Prethermalization and Conservation Laws in Quasi-Periodically Driven Quantum Systems

Abstract

We study conservation laws of a general class of quantum many-body systems subjected to an external time dependent quasi-periodic driving. When the frequency of the driving is large enough or the strength of the driving is small enough, we prove a Nekhoroshev-type stability result: we show that the system exhibits a prethermal state for stretched exponentially long times in the perturbative parameter. Moreover, we prove the quasi-conservation of the constants of motion of the unperturbed Hamiltonian and we analyze their physical meaning in examples of relevance to condensed matter and statistical physics.

1 Introduction

The investigation of physics of many-body systems out of thermal equilibrium has recently attained a considerable amount of attention. Due to new developments in manipulating quantum systems, fundamental questions related to the thermalization process can be investigated both theoretically and experimentally. Moreover, in the last few years, the establishment of the paradigm of prethermalization opened an avenue in the experimental realization of new, exotic out-of-equilibrium phases of matter. Mathematically rigorous investigations on this phenomenon have also been initiated, especially in the context of quantum many-body systems in presence of external drivings [1, 23, 25].

With this motivation, in the present paper we consider a class of quantum many-body systems on a lattice whose dynamics is generated by a time dependent Hamiltonian H(t) of the form

$$\begin{aligned} H(t)\;=\;H_0+P( \omega t)\,, \end{aligned}$$

(1.1)

where $\omega \in \mathbb R^m$ and $P(\omega t)$ is regarded as a perturbation of $H_0$. We consider perturbations that are quasi-periodic in time and either small in size, i.e. $P \sim \epsilon$, or having high frequency, i.e. $\omega \sim \epsilon ^{-1}$, $\epsilon \ll 1$. We focus on the case where the unperturbed operator admits the structure

$$\begin{aligned} H_0\;=\;\sum _{\alpha =1}^r J_\alpha N^{(\alpha )} \;=:\; J \cdot N\,, \end{aligned}$$

(1.2)

where $J_1, \dots , J_r \in \mathbb {R}$, and $N^{(1)}, \dots , N^{(r)}$ are local,¹ mutually commuting, extensive self-adjoint operators with integer spectrum. In this setting, in absence of the perturbation, the dynamics conserves each of the $N^{(\alpha )}$’s separately. Under suitable non resonance and strong locality hypotheses, we prove that, for the two classes of perturbations delined above, each of the $N^{(\alpha )}$’s is quasi-conserved for a stretched exponentially long time in the small perturbative parameter $\epsilon$. All our results hold uniformly in the size of the lattice. Moreover, we construct an effective time independent local Hamiltonian $H_\textrm{eff}$ whose flow approximates very well the dynamics of local observables in the following sense. If O is a local observable and U denotes the unitary operator representing the time evolution generated by H(t), then

• (i) for all $t \lesssim e^{\epsilon ^{-c}}$

  $$\begin{aligned} \Vert U^*(t) O U(t)-e^{-\textrm{i}H_\textrm{eff}t} O e^{\textrm{i}H_\textrm{eff}t} \Vert _{\textrm{op}} \;\lesssim \; \epsilon ^a \, , \end{aligned}$$

  (1.3)

• (ii) there exists an infinite sequence of times $\{t_j\}_{j \in \mathbb N}$, $t_j \sim j e^{f \epsilon ^{-c}}$ for which this vicinity of dynamics is actually stretched exponentially small, namely

  $$\begin{aligned} \Vert U^*(t_j) O U(t_j)-e^{-\textrm{i}H_\textrm{eff}t_j} O e^{\textrm{i}H_\textrm{eff}t_j} \Vert _{\textrm{op}} \;\lesssim \; j^{d+2} e^{-\epsilon ^{-c}} \, , \end{aligned}$$

  (1.4)

where d is the dimension of the lattice and $a, f, c>0$ are independent of $\epsilon$ and of the size of the lattice.

The conservation of each of the $N^{(\alpha )}$’s encodes information on the collective dynamical behavior of the model during the prethermal phase. Practically, this suggests that in the absence of other conserved quantities, a good candidate state for predictions in the prethermal phase is given by the Generalized Gibbs Ensemble $\rho _{GGE} = e^{-\sum _{j=1}^{r} \lambda _j N^{(j)}}/Z_{GGE}$ with $Z_{GGE}=\textrm{Tr}(\rho _{GGE})$ [39].

Paradigmatic models that fit the assumptions of our analysis are the quantum Ising chain in d spatial dimensions (with or without external magnetic field) and generalizations of the Fermi-Hubbard Hamiltonian where $H_0$ is given by the interaction and $P(\omega t)$ by the kinetic energy. Leaving a more detailed discussion on these two models to Subsection 2.4, here we briefly discuss the consequences of our result on the dynamics of the one-dimensional quantum Ising chain with an external magnetic field. The magnetic field has two components: a time independent one along the (3) axis, space-periodic with period 2 lattice sites, and a quasi-periodically time dependent one along the (1) axis. The Hamiltonian of the model is

$$\begin{aligned} H(t)\;=\; - J \sum _{x \in \Lambda } \sigma _x^{(3)} \sigma _{x+1}^{(3)} - J_{\textrm{odd}} \sum _{x \in \Lambda _{\textrm{odd}}} \sigma _x^{(3)} -J_{\textrm{even}} \sum _{x \in \Lambda _{\textrm{even}}} \sigma _x^{(3)} -\epsilon B(\omega _1 t,\omega _2 t) \sum _{x \in \Lambda } \sigma _x^{(1)} \, . \end{aligned}$$

(1.5)

When the perturbation is turned off ($\epsilon =0$) the number operators

$$\begin{aligned} N_{\textrm{odd}}:=\sum _{x \in \Lambda _\textrm{odd}} \sigma _x^{(3)} \, , \qquad N_{\textrm{even}}:=\sum _{x \in \Lambda _\textrm{even}} \sigma _x^{(3)}\,, \end{aligned}$$

(1.6)

which are the total magnetizations on even and odd sites, are conserved.

When the perturbation is turned on, $N_\textrm{odd}$ and $N_\textrm{even}$ are not conserved anymore. A naive analysis suggests that $N_\textrm{odd}$ and $N_\textrm{even}$ are quasi-conserved for times $t \lesssim \epsilon ^{-1}$. Our result improves significantly this bound: if the vector $(J,J_\textrm{odd},J_\textrm{even},\omega _1,\omega _2)$ is non resonant, then $N_\textrm{odd}$ and $N_\textrm{even}$ are quasi-conserved for stretched exponentially long times in $\epsilon$. In practice, this means that if in the initial state of the system all sites with spin up are in the odd sublattice, the number of sites with spin up on the even sublattice is $\sim \epsilon ^a |\Lambda |$ for times that are stretched exponentially long in $\epsilon$.

Perturbations of Hamiltonians $H_0$ of the form (1.2) were first studied by De Roeck and Verreet [23] in the small size, time periodic case, proving quasi-conservation of the $N^{(\alpha )}$ operators and the existence of an effective time dependent Hamiltonian $H_\textrm{eff}(t)$ whose flow approximates the evolution of local observables for stretched exponentially long time.

In our work we prove that the quasi-conservation of the $N^{(\alpha )}$’s persists under quasi-periodic perturbations. This extends what is found in [23] for the small size time periodic case and is completely new in the case of high frequency perturbations. Moreover, we conjugate the dynamics, up to exponentially small errors, to the one of a time independent Hamiltonian $H_\textrm{eff}$ quasi-commuting with all the operators $N^{(\alpha )}$’s, in contrast with the general expectations expressed in [23]. To obtain this result, we implement a new normal form procedure with respect to the ones developed in the previous works in the field [1, 23, 25], deeply based on the spectral properties of the operators $N^{(\alpha )}$. In addition, we prove that the phenomenon of stroboscopic times originally found by Abanin et al. [1] in the time periodic setting can also be observed in systems with quasi-periodic driving. Such a result is new both for fast forcing and for small size perturbations. It requires the combination of Lieb-Robinson bounds [42], in order to control the time evolution of local observables, with a refined analysis of the ergodization time on the torus, for which we apply the results of [15].

1.1 Prethermalization in Classical and Quantum Systems

In general, prethermalization refers to the fact that certain physical systems, when initialized far from equilibrium, quickly relax to long-living non-thermal states, before eventually reaching the thermal state on much longer timescales [14, 16, 20, 34, 37, 38, 43, 48, 52, 54, 62]. Often, during the prethermal dynamics, the time-evolution of the system is well-approximated by an effective Hamiltonian with some additional symmetries that are broken in the Hamiltonian of the system. This is the case, for example, of the first model on which prethermalization was observed: the Fermi-Pasta-Ulam-Tsingou chain [27]. For this nonintegrable model, the dynamics in the prethermal phase (in the past, this has often been referred to as the “metastable state”) has been related to the dynamics of two integrable systems: the Toda chain [9, 13, 26], and the Korteweg-de Vries hierarchy of equations [10, 30, 31, 66]. Rigorous prethermalization results for this model holding in the thermodynamic limit rely either on estimates of autocorrelation of the energies of Fourier modes [47] or on the quasi-conservation of bunches of Toda actions as adiabatic invariants [33]. In general, the dynamics of a model in its prethermal phase is not necessarily described by an integrable Hamiltonian. This is the case, for example, of certain systems subjected to external drivings, mathematically encoded in a time dependent Hamiltonian: in the cases when the system prethermalizes, during the prethermal state the dynamics is often well-approximated by the dynamics generated by a time independent Hamiltonian, but this in general guarantees an approximate conservation of energy only.

For many-body quantum systems, an effective protocol to produce prethermal states with certain controlled features is given by external drivings. It has been predicted theoretically that in absence of Many-Body Localization, the time dependent perturbation increases the energy of the system, which eventually reaches thermal equilibrium in a featureless, infinite-temperature Gibbs state [22, 41, 57]. In the particular case of a fast periodic driving, the heating rate of the system is suppressed and for a very long time, which is exponential in the frequency [1], the system remains in a prethermal state. This prethermal state often presents interesting, non trivial features, and it is now customary to refer to the phases arising in this way as Floquet phases of matter [24, 60, 65, 67, 68].

In the case of quasi-periodic drivings, recent theoretical works analyzed the possibility of creating prethermal states which exhibit non-trivial thermodynamic features that are different from the Floquet ones [40, 44, 45, 50, 51, 61, 69, 70]. Experimental works confirmed this possibility [18, 35, 49].

1.2 Main Novelties and Related Literature

As mentioned above, our work stems from a recent series of rigorous results on prethermalization for quantum many-body systems subjected to time periodic or time quasi-periodic driving. Abanin et al. in [1] exhibit one observable $H_\textrm{eff}$ which is quasi-conserved for stretched exponentially long times in systems with time periodic driving, and prove the existence of stroboscopic times $t_j \sim j$, $j \in \mathbb Z$, at which the dynamics generated by $H_\textrm{eff}$ approximates the time evolution of local observables up to exponentially small errors. Else, Ho and Dumitrescu in [25] extend the existence result of one quasi-conserved quantity $H_\textrm{eff}$ proven in [1] to the case of Hamiltonians of the form (1.1), with $P(\omega t)$ a fast quasi-periodic forcing and without any further assumption on the unperturbed Hamiltonian $H_0$. De Roeck and Verreet in [23] treat models of the form $H(t) = H_0 + \epsilon P(t)$, with $\epsilon P(t)$ a small size time periodic driving and $H_0$ as in (1.2). They prove quasi-conservation of the $N^{(\alpha )}$’s separately and the existence of an effective time dependent Hamiltonian $H_\textrm{eff}(t)$ approximating the evolution of local observables with polynomial errors for exponentially long times.

In our work we extend both the results of [1] and of [23], which hold for the time periodic case, showing that quasi-conservation of the $N^{(\alpha )}$’s, the existence of an effective Hamiltonian whose flow approximates time evolution of local observables, and the phenomenon of recurrence times all persist under time quasi-periodic perturbations, either small in size or fast forcing. We also extend the results in [25], in the sense that, assuming the more specific structure (1.2) for the unperturbed Hamiltonian, we exhibit the existence of several conserved quantities, instead of only one. Furthermore, we also treat the case of small size time quasi-periodic perturbations, which was not covered in the above papers. Up to our knowledge, ours is the first existence result of a prethermal Hamiltonian with perturbations in such class. We remark that none of the improvements that we present could be obtained with the normal form techniques of [1, 23, 25].

In finite dimensional quasi-integrable classical systems, non convergent normal forms have been largely used to prove confinement of the dynamics for exponentially long times. Among all, we only mention here the milestone result given by Nekhoroshev Theorem [55, 56], see also [46, 59], under assumptions of convexity or, more generally, steepness, of the unperturbed Hamiltonian $H_0$ in the actions (for a simplified exposition in the quadratic case see also [8, 32]), and the result in [12], which deals with the linear, non steep case $H_0 = J \cdot n$, where $n_1, \dots , n_r$ are classical action variables and $J \in \mathbb R^r$ is a non resonant vector.

We point out that, from a mathematical point of view, the main interest of the normal form we perform is that we require the smallness conditions on the perturbative parameter to be uniform with respect to the volume of the lattice $\Lambda$. The typical situation is that the Hamiltonian H(t) of a many-body system of particles on a lattice of volume $|\Lambda |$ has operator norm growing linearly with $|\Lambda |$; for such a reason, one cannot implement a normal form reducing at every step the size of the time dependent perturbation in operator norm, since this would yield empty results in the limit $|\Lambda | \rightarrow \infty$. For a prethermalization result on a quantum system without uniformity in the volume $|\Lambda |$, see the work [53]. Instead, in order to obtain uniformity in the parameter $|\Lambda |$, following the ideas in [1], we exploit the fact that H(t) can be written as the sum of local operators and all its local parts have operator norm independent of $|\Lambda |$. Then one defines a suitable norm, which takes into account these properties and turns out to be uniform in $|\Lambda |$ in all our applications. Similar attempts in this direction were previously made in [58] (see also [11, 63, 64]), where a KAM scheme is implemented for several classes of classical systems, among which arbitrary long chains of weakly coupled harmonic oscillators. The work [58] actually presents several choices of norms, suitably tuned to optimize the quantitative aspects of the theory with respect to the number of degrees of freedom of the system, with the same idea that here we pursue of “considering perturbations not as a single chunk but rather as composites of smaller pieces reflecting an underlying spatial structure”. However, none of the norms presented therein suites to our analysis, as they would not allow us to assume smallness of the perturbation uniformly in $|\Lambda |$. Again in the classical case, a very similar idea based on the decomposition of functions on a lattice in terms of their local parts is implemented in [19].

Finally, for finite and infinite dimensional KAM schemes dealing with fast quasi-periodic frequencies, we mention the works [2, 21, 28, 29].

Our results hold under suitable non resonance hypotheses on the energy levels J of the operators $N^{(\alpha )}$ and on the frequencies $\omega$ of the time forcing. Such conditions differ according to the class of perturbations we deal with. In the fast forced case, due to the fact that $|\omega | \gg |J|$, it is sufficient to require separately that the vectors J and $\omega$ are non resonant, and in particular Diophantine (see the definitions in (2.21a) and (2.21b)). Recall that this is not a restrictive assumption, since Diophantine vectors are a full measure set. Instead, in the case of small size perturbations we impose that the frequency of the forcing term does not resonate with the energy levels of the unperturbed Hamiltonian $H_0$, which amounts to require that $(J, \omega ) \in \mathbb R^{r+m}$ is Diophantine. For an example of a prethermalization result obtained in a specific case where the vector $(J, \omega )$ is instead resonant (and without the presence of an effective time independent Hamiltonian), see the work [36]. For a classical result in absence of non resonance conditions, we mention the work [4], see also [3], which deals with systems of high frequency harmonic oscillators coupled with a slow system. In these works the frequencies J of oscillation may actually be resonant; as a counterpart, differently from our case, nothing prevents energy exchanges among the harmonic oscillators in the fast system.

1.3 Structure of the Work

In Sect. 2 we present the functional setting of the problem and the main results of the paper: conservation of the number operators $N^{(\alpha )}$’s and the properties of the effective Hamiltonian. We then proceed to explicitly study the consequences of our results on a couple of physically relevant models: the Fermi-Hubbard model with next-to nearest neighbor interactions and the quantum Ising chain in a quasi-periodic transverse field.

In Sect. 3 we provide the statements of the normal form results, together with a description of the main ideas of the proof. In Sect. 4 we discuss the properties of strongly local operators and provide the solution(s) to the homological equations that is the key point in the proof of normal form results, presented in Sect. 5. Finally, in Sect. 6 we combine the normal form results, suitable Lieb-Robinson bounds and results from dynamical systems to deduce the main results of the paper.

1.4 Notation

All the notation we use is standard. Nevertheless, for clarity of the reader, we would like to specify the following

$$\begin{aligned} \begin{array}{ll} |k|_1 &{} \hbox {is the }\ell ^1\hbox { norm of the vector }k \in \mathbb {R}^q \\ |k|_\infty &{}\hbox { is the }\ell ^{\infty }\hbox { norm of the vector }k \in \mathbb {R}^q \\ \langle \omega \rangle &{}\hbox { if }\omega \in \mathbb {R}^m,\hbox { denotes the Japanese bracket: }\langle \omega \rangle =\max \{1,|\omega |_\infty \}. \\ \langle V \rangle &{}\hbox { instead, if }V\hbox { is an operator, is defined in (4.31)} \\ \langle \! \langle V \rangle \! \rangle &{}\hbox { if }V\hbox { is an operator, is defined in (4.45)} \\ \lfloor \lambda \rfloor &{} \hbox {is the integer part of }\lambda \in \mathbb {R}. \end{array} \end{aligned}$$

2 Setting and Main Results

We consider a system on a finite lattice $\Lambda =\mathbb {Z}^d \cap [-L,L]^d$, $L \gg 1$ in d spatial dimensions. The dynamics is generated by a quasi-periodic time dependent self-adjoint Hamiltonian $H(\omega t)$ acting on the Hilbert space $\mathcal {H}_\Lambda :=\bigotimes _{x \in \Lambda } \mathbb {C}^q \simeq (\mathbb {C}^q)^{\otimes |\Lambda |}$ for some $q \in \mathbb N$.

For our purposes, we need to recall the standard notion of locality. Let $\mathscr {B}(\mathcal {H}_\Lambda )$ be the algebra of bounded operators on $\mathcal {H}_\Lambda$ with the usual operator norm, namely

$$\begin{aligned} \Vert A \Vert _{\textrm{op}}\; := \; \sup _{\begin{array}{c} \psi \in \mathcal {H}_\Lambda \\ \Vert \psi \Vert _{\mathcal {H}_\Lambda }=1 \end{array}} \Vert A \psi \Vert _{\mathcal {H}_\Lambda } \, \end{aligned}$$

(2.1)

for any $A \in \mathscr {B}(\mathcal {H}_\Lambda )$. If $S \subset \Lambda$, we denote by $\mathcal {H}_S$ the Hilbert subspace obtained as $\mathcal {H}_S:=\bigotimes _{x \in S} \mathbb {C}^q$. An operator $A \in \mathscr {B}(\mathcal {H}_\Lambda )$ is said to be a local operator acting within $S \subset \Lambda$ if there exists an operator $A_S \in \mathscr {B}(\mathcal {H}_S)$ such that $A=\mathbb {1}_{\Lambda \setminus S} \otimes A_S$. We denote by $\mathcal {P}_c(\Lambda )$ the collection of all the connected subsets $S \subset \Lambda$. We define the set of quasi-local operators as the closure (with respect to the operator norm) of the set of finite linear combinations of local operators. A collection of operators $\{A_S\}_{S \in \mathcal {P}_c(\Lambda )}$ defines an operator $A=\sum _{S \in \mathcal {P}_c(\Lambda )} A_S {\in \mathscr {B}(\mathcal {H}_\Lambda )}$.

Following [1], given $\kappa \ge 0$, we define the norm

$$\begin{aligned} \Vert A \Vert _{\kappa } \;:=\; \sup _{x \in \Lambda } \sum _{\begin{array}{c} S \in \mathcal {P}_c(\Lambda ) \\ x \in S \end{array}} \Vert A_S \Vert _{\textrm{op}} \, e^{\kappa |S|} \, . \end{aligned}$$

(2.2)

If there exists a constant $C>0$, independent of $|\Lambda |$, such that $\Vert A \Vert _\kappa < C$, we say that the collection $\{A_S\}_S \in \mathcal {O}_\kappa$. By abuse of notation, we also simply say that $A \in \mathcal {O}_\kappa$.

Given two local operators $A_S$ and $B_{S'}$, we notice that if $S \cap S' = \varnothing$, then $[A_S,B_{S'}]=0$; otherwise if $S \cap S' \ne \varnothing$, there exists an operator $C_{S \cup S'} \in \mathscr {B}(\mathcal {H}_{S \cup S'})$ such that $[A_S,B_{S'}]=C_{S\cup S'} \otimes \mathbb {1}_{\Lambda {\setminus } (S \cup S')}$. Then, given two operators of the form $A=\sum _{S' \in \mathcal {P}_c(\Lambda )} A_{S'}$, $B=\sum _{S'' \in \mathcal {P}_c(\Lambda )} B_{S''}$ one can consider for their commutator [A, B] the following decomposition:

$$\begin{aligned} {[}A, B]\;=\;\sum _{S \in \mathcal {P}_c(\Lambda )} [A,B]_S\,, \quad [A,B]_S \;:=\; \sum _{\begin{array}{c} S',S'' \in \mathcal {P}_c(\Lambda ) \\ S' \cup S'' = S, \, S' \cap S'' \ne \varnothing \end{array}} [A_{S'},B_{S''}] \, . \end{aligned}$$

(2.3)

A time quasi-periodic family of quasi-local operators is a map $\mathbb R\ni t \mapsto \{A_S(\omega t)\}_{S \in \mathcal {P}_c(\Lambda )}$ such that $\omega \in \mathbb R^m$ for some $m\in \mathbb R$ and for any $S \in \mathcal {P}_c(\Lambda )$ the map $\mathbb T^m \ni \varphi \mapsto A_S(\varphi )$ is analytic. Then, one defines

$$\begin{aligned} A(\omega t) \;:=\; \sum _{S \in \mathcal {P}_c(\Lambda )} A_S(\omega t) \, . \end{aligned}$$

(2.4)

In the normal form procedure, we need to take into account the amount of regularity of the map $\varphi \mapsto A(\varphi )$. Thus, for $\kappa \ge 0$, $\rho \ge 0$, we introduce the norm

$$\begin{aligned} \Vert A \Vert _{\kappa ,\rho } \;:=\; \sup _{x \in \Lambda } \sum _{\begin{array}{c} S \in \mathcal {P}_c(\Lambda ) \\ x \in S \end{array}} \sum _{l \in \mathbb {Z}^m} \big \Vert (\widehat{A_S})_{l} \big \Vert _{\textrm{op}} \, e^{\kappa |S|} e^{\rho |l|} \, , \end{aligned}$$

(2.5)

where, for $l \in \mathbb {Z}^m$, $(\widehat{A_S})_{l}:= (2 \pi )^{-m} \int _{\mathbb {T}^m} e^{-\textrm{i}l \cdot \varphi } A_S(\varphi ) \, \textrm{d}\varphi$. Again, by the same abuse of notation, we say that $A \in \mathcal {O}_{\kappa ,\rho }$ if there exists $C>0$, independent of $|\Lambda |$, such that $\Vert A \Vert _{\kappa ,\rho } < C$.

We point out that the classes of operators $\mathcal {O}_\kappa$ and $\mathcal {O}_{\kappa ,\rho }$ contain a quite general class of physically relevant elements. Among them, translationally invariant Hamiltonians with finite range interaction, whose strength and range are independent of $|\Lambda |$. More generally, all operators $A \equiv \{ A_S\}_{S \in \mathcal {P}_c(\Lambda )}$ with

$$\begin{aligned} \sup _{S \in \mathcal {P}_c(\Lambda )} \Vert A_S\Vert _{\textrm{op}} \le C_A \quad \text{ and } \quad A_{S} \ne 0 \ \Rightarrow \ |S| \le \overline{s}\, \end{aligned}$$

(2.6)

for some positive $C_A, \overline{s}$ independent of $|\Lambda |$, are such that $A \in \mathcal {O}_{\kappa }$. Indeed, one has

$$\begin{aligned} \Vert A\Vert _{\kappa }= & {} \sup _{x \in \Lambda } \sum _{\begin{array}{c} S \in \mathcal {P}_{c}(\Lambda ) \\ S \ni x\,,\ |S| \le \overline{s} \end{array}} \Vert A_{S}\Vert _{\textrm{op}} e^{\kappa |S|} \le C_A \sup _{x \in \Lambda } \sum _{\begin{array}{c} S \in \mathcal {P}_{c}(\Lambda ) \\ S \ni x\,,\ |S| \le \overline{s} \end{array}} e^{\kappa |S|} \\= & {} C_A \sum _{\begin{array}{c} S \in \mathcal {P}_{c}(\Lambda ) \\ S \ni 0\,,\ |S| \le \overline{s} \end{array}} e^{\kappa |S|} \le C_A C_{\overline{s},\kappa , d}\, \end{aligned}$$

for some $C_{\overline{s}, \kappa , d}>0$. Note that the operators treated in Sect. 2.4 are in $\mathcal {O}_{\kappa }$, since they satisfy (2.6), while they have ${\Vert A\Vert _{\textrm{op}} \simeq |\Lambda |.}$

2.1 Hamiltonian of the Model

As stated in the Introduction, we consider perturbations of the following unperturbed Hamiltonian:

$$\begin{aligned} H_0 \;:=\; J \cdot N\; :=\; \sum _{\alpha =1}^r J_\alpha N^{(\alpha )}\,, \end{aligned}$$

(2.7)

where $J \in \mathbb {R}^r$ and $N=(N^{(1)},\dots ,N^{(r)})$. Each of the $N^{(\alpha )}$ has the following properties:

• (N.i) for any $\alpha \in \{1,\dots ,r\}$, $N^{(\alpha )}$ is self-adjoint on $\mathcal {H}_\Lambda$;

• (N.ii) there exists $\kappa >0$ such that $N^{(\alpha )} \in \mathcal {O}_{\kappa }$ for any $\alpha \in \{1,\dots ,r\}$;

• (N.iii) $\left[ N^{(\alpha )}_S, N^{(\alpha ')}_{S'}\right] = 0 \quad \forall \alpha , \alpha ' = 1, \dots , r,$ $\forall S, S' \in \mathcal {P}_c(\Lambda )$;

• (N.iv) $N^{(\alpha )}$ has integer spectrum $\forall \alpha = 1, \dots , r$.

Due to property (N.iv), the operators $N^{(\alpha )}$ are called number operators. Following [23], we say that $A=\sum _{S \in \mathcal {P}_c(\Lambda )} A_S$ is a strongly local operator (in symbols $A\in \mathcal {O}^\texttt {s}$) if for any $\alpha =1,\dots ,r$

$$\begin{aligned} \forall S' \in \mathcal {P}_c(\Lambda ) \quad \text {s.t.} \quad S' \nsubseteq S, \quad [A_{S}, N^{(\alpha )}_{S'}] = 0\,. \end{aligned}$$

(2.8)

Moreover, we write $A\in \mathcal {O}^\texttt {s} _\kappa$ if it is strongly local and there exists $\kappa >0$ such that $\Vert A\Vert _{\kappa }< \infty$. We write $A\in \mathcal {O}^\texttt {s} _{\kappa ,\rho }$ if there exists $\omega \in \mathbb R^m$ such that the map $t \mapsto A(\omega t)$ is quasi-periodic, for any $\varphi \in \mathbb {T}^m$, $A(\varphi )$ is strongly local, and there exist $\kappa , \rho >0$ such that $\Vert A\Vert _{\kappa , \rho }< \infty$ and (2.8) holds.

Here and in the following, given a family of self-adjoint operators H(t) with quasi-periodic dependence on time, we will denote by $U_H(t)$ the (unique) solution of the equation

$$\begin{aligned} U_H(t) \;=\; -\textrm{i}\int _0^t H(s) U_H(s) \, \textrm{d}s {\, + U_{H}(0)}\, , \qquad U_H(0)=\mathbb {1} \, . \end{aligned}$$

(2.9)

2.2 Small-Size Perturbations

The first class of perturbations we consider is composed by small in size and quasi-periodic operators; that is, we consider the self-adjoint time dependent Hamiltonian

$$\begin{aligned} H(\omega t)\;=\; J \cdot N + \varepsilon V(\omega t)\, , \qquad \varepsilon \ll 1\,, \end{aligned}$$

(2.10)

with $V \in \mathcal {O}^\texttt {s} _{\kappa ,\rho }$ for some $\kappa , \rho >0$, $J \in \mathbb R^r$ and $\omega \in \mathbb R^m$. We consider models for which the vector $(J,\omega ) \in \mathbb R^{r+m}$ is Diophantine, that is there exist $\gamma >0$, $\tau >r+m-1$ such that $(J, \omega )$ belongs to the set

$$\begin{aligned} \mathcal {D}_{\gamma , \tau }:= \left\{ (J, \omega ) \in \mathbb R^{m+r}\ \left| \ |J\cdot k + \omega \cdot l | \; > \; \frac{\gamma }{(|k|+|l|)^\tau } \, , \quad \forall (k,l) \in \mathbb {Z}^{m+r} \setminus \{ (0,0) \} \right. \right\} \,. \end{aligned}$$

(2.11)

Moreover, we fix $\Omega >0$ and $\mathcal {J}>0$ and we shall assume that $|\omega |\le \Omega$, $|J| \le \mathcal {J}$.

We will say that a quantity depends on the parameters of the system associated to (2.10) if it depends only on $d, r, m, \kappa , \rho , \gamma , \tau , \mathcal {J}, \Omega$, $\{\Vert N^{(\alpha )}\Vert _{0}\}_{\alpha = 1}^r$, $\{\Vert N^{(\alpha )}\Vert _{\kappa }\}_{\alpha = 1}^r$, $\Vert V\Vert _{\kappa , \rho }$. In particular, a quantity depending on the parameters of the system associated to (2.10) does not depend on $|\Lambda |$ or on $\varepsilon$.

