---
abstract: 'We study almost periodic orbits of quantum systems and prove that for periodic time-dependent Hamiltonians an orbit is almost periodic if, and only if, it is precompact. In the case of quasiperiodic time-dependence we present an example of a precompact orbit that is not almost periodic. Finally we discuss some simple conditions assuring dynamical stability for nonautonomous quantum system.'
address:
- 'Departamento de Matemática – UFSCar, São Carlos, SP, 13560-970 Brazil'
- 'Departamento de Matemática – UFSCar, São Carlos, SP, 13560-970 Brazil\'
author:
- 'César R. de Oliveira'
- 'Mariza S. Simsen'
title: 'Almost Periodic Orbits and Stability for Quantum Time-Dependent Hamiltonians'
---

[^1]

[^2]

[Keywords]{}: almost periodicity; quantum stability; time-dependent systems; precompact orbits.

Introduction {#IntroductionSection}
============

The time evolution of a quantum mechanical system with time-dependent Hamiltonians $H(t)$ is determined by the Schrödinger equation $$i\frac{d\psi(t)}{dt}=H(t)\psi(t),$$ where $H(t)$ is a family of self-adjoint operators in the Hilbert space $\mathcal{H}$ and $\psi(t)\in\mathcal{H}$ for all $t\in{\ensuremath{{\mathrm{I\!R}}}}$. The initial value problem $\psi(0)=\psi$ has a unique solution $$\psi(t)\doteq U(t,0)\psi,$$ under suitable conditions on $H(t)$ (see [@RS; @K; @K1; @I]) and the propagators, or time evolution operators $U(t,s)$, form a strongly continuous family of unitary operators acting on $\mathcal{H}$, such that $$U(t,r)U(r,s)=U(t,s),\qquad \forall r,s,t,\in{\ensuremath{{\mathrm{I\!R}}}}$$ $$U(t,t)=
{{\mathrm{I_d}}},\qquad \forall t.$$ ${{\mathrm{I_d}}}$ denotes the identity operator. If the Hamiltonian is time-periodic with period $T$, then $U(t+T,r+T)=U(t,r)$ and the Floquet operator at $s$ is defined by $U_F(s)\doteq U(s+T,s)$; $U_F(0)$ is simply called Floquet operator and denoted by $U_F$, and $U_F(s)$ is unitarily equivalent to $U_F(r)$, $\forall
r,s$. Let $$\mathcal{O}(\psi)\doteq\{ U(t,0)\psi:t\in{\ensuremath{{\mathrm{I\!R}}}}\}$$ be the orbit of a vector $\psi\in\mathcal{H}$.

If $H(t)=H$ is independent of $t$ the time evolution operators are $U(t,s)=e^{-iH(t-s)}$. In this case, it is a well-known fact that if $\psi$ is in the point subspace of $H$ then the quantum time evolution of the state $\psi$, $\psi(t)$, is almost periodic, since it can be expanded in terms of the eigenfunctions $\varphi_n$ of $H$, with eigenvalues $E_n$, $$\psi(t)=\sum_nc_ne^{-iE_nt}\varphi_n.$$ Reciprocally, if $\psi(t)$ is almost periodic then using the results in [@Kat] (Chapter VI) it holds true that $\mathcal{O}(\psi)$ is precompact and then $\psi$ is in the point subspace of $H$ (see Theorem \[teo.31\] ahead). In this work, we prove that this fact remains true in the periodic case, that is, $\psi$ is in the point subspace of $U_F$ if, and only if, $\psi(t)$ is almost periodic (see Theorem \[teo.31a\]).

In the studies of time-dependent systems it is common to consider the quasienergy operator, i.e., a self-adjoint operator formally given by $$K=-i\frac{d}{dt}+H(t)$$ acting in some enlarged Hilbert space. The quasienergy operator $K$ was previously defined for periodic Hamiltonians [@Y; @H] and then generalized for general time dependence in [@H1]. In the periodic case it was proved that $$e^{-iKT}\simeq {{\mathrm{I_d}}}\otimes U_F,$$ where $\simeq$ means unitary equivalence.

A natural framework for considering general time-dependent perturbations, which includes both periodic and the random potentials as special cases, is to write $H(t)$ in the form $$H(t)=H(g_t(\theta))=H_0+V(g_t(\theta)),$$ where $g_t:\Omega\rightarrow\Omega$ is an invertible flow on a compact manifold $\Omega$ with a probability ergodic measure $\mu$ and $H_0$ is the Hamiltonian of the isolated system (see [@JL; @BJL]). Again, under suitable conditions on $V$ there exists a unitary time evolution operator $U_{\theta}(t,s)$ and the generalized quasienergy operator is defined [@JL] on $L^2(\Omega,\mathcal{H},d\mu)$ by $$(e^{-i\tilde{K}t}f)_{\theta}=\mathcal{F}_{-t}U_{\theta}(t,0)f_{\theta}=
U_{\theta}(0,-t)\mathcal{F}_{-t}f_{\theta},$$ where $\mathcal{F}_{-t}f_{\theta}=f_{g_{-t}(\theta)};$ we refer to this construction as [*Jauslin-Lebowitz formulation.*]{} The operator $\tilde{K}$ acts as $$(\tilde{K}f)_{\theta}=i\frac{d}{dt}f_{g_{-t}(\theta)}\Big|_{t=0}+H_{\theta}f_{\theta}.$$ In the case of a periodic potential one has $\Omega=S^1\equiv[0,2\pi)$, $g_t(\theta)=\theta+\omega t$ and $d\mu=\frac{d\theta}{2\pi}$.

For quasiperiodic potentials with two incommensurate frequencies $\omega_1/\omega_2\notin{\ensuremath{{\mathrm{Q\hspace{-2.1mm}\rule{0.3mm}{2.6mm}\;}}}}$ the manifold $\Omega$ is $S^1\times S^1$, $g_t(\theta_1,\theta_2)=(\theta_1+\omega_1t,\theta_2+\omega_2t)$ and $d\mu=\frac{d\theta_1}{2\pi}\frac{d\theta_2}{2\pi}$. We denote the two periods by $T_j=\frac{2\pi}{\omega_j}$. In this case the generalized Floquet operator acting on $\mathcal{K}_1\doteq
L^2(S^1,\mathcal{H},\frac{d\theta_1}{2\pi})$ is defined by $$\label{FloquetGenEq}
U_{\mathrm{F}}=\mathcal{T}_{-T_2}u_1,$$ where $u_1(\theta_1)=U_{(\theta_1,0)}(T_2,0)$ ($\doteq$ monodromy operator) and $(\mathcal{T}_{-T_2}\phi)(\theta_1)=\phi(\theta_1-\omega_1T_2)$.

Let $A:{{\mathrm{dom}~}}A\subset\mathcal{H}\rightarrow\mathcal{H}$ be an unbounded positive self-adjoint operator with discrete spectrum which we call a [*probe operator*]{}. Assuming that if $\psi\in{{\mathrm{dom}~}}A$, then $U(t,0)\psi\in{{\mathrm{dom}~}}A$ for all $t\geq0$, a very interesting question is about the behavior of the expectation value of $A$, that is, $$E_{\psi}^{A}(t)\equiv\langle
U(t,0)\psi,AU(t,0)\psi\rangle.$$ We say the system is $A$-dynamically stable if $E_{\psi}^{A}(t)$ is a bounded function of time, and $A$-dynamically unstable otherwise. A particular case is when the Hamiltonian has the form $H(t)=H_0+V(t)$ and $A=H_0$. In this work we discuss some simple conditions assuring dynamical stability, mainly when either the Floquet or quasienergy operator has purely point spectrum; recall that in the periodic case it is known that continuous spectrum of the Floquet operator implies dynamical instability (see Section \[PreliminarSection\]).

Usually it is not a simple task to get results on dynamical (in)stability in the original Hilbert space $H$ through properties of $K$ or $\tilde{K}$ acting in the corresponding enlarged space. We present some theoretical results about this point in Section \[BoundedSection\]. An important result in the periodic case was proved in [@DSSV], i.e., that the applicability of the KAM method for the quasienergy operator $K$, which is a technique to find out a unitary operator $U$ such that $UKU^{-1}=D,$ where $D$ is pure point, gives a uniform bound at the expectation value of the energy for a class of time-periodic Hamiltonians of the form $H(t)=H_0+V(t)$ considered in [@DLSV].

