}_{mn}:=\left\langle\mathsf{H}\,{\bf e}_{n},{\bf
e}_{m}\right\rangle\ ,\qquad\forall m,n\in{\mathbb{N}}\ .$ (4.3)

######  Remark 4.4.

If $\mathsf{H}$ is selfadjoint, so is its matrix
$(\mathsf{H}_{mn})_{m,n\in{\mathbb{N}}}$, in particular
$\mathsf{H}_{mn}=\overline{\mathsf{H}_{nm}}$.

######  Definition 4.5 (Smooth Töplitz operators).

A linear operator $\mathsf{H}\in{\mathcal{L}}({\mathcal{H}})$ is said a
Töplitz operator if the entries of its matrix are constant along each
diagonal, i.e.

$\mathsf{H}_{m_{1}n_{1}}=\mathsf{H}_{m_{2}n_{2}},\quad\forall
m_{1},n_{1},m_{2},n_{2}\in{\mathbb{N}}\colon\ \ \ m_{1}-n_{1}=m_{2}-n_{2}\ .$
(4.4)

A Töplitz operator is said smooth if its matrix elements decay fast off
diagonal, i.e. $\forall N>0$, $\exists C_{N}>0$ such that

$\left|\mathsf{H}_{mn}\right|\leq\frac{C_{N}}{\left\langle
m-n\right\rangle^{N}}\ ,\qquad\forall m,n\in{\mathbb{N}}\ .$ (4.5)

######  Example 4.6.

The shift operators $S$ and its adjoint $S^{*}$ are defined on the Hermite
functions $\\{{\bf e}_{n}\\}_{n\geq 1}$ by

$S{\bf e}_{n}={\bf e}_{n+1}\ ,\quad\forall n\in{\mathbb{N}}\ ,\qquad S^{*}{\bf
e}_{n}=\begin{cases}0&\mbox{ if }n=1\\\ {\bf e}_{n-1}&\mbox{ if }n\geq
2\end{cases}\ .$ (4.6)

The action of $S$ (and of $S^{*}$) is extended on all ${\mathcal{H}}$ by
linearity, giving $S\psi=\sum_{n\geq 1}\psi_{n}{\bf e}_{n+1}$, where we
defined $\psi_{n}:=\left\langle\psi,{\bf e}_{n}\right\rangle$ for $n\geq 1$.
Their matrices are given by

$(S_{mn})_{m,n\in{\mathbb{N}}}=\begin{pmatrix}0&&&\\\ 1&0&&\\\ &1&0&\\\
&&\ddots&\ddots\end{pmatrix},\qquad(S^{*}_{mn})_{m,n\in{\mathbb{N}}}=\begin{pmatrix}0&1&&\\\
&0&1&\\\ &&0&1\\\ &&&\ddots\end{pmatrix},$

from which it is clear that both $S$ and $S^{*}$ are smooth Töplitz operators.

We prove in the following that any smooth Töplitz operator is actually a
pseudodifferential operator in ${\mathcal{A}}_{0}$, see Lemma 4.10.

As an application of the abstract theorems, we show that any smooth Töplitz
operator becomes a transporter for the Harmonic oscillator once it is
multiplied by an appropriate scalar time periodic function.

######  Theorem 4.7.

Let $\mathsf{V}(x,D)$ be a selfadjoint and smooth Töplitz operator (see
Definition 4.5). Take $m,n\in{\mathbb{N}}$, $m>n$, such that the matrix
element

$\mathsf{V}_{m-n}:=\left\langle\mathsf{V}(x,D)\,{\bf e}_{n},{\bf
e}_{m}\right\rangle\neq 0\ .$

Then

$V(t,x,D):=\cos((m-n)t)\,\mathsf{V}(x,D)$ (4.7)

is a transporter for (4.1). More precisely, $\forall r\geq 0$ there exist a
solution $\psi(t)\in{\mathcal{H}}^{r}$ of (4.1) and constants $C,T>0$ such
that

$\|\psi(t)\|_{r}\geq C\left\langle t\right\rangle^{r},\quad\forall t>T.$

The theorem follows applying Theorem 2.7. So we check that Assumptions I-III
are fulfilled. Regarding Assumption I, it is the usual Weyl calculus for
symbols in $S^{\rho}_{{\rm har}}$, see e.g. [40]. Concerning Assumption II,
one has $\sigma(K_{0})=\\{n-\frac{1}{2}\\}_{n\in{\mathbb{N}}}$. Furthermore
Egorov theorem for the Harmonic oscillator [27] states that the map $t\mapsto
e^{-{\rm i}tK_{0}}\mathsf{A}e^{{\rm i}tK_{0}}\in
C^{\infty}({\mathbb{T}},{\mathcal{A}}_{\rho})$ for any
$\mathsf{A}\in{\mathcal{A}}_{\rho}$ (use also the periodicity of the flow of
$K_{0}$). This can be seen e.g. by remarking that the symbol of $e^{-{\rm
i}tK_{0}}\mathsf{A}e^{{\rm i}tK_{0}}$ is $a\circ\phi_{{\rm har}}^{t}$, where
$a\in S^{\rho}_{\rm har}$ is the symbol of $\mathsf{A}$ and $\phi^{t}_{{\rm
har}}$ is the time $t$ flow of the harmonic oscillator; explicitly

$\displaystyle\left(a\circ\phi^{t}_{{\rm har}}\right)(x,\xi)=a(x\cos t+\xi\sin
t,-x\sin t+\xi\cos t)\ .$ (4.8)

Verification of Assumption III. First we show that smooth Töplitz operators
belong to ${\mathcal{A}}_{0}$. We exploit Chodosh’s characterization [11],
which we now recall. Define the discrete difference operator $\triangle$ on a
function $M\colon{\mathbb{N}}\times{\mathbb{N}}\to{\mathbb{C}}$ by

$(\triangle M)(m,n):=M(m+1,n+1)-M(m,n)\ ,$

and its powers $\triangle^{\gamma}$, $\gamma\in{\mathbb{N}}$, by $\triangle$
applied $\gamma$-times.

######  Definition 4.8 (Symbol matrix).

A function $M\colon{\mathbb{N}}\times{\mathbb{N}}\to{\mathbb{C}}$ will be said
to be a symbol matrix of order $\rho$ if for any $\gamma\in{\mathbb{N}}_{0}$,
$N\in{\mathbb{N}}$, there exists $C_{\gamma,N}>0$ such that

$\left|(\triangle^{\gamma}M)(m,n)\right|\leq
C_{\gamma,N}\frac{(1+m+n)^{\rho-|\gamma|}}{\left\langle m-n\right\rangle^{N}}\
,\quad\forall m,n\in{\mathbb{N}}\ .$ (4.9)

The connection between pseudodifferential operators and symbol matrices is
given by Chodosh’s characterization:

######  Theorem 4.9 ([11]).

An operator $\mathsf{H}$ belongs to ${\mathcal{A}}_{\rho}$ if and only if its
matrix $M^{(\mathsf{H})}(m,n):=\mathsf{H}_{mn}$ (as defined in (4.3)) is a
symbol matrix of order $\rho$.

As a direct consequence we have the following result:

######  Lemma 4.10.

Any smooth Töplitz operator is a pseudodifferential operator in
${\mathcal{A}}_{0}$.

###### Proof.

We use Theorem 4.9. Let $\mathsf{H}$ be smooth Töplitz and put
$M^{(\mathsf{H})}(m,n):=\mathsf{H}_{mn}$. Then (4.9) holds with
$\rho=\gamma=0$ by (4.5). By (4.4) one has $\triangle M^{(\mathsf{H})}=0$; so
(4.9) holds also $\forall\gamma\geq 1$. ∎

In particular $V(t,x,D)=\cos((m-n)t)\mathsf{V}(x,D)$ belongs to
$C^{\infty}({\mathbb{T}},{\mathcal{A}}_{0})$, which is the first required
property of Assumption III.

######  Remark 4.11.

The shift operators $S,S^{*}$, defined in (4.6), belong to ${\mathcal{A}}_{0}$
being smooth Töplitz. Also their (integer) powers $S^{k}$, $S^{*k}$, given for
$k\in{\mathbb{N}}$ by

$S^{k}{\bf e}_{n}={\bf e}_{n+k}\ ,\quad\forall n\in{\mathbb{N}}\ ,\qquad
S^{*k}{\bf e}_{n}=\begin{cases}0&\mbox{ if }n\leq k\\\ {\bf e}_{n-k}&\mbox{ if
}n\geq k+1\end{cases}\ $ (4.10)

are smooth Töplitz, so in ${\mathcal{A}}_{0}$.

Next we compute the resonant average of $V(t,x,D)$.

######  Lemma 4.12.

Let $V(t,x,D)$ as in (4.7). Its resonant average $\left\langle V\right\rangle$
(see (1.3)) is

$\left\langle
V\right\rangle=\frac{1}{2}\left(\mathsf{V}_{k}\,S^{k}+\overline{\mathsf{V}_{k}}\,S^{*k}\right)\
,\qquad k:=m-n\in{\mathbb{N}}\ ,$ (4.11)

where $S\in{\mathcal{A}}_{0}$ is defined in (4.6) and
$\mathsf{V}_{k}:=\mathsf{V}_{m-n}:=\left\langle\mathsf{V}\,{\bf e}_{n},{\bf
e}_{m}\right\rangle\in{\mathbb{C}}$.