Theorem 2.1

(Quasi-conservation laws for small perturbations) Let $H(\omega t)$ be as in (2.10) and let $\kappa , \rho , \gamma , \Omega >0$ and $\tau > r + m -1$ be such that the operators $\{N^{(\alpha )}\}_{\alpha =1}^r$ satisfy assumptions (N.i)–(N.iv), $V \in \mathcal {O}_{\kappa , \rho }^\texttt {s}$, and $(J, \omega ) \in \mathcal {D}_{\gamma , \tau }$ with $|J| \le \mathcal {J}$ and $|\omega | \le \Omega$. Then there exists $\varepsilon _0>0$ depending on the parameters of the system associated to (2.10) such that, if $0< \varepsilon <\varepsilon _0$, the following holds. For any $\alpha = 1, \dots , r$ one has

$$\begin{aligned} \frac{1}{|\Lambda |}\Vert U^*_{H}(t) N^{(\alpha )}U_{H}(t) -N^{(\alpha )}\Vert _{\textrm{op}} \le K \left( e^{-{\varepsilon ^{-2\texttt {b} }}} t + \varepsilon ^{\frac{1}{2}} \right) \, \qquad \forall t > 0\, , \end{aligned}$$

(2.12)

where

$$\begin{aligned} \begin{aligned} \texttt {b} := \frac{1}{4(\tau + r +2)}\,, \quad K:=C(\kappa , \rho ) \Vert V \Vert _{\kappa ,\rho } \max _{\alpha =1, \dots , r}\{\Vert N^{(\alpha )} \Vert _{\kappa } \}\,, \end{aligned} \end{aligned}$$

(2.13)

for some $C(\kappa ,\rho )>0$. As a consequence,

$$\begin{aligned} \frac{1}{|\Lambda |}\Vert U^*_{H_{}}(t) N^{(\alpha )}U_{H_{}}(t) -N^{(\alpha )}\Vert _{\textrm{op}} \le 2 K \varepsilon ^{\frac{1}{2}} \quad \forall 0< t < e^{{\varepsilon ^{-2\texttt {b} }}} \varepsilon ^{\frac{1}{2}}\,. \end{aligned}$$

(2.14)

Theorem 2.2

(Evolution of local observables for small perturbations) Let $H(\omega t)$ be as in (2.10) and let $\kappa , \rho , \gamma , \Omega >0$ and $\tau > r + m -1$ be such that the operators $\{N^{(\alpha )}\}_{\alpha =1}^r$ satisfy assumptions (N.i)–(N.iv), $V \in \mathcal {O}_{\kappa , \rho }^\texttt {s}$, and $(J, \omega ) \in \mathcal {D}_{\gamma , \tau }$ with $|J| \le \mathcal {J}$ and $|\omega | \le \Omega$. Then there exists $\varepsilon _0>0$ depending on the parameters of the system associated to (2.10) such that, if $0<\varepsilon < \varepsilon _0$ and $\kappa _*:= \frac{\min \{\kappa , \rho \}}{\sqrt{2}}$, the following holds. There exists a quasi-local, time independent effective Hamiltonian

$$\begin{aligned} H_{\textrm{eff}} \;:=\;J \cdot N + Z_{\textrm{eff}}\,, \qquad \Vert Z_{\textrm{eff}}\Vert _{\kappa _*} \le \frac{e}{e-1} \varepsilon ^{\frac{1}{2}} \Vert V\Vert _{\kappa , \rho }\,, \end{aligned}$$

(2.15)

such that for any $S \in \mathcal {P}_c(\Lambda )$ and any local operator O acting only within S, there exist $C_1(O)>0$ and $C_2(O)>0$, depending on the parameters of the system associated to (2.10) and on O only, such that

• (i) for any $t < \left( e^{{\varepsilon ^{-\texttt {b} }}} \varepsilon ^{\frac{1}{2}}\right) ^{\frac{1}{d+2}}$

  $$\begin{aligned} \Vert U_H^*(t) O U_H(t) - e^{-\textrm{i}H_{\textrm{eff}} t} O e^{\textrm{i}H_{\textrm{eff}} t} \Vert _{\textrm{op}} \le C_1(O) \varepsilon ^{\frac{1}{2}} \, , \end{aligned}$$

  (2.16)

  with $\texttt {b}$ defined in (2.13);

• (ii) there exists a collection of recurrence times $\{t_j\}_{j\in \mathbb {N}}$ such that for any $j \in \mathbb N$

  $$\begin{aligned} \Vert U_H^*(t_j) O U_H (t_j)- e^{-\textrm{i}H_{\textrm{eff}} t_j} O e^{\textrm{i}H_{\textrm{eff}} t_j} \Vert _{\textrm{op}} \le C_2(O) {j^{d+2}} \varepsilon ^{\frac{1-\textsf {f} }{2}} e^{-{\textsf {f} }\varepsilon ^{-\texttt {b} }}\,, \end{aligned}$$

  (2.17)

  where

  $$\begin{aligned} \textsf {f} :=\frac{1}{\tau (d+2)+1} \, . \end{aligned}$$

  (2.18)

  More precisely, there exists a constant $a_{\gamma , m, \tau }>0$ such that $\forall j \in \mathbb {N}$

  $$\begin{aligned} t_j\in [jT(\varepsilon ),(j+1)T(\varepsilon )] \, , \qquad T(\varepsilon ):={a_{\gamma , m, \tau }e^{{\textsf {f} }\tau \varepsilon ^{-\texttt {b} }}\varepsilon ^{\frac{\textsf {f} }{2}}}\,. \end{aligned}$$

  (2.19)

The proofs of Theorems 2.1 and 2.2 are in Sect. 6. We make the following comments:

• Theorem 2.1 and Item (i) of Theorem 2.2 generalize the result in [23], which holds for time periodic small size perturbations, to the time quasi-periodic case. Moreover, in [23] De Roeck and Verreet show the existence of a time dependent effective Hamiltonian $H_\textrm{eff}(t)$ such that (2.16) holds, whereas the Hamiltonian $H_\textrm{eff}$ in Theorem 2.2 is time independent.

• In [1] Abanin et al. study the case $H(t) = H_0 + P(\nu t)$, with P time periodic and $\nu \gg 1$ a fast frequency. They prove the existence of an effective Hamiltonian $H_\textrm{eff}$ and of stroboscopic times $t_j:= \frac{2\pi }{j}$, $j \in \mathbb N$, such that $U_H^*(t) O U_H(t)$ and $e^{-\textrm{i}H_\textrm{eff}t} O e^{\textrm{i}H_\textrm{eff}t}$ become exponentially close at times $t=t_j$. In Item (ii) of Theorem 2.2 we show that such a phenomenon is true also for quasi-periodic systems, replacing stroboscopic times with recurrence times $\{t_j\}_j$ (2.19).

• Theorem 2.1 is based on the non convergent normal form result of Proposition 3.1. Theorem 2.2 requires the combination of the normal form results of Proposition 3.1 with Lieb-Robinson bounds [42] and results from classical dynamical systems [15]. The same holds for the proof of Theorems 2.3 and 2.4 below, concerning the fast forced case.

2.3 Fast-Forcing Perturbations

The second class of perturbations we consider are non-small in size but high frequency and time quasi-periodic. We consider the self-adjoint time dependent Hamiltonian

$$\begin{aligned} H(\lambda \omega t)\;=\; J \cdot N + V(\lambda \omega t)\, , \qquad \lambda \gg 1\,, \end{aligned}$$

(2.20)

with $V \in \mathcal {O}^\texttt {s} _{\kappa ,\rho }$ for some $\kappa ,\rho >0$. For these models, we require $J \in \mathbb R^r$ and $\omega \in \mathbb R^m$ to be Diophantine separately: there exist $\gamma _\omega >0$ and $\gamma _J>0$, $\tau _\omega > m-1$ and $\tau _J > r-1$ such that $J \in \mathcal {D}_{\gamma _J, \tau _J}$ and $\omega \in \mathcal {D}_{\gamma _\omega , \tau _\omega }$, with

$$\begin{aligned} \mathcal {D}_{\gamma _J, \tau _J} := \left\{ J \in \mathbb {R}^r \ \Big |\ |J\cdot k| \ge \frac{\gamma _J}{|k|^{\tau _J}} \, , \qquad \forall k \in \mathbb {Z}^r \setminus \{0\} \right\} \, , \end{aligned}$$

(2.21a)

$$\begin{aligned} \mathcal {D}_{\gamma _\omega , \tau _\omega } := \left\{ \omega \in \mathbb {R}^m \ \Big |\ |\omega \cdot l| \ge \frac{\gamma _\omega }{|l|^{\tau _\omega }} \, , \qquad \forall l \in \mathbb {Z}^m \setminus \{0\} \right\} \, . \end{aligned}$$

(2.21b)

Furthermore, we shall fix $\mathcal {J}>0$ and require that $|J| \le \mathcal {J}$.

We will say that a quantity depends on the parameters of the system associated to (2.20) if it depends only on $d, r, m, \kappa , \rho , \tau _J, \tau _\omega , \gamma _J, \gamma _\omega , {\mathcal {J}}, \{\Vert N^{(\alpha )} \Vert _{0}\}_\alpha , \{\Vert N^{(\alpha )} \Vert _{\kappa }\}_\alpha , \Vert V\Vert _{\kappa , \rho }$. In particular, a quantity depending on the parameters of the system associated to (2.20) does not depend on $|\Lambda |$, or on $\lambda$.

Theorem 2.3

(Quasi-conservation laws for fast forcing perturbations) Let $H(\lambda \omega t)$ be as in (2.20) and let $\kappa , \rho , \gamma _J, \gamma _\omega , \mathcal {J}>0$, $\tau _J > r-1$ and $\tau _\omega > m-1$ be such that the operators $\{N^{(\alpha )}\}_{\alpha =1}^r$ satisfy assumptions (N.i)–(N.iv), $V \in \mathcal {O}^\texttt {s} _{\kappa , \rho }$, $J \in \mathcal {D}_{\gamma _J, \tau _J}$, $\omega \in \mathcal {D}_{\gamma _\omega , \tau _\omega }$ and $|J| \le \mathcal {J}$.

Then, there exists $\lambda _0>0$, depending on the parameters of the system associated to (2.20) only, such that for any $\lambda > \lambda _0$ and any $\alpha = 1, \dots , r$ the following holds.

$$\begin{aligned} \frac{1}{|\Lambda |}\Vert U^*_{H_{}}(t) N^{(\alpha )}U_{H_{}}(t) -N^{(\alpha )}\Vert _{\textrm{op}} \le K \left( e^{- \lambda ^{2\beta }} t + \lambda ^{-\frac{1}{8 \tau _\omega }} \right) \, \qquad \forall t > 0\,, \end{aligned}$$

(2.22)

where

$$\begin{aligned} \beta := \frac{1}{16 \tau _\omega ( \tau _J +r+2)}\,, \qquad K:=C(\kappa ,\rho ) \Vert V \Vert _{\kappa ,\rho } \max _{\alpha = 1, \dots , r}\{ \Vert N^{(\alpha )} \Vert _{\kappa }\}\, \end{aligned}$$

(2.23)

for some $C(\kappa ,\rho )>0$. As a consequence,

$$\begin{aligned} \frac{1}{|\Lambda |}\Vert U^*_{H_{}}(t) N^{(\alpha )}U_{H_{}}(t) -N^{(\alpha )}\Vert _{\textrm{op}} \le 2 K \lambda ^{-\frac{1}{8 \tau _\omega }} \quad \forall 0< t < e^{ \lambda ^{2\beta }} \lambda ^{-\frac{1}{8\tau _\omega } }\,. \end{aligned}$$

(2.24)

Theorem 2.4

(Evolution of local observables for fast forcing perturbations) Let $H(\lambda \omega t)$ be as in (2.20) and let $\kappa , \rho , \gamma _J, \gamma _\omega , \mathcal {J}>0$, $\tau _J > r-1$ and $\tau _\omega > m-1$ be such that the operators $\{N^{(\alpha )}\}_{\alpha =1}^r$ satisfy assumptions (N.i)–(N.iv), $V \in \mathcal {O}^\texttt {s} _{\kappa , \rho }$, $J \in \mathcal {D}_{\gamma _J, \tau _J}$, $\omega \in \mathcal {D}_{\gamma _\omega , \tau _\omega }$, and $|J| \le \mathcal {J}$. Then, there exists $\lambda _0>0$, depending only on the parameters of the system associated to (2.20) such that, if $\lambda \ge \lambda _0$ and $\kappa _*:= \frac{\min \{\kappa , \rho \}}{\sqrt{2}}$, the following holds. There exists a quasi-local time independent effective Hamiltonian

$$\begin{aligned} H_{\textrm{eff}} \;:=\;J \cdot N + Z_{\textrm{eff}}\,, \qquad \Vert Z_{\textrm{eff}}\Vert _{\kappa _*} \le \frac{e}{e-1} \lambda ^{-\frac{1}{8 \tau _\omega }} \Vert V\Vert _{\kappa , \rho }\,, \end{aligned}$$

(2.25)

such that for any $S \in \mathcal {P}_c(\Lambda )$ and any local operator O acting only within S, there exist $C_1(O)>0$ and $C_2(O)>0$, depending only on O and on the parameters of the systems associated to (2.20), such that

• (i) for any $t < \left( e^{ \lambda ^\beta } \lambda ^{-\frac{1}{8 \tau _\omega }}\right) ^{\frac{1}{d+2}}$

  $$\begin{aligned} \Vert U^*(t) O U (t)- e^{-\textrm{i}H_{\textrm{eff}} t} O e^{\textrm{i}H_{\textrm{eff}} t} \Vert _{\textrm{op}} \le C_1(O) \lambda ^{-\frac{1}{8\tau _\omega }} \,, \end{aligned}$$

  (2.26)

  where $\beta$ is defined as in (2.23);

• (ii) there exists a collection of recurrence times $\{t_j\}_{j\in \mathbb {N}}$ such that for any $j \in \mathbb N$

  $$\begin{aligned} \Vert U^*({t_j}) O U ({t_j})- e^{-\textrm{i}H_{\textrm{eff}} {t_j}} O e^{\textrm{i}H_{\textrm{eff}} {t_j}} \Vert _{\textrm{op}} \le C_2(O) {j^{d+2}} e^{- \textsf {g} \lambda ^\beta } \lambda ^{\frac{(8\tau _\omega -1)(1-\textsf {g} )}{8 \tau _\omega }}\,, \end{aligned}$$

  (2.27)

  where

  $$\begin{aligned} \textsf {g} := \frac{1}{\tau _\omega (d+2) + 1}\,. \end{aligned}$$

  (2.28)

  More precisely, there exists a constant $a_{\gamma , m, \tau _\omega } >0$ such that $\forall j \in \mathbb N$

  $$\begin{aligned} t_j \in [j T(\lambda ), (j+1) T(\lambda )]\,, \quad T(\lambda ) := a_{\gamma _\omega , m , \tau _\omega } e^{ \tau _\omega \textsf {g} \lambda ^{\beta }} \lambda ^{\frac{(8 \tau _\omega -1) \textsf {g} }{8}} \,. \end{aligned}$$

  (2.29)

We point out the following:

• In [25] Else, Ho and Dumitrescu treat the case of a quasi-periodically fast forced Hamiltonian H(t) as in (1.1) without any structural assumption on $H_0$, and they exhibit one quasi-conserved quantity for stretched exponentially long times. Theorem 2.3, under the stronger assumption that the unperturbed Hamiltonian has the form $H_0 = J \cdot N$, with $N^{(1)}, \dots , N^{(r)}$ number operators, exhibits r distinct quasi-conserved quantities over analogous time scales. We point out that this result would not be reachable with the normal form procedure of [25], through which energy exchanges among the number operators $N^{(\alpha )}$ cannot be excluded.

• Item (i) of Theorem 2.4 is not new, and here we recall it only to draw a complete picture also for the case of fast-forcing perturbation. Its proof is contained in [25], without assuming that $H_0$ has the structure in (1.2) (see Remark 3.4 below).

• Item (ii) of Theorem 2.4 is instead original, and it shows for the first time the existence of recurrence times for local observables, previously proved in [1] only for time periodic forcing, to the more delicate case of time quasi-periodic perturbations. As for item (i), the assumption on the special form (1.2) of $H_0$ is not required.

2.4 Applications: Fermi-Hubbard Model and Quantum Ising Chain

2.4.1 Generalized Fermi-Hubbard Model

We consider a model of $\frac{1}{2}$-spin fermions on a lattice composed of sites whose distance oscillates quasi-periodically in time. The Hamiltonian of the model is

$$\begin{aligned} H\;=\; \varepsilon \sum _{\begin{array}{c} x \sim y \\ \sigma \in \{\uparrow ,\downarrow \} \end{array}} K_{x,y}(\omega t) (a^+_{x,\sigma } a_{y, \sigma } + a^+_{y,\sigma } a_{x,\sigma }) + \sum _{\alpha =1}^r J_\alpha \sum _{x \in \Lambda }\sum _{\begin{array}{c} y \in \Lambda \\ |x-y|_1 = \alpha \end{array}} n_{x,\uparrow } n_{y,\downarrow }\,, \end{aligned}$$

(2.30)

where $\forall x \in \Lambda :=\mathbb {Z}^d \cap [-L,L]^d$ and $\sigma \in \{\uparrow , \downarrow \}$, $a_{x,\sigma }$ and $a^+_{x,\sigma }$ are fermionic annihilation and creation operators, $n_{x,\sigma }:=a^+_{x,\sigma } a_{x,\sigma }$, $\forall x, y \in \Lambda$ $K_{x, y} = K_{y, x}: \mathbb {T}^m \rightarrow \mathbb R$ are analytic functions uniformly bounded in x, y, $(J, \omega ) \in \mathbb R^{r+m}$ satisfies Diophantine condition (2.11), $\varepsilon \ll 1$, and $x \sim y$ denotes the sum over the couples $x, y \in \Lambda$ such that $|x-y|_1 = 1$. As number operators, we consider

$$\begin{aligned} N^{(\alpha )} \;:=\; \sum _{x \in \Lambda } \sum _{\begin{array}{c} y \in \Lambda \\ |x-y|_1=\alpha \end{array}} n_{x,\uparrow } n_{y,\downarrow }\,, \end{aligned}$$

(2.31)

which for any $\alpha$ count the number of couples of particles with opposite spin and occupying sites at distance $\alpha$. For example, for $\alpha = 1$, $N^{(1)}$ counts the number of particles with opposite spin and occupying nearest-neighbor sites. Each $N^{(\alpha )}$ admits the decomposition in local operators

$$\begin{aligned} N^{(\alpha )} = \sum _{x \in \Lambda } N^{(\alpha )}_{S_x^{(\alpha )}}\,, \quad S_x^{(\alpha )} := \{ y \in \Lambda \ | \ |x-y|_1 \le \alpha \}\,, \quad N^{(\alpha )}_{S_x^{(\alpha )}} := \sum _{\begin{array}{c} y \in \Lambda \\ |x-y|_1 = \alpha \end{array}} n_{x, \uparrow } n_{y, \downarrow }\,, \end{aligned}$$

(2.32)

and in order to ensure strong locality, it is convenient to decompose the perturbation as

$$\begin{aligned} \begin{aligned} \varepsilon V(\omega t) = \varepsilon \sum _{x \in \Lambda } V_{S'_x}(\omega t)\,, \quad S'_{x} := \{ y \in \Lambda \ |\ |x-y|_1 \le 2r +1\}\,, \\ V_{S'_x}(\omega t) = \sum _{\begin{array}{c} y \in \Lambda \\ y \sim x \end{array}} \sum _{\sigma \in \{\uparrow , \downarrow \}} K_{x, y}(\omega t) (a^+_{x,\sigma } a_{y, \sigma } + a^+_{y, \sigma } a_{x, \sigma })\,. \end{aligned} \end{aligned}$$

(2.33)

We point out that, since $\Vert n_{x,\sigma } \Vert _{\textrm{op}}=1$, each $N^{(\alpha )}$ satisfies (2.6) with $\Vert N^{(\alpha )}_{S^{(\alpha )}} \Vert _{\textrm{op}} \le C_{\alpha ,d}$ and $\bar{s}=\alpha ^d$ for certain constants $C_{\alpha ,d}>0$, and therefore $\Vert N^{(\alpha )} \Vert _{\kappa } < C_{\alpha ,\kappa ,d}$. With analogous argument, it is possible to show that, for a certain explicit constant $C_{d,\alpha }>0$, $\Vert N^{(\alpha )} \Vert _{\textrm{op}}=C_{d,\alpha } |\Lambda |$. Analogously, using that V has only nearest-neighbour interactions, the analyticity in time of $K_{x,y}(\omega t)$ and the boundedness of the fermionic creation and annihilation operators, we have $V \in \mathcal {O}_{\kappa ,\rho }$ for any $\kappa ,\rho > 0$.

If $\varepsilon =0$, then all $N^{(\alpha )}$’s are conserved. For $\varepsilon \ne 0$ and $K_{x,y}$ non trivial, one has $[N^{(\alpha )}, V(\omega t)] \ne 0$. Theorem 2.1 ensures that each $N^{(\alpha )}$ is quasi-conserved for exponentially long times in $\varepsilon$:

$$\begin{aligned} |\Lambda |^{-1} \Vert U^*_{H}(t) N^{(\alpha )} U_H(t) - N^{(\alpha )} \Vert _{\textrm{op}} \le C \varepsilon ^{\frac{1}{2}} \quad \forall 0< t < e^{ \varepsilon ^{-2\texttt {b} }} \varepsilon ^{\frac{1}{2}}\,. \end{aligned}$$

(2.34)

The dynamical consequences of these quasi-conservation laws are pictorically represented in Figs. 1 and 2. In particular, for an initial datum as in the first line of Figs. 1 and 2, Fig. 1 shows that the highlighted package can not disperse, while Fig. 2 shows that it can not get too close to the group of particles on the right, as it happens for hard spheres.

Furthermore, Theorem 2.2 ensures that there exists an effective time independent and local Hamiltonian $H_{\textrm{eff}}$ which approximates the dynamics of local observables for exponentially long times, according to (2.16) and (2.17). $H_\textrm{eff}$ is also explicitly computable: at the first order in $\varepsilon$, using (6.2), (5.28) and (4.45), one obtains

$$\begin{aligned} H_{\textrm{eff}}= & {} J\cdot N + Z^{(1)} +O(\varepsilon ^2), \\ Z^{(1)}= & {} \varepsilon \sum _{x \sim y, \sigma } {(\widehat{ K_{x,y}})_0} (a^+_{y,\sigma } a_{x,\sigma }+a^+_{x,\sigma } a_{y,\sigma }) \prod _{\alpha =1}^rP_{\mathcal {K}_{x,y,\alpha ,\sigma }}(0) \\{} & {} +\varepsilon \sum _{x \sim y, \sigma } \sum _{\begin{array}{c} l \in \mathbb {Z}^m \\ k \in \mathbb {Z}^r \end{array}}\frac{J\cdot k}{\omega \cdot l+J \cdot k} \left( ( \widehat{K_{x,y}} )_l a_{x,\sigma }^+a_{y,\sigma }+ ( \widehat{K_{x,y}} )_l^* a_{y,\sigma }^+a_{x,\sigma } \right) \prod _{\alpha =1}^rP_{\mathcal {K}_{x,y,\alpha ,\sigma }}(k_\alpha ) \end{aligned}$$

where $(\widehat{K_{x,y}})_l:= (2\pi )^{-m}\int _{\mathbb {T}^m} K_{x,y}(\varphi ) e^{-\textrm{i}l\cdot \varphi } \textrm{d}\varphi$, and $P_{\mathcal {K}_{x,y,\alpha ,\sigma }}(k_\alpha )$ denotes the projector onto the space $\mathcal {K}_{x,y,\alpha ,\sigma }(k_\alpha )$ defined as

$$\begin{aligned} \mathcal {K}_{x,y,\alpha ,\sigma }(k_\alpha ):=\ker \left( \sum _{\eta \, : \, |\eta -x|_1=\alpha } n_{\eta ,-\sigma }-\sum _{\eta \, : \, |\eta -y|_1=\alpha } n_{\eta ,-\sigma }-k_\alpha \right) \, . \end{aligned}$$

(2.35)

Fig. 1

Pictorial representation of the Fermi-Hubbard dynamics. Time evolution is represented vertically from top to bottom; at each level, solid dots represent the lattice sites. A particle is represented by a colored sphere above the site where it lies. The arrow inside the particle represents the spin. The highlighted package of particles cannot disperse, since this is not allowed by conservation laws. Indeed, given the initial datum on the first line, one has $N^{(1)}=6$ and $N^{(2)}=0$ while, after the first time-lapse one has $N^{(1)}=4$ and $N^{(2)}=2$

Fig. 2

Pictorial representation of the Fermi-Hubbard dynamics. Time evolution is represented as in Fig. 1. The picture shows that, according to conservation laws, the highlighted package of particles can translate rigidly but cannot get too close to the pair of particles on the right

2.4.2 Quantum Ising Model with Magnetic Field

Let us consider the quantum Ising model in one spatial dimension on a lattice $\Lambda =[-3L,3L-1] \cap \mathbb {Z}$ with periodic boundary conditions and in presence of an external magnetic field. We consider the case where the external magnetic field has two components. The one along the (3)-axis is space periodic of period 3 sites. The one along the (1)-axis is time quasi-periodic with Diophantine frequency $\omega$ satisfying (2.21b). To write the Hamiltonian of the model, it is convenient to introduce the sublattices $\Lambda _1$, $\Lambda _2$ and $\Lambda _3$ defined as

$$\begin{aligned} \Lambda _1= & {} \{-3L,-3L+3, \dots , 0,3,\dots \} \,, \end{aligned}$$

(2.36a)

$$\begin{aligned} \Lambda _2= & {} \{-3L+1,-3L+4,\dots ,1,4,\dots \} \,, \end{aligned}$$

(2.36b)

$$\begin{aligned} \Lambda _3= & {} \{-3L+2,-3L+5,\dots ,2,5,\dots \} \,. \end{aligned}$$

(2.36c)

The Hamiltonian of the model is

$$\begin{aligned} H\;=\; -J \sum _{x \in \Lambda } \sigma _{x}^{(3)} \sigma _{x+e_j}^{(3)}-\sum _{\alpha =1}^{3} h_\alpha \sum _{x \in \Lambda _\alpha } \sigma _x^{(3)}-B(\lambda \omega t) \sum _{x \in \Lambda } \sigma _x^{(1)} \end{aligned}$$

(2.37)

where $h_1,h_2,h_3 \in \mathbb {R}$ are the values of the magnetic field on the sublattices $\Lambda _1,\Lambda _2,\Lambda _3$ respectively, $\sigma ^{(a)}$ is the a-th Pauli matrix, $(h_1, h_2,h_3, J) \in \mathbb R^{4}$ satisfies the Diophantine condition (2.21a), $\lambda \gg 1$ and $B: \mathbb {T}^m \rightarrow \mathbb R$ is an analytic function. We consider as number operators:

$$ N^{(\alpha )}:=\sum _{x \in \Lambda _{\alpha }} \sigma _x^{(3)} \, {, \quad \alpha =1,2,3\,,} \qquad N^{(4)}:=\sum _{x \in \Lambda } \sigma _{x}^{(3)} \sigma _{x+1}^{(3)} \,. $$

The operators $N^{(\alpha )}$ are local, since they admit the decomposition

$$\begin{aligned} \begin{aligned} N^{(\alpha )} = \sum _{x \in \Lambda _\alpha } N^{(\alpha )}_{S_x}\,, \quad S_{x} := \{x\}\,, \quad N^{(\alpha )}_{S_x} = \sigma ^{(3)}_x\,, \quad \forall \alpha = 1, 2, 3\,,\\ N^{(4)} = \sum _{x \in \Lambda } N^{(4)}_{S'_x}\,, \quad S'_x := \{ x, x + 1 \}\,, \quad N^{(4)}_{S'_x} := \sigma ^{(3)}_{x} \sigma ^{(3)}_{x+1}\,. \end{aligned} \end{aligned}$$

(2.38)

For the perturbation, we consider the decomposition

$$\begin{aligned} \begin{aligned} B(\lambda \omega t) \sum _{x \in \Lambda } \sigma ^{(1)}_x = \sum _{x \in \Lambda } V_{S^{''}_x}(\lambda \omega t)\,, \\ S^{''}_x : = \{ y \in \Lambda \ |\ |x-y|_1 \le 2 \}\,, \quad V_{S^{''}_x}(\lambda \omega t) = B(\lambda \omega t) \sigma _{x}^{(1)}\,. \end{aligned} \end{aligned}$$

(2.39)

We remark that, using the notation of (2.6), for $\alpha =1,2,3$, $\Vert N^{(\alpha )}_{S_x} \Vert _{\textrm{op}} \le 1$ and $\bar{s}=1$. Therefore, $\Vert N^{(\alpha )} \Vert _{\kappa } \le e^{\kappa }$ and $N^{(\alpha )} \in \mathcal {O}_{\kappa }$ for any $\kappa >0$. Also, $\Vert N^{(\alpha )} \Vert _{\textrm{op}}=|\Lambda |$. For $\alpha =4$, $\Vert N_{S_x'}^{(4)} \Vert _{\textrm{op}}=1$, $\bar{s}=2$ and therefore $\Vert N^{(4)} \Vert _{\kappa } = 2 e^{2 \kappa }$. On the other hand, $\Vert N^{(4)} \Vert _{\textrm{op}} = |\Lambda |$.

Theorem 2.3 guarantees that each $N^{(\alpha )}$ is quasi-conserved for times $t \le \lambda ^{-\frac{1}{8\tau _\omega }} e^{\lambda ^{2\beta }}$. Note that, since the Hamiltonian (2.37) has a term with $\sigma _x^{(1)}$, then the total magnetization along the (3)-direction is not conserved. Each $N^{(\alpha )}$ for $\alpha =1,2,3$ represents the total magnetization along the direction (3) restricted on the sites $x \in \Lambda _\alpha$. In the case $\alpha = 4$, $N^{(4)}$ represents the number of domain walls, namely the number of pairs of consecutive particles with opposite spin along the (3) component. As a first consequence of Theorem 2.3, both the total magnetization of each $\Lambda _\alpha$ and the number of domain walls are quasi-conserved up to exponentially long times in $\lambda$.

Fig. 3

Pictorial representation of the dynamics of the quantum Ising model. Solid dots represent the lattice sites; above each dot, the arrow represents the (3)-component of the spin state of the particle on the related site. Two different time-lapses are represented vertically. Below them, a pictorial representation of the magnetic field is given (blue line). In each portion of lattice delimited by two vertical blue dashed lines there is (exactly) one site per element of the partition $\{\Lambda _{\alpha }\}$. In particular, from left to right, the first site belongs to $\Lambda _1$, the second site belongs to $\Lambda _2$, and the third site belongs to $\Lambda _3$. Starting from a configuration as in the first line, with only two spins up in two consecutive sites, conservation of the operators $N^{(\alpha )}$ implies that the two spin up packet has to move rigidly from one portion of the lattice $\Lambda$ delimited by two blue dashed lines to another

Let us now focus on the dynamical consequences of such quasi-conservation laws for particular initial states:

• If at time $t=0$ the system is in the state where all sites in the lattice $\Lambda _1$ have spin up and all the other spins are down, equation (2.24) states that, calling $N_{\uparrow }^{(2)}(t),N_{\uparrow }^{(3)}(t)$ the number of sites with spin up in the sublattices $\Lambda _2$ and $\Lambda _3$ respectively, $N_\downarrow ^{(1)}(t)$ the number of sites with spin down in the sublattice $\Lambda _1$, one has

  $$N_\downarrow ^{(1)}(t) \le 12 L K \lambda ^{-\frac{1}{8\tau _\omega }} \,, \qquad N_\uparrow ^{(2)}(t) \le 12 L K \lambda ^{-\frac{1}{8\tau _\omega }} \,, \qquad N_\uparrow ^{(3)}(t) \le 12 L K \lambda ^{-\frac{1}{8\tau _\omega }} \,, $$

  for times $t < e^{\lambda ^{2\beta }} \lambda ^{-\frac{1}{8\tau _\omega }}$.

• If at time $t=0$ the system is in the state where only two consecutive sites of the lattice have spin up and all the others have spin down, constraints on the dynamics are discussed in Fig. 3.

One can easily generalize this example to Ising models in higher spatial dimensions and more general structures of the sublattices $\Lambda _\alpha$.

3 Normal Form Results

A crucial role in the proof of Theorems 2.1 and 2.2 is played by the following normal form results.

Proposition 3.1

(Normal form for small perturbations) Let H(t) satisfy assumptions of Theorem 2.2. Then there exists $\varepsilon _0:= \varepsilon _0(r, \tau , \gamma , \kappa , \rho , \Omega , \Vert V\Vert _{\kappa , \rho }, \{ \Vert N^{(\alpha )}\Vert _{0}\}_{\alpha }) >0$ such that $\forall \ 0< \varepsilon < \varepsilon _0$, the following holds.