The study of precompacity (and related properties) of orbits of a time-dependent quantum system and their connection with spectral type and stability was carried out, e.g., in [@EV; @dOT; @dO; @BJLPN; @JL; @BJL]. In this work we prove that in the periodic case (including the autonomous case) the orbit $\mathcal{O}(\psi)$ is precompact if, and only if, $\psi(t)$ is an almost periodic function. Moreover, already in the quasiperiodic case we present an example with precompact orbits which are not almost periodic.

This paper is organized as follows. In Section \[PreliminarSection\] we recall some subspaces of $\mathcal{H}$ that were studied in the literature and the results that connect this subspaces with dynamical (in)stability and spectral properties of the Floquet or quasienergy operators. In Section \[PeneperiodicSection\] we present ours results about almost periodic orbits. In Section \[BoundedSection\] we discuss some simple conditions assuring dynamical stability; we pay special attention to connection between enlarged spaces and the original quantum Hilbert space. A number of known results are recalled in the text in order to make it as readable as possible.

Preliminaries {#PreliminarSection}
=============

In this section we present a short account of suitable subspaces and relations among them, in order to put our results in context.

Consider a time-dependent Hamiltonian $H(t)$ acting in a separable Hilbert space $\mathcal{H}$, which may be nonperiodic, and let $U(t,0)$ the corresponding propagators. Denote by $A:{{\mathrm{dom}~}}A\subset\mathcal{H}\rightarrow\mathcal{H}$ a probe operator, such that ${{\mathrm{dom}~}}A$ is invariant under time evolution $U(t,0)$. Let $F(A>E)$ be the spectral projection onto the closed space spanned by the eigenvectors of $A$ corresponding to the eigenvalues larger than $E\in{\ensuremath{{\mathrm{I\!R}}}}$. The relevant definitions are as follows [@EV; @dOT; @dO; @BJLPN].

\[def.26\] (i) $\mathcal{H}_{pc}\doteq\{\xi\in\mathcal{H}:
\mathcal{O}(\xi)\;\textrm{is precompact in}\;\mathcal{H}\}$. (ii) $\mathcal{H}_{\mathrm{f}}\doteq\big\{\xi\in\mathcal{H}:
\lim_{\tau\rightarrow\infty}\frac{1}{\tau}
\int_0^{\tau}\|CU(t,0)\xi\|dt=0\;\textrm{for any compact}$$
\textrm{operator}\; C\big\}$. (iii) $\mathcal{H}_{\mathrm{be}}\doteq\{0\neq\xi\in\mathcal{H}:
\lim_{E\rightarrow\infty}\sup_{t\in{\ensuremath{{\mathrm{I\!R}}}}}
\|F(A>E)U(t,0)\frac{\xi}{\|\xi\|}\|=0\}\cup\{0\}$. (iv) $\mathcal{H}_{\mathrm{ue}}\doteq\{0\neq\xi\in\mathcal{H}:
\lim_{E\rightarrow\infty}\sup_{t\in{\ensuremath{{\mathrm{I\!R}}}}}
\|F(A>E)U(t,0)\frac{\xi}{\|\xi\|}\|=1\}\cup\{0\}$. (v) $\mathcal{S}^{\mathrm{bd}}(A)\doteq\{\xi\in{{\mathrm{dom}~}}A:\text{the
function}\;t\mapsto E_{\xi}^A(t)\;\textrm{is bounded}\}$. (vi) $\mathcal{S}^{\mathrm{un}}(A)\doteq\{\xi\in{{\mathrm{dom}~}}A:\text{the
function}\;t\mapsto E_{\xi}^A(t)\;\textrm{is unbounded}\}$.

Important compact operators are the projections onto finite subspaces of $\mathcal{H}$, so that the elements of $\mathcal{H}_{\mathrm{f}}$ are interpreted as the vectors that under time evolution leave, on average, any finite-dimensional subspace of $\mathcal{H}$.

Some basic properties of the sets that appeared in the above definition are summarized ahead. For proofs we refer the reader to [@dO; @dOT; @EV; @BJLPN].

\[GeneralTheo\] Let $H(t)$ be a time-dependent Hamiltonian and $A$ as above; then: (a) $\mathcal{H}_{\mathrm{f}}$ and $\mathcal{H}_{\mathrm{pc}}$ are closed subspaces of $\mathcal{H}$. (b) $\mathcal{H}_{\mathrm{pc}}\perp\mathcal{H}_{\mathrm{f}}$. (c) $\mathcal{H}_{\mathrm{be}}=\mathcal{H}_{\mathrm{pc}}$ and $\mathcal{H}_{\mathrm{f}}\subset\mathcal{H}_{\mathrm{ue}}$. (d) If $\xi\in{{\mathrm{dom}~}}A$ and $\xi\notin\mathcal{H}_{\mathrm{pc}}$ then $\xi\in\mathcal{S}^{\mathrm{un}}(A)$, that is, $\mathcal{S}^{\mathrm{bd}}(A)\subset\mathcal{H}_{\mathrm{pc}}$. In particular, $({{\mathrm{dom}~}}A\cap\mathcal{H}_{\mathrm{f}})\setminus\{0\}
\subset\mathcal{S}^{\mathrm{un}}(A)$.

Note that if the Hamiltonian $H(t)$ has the form $H(t)=H_0+V(t)$ with $H_0$ an unbounded, positive, self-adjoint operator with discrete spectrum, then Theorem \[GeneralTheo\](d) holds true for $A=H_0$.

Periodic Case {#PeriodicSection}
-------------

If $H(t)$ is periodic of period $T$ and $U_F=U(T,0)$ is the corresponding Floquet operator, we denote by $\mathcal{H}_{\mathrm{p}}$ the point spectral subspace and by $\mathcal{H}_{\mathrm{c}}$ the continuous subspace of the Floquet operator $U_{\mathrm{F}}$. Recall the important

\[teo.31R\] Let $C:\mathcal{H}\rightarrow\mathcal{H}$ be a compact operator and $\xi\in\mathcal{H}_{\mathrm{c}}$, then $$\lim_{\tau\rightarrow\infty}\frac{1}{\tau}\int_0^{\tau}\|CU(t,0)\xi\|dt=0.$$

A detailed proof of Theorem \[teo.31R\] can be found in [@EV]; this result was firstly proved for the autonomous case (see, e.g., [@AG]). As a consequence of this theorem it follows that if $\xi\in\mathcal{H}_{\mathrm{c}}$ then $\xi\in\mathcal{H}_{\mathrm{f}}$, so by Theorem \[GeneralTheo\](d) it follows that $\langle
U(t,0)\xi,AU(t,0)\xi\rangle$ is unbounded. Thus, as it is well known, the presence of continuous spectrum for the Floquet operator is a signature of quantum instability. In principle, one would expect that a Floquet operator with purely point spectrum would imply quantum stability, however there are examples with purely point spectrum and dynamically unstable; see [@dRJLS; @JSS; @dOP] for examples in the autonomous case and [@dOS] for the time-periodic case.

Using the above theorem and a series of technical lemmas in [@dOT], one gets

\[teo.31\] If the Hamiltonian operator is periodic in time, then (a) $\mathcal{H}_{\mathrm{p}}=\mathcal{H}_{\mathrm{be}}=
\mathcal{H}_{\mathrm{pc}}$; (b) $\mathcal{H}_{\mathrm{c}}=\mathcal{H}_{\mathrm{ue}}=\mathcal{H}_{\mathrm{f}}$.

We observe that Theorem \[teo.31\] also holds in the autonomous case $H(t)=H$ and with $\mathcal{H}_{\mathrm{p}}$ and $\mathcal{H}_{\mathrm{c}}$ denoting, respectively, the point and continuous subspace of the Hamiltonian $H$.

According to the above-quoted results, for periodic systems we have $$\label{eq.22}
\mathcal{H}=\mathcal{H}_{\mathrm{pc}}\oplus\mathcal{H}_{\mathrm{f}}.$$ In [@dO] was presented an example for which relation (\[eq.22\]) does not hold for nonperiodic time dependence. It was defined the “unusual" subspace $\mathcal{H}_{\mathrm{a}}$ by the relation $$\mathcal{H}=\mathcal{H}_{\mathrm{pc}}\oplus\mathcal{H}_{\mathrm{f}}
\oplus\mathcal{H}_{\mathrm{a}},$$ and constructed a nonperiodic Hamiltonian such that $\mathcal{H}=\mathcal{H}_{\mathrm{a}}$. The example is given by the Floquet operator generated by the kicked Hamiltonian $$H(t)=p^2+x\sum_{n=1}^{\infty}\epsilon_n\delta(t-n),\qquad
x\in[0,2\pi),$$ acting on $\mathcal{H}=L^2(\mathbb{T})$ and $\epsilon_n\in\{-1,0,1\}$ adequately chosen. This example illustrates some possible unusual properties of nonstationary quantum systems.