###### Proof.

For $\ell\in{\mathbb{N}}$, denote by
$\Pi_{\ell}{\varphi}:=\left\langle{\varphi},{\bf e}_{\ell}\right\rangle\,{\bf
e}_{\ell}$ the projector on the Hermite function ${\bf e}_{\ell}$. Clearly

$e^{{\rm i}sK_{0}}\,\Pi_{\ell}=\Pi_{\ell}\,e^{{\rm i}sK_{0}}=e^{{\rm
i}s(\ell-\frac{1}{2})}\,\Pi_{\ell},\quad\forall\ell\in{\mathbb{N}}\ .$

From now on we simply write $\mathsf{V}\equiv\mathsf{V}(x,D)$. Using this
identity and writing ${\rm Id}=\sum_{\ell\geq 1}\Pi_{\ell}$ we get

$\displaystyle e^{{\rm i}sK_{0}}\,\mathsf{V}\,e^{-{\rm i}sK_{0}}$
$\displaystyle=\sum_{j,\ell\geq 1}e^{{\rm
i}s(j-\ell)}\Pi_{j}\,\mathsf{V}\,\Pi_{\ell}=\sum_{j,\ell\geq 1}e^{{\rm
i}s(j-\ell)}\,\left\langle\cdot,{\bf
e}_{\ell}\right\rangle\,\left\langle\mathsf{V}{\bf e}_{\ell},{\bf
e}_{j}\right\rangle\,{\bf e}_{j}\ .$

Now we compute, with $k:=m-n\in{\mathbb{N}}$,

$\displaystyle\left\langle V\right\rangle$
$\displaystyle=\frac{1}{2\pi}\int_{0}^{2\pi}\cos(ks)\,e^{{\rm
i}sK_{0}}\,\mathsf{V}\,e^{-{\rm i}sK_{0}}{\rm d}s=\sum_{j,\ell\geq
1}\left\langle\mathsf{V}{\bf e}_{\ell},{\bf
e}_{j}\right\rangle\,\left\langle\cdot,{\bf e}_{\ell}\right\rangle\,{\bf
e}_{j}\,\frac{1}{2\pi}\int_{0}^{2\pi}\cos(ks)\,e^{{\rm i}s(j-\ell)}\,{\rm d}s$
$\displaystyle=\frac{1}{2}\sum_{\ell\geq 1}\left\langle\mathsf{V}{\bf
e}_{\ell},{\bf e}_{\ell+k}\right\rangle\left\langle\cdot,{\bf
e}_{\ell}\right\rangle\,{\bf e}_{\ell+k}+\frac{1}{2}\sum_{\ell\geq
k+1}\left\langle\mathsf{V}{\bf e}_{\ell},{\bf
e}_{\ell-k}\right\rangle\left\langle\cdot,{\bf e}_{\ell}\right\rangle\,{\bf
e}_{\ell-k}=\frac{1}{2}\mathsf{V}_{k}\,S^{k}+\frac{1}{2}\overline{\mathsf{V}_{k}}\,S^{*k}$

where in the last line we used $\mathsf{V}_{-k}=\left\langle\mathsf{V}{\bf
e}_{\ell},{\bf e}_{\ell-k}\right\rangle=\overline{\left\langle\mathsf{V}{\bf
e}_{\ell-k},{\bf e}_{\ell}\right\rangle}=\overline{\mathsf{V}_{k}}$ being
$\mathsf{V}$ selfadjoint and smooth Töplitz (see Remark 4.4). ∎

Now define the selfadjoint operator

$A:=\frac{\mathsf{V}_{k}}{{\rm
i}}\,(K_{0}+\frac{1}{2})\,S^{k}-\frac{\overline{\mathsf{V}_{k}}}{{\rm
i}}\,S^{*k}\,(K_{0}+\frac{1}{2})-\frac{\overline{\mathsf{V}_{k}}}{{\rm
i}}\,(K_{0}+\frac{1}{2})\,S^{*k}+\frac{{\mathsf{V}_{k}}}{{\rm
i}}\,S^{k}\,(K_{0}+\frac{1}{2})\ ,$ (4.12)

which belongs to ${\mathcal{A}}_{1}$ by symbolic calculus as
$K_{0}\in{\mathcal{A}}_{1}$ and $S,S^{*}\in{\mathcal{A}}_{0}$ (see Remark
4.11).
The next lemma verifies Assumption III.

######  Lemma 4.13.

Assume that $\mathsf{V}_{k}\neq 0$. The following holds true:

  * (i)

The spectrum of the operator $H_{0}:=\left\langle V\right\rangle$ fulfills
$\sigma(H_{0})\supseteq\left[-|\mathsf{V}_{k}|,\,|\mathsf{V}_{k}|\right]$.

  * (ii)

For any interval
$I_{0}\subset\left[-|\mathsf{V}_{k}|,\,|\mathsf{V}_{k}|\right]$, any
$g_{I_{0}}\in C^{\infty}_{c}({\mathbb{R}},{\mathbb{R}}_{\geq 0})$ with
$g_{I_{0}}\equiv 1$ over $I_{0}$ and ${\rm
supp}\,g_{I_{0}}\subset\left[-|\mathsf{V}_{k}|,\,|\mathsf{V}_{k}|\right]$,
there exist $\theta>0$ and $\mathsf{K}$ compact operator such that

$g_{I_{0}}(H_{0})\,{\rm
i}[H_{0},A]\,g_{I_{0}}(H_{0})\geq\theta\,g_{I_{0}}(H_{0})^{2}+\mathsf{K}\ .$

Here $A$ is defined in (4.12).

###### Proof.

$(i)$ Let ${\mathtt{f}}(\rho):={\rm Re}(\mathsf{V}_{k}\,e^{-{\rm i}\rho k})$.
We shall prove that ${\mathtt{f}}(\rho)\in\sigma(H_{0})$
$\forall\rho\in{\mathbb{R}}$, from which the claim follows. As $H_{0}$ is
selfadjoint, it is enough to construct a Weyl sequence for
${\mathtt{f}}(\rho)$, i.e. a sequence $(\psi^{(n)})_{n\geq 1}$ with
$\|\psi^{(n)}\|_{0}=1$ $\,\forall n$ and
$\|(H_{0}-{\mathtt{f}}(\rho))\psi^{(n)}\|_{0}\to 0$ as $n\to\infty$. We put

$\psi^{(n)}:=\frac{1}{\sqrt{n}}\sum_{\ell=1}^{n}e^{{\rm i}\rho\ell}{\bf
e}_{\ell}\ .$

Then $\|\psi^{(n)}\|_{0}=1$ $\,\forall n$ and a direct computation shows that
for $n>k$

$H_{0}\psi^{(n)}=\frac{1}{\sqrt{n}}\frac{\overline{\mathsf{V}}_{k}}{2}e^{{\rm
i}\rho k}\sum_{m=1}^{k}e^{{\rm i}\rho m}\,{\bf
e}_{m}+\frac{1}{\sqrt{n}}{\mathtt{f}}(\rho)\sum_{m=k+1}^{n-k}e^{{\rm i}\rho
m}\,{\bf e}_{m}+\frac{1}{\sqrt{n}}\frac{\mathsf{V}_{k}}{2}e^{-{\rm i}\rho
k}\sum_{m=n-k+1}^{n+k}e^{{\rm i}\rho m}\,{\bf e}_{m}\ .$

Thus one finds a constant $C_{k}>0$ such that

$\|(H_{0}-{\mathtt{f}}(\rho))\psi^{(n)}\|_{0}\leq\frac{C_{k}}{\sqrt{n}}\to
0\quad\mbox{ as }n\to\infty\ ,$

proving that $\psi^{(n)}$ is a Weyl sequence; by Weyl criterium
${\mathtt{f}}(\rho)\in\sigma(H_{0})$.

$(ii)$ First note that, by (4.2) and (4.6), one has $\forall k\in{\mathbb{N}}$

$\displaystyle[S^{k},K_{0}]=-kS^{k},\qquad[S^{*k},K_{0}]=kS^{*k},\qquad[S^{*k},S^{k}]=\Pi_{\leq
k}$ (4.13) $\displaystyle S^{k}S^{*k}={\rm Id}-\Pi_{\leq k}\,\qquad
S^{*k}S^{k}={\rm Id}$ (4.14)

where $\Pi_{\leq k}:=\sum_{\ell=1}^{k}\Pi_{\ell}$ is the projector on the
Hermite modes with index $\leq k$. Using (4.13) a direct computation gives

$\displaystyle{\rm i}[H_{0},A]$
$\displaystyle=k\big{(}2|\mathsf{V}_{k}|^{2}-\mathsf{V}_{k}^{2}S^{2k}-\overline{\mathsf{V}}^{2}_{k}S^{*2k}-|\mathsf{V}_{k}|^{2}\Pi_{\leq
k}\big{)}+2|\mathsf{V}_{k}|^{2}(K_{0}+\frac{1}{2})\Pi_{\leq k}$
$\displaystyle=4k\big{(}|\mathsf{V}_{k}|^{2}-H_{0}^{2}\big{)}+2|\mathsf{V}_{k}|^{2}(K_{0}+\frac{1}{2}-k)\Pi_{\leq
k}\ .$

Clearly $\mathsf{K}:=2|\mathsf{V}_{k}|^{2}(K_{0}+\frac{1}{2}-k)\Pi_{\leq k}$
is compact, being finite rank.
Next put $\tilde{f}(\lambda)=4k(|\mathsf{V}_{k}|^{2}-\lambda^{2})$ getting
$\forall{\varphi}\in{\mathcal{H}}$

$\left\langle g_{I_{0}}(H_{0})\,{\rm
i}[H_{0},A]\,g_{I_{0}}(H_{0}){\varphi},{\varphi}\right\rangle=\left\langle
g_{I_{0}}(H_{0})\,\tilde{f}(H_{0})\,g_{I_{0}}(H_{0}){\varphi},{\varphi}\right\rangle+\left\langle
g_{I_{0}}(H_{0})\,\mathsf{K}\,g_{I_{0}}(H_{0}){\varphi},{\varphi}\right\rangle\
.$ (4.15)

Note that $\tilde{f}$ is strictly positive in the interior of
$\left[-|\mathsf{V}_{k}|,\,|\mathsf{V}_{k}|\right]$; we put