• (a) There exist a unitary time quasi-periodic transformation $Y_{\textrm{inv}}$ and a time quasi-periodic self-adjoint operator $H_{\textrm{inv}}$ such that

  $$\begin{aligned} U_{H_{\textrm{inv}}}(t) \;=\; Y_{\textrm{inv}}(\omega t) U_{H}(t) {Y_{{\textrm{inv}}}^*(0)} \end{aligned}$$

  (3.1)

  and

  $$\begin{aligned} H_{{\textrm{inv}}}(\omega t)=J \cdot N + Z_{\textrm{inv}}+V_{\textrm{inv}}(\omega t)\,, \end{aligned}$$

  (3.2)

  with

  • (a.i) $[Z_{\textrm{inv}}, N^{(\alpha )}] = 0\quad \forall \alpha = 1, \dots , r$;

  • (a.ii) $Z_{\textrm{inv}} \in \mathcal {O}^\texttt {s} _{\kappa _*}$, $\Vert Z_{\textrm{inv}} \Vert _{\kappa _*} \le {\frac{{e}}{e-1}} \varepsilon \Vert V \Vert _{\kappa , \rho }$;

  • (a.iii) $V_{\textrm{inv}} \in \mathcal {O}^\texttt {s} _{\kappa _*, \rho _*}$, with $\Vert V_{\textrm{inv}} \Vert _{\kappa _{*}, \rho _*} \le e^{-{\varepsilon ^{-2\texttt {b} }}} \Vert V \Vert _{\kappa ,\rho }\,$;

  • (a.iv) $\forall P \in \mathcal {O}_{\kappa _*, \rho _*},$ $\Vert Y_{\textrm{inv}} P Y_{\textrm{inv}}^* -P\Vert _{\kappa _{*}, \rho _*} \le {\frac{2{e}}{e-1}} \varepsilon ^{\frac{1}{2}} \Vert P\Vert _{\kappa _*, \rho _*}\,$;

• (b) There exist a unitary time quasi-periodic transformation $Y_{\textrm{obs}}$ and a time quasi-periodic self-adjoint operator $H_{\textrm{obs}}$ such that

  $$\begin{aligned} U_{H_{\textrm{obs}}}(t) \;=\; Y_{\textrm{obs}}(\omega t) U_{H}(t) \end{aligned}$$

  (3.3)

  and

  $$\begin{aligned} H_{{\textrm{obs}}}(\omega t)=J \cdot N + Z_{\textrm{obs}}+V_{\textrm{obs}}(\omega t)\,, \end{aligned}$$

  (3.4)

  with

  • (b.i) $Y_{\textrm{obs}}(0) = \mathbb {1}$;

  • (b.ii) $Z_{\textrm{obs}} \in \mathcal {O}^\texttt {s} _{\kappa _*}$, $\Vert Z_{\textrm{obs}} \Vert _{\kappa _*} \le {\frac{{e}}{e-1}} \varepsilon ^{\frac{1}{2}} \Vert V \Vert _{\kappa , \rho }$;

  • (b.iii) $V_{\textrm{obs}} \in \mathcal {O}^\texttt {s} _{\kappa _*, \rho _*}$, with $\Vert V_{\textrm{obs}} \Vert _{\kappa _{*}, \rho _*} \le e^{-{\varepsilon ^{-\texttt {b} }}} \Vert V \Vert _{\kappa ,\rho }\,$;

  • (b.iv) $\forall P \in \mathcal {O}_{\kappa _*, \rho _*},$ $\Vert Y_{\textrm{obs}} P Y_{\textrm{obs}}^* -P\Vert _{\kappa _{*}, \rho _*} \le {\frac{2{e}}{e-1}} \varepsilon ^{\frac{1}{2}} \Vert P\Vert _{\kappa _*, \rho _*}$. More precisely, there exist $n^* \in \mathbb N$ and self-adjoint operators $G^{(0)}_\textrm{obs}, \dots , G^{({n^*-1})}_\textrm{obs}\in \mathcal {O}^\texttt {s} _{\kappa _*, \rho _*}$ such that $Y_\textrm{obs}(\omega t) = e^{-\textrm{i}G^{({n^*-1})}_\textrm{obs}(\omega t)} \cdots e^{-\textrm{i}G^{(0)}_\textrm{obs}(\omega t)}$, with $\Vert G^{(n)}_\textrm{obs}\Vert _{\kappa _*, \rho _*} \le \varepsilon ^{\frac{1}{2}} e^{-n} \Vert V\Vert _{\kappa , \rho }$ for any n.

Here $\texttt {b}$ is the constant defined in (2.13) and

$$\begin{aligned} \kappa _* := \frac{\min \{\kappa , \rho \}}{\sqrt{2}}\,, \quad \rho _* := \frac{\min \{\kappa ,\rho \}}{8 \sqrt{2} r \max _\alpha \{\Vert N^{(\alpha )}\Vert _0\} }\,. \end{aligned}$$

(3.5)

Remark 3.2

The two normal form Hamiltonians $H_{\textrm{inv}}$ and $H_{\textrm{obs}}$ differ because of Item (a.i) and (b.i). In general, it is not possible to perform a unique normal form such that both Items (a.i) and (b.i) hold, as the case of stationary perturbations $V(\omega t) \equiv V$ easily shows. However, Item (a.i) plays a fundamental role in showing the quasi-invariance of the $N^{(\alpha )}$ operators (Theorem 2.1), while Item (b.i) is necessary to obtain the results on recurrence times in Theorem 2.2.

Analogously, a crucial role in the proof of Theorems 2.3 and 2.4 is played by the following:

Proposition 3.3

(Normal form for fast forcing perturbations) Let $H(\lambda \omega t)$ be as in (2.20) and suppose that $\gamma _J>0$, $\gamma _\omega >0$, $\tau _J> r-1$, $\tau _\omega > m-1$ and $\mathcal {J}>0$ are such that J satisfies (2.21a) with $|J| \le \mathcal {J}$ and $\omega$ satisfies (2.21b). There exists $\lambda _0 = \lambda _0(r, \tau _J, \tau _\omega , \gamma _J, \gamma _\omega , {\mathcal {J}}, \kappa , \rho , {\{\Vert N^{(\alpha )} \Vert _{0}\}_\alpha }, \Vert V\Vert _{\kappa , \rho }) >0$ such that $\forall \lambda > \lambda _0$, the following holds.

• (a) There exist a unitary time quasi-periodic transformation $Y_{\textrm{inv}}$ and a time quasi-periodic self-adjoint operator $H_{\textrm{inv}}$ such that

  $$\begin{aligned} U_{H_{\textrm{inv}}}(t) \;=\; Y_{\textrm{inv}}(\lambda \omega t) U_H(t) {Y_\textrm{inv}^*(0)} \end{aligned}$$

  (3.6)

  and

  $$\begin{aligned} H_{\textrm{inv}}(\lambda \omega t)=J \cdot N + Z_{\textrm{inv}}+V_{\textrm{inv}}(\lambda \omega t)\,, \end{aligned}$$

  (3.7)

  with

  • (a.i) $[Z_{\textrm{inv}}, N^{(\alpha )}] = 0\quad \forall \alpha = 1, \dots , r$;

  • (a.ii) $Z_{\textrm{inv}} \in \mathcal {O}^\texttt {s} _{\kappa _*}$, $\Vert Z_{\textrm{inv}} \Vert _{\kappa _*} \le {\frac{e}{e-1}} \lambda ^{-\frac{1}{4\tau _\omega }} \Vert V \Vert _{\kappa , \rho }$;

  • (a.iii) $V_{\textrm{inv}} \in \mathcal {O}^\texttt {s} _{\kappa _*, \rho _*}$, with $\Vert V_{\textrm{inv}} \Vert _{\kappa _{*}, \rho _*} \le e^{- \lambda ^{2\beta }} \Vert V \Vert _{\kappa ,\rho }$;

  • (a.iv) $\forall P \in \mathcal {O}_{\kappa _*, \rho _*},$ $\Vert Y_{\textrm{inv}}(\lambda \omega t) P Y_{\textrm{inv}}^*(\lambda \omega t)-P \Vert _{\kappa _{*}, \rho _*} \le {\frac{2e}{e-1}} \lambda ^{-\frac{1}{8 \tau _\omega }} \Vert P\Vert _{\kappa _*, \rho _*}$.

• (b) There exist a unitary time quasi-periodic transformation $Y_{\textrm{obs}}$ and a time quasi-periodic self-adjoint operator $H_{\textrm{obs}}$ such that

  $$\begin{aligned} U_{H_{\textrm{obs}}}(t) \;=\; Y_{\textrm{obs}}(\lambda \omega t) U_H(t) \end{aligned}$$

  (3.8)

  and

  $$\begin{aligned} H_{\textrm{obs}}(\lambda \omega t)=J \cdot N + Z_{\textrm{obs}}+V_{\textrm{obs}}(\lambda \omega t)\,, \end{aligned}$$

  (3.9)

  with

  • (b.i) $Y_{\textrm{obs}}(0) = \mathbb {1}$;

  • (b.ii) $Z_{\textrm{obs}} \in \mathcal {O}^\texttt {s} _{\kappa _*}$, $\Vert Z_{\textrm{obs}} \Vert _{\kappa _*} \le {\frac{{e}}{e-1}} \lambda ^{-\frac{1}{8 \tau _\omega }} \Vert V \Vert _{\kappa , \rho }$;

  • (b.iii) $V_{\textrm{obs}} \in \mathcal {O}^\texttt {s} _{\kappa _*, \rho _*}$, with $\Vert V_{\textrm{obs}} \Vert _{\kappa _{*}, \rho _*} \le e^{- \lambda ^{\beta }} \Vert V \Vert _{\kappa ,\rho },$;

  • (b.iv) $\forall P \in \mathcal {O}_{\kappa _*, \rho _*},$ $\Vert Y_{\textrm{obs}}(\lambda \omega t) P Y_{\textrm{obs}}^*(\lambda \omega t)-P \Vert _{\kappa _{*}, \rho _*} \le {\frac{2{e}}{e-1}} \lambda ^{-\frac{1}{8 \tau _\omega }} \Vert P\Vert _{\kappa _*, \rho _*}\,$. More precisely, there exist $n^\star \in \mathbb N$ and self-adjoint operators $G^{(0)}_\textrm{obs}, \dots , G^{({n^\star -1})}_\textrm{obs}\in \mathcal {O}^\texttt {s} _{\kappa _*, \rho _*}$ such that $Y_\textrm{obs}(\lambda \omega t) = e^{-\textrm{i}G^{({n^\star -1})}_\textrm{obs}(\lambda \omega t)} \cdots e^{-\textrm{i}G^{(0)}_\textrm{obs}(\lambda \omega t)}$, with $\Vert G^{(n)}_\textrm{obs}\Vert _{\kappa _*, \rho _*} \le \lambda ^{-\frac{1}{8 \tau _\omega }} e^{-n} \Vert V\Vert _{\kappa , \rho }$ for any n.

Here $\beta$ is the constant defined in (2.23) and $\kappa _*, \rho _*$ are defined as in (3.5).

Remark 3.4

The result in Item (b) of Proposition 3.3 does not depend on the particular structure $J \cdot N$ of the unperturbed Hamiltonian; furthermore, with minor changes in the regularity parameters, it is proven in [25]. Here we state it in the particular case of $H_0 = J \cdot N$ only for self-consistency. The same does not hold true for Item (b) of Proposition 3.1, which instead explicitly relies on the structure of $H_0$ and is one of the novelties of the present work.

3.1 Ideas of the Normal Form Proofs

The class of models we deal with is described by the Hamiltonian

$$\begin{aligned} H(\nu t) \;=\; J \cdot N + W(\nu t) \end{aligned}$$

(3.10)

where $\nu \in \mathbb R^m$ and $W(\nu t)$ can be either a small perturbation (i.e. $W(\nu t)=\varepsilon V(\omega t)$, $\varepsilon \ll 1$) or a fast-forcing one (i.e. $W(\nu t)=V(\lambda \omega t)$, $\lambda \gg 1$).

We look for a coordinate transformation of the form $Y_1(\nu t)=e^{\textrm{i}G_0(\nu t)}$ generated by a quasi-local self-adjoint operator $G_0(\nu t) \in \mathcal {O}^\texttt {s}$ which reduces the size of the perturbation W. If $G_0$ is small (which will be the case), denoting by $H_1$ the transformed Hamiltonian (see Lemma 5.2), we have

$$\begin{aligned} H_1(\varphi )=J\cdot N +\nu \cdot \partial _\varphi G_0(\varphi ) + W(\varphi ) + \text {h.o.t.} \end{aligned}$$

(3.11)

with $\varphi :=\nu t$, where h.o.t. denotes terms of higher order in the perturbative parameter. The core of the procedure is to solve for all $\varphi$ the so-called homological equation:

$$\begin{aligned} \textrm{i}[J\cdot N,G_0(\varphi )]+ \nu \cdot \partial _\varphi G_0(\varphi )+ W(\varphi )-Z_1=0 \end{aligned}$$

(3.12)

in the unknowns $G_0(\varphi )$ and $Z_1$, for a suitable time independent operator $Z_1$ (thus, independent of $\varphi$).

For our scopes, we have two possible solutions, depending on an additional requirement on the solution to the homological equation: either we require $[Z_1,N^{(\alpha )}]=0$ for any $\alpha = 1, \dots , r$, which leads to the normal form $H_\textrm{inv}$, or we require $G(0)=0$, and this leads to the normal form $H_\textrm{obs}$. (Both requirements are not compatible, in general, see Remark 3.2.)

To solve equation (3.12) we follow the ideas in [5,6,7], and we add an additional parametric dependence to our operators: we define

$$\begin{aligned} \mathcal {G}_0(\theta ,\varphi )\;:=\; e^{\textrm{i}\theta \cdot N} G_0(\varphi ) e^{-\textrm{i}\theta \cdot N} \, , \qquad \mathcal {W}(\theta ,\varphi ) \;:=\; e^{\textrm{i}\theta \cdot N} W(\varphi ) e^{-\textrm{i}\theta \cdot N} \, , \end{aligned}$$

(3.13)

and we look for a solution of

$$\begin{aligned} \textrm{i}[J\cdot N,\mathcal {G}_0(\theta ,\varphi )]+ \nu \cdot \partial _\varphi \mathcal {G}_0(\theta ,\varphi )+ \mathcal {W}(\theta ,\varphi )-Z_1=0 \end{aligned}$$

(3.14)

for any $\theta \in \mathbb {T}^r$. Clearly, the solution $\mathcal {G}_0(\theta ,\varphi )$ we find for any $\theta$, is a solution to (3.13) once evaluated at $\theta =0$. Since the spectrum of each of the number operators is integer, i.e. $\sigma (N^{(\alpha )})\subset \mathbb {Z}$ $\forall \alpha$, the dependence on $\theta$ of all the operators is quasi-periodic, allowing us to introduce their Fourier series based on the local structures as

$$\begin{aligned} \mathcal {G}_0(\theta ,\varphi )=\sum _{S \in \mathcal {P}_c(\Lambda )} \sum _{l \in \mathbb {Z}^m} \sum _{k \in \mathbb {Z}^r } \big ( \widehat{\mathcal {G}_{0, S}} \big )_{l,k} \, e^{\textrm{i}k \cdot \theta } e^{\textrm{i}l \cdot \varphi } \, , \end{aligned}$$

(3.15)

and analogously for $\mathcal {W}$. Then one solves (3.14) exploiting that $\textrm{i}[J\cdot N, \mathcal {G}_0(\theta ,\varphi )] = J \cdot \partial _\theta \mathcal {G}_0(\theta ,\varphi )$ and passing to Fourier coefficients. In the solution of (3.14), one has to deal with small denominators which are controlled differently in the small perturbation and fast forcing case. As mentioned in the Introduction, in the small perturbation case one imposes a Diophantine condition on the vector $(J,\omega )$ while on the fast-forcing case, one requires that J and $\omega$ are Diophantine separately, and that $\lambda$ is large enough to separate the energy scales of the perturbation and the ones of the unperturbed Hamiltonian. Once equation (3.12) has been solved, we are left with $H_1(t) = J \cdot N + Z_1 + W_1(\nu t)$, with $W_1(\nu t)$ smaller in size, and we iterate the procedure a finite number of times to further reduce the size of the time dependent part.

4 Strongly Local Operators and Their Properties

4.1 General Properties

We start with the following result, which relates operator and local norms:

Lemma 4.1

Let $t \mapsto A(\omega t)$ be such that $A \in \mathcal {O}_{0, 0}$. Then

$$\begin{aligned} \sup _{t \in \mathbb R}\Vert A(\omega t)\Vert _{\textrm{op}} \le |\Lambda | \Vert A\Vert _{0,0}\,. \end{aligned}$$

(4.1)

Furthermore, if $\forall t$ $A(\omega t) = \sum _{S'\subseteq S} A_{S'}(\omega t)$ for some $S \in \mathcal {P}_c(\Lambda )$, one has

$$\begin{aligned} \sup _{t \in \mathbb R} \Vert A(\omega t)\Vert _{\textrm{op}} \le |S| \Vert A\Vert _{0,0}\,. \end{aligned}$$

(4.2)

Proof

If $A \in \mathcal {O}_{0,0}$, one has

$$\begin{aligned} \sup _{t\in \mathbb R} \Vert A(\omega t)\Vert _{\textrm{op}}&= \sup _{t \in \mathbb R} \left\| \sum _{S \in \mathcal {P}_c(\Lambda )} \sum _{l \in \mathbb Z^m} (\widehat{A}_S)_l e^{\textrm{i}l \cdot \omega t}\right\| _{\textrm{op}} \le \sum _{x \in \Lambda } \sum _{\begin{array}{c} S \in \mathcal {P}_c(\Lambda ) \\ S \ni x \end{array}} \sum _{l \in \mathbb Z^m} \Vert (\widehat{A}_S)_l\Vert _{\textrm{op}}\\&\le |\Lambda | \sup _{x \in \Lambda } \sum _{\begin{array}{c} S \in \mathcal {P}_c(\Lambda ) \\ S \ni x \end{array}} \sum _{l \in \mathbb Z^m} \Vert (\widehat{A}_S)_l\Vert _{\textrm{op}} = |\Lambda | \Vert A\Vert _{0,0}\,. \end{aligned}$$

Suppose now $A = \sum _{S' \subseteq S} A_{S'}$; one has

$$ \begin{aligned} \sup _{t\in \mathbb R} \Vert A(\omega t)\Vert _{\textrm{op}}&= \sup _{t \in \mathbb R} \left\| \sum _{\begin{array}{c} S' \in \mathcal {P}_c(\Lambda ) \\ S' \subseteq S \end{array}} \sum _{l \in \mathbb Z^m} (\widehat{A}_{S'})_l e^{\textrm{i}l \cdot \omega t}\right\| _{\textrm{op}} \le \sum _{x \in S} \sum _{\begin{array}{c} S' \subseteq S \\ S' \ni x \end{array}} \sum _{l \in \mathbb Z^m} \Vert (\widehat{A}_{S'})_l\Vert _{\textrm{op}}\\&\le |S| \sup _{x \in \Lambda } \sum _{\begin{array}{c} S' \in \mathcal {P}_c(\Lambda ) \\ S' \ni x \end{array}} \sum _{l \in \mathbb Z^m} \Vert (\widehat{A}_{S'})_l\Vert _{\textrm{op}} = |S| \Vert A\Vert _{0,0}\,. \end{aligned} $$

$\square$

The normal form procedure described in Sect. 3.1 consists in a conjugation of the Hamiltonian with coordinate transformations of the form $e^{\textrm{i}G(\nu t)}$, with G being a family of strongly exponentially localized self-adjoint operators. Thus, considering a generic $A \in \mathcal {O}^\texttt {s} _{\kappa ,\rho }$, we need to control locality properties of the quantity

$$\begin{aligned} e^{\textrm{i}G(\nu t)} A (\nu t) e^{-\textrm{i}G(\nu t)} \;=\; \sum _{q=0}^\infty \frac{(-\textrm{i})^q}{q!} \textrm{Ad}_{ G(\nu t)}^q A(\nu t) \, , \end{aligned}$$

(4.3)

where we have defined

$$\begin{aligned} \textrm{Ad}^0_{B} A := A \, ,\qquad \textrm{Ad}^q_{B} A := [\textrm{Ad}^{q-1}_B A , B] \qquad \forall q \ge 1, \quad \forall A,B \in \mathcal {O}_{\kappa ,\rho }\, . \end{aligned}$$

(4.4)

This is done in Lemma 4.2 below. We point out that an analogous result to Lemma 4.2, with different quantitative estimates, was given in [1] for time periodic operators. For this reason, here we only state it and we postpone a detailed proof of it, suited to our setting, to Appendix A.

Lemma 4.2

(see also Lemma 4.1 of [1]) Let $A, B \in \mathcal {O}^\texttt {s} _{\kappa + \sigma , \rho }$ for some $\kappa , \rho \ge 0$ and $\sigma >0$, and assume that there exists $\eta \in (0, 1)$ such that

$$\begin{aligned} \frac{4 e^{-\kappa }\Vert A\Vert _{\kappa + \sigma , \rho }}{\sigma } \le \eta \,. \end{aligned}$$

(4.5)

Then,

$$\begin{aligned} \Big \Vert e^{A} B e^{-A}-B\Big \Vert _{\kappa , \rho } \le C e^{-\kappa } \frac{1}{\sigma } \Vert A\Vert _{\kappa + \sigma , \rho } \Vert B \Vert _{\kappa + \sigma , \rho }\,, \end{aligned}$$

(4.6)

$$\begin{aligned} \Big \Vert e^{A} B e^{-A}-B-[A,B] \Big \Vert _{\kappa , \rho } \le 4 C e^{-2\kappa } \frac{1}{\sigma ^2} \Vert A\Vert ^2_{\kappa + \sigma , \rho } \Vert B \Vert _{\kappa + \sigma , \rho }\,, \end{aligned}$$

(4.7)

with $C=\frac{4}{1-\eta }>0$.

4.2 Conjugation with the Number Operators

In the normal form construction we need to exploit the notion of strong locality to obtain that for a strongly local operator A,

$$\begin{aligned} \big (e^{\textrm{i}\theta \cdot N} A e^{-\textrm{i}\theta \cdot N} \big )_{S}\;=\;e^{\textrm{i}\theta \cdot N} A_S e^{-\textrm{i}\theta \cdot N} \, . \end{aligned}$$

(4.8)

The aim of this Subsection is to prove (4.8) and some properties of $e^{\textrm{i}\theta \cdot N} A e^{-\textrm{i}\theta \cdot N}$. First we prove that the series $\sum _{q} \frac{(-\textrm{i})^q}{q!} \textrm{Ad}_{\theta \cdot N}^q A$ converges uniformly in $\theta$ (but with a bad dependence on the volume $|\Lambda |$):

Lemma 4.3

Fix $\alpha \in \{1, \dots , r\}$ and suppose that $A = A_S$ is a strongly local operator. Then $\forall q \in \mathbb N$ ${\text {Ad}}^q_{N^{(\alpha )}} A_S$ is a strongly local operator, supported on S only, and $\forall \alpha$

$$\begin{aligned} \left\| {\text {Ad}}_{N^{(\alpha )}}^q A_S \right\| _{\textrm{op}} = \left\| \left( {\text {Ad}}_{N^{(\alpha )}}^q A_S\right) _S\right\| _{\textrm{op}}\le 2^q |S|^q \Vert N^{(\alpha )}\Vert ^q_{0} \Vert {A_S}\Vert _{\textrm{op}} \le 2^q |\Lambda |^q \Vert N^{(\alpha )}\Vert ^q_{0} \Vert {A_S}\Vert _{\textrm{op}}\,. \end{aligned}$$

(4.9)

Proof

We inductively prove that the first inequality in (4.9) holds, with the second one trivially following from $|S| \le |\Lambda |.$ If $q=1$, one has

$$\begin{aligned} {\text {Ad}}^1_{N^{(\alpha )}} A_S = \left[ A_S, N^{(\alpha )}\right] = \sum _{S' \subseteq S} \left[ A_S, N^{(\alpha )}_{S'}\right] \,, \end{aligned}$$

(4.10)

thus $\text {Ad}^1_{N^{(\alpha )}} A_S$ is localized on S. We now observe that, by (4.10), one also has

$$\begin{aligned} \begin{aligned} \Vert {\text {Ad}}^1_{N^{(\alpha )}} A_S\Vert _{\textrm{op}}&\le 2 \sum _{ S' \subseteq S} \Vert A_S\Vert _{\textrm{op}} \Vert N^{(\alpha )}_{S'}\Vert _{\textrm{op}}\le 2 \Vert A_S\Vert _{\textrm{op}} \sum _{S' \subseteq S} \Vert N^{(\alpha )}_{S'}\Vert _{\textrm{op}}\\&\le 2 \Vert A_S\Vert _{\textrm{op}} \sum _{x \in S} \sum _{\begin{array}{c} S' \ni x \\ S' \subseteq S \end{array}} \Vert N^{(\alpha )}_{S'}\Vert _{\textrm{op}}\\&\le 2 \Vert A_S\Vert _{\textrm{op}} |S| \sup _{x \in \Lambda } \sum _{S' \ni x} \Vert N^{(\alpha )}_{S'}\Vert _{\textrm{op}} = 2 \Vert A_S\Vert _{\textrm{op}} |S| \Vert N^{(\alpha )}\Vert _{0}\,, \end{aligned} \end{aligned}$$

which gives (4.9) for $q=1$. Now let us suppose that ${\text {Ad}}^{q}_{N^{(\alpha )}} A_S$ is a local operator acting within S and that (4.9) holds for some q; then one has

$$ \begin{aligned} {\text {Ad}}^{q+1}_{N^{(\alpha )}} A_S = \left[ {\text {Ad}}^{q}_{N^{(\alpha )}} A_S, N^{(\alpha )}\right] =\sum _{S' \subseteq S} \left[ {\text {Ad}}^{q}_{N^{(\alpha )}} A_S, N^{(\alpha )}_{S'}\right] \,, \end{aligned} $$

which shows that ${\text {Ad}}^{q+1}_{N^{(\alpha )}} A_S$ is a local operator acting on S; furthermore, arguing as above one obtains

$$ \begin{aligned} \Vert {\text {Ad}}^{q+1}_{N^{(\alpha )}} A_S\Vert _{\textrm{op}}&\le 2 \left\| {\text {Ad}}^{q}_{N^{(\alpha )}} A_S \right\| _{\textrm{op}} |S| \Vert N^{(\alpha )}\Vert _{0}\\&\le 2^{q+1} |S|^{q+1} \Vert N^{(\alpha )}\Vert ^{q+1}_{0} \Vert A_S\Vert _{\textrm{op}}\,, \end{aligned} $$

which gives (4.9) for $q+1$. The fact that $\textrm{Ad}_{N^{(\alpha )}}^q A_S$ is strongly local follows as in the proof of Lemma 4.2, using the fact that the operators $N^{(\alpha )}$ are strongly local. $\square$

As a second step, we have the following.

Lemma 4.4

Suppose that A is a quasi-periodic family of strongly local operators, with

$$ A(\varphi ) = \sum _{S \in \mathcal {P}_c(\Lambda )} A_S(\varphi )\,, $$

then $\forall \theta \in \mathbb R^r$

$$\begin{aligned} \mathcal {A}(\theta , \varphi ):= e^{\textrm{i}\theta \cdot N} A(\varphi ) e^{-\textrm{i}\theta \cdot N} = \sum _{S \in \mathcal {P}_c(\Lambda )} \left( e^{\textrm{i}\theta \cdot N} A(\varphi ) e^{-\textrm{i}\theta \cdot N}\right) _S\, \end{aligned}$$

(4.11)

is a strongly local operator and $\forall S$

$$\begin{aligned} \left( e^{\textrm{i}\theta \cdot N} A(\varphi ) e^{-\textrm{i}\theta \cdot N}\right) _S = e^{\textrm{i}\theta \cdot N} \left( A(\varphi )\right) _S e^{-\textrm{i}\theta \cdot N} \, . \end{aligned}$$

(4.12)

Proof

Since the operators $\{N^{(\alpha )}\}_{\alpha = 1}^r$ mutually commute, we observe that

$$ e^{\textrm{i}\theta \cdot N} A(\varphi ) e^{-\textrm{i}\theta \cdot N} = e^{\textrm{i}\theta _r N^{(r)}} \cdots e^{\textrm{i}\theta _1 N^{(1)}} A(\varphi ) e^{-\textrm{i}\theta _1 N^{(1)}} \cdots e^{-\textrm{i}\theta _r N^{(r)}}\,. $$

Define for any $\alpha = 1, \dots , r$ $\mathcal {A}_{\alpha }(\theta , \varphi ):= e^{\textrm{i}\theta _\alpha N^{(\alpha )}} \mathcal {A}_{\alpha -1}(\varphi ) e^{-\textrm{i}\theta _\alpha N^{(\alpha )}}$, $\mathcal {A}_{0}(\theta , \varphi ) = A(\varphi )$, so that $\mathcal {A}_r(\theta , \varphi ) = \mathcal {A}(\varphi )$; we now use Lemma 4.3 to show inductively on $\alpha$ that

$$\begin{aligned} \begin{aligned} \mathcal {A}_{\alpha }(\theta , \varphi )&= \sum _{S \in \mathcal {P}_c(\Lambda )} \left( \mathcal {A}_{\alpha }(\theta , \varphi )\right) _S \,,\\ \left( \mathcal {A}_{\alpha }(\theta , \varphi )\right) _S&= e^{\textrm{i}\theta _\alpha N^{(\alpha )}} \left( \mathcal {A}_{\alpha -1}(\theta ,\varphi )\right) _S e^{-\textrm{i}\theta _\alpha N^{(\alpha )}} \in \mathscr {B}(\mathcal {H}_\Lambda ) \quad \forall S \in \mathcal {P}_c(\Lambda )\,, \end{aligned} \end{aligned}$$

(4.13)

and that $\forall S \in \mathcal {P}_c(\Lambda )$, $\left( \mathcal {A}_{\alpha }(\theta , \varphi )\right) _S$ is a strongly local operator acting within S. Once (4.13) has been proven, one recursively deduces that

$$ e^{\textrm{i}\theta \cdot N} A(\theta , \varphi ) e^{-\textrm{i}\theta \cdot N}= \mathcal {A}_{r}(\theta , \varphi ) = \sum _{S \in \mathcal {P}_c(\Lambda )} \left( \mathcal {A}_{r}(\theta , \varphi )\right) _S = \sum _{S \in \mathcal {P}_c(\Lambda )} e^{\textrm{i}\theta \cdot N} \left( A(\varphi )\right) _S e^{-\textrm{i}\theta \cdot N}\,, $$

with the operators $e^{\textrm{i}\theta \cdot N} \left( A(\varphi )\right) _S e^{-\textrm{i}\theta \cdot N} \in \mathscr {B}(\mathcal {H}_\Lambda )$ being strongly local operators acting within S, which gives the thesis.

We now prove (4.13). First of all, we observe that

$$\begin{aligned} \begin{aligned} \mathcal {A}_{\alpha }(\theta , \varphi )&= e^{\textrm{i}\theta _\alpha N^{(\alpha )}} \mathcal {A}_{\alpha -1}(\theta , \varphi ) e^{-\textrm{i}\theta _\alpha N^{(\alpha )}}\\&= \sum _{q \ge 0} \frac{(-\textrm{i}\theta _\alpha )^q}{q!} {\text {Ad}}^q_{N^{(\alpha )}} \mathcal {A}_{\alpha -1}(\theta , \varphi ) \\&= \sum _{q \ge 0} \frac{(-\textrm{i}\theta _\alpha )^q}{q!} \sum _{S \in \mathcal {P}_c(\Lambda )} {\text {Ad}}^q_{N^{(\alpha )}} \left( \mathcal {A}_{\alpha -1}(\theta , \varphi )\right) _S\,. \end{aligned} \end{aligned}$$

(4.14)

Since by inductive hypothesis $\left( \mathcal {A}_{\alpha -1}(\theta , \varphi )\right) _S \in \mathscr {B}(\mathcal {H}_\Lambda )$ is a strongly local operator acting within S, by Lemma 4.3 with A replaced by $\left( \mathcal {A}_{\alpha -1}(\theta , \varphi )\right) _S$ one has, $\forall S$

$$\begin{aligned} e^{\textrm{i}\theta _\alpha N^{(\alpha )}} \left( \mathcal {A}_{\alpha -1}(\theta , \varphi )\right) _S e^{-\textrm{i}\theta _\alpha N^{(\alpha )}} =\sum _{q \ge 0} \frac{(-\textrm{i}\theta _\alpha )^q}{q!} {\text {Ad}}^q_{N^{(\alpha )}} \left( \mathcal {A}_{\alpha -1}(\theta , \varphi )\right) _S \in \mathscr {B}(\mathcal {H}_\Lambda )\,, \end{aligned}$$

(4.15)

and $e^{\textrm{i}\theta _\alpha N^{(\alpha )}} \left( \mathcal {A}_{\alpha -1}(\theta , \varphi )\right) _S e^{-\textrm{i}\theta _\alpha N^{(\alpha )}}$ is a strongly local operator acting within S, since by Lemma 4.3 the operators ${\text {Ad}}^q_{N^{(\alpha )}} \left( \mathcal {A}_{\alpha -1}(\theta , \varphi )\right) _S$ are strongly local and act within S, and the series in (4.15) is convergent, again due to Lemma 4.3. We are allowed to exchange the sum over S and over q in (4.14) and then (4.13) follows immediately. $\square$

As a consequence, the following is the main technical result we are using in the analysis of the homological equation.