Quasiperiodic Case {#QuasiperiodicSection}
------------------

In this case we have the generalized Floquet operator $U_{\mathrm{F}}$ as defined in (\[FloquetGenEq\]), acting on the enlarged space $\mathcal{K}_1=L^2(S^1,\mathcal{H},\frac{d\theta_1}{2\pi})$, and the generalized quasienergy operator $\tilde{K}$ acting in $L^2(S^1\times
S^1,\mathcal{H},\frac{d\theta_1}{2\pi}\frac{d\theta_2}{2\pi})$. We denote, respectively, by $\mathcal{K}_{1,p}$ and $\mathcal{K}_{1,\mathrm{c}}$ the point and continuous subspace of the generalized Floquet operator $U_{\mathrm{F}}$.

For each fixed $t$ let the unitary operator $U(t):\mathcal{K}_1\rightarrow\mathcal{K}_1$ be given by $(U(t)\psi)(\theta_1)=U_{(\theta_1,0)}(t,0)\psi(\theta_1)$, that is, $$U(t)=\int_{S^1}^{\oplus}U_{(\theta_1,0)}(t,0)\frac{d\theta_1}{2\pi},$$ and given $\psi\in\mathcal{K}_1$ let $\tilde{\mathcal{O}}(\psi)=\{U(t)\psi:t\in{\ensuremath{{\mathrm{I\!R}}}}\}$ be the orbit of $\psi$ in the enlarged space $\mathcal{K}_1$.

Let $A:{{\mathrm{dom}~}}A\subset\mathcal{K}_1\rightarrow\mathcal{K}_1$ be a probe operator with $U(t){{\mathrm{dom}~}}A\subset{{\mathrm{dom}~}}A$ and $F(A>E)$ as before. The relevant definitions are as follows [@JL; @BJL; @dOT]:

\[def.33\] (a) $\mathcal{K}_{1,\mathrm{f}}\doteq\big\{\psi\in
\mathcal{K}_1:\lim_{\tau\rightarrow\infty}\frac{1}{\tau}\int_0^{\tau}
\|CU(t)\psi\|_{\mathcal{K}_1}dt=0\;\textrm{for any}$ $\textrm{compact operator}\; C\;\textrm{in}\;\mathcal{K}_1\big\}$. (b) $\mathcal{K}_{1,\mathrm{pc}}=\{\psi\in\mathcal{K}_1:\tilde{\mathcal{O}}(\psi)\;
\textrm{is precompact in}\;\mathcal{K}_1\}$. (c) $\mathcal{K}_{1,\mathrm{be}}\doteq\{0\neq\psi\in\mathcal{K}_1:
\lim_{E\rightarrow\infty}\sup_{t\in{\ensuremath{{\mathrm{I\!R}}}}}
\|F(A>E)U(t)\frac{\psi}{\|\psi\|}\|=0\}\cup\{0\}$. (d) $\mathcal{K}_{1,\mathrm{ue}}\doteq\{0\neq\psi\in\mathcal{K}_1:
\lim_{E\rightarrow\infty}\sup_{t\in{\ensuremath{{\mathrm{I\!R}}}}}
\|F(A>E)U(t)\frac{\psi}{\|\psi\|}\|=1\}\cup\{0\}$.

In [@JL] it was proved the analog of the RAGE Theorem for the quasiperiodic case. The proof is an adaptation of the similar statement in the periodic case discussed in [@EV]. As in the periodic case one has:

\[teo.35\] If the Hamiltonian operator is quasiperiodic in time, then (a) $\mathcal{K}_{1,p}=\mathcal{K}_{1,\mathrm{pc}}=\mathcal{K}_{1,\mathrm{be}}$; (b) $\mathcal{K}_{1,\mathrm{c}}=\mathcal{K}_{1,\mathrm{ue}}=\mathcal{K}_{1,\mathrm{f}}$.

It is worth mentioning that the relation between the energy growth and the characterizations in Definition \[def.33\] is not as direct as in the case of periodic and autonomous potentials. The above theorem holds on the enlarged space $\mathcal{K}_1$ so that a generalized operator with continuous spectrum does not ensure unbounded energy growth in the original Hilbert space $\mathcal{H}$, although it does in $\mathcal{K}_1$. See [@JL; @BJL] for interesting examples on systems with time-quasiperiodic dependence.

Almost Periodic Orbits {#PeneperiodicSection}
======================

Let $\mathcal{B}$ be a Banach space. A continuous function $f:{\ensuremath{{\mathrm{I\!R}}}}\rightarrow\mathcal{B}$ is called *almost periodic* if for any number $\epsilon>0$, one can find a number $l(\epsilon)>0$ such that any interval of the real line of length $l(\epsilon)$ contains at least one point $\tau$ with the property that $$\|f(t+\tau)-f(t)\|<\epsilon, \qquad\qquad \forall t\in{\ensuremath{{\mathrm{I\!R}}}}.$$ For properties of almost periodic functions we refer the reader to [@COR; @Kat]. Now we introduce the following subset of $\mathcal{H}$: $$\mathcal{H}_{\mathrm{ap}}\doteq\{\xi\in\mathcal{H}:
\textrm{the function}\;{\ensuremath{{\mathrm{I\!R}}}}\ni t\mapsto\xi(t)=U(t,0)\xi\;\textrm{is
almost periodic}\}.$$ By abuse of language sometimes we say that the orbit $\mathcal{O}(\xi)$ is almost periodic.

For general time dependence one has

\[prop.27a\] $\mathcal{H}_{\mathrm{ap}}$ is a closed subspace of $\mathcal{H}$ and $\mathcal{H}_{\mathrm{ap}}\subset\mathcal{H}_{\mathrm{pc}}$.

Clearly $0\in\mathcal{H}_{\mathrm{ap}}$. If $\xi,\psi\in\mathcal{H}_{\mathrm{ap}}$ then $\xi(t)=U(t,0)\xi$ and $\psi(t)=U(t,0)\psi$ are almost periodic functions. Since the sum of two almost periodic functions with values in $\mathcal{H}$ is an almost periodic function, it follows that $\xi(t)+\psi(t)=U(t,0)\xi+U(t,0)\psi=U(t,0)(\xi+\psi)=(\xi+\psi)(t)$ is an almost periodic function. So $\xi+\psi\in\mathcal{H}_{\mathrm{ap}}$. Now, let $\xi\in\mathcal{H}_{\mathrm{ap}}$ and $\lambda$ a complex number, then $\xi(t)=U(t,0)\xi$ is an almost periodic function. Since $\lambda\xi(t)=\lambda U(t,0)\xi=U(t,0)(\lambda\xi)$ is an almost periodic function, it follows that $\lambda\xi\in\mathcal{H}_{\mathrm{ap}}$. So $\mathcal{H}_{\mathrm{ap}}$ is a vector subspace of $\mathcal{H}$.

Suppose that $\{\xi_j\}\subset\mathcal{H}_{\mathrm{ap}}$ and $\lim_{j\rightarrow\infty}\xi_j=\xi$. Given $\epsilon>0$ there exists $N\in{\ensuremath{{\mathrm{I\!N}}}}$ such that $\|\xi_j-\xi\|<\epsilon$ for all $j\geq
N$; thus, there exists $N$ as above such that $j\geq N$ implies that $\forall\;t\in{\ensuremath{{\mathrm{I\!R}}}}$ $$\|\xi(t)-\xi_j(t)\|=\|U(t,0)\xi-U(t,0)\xi_j\|\leq\|\xi-\xi_j\|<\epsilon.$$ So $\xi_j(t)\rightarrow\xi(t)$ uniformly in ${\ensuremath{{\mathrm{I\!R}}}}$ in the sense of convergence in the norm. Since each $\xi_j(t)$ is an almost periodic function, it follows that $\xi(t)$ is an almost periodic function (Theorem 6.4 in [@COR]) and $\xi\in\mathcal{H}_{\mathrm{ap}}$, which shows that $\mathcal{H}_{\mathrm{ap}}$ is a closed vector subspace of $\mathcal{H}$.