$\theta:=\inf\\{\tilde{f}(\lambda)\colon\lambda\in{\rm supp}\,g_{I_{0}}\\}>0\
.$

With this information we apply the spectral theorem and get

$\displaystyle\left\langle
g_{I_{0}}(H_{0})\,\tilde{f}(H_{0})\,g_{I_{0}}(H_{0}){\varphi},{\varphi}\right\rangle$
$\displaystyle=\int\limits_{\lambda\in\sigma(H_{0})}g_{I_{0}}(\lambda)^{2}\,\tilde{f}(\lambda)\,{\rm
d}m_{\varphi}(\lambda)$
$\displaystyle\geq\theta\int\limits_{\lambda\in\sigma(H_{0})}g_{I_{0}}(\lambda)^{2}\,\,{\rm
d}m_{\varphi}(\lambda)=\theta\|g_{I_{0}}(H_{0}){\varphi}\|_{0}^{2}\ .$

This estimate and (4.15) proves that $H_{0}$ fulfills a Mourre estimate over
$I_{0}$. ∎

To conclude this section, we recall that in [34] it is proved that the
pseudodifferential operator

$\displaystyle V(t):=e^{-{\rm i}tK_{0}}\,(S+S^{*})\,e^{{\rm i}tK_{0}}$ (4.16)

is a universal transporter (see Definition 1.1). Using the abstract Theorem
2.8 we prove its stability under perturbations of class
$C^{\infty}({\mathbb{T}},{\mathcal{A}}_{0})$:

######  Theorem 4.14.

Consider equation (4.1) with $V(t)$ defined in (4.16). There exist
$\epsilon_{0},{\mathtt{M}}>0$ such that $\forall W\in
C^{\infty}({\mathbb{T}},{\mathcal{A}}_{0})$ with
$\sup_{t}\wp^{0}_{\mathtt{M}}(W(t))\leq\epsilon_{0}$, the operator $V+\epsilon
W$ is a transporter. More precisely $\forall r>0$ there exist a solution
$\psi(t)\in{\mathcal{H}}^{r}$ of ${\rm
i}\partial_{t}\psi=\big{(}\frac{-\partial_{x}^{2}+x^{2}}{2}+V(t)+W(t)\big{)}\psi$
and constants $C,T>0$ such that

$\|\psi(t)\|_{r}\geq C\left\langle t\right\rangle^{r},\quad\forall t\geq T\ .$

###### Proof.

Again we verify Assumption III. It is clear that $V(t)\in
C^{\infty}({\mathbb{T}},{\mathcal{A}}_{0})$ and that $\left\langle
V\right\rangle=S+S^{*}$, so it has the form (4.11) with $k=1$ and
$\mathsf{V}_{1}=2$. Then Lemma 4.13 implies that $\left\langle V\right\rangle$
fulfills a Mourre estimate. ∎

###  4.2 Half-wave equation on ${\mathbb{T}}$

The half-wave equation on ${\mathbb{T}}$ is given by

${\rm i}\partial_{t}\psi=|D|\psi+V(t,x,D)\psi\ ,\qquad x\in{\mathbb{T}}\ .$
(4.17)

Here $|D|$ is the Fourier multiplier defined by

$|D|\psi:=\sum_{j\in{\mathbb{Z}}}|j|\,\psi_{j}\,e^{{\rm i}jx}\
,\qquad\psi_{j}:=\frac{1}{2\pi}\int_{\mathbb{T}}\psi(x)e^{-{\rm i}jx}{\rm d}x\
,$

whereas $V(t,x,D)$ is a pseudodifferential operator of order 0. In this case
$K_{0}:=|D|+1$, the scale of Hilbert spaces defined as ${\mathcal{H}}^{r}={\rm
Dom}\left(K_{0}^{r}\right)$ coincides with standard Sobolev spaces on the
torus $H^{r}({\mathbb{T}})$, and the base space
$({\mathcal{H}}^{0},\left\langle\cdot,\cdot\right\rangle)$ is
$L^{2}({\mathbb{T}},{\mathbb{C}})$ with its standard scalar product. In this
setting we shall use pseudodifferential operators with periodic symbols,
belonging to the following class:

######  Definition 4.15.

A function $a(x,\xi)$ is a periodic symbol of order $\rho\in{\mathbb{R}}$ if
$a\in C^{\infty}({\mathbb{T}}_{x}\times{\mathbb{R}}_{\xi},{\mathbb{C}})$ and
for any $\alpha,\beta\in{\mathbb{N}}_{0}$, there exists a constant
$C_{\alpha\beta}>0$ such that

$\left|\partial_{x}^{\alpha}\,\partial_{\xi}^{\beta}\,a(x,\xi)\right|\leq
C_{\alpha\beta}\,\left\langle\xi\right\rangle^{\rho-\beta},\quad\forall
x\in{\mathbb{T}},\,\forall\xi\in{\mathbb{R}}\,.$ (4.18)

We will write $a\in S^{\rho}_{{\rm per}}$. We also put $S^{-\infty}_{\rm
per}:=\bigcap_{\rho\in{\mathbb{R}}}S^{\rho}_{\rm per}$ the class of smoothing
symbols.

We endow $S^{\rho}_{\rm per}$ with the family of seminorms

$\wp^{\rho}_{j}(a):=\sum_{|\alpha|+|\beta|\leq j}\ \
\sup_{(x,\xi)\in{\mathbb{T}}\times{\mathbb{R}}}{\left|\partial_{x}^{\alpha}\,\partial_{\xi}^{\beta}a(x,\xi)\right|\,\left\langle\xi\right\rangle^{-\rho+\beta}}\
,\qquad j\in{\mathbb{N}}_{0}\ .$ (4.19)

Such seminorms turn $S^{\rho}_{\rm per}$ into a Fréchet space. If a symbol $a$
depends on additional parameters (e.g. it is time dependent), we ask that all
the seminorms are uniform w.r.t. such parameters.
To a symbol $a\in S^{\rho}_{\rm per}$ we associate its quantization $a(x,D)$
acting on a $2\pi$-periodic function $u(x)=\sum_{j\in{\mathbb{Z}}}u_{j}e^{{\rm
i}jx}$ as

$a(x,D)u:={\rm Op}{(a)}[u]:=\sum_{j\in{\mathbb{Z}}}\,a(x,j)\,u_{j}\,e^{{\rm
i}jx}\,.$ (4.20)

######  Remark 4.16.

Given a symbol $a(\xi)$ independent of $x$, then ${\rm Op}{(a)}$ is the
Fourier multiplier operator
$a(D)u=\sum_{j\in{\mathbb{Z}}}\,a(j)\,u_{j}\,e^{{\rm i}jx}$. If instead the
symbol $a(x)$ is independent of $\xi$, then ${\rm Op}{(a)}$ is the
multiplication operator ${\rm Op}{(a)}u=a(x)u$.

######  Definition 4.17.

We say that $A\in{\mathcal{A}}_{\rho}$ if $A={\rm Op}(a)$ with $a\in
S^{\rho}_{{\rm per}}$.

######  Example 4.18.

The operator $|D|\in{\mathcal{A}}_{1}$ with symbol given by
${\mathtt{d}}(\xi):=|\xi|\chi(\xi)$ where $\chi$ is an even, positive smooth
cut-off function satisfying $\chi(\xi)=0$ for $|\xi|\leq\frac{1}{5}$,
$\chi(\xi)=1$ for $|\xi|\geq\frac{2}{5}$ and $\partial_{\xi}\chi(\xi)>0$
$\,\forall\xi\in(\frac{1}{5},\frac{2}{5})$.
Also the Fourier projectors $\Pi_{\pm}$ and $\Pi_{0}$ defined by

$\Pi_{+}u:=\sum_{j\geq 1}u_{j}\,e^{{\rm
i}jx},\qquad\Pi_{-}u:=\sum_{j\leq-1}u_{j}\,e^{{\rm
i}jx},\qquad\Pi_{0}u:=u_{0}$ (4.21)

are pseudodifferential operators. In particular $\Pi_{\pm}={\rm
Op}\left(\pi_{\pm}\right)\in{\mathcal{A}}_{0}$ and $\Pi_{0}={\rm
Op}\left(\pi_{0}\right)\in{\mathcal{A}}_{-\infty}$, where $\pi_{\pm},\pi_{0}$
are a smooth partition of unity, $\pi_{+}(\xi)+\pi_{-}(\xi)+\pi_{0}(\xi)=1$
$\,\forall\xi$, fulfilling

$\pi_{+}(\xi)=\begin{cases}1&\mbox{ if }\xi\geq\frac{4}{5}\\\ 0&\mbox{ if
}\xi\leq\frac{3}{5}\end{cases}\ ,\quad\pi_{-}(\xi)=\begin{cases}1&\mbox{ if
}\xi\leq-\frac{4}{5}\\\ 0&\mbox{ if }\xi\geq-\frac{3}{5}\end{cases}\
,\quad\pi_{0}(\xi)=\begin{cases}1&\mbox{ if }|\xi|\leq\frac{3}{5}\\\ 0&\mbox{
if }|\xi|\geq\frac{1}{5}\end{cases}\ .$ (4.22)

In this setting we prove that any multiplication operator, multiplied by an
appropriate time periodic function, becomes a transporter. Here the result.

######  Theorem 4.19.

Let $v\in C^{\infty}({\mathbb{T}},{\mathbb{R}})$. Choose
$j\in{\mathbb{Z}}\setminus\\{0\\}$ such that the Fourier coefficient
$v_{j}\neq 0$. Then the selfadjoint operator

$V(t,x):=\cos(jt)\,v(x)$ (4.23)

is a transporter. More precisely, $\forall r>0$ there exist a solution
$\psi(t)\in{\mathcal{H}}^{r}$ of ${\rm
i}\partial_{t}\psi=\big{(}|D|+V(t,x)\big{)}\psi$ and constants $C,T>0$ such
that

$\|\psi(t)\|_{r}\geq C\left\langle t\right\rangle^{r},\quad\forall t>T.$

The theorem follows from Theorem 2.7. So first we put ourselves in the setting
of the abstract theorem and rewrite (4.17) as

${\rm
i}\partial_{t}\psi=K_{0}\psi+{\widetilde{V}}(t,x)\psi,\qquad{\widetilde{V}}(t,x):=\cos(jt)v(x)-1\in{\mathcal{A}}_{0}\
.$ (4.24)

Again we check Assumptions I-III. Regarding Assumption I, it is the usual
pseudodifferential calculus for periodic symbols, see e.g. [39].