Lemma 4.5

Given $\kappa , \sigma , \rho >0$ and $A \in \mathcal {O}^{\texttt {s} }_{\kappa + \sigma , \rho }$, let

$$\begin{aligned} \mathcal {A}(\theta , \varphi ) := e^{\textrm{i}\theta \cdot N} A(\varphi ) e^{-\textrm{i}\theta \cdot N}\,, \quad \theta \in \mathbb {R}^r\,. \end{aligned}$$

(4.16)

Then

• (i) $\forall \theta \in \mathbb R^r$, $\mathcal {A}(\theta , \cdot ) \in \mathcal {O}^\texttt {s} _{\kappa + \sigma , \rho }$, with $\Vert \mathcal {A}(\theta , \cdot )\Vert _{\kappa + \sigma , \rho } = \Vert A\Vert _{\kappa + \sigma , \rho }$.

• (ii) $\forall S \in \mathcal {P}_c(\Lambda )$, $\mathcal {A}_S$ is periodic w.r.t. $\theta ,$ namely $\mathcal {A}_S(\theta + 2\pi e_j, \varphi ) = \mathcal {A}_S(\theta , \varphi )$ $\forall j =1, \dots , r$. In particular, $\mathcal {A}$ admits the representation

  $$\begin{aligned} \begin{aligned} \mathcal {A}(\theta , \varphi )&= \sum _{S \in \mathcal {P}_c(\Lambda )} \sum _{k \in \mathbb Z^r} \sum _{l \in \mathbb Z^m} (\widehat{\mathcal {A}_S})_{l,k} \, e^{\textrm{i}k \cdot \theta } e^{\textrm{i}l \cdot \varphi }\,, \\ (\widehat{\mathcal {A}_S})_{l,k}&:= \frac{1}{(2\pi )^{r+m}} \int _{\mathbb {T}^m}\int _{\mathbb {T}^r} \big (\mathcal {A}(\theta , \varphi )\big )_{S} e^{-\textrm{i}\varphi \cdot l} e^{-\textrm{i}\theta \cdot k} \, \textrm{d}\theta \textrm{d}\varphi . \end{aligned} \end{aligned}$$

  (4.17)

  for all $S \in \mathcal {P}_c(\Lambda ), l \in \mathbb Z^r, k \in \mathbb Z^m.$

• (iii) Defining

  $$\begin{aligned} \Vert \mathcal {A}\Vert _{\kappa , \rho , \zeta } := \sup _{x \in \Lambda } \sum _{S \in \mathcal {P}_c(\Lambda )} \sum _{k \in \mathbb Z^r} \sum _{l \in \mathbb Z^m} \big \Vert (\widehat{\mathcal {A}_S})_{l,k}\big \Vert _{\textrm{op}} e^{\zeta |k|} e^{\rho |l|} e^{\kappa |S|}\,, \end{aligned}$$

  (4.18)

  $\forall \zeta >0$ such that

  $$\begin{aligned} \frac{4 \zeta \max _{\alpha } \Vert N^{(\alpha )}\Vert _{0}}{\sigma } \le \frac{1}{2}\,, \end{aligned}$$

  (4.19)

  one has

  $$\begin{aligned} \Vert \mathcal {A}\Vert _{\kappa , \rho , \zeta } \le 2^{2r}\left( { \frac{1+\zeta }{\zeta }}\right) ^r\Vert A\Vert _{\kappa + r\sigma , \rho }\,, \end{aligned}$$

  (4.20)

  and

  $$\begin{aligned} \Vert A\Vert _{\kappa , \rho } \le \Vert \mathcal {A}\Vert _{\kappa , \rho , 0} \le \Vert \mathcal {A}\Vert _{\kappa , \rho , \zeta }\,. \end{aligned}$$

  (4.21)

In the following, given $\kappa , \rho , \zeta \ge 0$, we will denote

$$\begin{aligned} \mathcal {O}^\texttt {s} _{\kappa , \rho , \zeta } :=\big \{ \mathcal {A} : \mathbb {T}^r\times \mathbb {T}^m \rightarrow \mathscr {B}(\mathcal {H}_\Lambda )\ |\ \forall (\theta , \varphi )\quad \mathcal {A}(\theta , \varphi ) \in \mathcal {O}^\texttt {s} _\kappa \ \text { and }\Vert \mathcal {A}\Vert _{\kappa , \rho , \zeta } < \infty \big \}\,, \end{aligned}$$

(4.22)

where $\Vert \cdot \Vert _{\kappa , \rho , \zeta }$ is defined as in (4.18).

Proof of Lemma 4.5

By Lemma 4.4, one has $\mathcal {A}(\theta , \varphi ) = \sum _{S \in \mathcal {P}_c(\Lambda ) } \left( \mathcal {A}(\theta , \varphi )\right) _S$,

$$\begin{aligned} \begin{aligned} \left( \mathcal {A}(\theta , \varphi )\right) _S&= e^{\textrm{i}\theta \cdot N} \left( A(\varphi )\right) _S e^{-\textrm{i}\theta \cdot N} \quad \forall S \in \mathcal {P}_c(\Lambda )\,, \end{aligned} \end{aligned}$$

(4.23)

with $\left( \mathcal {A}(\theta , \varphi )\right) _S$ strongly local operators acting within S. Since the operators $N^{(\alpha )}$ are independent of $\varphi$, one has that the operators $\left( \mathcal {A}(\theta , \cdot )\right) _S$ are still periodic w.r.t $\varphi$, and

$$\begin{aligned} (\widehat{(\mathcal {A}(\theta , \cdot ))_S})_{l} = e^{\textrm{i}\theta \cdot N} (\widehat{A_S})_{l} e^{-\textrm{i}\theta \cdot N} \quad \forall l \in \mathbb Z^m\,. \end{aligned}$$

(4.24)

Thus, one has $\forall \theta \in \mathbb R^r$

and Item (i) is proven.

Item (ii) follows from spectral theorem and from the fact that the spectrum of the operators $N^{(\alpha )}$ is integer.

It remains to prove Item (iii). First note that, since $A(\,\cdot \,) = \mathcal {A}(0,\,\cdot \,)$ and $\sup _{\theta \in \mathbb T^r} \Vert \mathcal {A}(\theta , \cdot ) \Vert _{\kappa , \rho } \le \Vert \mathcal {A}\Vert _{\kappa , \rho , 0}$, one has $\Vert A\Vert _{\kappa , \rho } = \Vert \mathcal {A}(0, \cdot )\Vert _{\kappa , \rho } \le \Vert \mathcal {A}\Vert _{\kappa , \rho , 0}$. Thus, using the monotonicity of $\Vert \cdot \Vert _{\kappa , \rho , \zeta }$ in $\zeta$, (4.21) is proven. We now prove (4.20).

We observe that, by (4.17), $\forall k \in \mathbb {Z}^r$ one has

$$ (\widehat{\mathcal {A}_S})_{l,k} = \frac{1}{(2\pi )^r}\int _{\mathbb {T}^r} e^{\textrm{i}\theta \cdot N} (\widehat{A_S})_{l}\, e^{-\textrm{i}\theta \cdot N} e^{-\textrm{i}\theta \cdot k} \, \textrm{d}\theta \,. $$

We first claim that, $\forall k \in \mathbb {Z}^r$, $\forall l \in \mathbb {Z}^m$ and $\forall S \in \mathcal {P}_c(\Lambda )$ and $\forall \zeta > 0$ one has

$$\begin{aligned} \Vert (\mathcal {A}_S)_{l,k} \Vert _{\textrm{op}}\le & {} e^{-2 \zeta |k|_1} \sup _{\upsilon \in \mathbb Z^r, |\upsilon |_\infty {\le 1}} \left\{ \left\| e^{2\zeta \upsilon \cdot N} (\widehat{{A}_S})_{l} e^{-2\zeta \upsilon \cdot N}\right\| _{\textrm{op}} \right\} \end{aligned}$$

(4.25)

$$\begin{aligned}\le & {} e^{-2 \zeta |k|_1} 2^r \Vert (\widehat{A_S})_l \Vert _{\textrm{op}} e^{r \sigma |S|} \,. \end{aligned}$$

(4.26)

Then, one concludes as follows:

$$\begin{aligned} \begin{aligned} \left\| \mathcal {A}\right\| _{\kappa , \rho , \zeta }&= \sup _{x \in \Lambda } \sum _{\begin{array}{c} S \in \mathcal {P}_c(\Lambda ) \\ x \in S \end{array}} \sum _{k \in \mathbb Z^r} \sum _{l \in \mathbb Z^m} \Vert (\widehat{\mathcal {A}_S})_{l,k} \Vert _{\textrm{op}} e^{\zeta |k|} e^{\rho |l|} e^{\kappa |S|}\\&\le \sup _{x \in \Lambda } \sum _{\begin{array}{c} S \in \mathcal {P}_c(\Lambda ) \\ x \in S \end{array}} \sum _{k \in \mathbb Z^r} \sum _{l \in \mathbb Z^m} 2^r \Vert (\widehat{{A}_S})_{l}\Vert _{\textrm{op}} e^{-2\zeta |k|_1 + \zeta |k|} e^{\rho |l|} e^{(r \sigma + \kappa ) |S|}\\&\le \sup _{x \in \Lambda } \sum _{\begin{array}{c} S \in \mathcal {P}_c(\Lambda ) \\ x \in S \end{array}} \sum _{k \in \mathbb Z^r} \sum _{l \in \mathbb Z^m} 2^r \Vert (\widehat{{A}_S})_{l}\Vert _{\textrm{op}} e^{-\zeta |k|_1} e^{\rho |l|} e^{(r \sigma + \kappa ) |S|}\\&= \frac{2^{2 r}}{(1-e^{-\zeta })^r}\Vert A\Vert _{\kappa + r \sigma , \rho } \le 2^{2r} \left( \frac{\zeta +1}{\zeta }\right) ^r \Vert A \Vert _{\kappa +r\sigma ,\rho }\,. \end{aligned} \end{aligned}$$

It remains to prove (4.25) and (4.26).

To prove (4.25), one has to estimate in operator norm

$$ (\widehat{\mathcal {A}_S})_{l,k} = \frac{1}{(2\pi )^r} \int _0^{2\pi } \dots \int _0^{2\pi } e^{\textrm{i}\theta \cdot N} (\widehat{A_S})_{l} e^{-\textrm{i}\theta \cdot N} e^{-\textrm{i}\theta \cdot k} \, \textrm{d}\theta _1 \dots \textrm{d}\theta _r \,. $$

This is done using the fact that $(\widehat{\mathcal {A}_S})_{l,k}$ is analytic on a strip of width $2 \zeta$ around the real segment $[0,2\pi ]$ (with periodic boundary conditions) and using Cauchy Theorem on integration of analytic functions. For simplicity let us consider the case $r=1$ and suppose $k_1>0$, one has

$$\begin{aligned} (\widehat{\mathcal {A}_S})_{l,k}&= \frac{1}{2\pi } \int _{0}^{2\pi } e^{\textrm{i}\theta _1 N_1} (\widehat{A_S})_{l} e^{-\textrm{i}\theta _1 N_1} e^{-\textrm{i}\theta _1 k_1} \,\textrm{d}\theta _1 \\&= \frac{1}{2\pi } \int _{-2\textrm{i}\zeta }^{2\pi -2\textrm{i}\zeta } e^{\textrm{i}\theta _1 N_1} (\widehat{A_S})_{l} e^{-\textrm{i}\theta _1N_1} e^{-\textrm{i}\theta _1 k_1} \,\textrm{d}\theta _1\\&= \frac{1}{2\pi } \int _{0}^{2\pi } e^{\textrm{i}(\theta _1 - 2\textrm{i}\zeta ) N_1} (\widehat{A_S})_{l} e^{-\textrm{i}(\theta _1 - 2\textrm{i}\zeta ) N_1} e^{-\textrm{i}(\theta _1 - 2\textrm{i}\zeta ) k_1} \,\textrm{d}\theta _1 \, . \end{aligned}$$

Taking $\Vert \cdot \Vert _{\textrm{op}}$ norms on both sides, one gets

$$\begin{aligned} \left\| (\widehat{\mathcal {A}_S})_{l,k} \right\| _{\textrm{op}}&\le \frac{1}{2\pi } e^{- 2\zeta k_1} \int _{\mathbb {T}} \left\| e^{\textrm{i}\theta _1 N_1} e^{ 2 \zeta N_1} (\widehat{A_S})_{l} e^{-2 \zeta N_1} e^{-\textrm{i}\theta _1 N_1}\right\| _{\textrm{op}} \,\textrm{d}\theta _1\\&= \frac{1}{2\pi } e^{- 2\zeta k_1} \int _{\mathbb {T}^r} \left\| e^{ 2 \zeta N_1} (\widehat{A_S})_{l} e^{ -2 \zeta N_1} \right\| _{\textrm{op}} \,\textrm{d}\theta _1\\&\le e^{-2\zeta k_1} \left\| e^{ 2\zeta N_1} (\widehat{A_S})_{l} e^{ -2 \zeta N_1} \right\| _{\textrm{op}}\,. \end{aligned}$$

Repeating the argument in the general case yields

$$ \left\| (\widehat{\mathcal {A}_S})_{l,k} \right\| _{\textrm{op}} \le e^{-2\zeta |k|_1} \left\| e^{ 2\text {sign}(k) \zeta \cdot N} (\widehat{A_S})_{l} e^{- 2\text {sign}(k) \zeta \cdot N} \right\| _{\textrm{op}}\,, $$

where, for $a \in \mathbb R^r$, $\mathbb R^r \ni \text {sign}(a):= (\text {sign}(a_1), \dots , \text {sign}(a_r))$. Noting that $\upsilon :=\textrm{sign}(a)$ has $|\upsilon |_\infty \le 1$, this proves (4.25).

To prove (4.26), we argue as in the proof of Lemma 4.3. For any $\upsilon , S$ and l and any $\alpha = 0, \dots , r$ we define $\mathscr {A}_{\alpha }:= e^{2\zeta \upsilon _\alpha N^{(\alpha )}} \mathscr {A}_{\alpha -1} e^{-2\zeta \upsilon _\alpha N^{(\alpha )}}$, $\mathscr {A}_{0}:= (\widehat{{A}_S})_{l}$ (note that, as a consequence of Lemma 4.3, the $\mathscr {A}_{\alpha }$’s are strongly local operators acting within a certain $S \in \mathcal {P}_c(\Lambda )$ and they are not the operators $\mathcal {A}$ defined in the proof of Lemma 4.4) and we prove that

$$\begin{aligned} \left\| \mathscr {A}_{\alpha }\right\| _{\textrm{op}} \le 2 e^{\sigma |S|} \left\| \mathscr {A}_{\alpha -1}\right\| _{\textrm{op}} \,, \end{aligned}$$

(4.27)

and that $\mathscr {A}_\alpha$ is a local operator acting within S. Then, (4.26) will follow observing that $e^{2\zeta \upsilon \cdot N} (\widehat{{A}_S})_{l} e^{-2\zeta \upsilon \cdot N} = \mathscr {A}_{r}$, applying recursively (4.27) and taking the supremum over $\upsilon$.

To prove that (4.27) holds, for any $\alpha = 1, \dots , r$ we observe that

$$\begin{aligned} \begin{aligned} \left\| \mathscr {A}_{\alpha } \right\| _{\textrm{op}}&= \left\| e^{2\zeta \upsilon _\alpha N^{(\alpha )}} \mathscr {A}_{\alpha -1} e^{-2\zeta \upsilon _\alpha N^{(\alpha )}} \right\| _{\textrm{op}} = \Big \Vert \sum _{q \ge 0} \frac{(2\zeta \upsilon _\alpha )^q}{q!} {\text {Ad}}_{N^{(\alpha )}}^q \mathscr {A}_{\alpha -1} \Big \Vert _{\textrm{op}} \end{aligned} \end{aligned}$$

(4.28)

and, by Lemma 4.3, since $\mathscr {A}_{\alpha -1}$ is a local operator acting only within S, one has

$$\begin{aligned} \begin{aligned} \left\| \mathscr {A}_{\alpha } \right\| _{\textrm{op}}&\le \sum _{q \ge 0} \frac{(2\zeta \upsilon _\alpha )^q}{q!} (2 |S| \Vert N^{(\alpha )}\Vert _{0})^q \left\| \mathscr {A}_{\alpha -1}\right\| _{\textrm{op}}\\&\le \sum _{q \ge 0} \frac{(4\zeta \Vert N^{(\alpha )}\Vert _{0})^q}{q!} \left( \frac{q}{e\sigma }\right) ^q \left\| \mathscr {A}_{\alpha -1}\right\| _{\textrm{op}} e^{\sigma |S|}\\&\le \sum _{q \ge 0} \left( \frac{4\zeta \Vert N^{(\alpha )}\Vert _{0}}{\sigma }\right) ^q \left\| \mathscr {A}_{\alpha -1}\right\| _{\textrm{op}} e^{\sigma |S|}\le 2 \left\| \mathscr {A}_{\alpha -1}\right\| _{\textrm{op}} e^{\sigma |S|}\,, \end{aligned} \end{aligned}$$

(4.29)

where we have used Cauchy estimate, Stirling bound $q! \ge \left( \frac{q}{e}\right) ^q \sqrt{2\pi q}$ and the smallness assumption (4.19) on $\zeta$. Furthermore, again by Lemma 4.3 one has that ${\text {Ad}}_{N^{(\alpha )}}^q \mathscr {A}_{\alpha -1}$ is a strongly local operator acting within S, and since by (4.29) the series in (4.28) is convergent, also $\mathscr {A}_{\alpha }$ is a strongly local operator acting within S. $\square$

Lemma 4.6

An operator $\mathcal {A}(\theta ,\varphi )=\sum _{S \in \mathcal {P}_c(\Lambda )} \sum _{(k,l) \in \mathbb {Z}^{r+m}} (\widehat{\mathcal {A}_S})_{l,k} e^{\textrm{i}k \cdot \theta } e^{\textrm{i}l \cdot \varphi }$ is of the form $\mathcal {A}(\theta ,\varphi )=e^{\textrm{i}\theta \cdot N} A(\varphi ) e^{-\textrm{i}\theta \cdot N}$ if and only if for any $S \in \mathcal {P}_c(\Lambda )$, $l \in \mathbb {Z}^m$, $k \in \mathbb {Z}^r$ and $\alpha = 1, \dots , r$ one has

$$\begin{aligned}{}[N^{(\alpha )},(\widehat{\mathcal {A}_S})_{l,k}]=k_\alpha (\widehat{\mathcal {A}_S})_{l,k} \, . \end{aligned}$$

(4.30)

Proof

This lemma is proved by observing that $\mathcal {A}(\theta ,\varphi )$ is of the form $\mathcal {A}(\theta ,\varphi )=e^{\textrm{i}\theta \cdot N} A(\varphi ) e^{-\textrm{i}\theta \cdot N}$ if and only if for any $\alpha =1,\dots ,r$, one has $\frac{\textrm{d}}{\textrm{d}\theta _\alpha } (e^{-\textrm{i}\theta \cdot N} \mathcal {A}(\theta ,\varphi ) e^{\textrm{i}\theta \cdot N})=0$. $\square$

Since we are going to solve the homological equation using the Fourier representation of the operators, we need the guarantee that strong locality can be reconstructed back with anti-Fourier transform.

Lemma 4.7

Let $\mathcal {A}:\mathbb {T}^r \times \mathbb {T}^m \rightarrow \mathcal {A}(\theta , \varphi )$, with $\mathcal {A}(\theta , \varphi ) \in \mathcal {O}_{0,0,0}$ as in (4.17). Then $\mathcal {A}(\theta , \varphi )$ is strongly local $\forall (\theta , \varphi ) \in \mathbb {T}^{r+m}$ if and only if $(\widehat{\mathcal {A}_S})_{l,k}$ is strongly local $\forall (k, l) \in \mathbb Z^{r+m}$ $\forall S \in \mathcal {P}_c(\Lambda )$.

Proof

If $\forall (l,k) \in \mathbb Z^{r+m}$ $(\mathcal {\widehat{A}_S})_{l,k}$ is strongly local, then $\forall (\theta , \varphi )$ $\mathcal {A}$ is a convergent series of strongly local operators, and $\forall S' \nsubseteq S$

$$\begin{aligned}\left[ \mathcal {A}_{S} (\theta , \varphi ), N^{(\alpha )}_{S'}\right] = \left[ \sum _{k \in \mathbb Z^{r}, l \in \mathbb Z^{m}} (\widehat{\mathcal {A}_{S}})_{l,k} e^{\textrm{i}k \cdot \theta } e^{\textrm{i}l \cdot \varphi }, N^{(\alpha )}_{S'}\right] \\ = \sum _{k \in \mathbb Z^{r}, l \in \mathbb Z^m}\left[ (\widehat{\mathcal {A}_{S}})_{l,k}, N^{(\alpha )}_{S'}\right] e^{\textrm{i}k \cdot \theta } e^{\textrm{i}l \cdot \varphi } = 0\,. \end{aligned}$$

On the contrary, suppose that $\mathcal {A}(\theta , \varphi )$ is strongly local $\forall (\theta , \varphi )$ and let $S' \nsubseteq S$. One has

$$\begin{aligned} \left[ (\widehat{\mathcal {A}(\theta , \cdot )})_{S,l}, N^{(\alpha )}_{S'} \right]&= \frac{1}{(2\pi )^m}\int _{\mathbb {T}^m} [\mathcal {A}_S(\theta , \varphi ), N^{(\alpha )}_{S'}] e^{-\textrm{i}l \cdot \varphi }\,\textrm{d}\varphi = 0\,, \end{aligned}$$

and

$$ \begin{aligned} \left[ (\widehat{\mathcal {A}_S})_{l,k}, N^{(\alpha )}_{S'} \right]&= \frac{1}{(2\pi )^r}\int _{\mathbb {T}^r} \left[ (\widehat{(\mathcal {A}(\theta , \cdot ))_S})_{l}, N^{(\alpha )}_{S'}\right] e^{-\textrm{i}\theta \cdot k}\,\textrm{d}\theta = 0\,, \end{aligned} $$

which proves that $(\widehat{\mathcal {A}_S})_{l,k}$ is strongly local $\forall k, l$. $\square$

Last, let us see that the average of the operator $\mathcal {A}(\theta , \varphi )$ defined in (4.11) over $\varphi$ and $\theta$ commutes with all the number operators.

Lemma 4.8

Given $A \in \mathcal {O}^{\texttt{s}}_{\kappa , \rho }$ for some $\kappa , \rho >0$, let

$$\begin{aligned} \langle A \rangle := \frac{1}{(2\pi )^{r+m}} \int _{\mathbb {T}^r} \int _{\mathbb {T}^m} e^{\textrm{i}\theta \cdot N} A(\varphi ) e^{-\textrm{i}\theta \cdot N}\,\textrm{d}\varphi \textrm{d}\theta \,. \end{aligned}$$

(4.31)

Then, $\forall \alpha = 1, \dots , r$ one has $[\langle A \rangle , N^{(\alpha )}] = 0$ and

$$\begin{aligned} \Vert \langle A \rangle \Vert _{\kappa } \le \Vert A\Vert _{\kappa , \rho }\,. \end{aligned}$$

(4.32)

Proof

Given $\alpha = 1, \dots , r$, consider the function $\mathbb R\ni s_\alpha \mapsto e^{\textrm{i}s_\alpha N^{(\alpha )}} \langle A \rangle e^{-\textrm{i}s_\alpha N^{(\alpha )}}$. By reparametrizing $\theta _\alpha \mapsto \theta _\alpha +s_\alpha$ in (4.31), one has $e^{\textrm{i}s_\alpha N^{(\alpha )}} \langle A \rangle e^{-\textrm{i}s_\alpha N^{(\alpha )}}=\langle A \rangle$, which is the first part of the thesis.

Finally, to prove equation (4.32), one observes

$$\begin{aligned} \begin{aligned} \Vert \langle A \rangle \Vert _{\kappa }&\le \sup _{x \in \Lambda } \sum _{\begin{array}{c} S \in \mathcal {P}_c(\Lambda ) \\ x \in S \end{array}} \sup _{\theta \in \mathbb {R}^r} \Vert e^{\textrm{i}\theta \cdot N} (\widehat{A_S})_{0} e^{-\textrm{i}\theta \cdot N} \Vert _{\text {op}} e^{\kappa |S|} \\&=\sup _{x \in \Lambda } \sum _{\begin{array}{c} S \in \mathcal {P}_c(\Lambda ) \\ x \in S \end{array}} \Vert (\widehat{A_S})_{0} \Vert _{\text {op}} e^{\kappa |S|} \le \Vert S \Vert _{\kappa ,0}\,. \end{aligned} \end{aligned}$$

$\square$

4.3 Homological Equations for Small Perturbations

In this Subsection we study the solution to the homological equation (3.14) in the case of small perturbations. That is, given an operator $P \in \mathcal {O}^\texttt {s} _{\kappa , \rho }$ for some $\kappa , \rho >0$ with $\Vert P\Vert _{\kappa , \rho } \ll 1$, we solve

$$\begin{aligned} \textrm{i}[J \cdot N, G(\varphi )] + \omega \cdot \partial _\varphi G(\varphi ) + P(\varphi ) = Z\,, \end{aligned}$$

(4.33)

for suitable operators $G \in \mathcal {O}^\texttt {s} _{\kappa ', \rho '}$ and $Z \in \mathcal {O}^\texttt {s} _{\kappa '}$, with $\kappa ' \le \kappa$, $\rho ' \le \rho$. Following the strategy presented in Subsection 3.1, as a first step we define

$$\begin{aligned} \mathcal {G}(\theta , \varphi ) := e^{\textrm{i}\theta \cdot N} G(\varphi ) e^{-\textrm{i}\theta \cdot N}\,, \quad \mathcal {P}(\theta , \varphi ) = e^{\textrm{i}\theta \cdot N} P(\varphi ) e^{-\textrm{i}\theta \cdot N}\,, \quad \theta \in \mathbb {T}^r\,, \end{aligned}$$

(4.34)

and we solve $\forall \theta \in \mathbb {T}^r$ an equation of the following form:

$$\begin{aligned} \textrm{i}[J \cdot N, \mathcal {G}(\theta , \varphi )] + \omega \cdot \partial _\varphi \mathcal {G}(\theta , \varphi ) + \mathcal {P}(\theta , \varphi ) = \langle P\rangle \,, \end{aligned}$$

(4.35)

with $\langle P \rangle$ defined in (4.31).

Lemma 4.9

($\theta$-dependent homological equation) Let $\kappa , \rho , \zeta >0$ and $\mathcal {P} \in \mathcal {O}^\texttt {s} _{\kappa , \rho , \zeta }$. There exists $\mathcal {G} \in \mathcal {O}^{\texttt {s} }_{\kappa , \rho -\delta , \zeta -\delta }$ $\forall 0< \delta \le \min \{\rho , \zeta \}$ which solves (4.35) and the following estimate holds:

$$\begin{aligned} \Vert \mathcal {G}\Vert _{\kappa , \rho -\delta , \zeta -\delta } \le \frac{\tau ^\tau }{e^\tau \delta ^\tau \gamma } \Vert \mathcal {P}\Vert _{\kappa , \rho , \zeta }\,. \end{aligned}$$

(4.36)

Proof

We observe that

$$ \textrm{i}[J \cdot N, \mathcal {G}(\theta , \varphi )] = (J \cdot \partial _{\theta }) \mathcal {G}(\theta , \varphi ) = \sum _{S \in \mathcal {P}_c(\Lambda )} \sum _{l \in \mathbb Z^m} \sum _{k \in \mathbb Z^r} \textrm{i}J \cdot k (\widehat{\mathcal {G}_S})_{l,k} e^{\textrm{i}k \cdot \theta } e^{\textrm{i}l \cdot \varphi }\,. $$

Passing to Fourier coefficients, (4.35) is equivalent to

$$ \textrm{i}\left( J \cdot k + \omega \cdot l\right) (\widehat{\mathcal {G}_{S}})_{l,k} + (\widehat{\mathcal {P}_S})_{l,k} = 0 \quad \forall (l,k) \ne (0,0)\,, $$

observing that $\langle P \rangle = \sum _{S \in \mathcal {P}_c(\Lambda )} (\widehat{\mathcal {P}_S})_{0,0}$. Then $\forall S \in \mathcal {P}_c(\Lambda )$ one sets

$$\begin{aligned} (\widehat{\mathcal {G}_{S}})_{l,k} = -\frac{(\widehat{\mathcal {P}_S})_{l,k}}{\textrm{i}\left( J \cdot k + \omega \cdot l\right) } \quad \forall (l,k) \ne 0\,, \quad (\widehat{\mathcal {G}_{S}})_{0,0} = 0\,, \end{aligned}$$

(4.37)

and $\forall 0<\delta \le \min \{\zeta , \rho \}$ has

$$\begin{aligned} \Vert \mathcal {G}\Vert _{\kappa , \rho -\delta , \zeta -\delta }&\le \gamma ^{-1} \sup _{x \in \Lambda } \sum _{S\in \mathcal {P}_c(\Lambda )} \sum _{l \in \mathbb Z^m} \sum _{k \in \mathbb Z^r} \Vert (\widehat{\mathcal {P}_S})_{l,k}\Vert _{\textrm{op}} (|k| + |l|)^{\tau } e^{-(|k|+ |l|)\delta }e^{|k|\zeta } e^{\rho |l|} e^{|S|\kappa }\\&\le \frac{\tau ^\tau }{e^\tau \delta ^\tau \gamma } \sup _{x \in \Lambda } \sum _{S\in \mathcal {P}_c(\Lambda )} \sum _{l \in \mathbb Z^m} \sum _{k \in \mathbb Z^r} \Vert (\widehat{\mathcal {P}_S})_{l,k}\Vert _{\textrm{op}} e^{|k|\zeta } e^{\rho |l|} e^{|S|\kappa } =\frac{\tau ^\tau }{e^\tau \delta ^\tau \gamma } \Vert \mathcal {P}\Vert _{\kappa , \rho , \zeta }\,, \end{aligned}$$

where we have used the fact that $(J, \omega )$ satisfies the Diophantine condition (2.11). It remains to prove that $\mathcal {G}(\theta , \varphi )$ is strongly local $\forall (\theta , \varphi )$. This follows from the fact that $\mathcal {P}(\theta , \varphi )$ is strongly local, from Lemma 4.7 and from the fact that, by (4.37), the Fourier coefficients of $\mathcal {G}_S(\theta , \varphi )$ are strongly local if and only if the Fourier coefficients of $\mathcal {P}_S(\theta , \varphi )$ are strongly local. $\square$

From Lemma 4.9 we deduce the two following results:

Lemma 4.10

(Homological equation for $H_\textrm{inv}$) Let $\kappa , \rho >0$ and $P \in \mathcal {O}^\texttt {s} _{\kappa , \rho }$ and let

$$\begin{aligned} \texttt {c} := 8 r \max _{\alpha } \Vert N^{(\alpha )}\Vert _0\,. \end{aligned}$$

(4.38)

There exist $G:= G_\textrm{inv}\in \mathcal {O}^\texttt {s} _{\kappa - \texttt {c} \delta , \rho - \delta }$ and $Z:=Z_\textrm{inv}\in \mathcal {O}^\texttt {s} _{\kappa - \texttt {c} \delta , \rho }$ for any $0<\delta < \min \{\rho , \texttt {c} ^{-1} \kappa \}$ such that (4.33) holds, with

$$\begin{aligned} \left[ Z_\textrm{inv}, N^{(\alpha )}\right] = 0 \quad \forall \alpha = 1, \dots , r\,. \end{aligned}$$

(4.39)

Moreover, $Z_\textrm{inv}=\langle P \rangle$ and $\forall \delta >0$ one has

$$\begin{aligned} \Vert G_\textrm{inv}\Vert _{\kappa - \texttt {c} \delta , \rho -\delta } \le \frac{4^r \tau ^{\tau }}{e^{\tau } \gamma \delta ^{\tau +r}}(1+\delta )^r \Vert P\Vert _{\kappa , \rho }\,. \end{aligned}$$