Since the set of values of an almost periodic function with values in $\mathcal{H}$ is precompact in $\mathcal{H}$ (Theorem 6.5 in [@COR]), it follows that $\mathcal{H}_{\mathrm{ap}}\subset\mathcal{H}_{\mathrm{pc}}$.

Periodic Systems
----------------

If the Hamiltonian time dependence is periodic (or autonomous) more can be said.

\[EigenAlmostProp\] If the Hamiltonian operator is periodic in time and $\xi\in\mathcal{H}_{\mathrm{p}}$ is an eigenvector of $U_{\mathrm{F}}$, that is, $U_{\mathrm{F}}\xi=e^{-i\alpha}\xi$, $\alpha\in{\ensuremath{{\mathrm{I\!R}}}}$, then $\xi\in\mathcal{H}_{\mathrm{ap}}\subset\mathcal{H}_{\mathrm{pc}}$.

Since $U(t,0)$ is strongly continuous the map $t\mapsto\xi(t)$ is continuous.

Any $t\in{\ensuremath{{\mathrm{I\!R}}}}$ can be written in the form $t=nT+s$, with $n\in{\ensuremath{{\mathsf{Z\!\!Z}}}}$ and $0\leq s<T$. We have $U_{\mathrm{F}}\xi=e^{-i\alpha}\xi$ and $U_{\mathrm{F}}^{-1}\xi=e^{i\alpha}\xi$. Since for $t\geq0$ ($n\geq0$) $$\begin{aligned}
U(t,0)\xi &=& U(s+nT,nT)U(nT,(n-1)T)\ldots U(T,0)\xi\\ &=&
U(s,0)\underbrace{U(T,0)\ldots
U(T,0)}_{n\;\textrm{factors}}\xi=U(s,0)e^{-in\alpha}\xi,\end{aligned}$$ and for $t<0$ ($n<0$) $$\begin{aligned}
U(t,0)\xi &=& U(s+nT,nT)U(nT,(n+1)T)\ldots U(-T,0)\xi\\ &=&
U(s,0)\underbrace{U(T,0)^{-1}\ldots
U(T,0)^{-1}}_{n\;\textrm{factors}}\xi=U(s,0)e^{-in\alpha}\xi,\end{aligned}$$ it follows that $$U(t,0)\xi=U(s,0)e^{-in\alpha}\xi,$$ for $t=nT+s\in{\ensuremath{{\mathrm{I\!R}}}}$, $n\in{\ensuremath{{\mathsf{Z\!\!Z}}}}$ and $0\leq s<T$. So for each $t=nT+s\in{\ensuremath{{\mathrm{I\!R}}}}$ $$\begin{aligned}
\xi(t+T)&=&U(t+T,0)\xi=U(s,0)e^{-i(n+1)\alpha}\xi\\&=&
e^{-i\alpha}U(s,0)e^{-in\alpha}\xi=e^{-i\alpha}\xi
U(t,0)\xi=e^{-i\alpha}\xi(t),\end{aligned}$$ so $t\rightarrow\xi(t)$ is an almost periodic function and the result is proved.

Summing up, we conclude:

\[teo.31a\] If the Hamiltonian operator is periodic in time, then (a) $\mathcal{H}_{\mathrm{p}}=\mathcal{H}_{\mathrm{be}}=
\mathcal{H}_{\mathrm{pc}}=\mathcal{H}_{\mathrm{ap}}$; (b) $\mathcal{H}_{\mathrm{c}}=\mathcal{H}_{\mathrm{ue}}=
\mathcal{H}_{\mathrm{f}}$.

It is enough to prove that $\mathcal{H}_{\mathrm{pc}}=\mathcal{H}_{\mathrm{ap}}$. The inclusion $\mathcal{H}_{\mathrm{ap}}\subset\mathcal{H}_{\mathrm{pc}}$ was proved in Proposition \[prop.27a\]. On the other hand, it is a consequence of Propositions  \[prop.27a\] and \[EigenAlmostProp\] that $\mathcal{H}_{\mathrm{p}}\subset\mathcal{H}_{\mathrm{ap}}$. Since $\mathcal{H}_{\mathrm{p}}=\mathcal{H}_{\mathrm{pc}}$, it follows that $\mathcal{H}_{\mathrm{pc}}\subset\mathcal{H}_{\mathrm{ap}}$.

Theorem \[teo.31a\] holds also for autonomous Hamiltonians.

Quasiperiodic Systems
---------------------

In the above theorem we proved that for time-periodic Hamiltonians an orbit $\mathcal{O}(\xi)$ is precompact if, and only if, $t\mapsto\xi(t)$ is almost periodic. Now we construct an example showing that already in the case of time-quasiperiodic Hamiltonians there are precompact orbits that are not almost periodic.

[**Example**]{} Given the matrix $$u_1(\theta_1)=
\left(\begin{array}{cc} e^{i\theta_1} & 0\\ 0 &
e^{-i\theta_1}\end{array}\right),$$ it is known (see Lemma 5.1 in [@BJL]) that there exists a quasiperiodic Hamiltonian $H_{\theta}(t)$, $\theta=(\theta_1,\theta_2)$, acting on $\mathcal{H}={\ensuremath{{\mathrm{C\hspace{-1.7mm}\rule{0.3mm}{2.6mm}\;}}}}^2$, of the form $$\label{eq.25}
H_{\theta}(t)=h_0(t){{\mathrm{I_d}}}+\sum_{j=1}^3h_j(t)\sigma_j,$$ where $\sigma_j$ are the Pauli matrices, and $h_j(t)$ are real quasiperiodic functions, i.e., $h_j(t)=
\bar{h}_j(\omega_1t+\theta_1,\omega_2t+\theta_2)$, where $\bar{h}_j(\theta_1,\theta_2)$ are continuous and $2\pi$-periodic in the two arguments $\theta_1,\;\theta_2\in
S^1,$ and $\omega_1,\;\omega_2$ are positive real numbers so that $u_1(\theta_1)=U_{(\theta_1,0)}(T_2,0)$ is the corresponding monodromy operator. Moreover, the corresponding generalized Floquet operator $U_F=\mathcal{T}_{-T_2}u_1$ has absolutely continuous spectrum for any irrational $\alpha\doteq\frac{\omega_1}{\omega_2}$.

By the construction in the proof of Lemma 5.1 in [@BJL], it is found that for $k\in{\ensuremath{{\mathsf{Z\!\!Z}}}}$, $k>0$, $$\begin{aligned}
U_{(\theta_1,0)}(kT_2,0) &=& u_1(\theta_1+(k-1)2\pi\alpha)\ldots
u_1(\theta_1+2\pi\alpha)u_1(\theta_1)\\ &=&
\left(\begin{array}{cc} e^{i(\theta_1+(k-1)2\pi\alpha)}& 0 \\ 0 &
e^{-i(\theta_1+(k-1)2\pi\alpha)}\\
\end{array}\right)\ldots\left(\begin{array}{cc}
e^{i\theta_1}& 0 \\ 0 & e^{-i\theta_1}\\
\end{array}\right)\\ &=& \left(\begin{array}{cc}
e^{i(\theta_1+(k-1)2\pi\alpha)}\ldots e^{i\theta_1} & 0
\\ 0 &
e^{-i(\theta_1+(k-1)2\pi\alpha)}\ldots e^{-i\theta_1}\\
\end{array}\right)\\ &=& \left(\begin{array}{cc}
e^{i(k\theta_1+(1+2+\ldots(k-1))2\pi\alpha)} & 0
\\ 0 & e^{-i(k\theta_1+(1+2+\ldots(k-1))2\pi\alpha)} \\
\end{array}\right)\\ &=& \left(\begin{array}{cc}
e^{ik(\theta_1+(k-1)\pi\alpha)} & 0
\\ 0 & e^{-ik(\theta_1+(k-1)\pi\alpha)} \\
\end{array}\right);\end{aligned}$$ for $k<0$ the same expression is found. Therefore, for all $k\in{\ensuremath{{\mathsf{Z\!\!Z}}}}$ $$U_{(\theta_1,0)}(kT_2,0)=\left(\begin{array}{cc}
e^{ik(\theta_1+(k-1)\pi\alpha)} & 0
\\ 0 & e^{-ik(\theta_1+(k-1)\pi\alpha)} \\
\end{array}\right).$$ Moreover, for $\theta_1\in S^1$, $0\leq t\leq T_2$, define $$v(t;\theta_1)=\left(\begin{array}{cc}
e^{i\frac{t}{T_2}(\theta_1+(\frac{t}{T_2}-1)\pi\alpha)} & 0
\\ 0 & e^{-i\frac{t}{T_2}(\theta_1+(\frac{t}{T_2}-1)\pi\alpha)} \\
\end{array}\right),$$ which is differentiable with respect to $t$ and satisfies $$v(0;\theta_1)=\left(\begin{array}{cc}
1 & 0 \\ 0 & 1 \\ \end{array}\right)={{\mathrm{I_d}}},\qquad
v(T_2;\theta_1)=\left(\begin{array}{cc} e^{i\theta_1} & 0 \\ 0 & e^{-i\theta_1} \\
\end{array}\right)=u_1(\theta_1).$$ So for $t\in{\ensuremath{{\mathrm{I\!R}}}}$, $t=kT_2+\delta_t$, $0\leq\delta_t\leq T_2,$ one has $$U_{(\theta_1,0)}(t,0)=v(\delta_t;\theta_1+k2\pi\alpha)U_{(\theta_1,0)}(kT_2,0).$$