Verification of Assumption II. One has
$\sigma(K_{0})=\\{n\\}_{n\in{\mathbb{N}}}$. To prove Assumption II $(ii)$ we
use the identity $e^{-{\rm i}tK_{0}}\mathsf{A}\,e^{{\rm i}tK_{0}}=e^{-{\rm
i}t|D|}\mathsf{A}\,e^{{\rm i}t|D|}$ and Egorov theorem for $|D|$, see e.g.
[43, Theorem 4.3.6]. Actually we need also the following version of Egorov
theorem.

######  Lemma 4.20.

Let $a\in S^{\rho}_{\rm per}$, $\rho\in{\mathbb{R}}$. Then

$e^{{\rm i}t|D|}\,{\rm Op}\left(a\right)\,e^{-{\rm i}t|D|}={\rm
Op}\left(a(x+t,\xi)\right)\Pi_{+}+{\rm Op}\left(a(x-t,\xi)\right)\Pi_{-}+R(t)$
(4.25)

where $\Pi_{\pm}$ are defined in (4.21) and $R(t)\in
C^{\infty}({\mathbb{T}},{\mathcal{A}}_{\rho-1})$.
If ${\rm Op}\left(a\right)$ is selfadjoint, so is $e^{{\rm i}t|D|}{\rm
Op}\left(a\right)\,e^{-{\rm i}t|D|},$ $\,\forall t$.

###### Proof.

The classical Egorov theorem for the half-Laplacian $|D|$ says that

$e^{{\rm i}t|D|}\,{\rm Op}\left(a\right)\,e^{-{\rm i}t|D|}={\rm
Op}\left(a\circ\phi_{\mathtt{d}}^{t}(x,\xi)\right)+{\widetilde{R}}(t)$

where $\phi_{\mathtt{d}}^{t}(x,\xi)$ is the time $t$ flow of the classical
Hamiltonian ${\mathtt{d}}(\xi)=|\xi|\chi(\xi)$ (the symbol of $|D|$) and
${\widetilde{R}}(t)\in C^{\infty}({\mathbb{R}},{\mathcal{A}}_{\rho-1})$, see
e.g. [43, Theorem 4.3.6].
We compute more explicitly $a\circ\phi_{\mathtt{d}}^{t}(x,\xi)$. The
Hamiltonian equations of ${\mathtt{d}}(\xi)$ and its flow
$\phi^{t}_{\mathtt{d}}$ are given by

$\begin{cases}\dot{x}=\partial_{\xi}{\mathtt{d}}(\xi)={\mathtt{d}}^{\prime}(\xi)\\\
\dot{\xi}=-\partial_{x}{\mathtt{d}}(\xi)=0\end{cases}\
,\qquad\phi^{t}_{\mathtt{d}}(x,\xi)=(x+t{\mathtt{d}}^{\prime}(\xi),\,\xi)\ .$

As ${\mathtt{d}}^{\prime}(\xi)=1$ for $\xi\geq\frac{2}{5}$ and
${\mathtt{d}}^{\prime}(\xi)=-1$ for $\xi\leq-\frac{2}{5}$, we write

$(a\circ\phi_{\mathtt{d}}^{t})(x,\xi)=a(x+t,\xi)\,\pi_{+}(\xi)+a(x-t,\xi)\,\pi_{-}(\xi)+a(x+t{\mathtt{d}}^{\prime}(\xi),\xi)\,\pi_{0}(\xi)\
.$

As $\pi_{0}\in S^{-\infty}_{\rm per}$, the operator ${\rm
Op}\left(a(x+t{\mathtt{d}}^{\prime}(\xi),\xi)\,\pi_{0}(\xi)\right)\in
C^{\infty}({\mathbb{R}},{\mathcal{A}}_{-\infty})$. Moreover by symbolic
calculus

${\rm Op}\left(a(x\pm t,\xi)\,\pi_{\pm}(\xi)\right)={\rm Op}\left(a(x\pm
t,\xi)\right)\Pi_{\pm}+R_{\pm}(t),\quad R_{\pm}(t)\in
C^{\infty}({\mathbb{R}},{\mathcal{A}}_{\rho-1})\ .$

Formula (4.25) follows with $R(t):={\widetilde{R}}(t)+R_{+}(t)+R_{-}(t)+{\rm
Op}\left(a(x+t{\mathtt{d}}^{\prime}(\xi),\xi)\,\pi_{0}(\xi)\right)$. We claim
that $R(t)$ is periodic in time. This follows by difference since both
$e^{{\rm i}t|D|}\,{\rm Op}\left(a\right)\,e^{-{\rm i}t|D|}$ and ${\rm
Op}\left(a(x\pm t,\xi)\right)\Pi_{\pm}$ are periodic in $t$ (recall that the
symbol $a(x,\xi)$ is periodic in $x$).
Finally as $e^{\pm{\rm i}t|D|}$ are unitary, the claim on the selfadjointness
of $e^{{\rm i}t|D|}\,{\rm Op}\left(a\right)\,e^{-{\rm i}t|D|}$ follows. ∎

Verification of Assumption III. First we compute
$\langle{\widetilde{V}}\rangle$.

######  Lemma 4.21.

The resonant average $\langle{\widetilde{V}}\rangle\in{\mathcal{A}}_{0}$ of
${\widetilde{V}}$ (defined in (4.24)) is given by

$\langle{\widetilde{V}}\rangle={\mathtt{v}}(x)-1+R,\qquad{\mathtt{v}}(x):={\rm
Re}(v_{j}e^{{\rm i}jx})$ (4.26)

and $R\in{\mathcal{A}}_{-1}$ is selfadjoint.

###### Proof.

First remark that, as $e^{{\rm i}tK_{0}}=e^{{\rm i}t|D|}e^{{\rm i}t}$,

$\displaystyle\langle{\widetilde{V}}\rangle$
$\displaystyle=\frac{1}{2\pi}\int_{0}^{2\pi}e^{{\rm
i}s|D|}\,{\widetilde{V}}(s)\,e^{-{\rm i}s|D|}{\rm
d}s=\frac{1}{2\pi}\int_{0}^{2\pi}\cos(js)\,e^{{\rm i}s|D|}\,v(x)\,e^{-{\rm
i}s|D|}\,{\rm d}s-1\ .$ (4.27)

We compute $e^{{\rm i}s|D|}\,v(x)\,e^{-{\rm i}s|D|}$ with the aid of Lemma
4.20, getting

$e^{{\rm i}s|D|}\,v(x)\,e^{-{\rm
i}s|D|}=v(x+s)\,\Pi_{+}+v(x-s)\,\Pi_{-}+{\widetilde{R}}(s),$ (4.28)

where ${\widetilde{R}}(s)\in C^{\infty}({\mathbb{T}},{\mathcal{A}}_{-1})$.
Then, recalling that $v_{j}=\overline{v_{-j}}$ being $v(x)$ real valued,

$\displaystyle\frac{1}{2\pi}\int_{0}^{2\pi}\cos(js)\,e^{{\rm
i}s|D|}\,v(x)\,e^{-{\rm i}s|D|}\,{\rm d}s$
$\displaystyle\stackrel{{\scriptstyle\eqref{eg.per}}}{{=}}\sum_{\sigma=\pm}\frac{1}{2\pi}\int_{0}^{2\pi}\cos(js)\,v(x\sigma
s)\,{\rm
d}s\,\Pi_{\sigma}+\frac{1}{2\pi}\int_{0}^{2\pi}\cos(js){\widetilde{R}}(s){\rm
d}s$ $\displaystyle={\rm Re}\left(v_{j}e^{{\rm
i}jx}\right)\,(\Pi_{+}+\Pi_{-})+\frac{1}{2\pi}\int_{0}^{2\pi}\cos(js){\widetilde{R}}(s){\rm
d}s$ $\displaystyle={\rm Re}\left(v_{j}e^{{\rm i}jx}\right)+R$

where $R:=\frac{1}{2\pi}\int_{0}^{2\pi}\cos(js){\widetilde{R}}(s){\rm d}s-{\rm
Re}\left(v_{j}e^{{\rm i}jx}\right)\Pi_{0}\in{\mathcal{A}}_{-1}$. Together with
(4.27), this proves (4.26). Finally $R$ is selfadjoint by difference, since
both $\langle{\widetilde{V}}\rangle$ and ${\mathtt{v}}(x)-1$ are selfadjoint
operators. ∎

Define the selfadjoint operator

$\displaystyle A:={\mathtt{w}}(x)\,\frac{\partial_{x}}{{\rm
i}}+\frac{\partial_{x}}{{\rm i}}\,{\mathtt{w}}(x)\
,\qquad{\mathtt{w}}(x):={\rm Im}(v_{j}e^{{\rm i}jx})\ $ (4.29)

belonging to ${\mathcal{A}}_{1}$. The next lemma verifies Assumption III.

######  Lemma 4.22.

Assume that $v_{j}\neq 0$. The following holds true:

  * (i)

The operator $H_{0}:=\langle{\widetilde{V}}\rangle$ has spectrum
$\sigma(H_{0})\supseteq[-|v_{j}|-1,\,|v_{j}|-1]=:I$.