(4.40)

Proof

Let $\zeta :=\delta$ and $\sigma := 8 \max _{\alpha } \Vert N^{(\alpha )}\Vert _0 \delta$; by Item (iii) of Lemma 4.5, one has that, defining for any $\theta \in \mathbb {T}^r$ and for any $\varphi \in \mathbb {T}^m$ $\mathcal {P}(\theta , \varphi )$ as in (4.34), $\mathcal {P} \in \mathcal {O}^\texttt {s} _{\kappa - r \sigma , \rho , \zeta }$, with

$$\begin{aligned} \Vert \mathcal {P}\Vert _{\kappa -r \sigma , \rho , \zeta } \le 2^{2r} \left( \frac{\zeta +1}{\zeta }\right) ^r \Vert P\Vert _{\kappa , \rho }\,. \end{aligned}$$

(4.41)

Then by Lemma 4.9 there exists $\mathcal {G} \in \mathcal {O}^\texttt {s} _{\kappa , \rho -\delta , 0}$ solving (4.35) and

$$\begin{aligned} \Vert \mathcal {G}\Vert _{\kappa -r \sigma , \rho -\delta , 0} \le \frac{\tau ^{\tau }}{e^{\tau } \gamma \delta ^{\tau }} \Vert \mathcal {P}\Vert _{\kappa - r \sigma , \rho , \delta }\,. \end{aligned}$$

(4.42)

Comparing (4.35), (4.33) and (4.31) one gets $Z_{\textrm{inv}}=\langle P \rangle$. Furthermore we have that $G_\textrm{inv}(\varphi ):= \mathcal {G}(0, \varphi )$ solves (4.35) for $\theta = 0$. Moreover, again by Item (iii) of Lemma 4.5, one has

$$\begin{aligned} \Vert G_\textrm{inv}\Vert _{\kappa -r \sigma , \rho - \delta } \le \Vert \mathcal {G}\Vert _{\kappa - r \sigma , \rho - \delta , 0}\,. \end{aligned}$$

(4.43)

Combining (4.38), (4.41), (4.42) and (4.43), one gets

$$ \Vert G_\textrm{inv}\Vert _{\kappa -r \sigma , \rho - \delta } = \Vert G_\textrm{inv}\Vert _{\kappa -\texttt {c} \delta , \rho - \delta } \le \frac{2^{2r}\tau ^{\tau }}{e^{\tau } \gamma \delta ^{\tau +r}}(1+\delta )^r \Vert P\Vert _{\kappa , \rho }\,, $$

which gives (4.40). $\square$

Lemma 4.11

(Homological equation for $H_\textrm{obs}$) Let $\kappa ,\rho >0$ and $P \in \mathcal {O}^\texttt {s} _{\kappa ,\rho }$, there exist $G:= G_\textrm{obs}\in \mathcal {O}^\texttt {s} _{\kappa -\texttt{c} \delta , \rho -\delta }$ and $Z:= Z_\textrm{obs}\in \mathcal {O}^\texttt {s} _{\kappa -\texttt{c}\delta }$ for any $0<\delta <\min \{\texttt{c}^{-1}\kappa ,\rho \}$ such that (4.33) holds with $G_\textrm{obs}(0) = 0$. One has

$$\begin{aligned} \begin{aligned} \Vert G_\textrm{obs}\Vert _{\kappa -\texttt{c}\delta , \rho -\delta }&\le \frac{2^{2r+1} \tau ^{\tau }}{e^{\tau } \gamma \delta ^{\tau +r}}(1+\delta )^r \Vert P\Vert _{\kappa , \rho }\,, \qquad \\ \Vert Z_\textrm{obs}\Vert _{\kappa - \texttt{c} \delta }&\le \frac{\mathcal {J} 2 ^{2r}(1 +\delta )^r \tau ^\tau }{\gamma e^\tau \delta ^{\tau + r + 1}} \Vert P\Vert _{\kappa , \rho } + \Vert P\Vert _{\kappa , \rho }\,. \end{aligned} \end{aligned}$$

(4.44)

In particular,

$$\begin{aligned} Z_\textrm{obs}= \langle \! \langle P \rangle \! \rangle \,, \qquad \langle \! \langle P \rangle \! \rangle := \langle P \rangle + \sum _{S \in \mathcal {P}_c(\Lambda )} \sum _{\begin{array}{c} (k, l) \ne (0,0) \\ k \in \mathbb Z^{r}, l \in \mathbb Z^m \end{array}} \frac{(J \cdot k) (\widehat{\mathcal {P}_S})_{l,k}}{\omega \cdot l + J \cdot k}\,. \end{aligned}$$

(4.45)

Proof

Since by Lemma 4.10$\mathcal {G}(0, \varphi )$ solves (4.33) with $Z = \langle P \rangle$, then $G_\textrm{obs}(\varphi ):= \mathcal {G}(0, \varphi ) - \mathcal {G}(0,0)$ solves (4.33) with $Z = \langle P \rangle - \textrm{i}[J \cdot N, \mathcal {G}(0,0)]$ and by construction it satisfies $G_\textrm{obs}(0) = 0$. Using Lemma 4.6, one obtains that $Z_{\textrm{obs}}$ is given by the expression in (4.45). The first estimate in (4.44) follows from the estimate (4.40) on $\mathcal {G}(0, \varphi )$, while to obtain the second estimate in (4.44) we observe that $\Vert \langle \! \langle P \rangle \! \rangle \Vert _{\kappa - \texttt{c} \delta } \le \Vert \langle P \rangle \Vert _{\kappa - \texttt{c}\delta } + \Vert [J \cdot N, \mathcal {G}(0,0)]\Vert _{\kappa - \texttt{c}\delta }$, with

$$ \Vert [J \cdot N,\ \mathcal {G}(0,0)]\Vert _{\kappa - \texttt{c}\delta } \le \sup _{x \in \Lambda } \sum _{\begin{array}{c} S \ni x \\ S \in \mathcal {P}_c(\Lambda ) \end{array}} \sum _{(k, l) \ne (0,0)} \frac{|J \cdot k|}{|\omega \cdot l + J \cdot k|} \Vert (\widehat{\mathcal {P}_S})_{l,k} \Vert _{\textrm{op}} e^{(\kappa - \texttt{c}\delta )|S|}\,. $$

Then the second estimate in (4.44) follows using Diophantine condition (2.11), Cauchy inequality, equation (4.20), and estimate (4.32) on $\Vert \langle P \rangle \Vert _{\kappa - \texttt{c}\delta }$. $\square$

4.4 Homological Equation for Fast Forcing Perturbations

In this Subsection, we study the solution to the homological equation (3.14) in the fast-forcing case, that is, given $\lambda \gg 1$ and $P \in \mathcal {O}^\texttt {s} _{\kappa , \rho }$ for some $\kappa , \rho >0$, we want to solve the equation

$$\begin{aligned} \textrm{i}[J \cdot N, G(\varphi )] + \lambda \omega \cdot \partial _\varphi G(\varphi ) + P(\varphi ) = Z\,, \end{aligned}$$

(4.46)

for some operators $G \in \mathcal {O}^\texttt {s} _{\kappa ', \rho '}$ and $Z \in \mathcal {O}^\texttt {s} _{\kappa '}$ with $\kappa ' \le \kappa$, $\rho ' \le \rho$. In particular, we set $Z:= \langle P \rangle$, with $\langle P \rangle$ defined in (4.31) and, after having defined $\mathcal {G}(\theta ,\varphi )$ and $\mathcal {P}(\theta ,\varphi )$ as in (4.34), we first solve $\forall \theta \in \mathbb {T}^r$ the following equation:

$$\begin{aligned} \textrm{i}[J \cdot N, \mathcal {G}(\theta , \varphi )] + \lambda \omega \cdot \partial _\varphi \mathcal {G}(\theta , \varphi ) + \mathcal {P}(\theta , \varphi ) = \langle P \rangle \,. \end{aligned}$$

(4.47)

The strategy we use to solve this equation (up to higher order terms in $\lambda ^{-1}$) makes use of the splitting of Fourier modes into $\texttt {uv}$/$\texttt {ir}$ regions. That is, given

$$\begin{aligned} K := \frac{\gamma _\omega }{2|J|}\lambda ^{\frac{1}{2}}\,, \quad L := \lambda ^{\frac{1}{2\tau _\omega }}\,, \end{aligned}$$

(4.48)

we define the following sets:

$$\begin{aligned} \mathcal {K}_{\texttt {uv} }:= & {} \left\{ k \in \mathbb Z^r\ \left| \ |k| > K\right. \right\} \,, \quad \mathcal {K}_{\texttt {ir} }:= \left\{ k \in \mathbb Z^r\ \left| \ |k| \le K\right. \right\} \,, \end{aligned}$$

(4.49)

$$\begin{aligned} \mathcal {L}_{\texttt {uv} }:= & {} \left\{ l \in \mathbb Z^m\ \left| \ |l| > L\right. \right\} \,, \quad \mathcal {L}_{\texttt {ir} }:= \left\{ l \in \mathbb Z^m\ \left| \ |l| \le L\right. \right\} \,, \end{aligned}$$

(4.50)

and for $\texttt {i} , \texttt {j} \in \{\texttt {uv} , \texttt {ir} \}$,

$$\begin{aligned} \mathcal {P}^{\texttt {i} , \texttt {j} } (\theta , \varphi ) := \sum _{S \in \mathcal {P}_c(\Lambda ) } \sum _{k \in \mathcal {K}_{\texttt{i}}} \sum _{l \in \mathcal {L}_{\texttt{j}}} (\widehat{\mathcal {P}_S})_{l, k} e^{\textrm{i}k \cdot \theta } e^{\textrm{i}l \cdot \varphi }\,. \end{aligned}$$

(4.51)

Consequently, we also define

$$\begin{aligned} {P}^{\texttt {i} , \texttt {j} } (\varphi ) := \mathcal {P}^{\texttt {i} , \texttt {j} } (0, \varphi )\,. \end{aligned}$$

(4.52)

Also, for simplicity, we define

$$\begin{aligned} \begin{aligned} \mathcal {P}^{\texttt {uv} } := \mathcal {P}^{\texttt {uv} , \texttt {ir} } + \mathcal {P}^{\texttt {uv} , \texttt {uv} } + \mathcal {P}^{\texttt {ir} , \texttt {uv} }\,, \quad \mathcal {P}^{\texttt {ir} } := \mathcal {P}^{\texttt {ir} , \texttt {ir} }\,,\\ P^{\texttt {uv} } := \mathcal {P}^{\texttt {uv} }(0 ,\cdot )\,, \quad P^{\texttt {ir} } := \mathcal {P}^{\texttt {ir} }(0 ,\cdot )\,. \end{aligned} \end{aligned}$$

(4.53)

First, we prove that the $\texttt {uv}$ terms can be treated as remainders in the solution of the homological equation:

Lemma 4.12

Let $\kappa , \rho , \zeta , \sigma , \eta >0$ and $\mathcal {P} \in \mathcal {O}^\texttt {s} _{\kappa , \rho + \sigma , \zeta + \eta }$. One has

$$\begin{aligned} \Vert \mathcal {P}^{\texttt {uv} }\Vert _{\kappa , \rho , \zeta }\le & {} \frac{2}{e} \max \left\{ \frac{1}{ \eta K}, \frac{1}{\sigma L}\right\} \Vert \mathcal {P}\Vert _{\kappa , \rho + \sigma , \zeta + \eta }\,, \end{aligned}$$

(4.54)

$$\begin{aligned} \Vert \mathcal {P}^{\texttt {uv} }\Vert _{\kappa , \rho , \zeta }\le & {} \Vert \mathcal {P}\Vert _{\kappa , \rho , \zeta }\,, \quad \Vert \mathcal {P}^{\texttt {ir} }\Vert _{\kappa , \rho , \zeta } \le \Vert \mathcal {P}\Vert _{\kappa , \rho , \zeta }\,. \end{aligned}$$

(4.55)

Proof

Estimates (4.55) immediately follow from the definition of $\mathcal {P}^{\texttt {uv} }, \mathcal {P}^\texttt {ir}$. Concerning (4.54), one has

$$\begin{aligned} \left\| \mathcal {P}^{\texttt {uv} }\right\| _{\kappa , \rho , \zeta }&\le \left\| \mathcal {P}^{\texttt {uv} , \texttt {ir} } + \mathcal {P}^{\texttt {uv} , \texttt {uv} }\right\| _{\kappa , \rho , \zeta } + \left\| \mathcal {P}^{\texttt {ir} , \texttt {uv} } \right\| _{\kappa , \rho , \zeta }\,. \end{aligned}$$

Using Cauchy estimate, the first summand is estimated by

$$\begin{aligned} \left\| \mathcal {P}^{\texttt {uv} , \texttt {ir} } + \mathcal {P}^{\texttt {uv} , \texttt {uv} }\right\| _{\kappa , \rho , \zeta }&\le \frac{1}{K} \sup _{x \in \Lambda } \sum _{S \ni x} \sum _{|k| > K} \sum _{l \in \mathbb Z^m} \Big \Vert (\widehat{\mathcal {P_S}})_{l,k} \Big \Vert _{\textrm{op}} |k| e^{\zeta |k|} e^{\rho |l|} e^{\kappa |S|}\\&\le \frac{1}{e \eta K} \Vert \mathcal {P}\Vert _{\kappa , \rho , \zeta + \eta } \,, \end{aligned}$$

and an analogous estimate entails $\left\| \mathcal {P}^{\texttt {ir} , \texttt {uv} } \right\| _{\kappa , \rho , \zeta } \le \frac{1}{e \sigma L} \Vert \mathcal {P}\Vert _{\kappa , \rho + \sigma , \zeta }$. $\square$

Corollary 4.13

Let $\kappa , \rho >0$ and $P \in \mathcal {O}^\texttt {s} _{\kappa , \rho }$. Defining $\texttt {c}$ as in (4.38), $\forall 0<\delta < \min \{\rho , {\texttt {c} }^{-1} \kappa \}$ one has $P^{\texttt {uv} } \in \mathcal {O}^\texttt {s} _{\kappa - \texttt {c} \delta , \rho - \delta }$, with

$$\begin{aligned} \Vert P^{\texttt {uv} }\Vert _{\kappa - \texttt {c} \delta , \rho - \delta } \le \frac{2^{2r+1}}{e\delta ^{r+1}} (1+\delta )^r\max \left\{ \frac{1}{K}, \frac{1}{L}\right\} \Vert P\Vert _{\kappa , \rho }\,. \end{aligned}$$

(4.56)

Proof

Define $\zeta := \delta$ and $\sigma := 8 \max _\alpha \Vert N^{(\alpha )}\Vert _{0}$; then estimate (4.19) is fulfilled and by Item (iii) of Lemma 4.5 one has

$$ \Vert \mathcal {P}\Vert _{\kappa - r \sigma , \rho , \zeta } = \Vert \mathcal {P}\Vert _{\kappa - \texttt {c} \delta , \rho , \delta } \le 2^{2r}\left( \frac{1+\delta }{\delta }\right) ^r \Vert P\Vert _{\kappa , \rho }\,. $$

Then one applies Lemma 4.12 with $\zeta =\eta = \sigma = \delta$ to obtain

$$ \Vert \mathcal {P}^{\texttt {uv} }\Vert _{\kappa - \texttt {c} \delta , \rho - \delta , 0} \le \frac{2}{e \delta } \Vert \mathcal {P}\Vert _{\kappa - \texttt {c} \delta , \rho , \delta }\,, $$

and (4.56) follow again by Item (iii) of Lemma 4.5. $\square$

Given $\kappa ,\zeta ,\rho > 0$, $\mathcal {P} \in \mathcal {O}^\texttt {s} _{\kappa ,\rho ,\zeta }$ we define

$$\begin{aligned} \langle \mathcal {P} \rangle _{\mathbb {T}^m}(\theta ) \;:=\; \frac{1}{(2 \pi )^m} \int _{\mathbb {T}^m} \mathcal {P}(\theta ,\varphi ) \, \textrm{d}\varphi \, . \end{aligned}$$

(4.57)

We are now ready to solve an equation of the form of (4.47), with $\mathcal {P} = \mathcal {P}^\texttt {ir}$:

Lemma 4.14

($\theta$-dependent homological equation) Let $\kappa ,\rho ,\zeta >0$ and $\mathcal {P} \in \mathcal {O}^\texttt {s} _{\kappa ,\rho ,\zeta }$, there exists $\mathcal {G} \in \mathcal {O}^{\texttt {s} }_{\kappa ,\rho ,\zeta -\delta }$ for any $0< \delta \le \zeta$ which solves

$$\begin{aligned} \textrm{i}[J\cdot N, \mathcal {G}]+\mathcal {P}^\texttt {ir} + \lambda \omega \cdot \partial _\varphi \mathcal {G} = \langle P \rangle \,, \end{aligned}$$

(4.58)

with $\langle P \rangle$ as in (4.31) and

$$\begin{aligned} \Vert \mathcal {G} \Vert _{\kappa , \rho ,\zeta -\delta } \; \le \; \frac{4}{\gamma _\omega } \frac{1}{\lambda ^{1/2}} \Vert \mathcal {P}-\langle \mathcal {P} \rangle _{\mathbb {T}^m} \Vert _{\kappa ,\rho ,\zeta } + \frac{2 \tau _J^{\tau _J}}{\delta ^{\tau _J} e^{\tau _J} \gamma _J} \Vert \langle \mathcal {P} \rangle _{\mathbb {T}^m} \Vert _{\kappa ,0,\zeta } \, . \end{aligned}$$

(4.59)

Proof

We observe that

$$ \textrm{i}[J \cdot N, \mathcal {G}(\theta , \varphi )] = (J \cdot \partial _{\theta }) \mathcal {G}(\theta , \varphi ) = \sum _{S \in \mathcal {P}_c(\Lambda )} \sum _{l \in \mathbb Z^m} \sum _{k \in \mathbb Z^r} \textrm{i}J \cdot k \, (\widehat{\mathcal {G}_S})_{l,k} e^{\textrm{i}k \cdot \theta } e^{\textrm{i}l \cdot \varphi }\,. $$

The first equation in (4.58) is equivalent to

$$\begin{aligned} (J \cdot \partial _\theta ) \mathcal {G} + \lambda \omega \cdot \partial _\varphi \mathcal {G} = -\mathcal {P}^\texttt {ir} + \langle P \rangle \,, \end{aligned}$$

(4.60)

and, passing to Fourier coefficients,

$$ \textrm{i}\left( J \cdot k + \omega \cdot l\right) (\widehat{\mathcal {G}_{S}})_{l,k} + (\widehat{\mathcal {P}_S})_{l,k} = 0 \quad \forall (l,k) \ne (0,0)\,, \quad (l,k) \in \mathcal {K}_{\texttt {ir} } \times \mathcal {L}_{\texttt {ir} }\,. $$

Then $\forall S \in \mathcal {P}_c(\Lambda )$ one sets

$$\begin{aligned} (\widehat{\mathcal {G}_{S}})_{l,k} = {\left\{ \begin{array}{ll} -\dfrac{(\widehat{\mathcal {P}_S})_{l,k}}{\textrm{i}\left( J \cdot k + \omega \cdot l\right) } &{}\forall (l,k) \in \mathcal {K}_\texttt {ir} \times \mathcal {L}_\texttt {ir} \setminus \{(0,0)\}\,,\\ 0 &{} \text {otherwise}\,. \end{array}\right. } \end{aligned}$$

(4.61)

We now provide an estimate on the small divisors appearing in (4.61). We estimate them differently according if $l = 0$ or not. Due to the choice of K, L as in (4.48) and to the fact that $\omega$ is Diophantine (2.11), if $l \ne 0$ one has

$$\begin{aligned} |J \cdot k + \lambda \omega \cdot l| \ge \lambda |\omega \cdot l| - |J| |k| \ge \lambda \frac{\gamma _\omega }{ |l|^{\tau _\omega }} - |J||k| \ge \lambda \frac{\gamma _\omega }{L^{\tau _\omega }} - |J| K \ge \frac{\gamma _\omega }{2} \lambda ^{\frac{1}{2}}\,. \end{aligned}$$

(4.62)

On the other hand, if $l=0$ due to the fact that J is Diophantine (2.11) we have $|J \cdot k | \ge \frac{\gamma _J}{|k|^{\tau _J}}.$ Then $\forall 0<\delta \le \zeta$ one has $\Vert \mathcal {G}\Vert _{\kappa , \rho , \zeta -\delta } \le T_1 + T_2$, with

and

where in the last step we have used the fact that

$$ \mathcal {P} - \langle \mathcal {P}\rangle _{\mathbb {T}^m} = \sum _{S \in \mathcal {P}_c(\Lambda )} \sum _{k \in \mathbb Z^r, l \in \mathbb Z^m} (\widehat{\mathcal {P}_S})_{l,k}- \sum _{S\in \mathcal {P}_c(\Lambda )} \sum _{k \in \mathbb Z^r} (\widehat{{\mathcal {P}}_{S}})_{k,0} = \sum _{S \in \mathcal {P}_c(\Lambda )} \sum _{k \in \mathbb Z^r, 0 \ne l \in \mathbb Z^m} (\widehat{{\mathcal {P}}_{S}})_{l,k}\,. $$

It remains to prove that $\mathcal {G}(\theta , \varphi )$ is strongly local $\forall (\theta , \varphi )$. This follows from the fact that $\mathcal {P}(\theta , \varphi )$ is strongly local, from Lemma 4.7 and from the fact that, by (4.37), the Fourier coefficients of $\mathcal {G}_S(\theta , \varphi )$ are strongly local if and only if the Fourier coefficients of $\mathcal {P}_S(\theta , \varphi )$ are strongly local. $\square$

Corollary 4.15

Let $\kappa , \rho >0$ and $P \in \mathcal {O}^\texttt {s} _{\kappa , \rho }.$ Let $\texttt {c}$ be as in (4.38), there exists $G \in \mathcal {O}^\texttt {s} _{\kappa - \texttt {c} \delta , \rho }$ for any $0<\delta < \min \{\rho , \texttt {c} ^{-1} \kappa \}$ such that

$$\begin{aligned} \textrm{i}[J \cdot N, G] + \lambda \omega \cdot \partial _\varphi G + P^{\texttt {ir} } = \langle P \rangle \,. \end{aligned}$$

(4.63)

Moreover, $\forall \delta >0$ one has

$$\begin{aligned} \Vert G\Vert _{\kappa - \texttt {c} \delta , \rho } \le 2^{2r} \left( \frac{\delta +1}{\delta }\right) ^r \left( \frac{4}{\gamma _\omega \lambda ^{1/2}} \Vert P - \langle P \rangle _{\mathbb {T}^m} \Vert _{\kappa ,\rho } + \frac{2 \tau _J^{\tau _J}}{\gamma _J e^{\tau _J} \delta ^{\tau _J}} \Vert \langle P \rangle _{\mathbb {T}^m} \Vert _{\kappa ,0} \right) \,. \end{aligned}$$

(4.64)

Proof

Let $\zeta :=\delta$ and $\sigma := 8 \max _{\alpha } \Vert N^{\alpha }\Vert _0 \delta$; by Item (iii) of Lemma 4.5, one has that, defining $\forall (\theta , \varphi )$ $\mathcal {P}(\theta , \varphi )$ as in (4.34), $\mathcal {P} \in \mathcal {O}^\texttt {s} _{\kappa - r \sigma , \rho , \zeta }$, with

$$\begin{aligned} \Vert \mathcal {P}\Vert _{\kappa -r \sigma , \rho , \zeta } \le 2^{2r}\left( \frac{\zeta +1}{\zeta }\right) ^r \Vert P\Vert _{\kappa , \rho }\,. \end{aligned}$$

(4.65)

Then, by Lemma 4.14 there exists $\mathcal {G} \in \mathcal {O}^\texttt {s} _{\kappa -r \sigma , \rho , 0}$ solving (4.58) with $\langle P \rangle$ as in (4.31) and

$$\begin{aligned} \Vert \mathcal {G}\Vert _{\kappa -r \sigma , \rho , 0} \le \frac{4}{\gamma _\omega \lambda ^{1/2}} \Vert \mathcal {P} - \langle \mathcal {P}\rangle _{\mathbb {T}^m} \Vert _{\kappa ,\rho ,\delta } + \frac{2\tau _J^{\tau _J}}{\gamma _J e^{\tau _J} \delta ^{\tau _J}} \Vert \langle \mathcal {P} \rangle _{\mathbb {T}^m} \Vert _{\kappa ,0,\delta }\,. \end{aligned}$$

(4.66)

Furthermore, we have that $G(\varphi ):= \mathcal {G}(0, \varphi )$ solves (4.58) for $\theta = 0$, namely (4.63). Moreover, again by Item (iii) of Lemma 4.5, one has

$$\begin{aligned} \Vert G \Vert _{\kappa -r \sigma , \rho } \le \Vert \mathcal {G}\Vert _{\kappa - r \sigma , \rho , 0}\,. \end{aligned}$$

(4.67)

Combining (4.65), (4.66) and (4.67), one gets

$$ \Vert {G}\Vert _{\kappa -r \sigma , \rho } \le 2^{2r}\left( \frac{\delta +1}{\delta }\right) ^r \left( \frac{4}{\gamma _\omega \lambda ^{1/2}} \Vert {P} - \langle {P}\rangle _{\mathbb {T}^m} \Vert _{\kappa ,\rho } + \frac{2\tau _J^{\tau _J}}{\gamma _J e^{\tau _J} \delta ^{\tau _J}} \Vert \langle {P} \rangle _{\mathbb {T}^m} \Vert _{\kappa ,0}\right) \,, $$

which gives (4.40). $\square$

5 Proof of Normal Form Results

In this Section we use the estimates of the previous sections to prove the rigorous normal form results, as stated in Propositions 3.1 and 3.3.

5.1 Small Perturbations

Item (a) of Proposition 3.1 immediately follows from the following lemma.

Lemma 5.1

Under the assumptions of Proposition 3.1, define

$$\begin{aligned} n_* := \left\lfloor \varepsilon ^{-2 \texttt {b} }\right\rfloor \,, \quad \kappa _0 := \min \{\kappa , \rho \}\,, \quad \rho _0 := 2(\texttt {c} + 1)^{-1} \kappa _0\,, \end{aligned}$$

(5.1)

with $\texttt {c}$ as in (4.38) and $\texttt {b}$ as in (2.13), and define $\forall n = 1, \dots , n_*$

$$\begin{aligned} \kappa _n := {\kappa _0}\sqrt{1 - b n}\,, \quad \rho _n := {\rho _0}\sqrt{1 - b n}\,,\quad b := \frac{1}{2 n_*}\,, \end{aligned}$$

(5.2)

and $\forall n = 0, \dots , n_*$

$$\begin{aligned} \varepsilon _n := \left( \frac{1}{e}\right) ^n \varepsilon \Vert V\Vert _{\kappa _0, \rho _0}\,. \end{aligned}$$

(5.3)

Then $\forall n = 0, \dots , n_*$ there exist a time quasi-periodic family of unitary maps $Y^{(n)}_{\textrm{inv}}(\omega t)$, a self-adjoint operator $Z^{(n)}_{\textrm{inv}}$ and a time quasi-periodic family of self-adjoint operators $V^{(n)}_{\textrm{inv}}(\omega t)$ such that, defining

$$\begin{aligned} H^{(n)}_{\textrm{inv}}(\omega t):= J \cdot N + Z^{(n)}_{\textrm{inv}} + V^{(n)}_{\textrm{inv}}( \omega t) \,, \end{aligned}$$

(5.4)

one has

$$\begin{aligned} U_{H^{(n)}_{\textrm{inv}}}(t) = Y^{(n)}_{\textrm{inv}}( \omega t) U_{H}(t) (Y^{(n)}_{\textrm{inv}})^*(0)\,, \end{aligned}$$

(5.5)

and the following are satisfied:

• 1. $Z^{(n)}_{\textrm{inv}} \in \mathcal {O}^\texttt {s} _{\kappa _n}$ is time independent, with $Z^{(0)}=0$, and for $n>0$

  $$\begin{aligned}{}[Z^{(n)}_{\textrm{inv}}, N^{(\alpha )}] = 0 \quad \forall \alpha = 1, \dots , r\,, \quad \Vert Z^{(n)}_{\textrm{inv}}\Vert _{\kappa _n} \le \sum _{n'=0}^{{n-1}} \varepsilon _{{n'}}\,; \end{aligned}$$

  (5.6)

• 2. $V^{(n)}_{\textrm{inv}} \in \mathcal {O}^\texttt {s} _{\kappa _n, \rho _n}$, and

  $$\begin{aligned} \Vert V^{(n)}_{\textrm{inv}}\Vert _{\kappa _n, \rho _n} \le \varepsilon _n\,; \end{aligned}$$

  (5.7)

• 3. The maps $Y^{(n)}_{\textrm{inv}}(\omega t)$ are close to the identity, in the sense that $Y_\textrm{inv}^{(0)}=\mathbb {1}$, and for $n>0$

  $$\begin{aligned} \Vert Y^{(n)}_{\textrm{inv}} P (Y^{(n)}_{\textrm{inv}})^*- P\Vert _{\kappa _n, \rho _n} \le 2 \varepsilon ^{\frac{1}{2}} \sum _{n'=0}^{{n-1}} \left( \frac{1}{e}\right) ^{{n'}}\Vert V\Vert _{\kappa , \rho } \Vert P\Vert _{\kappa , \rho } \quad \forall P \in \mathcal {O}_{\kappa , \rho }\,. \end{aligned}$$

  (5.8)

To prove the lemma, we recall the following well-known result:

Lemma 5.2

(Lemma 3.1 of [5]) Given G(t) and H(t) two families of self-adjoint operators with smooth dependence on time, define $\widetilde{U}(t):= e^{-\textrm{i}G(t)} U_{H}(t) e^{\textrm{i}G(0)}$. Then $\forall t$ $\widetilde{U}(t) = U_{H^\prime }(t)$, with

$$\begin{aligned} H^\prime (t) =e^{-\textrm{i}G(t)} H(t) e^{\textrm{i}G(t)} + \int _0^1 e^{-\textrm{i}G(t)s} \partial _t{G}(t) e^{\textrm{i}G(t)s}\,ds\,. \end{aligned}$$

(5.9)

Proof of Lemma 5.1

If $n=0$, the result holds true with $Y^{(0)}_\textrm{inv}= \mathbb {1}$, $Z^{(0)}_\textrm{inv}= 0$, and $V^{(0)}_\textrm{inv}= \varepsilon V$. Let us assume the thesis has been proven for $n' = 0, \dots , n$; we are now going to prove that it holds also for $n' = n+1$. We look for a change of variables of the form

$$ \widetilde{U}^{(n)}_{\textrm{inv}}(t) = e^{-\textrm{i}G^{(n)}_{\textrm{inv}}(\omega t)} U_{H^{(n)}_{\textrm{inv}}}(t) e^{\textrm{i}G^{(n)}_{\textrm{inv}}(0)}\,, $$

where $G^{(n)}_{\textrm{inv}}$ solves

$$\begin{aligned} \textrm{i}\left[ J \cdot N, G^{(n)}_{\textrm{inv}}(\varphi )\right] + \omega \cdot \partial _\varphi {G}^{(n)}_{\textrm{inv}}(\varphi ) + V^{(n)}_{\textrm{inv}}( \varphi ) - \langle V^{(n)}_{\textrm{inv}} \rangle = 0\,, \end{aligned}$$

(5.10)

and $\langle \cdot \rangle$ is defined in (4.31). Recalling that $H^{(n)}_{\textrm{inv}} = J \cdot N + Z^{(n)}_{\textrm{inv}} + V^{(n)}_{\textrm{inv}}$, by Lemma 5.2 one has $\widetilde{U}^{(n)}_{\textrm{inv}}(t) = U_{H^{(n+1)}_{\textrm{inv}}}(t)$, with $H^{(n+1)}_{\textrm{inv}}(\varphi )$ given by (5.9):

$$\begin{aligned} H^{(n+1)}_{\textrm{inv}}(\varphi )&= J \cdot N + Z^{(n)}_{\textrm{inv}} + \langle V^{(n)}_{\textrm{inv}} \rangle \end{aligned}$$