Therefore, for $\xi\in\mathcal{H}={\ensuremath{{\mathrm{C\hspace{-1.7mm}\rule{0.3mm}{2.6mm}\;}}}}^2$, $\xi=\left(\begin{array}{cc}
\xi_1 \\ \xi_2 \\
\end{array}\right),$ we have $$U_{(\theta_1,0)}(t,0)\xi=\left(\begin{array}{cc}
e^{i\frac{t}{T_2}(\theta_1+(\frac{t}{T_2}-1)\pi\alpha)}\xi_1 \\
e^{-i\frac{t}{T_2}(\theta_1+(\frac{t}{T_2}-1)\pi\alpha)}\xi_2 \\
\end{array}\right).$$ Since the map, for $0\neq
a\in{\ensuremath{{\mathrm{I\!R}}}}$, $t\mapsto\sin at^2$ is not almost periodic, because it is not uniformly continuous, we conclude that the map $t\mapsto e^{iat^2}$ is not almost periodic. Thus, $$t\mapsto
g(t)=e^{i\frac{t}{T_2}(\theta_1+(\frac{t}{T_2}-1)\pi\alpha)}=e^{i\frac{t}{T_2}\theta_1}
e^{it^2\frac{\omega_1\omega_2}{4\pi}}e^{-it\frac{\omega_1}{2}}$$ is not almost periodic, because on the contrary the map $$e^{-i\frac{t}{T_2}\theta_1}g(t)e^{it\frac{\omega_1}{2}}=
e^{it^2\frac{\omega_1\omega_2}{4\pi}}$$ would be almost periodic.

Therefore, if $\xi\neq0$ then the map $t\mapsto
U_{(\theta_1,0)}(t,0)\xi$ is not almost periodic for all $\theta_1\in S^1$. Hence we have got an example of precompact orbit (a closed and bounded set on ${\ensuremath{{\mathrm{C\hspace{-1.7mm}\rule{0.3mm}{2.6mm}\;}}}}^2$ is compact) which is not almost periodic. This finishes the example.

The above example can be extend to the infinite dimensional Hilbert space $\mathcal{H}=\bigoplus_{n\in{\ensuremath{{\mathrm{I\!N}}}}}{\ensuremath{{\mathrm{C\hspace{-1.7mm}\rule{0.3mm}{2.6mm}\;}}}}^2$ of the elements $\xi=(\xi_n)_{n\in{\ensuremath{{\mathrm{I\!N}}}}}$ with $\xi_n\in{\ensuremath{{\mathrm{C\hspace{-1.7mm}\rule{0.3mm}{2.6mm}\;}}}}^2$ and $\sum_n|\xi_n|^2<\infty$. Denote $$\tilde{u_1}(\theta_1)=\left(\begin{array}{cc} e^{i\theta_1} & 0\\
0 & e^{-i\theta_1}\\
\end{array}\right);$$ we know that there exists a quasiperiodic $\tilde{H}_{\theta}(t)$ such that $\tilde{u_1}(\theta_1)$ is the corresponding monodromy operator. Moreover, $\sigma(\tilde{U_{\mathrm{F}}})$ is absolutely continuous for all irrational $\alpha$. Let $$u_1(\theta_1)=\left(\begin{array}{cccc}
\left(\begin{array}{cc} e^{i\theta_1} & 0\\
0 & e^{-i\theta_1}\\
\end{array}\right) &  &  &  \\
  & \left(\begin{array}{cc} e^{i\theta_1} & 0\\
0 & e^{-i\theta_1}\\
\end{array}\right) & &  \\
 & & \left(\begin{array}{cc} e^{i\theta_1} & 0\\
0 & e^{-i\theta_1}\\
\end{array}\right) & \\
 & & & \ddots \\
\end{array}\right)$$ or, writing in the another way, $u_1(\theta_1)=\bigoplus\tilde{u_1}(\theta_1)$. For $\xi\in\mathcal{H}$ one has $u_1(\theta_1)\xi=\bigoplus\tilde{u_1}(\theta_1)\xi_n$. The Floquet operator corresponding to $u_1(\theta_1)$, $U_{\mathrm{F}}=\mathcal{T}_{-T_2}u_1:L^2(S^1,\mathcal{H},
\frac{d\theta_1}{2\pi})\rightarrow
L^2(S^1,\mathcal{H},\frac{d\theta_1}{2\pi})$ has absolutely continuous spectrum for all irrational $\alpha$.

If $H_{\theta}(t)=\bigoplus_{n\in{\ensuremath{{\mathrm{I\!N}}}}}\tilde{H}_{\theta}(t)$ then the propagator of $H_{\theta}(t)$ is $U_{\theta}(t,0)=\break\bigoplus\tilde{U}_{\theta}(t,0)$. Thus, $H_{\theta}(t)$ has $u_1(\theta_1)$ as the corresponding monodromy operator, and given $0\neq\xi\in\mathcal{H}$ and $\theta=(\theta_1,0)\in S^1\times S^1$ one has $$\begin{aligned}
U_{(\theta_1,0)}(t,0)\xi &=& \bigoplus_n\left(\begin{array}{cc}
e^{i\frac{t}{T_2}(\theta_1+(\frac{t}{T_2}-1)\pi\alpha)} & 0\\
0 & e^{-i\frac{t}{T_2}(\theta_1+(\frac{t}{T_2}-1)\pi\alpha)}\\
\end{array}\right) \xi_n \\ &=&
\bigoplus_n\left(\begin{array}{c}
e^{i\frac{t}{T_2}(\theta_1+(\frac{t}{T_2}-1)\pi\alpha)}\xi_n^1\\
e^{-i\frac{t}{T_2}(\theta_1+(\frac{t}{T_2}-1)\pi\alpha)}\xi_n^2\\
\end{array}\right).\end{aligned}$$ So $t\mapsto U_{(\theta_1,0)}(t,0)\xi$ is not almost periodic. If $\xi$ satisfies $\xi=\oplus\xi_n$ with $\xi_n\neq0$ if, and only if, $n=l$, then $$U_{(\theta_1,0)}(t,0)\xi=\left(\begin{array}{c}
e^{i\frac{t}{T_2}(\theta_1+(\frac{t}{T_2}-1)\pi\alpha)}\xi_l^1\\
e^{-i\frac{t}{T_2}(\theta_1+(\frac{t}{T_2}-1)\pi\alpha)}\xi_l^2\\
\end{array}\right),$$ and the orbit is precompact since it lives in a finite dimension subspace. In the same way, if $\xi$ is of the form $\xi=\oplus\xi_n$ with $\xi_n\neq0$ only for finitely many indices $n$, we have an example of a theoretical quantum model with precompact orbits which are not almost periodic.