  * (ii)

For any interval $I_{0}\subset I$, any $g_{I_{0}}\in
C^{\infty}_{c}({\mathbb{R}},{\mathbb{R}}_{\geq 0})$ with $g_{I_{0}}\equiv 1$
over $I_{0}$ and ${\rm supp}\,g_{I_{0}}\subset I$, there exist $\theta>0$ and
a compact operator $\mathsf{K}$ such that

$g_{I_{0}}(H_{0})\,{\rm
i}[H_{0},A]\,g_{I_{0}}(H_{0})\geq\theta\,g_{I_{0}}(H_{0})^{2}+\mathsf{K}\ .$

Here $A$ is defined in (4.29).

###### Proof.

During the proof we shall use that any operator in ${\mathcal{A}}_{-1}$ is
compact. Moreover we shall simply denote any compact operator by $\mathsf{K}$,
which can change from line to line.
$(i)$ By Lemma 4.21, $H_{0}$ is a compact perturbation of the multiplication
operator by ${\mathtt{v}}(x)-1$, whose spectrum coincides with $I$. Then by
Weyl’s theorem

$\sigma(H_{0})\supseteq\sigma_{ess}(H_{0})=\sigma_{ess}({\mathtt{v}}(x)-1)=I\
.$

$(ii)$ First notice that, as ${\mathtt{v}}(x)={\rm Re}(v_{j}e^{{\rm i}jx})$
and ${\mathtt{w}}(x)={\rm Im}(v_{j}e^{{\rm i}jx})$, one has the identities

${\mathtt{v}}(x)^{2}+{\mathtt{w}}(x)^{2}=|v_{j}|^{2},\qquad{\mathtt{v}}^{\prime}(x)=-j\,{\mathtt{w}}(x)\
.$ (4.30)

Next we compute

$\displaystyle{\rm i}[H_{0},A]$ $\displaystyle={\rm
i}[{\mathtt{v}}(x)-1+R,A]=-2{\mathtt{w}}(x)\,{\mathtt{v}}^{\prime}(x)+{\rm
i}[R,A]$
$\displaystyle\stackrel{{\scriptstyle\eqref{vw}}}{{=}}2j\big{(}|v_{j}|^{2}-{\mathtt{v}}(x)^{2}\big{)}+\mathsf{K}=2j\big{(}|v_{j}|^{2}-(H_{0}+1-R)^{2}\big{)}+\mathsf{K}$
$\displaystyle=2j\big{(}|v_{j}|^{2}-(H_{0}+1)^{2}\big{)}+\mathsf{K}\ .$

Putting $f(\lambda):=2j\big{(}|v_{j}|^{2}-(\lambda+1)^{2}\big{)}$, we get
$\forall{\varphi}\in{\mathcal{H}}$

$\displaystyle\left\langle g_{I_{0}}(H_{0})\,{\rm
i}[H_{0},A]\,g_{I_{0}}(H_{0}){\varphi},{\varphi}\right\rangle=\left\langle
g_{I_{0}}(H_{0})\,f(H_{0})\,g_{I_{0}}(H_{0}){\varphi},{\varphi}\right\rangle+\left\langle\mathsf{K}{\varphi},{\varphi}\right\rangle.$
(4.31)

Now we notice that $f(\lambda)$ is positive in the interior of $I$; so we put

$\theta:=\inf\left\\{f(\lambda)\colon\ \ \lambda\in{\rm
supp}\,g_{I_{0}}\right\\}>0\ .$

With this information we apply the spectral theorem, getting, as in the
previous section,

$\displaystyle\left\langle
g_{I_{0}}(H_{0})\,f(H_{0})\,g_{I_{0}}(H_{0}){\varphi},{\varphi}\right\rangle$
$\displaystyle\geq\theta\int\limits_{\lambda\in
I}g_{I_{0}}(\lambda)^{2}\,\,{\rm
d}m_{\varphi}(\lambda)=\theta\|g_{I_{0}}(H_{0}){\varphi}\|_{0}^{2}\ .$ (4.32)

This together with (4.31) establishes the Mourre estimate over $I_{0}$. ∎

##  Appendix A Flows of pseudodifferential operators

In this appendix we collect some known results about the flow generated by
pseudodifferential operators belonging to the algebra ${\mathcal{A}}$. The
setting is the same as [5] and we refer to that paper for the proofs. The
first result describes how a Schrödinger equation is changed under a change of
variables induced by the flow of a pseudodifferential operator, see Lemma 3.1
of [5]:

######  Lemma A.1.

Let $H(t)$ be a time dependent selfadjoint operator, and $X(t)$ be a
selfadjoint family of operators. Assume that $\psi(t)=e^{-{\rm
i}X(t)}{\varphi}(t)$ then

${\rm i}\partial_{t}\psi=H(t)\psi\ \quad\iff\quad{\rm
i}\partial_{t}{\varphi}=H^{+}(t){\varphi}$ (A.1)

where

$\displaystyle H^{+}(t):=e^{{\rm i}X(t)}\,H(t)\,e^{-{\rm
i}X(t)}-\int_{0}^{1}e^{{\rm i}sX(t)}\,(\partial_{t}X(t))\,e^{-{\rm i}sX(t)}\
{\rm d}s\ .$ (A.2)

The next property we shall need is the Lie expansion of $e^{{\rm
i}X}\,A\,e^{-{\rm i}X}$ in operators of decreasing order, see Lemma 3.2 of
[5]:

######  Lemma A.2.

Let $X\in{\mathcal{A}}_{\rho}$ with $\rho<1$ be a symmetric operator. Let
$A\in{\mathcal{A}}_{m}$ with $m\in{\mathbb{R}}$. Then $e^{{\rm i}\tau
X}\,A\,e^{-{\rm i}\tau X}$ is selfadjoint and for any $M\geq 1$ we have777in
[5] we have defined ${\rm ad}_{X}(A)={\rm i}[X,A]$ rather than (3.21); so we
formulate the next result with the current notation

$e^{{\rm i}\tau X}\,A\,e^{-{\rm i}\tau
X}=\sum_{\ell=0}^{M}\frac{\tau^{\ell}}{{\rm i}^{\ell}\,\ell!}{\rm
ad}_{X}^{\ell}(A)+R_{M}(\tau,X,A)\ ,\qquad\forall\tau\in{\mathbb{R}}\ ,$ (A.3)

where $R_{M}(\tau,X,A)\in{\mathcal{A}}_{m-(M+1)(1-\rho)}$.
In particular ${\rm ad}_{X}^{\ell}(A)\in{\mathcal{A}}_{m-\ell(1-\rho)}$ and
$e^{{\rm i}\tau X}\,A\,e^{-{\rm i}\tau X}\in{\mathcal{A}}_{m}$,
$\forall\tau\in{\mathbb{R}}$.

The last result concerns boundedness properties of the operator $e^{-{\rm
i}\tau X}$, see Lemma 3.3 of [5]:

######  Lemma A.3.

Assume that $X(t)$ is a family of selfadjoint operators in ${\mathcal{A}}_{1}$
s.t.

$\sup_{t\in{\mathbb{R}}}\wp^{1}_{j}(X(t))<\infty\ ,\quad\forall j\geq 1\ .$
(A.4)

Then $e^{-{\rm i}\tau X(t)}$ extends to an operator in
${\mathcal{L}}({\mathcal{H}}^{r})$ $\,\forall r\in{\mathbb{R}}$, and moreover
there exist $c_{r},C_{r}>0$ s.t.

$c_{r}\|\psi\|_{r}\leq\|e^{-{\rm i}\tau X(t)}\psi\|_{r}\leq C_{r}\|\psi\|_{r}\
,\qquad\forall t\in{\mathbb{R}}\ ,\quad\forall\tau\in[0,1]\ .$ (A.5)

##  Appendix B Functional calculus

In this section we collect some known results about functional calculus of
selfadjoint operators which are used thorough the paper. We begin recalling
Helffer-Sjöstrand formula [26], following the presentation of [15].

######  Definition B.1.

A function $f\in C^{\infty}({\mathbb{R}},{\mathbb{C}})$ will be said to belong
to the class $S^{\rho}$, $\rho\in{\mathbb{R}}$, if $\forall
m\in{\mathbb{N}}_{0}$, $\exists C_{m}>0$ such that

$\left|\frac{{\rm d}^{m}}{{\rm d}x^{m}}f(x)\right|\leq C_{m}\left\langle
x\right\rangle^{\rho-m},\quad\forall x\in{\mathbb{R}}\ .$

As usual we set the seminorms

$\wp^{\rho}_{m}(f):=\sum_{0\leq j\leq m}\
\sup_{x\in{\mathbb{R}}}\left|\frac{{\rm d}^{m}f(x)}{{\rm
d}x^{m}}\right|\,\left\langle x\right\rangle^{-\rho+m}\ ,\qquad
m\in{\mathbb{N}}_{0}\ .$

Given $f\in S^{\rho}$, we define its almost analytic extension as follows: for
any $N\in{\mathbb{N}}$, put

${\widetilde{f}}_{N}\colon{\mathbb{R}}^{2}\to{\mathbb{C}},\qquad{\widetilde{f}}_{N}(x,y):=\left(\sum_{\ell=0}^{N}f^{(\ell)}(x)\frac{({\rm
i}y)^{\ell}}{\ell!}\right)\,\tau\left(\frac{y}{\left\langle
x\right\rangle}\right)$ (B.1)

where $\tau\in C^{\infty}({\mathbb{R}},{\mathbb{R}}_{\geq 0})$ is a cut-off
function fulfilling $\tau(s)=1$ for $|s|\leq 1$ and $\tau(s)=0$ for $|s|\geq
2$. It is well known [15] that the choice of $N$ and of the cut-off function
$\tau$ are by no means critical, and even other choices of $\widetilde{f}_{N}$
are possible (see e.g. [18]). The following properties are true [15]: let
$f\in S^{\rho}$ with $\rho<0$, then

$\displaystyle{\widetilde{f}}_{N}|_{\mathbb{R}}=f,\qquad{\rm
supp}\,{\widetilde{f}}_{N}\subset\left\\{x+{\rm i}y\ \colon\ \ \ x\in{\rm
supp}\,f,\quad|y|\leq 2\left\langle x\right\rangle\right\\}\ ,$ (B.2)
$\displaystyle\left|\frac{\partial{\widetilde{f}}_{N}(x,y)}{\partial{\overline{z}}}\right|\leq
C_{N}\left\langle
x\right\rangle^{\rho-N-1}\,|y|^{N},\qquad\frac{\partial{\widetilde{f}}_{N}}{\partial{\overline{z}}}:=\left(\frac{\partial{\widetilde{f}}_{N}}{\partial
x}+{\rm i}\frac{\partial{\widetilde{f}}_{N}}{\partial y}\right)$ (B.3)
$\displaystyle\int_{{\mathbb{R}}^{2}}\left|\frac{\partial{\widetilde{f}}_{N}(z)}{\partial{\overline{z}}}\right|\,\left|{\rm
Im}\,(z)\right|^{-p-1}{\rm d}\overline{z}\wedge{\rm d}z\leq
C_{N}\,\wp^{\rho}_{N+2}(f),\qquad\forall p=0,\ldots,N,$ (B.4)

where $z=x+{\rm i}y$ and ${\rm d}\overline{z}\wedge{\rm d}z$ is the Lebesgue
measure on ${\mathbb{C}}$.