(5.11a)

$$\begin{aligned}&+ \textrm{i}\left[ J \cdot N, G^{(n)}_{\textrm{inv}}(\varphi )\right] + \omega \cdot \partial _\varphi {G}^{(n)}_{\textrm{inv}}(\varphi ) + V^{(n)}_{\textrm{inv}}( \varphi ) - \langle V^{(n)}_{\textrm{inv}} \rangle \end{aligned}$$

(5.11b)

$$\begin{aligned}&+ e^{-\textrm{i}G^{(n)}_{\textrm{inv}}(\varphi )} (J \cdot N) e^{\textrm{i}G^{(n)}_{\textrm{inv}}(\varphi )} - J \cdot N - \textrm{i}[J \cdot N, G^{(n)}_{\textrm{inv}}(\varphi )] \end{aligned}$$

(5.11c)

$$\begin{aligned}&+ e^{-\textrm{i}G^{(n)}_{\textrm{inv}}(\varphi )} V^{(n)}_{\textrm{inv}}( \varphi ) e^{\textrm{i}G^{(n)}_{\textrm{inv}}(\varphi )} - V^{(n)}_{\textrm{inv}}( \varphi ) \end{aligned}$$

(5.11d)

$$\begin{aligned}&+ e^{-\textrm{i}G^{(n)}_{\textrm{inv}}( \varphi )} Z^{(n)}_{\textrm{inv}} e^{\textrm{i}G^{(n)}_{\textrm{inv}}(\varphi )} - Z^{(n)}_{\textrm{inv}} \end{aligned}$$

(5.11e)

$$\begin{aligned}&+ \int _0^1 e^{-\textrm{i}G^{(n)}_{\textrm{inv}}(\varphi )s} \omega \cdot \partial _\varphi G^{(n)}_{\textrm{inv}}( \varphi ) e^{\textrm{i}G^{(n)}_{\textrm{inv}}(\varphi )s }\,\textrm{d}s - \omega \cdot \partial _\varphi {G}^{(n)}_{\textrm{inv}}( \varphi )\,. \end{aligned}$$

(5.11f)

We set

$$\begin{aligned} Z^{(n+1)}_{\textrm{inv}} := Z^{(n)}_{\textrm{inv}} + \langle V^{(n)}_{\textrm{inv}} \rangle \,, \quad V^{(n+1)}_{\textrm{inv}} := \text {(5.11c)} + \text {(5.11d)} + \text {(5.11e)} + \text {(5.11f)} \,. \end{aligned}$$

(5.12)

We observe that, due to our choice of the parameters $\{\kappa _n\}_{n=1}^{n_*}$, $\{\rho _n\}_{n=1}^{n_*}$, one has $\kappa _n = {\frac{(\texttt {c} + 1)}{2}} \rho _n$ $\forall n$, thus defining $\delta _n:=\frac{\rho _n-\rho _{n+1}}{2}$, also

$$ \kappa _{n} - \kappa _{n+1} = {\frac{( \texttt {c} + 1)}{2}} (\rho _n -\rho _{n+1}) = ( \texttt {c} + 1) \delta _n $$

and $\kappa _{n} - \texttt {c} \delta _n = \kappa _{n+1} + \delta _n$. Using Lemma 4.10, we find $G^{(n)}_{\textrm{inv}}$ such that (5.10) holds and, using the inductive hypothesis on $V^{(n)}_{\textrm{inv}}$,

$$\begin{aligned} \begin{aligned} \Vert G^{(n)}_{\textrm{inv}}\Vert _{\kappa _{n+1} + \delta _{n}, \rho _{n+1}+\delta _n}&\le \frac{2^{2r}\tau ^{\tau }(1+\delta _n)^r}{e^{\tau } \gamma \delta _{n}^{\tau +r}} \Vert V^{(n)}_{\textrm{inv}}\Vert _{\kappa _n, \rho _n} \le \frac{2^{2r}\tau ^{\tau }(\delta _n+1)^r}{e^{\tau } \gamma \delta _{n}^{\tau +r}} \varepsilon _n\,, \\ \Vert \langle V^{(n)}_{\textrm{inv}} \rangle \Vert _{\kappa _{n+1} + \delta _n}&\le \Vert V^{(n)}_{\textrm{inv}}\Vert _{\kappa _{n}, \rho _n} \le \varepsilon _n\,. \end{aligned} \end{aligned}$$

(5.13)

In particular, the second estimate in (5.13) implies (5.6) with n replaced by $n+1$. Furthermore, since by Lemma 4.8$\langle V^{(n+1)}_{\textrm{inv}}\rangle$ commutes with all the $N^{(\alpha )}$’s, also the first condition in (5.6) is still satisfied also at $n+1$. We now estimate $V^{(n+1)}_{\textrm{inv}}$. By Lemma 4.2, if

$$\begin{aligned} \frac{4 e^{-\kappa _{n+1}}}{\delta _{n}}\Vert G^{(n)}_{\textrm{inv}}\Vert _{\kappa _{n+1} + \delta _n, \rho _{n+1}+\delta _n} < \frac{1}{2}\,, \end{aligned}$$

(5.14)

we have

$$\begin{aligned} \begin{aligned} \Vert \, \text {(5.11d)} \, \Vert _{\kappa _{n+1},\rho _{n+1}}&\le \frac{8 e^{-\kappa _{n+1}}}{\delta _n} \Vert G^{(n)}_{\textrm{inv}} \Vert _{\kappa _{n+1}+\delta _n,\rho _{n+1}+\delta _n} \Vert V^{(n)}_{\textrm{inv}} \Vert _{\kappa _{n+1}+\delta _n,\rho _{n+1}+\delta _n} \, , \\ \Vert \, \text {(5.11e)} \, \Vert _{\kappa _{n+1},\rho _{n+1}}&\le \frac{8 e^{-\kappa _{n+1}}}{\delta _n} \Vert G^{(n)}_{\textrm{inv}} \Vert _{\kappa _{n+1}+\delta _n,\rho _{n+1}+\delta _n} \Vert Z^{(n)}_{\textrm{inv}} \Vert _{\kappa _{n+1}+\delta _n} \, , \\ \Vert \, \text {(5.11f)} \, \Vert _{\kappa _{n+1},\rho _{n+1}}&\le \frac{8 |\omega |e^{-\kappa _{n+1}}}{\delta _n^2 } \Vert G^{(n)}_{\textrm{inv}} \Vert _{\kappa _{n+1}+\delta _n,\rho _{n+1}+\delta _n}^2 \, , \\ \Vert \, \text {(5.11c)}\, \Vert _{\kappa _{n+1}, \rho _{n+1}}&\le \frac{8 e^{-\kappa _{n+1}}}{\delta _n} \Vert G^{(n)}_{\textrm{inv}} \Vert _{\kappa _{n+1}+\delta _n,\rho _{n+1}+\delta _n} \Vert \omega \cdot \partial _\varphi G^{(n)}_{\textrm{inv}}+V^{(n)}_{\textrm{inv}}- \langle V^{(n)}_{\textrm{inv}} \rangle \Vert _{\kappa _{n+1}+\delta _n,\rho _{n+1}} \, , \end{aligned} \end{aligned}$$

(5.15)

where to obtain the last inequality we have used the fact that $-\textrm{i}[J \cdot N, G^{(n)}_{\textrm{inv}}(\varphi )] = \omega \cdot \partial _\varphi G^{(n)}_{\textrm{inv}} + V^{(n)}_{\textrm{inv}} - \langle V^{(n)}_{\textrm{inv}} \rangle$, since $G^{(n)}_{\textrm{inv}}$ solves (5.10). Then the inductive estimate on $V^{(n)}_n$ follows if all the terms in (5.15) can be bounded by $\frac{\varepsilon _{n+1}}{4}$. Observing that

$$\begin{aligned} \begin{aligned} \delta _{n} =&\frac{\rho _{n} - \rho _{n+1}}{2} = \frac{1}{2}\rho _0 \sqrt{ 1 - b n}\left( 1 - \sqrt{1-\frac{b}{1-bn}}\right) \\ \ge&\frac{\rho _0}{2\sqrt{ 2}} \left( 1-\sqrt{1-b}\right) \ge \frac{\rho _0}{\sqrt{ 2}} \frac{b}{4} = \frac{\rho _0}{ 8 \sqrt{2} n_*}\,, \end{aligned} \end{aligned}$$

(5.16)

both the $\frac{\varepsilon _{n+1}}{4}$ bounds and condition (5.14) are satisfied, choosing $n_*$ as in (5.1) and $\varepsilon < \varepsilon _0$ for some $\varepsilon _0$ depending only on $r, \tau , \gamma , \kappa , \rho , \Omega , \Vert V\Vert _{\kappa , \rho }, \{ \Vert N^{(\alpha )}\Vert _{0}\}_{\alpha }$.

Last, we show that

$$ Y^{(n+1)}_{\textrm{inv}}(\varphi ):= e^{-\textrm{i}G^{(n)}_{\textrm{inv}}( \varphi )} Y^{(n)}_{\textrm{inv}}( \varphi ) $$

satisfies (5.8). To this aim we observe that (5.13), (5.16) and (5.1) imply

$$\begin{aligned} \frac{8 e^{-\kappa _{n+1}}}{\delta _{n}}\Vert G^{(n)}_{\textrm{inv}}\Vert _{\kappa _{n+1} + \delta _n, \rho _{n+1}} < \varepsilon ^{\frac{1}{2}} \Vert V\Vert _{\kappa _0, \rho _0} \left( {\frac{1}{e}}\right) ^n\,. \end{aligned}$$

(5.17)

Thus, due to (5.17), by Lemma 4.2 one has $\Vert e^{\textrm{i}G^{(n)}_{\textrm{inv}}} P e^{-\textrm{i}G^{(n)}_{\textrm{inv}}} - P\Vert _{\kappa _{n+1}, \rho _{n+1}} \le \varepsilon ^{\frac{1}{2}} \left( {\frac{1}{e}}\right) ^n \Vert V\Vert _{\kappa _0, \rho _0} \Vert P\Vert _{\kappa _{0}, \rho _0}$. Then (5.8) follows estimating

$$\begin{aligned} \Vert Y_\textrm{inv}^{(n+1)} P (Y_\textrm{inv}^{(n+1)})^* \Vert _{\kappa _{n+1},\rho _{n+1}}&=\Vert e^{-\textrm{i}G_\textrm{inv}^{(n)}} Y_\textrm{inv}^{(n)} P (Y_\textrm{inv}^{(n)})^* e^{\textrm{i}G_\textrm{inv}^{(n)}}-P \Vert _{\kappa _{n+1},\rho _{n+1}} \\&\le \Vert e^{-\textrm{i}G_\textrm{inv}^{(n)}} (Y^{(n)}_\textrm{inv}P (Y_\textrm{inv}^{(n)})^*-P) e^{\textrm{i}G^{(n)}_\textrm{inv}} - (Y_\textrm{inv}^{(n)} P (Y_\textrm{inv}^{(n)})^*-P) \Vert _{\kappa _{n+1},\rho _{n+1}} \\&\quad +\Vert Y^{(n)}_\textrm{inv}P (Y^{(n)}_\textrm{inv})^*-P \Vert _{\kappa _{n+1},\rho _{n+1}} + \Vert e^{-\textrm{i}G^{(n)}_\textrm{inv}} P e^{\textrm{i}G^{(n)}_\textrm{inv}} - P \Vert _{\kappa _{n+1},\rho _{n+1}} \\&\le \varepsilon ^{\frac{1}{2}} \Vert V\Vert _{\kappa _0, \rho _0} e^{-n} \left( 2 \varepsilon ^{\frac{1}{2}} \Vert V\Vert _{\kappa _0,\rho _0} \sum _{n'=0}^{n-1} e^{-n'} \Vert P \Vert _{\kappa _0,\rho _0} \right) \\&\quad + 2 \varepsilon ^{\frac{1}{2}} \Vert V \Vert _{\kappa _0,\rho _0} \sum _{n'=0}^{n-1} e^{-n'} \Vert P \Vert _{\kappa _0,\rho _0} + \varepsilon ^{\frac{1}{2}} e^{-n} \Vert V \Vert _{\kappa _0,\rho _0} \Vert P \Vert _{\kappa _0,\rho _0} \\&\le 2 \varepsilon ^{\frac{1}{2}} \Vert V \Vert _{\kappa _0,\rho _0} \sum _{n'=0}^{n} e^{-n'} \Vert P \Vert _{\kappa _0,\rho _0} \end{aligned}$$

where in the last step we have assumed ${2 \varepsilon ^{\frac{1}{2}}\Vert V\Vert _{\kappa _0, \rho _0}} \sum _{n'=0}^{{n-1}} e^{{-n'}} \le 1$, which is satisfied provided $\varepsilon \le \left( {\frac{2e\Vert V\Vert _{\kappa _0, \rho _0}}{e-1}}\right) ^{-2}$. $\square$

Analogously, Item (b) of Proposition 3.1 immediately follows from the following lemma.

Lemma 5.3

Under the assumptions of Proposition 3.1, define

$$\begin{aligned} n^* := \left\lfloor \varepsilon ^{-\texttt {b} }\right\rfloor \,, \end{aligned}$$

(5.18)

with $\texttt {b}$ as in (2.13), and $\forall n = 1, \dots , n^*$ define

$$\begin{aligned} \kappa _n := {\kappa _0}\sqrt{1 - b n}\,, \quad \rho _n := {\rho _0}\sqrt{1 - b n}\,,\quad b := \frac{1}{2 n^*}\,, \end{aligned}$$

(5.19)

with $\kappa _0, \rho _0$ as in (5.1) and $\texttt {c}$ as in (4.38). Let $\{\varepsilon _n\}_{n= 0}^{n^*}$ be defined with $\varepsilon _n$ as in (5.3), then $\forall n = 0, \dots , n^*$ there exist a time quasi-periodic family of unitary maps $Y^{(n)}_{\textrm{obs}}(\omega t)$, a self-adjoint operator $Z^{(n)}_{\textrm{obs}}$ and a time quasi-periodic family of self-adjoint operators $V^{(n)}_{\textrm{obs}}(\omega t)$ such that, defining

$$\begin{aligned} H^{(n)}_{\textrm{obs}}(\omega t):= J \cdot N + Z^{(n)}_{\textrm{obs}} + V^{(n)}_{\textrm{obs}}( \omega t) \,, \end{aligned}$$

(5.20)

one has

$$\begin{aligned} U_{H^{(n)}_{\textrm{obs}}}(t) = Y^{(n)}_{\textrm{obs}}( \omega t) U_{H}(t)\,, \end{aligned}$$

(5.21)

and the following are satisfied:

• 1. $Z^{(n)}_{\textrm{obs}} \in \mathcal {O}^\texttt {s} _{\kappa _n}$ is time independent, with $Z^{(0)}_\textrm{obs}=0$ and for $n>0$

  $$\begin{aligned} \Vert Z^{(n)}_{\textrm{obs}}\Vert _{\kappa _n} \le \varepsilon ^{-\frac{1}{2}}\sum _{n'=0}^{{n-1}} \varepsilon _{{n'}}\,; \end{aligned}$$

  (5.22)

• 2. $V^{(n)}_{\textrm{obs}} \in \mathcal {O}^\texttt {s} _{\kappa _n, \rho _n}$, and

  $$\begin{aligned} \Vert V^{(n)}_{\textrm{obs}}\Vert _{\kappa _n, \rho _n} \le \varepsilon _n\,; \end{aligned}$$

  (5.23)

• 3. The maps $Y^{(n)}_{\textrm{obs}}(\omega t)$ are such that $Y^{(n)}_{\textrm{obs}}(0) = \mathbb {1}$ and they are close to the identity, in the sense that $Y_\textrm{obs}^{(0)}(\omega t)\equiv \mathbb {1}$ and for $n>0$

  $$\begin{aligned} \Vert Y^{(n)}_{\textrm{obs}} P (Y^{(n)}_{\textrm{obs}})^*- P\Vert _{\kappa _n, \rho _n} \le 2 \varepsilon ^{\frac{1}{2}} \sum _{n'=0}^{{n-1}} e^{{-n'}}\Vert V\Vert _{\kappa , \rho } \Vert P\Vert _{\kappa , \rho } \quad \forall P \in \mathcal {O}_{\kappa , \rho }\,. \end{aligned}$$

  (5.24)

  Moreover, for $n>0$ they are of the form $Y^{(n)}_\textrm{obs}(\omega t) = e^{-\textrm{i}G^{({n-1})}_\textrm{obs}(\omega t)} \cdots e^{-\textrm{i}G^{(0)}_\textrm{obs}(\omega t)}$, with

  $$\begin{aligned} \Vert G^{(n)}_\textrm{obs}\Vert _{\kappa _{n+1}, \rho _{n+1}} \le \varepsilon ^{\frac{1}{2}} e^{-n} \Vert V\Vert _{\kappa _0, \rho _0}\,. \end{aligned}$$

  (5.25)

Proof

The proof follows the same scheme of the proof of Lemma 5.1, thus we only sketch the procedure highlighting the differences. At any step n one looks for a change of coordinates of the form

$$ \widetilde{U}^{(n)}_{\textrm{obs}}(t) = e^{-\textrm{i}G^{(n)}_{\textrm{obs}}(\omega t)} U_{H^{(n)}_{\textrm{obs}}}(t)\,, $$

where $G^{(n)}_{\textrm{obs}}$ solves

$$\begin{aligned} \textrm{i}\left[ J \cdot N, G^{(n)}_{\textrm{obs}}(\varphi )\right] + \omega \cdot \partial _\varphi {G}^{(n)}_{\textrm{obs}}(\varphi ) + V^{(n)}_{\textrm{obs}}( \varphi ) - \langle \! \langle V^{(n)}_{\textrm{obs}} \rangle \! \rangle = 0\,, \end{aligned}$$

(5.26)

with $\langle \! \langle V^{(n)}_\textrm{obs}\rangle \! \rangle$ as in (4.45). Then by Lemma 5.2 one has

$$\begin{aligned} H^{(n+1)}_{\textrm{obs}}(\varphi )&= J \cdot N + Z^{(n)}_{\textrm{obs}} + \langle \! \langle V^{(n)}_{\textrm{obs}} \rangle \! \rangle \nonumber \\&\qquad + \textrm{i}\left[ J \cdot N, G^{(n)}_{\textrm{obs}}(\varphi )\right] + \omega \cdot \partial _\varphi {G}^{(n)}_{\textrm{obs}}(\varphi ) + V^{(n)}_{\textrm{obs}}( \varphi ) - \langle \! \langle V^{(n)}_{\textrm{obs}} \rangle \! \rangle \end{aligned}$$

(5.27a)

$$\begin{aligned}&\qquad + e^{-\textrm{i}G^{(n)}_{\textrm{obs}}(\varphi )} (J \cdot N) e^{\textrm{i}G^{(n)}_{\textrm{obs}}(\varphi )} - J \cdot N - \textrm{i}[J \cdot N, G^{(n)}_{\textrm{obs}}(\varphi )] \end{aligned}$$

(5.27b)

$$\begin{aligned}&\qquad + e^{-\textrm{i}G^{(n)}_{\textrm{obs}}(\varphi )} V^{(n)}_{\textrm{obs}}( \varphi ) e^{\textrm{i}G^{(n)}_{\textrm{obs}}(\varphi )} - V^{(n)}_{\textrm{obs}}( \varphi ) \end{aligned}$$

(5.27c)

$$\begin{aligned}&\qquad + e^{-\textrm{i}G^{(n)}_{\textrm{obs}}( \varphi )} Z^{(n)}_{\textrm{obs}} e^{\textrm{i}G^{(n)}_{\textrm{obs}}(\varphi )} - Z^{(n)}_{\textrm{obs}} \end{aligned}$$

(5.27d)

$$\begin{aligned}&\qquad + \int _0^1 e^{-\textrm{i}G^{(n)}_{\textrm{obs}}(\varphi )s} \omega \cdot \partial _\varphi G^{(n)}_{\textrm{obs}}( \varphi ) e^{\textrm{i}G^{(n)}_{\textrm{obs}}(\varphi )s }\,\textrm{d}s - \omega \cdot \partial _\varphi {G}^{(n)}_{\textrm{obs}}( \varphi )\,. \end{aligned}$$

(5.27e)

Then one defines

$$\begin{aligned} Z_{\textrm{obs}}^{(n+1)} := Z_{\textrm{obs}}^{(n)} + \langle \! \langle V^{(n)}_{\textrm{obs}} \rangle \! \rangle \,, \quad V_{\textrm{obs}}^{(n+1)} = \text {(5.27b)} + \text {(5.27c)}+ \text {(5.27d)} + \text {(5.27e)} \,, \end{aligned}$$

(5.28)

and defining $\delta _n:= \frac{\rho _n - \rho _{n+1}}{2}$, Lemma 4.11 implies

$$\begin{aligned} \begin{aligned} \Vert G^{(n)}_\textrm{obs}\Vert _{\kappa _n -\texttt{c}\delta _n, \rho _n-\delta _n} \le \frac{2^{2r+1} \tau ^{\tau }}{e^{\tau } \gamma \delta _n^{\tau +r}}(1+\delta _{{n}})^r \Vert V^{(n)}_\textrm{obs}\Vert _{\kappa , \rho }\,, \\ \Vert \langle \! \langle V^{(n)}_\textrm{obs}\rangle \! \rangle \Vert _{\kappa _n - \texttt{c} \delta _n} \le \left( \frac{\mathcal {J} 2 ^{2r}(1 +\delta _n)^r \tau ^\tau }{\gamma e^\tau \delta _n^{\tau + r + 1}} + 1 \right) \Vert V^{(n)}_\textrm{obs}\Vert _{\kappa _n, \rho _n}\,. \end{aligned} \end{aligned}$$

(5.29)

Then one argues as in the proof of Lemma 5.1, with the only difference that for any $n = 0, \dots , n^*$ one has

$$\begin{aligned} \Vert \langle \! \langle V^{(n)}_\textrm{obs}\rangle \! \rangle \Vert _{\kappa _n - \texttt{c} \delta _n} \le \left( \frac{\mathcal {J} 2 ^{2r}(1 +\delta _n)^r \tau ^\tau }{\gamma e^\tau \delta _n^{\tau + r + 1}} + 1 \right) \varepsilon _n \le \frac{\mathcal {J} 2 ^{4r + 1}\tau ^\tau }{\gamma e^\tau } \left( \frac{8 \sqrt{2} n^*}{\rho _0}\right) ^{\tau + r + 1} \varepsilon _n \le \varepsilon ^{-\frac{1}{2}} \varepsilon _n\,, \end{aligned}$$

(5.30)

provided $\varepsilon \le \varepsilon _0$ with $\varepsilon _0$ small enough and depending on $\mathcal {J}, r, \gamma , \tau , \rho _0$. Summing over n and recalling the definition of $Z^{(n+1)}_\textrm{obs}$ as in (5.28), by estimate (5.30) one gets (5.22). Equation (5.25) is verified using (5.29) together with the inductive assumption (5.23) on $V^{(n)}_\textrm{obs}$, and (5.24) is proven defining recursively $Y_\textrm{obs}^{(n+1)}(\varphi ):= e^{-\textrm{i}G^{(n)}_\textrm{obs}(\varphi )} Y^{(n)}_{\textrm{obs}}(\varphi )$ and arguing as in the proof of Lemma 5.1. $\square$

Proof of Proposition 3.1

In order to obtain Item (a), it is sufficient to take $Z_\textrm{inv}:= Z^{(n_*)}_\textrm{inv}$, $V_\textrm{inv}:= V^{(n_*)}_\textrm{inv}$, $H_\textrm{inv}:= H^{(n_*)}_\textrm{inv}$ and $Y_\textrm{inv}:= Y^{(n_*)}_\textrm{inv}$, with $Z^{(n_*)}_\textrm{inv},\ V^{(n_*)}_\textrm{inv}\, H^{(n_*)}_\textrm{inv},\ Y^{(n_*)}_\textrm{inv}$ and $n_*$ as in Lemma 5.1. Analogously, Item (b) is obtained taking $Z_\textrm{obs}:= Z^{(n^*)}_\textrm{obs}$, $V_\textrm{obs}:= V^{(n^*)}_\textrm{obs}$, $H_\textrm{obs}:= H^{(n^*)}_\textrm{obs}$ and $Y_\textrm{obs}:= Y^{(n^*)}_\textrm{obs}$, with $Z^{(n^*)}_\textrm{obs},\ V^{(n^*)}_\textrm{obs}\, H^{(n^*)}_\textrm{obs},\ Y^{(n^*)}_\textrm{obs}$ and $n^*$ as in Lemma 5.3. $\square$

5.2 Fast Forcing Perturbations

Lemma 5.4

Under the assumptions of Proposition 3.3, define

$$\begin{aligned} n_\star := \left\lfloor \lambda ^{2\beta }\right\rfloor \,, \quad \kappa _0:= \min \{\kappa , \rho \}\,, \quad \rho _0:=2 (\texttt{c} + 1)^{-1} \kappa _0\,, \end{aligned}$$

(5.31)

with $\texttt{c}$ as in (4.38) and $\beta$ as in (2.23), and define $\forall n = 1, \dots , n_\star$

$$\begin{aligned} \kappa _n := {\kappa _0}\sqrt{1 - b n}\,, \quad \rho _n := {\rho _0}\sqrt{1 - b n}\,,\quad b = \frac{1}{2 n_\star }\,, \end{aligned}$$

(5.32)

and $\forall n = 0, \dots , n_\star$

$$\begin{aligned} \varepsilon _n := \left( \frac{1}{e}\right) ^{n-1} \lambda ^{-\frac{1}{4\tau _\omega }} \Vert V\Vert _{\kappa _0, \rho _0}\,. \end{aligned}$$

(5.33)

Then, for any $n = 0, \dots , n_\star$ there exist a time quasi-periodic family of unitary maps $Y^{(n)}_\textrm{inv}(\lambda \omega t)$, a self-adjoint operator $Z^{(n)}_\textrm{inv}$ and a time quasi-periodic family of self-adjoint operators $V_{\textrm{inv}}^{(n)}(\lambda \omega t)$ such that, defining

$$\begin{aligned} H^{(n)}_\textrm{inv}(t):= J \cdot N + Z^{(n)}_\textrm{inv}+ V^{(n)}_\textrm{inv}( \lambda \omega t) \,, \end{aligned}$$

(5.34)

one has

$$\begin{aligned} U_{H^{(n)}_\textrm{inv}}(t) = Y^{(n)}_{\textrm{inv}}(\lambda \omega t) U_{H}(t) (Y^{(n)}_\textrm{inv})^*(0)\,, \end{aligned}$$

(5.35)

and the following are satisfied:

• 1. $Z^{(n)}_\textrm{inv}\in \mathcal {O}^\texttt {s} _{\kappa _n}$ is time independent, $Z^{(0)}_\textrm{inv}=0$, and for any $n>0$

  $$\begin{aligned}{}[Z^{(n)}_\textrm{inv}, N^{(\alpha )}] = 0 \quad \forall \alpha = 1, \dots , r\,, \quad \Vert Z^{(n)}_\textrm{inv}\Vert _{\kappa _n} \le \sum _{n'=0}^{{n-1}} \varepsilon _{{n'}}\,; \end{aligned}$$

  (5.36)

• 2. $V^{(n)}_\textrm{inv}\in \mathcal {O}^\texttt {s} _{\kappa _n, \rho _n}$, and

  $$\begin{aligned} \Vert V^{(n)}_\textrm{inv}\Vert _{\kappa _n, \rho _n} \le \left\{ \begin{array}{lll} \varepsilon _{n}\,, &{} \qquad &{} n \ge 1 \\ &{} &{} \\ \Vert V \Vert _{\kappa _0,\rho _0} \, , &{} &{}n=0 \end{array} \right. ; \end{aligned}$$

  (5.37)

• 3. The maps $Y^{(n)}_\textrm{inv}(\lambda \omega t)$ are close to the identity, in the sense that $Y^{(0)}_\textrm{inv}(\lambda \omega t)\equiv \mathbb {1}$, and for any $n>0$

  $$\begin{aligned} \Vert Y^{(n)}_\textrm{inv}P (Y^{(n)}_\textrm{inv})^*- P\Vert _{\kappa _n, \rho _n} \le 2 \lambda ^{\frac{1}{8 \tau _\omega }} \sum _{n'=0}^{n-1}\varepsilon _{n'} \Vert P\Vert _{\kappa _n, \rho _n} \quad \forall P \in \mathcal {O}_{\kappa _n, \rho _n}\,. \end{aligned}$$

  (5.38)

Proof

The proof is analogous to the proof of Lemma 5.1 with some differences that we highlight here. Instead of (5.10), $G^{(n)}_\textrm{inv}$ solves

$$\begin{aligned} \textrm{i}\left[ J \cdot N, G^{(n)}_\textrm{inv}(\varphi )\right] + \lambda \omega \cdot \partial _\varphi {G}^{(n)}_\textrm{inv}(\varphi ) + (V^{(n)}_{\textrm{inv}})^\texttt {ir} ( \varphi ) - \langle V^{(n)}_\textrm{inv}\rangle = 0\,, \end{aligned}$$

(5.39)

with $\langle V^{(n)}_\textrm{inv}\rangle$ defined in (4.31). In the transformed Hamiltonian, we also split the $\texttt {uv}$/$\texttt {ir}$ parts and we have to control the following terms

$$\begin{aligned} H^{(n+1)}_\textrm{inv}&= J \cdot N + Z^{(n)}_\textrm{inv}+ \langle V^{(n)}_\textrm{inv}\rangle \nonumber \\&\qquad + \textrm{i}\left[ J \cdot N, G^{(n)}_\textrm{inv}\right] + \lambda \omega \cdot \partial _\varphi {G}^{(n)}_\textrm{inv}+ (V_\textrm{inv}^{(n)})^\texttt {ir} - \langle V^{(n)}_\textrm{inv}\rangle \end{aligned}$$

(5.40a)

$$\begin{aligned}&\qquad + e^{-\textrm{i}G^{(n)}_\textrm{inv}} (J \cdot N) e^{\textrm{i}G^{(n)}_\textrm{inv}} - J \cdot N - [J \cdot N, G^{(n)}_\textrm{inv}] \end{aligned}$$

(5.40b)

$$\begin{aligned}&\qquad + e^{-\textrm{i}G^{(n)}_\textrm{inv}} V^{(n)}_\textrm{inv}e^{\textrm{i}G^{(n)}_\textrm{inv}} - V^{(n)}_\textrm{inv}\end{aligned}$$

(5.40c)

$$\begin{aligned}&\qquad + e^{-\textrm{i}G^{(n)}_\textrm{inv}} Z^{(n)}_\textrm{inv}e^{\textrm{i}G_n} - Z^{(n)}_\textrm{inv}\end{aligned}$$

(5.40d)

$$\begin{aligned}&\qquad + \int _0^1 e^{-\textrm{i}G^{(n)}_\textrm{inv}s}\lambda (\omega \cdot \partial _\varphi ) (G^{(n)}_\textrm{inv}) e^{\textrm{i}G^{(n)}_\textrm{inv}s}\,\textrm{d}s - \lambda \omega \cdot \partial _\varphi G^{(n)}_\textrm{inv}\,. \end{aligned}$$

(5.40e)

One then sets $Z_\textrm{inv}^{(n+1)}:=Z_\textrm{inv}^{(n)}+\langle V^{(n)}_\textrm{inv}\rangle$ and, differently from (5.12), one has to take into account the term (5.40a$^{\prime }$):

$$\begin{aligned} V^{(n+1)}_\textrm{inv}:= (5.40\hbox {a}^\prime ) + \text {(5.40b)} + \text {(5.40c)} + \text {(5.40d)} + \text {(5.40e)}\,. \end{aligned}$$

(5.41)

We observe that, due to our choice of the parameters $\{\kappa _n\}_{n=1}^{n_\star }$, $\{\rho _n\}_{n=1}^{n_\star }$, one has $\kappa _n = (\texttt {c} + 1) \rho _n$ $\forall n$, thus also

$$ \kappa _{n} - \kappa _{n+1} = ( \texttt {c} + 2) (\rho _n -\rho _{n+1}) = ( \texttt {c} + 2) \delta _n $$

and $\kappa _{n} - \texttt {c} \delta _n = \kappa _{n+1} + 2 \delta _n$. Using Corollary 4.15 we find $G^{(n)}_\textrm{inv}$ solving (5.39), and the operators $G^{(n)}_\textrm{inv}, \langle V^{(n)}_\textrm{inv}\rangle$ satisfy

$$\begin{aligned} \begin{aligned} \begin{aligned} \Vert G^{(n)}_\textrm{inv}\Vert _{\kappa _{n+1} + 2 \delta _n, \rho _n}&\le \frac{2^{2r+2}}{\gamma _\omega \lambda ^{1/2}} \left( \frac{1+\delta _n}{\delta _n}\right) ^r \Vert V^{(n)}_\textrm{inv}- \langle V^{(n)}_\textrm{inv}\rangle _{\mathbb {T}^m} \Vert _{\kappa _n,\rho _n}\\&\quad + \frac{2^{2r+1}\tau _J^{\tau _J}}{\gamma _J e^{\tau _J} \delta _n^{\tau _J}}\left( \frac{1+\delta _n}{\delta _n}\right) ^r \Vert \langle V^{(n)}_\textrm{inv}\rangle _{\mathbb {T}^m} \Vert _{\kappa _n ,0}\\&\le \frac{2^{2r+2}}{\gamma _\omega \lambda ^{1/2}}\left( \frac{1+\delta _n}{\delta _n}\right) ^r \varepsilon _n + \frac{2^{2r+1}\tau _J^{\tau _J}}{\gamma _J e^{\tau _J} \delta _n^{\tau _J}} \left( \frac{1+\delta _n}{\delta _n}\right) ^r \varepsilon _n \quad \text {if} \quad n \ge 1\,, \end{aligned}\\ \Vert G^{(0)}_\textrm{inv}\Vert _{\kappa _{1} + 2\delta _0, \rho _0} \le \frac{2^{2r+2}}{\gamma _\omega \lambda ^{1/2}}\left( \frac{1+\delta _0}{\delta _0}\right) ^r \Vert V^{(0)}_\textrm{inv}\Vert _{\kappa _0,\rho _0} \le \frac{2^{2r+3}}{\gamma _\omega \lambda ^{1/2}} \left( \frac{1+\delta _0}{\delta _0}\right) ^r\Vert V\Vert _{\kappa _0, \rho _0}\,,\\ \Vert \langle V^{(n)}_\textrm{inv}\rangle \Vert _{\kappa _{n+1} +2 \delta _n} \le \Vert V^{(n)}_\textrm{inv}\Vert _{\kappa _n, \rho _n} \le \varepsilon _n \quad \text {if} \quad n \ge 1\,,\\ \Vert \langle V^{(0)}_\textrm{inv}\rangle \Vert _{\kappa _{1} + 2\delta _0} = 0\,, \end{aligned} \end{aligned}$$

(5.42)

where the case $n=0$ is treated differently from all other cases $n\ge 1$ since $\langle V \rangle _{\mathbb {T}^m} =0$. The terms (5.40b), (5.40c), (5.40d) are treated analogously as in the proof of Proposition 5.1 (with obvious adaptation of the norms), while for (5.40a$^{\prime }$) we use Corollary 4.13 and for (5.40e) we use the fact that $\lambda \omega \cdot \partial _\varphi G^{(n)}_\textrm{inv}$ solves the homological equation (5.39), thus obtaining

$$\begin{aligned} \Vert (5.40\hbox {a}^{\prime }) \Vert _{\kappa _{n+1},\rho _{n+1}}\le & {} \frac{2^{2r+1}}{e \delta _n^{r+1}}(1+\delta _n)^r \lambda ^{-\frac{1}{2\tau _\omega }} \Vert V^{(n)}_\textrm{inv}\Vert _{\kappa _n,\rho _n} \nonumber \\ \Vert \hbox {(5.40e)} \Vert _{\kappa _{n+1},\rho _{n+1}}\le & {} \frac{8 e^{-\kappa _{n+1}}}{\delta _{n}}\Vert G^{(n)}_\textrm{inv}\Vert _{\kappa _{n+1} + 2\delta _n, \rho _{n+1}} \Vert \textrm{i}[J \cdot N, G^{(n)}_\textrm{inv}] + V^{(n)}_\textrm{inv}- \langle V^{(n)}_\textrm{inv}\rangle \Vert _{\kappa _{n+1} + \delta _n, \rho _{n+1}}. \end{aligned}$$

(5.43)

Then, the inductive hypothesis follows if each of the terms (5.40a$^{\prime }$)-(5.40e) can be bounded by $\frac{\varepsilon _{n+1}}{5}$. Then, repeating with straightforward adaptations the proof of Lemma 5.1, all the bounds are satisfied by choosing $n_\star$ as in (5.31) with $\lambda > \lambda _0$ depending on r, $\tau _\omega$, $\tau _J$, $\gamma _\omega$, $\gamma _J$, $\kappa$, $\rho$, $\Vert H_0\Vert _\kappa$, $\Vert V \Vert _{\kappa ,\rho }$.