Quasienergy Operator and Almost Periodic Orbits
-----------------------------------------------

Let $H(t)$ be a general time-dependent Hamiltonian in a Hilbert space $\mathcal{H}$ such that the propagator $U(t,s)$ is well defined. In this case we have defined the quasienergy operator $K=-i\frac{d}{dt}+H(t)$ acting in the extended Hilbert space $\mathcal{K}=L^2({\ensuremath{{\mathrm{I\!R}}}},\mathcal{H},dt)$. It is known [@H; @H1] that the quasienergy operator and the propagator are connected by the relation $$\label{RelEq}
(e^{-iK\sigma}f)(t)=U(t,t-\sigma)f(t-\sigma).$$ Let $\mathcal{K}_{\mathrm{p}}(K)$ and $\mathcal{K}_{\mathrm{c}}(K)$ denote, respectively, the point and continuous subspaces of $K$. We get the following result:

\[prop.31\] Let $\xi\in\mathcal{H}$ be such that $1\otimes\xi\in\mathcal{K}_{\mathrm{p}}(K)$. Then:

-   The map $t\mapsto U(t,0)^{-1}\xi$ is almost periodic.

-   If the eigenvectors of $K$ have the form $\psi_m=1\otimes\xi_m$, with $\xi_m\in\mathcal{H}$, then $\xi\in\mathcal{H}_{\mathrm{ap}}$.

If $1\otimes\xi\in\mathcal{K}_{\mathrm{p}}(K)$ then $1\otimes\xi=\sum_mc_m\psi_m,$ with $K\psi_m=\lambda_m\psi_m$. So $$e^{iK\sigma}(1\otimes\xi)=\sum_mc_me^{i\lambda_m\sigma}\psi_m,$$ therefore by (\[RelEq\]) for each $t\in{\ensuremath{{\mathrm{I\!R}}}},$ $$U(t,t+\sigma)\xi=(e^{iK\sigma}(1\otimes\xi))(t)=
\sum_mc_me^{-i\lambda_m\sigma}\psi_m(t)$$ and we conclude that, for each fixed $t$, the map $\sigma\mapsto
U(t,t+\sigma)\xi$ is almost periodic. In particular taking $t=0$ we obtain that $\sigma\mapsto U(0,\sigma)\xi$ is almost periodic and i) is proved.

Now, if the eigenvectors of $K$ have the form $\psi_m=1\otimes\xi_m$, then $$\begin{aligned}
\xi(t)&=&U(t,0)\xi=(e^{-iKt}(1\otimes\xi))(t)\\&=&
\sum_mc_me^{-i\lambda_mt}\psi_m(t)\\&=&
\sum_mc_me^{-i\lambda_mt}\xi_m.\end{aligned}$$ If the sum is finite the map $t\mapsto\xi(t)$ is almost periodic since it is a trigonometric polynomial. If the sum is infinite then $\sum_{m=1}^kc_me^{-i\lambda_mt}\xi_m\rightarrow\sum_{m=1}^{\infty}c_me^{-i\lambda_mt}\xi_m$ uniformly as $k\rightarrow\infty$ and so the map $t\mapsto\xi(t)$ is almost periodic, that is, $\xi\in\mathcal{H}_{\mathrm{ap}}$, which is ii).

Bounded Energy {#BoundedSection}
==============

In this section we consider time-dependent Hamiltonians $H(t)=H_0+V(t)$ for which $H_0$ is a probe operator.

If $\psi_0\in{{\mathrm{dom}~}}H_0$ and $\psi(t)=U(t,0)\psi_0$ is the solution of the Schrödinger equation, under which conditions $$E_{\psi_0}^0(t)=\langle\psi(t),H_0\psi(t)\rangle$$ is a bounded function on $t$? Also, when $$E_{\psi_0}(t)=\langle\psi(t),H(t)\psi(t)\rangle$$ is a bounded function? Next we present a set of simple general conditions related to the boundedness of such energy functions.

General Systems
---------------

\[prop.38\] If $V(t)$ is an uniformly bounded family of operators, that is, $\sup_t\|V(t)\|<\infty$, then $E_{\psi_0}^0(t)$ is bounded if, and only if, $E_{\psi_0}(t)$ is bounded.

It is sufficient to note that $$E_{\psi_0}(t) =
\langle\psi(t),H(t)\psi(t)\rangle =
E_{\psi_0}^0(t)+\langle\psi(t),V(t)\psi(t)\rangle$$ and $$\sup_t|\langle\psi(t),V(t)\psi(t)\rangle|\leq\sup_t\|\psi(t)\|^2\|V(t)\|=\sup_t
\|\psi_0\|^2\|V(t)\|<\infty.$$

\[prop.39\] If $\psi(t)\in
C^1({\ensuremath{{\mathrm{I\!R}}}};\mathcal{H})$ is almost periodic and $\psi'(t)$ is uniformly continuous, then $E_{\psi_0}(t)$ is bounded.

For each $n\in{\ensuremath{{\mathrm{I\!N}}}}^*$ define $$f_n(t)=n\left[\psi\left(t+\frac{1}{n}\right)-\psi(t)\right]=
n\int_{t}^{t+\frac{1}{n}}\psi'(s)ds.$$ Since $\psi$ is almost periodic it follows that $f_n$ is almost periodic for each $n$. As $\psi'(t)$ is uniformly continuous, for each $\epsilon>0$ there exists $\delta>0$ such that $|s-t|<\delta$ implies $\|\psi'(t)-\psi'(s)\|<\epsilon$. Given $\epsilon>0$ let $N(\epsilon)$ the smallest integer larger or equal to $\frac{1}{\delta}$; then for all $n>N(\epsilon)$ and $t\in{\ensuremath{{\mathrm{I\!R}}}}$ $$\begin{aligned}
\left\|f_n(t)-\psi'(t)\right\| &=&
\left\|n\int_{t}^{t+\frac{1}{n}}(\psi'(s)-\psi'(t))ds\right\|\\ &\leq&
n\int_{t}^{t+\frac{1}{n}}\left\|\psi'(s)-\psi'(t)\right\|ds<\epsilon.\end{aligned}$$ So $f_n\rightarrow\psi'$ uniformly and therefore $\psi'(t)$ is almost periodic. Hence $i\psi'(t)$ and $\psi(t)$ are bounded maps. Since $$E_{\psi_0}(t)=\langle\psi(t),H(t)\psi(t)\rangle=
\langle\psi(t),i\frac{d\psi}{dt}(t)\rangle$$ the result follows.

Note that the boundedness of energy follows if $t\mapsto\psi(t)$ and $t\mapsto\psi'(t)$ are bounded maps. Though well known, it is worth mentioning Proposition \[prop.40\] in this set of conditions.

\[prop.40\] If $t\mapsto V(t)$ is strongly $C^1$ and $\psi'(t)\in{{\mathrm{dom}~}}H(t)$ for all $t$, then:

-   The map $t\mapsto E_{\psi}(t)$ is differentiable and $$\frac{d}{dt}E_{\psi}(t)=\langle\psi(t),V'(t)\psi(t)\rangle.$$

-   $\left|E_{\psi}(t)-E_{\psi}(0)\right|\leq t\times\sup_s\|V'(s)\|$.

-   If there are $C>0, a>1$ so that $\|V'(t)\|\leq\frac{C}{(1+|t|)^a}$, then $E_{\psi}(t)$ and $E_{\psi}^0(t)$ are bounded functions.

\(a) $E_{\psi}(t)=\langle\psi(t),(H_0+V(t))\psi(t)\rangle$ and so $$\begin{aligned}
\frac{d}{dt}E_{\psi}(t) &=& \langle\psi'(t),H(t)\psi(t)\rangle +
\langle\psi(t),H(t)\psi'(t)\rangle +
\langle\psi(t),V'(t)\psi(t)\rangle\\ &=&
\langle\psi'(t),i\psi'(t)\rangle + \langle i
\psi'(t),\psi'(t)\rangle + \langle\psi(t),V'(t)\psi(t)\rangle \\
&=& \langle\psi(t),V'(t)\psi(t)\rangle.\end{aligned}$$ (b) Since $$E_{\psi}(t)-E_{\psi}(0)=\int_0^t\frac{d}{ds}E_{\psi}(s)ds=
\int_0^t\langle\psi(s),V'(s)\psi(s)\rangle ds$$ the result follows. (c) Similar to (b).

A possibility for the proposition above is $V(t)=B_1\sin
t+\frac{B_2}{(1+|t|)^2}$ with $B_1,\;B_2\in B(\mathcal{H})$ and self-adjoint. From this we see that certainly the choices of $\psi$ depend on $B_1,B_2$, since $B_1\psi$ and $B_2\psi$ must be kept in suitable domains so that $E_{\psi}(t)$ is meaningful.