Given $\mathsf{H}$ a selfadjoint operator and $f\in S^{\rho}$, $\rho<0$, the
Helffer-Sjöstrand formula defines $f(\mathsf{H})$ as

$f(\mathsf{H}):=\frac{{\rm
i}}{2\pi}\int_{{\mathbb{R}}^{2}}\frac{\partial{\widetilde{f}}_{N}(z)}{\partial{\overline{z}}}\,(z-\mathsf{H})^{-1}{\rm
d}\overline{z}\wedge{\rm
d}z=-\frac{1}{\pi}\int_{{\mathbb{R}}^{2}}\frac{\partial{\widetilde{f}}_{N}(z)}{\partial{\overline{z}}}\,(z-\mathsf{H})^{-1}{\rm
d}x\,{\rm d}y\ .$ (B.5)

######  Theorem B.2 ([15]).

Let $f\in S^{\rho}$, $g\in S^{\mu}$ with $\rho,\mu<0$ and $\mathsf{H}$ a
selfadjoint operator. Then

  * (i)

The operator $f(\mathsf{H})$ is independent of $N$ and of the cut-off function
$\tau$.

  * (ii)

The integral in (B.5) is norm convergent and
$\|f(\mathsf{H})\|_{{\mathcal{L}}({\mathcal{H}})}\leq\|f\|_{L^{\infty}}$.

  * (iii)

$f(\mathsf{H})\,g(\mathsf{H})=(fg)(\mathsf{H})$.

  * (iv)

$\overline{f}(\mathsf{H})=f(\mathsf{H})^{*}$.

  * (v)

If $f\in C^{\infty}_{c}$ has support disjoint from $\sigma(\mathsf{H})$, then
$f(\mathsf{H})=0$.

  * (vi)

If $z\notin{\mathbb{R}}$ and $f_{z}(x):=(z-x)^{-1}$ for all
$x\in{\mathbb{R}}$, then $f_{z}\in S^{-1}$ and
$f_{z}(\mathsf{H})=(z-\mathsf{H})^{-1}$.

######  Remark B.3.

Given $f\in S^{\rho}$, $\rho<0$ and $\mathsf{H}$ selfadjoint, the operator
$f(\mathsf{H})$ defined via Helffer-Sjöstrand formula coincides with the
classical definition given by the spectral theorem, namely

$f(\mathsf{H})=\int_{{\mathbb{R}}}f(\lambda)\,{\rm d}E(\lambda)$

where ${\rm d}E(\lambda)$ is the spectral resolution of $\mathsf{H}$. For a
proof, see e.g. [19], Theorem 8.1.

Next we recall expansion formulas for commutators. We start from the basic
identities

$\displaystyle{\rm ad}^{n}_{\mathsf{A}}(\mathsf{P}\mathsf{Q})$
$\displaystyle=\sum_{k=0}^{n}\binom{n}{k}\,{\rm
ad}^{n-k}_{\mathsf{A}}(\mathsf{P})\,{\rm
ad}^{k}_{\mathsf{A}}(\mathsf{Q}),\qquad[\mathsf{P},\mathsf{A}^{n}]=\sum_{j=1}^{n}c_{n,j}\,{\rm
ad}^{j}_{\mathsf{A}}(\mathsf{P})\,\mathsf{A}^{n-j}\ .$ (B.6)

For the next lemma see e.g. [18, Lemma C.3.1] or [29, Appendix B].

######  Lemma B.4 (Commutator expansion formula).

Let $k\in{\mathbb{N}}$ and $\mathsf{A},\mathsf{B}$ selfadjoint operators with

$\|{\rm
ad}_{\mathsf{A}}^{j}(\mathsf{B})\|_{{\mathcal{L}}({\mathcal{H}})}<\infty,\qquad\forall\,1\leq
j\leq k\ .$

Let $f\in S^{\rho}$ with $\rho<0$, then one has the right and left commutator
expansions

$\displaystyle[\mathsf{B},f(\mathsf{A})]$
$\displaystyle=\sum_{j=1}^{k-1}\frac{1}{j!}\,f^{(j)}(\mathsf{A})\,{\rm
ad}_{\mathsf{A}}^{j}(\mathsf{B})+R_{k}(f,\mathsf{A},\mathsf{B})$ (B.7)
$\displaystyle=\sum_{j=1}^{k-1}\frac{(-1)^{j-1}}{j!}\,{\rm
ad}_{\mathsf{A}}^{j}(\mathsf{B})\,f^{(j)}(\mathsf{A})\,+{\widetilde{R}}_{k}(f,\mathsf{A},\mathsf{B})$
(B.8)

where the operators $R_{k},{\widetilde{R}}_{k}$ fulfill

$\|R_{k}(f,\mathsf{A},\mathsf{B})\|_{{\mathcal{L}}({\mathcal{H}})},\quad\|{\widetilde{R}}_{k}(f,\mathsf{A},\mathsf{B})\|_{{\mathcal{L}}({\mathcal{H}})}\leq\,C_{N}\,\wp^{\rho}_{k+2}(f)\,\|{\rm
ad}_{\mathsf{A}}^{k}(\mathsf{B})\|_{{\mathcal{L}}({\mathcal{H}})}\ .$ (B.9)

######  Lemma B.5.

Let $k\in{\mathbb{N}}$ and $\mathsf{A},\mathsf{H}$ selfadjoint operators such
that

$\|{\rm
ad}_{\mathsf{A}}^{j}(\mathsf{H})\|_{{\mathcal{L}}({\mathcal{H}})}<\infty,\qquad\forall\,1\leq
j\leq k\ .$ (B.10)

Let $g\in S^{\rho}$ with $\rho<0$. Then

$\|{\rm
ad}_{\mathsf{A}}^{j}(g(\mathsf{H}))\|_{{\mathcal{L}}({\mathcal{H}})}<\infty\,\quad\forall
1\leq j\leq k\ .$

###### Proof.

Take $N\geq k$ and use Helffer-Sjöstrand formula to write

${\rm ad}^{j}_{\mathsf{A}}(g(\mathsf{H}))=\frac{{\rm
i}}{2\pi}\int_{{\mathbb{R}}^{2}}\frac{\partial{\widetilde{g}}_{N}(z)}{\partial\overline{z}}\,{\rm
ad}^{j}_{\mathsf{A}}\big{(}(z-\mathsf{H})^{-1}\big{)}\,{\rm
d}\overline{z}\wedge{\rm d}z.$ (B.11)

As ${\rm
ad}_{\mathsf{A}}\big{(}(z-\mathsf{H})^{-1}\big{)}=(z-\mathsf{H})^{-1}\,{\rm
ad}_{\mathsf{A}}(\mathsf{H})\,(z-\mathsf{H})^{-1}$, by induction one gets for
$j=1,\ldots,k$

${\rm
ad}_{\mathsf{A}}^{j}\big{(}(z-\mathsf{H})^{-1}\big{)}=\sum_{\ell=1}^{j}\sum_{k_{1}+\cdots+k_{\ell}=j\atop
k_{1},\ldots,k_{\ell}\geq 1}c_{k_{1}\cdots
k_{\ell}}^{\ell,j}\,(z-\mathsf{H})^{-1}\,{\rm
ad}_{\mathsf{A}}^{k_{1}}(\mathsf{H})\,(z-\mathsf{H})^{-1}{\rm
ad}_{\mathsf{A}}^{k_{2}}(\mathsf{H})\cdots(z-\mathsf{H})^{-1}\,{\rm
ad}_{\mathsf{A}}^{k_{\ell}}(\mathsf{H})\,(z-\mathsf{H})^{-1}$

Using (B.10) and the estimate
$\|(z-\mathsf{H})^{-1}\|_{{\mathcal{L}}({\mathcal{H}})}\leq{\left|{\rm
Im}\,(z)\right|^{-1}}$, $\forall z\in{\mathbb{C}}\setminus{\mathbb{R}}$, one
has for $j=1,\ldots,k$

$\|{\rm
ad}_{\mathsf{A}}^{j}\big{(}(z-\mathsf{H})^{-1}\big{)}\|_{{\mathcal{L}}({\mathcal{H}})}\leq\sum_{\ell=1}^{j}C_{\ell}\,{\left|{\rm
Im}\,(z)\right|}^{-\ell-1},\quad\forall
z\in{\mathbb{C}}\setminus{\mathbb{R}}.$

Inserting this estimate into (B.11) and using (B.4) we bound for any
$j=1,\ldots,k$

$\|{\rm
ad}^{j}_{\mathsf{A}}(g_{J}(\mathsf{H}))\|_{{\mathcal{L}}({\mathcal{H}})}\lesssim\sum_{\ell=1}^{j}\int_{{\mathbb{R}}^{2}}\left|\frac{\partial{\widetilde{g}}_{N}(z)}{\partial\overline{z}}\right|\,\,{\left|{\rm
Im}\,(z)\right|}^{-\ell-1}{\rm d}\overline{z}\wedge{\rm
d}z\lesssim\wp^{\rho}_{N+2}(g)<\infty\ .$

∎

######  Lemma B.6.

Let $g\in C^{\infty}_{c}({\mathbb{R}},{\mathbb{R}})$. Let
$\mathsf{H},\mathsf{B}\in{\mathcal{L}}({\mathcal{H}})$ be selfadjoint. Then
$\exists\,C>0$ such that

$\|g(\mathsf{H}+\mathsf{B})-g(\mathsf{H})\|_{{\mathcal{L}}({\mathcal{H}})}\leq
C\|\mathsf{B}\|_{{\mathcal{L}}({\mathcal{H}})}\ .$ (B.12)

If $\mathsf{B}$ is compact on ${\mathcal{H}}$, so is
$g(\mathsf{H}+\mathsf{B})-g(\mathsf{H})$.