To prove (5.38), one uses the same argument of Lemma 5.1 substituting $\varepsilon ^{\frac{1}{2}} \mapsto \lambda ^{-\frac{1}{8\tau _\omega }}$. $\square$

Lemma 5.5

Under the assumptions of Proposition 3.3, define

$$\begin{aligned} n^{\star } := \left\lfloor \lambda ^{\beta }\right\rfloor \,, \end{aligned}$$

(5.44)

with $\beta$ as in (2.23), and $\forall n = 1, \dots , n^{\star }$ define

$$\begin{aligned} \kappa _n := {\kappa _0}\sqrt{1 - b n}\,, \quad \rho _n := {\rho _0}\sqrt{1 - b n}\,,\quad b := \frac{1}{2 n^{\star }}\,, \end{aligned}$$

(5.45)

with $\kappa _0, \rho _0$ as in (5.31) and $\texttt {c}$ as in (4.38). Let $\{\varepsilon _n\}_{n= 0}^{n^{\star }}$ be defined with $\varepsilon _n$ as in (5.33), then $\forall n = 0, \dots , n^{\star }$ there exist a time quasi-periodic family of unitary maps $Y^{(n)}_{\textrm{obs}}(\lambda \omega t)$, a self-adjoint operator $Z^{(n)}_{\textrm{obs}}$ and a time quasi-periodic family of self-adjoint operators $V^{(n)}_{\textrm{obs}}(\lambda \omega t)$ such that, defining

$$\begin{aligned} H^{(n)}_{\textrm{obs}}(\lambda \omega t):= J \cdot N + Z^{(n)}_{\textrm{obs}} + V^{(n)}_{\textrm{obs}}( \lambda \omega t) \,, \end{aligned}$$

(5.46)

one has

$$\begin{aligned} U_{H^{(n)}_{\textrm{obs}}}(t) = Y^{(n)}_{\textrm{obs}}(\lambda \omega t) U_{H}(t)\,, \end{aligned}$$

(5.47)

and the following are satisfied:

• 1. $Z^{(n)}_{\textrm{obs}} \in \mathcal {O}^\texttt {s} _{\kappa _n}$ is time independent, $Z^{(0)}_\textrm{obs}= 0$, and for any $n >0$

  $$\begin{aligned} \Vert Z^{(n)}_{\textrm{obs}}\Vert _{\kappa _n} \le \lambda ^{\frac{1}{8 \tau _\omega }}\sum _{n'=0}^{{n-1}} \varepsilon _{n'}\,; \end{aligned}$$

  (5.48)

• 2. $V^{(n)}_{\textrm{obs}} \in \mathcal {O}^\texttt {s} _{\kappa _n, \rho _n}$, and

  $$\begin{aligned} \Vert V^{(n)}_{\textrm{obs}}\Vert _{\kappa _n, \rho _n} \le \varepsilon _n\,; \end{aligned}$$

  (5.49)

• 3. The maps $Y^{(n)}_{\textrm{obs}}(\lambda \omega t)$ are such that $Y^{(n)}_{\textrm{obs}}(0) = \mathbb {1}$ and they are close to the identity, in the sense that $Y^{(0)}_\textrm{obs}(\lambda \omega t) \equiv \mathbb {1}$, and $\forall n >0$

  $$\begin{aligned} \Vert Y^{(n)}_{\textrm{obs}} P (Y^{(n)}_{\textrm{obs}})^*- P\Vert _{\kappa _n, \rho _n} \le 2 \lambda ^{-\frac{1}{8\tau _\omega }} \sum _{n'=0}^{{n-1}} e^{{-n'}}\Vert V\Vert _{\kappa , \rho } \Vert P\Vert _{\kappa , \rho } \quad \forall P \in \mathcal {O}_{\kappa , \rho }\,. \end{aligned}$$

  (5.50)

  Moreover, they are of the form $Y^{(n)}_\textrm{obs}(\omega t) = e^{-\textrm{i}G^{({n-1})}_\textrm{obs}(\omega t)} \cdots e^{-\textrm{i}G^{(0)}_\textrm{obs}(\omega t)}$ $\forall n >0$, with

  $$\begin{aligned} \Vert G^{(n)}_\textrm{obs}\Vert _{\kappa _{n+1}, \rho _{n+1}} \le \lambda ^{-\frac{1}{8\tau _\omega }} e^{-n} \Vert V\Vert _{\kappa _0, \rho _0}\,. \end{aligned}$$

  (5.51)

Proof

The algebraic scheme of the proof is contained in Appendix H of [25]. For the quantitative estimates, one argues as in the proof of Lemma 5.3 and 5.4. $\square$

Proof of Proposition 3.3

In order to obtain Item (a), it is sufficient to take $Z_\textrm{obs}:= Z^{(n_\star )}_\textrm{inv}$, $V_\textrm{inv}:= V^{(n_\star )}_\textrm{inv}$, $H_\textrm{inv}:= H^{(n_\star )}_\textrm{inv}$ and $Y_\textrm{inv}:= Y^{(n_\star )}_\textrm{inv}$, with $Z^{(n_\star )}_\textrm{inv},\ V^{(n_\star )}_\textrm{inv},\ H^{(n_\star )}_\textrm{inv},\ Y^{(n_\star )}_\textrm{inv}$ and $n_\star$ as in Lemma 5.4. Analogously, Item (b) is obtained taking $Z_\textrm{obs}:= Z^{(n^\star )}_\textrm{obs}$, $V_\textrm{obs}:= V^{(n^\star )}_\textrm{obs}$, $H_\textrm{obs}:= H^{(n^\star )}_\textrm{obs}$ and $Y_\textrm{obs}:= Y^{(n^\star )}_\textrm{obs}$, with $Z^{(n^\star )}_\textrm{obs},\ V^{(n^\star )}_\textrm{obs},\ H^{(n^\star )}_\textrm{obs},\ Y^{(n^\star )}_\textrm{obs}$ and $n^\star$ as in Lemma 5.5. $\square$

6 Proof of Dynamical Consequences

6.1 Small Perturbations

Proof of Theorem 2.1

Let $H_\textrm{inv}, Z_\textrm{inv}, V_\textrm{inv}$ and $Y_\textrm{inv}$ be defined as in Proposition 3.1. Using (3.1), we get

$$ \begin{aligned} \Vert U_H^*(t) N^{(\alpha )} U_H(t)-N^{(\alpha )}\Vert _{\textrm{op}}&\le \Vert U_H^*(t) N^{(\alpha )} U_H(t)-Y_\textrm{inv}^*(0) N^{(\alpha )} Y_\textrm{inv}(0)\Vert _{\textrm{op}} \\&\quad + \Vert Y_\textrm{inv}^*(0) N^{(\alpha )} Y_\textrm{inv}(0)-N^{(\alpha )}\Vert _{\textrm{op}} \\&\le \Vert U_{H_\textrm{inv}}^*(t) Y_\textrm{inv}(\omega t) N^{(\alpha )} Y_\textrm{inv}^*(\omega t) U_{H_\textrm{inv}}(t)- N^{(\alpha )} \Vert _{\textrm{op}}\\&\quad + \Vert Y_\textrm{inv}^*(0) N^{(\alpha )} Y_\textrm{inv}(0)-N^{(\alpha )}\Vert _{\textrm{op}} \\&\le \Vert U_{H_\textrm{inv}}^*(t) Y_\textrm{inv}^*(\omega t) N^{(\alpha )} Y_\textrm{inv}(\omega t) U_{H_\textrm{inv}}(t)-U_{H_\textrm{inv}}^*(t) N^{(\alpha )}U_{H_\textrm{inv}}(t) \Vert _{\textrm{op}} \\&\quad + \Vert Y^*_\textrm{inv}(0) N^{(\alpha )} Y_\textrm{inv}(0)-N^{(\alpha )}\Vert _{\textrm{op}}\\&\quad +\Vert U_{H_\textrm{inv}}^*(t) N^{(\alpha )} U_{H_\textrm{inv}}(t)-N^{(\alpha )} \Vert _{\textrm{op}} \, . \end{aligned} $$

Now we estimate the three terms separately. For the first and the second term, we use Lemma 4.1 and Item (a.iv) of Proposition 3.1. To estimate the third one, we use Duhamel together with the fact that, by Item (a.i) of Proposition 3.1, $[H_{\textrm{inv}},N^{(\alpha )}]=0$, as follows:

$$\begin{aligned} \begin{aligned} U_{H_\textrm{inv}}(t) N^{(\alpha )} U_{H_\textrm{inv}}^*(t)-N^{(\alpha )}&= \textrm{i}\int _0^t U_{H_\textrm{inv}}(s) [H_\textrm{inv}(\omega s),N^{(\alpha )}] U_{H_\textrm{inv}}^*(s) \, \textrm{d}s \\&=\textrm{i}\int _0^t U_{H_\textrm{inv}}(s) [V_\textrm{inv}(\omega s),N^{(\alpha )}] U_{H_\textrm{inv}}^*(s) \, \textrm{d}s \end{aligned} \end{aligned}$$

(6.1)

and we use again Lemma 4.1 and Item (a.iii) of Proposition 3.1. $\square$

In the remaining part of the subsection, we prove Theorem 2.2. The heart of the proof goes as follows: let

$$\begin{aligned} H_{\text {eff}} := J \cdot N + Z_{\textrm{obs}}\,, \end{aligned}$$

(6.2)

with $Z_\textrm{obs}$ defined in Item (b) Theorem 3.1. For any local observable O, one bounds

$$ \left\| U_H^*(t) O U_H(t) - e^{-\textrm{i}H_{\textrm{eff}} t} O e^{\textrm{i}H_{\textrm{eff}}t} \right\| _{\textrm{op}}\, $$

by using triangular inequality and estimates separately

$$\begin{aligned} \left\| U_{H_{\textrm{obs}}}^*(t) O U_{H_\textrm{obs}}(t) - e^{-\textrm{i}H_{\textrm{eff}} t} O e^{\textrm{i}H_{\textrm{eff}}t} \right\| _{\textrm{op}}\,, \quad \left\| U_H^*(t) O U_H(t) - U^*_{H_{\textrm{obs}}}(t) O U_{H_{\textrm{obs}}}(t) \right\| _{\textrm{op}}\,. \end{aligned}$$

(6.3)

Lemma 6.1

(Lieb-Robinson bound [42]) Let $Z=\sum _{S \in \mathcal {P}_c(\Lambda )} Z_S$ be a self-adjoint operator with $\Vert Z \Vert _{2\kappa } < +\infty$ for some $\kappa > 0$. Let the operators A, B act within $S_A,S_B \subset \Lambda$ respectively. Then,

$$\begin{aligned} \Vert [A, e^{\textrm{i}t Z} B e^{-\textrm{i}t Z} ] \Vert _{\textrm{op}} \; \le \; \Vert A \Vert _{\textrm{op}} \Vert B \Vert _{\textrm{op}} e^{-\kappa (d(S_A,S_B)-vt)} \min (|S_A|,|S_B|) \end{aligned}$$

(6.4)

with Lieb-Robinson speed $v=v(Z,\kappa ):=C(d)(\kappa ^{-(d+2)} e^\kappa ) \Vert Z \Vert _{2 \kappa }$ and C(d) only depending on the spatial dimension d.

The following result is proven in [1] for time-periodic operators (see Section 5.1 therein). For completeness, we present a proof of it in Appendix A.

Lemma 6.2

(See also [1]) Let O be a local observable acting within $S_O$, $A \in \mathcal {O}_{\kappa ,0}$ and $Z \in \mathcal {O}_{2 \kappa }$. Then there exists a positive constant ${C}(|S_O|,d,\kappa )$ such that

$$\begin{aligned} \hspace{-0.09cm}\int _0^t \textrm{d}\tau \Vert [A(\omega \tau ),e^{-\textrm{i}\tau Z} O e^{\textrm{i}\tau Z} ] \Vert _{\textrm{op}} \le C(|S_O|,d,\kappa ) \langle \Vert Z \Vert _{2 \kappa } \rangle ^{d+1} \langle t \rangle ^{d+2} \Vert O \Vert _{\textrm{op}} \Vert A \Vert _{\kappa ,0} \, . \end{aligned}$$

(6.5)

In the following, we will define

$$\begin{aligned} \kappa _{\min } \;:=\; \frac{\kappa _{*}}{2} \;=\; \frac{\min \{\kappa , \rho \}}{2 \sqrt{2}}\,, \end{aligned}$$

(6.6)

with $\kappa _*$ defined as in (3.5), so that the operators $Z_{\textrm{obs}}$, $V_{\textrm{obs}}$ and $\{G^{(n)}_{\textrm{obs}}\}_{n=1}^{n^*}$ defined in Proposition 5.1 satisfy

$$\begin{aligned} Z_{\textrm{obs}} \in \mathcal {O}_{2\kappa _{\min }}\,, \quad V_{\textrm{obs}} \in \mathcal {O}_{2 \kappa _{\min }, 0}\,, \quad G^{(n)}_{\textrm{obs}} \in \mathcal {O}_{\kappa _n, \rho _n} \subset \mathcal {O}_{2\kappa _{\min }, \rho _*}\,. \end{aligned}$$

(6.7)

Corollary 6.3

Let O be a local observable supported on $S_O$ and $G \in \mathcal {O}_{2\kappa _{\min },0}$, with $\Vert G \Vert _{2\kappa _{\min },0} \le 1$, then $\exists C(|S_O|,d,\kappa , \rho )>0$ such that $\forall t>0$

$$\begin{aligned} \Vert e^{-\textrm{i}G(\omega t)} O e^{\textrm{i}G(\omega t)} - O \Vert _{\textrm{op}} \le C(|S_O|,d, \kappa , \rho ) \Vert O \Vert _{\textrm{op}} {\Vert G(\omega t) \Vert _{\kappa }}\,. \end{aligned}$$

(6.8)

Proof

For any fixed $t_0:=t$, we write

$$ \begin{aligned} \Vert e^{-\textrm{i}G(\omega t_0)} O e^{\textrm{i}G(\omega t_0)} - O \Vert _{\textrm{op}}&\le \int _0^1 \textrm{d}\tau \Vert [ G(\omega t_0),e^{-\textrm{i}\tau G(\omega t_0)} O e^{\textrm{i}\tau G(\omega t_0)}] \Vert _{\textrm{op}}\, \end{aligned} $$

and we apply Lemma 6.2 with $Z=G(\omega t_0)$ and $t=1$. This yields

$$ \begin{aligned} \Vert e^{-\textrm{i}G(\omega t_0)} O e^{\textrm{i}G(\omega t_0)} - O \Vert _{\textrm{op}}&\;\le \; C(|S_O|,d,\kappa _{\min }) \langle \Vert G(\omega t) \Vert _{2 \kappa _{\min }} \rangle ^{d+1} \Vert O\Vert _{\textrm{op}} \Vert G(\omega t_0)\Vert _{\kappa _{\min }} \, , \end{aligned} $$

which is the thesis. $\square$

Lemma 6.4

Let $H_{\textrm{obs}}(\omega t) = H_0 + Z_{\textrm{obs}} + V_{\textrm{obs}}( \omega t)$ and $\varepsilon$ as in Proposition 3.1, and $H_{\textrm{eff}}$ as in (6.2). For any local observable O, there exists a constant $C = C(O, d, \kappa , \rho , \Vert H_0\Vert _{\kappa }, \Vert V\Vert _{\kappa , \rho })$ such that, for any $t \in \mathbb {R}$ we have

$$\begin{aligned} \Vert {U}^*_{H_{\textrm{obs}}}(t) O {U}_{H_{\textrm{obs}}}(t) - e^{-\textrm{i}H_{\textrm{eff}} t} O e^{\textrm{i}H_{\textrm{eff}} t} \Vert _{\textrm{op}} \; \le \; C {e^{-{ \varepsilon ^{-\texttt {b} }}} \langle t\rangle ^{d+2}}\,. \end{aligned}$$

(6.9)

Proof

By Proposition 3.1, provided $\varepsilon$ is small enough, one has

$$\begin{aligned} \frac{1}{2} \Vert H_0\Vert _{\kappa } \le \Vert H_{\textrm{eff}}\Vert _{2 \kappa _{\min }} \le 2 \Vert H_0\Vert _{\kappa }\,. \end{aligned}$$

(6.10)

Let furthermore $W(\tau )={U}_{H_{\textrm{obs}}}(\tau )^{-1} {U}_{H_{\textrm{obs}}}(t)$, then by Duhamel formula we have

$$ {U}_{H_{\textrm{obs}}}^*(t) O {U}_{H_{\textrm{obs}}}(t)-e^{-\textrm{i}H_{\textrm{eff}} t} O e^{\textrm{i}H_{\textrm{eff}} t} = \int _0^t W^*(\tau ) [{V}_{\textrm{obs}}(\omega \tau ), e^{-\textrm{i}\tau H_{\textrm{eff}}} O e^{\textrm{i}\tau H_{\textrm{eff}}}] W(\tau ) \textrm{d}\tau \,. $$

Taking the operator norm on both sides, one can use Lemma 6.2 to get

$$\begin{aligned} \begin{aligned} \left\| \int _0^t W^*(\tau ) [{V}_{\textrm{obs}}(\tau ), e^{-\textrm{i}\tau H_{\textrm{eff}}} O e^{\textrm{i}\tau H_{\textrm{eff}}}] W(\tau ) \textrm{d}\tau \right\| _{\textrm{op}} \le C(|S_O|,d,\kappa ), \rho )\cdot \\ \cdot \langle \Vert H_{\textrm{eff}} \Vert _{2 \kappa _{\min }} \rangle ^{d+1} \langle t \rangle ^{d+2} \Vert V_{\textrm{obs}} \Vert _{\kappa _{\min },0} \Vert O \Vert _{\textrm{op}}\,. \end{aligned} \end{aligned}$$

(6.11)

Then the thesis follows combining (6.11) with $\sup _{t \in \mathbb R}\Vert V_{\textrm{obs}}(\omega t)\Vert _{\kappa _{\min }} \le e^{-{ \varepsilon ^{-\texttt {b} }}} \Vert V\Vert _{\kappa , \rho }$, due to Item (b.iii) of Theorem 3.1. $\square$

Lemma 6.5

For any local observable O, there exist two constants $C_1 = C_1(O, d, \kappa , \rho , \Vert V\Vert _{\kappa , \rho })$ and $C_2 =( O, d, r, \kappa , \rho , \{\Vert N^{(\alpha )}\Vert _{0}\}_{\alpha = 1}^r, \Vert V\Vert _{\kappa , \rho })$ such that

• (i)
  $$\begin{aligned} \Vert U_H^*(t) O U_H(t) - {U}_{H_\textrm{obs}}^*(t) O {U}_{H_\textrm{obs}}(t) \Vert _{\textrm{op}} \le {C_1} \varepsilon ^{\frac{1}{2}} \, \qquad \forall t \in \mathbb {R} \end{aligned}$$

  (6.12)
• (ii)
  $$\begin{aligned} \Vert U_H^*(t) O U_H(t) - {U}^*_{H_\textrm{obs}}(t) O {U}_{H_\textrm{obs}}(t) \Vert _{\textrm{op}} \le C_2 \varepsilon ^{\frac{1}{2}} |\omega t|_{\mathbb {T}^m} \, \qquad \forall t \in \mathbb {R} \, . \end{aligned}$$

  (6.13)

Proof

By Item (b.iv) of Proposition 3.1, one has

$$ \begin{aligned}&\Vert U_H^*(t) O U_H(t) - {U}_{H_\textrm{obs}}^*(t) O {U}_{H_\textrm{obs}}(t) \Vert _{\textrm{op}} \\&\quad = \Vert U_H^*(t) (O-Y_\textrm{obs}^*(\omega t)OY_\textrm{obs}(\omega t)) U_H(t) \Vert _{\textrm{op}} \\&\quad = \Vert O - Y_\textrm{obs}^*(\omega t) O Y_\textrm{obs}(\omega t) \Vert _{\textrm{op}} \\&\quad \le \sum _{n={1}}^{n^*} \Vert (Y_\textrm{obs}^{(n)})^*(\omega t) O Y^{(n)}_\textrm{obs}(\omega t) - (Y_\textrm{obs}^{(n-1)})^*(\omega t) O Y_\textrm{obs}^{(n-1)}(\omega t) \Vert _{\textrm{op}} \\&\quad =\sum _{n={1}}^{n^*} \Vert (Y_\textrm{obs}^{(n-1)})^*(\omega t) (e^{-\textrm{i}G^{(n)}_\textrm{obs}(\omega t)} O e^{\textrm{i}G^{(n)}_\textrm{obs}(\omega t)}-O) Y_\textrm{obs}^{(n-1)}(\omega t) \Vert _{\textrm{op}} \\&\quad =\sum _{n={1}}^{n^*} \Vert e^{-\textrm{i}G^{(n)}_\textrm{obs}(\omega t)} O e^{\textrm{i}G^{(n)}_\textrm{obs}(\omega t)} - O \Vert _{\textrm{op}}\,. \end{aligned} $$

We are thus left with finding a good bound for $\Vert e^{-\textrm{i}G^{(n)}_\textrm{obs}(\omega t)} O e^{\textrm{i}G^{(n)}_\textrm{obs}(\omega t)} - O \Vert _{\textrm{op}}$. To this end, we use Corollary 6.3 to get

$$\begin{aligned} \Vert e^{-\textrm{i}G^{(n)}_\textrm{obs}(\omega t)} O e^{\textrm{i}G^{(n)}_\textrm{obs}(\omega t)} - O \Vert _{\textrm{op}} \le C(|S_O|,d, \kappa , \rho ) \Vert G^{(n)}_\textrm{obs}(\omega t) \Vert _{\kappa _{\min }} \Vert O \Vert _{\textrm{op}} \, . \end{aligned}$$

(6.14)

Now, using (5.25) and recalling $\kappa _{\min } \le \kappa _n \le \kappa$, one has

$$\begin{aligned} \Vert G^{(n)}_\textrm{obs}(\omega t)\Vert _{\kappa _{\min }} \le \Vert G_\textrm{obs}^{(n)} \Vert _{\kappa _n, \rho _n} \le \Vert V \Vert _{\kappa , \rho } \varepsilon ^{\frac{1}{2}} e^{-n}\,, \end{aligned}$$

(6.15)

whence

$$ \begin{aligned} \sum _{n=0}^{n^*-1} \Vert e^{-\textrm{i}G^{(n)}_\textrm{obs}(\omega t)} O e^{\textrm{i}G^{(n)}_\textrm{obs}(\omega t)}-O \Vert _{\textrm{op}}&\le C(|S_O|,d,\kappa , \rho ) \Vert O \Vert _{\textrm{op}} \Vert V \Vert _{\kappa , \rho } \varepsilon ^{\frac{1}{2}} \sum _{n=0}^{n^*-1} e^{-n} \\&\le C(|S_O|,d,\kappa , \rho ) \Vert O \Vert _{\textrm{op}} \frac{{e}}{e-1} \Vert V \Vert _{\kappa , \rho } \varepsilon ^{\frac{1}{2}} \, . \end{aligned} $$

This proves (6.12). Concerning (6.13), for any t, one has

$$ \begin{aligned} \Vert G^{(n)}_\textrm{obs}(\omega t) - G^{(n)}_\textrm{obs}(0) \Vert _{\kappa _n}&\le \sup _{\varphi \in \mathbb {T}^m}\max _{j=1,\dots ,m} \Vert \partial _{\varphi _j} G^{(n)}_\textrm{obs}(\varphi ) \Vert _{\kappa _n} |\varphi |_{\mathbb {T}^m} \\&\le \frac{1}{\rho _n} \Vert G^{(n)}_\textrm{obs}\Vert _{\kappa _n, \rho _n} |\varphi |_{\mathbb {T}^m} \le \frac{1}{\rho _*} \Vert G^{(n)}_\textrm{obs}\Vert _{\kappa _n, \rho _n} |\varphi |_{\mathbb {T}^m}\,. \end{aligned} $$

Thus, recalling that by (3.5) $\rho _*= \rho _*(\kappa , \rho , r, \{\Vert N^{(\alpha )}\Vert _0\}_\alpha )$, by (6.15) one has, for a positive constant $C(r, \kappa ,\rho ,\Vert V \Vert _{\kappa ,\rho },\Vert N^{(\alpha )} \Vert _{0})$,

$$ \Vert G^{(n)}_\textrm{obs}(\omega t) \Vert _{\kappa _n}=\Vert G^{(n)}_\textrm{obs}(\omega t) - G^{(n)}_\textrm{obs}(0) \Vert _{\kappa _n} \le C(r, \kappa ,\rho , \Vert V \Vert _{\kappa ,\rho },\Vert N^{(\alpha )} \Vert _0) \varepsilon ^{\frac{1}{2}}e^{-n} |\omega t|_{\mathbb {T}^m} \,. $$

which, combined with (6.14), yields (6.13). $\square$

Given $\omega \in \mathbb R^m$, $\delta >0$ and $\theta \in \mathbb {T}^m$, we define the $\delta -$ergodization time as

$$\begin{aligned} T_{\omega , \delta , \theta } := \inf \;\left\{ t \in \mathbb R_+\ |\ \forall \varphi \in \mathbb {T}^m \quad \Vert \varphi - (\theta + [0, t]\omega )\Vert _{\mathbb {T}^m} \le \delta \right\} \,. \end{aligned}$$

(6.16)

Note that $T_{\omega , \delta , \theta }$ is clearly independent of $\theta$, thus from now on we will omit the subscript $\theta$ and we will simply write $T_{\omega , \delta }$. The following result is due to [15, 17]:

Theorem 6.6

(Theorem 4.1 of [15]) $\forall m \in \mathbb N$ $\exists a_m>0$ such that $\forall \omega \in \mathbb R^m$ satisfying (2.11), $\forall \delta >0$ one has

$$\begin{aligned} T_{\omega ,\delta } \le \frac{1}{\gamma } \left( \frac{a_m}{\delta }\right) ^{\tau } =: \widetilde{T}_{\omega ,\delta }\,. \end{aligned}$$

(6.17)

This means that, starting from any point $\theta \in \mathbb {T}^m$, after a time $\widetilde{T}_{\omega , \delta }$ the trajectory of the dynamical system $\dot{\varphi }= \omega$, $\varphi (0) = \theta$, has visited a neighborhood of size $\delta$ of any point $\varphi \in \mathbb {T}^m$. In particular, we will be interested in neighborhoods of $\varphi = 0$.

Thus, Theorem 6.6 guarantees that there exists at least a time $t_0 \in [0, \widetilde{T}_{\omega , \delta }]$ such that $\Vert \omega t_0 -0\Vert _{\mathbb {T}^m} < \delta$. Furthermore, choosing $\theta = \omega \widetilde{T}_{\omega , \delta }$ and $\varphi = 0$, there exists a time $t_1 \in [\widetilde{T}_{\omega , \delta }, 2 \widetilde{T}_{\omega , \delta }]$ such that $\Vert \omega t_1\Vert _{\mathbb {T}^m} < \delta$. Iterating this construction, one obtains that $\forall j \in \mathbb N$ there exists a time

$$\begin{aligned} t_j \in [j \widetilde{T}_{\omega , \delta }, (j+1) \widetilde{T}_{\omega , \delta }] \quad \text {s.t.} \quad \Vert \omega t_j \Vert _{\mathbb {T}^m} < \delta \quad \forall j\,. \end{aligned}$$

(6.18)

Proof of Theorem 2.2

We define $H_{\text {eff}}=J\cdot N + Z_{\textrm{eff}}$ as in (6.2) and we observe that, by Item (b.ii) of Proposition 3.1, (2.15) is satisfied. We now prove that Items (i) and (ii) hold.