Purely Point Systems
--------------------

The next result is restricted to periodic time dependence and Floquet operators with nonempty point spectrum (see [@DSSV]).

\[prop.41\] Let $V$ be periodic with period $T$. If the subset $\{\xi_1,\ldots,\xi_n\}$ of eigenvectors of $U_{\mathrm{F}}$ is in ${{\mathrm{dom}~}}H_0$ and $t\mapsto\xi_j(t)$ are $C^1$ maps, then for $\psi=\sum_{j=1}^na_j\xi_j$, where $a_j\in{\ensuremath{{\mathrm{C\hspace{-1.7mm}\rule{0.3mm}{2.6mm}\;}}}},\;j=1,\cdots,n$, the map $E_{\psi}(t)$ is bounded. If, moreover, $V(t)$ are bounded operators and $\sup_t\|V(t)\|<\infty$, then $E_{\psi}^0(t)$ is also bounded.

Suppose $U_{\mathrm{F}}\xi_j=e^{i\lambda_j}\xi_j$ with $\lambda_j\in{\ensuremath{{\mathrm{I\!R}}}}$, $1\leq j\leq n$. We have $$E_{\xi_j,\xi_k}(t)\doteq\langle\xi_j(t),H(t)\xi_k(t)\rangle=
\langle\xi_j(t),i\frac{d}{dt}\xi_k(t)\rangle$$ and so $t\mapsto E_{\xi_j,\xi_k}(t)$ is continuous. Now $$\begin{aligned}
E_{\xi_j,\xi_k}(t+T) &=& \langle
U(t+T,0)\xi_j,H(t+T)U(t+T,0)\xi_k\rangle\\ &=& \langle
U(t+T,T)U_{\mathrm{F}}\xi_j,H(t)U(t+T,T)U_{\mathrm{F}}\xi_k\rangle\\ &=& e^{-i\lambda_j}
e^{i\lambda_k}\langle U(t,0)\xi_j,H(t)U(t,0)\xi_k\rangle\\ &=&
e^{i(\lambda_k-\lambda_j)}E_{\xi_j,\xi_k}(t)\end{aligned}$$ and then $t\mapsto E_{\xi_j,\xi_k}(t)$ is an almost periodic function. Since for $\psi=\sum_{j=1}^na_j\xi_j$ we have $E_{\psi}(t)=\sum_{j,k=1}^n\bar{a_j}a_kE_{\xi_j,\xi_k}(t)$ it follows that $E_{\psi}(t)$ is almost periodic and so bounded. The second statement follows by Proposition \[prop.38\].

According to Proposition \[prop.41\], in order to get dynamical stability in the periodic case we need conditions assuring the eigenvectors of $U_{\mathrm{F}}$ are in ${{\mathrm{dom}~}}H_0$ and $t\mapsto\xi_j(t)$ to be $C^1$ functions. We present some sufficient conditions in terms of the quasienergy operator $K$.

\[lema.42\] Let $\xi\in\mathcal{H}$ be such that $H(t)U(t,s)\xi$ is well defined. Then the map $t\mapsto
H(t)U(t,s)\xi$ is a $C^r$ function if, and only if, $t\mapsto
e^{i\lambda(t-s)}U(t,s)\xi$ is a $C^{r+1}$ function for fixed $\lambda,\;s\in{\ensuremath{{\mathrm{I\!R}}}}$.

Note that $$\frac{d}{dt}(e^{i\lambda(t-s)}U(t,s)\xi)=i\lambda
e^{i\lambda(t-s)}U(t,s)\xi- ie^{i\lambda(t-s)}H(t)U(t,s)\xi.$$ Thus, if $t\mapsto H(t)U(t,s)\xi$ is $C^r$ then $t\mapsto
e^{i\lambda(t-s)}U(t,s)\xi$ is $C^{r+1}$ and reciprocally if $t\mapsto e^{i\lambda(t-s)}U(t,s)\xi$ is $C^{r+1}$ then $t\mapsto
H(t)U(t,s)\xi$ is $C^r$.

\[cor.43\] If $f^{(\lambda)}$ is an eigenvector of $K$, $Kf^{(\lambda)}=\lambda f^{(\lambda)}$, then the map $t\mapsto f^{(\lambda)}(t)$ is $C^r$ if, and only if, there exists $s\in{\ensuremath{{\mathrm{I\!R}}}}$ so that $t\mapsto
H(t)U(t,s)f^{(\lambda)}(s)$ is $C^{r-1}$.

If $Kf^{(\lambda)}=\lambda f^{(\lambda)}$, then by relation (\[RelEq\]), $$e^{-i\lambda\sigma}f^{(\lambda)}(t)=U(t,t-\sigma)f^{(\lambda)}(t-\sigma);$$ so $f^{(\lambda)}(t)=e^{i\lambda\sigma}U(t,t-\sigma)f^{(\lambda)}(t-\sigma)$ for all $\sigma\in{\ensuremath{{\mathrm{I\!R}}}}$. Denoting $t-\sigma=s$ it follows that $f^{(\lambda)}(t)=e^{i\lambda(t-s)}U(t,s)f^{(\lambda)}(s)$ and the result follows by Lemma \[lema.42\].

By using relation (\[RelEq\]) one can easily show

\[Lemma12\] For periodic systems with period $T$, one has:

-   If $Kf=\lambda f$ then $U_{\mathrm{F}}(s)f(s)=e^{-i\lambda T}f(s)$, $\forall\;s\in{\ensuremath{{\mathrm{I\!R}}}}$.

-   If $U_{\mathrm{F}}(s)\xi_s=e^{-i\lambda T}\xi_s$, $\xi_s\in\mathcal{H}$, $\forall\;s$, then $$f_{\xi}(t)\doteq \break e^{i\lambda(t-s)}U(t,s)\xi_s\in{{\mathrm{dom}~}}K$$ and $Kf_{\xi}=\lambda f_{\xi}$.

\[cor.44\] (a) If $H(t+T)=H(t)$, and $\xi^{(\lambda)}$ is an eigenvector of $U_{\mathrm{F}}(s)$, $U_{\mathrm{F}}(s)\xi^{(\lambda)}=e^{-i\lambda T}\xi^{(\lambda)}$, then $\xi^{(\lambda)}\in{{\mathrm{dom}~}}H(s)$ if, and only if, there exists an eigenvector $f_{\xi^{(\lambda)}}$ of $K$, $Kf_{\xi^{(\lambda)}}=\lambda f_{\xi^{(\lambda)}}$, with $t\mapsto
f_{\xi^{(\lambda)}}(t)$ continuous and differentiable. (b) In particular, $U_{\mathrm{F}}(s)$ has a basis of eigenvectors in ${{\mathrm{dom}~}}H(s)$ if, and only if, $K$ has a basis of eigenvectors $\{f_j\}$ such that $t\mapsto f_j(t)$ is continuous and differentiable for each $j$.

\(a) Suppose that $\xi^{(\lambda)}\in{{\mathrm{dom}~}}H(s)$. By Lemma \[Lemma12\] $f_{\xi^{\lambda}}(t)=\break
e^{i\lambda(t-s)}U(t,s)\xi^{\lambda}\in{{\mathrm{dom}~}}K$ and $Kf_{\xi^{\lambda}}=\lambda f_{\xi^{\lambda}}$. Since $\xi^{(\lambda)}\in{{\mathrm{dom}~}}H(s)$ it follows that $U(t,s)\xi^{\lambda}\in{{\mathrm{dom}~}}H(t)$ and $i\partial_tU(t,s)\xi^{\lambda}=H(t)U(t,s)\xi^{\lambda}$. Thus, $t\mapsto f_{\xi^{(\lambda)}}(t)$ is continuous and differentiable.

Reciprocally, it there exists an eigenvector $f_{\xi^{(\lambda)}}$ of $K$ with $t\mapsto f_{\xi^{(\lambda)}}(t)$ continuous and differentiable, then $f_{\xi^{\lambda}}(t)=e^{i\lambda(t-s)}U(t,s)\xi^{\lambda}$ and $Kf_{\xi^{(\lambda)}}=\lambda f_{\xi^{(\lambda)}}$ implies $-i\partial_tf_{\xi^{\lambda}}(t)+H(t)f_{\xi^{\lambda}}(t)=\lambda
f_{\xi^{\lambda}}(t)$; therefore, $\xi^{\lambda}\in{{\mathrm{dom}~}}H(s)$.