###### Proof.

Take $N\geq 1$. Using Helffer-Sjöstrand formula and the resolvent identity we
obtain

$g(\mathsf{H}+\mathsf{B})-g(\mathsf{H})=\frac{{\rm
i}}{2\pi}\int_{{\mathbb{R}}^{2}}\frac{\partial{\widetilde{g}}_{N}(z)}{\partial\overline{z}}\,\big{(}z-(\mathsf{H}+\mathsf{B})\big{)}^{-1}\,\mathsf{B}\,(z-\mathsf{H})^{-1}\,{\rm
d}\overline{z}\wedge{\rm d}z\ .$

Then use
$\|\big{(}z-(\mathsf{H}+\mathsf{B})\big{)}^{-1}\|_{{\mathcal{L}}({\mathcal{H}})}$,
$\|(z-\mathsf{H})^{-1}\|_{{\mathcal{L}}({\mathcal{H}})}\leq\left|{\rm
Im}(z)\right|^{-1}$ for $z\in{\mathbb{C}}\setminus{\mathbb{R}}$ and (B.4).
If $\mathsf{B}$ is compact then
$\big{(}z-(\mathsf{H}+\mathsf{B})\big{)}^{-1}\,\mathsf{B}\,(z-\mathsf{H})^{-1}$
is a compact operator for any $z\in{\mathbb{C}}\setminus{\mathbb{R}}$. ∎

##  Appendix C Local energy decay estimates

In this section we prove a local energy decay estimate starting from Mourre
estimate. The result is essentially known but we could not find in the
literature a statement exactly as the one we use in the paper, so we include
here a proof, which follows closely the one of Lemma 4.1 of [24]. In this part
we do not require pseudodifferential properties of the operators. We shall
assume conditions (M1) and (M2) at page (M1).

######  Theorem C.1 (Local energy decay estimate).

Fix $k\in{\mathbb{N}}$ and assume (M1)–(M2) with ${\mathtt{N}}\geq 4k+2$ and
$\mathsf{K}=0$. Then for any interval $J\subset I$, any function $g_{J}\in
C^{\infty}_{c}({\mathbb{R}},{\mathbb{R}}_{\geq 0})$ with ${\rm
supp}\,g_{J}\subset I$, $g_{J}=1$ on $J$, there exists $C>0$ such that

$\|\left\langle\mathsf{A}\right\rangle^{-k}\,e^{-{\rm
i}\mathsf{H}t}\,g_{J}(\mathsf{H})\,\psi\|_{0}\leq C\left\langle
t\right\rangle^{-k}\|\left\langle\mathsf{A}\right\rangle^{k}\,g_{J}(\mathsf{H})\psi\|_{0},\quad\forall
t\in{\mathbb{R}},$ (C.1)

for any $\psi$ such that the r.h.s. is finite.

###### Proof.

Take $\chi(\xi):=\frac{1}{2}(1-\tanh\xi)$. Put
$\eta(\xi):=\frac{1}{\sqrt{2}\cosh\xi}$ and note that

$\chi^{\prime}=-\eta^{2},\qquad\left|\eta^{(m)}(\xi)\right|\leq
C_{m}\,\eta(\xi),\quad\forall\xi\in{\mathbb{R}},\ \ \forall m\in{\mathbb{N}}.$
(C.2)

Next we set for $a\in{\mathbb{R}}$, $s\geq 1$ and
$\vartheta:=\frac{\theta}{2}$ (with $\theta$ of (M2) )

$\mathsf{A}_{t,s}:=\frac{1}{s}\big{(}\mathsf{A}-a-\vartheta t\big{)}$

and define via functional calculus the operators $\chi(\mathsf{A}_{t,s})$ and
$\eta(\mathsf{A}_{t,s})$; both are bounded and selfadjoint on ${\mathcal{H}}$.
To shorten the notation, from now on we write
$\chi_{t,s}\equiv\chi(\mathsf{A}_{t,s})$,
$\eta_{t,s}\equiv\eta(\mathsf{A}_{t,s})$, $g_{J}\equiv g_{J}(\mathsf{H})$ and
$\psi_{t}:=e^{-{\rm i}t\mathsf{H}}\psi$. Note that $e^{-{\rm
i}\mathsf{H}t}g_{J}(\mathsf{H})\psi=g_{J}(\mathsf{H})e^{-{\rm
i}\mathsf{H}t}\psi\equiv g_{J}\psi_{t}$.
The starting point of the proof is an energy estimate for the quantity
$\|(\chi_{t,s})^{\frac{1}{2}}\,g_{J}\psi_{t}\|_{0}$. We have

$\frac{d}{dt}\|(\chi_{t,s})^{\frac{1}{2}}\,g_{J}\psi_{t}\|_{0}^{2}=\frac{d}{dt}\left\langle\chi_{t,s}g_{J}\psi_{t},\,g_{J}\psi_{t}\right\rangle=\frac{\vartheta}{s}\|\eta_{t,s}\,g_{J}\,\psi_{t}\|_{0}^{2}+\left\langle{\rm
i}[\mathsf{H},\chi_{t,s}]\,g_{J}\psi_{t},\,g_{J}\psi_{t}\right\rangle.$ (C.3)

To evaluate the right hand side we shall use the commutator formulas in Lemma
B.4, the identity

${\rm ad}^{j}_{\mathsf{A}_{t,s}}(\mathsf{H})=\frac{1}{s^{j}}{\rm
ad}^{j}_{\mathsf{A}}(\mathsf{H})\ ,\quad\forall s\geq 1,\ \ \forall 1\leq
j\leq{\mathtt{N}}$ (C.4)

and the fact that all the operators ${\rm ad}^{j}_{\mathsf{A}}(\mathsf{H})$
are bounded $\forall 1\leq j\leq{\mathtt{N}}$ by (M1). The goal now is to
estimate the second term in the right hand side of (C.3). For an arbitrary
$f\in{\mathcal{H}}$ we write

$\left\langle{\rm
i}[\mathsf{H},\chi_{t,s}]f,f\right\rangle\stackrel{{\scriptstyle\eqref{r.exp},\eqref{ad.Ats}}}{{=}}-\frac{1}{s}\left\langle\eta_{t,s}^{2}{\rm
i}[\mathsf{H},\mathsf{A}]f,f\right\rangle+\sum_{j=2}^{{\mathtt{N}}-1}\frac{1}{j!}\frac{1}{s^{j}}\left\langle\chi_{t,s}^{(j)}\,{\rm
i}\,{\rm
ad}_{\mathsf{A}}^{j}(\mathsf{H})f,f\right\rangle+\frac{1}{s^{\mathtt{N}}}\left\langle
R_{\mathtt{N}}f,f\right\rangle$ (C.5)

where $\chi_{t,s}^{(j)}:=\chi^{(j)}(\mathsf{A}_{t,s})$ and the remainder
$R_{\mathtt{N}}$ fulfills the estimate (see (B.9))

$\|R_{\mathtt{N}}\|_{{\mathcal{L}}({\mathcal{H}})}\leq C_{\mathtt{N}}\,\|{\rm
ad}_{\mathsf{A}}^{\mathtt{N}}(\mathsf{H})\|\leq C_{{\mathtt{N}}}.$ (C.6)

Note that the constant $C_{\mathtt{N}}>0$ in the previous estimate is uniform
in $a\in{\mathbb{R}}$. In the following we shall simply denote by
$R_{\mathtt{N}}$ any bounded operator fulfilling an estimate like (C.6).
Consider now the first term in the expansion (C.5) above. This time we use the
left expansion (B.8) and write

$\displaystyle\frac{1}{s}\left\langle\eta_{t,s}^{2}\,{\rm
i}[\mathsf{H},\mathsf{A}]f,f\right\rangle$
$\displaystyle=\frac{1}{s}\left\langle\,{\rm
i}[\mathsf{H},\mathsf{A}]\,\eta_{t,s}f,\,\eta_{t,s}f\right\rangle$ (C.7)
$\displaystyle\quad+\sum_{j=2}^{{\mathtt{N}}-1}\frac{(-1)^{j-1}}{(j-1)!}\frac{1}{s^{j}}\langle\,{\rm
i}\,{\rm
ad}_{\mathsf{A}}^{j}(\mathsf{H})\,\eta_{t,s}^{(j-1)}f,\eta_{t,s}f\rangle+\frac{1}{s^{\mathtt{N}}}\left\langle
R_{\mathtt{N}}f,f\right\rangle$ (C.8)

where $R_{\mathtt{N}}$ is estimated as in (C.6). Consider now the second term
in (C.5). From $\chi^{\prime}=-\eta^{2}$, we have by functional calculus
$\chi^{(j)}(\mathsf{A}_{t,s})=\sum_{\ell=1}^{j}c_{\ell
j}\,\eta^{(j-\ell)}(\mathsf{A}_{t,s})\,\eta^{(\ell)}(\mathsf{A}_{t,s})\,.$
Thus we get that

$\displaystyle\frac{1}{s^{j}}\langle\chi_{t,s}^{(j)}\,{\rm i}\,{\rm
ad}_{\mathsf{A}}^{j}(\mathsf{H})f,f\rangle$
$\displaystyle\stackrel{{\scriptstyle\eqref{l.exp}}}{{=}}\frac{1}{s^{j}}\sum_{\ell=1}^{j}c_{\ell
j}\langle\,{\rm i}\,{\rm
ad}_{\mathsf{A}}^{j}(\mathsf{H})\,\eta_{t,s}^{(\ell)}f,\eta_{t,s}^{(j-\ell)}f\rangle$
(C.9)
$\displaystyle\quad+\sum_{\ell=1}^{j}\sum_{n=1}^{{\mathtt{N}}-j-1}\frac{c_{\ell
jn}}{s^{j+n}}\langle\,{\rm i}\,{\rm
ad}_{\mathsf{A}}^{j+n}(\mathsf{H})\,\eta_{t,s}^{(\ell+n)}f,\,\eta_{t,s}^{(j-\ell)}f\rangle+\frac{1}{s^{{\mathtt{N}}}}\left\langle
R_{\mathtt{N}}f,f\right\rangle$ (C.10)

By (C.5), (C.7), (C.9) we have found that $\left\langle{\rm
i}[\mathsf{H},\chi_{t,s}]f,f\right\rangle$ is a sum of terms of the form

$\displaystyle\left\langle{\rm
i}[\mathsf{H},\chi_{t,s}]f,f\right\rangle=-\frac{1}{s}\left\langle\,{\rm
i}[\mathsf{H},\mathsf{A}]\,\eta_{t,s}f,\,\eta_{t,s}f\right\rangle+\sum_{j=2}^{{\mathtt{N}}-1}\frac{1}{s^{j}}\sum_{n,\ell,m}\langle\,R_{n}\,\eta_{t,s}^{(\ell)}f,\eta_{t,s}^{(m)}f\rangle+\frac{1}{s^{\mathtt{N}}}\left\langle
R_{\mathtt{N}}f,f\right\rangle$

where $R_{n},R_{\mathtt{N}}$ are bounded operators. Furthermore, from the
second of (C.2) and the spectral theorem, we bound

$\left|\langle\,R_{n}\,\eta_{t,s}^{(\ell)}f,\eta_{t,s}^{(m)}f\rangle\right|\leq
C\,\|\eta_{t,s}f\|_{0}^{2}\ .$ (C.11)

We thus obtain, for any $f\in{\mathcal{H}}$ and $s\geq 1$, the estimate

$\left\langle{\rm
i}[\mathsf{H},\chi_{t,s}]f,f\right\rangle\leq-\frac{1}{s}\left\langle\,{\rm
i}[\mathsf{H},\mathsf{A}]\,\eta_{t,s}f,\,\eta_{t,s}f\right\rangle+\frac{C_{\mathtt{N}}}{s^{2}}\|\eta_{t,s}f\|_{0}^{2}+\frac{C_{\mathtt{N}}}{s^{\mathtt{N}}}\|f\|_{0}^{2}\
.$ (C.12)

Now we evaluate such inequality at $f=g_{J}\psi_{t}$, getting

$\left\langle{\rm
i}[\mathsf{H},\chi_{t,s}]g_{J}\psi_{t},\,g_{J}\psi_{t}\right\rangle\leq-\frac{1}{s}\left\langle\,{\rm
i}[\mathsf{H},\mathsf{A}]\,\eta_{t,s}g_{J}\psi_{t},\,\eta_{t,s}g_{J}\psi_{t}\right\rangle+\frac{C_{\mathtt{N}}}{s^{2}}\|\eta_{t,s}g_{J}\psi_{t}\|_{0}^{2}+\frac{C_{\mathtt{N}}}{s^{\mathtt{N}}}\|g_{J}\psi_{t}\|_{0}^{2}\
.$ (C.13)

The next step is to prove that the first term in the right hand side above has
a sign, up to higher order terms in $s^{-j}$. This is the point where the
Mourre estimate (M2) comes into play. To see this, we analyze

$\displaystyle\left\langle\,{\rm
i}[\mathsf{H},\mathsf{A}]\,\eta_{t,s}\,g_{J}\psi_{t},\,\eta_{t,s}\,g_{J}\psi_{t}\right\rangle\equiv\left\langle\,{\rm
i}[\mathsf{H},\mathsf{A}]\,\eta_{t,s}\,g_{I}\,g_{J}\psi_{t},\,\eta_{t,s}g_{I}\,g_{J}\psi_{t}\right\rangle$
(C.14)

where we used that $g_{J}g_{I}=g_{J}$. Next we commute and expand in
commutators $\eta_{t,s}g_{I}$:

$\displaystyle\eta_{t,s}g_{I}=g_{I}\eta_{t,s}+[\eta_{t,s},g_{I}]\stackrel{{\scriptstyle\eqref{l.exp}}}{{=}}g_{I}\,\eta_{t,s}+\sum_{j=1}^{{\mathtt{N}}-2}\frac{c_{j}}{s^{j}}\,{\rm
ad}_{\mathsf{A}}^{j}(g_{I}(\mathsf{H}))\,\eta_{t,s}^{(j)}+\frac{1}{s^{{\mathtt{N}}-1}}{\widetilde{R}}_{{\mathtt{N}}-1}\
;$ (C.15)

note that Lemma B.5 assures that the operators ${\rm
ad}_{\mathsf{A}}^{j}(g_{I}(\mathsf{H}))$ are bounded $\forall
j=1,\ldots,{\mathtt{N}}$, so is the operator
${\widetilde{R}}_{{\mathtt{N}}-1}$ which fulfills

$\|{\widetilde{R}}_{{\mathtt{N}}-1}\|_{{\mathcal{L}}({\mathcal{H}})}\leq
C_{\mathtt{N}}\,{\rm
ad}_{\mathsf{A}}^{{\mathtt{N}}-1}(g_{I}(\mathsf{H}))<\infty\ .$ (C.16)

Again in the following we shall denote by ${\widetilde{R}}_{{\mathtt{N}}-1}$
any operator fulfilling an estimate like (C.16). Inserting the expansion
(C.15) into (C.14) one gets, with $w:=g_{J}\psi_{t}$,

$\displaystyle\eqref{HAeta}=\left\langle{\rm
i}[\mathsf{H},\mathsf{A}]\,g_{I}\,\eta_{t,s}w,\
g_{I}\,\eta_{t,s}w\right\rangle+\sum_{j=1}^{{\mathtt{N}}-2}\frac{c_{j}}{s^{j}}\sum_{n,\ell,m}\langle
R_{n}\,\eta^{(\ell)}_{t,s}w,\
\eta^{(m)}_{t,s}w\rangle+\frac{1}{s^{{\mathtt{N}}-1}}\langle{\widetilde{R}}_{{\mathtt{N}}-1}w,w\rangle$

where each term of the form $\langle R_{n}\,\eta^{(\ell)}_{t,s}w,\
\eta^{(m)}_{t,s}w\rangle$ fulfills an estimate like (C.11).
It is finally time to use the strict Mourre estimate: by assumption (M2) we
have for $s\geq 1$

$\displaystyle\left\langle{\rm i}[\mathsf{H},\mathsf{A}]\,g_{I}\,\eta_{t,s}w,\
g_{I}\,\eta_{t,s}w\right\rangle$
$\displaystyle\geq\theta\|g_{I}\,\eta_{t,s}w\|_{0}^{2}\ .$ (C.17)

Using again the expansion (C.15) and estimates (C.11), (C.16) we get therefore

$\displaystyle\left\langle{\rm i}[\mathsf{H},\mathsf{A}]\,g_{I}\,\eta_{t,s}w,\
g_{I}\,\eta_{t,s}w\right\rangle$
$\displaystyle\geq\theta\|\,\eta_{t,s}g_{I}w\|_{0}^{2}-\frac{C_{\mathtt{N}}}{s}\|\eta_{t,s}w\|_{0}^{2}-\frac{C_{\mathtt{N}}}{s^{{\mathtt{N}}-1}}\|w\|_{0}^{2}.$
(C.18)

This proves that the first term in the right hand side of (C.13) has a sign;
we proceed from (C.13) and using inequality (C.18) (recall $w=g_{J}\psi_{t})$
we get

$\left\langle{\rm
i}[\mathsf{H},\chi_{t,s}]g_{J}\psi_{t},\,g_{J}\psi_{t}\right\rangle\leq-\frac{\theta}{s}\|\eta_{t,s}\,g_{J}\psi_{t}\|_{0}^{2}+\frac{C_{\mathtt{N}}}{s^{2}}\|\eta_{t,s}\,g_{J}\psi_{t}\|_{0}^{2}+\frac{C_{\mathtt{N}}}{s^{\mathtt{N}}}\|\,g_{J}\psi_{t}\|_{0}^{2}\
.$ (C.19)

We come back to the estimate (C.3) of
$\|(\chi_{t,s})^{\frac{1}{2}}\,g_{J}\psi_{t}\|_{0}$. We finally obtain, with
$\vartheta=\frac{\theta}{2}$ and $s\geq 1$ sufficiently large,

$\displaystyle\frac{d}{dt}\|(\chi_{t,s})^{\frac{1}{2}}\,g_{J}\psi_{t}\|_{0}^{2}$
$\displaystyle\stackrel{{\scriptstyle\eqref{min53}}}{{\leq}}\frac{1}{s}\left(\vartheta-\theta\right)\|\eta_{t,s}\,g_{J}\psi_{t}\|_{0}^{2}+\frac{C_{\mathtt{N}}}{s^{2}}\|\eta_{t,s}\,g_{J}\psi_{t}\|_{0}^{2}+\frac{C_{\mathtt{N}}}{s^{\mathtt{N}}}\|\,g_{J}\psi_{t}\|_{0}^{2}$