\(b) It is a directly consequence of (a).

Jauslin-Lebowitz Formulation
----------------------------

We want to study an analogue of the expectation value of probe operators $A:{{\mathrm{dom}~}}A\subset\mathcal{H}\rightarrow\mathcal{H}$ on the formulation presented by Jauslin and Lebowitz [@JL; @BJL] briefly recalled in the Introduction. If the generalized quasienergy operator $\tilde{K}$ has pure point spectrum, there exists an orthonormal basis $B\doteq\{f_n\}_{n=1}^{\infty}$ of $\tilde{\mathcal{K}}$ with $\tilde{K}f_n=\lambda_nf_n$. By Theorem 4.2 in [@JL], if $f=1\otimes\xi$ is in the point subspace of $\tilde{K}$ the function $t\mapsto U_{\theta}(t,0)\xi$ is almost periodic a.e. $\theta$ with respect to the ergodic measure $\mu$ on the compact manifold $\Omega$ (see Section \[IntroductionSection\]).

Denote $$B_{n,m}(A)\doteq\int_{\Omega}\left\langle
f_n(\theta),Af_m(\theta)\right\rangle_{\mathcal{H}}d\mu(\theta)=\left\langle
f_n,(1\otimes A)f_m\right\rangle_{\tilde{\mathcal{K}}}.$$ If $f\in\tilde{\mathcal{K}}$ then $f=\sum_na_nf_n$, with $\sum_n|a_n|^2=\|f\|^2_{\tilde{\mathcal{K}}}$. For each time $t$, consider the average over $\Omega$ of the expectation value of $A$, that is, $$\begin{aligned}
A_f(t) &\doteq& \int_{\Omega}\left\langle
U_{\theta}(t,0)f(\theta),AU_{\theta}(t,0)f(\theta)\right\rangle_{\mathcal{H}}d\mu(\theta)\\
&=& \int_{\Omega}\langle
(\mathcal{F}_te^{-i\tilde{K}t}f)(\theta),A(\mathcal{F}_te^{-i\tilde{K}t}f)(\theta)
\rangle_{\mathcal{H}}d\mu(\theta)\end{aligned}$$ $$\begin{aligned}
&=& \left\langle\mathcal{F}_te^{-i\tilde{K}t}f,(1\otimes
A)\mathcal{F}_te^{-i\tilde{K}t}f\right\rangle_{\tilde{\mathcal{K}}}\\
&=& \left\langle e^{-i\tilde{K}t}f,(1\otimes
A)e^{-i\tilde{K}t}f\right\rangle_{\tilde{\mathcal{K}}}\\ &=&
\sum_{n,m}\overline{a_n}a_me^{-it(\lambda_m-\lambda_n)}\left\langle
f_n,(1\otimes A)f_m\right\rangle_{\tilde{\mathcal{K}}}\\ &=&
\sum_{n,m}\overline{a_n}a_me^{-it(\lambda_m-\lambda_n)}B_{n,m}(A).\end{aligned}$$

Note that if this sum is absolutely convergent then $A_f(t)$ is a bounded and almost periodic function of $t$, and $$t\mapsto\left\langle
U_{\theta}(t,0)f(\theta),AU_{\theta}(t,0)f(\theta)\right\rangle_{\mathcal{H}}$$ is bounded a.e. $\theta$. We conclude

\[cor.45\] If $f=\sum_{j=1}^{m}a_jf_j$, where $f_j$ are eigenvectors of $\tilde{K}$ and $f_j(\theta)\in{{\mathrm{dom}~}}A$, for all $\theta$, then $t\mapsto A_f(t)$ is a bounded and almost periodic function. Moreover, $$t\mapsto\left\langle
U_{\theta}(t,0)f(\theta),AU_{\theta}(t,0)f(\theta)\right\rangle_{\mathcal{H}}$$ is bounded for almost every $\theta$.

More generally we obtain the following result:

\[teo.47\] Suppose that $\Omega$ is a compact manifold, $g_t:\Omega\rightarrow\Omega$ a $C^1$ flow with $\sup_{t,\theta}\|\partial_tg_t(\theta)\|<\infty$, and $\tilde{K}f^{(\lambda)}=\lambda f^{(\lambda)}$ with $\theta\mapsto
f^{(\lambda)}(\theta)$ a $C^1$ map. Then for $\mu$ almost every $\theta$ one has $U_{\theta}(t,0)f^{(\lambda)}(\theta)\in{{\mathrm{dom}~}}H_{\theta}(t)$ and $$\left\langle U_{\theta}(t,0)f^{(\lambda)}(\theta),H_{\theta}(t)
U_{\theta}(t,0)f^{(\lambda)}(\theta)\right\rangle$$ is a bounded function of $t$. Moreover, if $H_{\theta}(t)=H_0+V(g_t(\theta))$ with $V(g_t(\theta))$ bounded and $\sup_{t,\theta}\|V(g_t(\theta))\|<\infty$, then the energy expectation $$\left\langle
U_{\theta}(t,0)f^{(\lambda)}(\theta),H_0
U_{\theta}(t,0)f^{(\lambda)}(\theta)\right\rangle$$ is also bounded.

Since $\tilde{K}f^{(\lambda)}=\lambda f^{(\lambda)}$ then $f^{(\lambda)}(\theta)\in{{\mathrm{dom}~}}H_{\theta}(0)$ a.e. $\theta$ and therefore $U_{\theta}(t,0)f^{(\lambda)}(\theta)\in{{\mathrm{dom}~}}H_{\theta}(t)$ a.e. $\theta$. On the other hand $$U_{\theta}(t,0)f^{(\lambda)}(\theta)=
\mathcal{F}_te^{-i\tilde{K}t}f^{(\lambda)}(\theta)=
\mathcal{F}_te^{-i\lambda t}f^{(\lambda)}(\theta)=e^{-i\lambda
t}f^{(\lambda)}(g_t(\theta))$$ and from the differentiability hypothesis it follows that $$i\frac{\partial}{\partial
t}U_{\theta}(t,0)f^{(\lambda)}(\theta)=\lambda e^{-i\lambda
t}f^{(\lambda)}(g_t(\theta))+ie^{-i\lambda
t}\frac{d}{d\theta}f^{(\lambda)}(g_t(\theta))\frac{d}{dt}g_t(\theta),$$ which implies that $$i\frac{\partial}{\partial
t}U_{\theta}(t,0)f^{(\lambda)}(\theta)=H_{\theta}(t)
U_{\theta}(t,0)f^{(\lambda)}(\theta)$$ is bounded and the first part of the result is proved. The second one follows as in Proposition \[prop.38\].

\[cor.48\] Suppose the hypotheses of the above theorem hold and that for each eigenvector $f^{(\lambda_n)}\in\tilde{\mathcal{K}}$ the function $\theta\mapsto f^{(\lambda_n)}(\theta)$ is $C^1$. Then for $\mu$ almost every $\theta$ and for all vectors $\xi\in\mathcal{H}$ of the form $$\xi=a_1f^{(\lambda_1)}(\theta)+\ldots+a_kf^{(\lambda_k)}(\theta),$$ the expectation value of the energy $$\left\langle
U_{\theta}(t,0)\xi,H_{\theta}(t) U_{\theta}(t,0)\xi\right\rangle$$ is a bounded function.

In case $\xi=\sum_{n=1}^{\infty}a_nf^{(\lambda_m)}(\theta)$ with $\sum|a_n|^2<\infty$, a sufficient condition for $U_{\theta}(t,0)\xi\in{{\mathrm{dom}~}}H_{\theta}(t)$ and bounded energy is $$\sum_{j=1}^{\infty}|a_j|\left(|\lambda_j|+
\sup_{\theta}\|\partial_{\theta}f^{\lambda_j}(\theta)\|\right)<\infty,$$ since this implies that $$t\mapsto
U_{\theta}(t,0)\xi=\sum_{j=1}^{\infty}a_je^{-i\lambda_jt}f^{\lambda_j}(g_t(\theta))$$ is a $C^1$ function and $i\partial_tU_{\theta}(t,0)$ is bounded.

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[^1]: CRdeO was partially supported by CNPq (Brazil).

[^2]: MSS was supported by CAPES (Brazil).