(i) By (6.9) and (6.12), $\forall t \in \mathbb R$ one has

$$ \begin{aligned} \hspace{-7pt}\Vert U_H^*(t) O U_H(t) - e^{-\textrm{i}H_{\textrm{eff}} t} O e^{\textrm{i}H_{\textrm{eff}} t} \Vert _{\textrm{op}}&\le \Vert {U}_{H_{\textrm{obs}}}^*(t) O {U}_{H_\textrm{obs}}(t) -e^{-\textrm{i}H_{\textrm{eff}} t} O e^{\textrm{i}H_{\textrm{eff}} t} \Vert _{\textrm{op}}\\&+\Vert U_H^*(t) O U_H(t)- {U}_{H_{\textrm{obs}}}^*(t) O {U}_{H_{\textrm{obs}}}(t)\Vert _{\textrm{op}} \\&\le C_1 t^{d+2} e^{-{ \varepsilon ^{-\texttt {b} }}} + C_2 \varepsilon ^{\frac{1}{2}}\,, \end{aligned} $$

where $C_1:= C_1(O, d, \kappa , \rho , \Vert H_0\Vert _{\kappa }, \Vert V\Vert _{\kappa , \rho })>0$ and $C_2:= C_2(O, d, \kappa , \rho , \Vert V\Vert _{\kappa , \rho })>0$.

(ii) Given $\delta >0$, let $\{t_j\}_{j \in \mathbb N}$ be defined as in (6.18). By (6.9) and (6.13), one has

$$ \begin{aligned} \hspace{-7pt}\Vert U_H^*(t_j) O U_H(t_j) - e^{-\textrm{i}H_{\textrm{eff}} t_j} O e^{\textrm{i}H_{\textrm{eff}} t_j} \Vert _{\textrm{op}}&\le \Vert {U}_{H_\textrm{obs}}^*(t_j) O {U}_{H_\textrm{obs}}(t_j) -e^{-\textrm{i}H_{\textrm{eff}} t_j} O e^{\textrm{i}H_{\textrm{eff}} t_j} \Vert _{\textrm{op}}\\&+\Vert U_H^*(t_j) O U_H(t_j)- {U}_{H_\textrm{obs}}^*(t_j) O {U}_{H_\textrm{obs}}(t_j)\Vert _{\textrm{op}} \\&\le C_1 t_j^{d+2} e^{-{ \varepsilon ^{-\textsf {b} }}} + C_3 \varepsilon ^{\frac{1}{2}} \delta \, \end{aligned} $$

with $C_3:=C_3(O, d, r, \kappa , \rho , \Vert V\Vert _{\kappa , \rho }, \{\Vert N^{(\alpha )}\Vert _0\})>0$. Then in particular ${j \widetilde{T}_{\omega , \delta }} \le t_j \le {(j+1)\widetilde{T}_{\omega , \delta }}$ with $\widetilde{T}_{\omega , \delta }= \gamma a_{m}^\tau \delta ^{-\tau }$. Thus one has

$$ \begin{aligned} \Vert U_{H}^*(t_j) O U_H(t_j) - e^{-\textrm{i}H_{\textrm{eff}} t_j} O e^{\textrm{i}H_{\textrm{eff}} t_j} \Vert _{\textrm{op}}&\le C_1 \left( \gamma a_m^\tau (j+1)\right) ^{d+2} \delta ^{-\tau (d+2)} e^{-{\varepsilon ^{-\texttt {b} }}} + C_3 \varepsilon ^{\frac{1}{2}}\delta \,. \end{aligned} $$

Now take

$$ \delta = e^{-{\textsf {f} } \varepsilon ^{-\textsf {b} }} \varepsilon ^{-\frac{\textsf {f} }{2}}\, $$

with $\textsf {f}$ defined in (2.18). One has

$$ \Vert U_{H}^*(t_j) O U_H(t_j) - e^{-\textrm{i}H_{\textrm{eff}} t_j} O e^{\textrm{i}H_{\textrm{eff}} t_j} \Vert _{\textrm{op}} \le \left( C_1 \left( \gamma a_m^\tau (j+1)\right) ^{d+1} + C_3\right) \varepsilon ^{\frac{1-\textsf {f} }{2}} e^{-{\textsf {f} } \varepsilon ^{-\textsf {b} }}\,, $$

which gives the thesis. $\square$

6.2 Fast Forcing Perturbations

The proof of Theorem 2.3 goes exactly as in the case of small perturbations (see the proof of Theorem 2.1). Concerning the proof of Theorem 2.4, one uses the same arguments that are used to prove Theorem 2.2 in the case of small perturbations, except for the fact that Lemmas 6.4, 6.5 are substituted by the following:

Lemma 6.7

Let $H_\textrm{obs}(\lambda \omega t) = J \cdot N + Z_\textrm{obs}+ V_\textrm{obs}(\lambda \omega t)$ be defined as in Proposition 3.1. There exists $\lambda _0 = \lambda _0(\Vert H_0\Vert _{\kappa }, \Vert V\Vert _{\kappa , \rho }, r)>0$ such that $\forall \lambda < \lambda _0$ and for any local observable O, there exists a constant $C = C(O, d, \kappa , \rho , \Vert H_0\Vert _{\kappa }, \Vert V\Vert _{\kappa , \rho })$ such that, for any $t \in \mathbb {R}$ we have

$$\begin{aligned} \Vert {U}^*_{H_\textrm{obs}}(t) O {U}_{H_\textrm{obs}}(t) - e^{-\textrm{i}H_{\textrm{eff}} t} O e^{\textrm{i}H_{\textrm{eff}} t} \Vert _{\textrm{op}} \; \le \; C {e^{-{ \lambda ^\beta }} \langle t\rangle ^{d+2}}\,. \end{aligned}$$

(6.19)

Lemma 6.8

For any local observable O, there exist two constants $C_1:= C_1(O, d, \kappa , \rho , \Vert V\Vert _{\kappa , \rho })>0$ and $C_2:=(O, d, r, \kappa , \rho , \{\Vert N^{(\alpha )}\Vert _{0}\}_{\alpha = 1}^r, \Vert V\Vert _{\kappa , \rho })>0$ such that

• (i)
  $$\begin{aligned} \Vert U_H^*(t) O U_H(t) - {U}_{H_\textrm{obs}}^*(t) O {U}_{H_\textrm{obs}}(t) \Vert _{\textrm{op}} \le {C_1} \lambda ^{-\frac{1}{8\tau _\omega }} \, \qquad \forall t \in \mathbb {R} \end{aligned}$$

  (6.20)
• (ii)
  $$\begin{aligned} \Vert U_H^*(t) O U_H(t) - {U}^*_{H_\textrm{obs}}(t) O {U}_{H_\textrm{obs}}(t) \Vert _{\textrm{op}} \le C_2 \lambda ^{-\frac{1}{8 \tau _\omega }+1} |\omega t|_{\mathbb {T}^m} \, \qquad \forall t \in \mathbb {R} \, . \end{aligned}$$

  (6.21)

Note the presence of a factor $\lambda$ in (6.21) due to the fast forcing.

Proof of Theorem 2.4

The proof of Item (i) goes as the one of Item (i) of Theorem 2.2. We now prove Item (ii).

Given $\delta >0$, let $\{t_j\}_{j \in \mathbb N}$ be defined as in (6.18). By (6.19) and (6.21), one has

$$ \begin{aligned} \hspace{-7pt}\Vert U_H^*(t_j) O U_H(t_j) - e^{-\textrm{i}H_{\textrm{eff}} t_j} O e^{\textrm{i}H_{\textrm{eff}} t_j} \Vert _{\textrm{op}}&\le \Vert {U}_{H_\textrm{obs}}^*(t_j) O {U}_{H_\textrm{obs}}(t_j) -e^{-\textrm{i}H_{\textrm{eff}} t_j} O e^{\textrm{i}H_{\textrm{eff}} t_j} \Vert _{\textrm{op}}\\&+\Vert U_H^*(t_j) O U_H(t_j)- {U}_{H_\textrm{obs}}^*(t_j) O {U}_{H_\textrm{obs}}(t_j)\Vert _{\textrm{op}} \\&\le C_1 t_j^{d+2} e^{-{\lambda ^\beta }} + C_2 \lambda ^{-\frac{1}{8\tau _\omega }+1} \delta \,, \end{aligned} $$

with $C_1:= C_1(O, d, \kappa , \rho , \Vert H_0\Vert _{\kappa }, \Vert V\Vert _{\kappa , \rho })>0$ and $C_2:= C_2(O, d, r, \kappa , \rho , \{\Vert N^{(\alpha )}\Vert _0\}_\alpha , \Vert V\Vert _{\kappa , \rho })>0$.

Then in particular ${j \widetilde{T}_{\omega , \delta }} \le t_j \le {(j+1)\widetilde{T}_{\omega , \delta }}$ with $\widetilde{T}_{\omega , \delta }= \gamma a_{m}^{\tau _\omega } \delta ^{-\tau _\omega }$. Thus one has

$$ \begin{aligned}&\Vert U_{H}^*(t_j) O U_H(t_j) - e^{-\textrm{i}H_{\textrm{eff}} t_j} O e^{\textrm{i}H_{\textrm{eff}} t_j} \Vert _{\textrm{op}}\\&\le C_1 \left( \gamma a_m^{\tau _\omega } (j+1)\right) ^{d+2} \delta ^{-\tau _\omega (d+2)} e^{-{ \lambda ^\beta }} + C_2 \lambda ^{1-\frac{1}{8\tau _\omega }}\delta \,. \end{aligned} $$

Then the thesis follows taking

$$ \delta :=e^{-{\textsf {g} } \lambda ^\beta } \lambda ^{\frac{(1-8\tau _\omega )\textsf {g} }{8 \tau _\omega }}\,, $$

with $\textsf {g}$ defined as in (2.28). $\square$

Proof of Lemmas 4.2 and 6.2

1.1 Proof of Lemma 4.2

We are going to control the locality properties of $e^{\textrm{i}G(\nu t)} A (\nu t) e^{-\textrm{i}G(\nu t)}$ for $A \in \mathcal {O}^\texttt {s} _{\kappa , \rho }$ and $G \in \mathcal {O}^\texttt {s} _{\kappa , \rho }$, using the commutator expansion (4.3). An important part of this procedure is to control the locality property of the operators $\text {Ad}^q_{G} A$. Indeed, considering two operators A and B supported on $S_A$ and $S_B$ respectively, from (2.3) one sees that if $S_A \cap S_B \ne \varnothing$, then the support of [A, B] is $S_A \cup S_B$. Taking several commutators, in fact, enlarges the support of each local term of $\textrm{Ad}^q_A B$. This phenomenon is what is responsible for the loss of regularity in the parameter $\kappa$ in Lemma 4.2. To control the smearing out of the supports of the $\textrm{Ad}^q_A B$’s, we use the following Lemma

Lemma A.1

$S_1,\dots ,S_q \in \mathcal {P}_c(\Lambda )$ and let $f(S_1,\dots ,S_q)$ be a positive function. Then

$$\begin{aligned} \begin{aligned} \sup _{x \in \Lambda }&\sum _{\begin{array}{c} S_1,\dots ,S_q \in \mathcal {P}_c(\Lambda )\ \\ mytext{s.t. } x \in \bigcup _{j=1}^q S_j \\ S_j \cap (\bigcup _{r=j+1}^q S_r) \ne \varnothing \, \forall j=1,\dots ,q-1 \end{array}} f(S_1,\dots ,S_q) \\ {}&\le \max _{p \in \mathfrak {S}_q} \sup _{x_{p(1)} \in \Lambda } \sum _{\begin{array}{c} S_{1} \, s.t. \\ x_{p(1)} \in S_1 \end{array}} \dots \sup _{x_{p(q)} \in \Lambda } \sum _{\begin{array}{c} S_q \, s.t. \\ x_{p(q)} \in S_q \end{array}} 2^{q-1} \left( |S_1|+\dots + |S_q| \right) ^{q-1} f(S_1,\dots ,S_q) \end{aligned} \end{aligned}$$

(A.1)

where $\mathfrak {S}_q$ denotes the set of permutation of q elements.

Proof of Lemma A.1

To avoid heavy notation, it is understood that each of the $S_j$ is such that $S_j \in \mathcal {P}_c(\Lambda )$. The proof goes by induction on the number of sets. Precisely, we prove the inductive estimate

$$\begin{aligned}{} & {} \sum _{\begin{array}{c} S_j,\dots ,S_{q}\\ x \in \bigcup _{r=j}^q S_r \\ S_r \cap \left( \bigcup _{l=r+1}^q S_l \right) \ne \varnothing \\ \forall r=j, \cdots q-1 \end{array}} f(S_1,\dots ,S_q) \; \\{} & {} \quad \le \; 2^{q-j} \max _{p \in \mathfrak {S}_{[j,q]}} \sup _{x_{p(j)} \in \Lambda } \sum _{\begin{array}{c} S_j\, s.t. \\ S_j \ni x_{p(j)} \end{array}} \dots \sup _{x_{p(q)} \in \Lambda } \sum _{\begin{array}{c} S_q\, s.t. \\ S_q \ni x_{p(q)} \end{array}} (|S_j|+\dots +|S_q|)^{q-j} f(S_1,\dots ,S_q) \end{aligned}$$

where $\mathfrak {S}_{[j,q]}$ denotes the set of permutations of $\{j,\dots ,q\}$. First we consider the case with only $S_{q-1}$ and $S_q$. We have to prove that

$$\begin{aligned}{} & {} \sup _{x \in \Lambda } \sum _{\begin{array}{c} S_{q-1},S_q \text { s.t.} \\ x \in S_{q-1} \cup S_q \\ S_{q-1} \cap S_q \ne \varnothing \end{array}} f(S_{q-1},S_q) \\{} & {} \quad \le \max _{p \in \mathfrak {S}_{[q-1,q]}} \sup _{x_{p(q-1)} \in \Lambda } \sum _{\begin{array}{c} S_{q-1} \, \text {s.t.} \\ x_{p(q-1)} \in S_{q-1} \end{array}} \sup _{x_{p(q)} \in \Lambda } \sum _{\begin{array}{c} S_q \, \text {s.t.} \\ x_{p(q)} \in S_q \end{array}} 2 (|S_{q-1}|+|S_q|) f(S_{q-1},S_q) \,. \end{aligned}$$

Starting from l.h.s., we notice that $x \in S_{q-1} \cup S_q$ means that either $x \in S_{q-1}$ or $x \in S_q$. Thus we can bound l.h.s. as

$$ \begin{aligned} \sup _{x \in \Lambda } \sum _{\begin{array}{c} S_{q-1},S_q \text { s.t.} \\ \, x \in S_{q-1} \cup S_q \, , \\ S_{q-1} \cap S_q \ne \varnothing \end{array}} f(S_{q-1},S_q)&\le \sup _{x \in \Lambda } \Bigg [\sum _{\begin{array}{c} S_{q-1},S_q \text { s.t.} \\ x \in S_{q-1} \, ,\\ S_{q-1} \cap S_q \ne \varnothing \end{array}} f(S_{q-1},S_q)+ \sum _{\begin{array}{c} S_q,S_{q-1} \text { s.t.}\\ \, x \in S_q ,\\ S_{q-1} \cap S_q \ne \varnothing \, . \end{array}} f(S_{q-1},S_q)\Bigg ]\;=:\; (\star ) \end{aligned} $$

Fix $S_q$, to bound the sums it is convenient to define

$$ \begin{aligned} \mathscr {A}(S_q) \;&:=\; \big \{ S_{q-1} \, \big | \, \exists y \in S_q \cap S_{q-1} \big \} \subseteq \; \bigcup _{y \in S_q} \big \{ S_{q-1} \, \big | \, y \in S_{q-1} \big \} =\;\bigcup _{y \in S_q} \mathscr {A}_y(S_q) \end{aligned} $$

which permits us to get

$$ \begin{aligned} (\star ) \;&\le \; \sup _{x \in \Lambda } \Bigg [ \sum _{\begin{array}{c} S_q \text { s.t.} \\ \, x \in S_q \end{array}} \sum _{y \in S_q} \sum _{S_{q-1} \in \mathscr {A}_y(S_q)} f(S_{q-1},S_q)+ \sum _{\begin{array}{c} S_{q-1} \text { s.t.}\\ \, x \in S_{q-1} \end{array}} \sum _{y \in S_{q-1}} \sum _{S_q \in \mathscr {A}_y(S_{q-1})} f(S_{q-1},S_q) \Bigg ] \\&\le \; \sup _{x \in \Lambda } \Bigg [ \sum _{\begin{array}{c} S_q \text { s.t.} \\ x \in S_q \end{array}} |S_q| \sup _{y \in S_q} \sum _{S_{q-1} \in \mathscr {A}_y(S_q)} f(S_{q-1},S_q)+ \sum _{\begin{array}{c} S_{q-1} \text { s.t.} \\ x \in S_{q-1} \end{array}} |S_{q-1}| \sup _{y \in S_{q-1}} \sum _{S_q \in \mathscr {A}_y(S_{q-1})} f(S_{q-1},S_q) \Bigg ] \\&\le \; \sup _{x \in \Lambda } \Bigg [ \sum _{\begin{array}{c} S_q \text { s.t.} \\ \, x \in S_q \end{array}} (|S_q|+|S_{q-1}|) \sup _{y \in S_q} \sum _{S_{q-1} \in \mathscr {A}_y(S_q)} f(S_{q-1},S_q)\\&\qquad +\sum _{\begin{array}{c} S_{q-1} \text { s.t.} \\ x \in S_{q-1} \end{array}} (|S_q|+|S_{q-1}|) \sup _{y \in S_{q-1}} \sum _{S_q \in \mathscr {A}_y(S_{q-1})} f(S_{q-1},S_q) \Bigg ] \\&\le \; \sup _{x \in \Lambda } \Bigg [ \Bigg ( \sum _{\begin{array}{c} S_q \text { s.t.}\\ x \in S_q \end{array}} \sup _{y \in \Lambda } \sum _{S_{q-1} \in \mathscr {A}_y(S_{q-1})}+ \sum _{\begin{array}{c} S_{q-1}\text { s.t.} \\ x \in S_q \end{array}} \sup _{y \in \Lambda } \sum _{S_{q} \in \mathscr {A}_y(S_{q})} \Bigg ) (|S_q|+|S_{q-1}|) f(S_{q-1},S_q) \Bigg ] \\&\le \max _{p \in \mathfrak {S}_2} \sup _{x_{p(1)} \in \Lambda } \sum _{\begin{array}{c} S_q \text { s.t.} \\ x_{p(1)} \in S_q \end{array}} \sup _{x_{p(2)} \in \Lambda } \sum _{\begin{array}{c} S_{q-1} \text { s.t.}\\ x_{p(2)} \in S_{q-1} \end{array}} 2 (|S_q|+|S_{q-1}|) f(S_{q-1},S_q)\,. \end{aligned} $$

This proves the thesis for the function of $S_{q-1},S_q$. Now we proceed iteratively: we suppose that the thesis is true for $S_{j+1},\dots ,S_q$ and we prove it is true also for $S_j,\dots ,S_q$.

$$ \begin{aligned} \sum _{\begin{array}{c} S_j,\dots ,S_{q} \text { s.t.}\\ x \in \bigcup _{r=j}^q S_r, \\ S_r \cap \left( \bigcup _{l=r+1}^q S_l \right) \ne \varnothing \\ \forall r=j, \cdots q-1 \end{array}} f(S_1,\dots ,S_q)&\le \; \sup _{x \in \Lambda } \Bigg \{ \sum _{\begin{array}{c} S_j \text { s.t.} \\ x \in S_j \end{array}} \sum _{\begin{array}{c} S_{j+1},\dots ,S_q \text { s.t.}\\ S_r \cap (\bigcup _{l=r+1}^q S_l) \ne \varnothing , \\ \forall r=j+1,\dots , q-1 \\ S_j \cap \left( \bigcup _{l=j+1}^qS_l \right) \ne \varnothing \end{array}} f(S_j,\dots ,S_q) \\&\qquad +\sum _{\begin{array}{c} S_{j+1},\dots ,S_q \text { s.t.}\\ x \in \bigcup _{r={j+1}}^q S_r, \\ S_r \cap (\bigcup _{l={r+1}}^q S_l) \ne \varnothing \\ \forall r=j+1,\dots ,q-1 \end{array}} \sum _{\begin{array}{c} S_j \text { s.t.} \\ S_j \cap (\bigcup _{r=j+1}^q S_r) \ne \varnothing \end{array}} f(S_j,\dots ,S_q) \Bigg \} \\ =: (\blacksquare )+(\clubsuit ) \, . \end{aligned} $$

We have now to estimate the two terms separately. For the first one, let us fix $S_j$ and consider

$$ \begin{aligned} \mathscr {A}(S_j) \;&:=\;\Big \{(S_{j+1},\dots ,S_q) \, \Big | \, S_j \cap \Big (\bigcup _{r=j+1}^q S_r \Big ) \ne \varnothing \Big \} \subseteq \bigcup _{y \in S_j} \Big \{\bigcup _{r={j+1}}^q S_r \, \Big | \, \bigcup _{r={j+1}}^q S_r \ni y \Big \} \\&= \bigcup _{y \in S_j} \mathscr {A}_y(S_j) \, . \end{aligned} $$

For the second one, fix $S_{j+1},\dots ,S_q$ and consider

$$ \begin{aligned} \mathscr {A}'(S_{j+1},\dots ,S_q)\;&=\; \Big \{S_j \,\Big |\, S_j \cap \Big (\bigcup _{r=j+1}^q S_r\Big ) \ne \varnothing \Big \} \subseteq \bigcup _{y \in \bigcup _{r={j+1}}^q S_r} \left\{ S_j \, | \, y \in S_j \right\} \;\\&=\; \mathscr {A}'_y(S_{j+1},\dots ,S_q) \, . \end{aligned} $$

Thus,

$$\begin{aligned} (\blacksquare )&\le \sup _{x \in \Lambda } \sum _{\begin{array}{c} S_j \text { s.t.}\\ x \in S_j \end{array}}\, \sum _{y \in S_j} \sum _{\begin{array}{c} S_{j+1},\dots ,S_q \text { s.t.} \\ y \in \bigcup _{r=j+1}^q S_r, \\ S_r \cap (\bigcup _{l={r+1}}^q S_l) \ne \varnothing , \\ \forall r=j+2,\dots ,q-1 \end{array}} f(S_j,\dots ,S_q) \\&\le \; \sup _{x \in \Lambda } \sum _{\begin{array}{c} S_j \text { s.t.} \\ x \in S_j \end{array}} |S_j| \sup _{y \in \Lambda } \sum _{\begin{array}{c} S_{j+1},\dots ,S_q \text { s.t.}\\ y \in \bigcup _{r=j+1}^q S_r, \\ S_r \cap (\bigcup _{l={r+1}}^q S_l) \ne \varnothing \\ \forall r=j+2,\dots ,q-1 \end{array}} f(S_j,\dots ,S_q) \\&\!\!\!\!\!\!\!\!\!\!\overset{\text {inductive hyp}}{\le } \sup _{x \in \Lambda } \sum _{\begin{array}{c} S_j \text { s.t.} \\ x \in S_j \end{array}} |S_j| 2^{q-j-1} \max _{p \in \mathfrak {S}_{[j+1, q]}} \sup _{x_{p(j+1)} \in \Lambda } \sum _{\begin{array}{c} S_{j+1} \text { s.t.} \\ x_{p(j+1)} \in S_{j+1} \end{array}} \dots \\&\qquad \qquad \qquad \dots \sup _{x_{p(q)} \in \Lambda } \sum _{\begin{array}{c} S_q \text { s.t.}\\ x_{p(q)} \in S_q \end{array}} (|S_{j+1}|+ \dots + |S_q|)^{q-j-1} f(S_j,\dots ,S_q) \\&\le 2^{q-j-1} \sup _{x \in \Lambda } \sum _{\begin{array}{c} S_j \text { s.t.} \\ x \in S_j \end{array}} \max _{p \in \mathfrak {S}_{[j+1,q}}\sup _{x_{p(j+1)} \in \Lambda } \sum _{\begin{array}{c} S_{j+1} \text {s.t. } x_{p(j+1)} \in S_{j+1} \end{array}} \dots \\ {}&\qquad \dots \qquad \sup _{x_{p(q)}\in \Lambda } \sum _{\begin{array}{c} S_q\text { s.t.} \\ x_{p(q)} \in S_q \end{array}}(|S_j|+\dots +|S_q|)^{q-j} f(S_j,\dots ,S_q) \, \\&\le 2^{q-j-1} \max _{p \in \mathfrak {S}_{[j,q]}} \sup _{x_{p(j)} \in \Lambda } \sum _{\begin{array}{c} S_j \text { s.t.}\\ x_{p(j)} \in S_j \end{array}} \dots \sup _{x_{p(q)} \in \Lambda } \sum _{\begin{array}{c} S_q \text { s.t.}\\ x_{p(q)} \in S_q \end{array}} (|S_j|+\dots +|S_q|)^{q-j} f(S_j,\dots ,S_q)\,. \end{aligned}$$

Concerning the second piece,

$$ \begin{aligned} (\clubsuit )&\le \sup _{x \in \Lambda } \sum _{\begin{array}{c} S_{j+1},\dots ,S_q \text { s.t.}\\ x \in S_{j+1} \cup \dots \cup S_q , \\ S_r \cap (\bigcup _{l=r+1}^q S_l) \ne \varnothing \\ \forall r={j+1},\dots ,q-1 \end{array}} \sum _{y \in S_{j+1} \cup \dots \cup S_q} \sum _{\begin{array}{c} S_j \text { s.t.} \\ y \in S_j \end{array}} f(S_j,\dots ,S_q) \\&\le \sup _{x \in \Lambda } \sum _{\begin{array}{c} S_{j+1},\dots ,S_q \\ x \in S_{j+1} \cup \dots \cup S_q \\ S_r \cap (\bigcup _{l=r+1}^q S_l) \ne \varnothing \\ \forall r={j+1},\dots ,q-1 \end{array}} (|S_{j+1}|+\dots +|S_q|) \sup _{y \in \Lambda }\sum _{\begin{array}{c} S_j \text { s.t.} \\ S_j \ni y \end{array}} f(S_j,\dots ,S_q) \\&\le \max _{p \in \mathfrak {S}_{[j+1,q]}} \sup _{x_{p(j+1)} \in \Lambda } \hspace{-7pt} \sum _{\begin{array}{c} S_{j+1} \text { s.t.} \\ x_{p(j+1)} \in S_{j+1} \end{array}}\hspace{-20pt} \dots \sup _{x_{p(q)} \in \Lambda } \hspace{-5pt} \sum _{\begin{array}{c} S_q \text { s.t.} \\ x_{p(q)} \in S_q \end{array}}2^{q-j-1}(|S_{j+1}|+ \dots + |S_q|)^{q-j-1} \\&\quad \sup _{y \in \Lambda } \sum _{\begin{array}{c} S_j \text { s.t.} \\ y \in S_j \end{array}}f(S_j,\dots ,S_q) \end{aligned} $$

$$ \begin{aligned}&\le \max _{p \in \mathfrak {S}_{[j,q]}} 2^{q-j-1} \sup _{x_{p(j)} \in \Lambda } \sum _{\begin{array}{c} S_j \text{ s.t. } \\ x_{p(j)} \in S_j \end{array}} \dots \sup _{x_{p(q)} \in \Lambda }\sum _{\begin{array}{c} S_q \text { s.t.}\\ x_{p(q)} \in S_q \end{array}} 2^{q-j-1} (|S_{j}|+\dots +|S_q|)^{q-j} f(S_j,\dots ,S_q) \, . \end{aligned} $$

At this point, combining the estimates for $(\clubsuit )$ and $(\blacksquare )$ we get

$$ (\blacksquare )+(\clubsuit ) \le \max _{p \in \mathfrak {S}_{[j,q]}} \sup _{x_{p(j)} \in \Lambda } \sum _{\begin{array}{c} S_{j}\, s.t. \\ S_j \ni x_{p(j)} \end{array}} \dots \sup _{x_{p(q)} \in \Lambda } \sum _{\begin{array}{c} S_q\, s.t. \\ S_q \ni x_{p(q)} \end{array}} 2^{q-j} \left( |S_j|+\dots + |S_q| \right) ^{q-j} f(S_j,\dots ,S_q) $$

proving the inductive thesis and thus the technical Lemma. $\square$

To understand quantitatively how to control the norm of $\textrm{Ad}_A^q B$, let us prove first the case of a single commutator.

Lemma A.2

Let $\kappa , \rho >0$. Then $\forall \sigma >0$ and $\forall A, B \in \mathcal {O}^\texttt {s} _{\kappa + \sigma , \rho }$ one has $[A, B] \in \mathcal {O}^\texttt {s} _{\kappa , \rho }$, with

$$\begin{aligned} \Vert [A, B]\Vert _{\kappa , \rho } \le \frac{4 e^{-(\kappa +1)}}{\sigma } \Vert A\Vert _{\kappa + \sigma , \rho } \Vert B\Vert _{\kappa + \sigma , \rho }\,. \end{aligned}$$

(A.2)

Proof

Using the definition of local parts of a commutator given by (2.3) and the fact that $A,B \in \mathcal {O}_{\kappa +\sigma ,\rho }$, one has

$$\begin{aligned} \Vert [A, B] \Vert _{\kappa , \rho }&\;=\; \sup _{x \in \Lambda } \sum _{S \ni x} \sum _{l \in \mathbb {Z}^m} e^{\kappa |S|} e^{\rho |l|} \Vert (\widehat{[A, B]_S})_{l} \Vert _{\textrm{op}}\\&\le 2 \sup _{x \in \Lambda } \sum _{S \ni x} \sum _{\begin{array}{c} S_1, S_2 : S_1 \cup S_2 = S\\ S_1 \cap S_2 \ne \emptyset \end{array}} \sum _{k \in \mathbb {Z}^m} \sum _{l' \in \mathbb {Z}^m} e^{\kappa |S|} e^{\rho |l-l'|} \Vert (\widehat{A_{S_1}})_{l-l'}\Vert _{\textrm{op}} \Vert (\widehat{B_{S_2}})_{l'}\Vert _{\textrm{op}}\\&\le 2 e^{-\kappa } \sup _{x \in \Lambda } \sum _{\begin{array}{c} S_1, S_2 : S_1 \cup S_2 \ni x \\ S_1 \cap S_2 \ne \emptyset \end{array}} e^{\kappa \left( |S_1| + |S_2|\right) } F_{S_1, S_2}(A,B)\,, \end{aligned}$$

with

$$\begin{aligned} F_{S_1, S_2}(A, B)&:= \sum _{l\in \mathbb {Z}^m} \sum _{l' \in \mathbb {Z}^m} e^{\rho |l-l'|} e^{\rho |l'|} \Vert (\widehat{A_{S_1}})_{l - l'}\Vert _{\textrm{op}} \Vert (\widehat{B_{S_2}})_{l'}\Vert _{\textrm{op}} \\&= \sum _{l' \in \mathbb {Z}^m} e^{\rho |l'|} \Vert (\widehat{B_{S_2}})_{l'}\Vert _{\textrm{op}} \sum _{l \in \mathbb {Z}^m} e^{\rho |l|} \Vert (\widehat{A_{S_1}})_{l}\Vert _{\textrm{op}}\,. \end{aligned}$$

By Lemma A.1 with $q = 1$, one has

$$\begin{aligned} \Vert [A, B] \Vert _{\kappa ,\rho }&\le 4 e^{-\kappa } \max _{p \in \mathfrak {S}_2} \sup _{x_{p(1)} \in \Lambda } \sum _{S_1 \ni x_{p(1)}} \sup _{x_{p(2)} \in \Lambda } \sum _{S_2 \ni x_{p(2)}} e^{\kappa (|S_1| + |S_2|)} (|S_1| + |S_2|) F_{S_1, S_2}(A, B)\\&\le 4 e^{-(\kappa + 1)} \sigma ^{-1} \max _{p \in \mathfrak {S}_2} \sup _{x_{p(1)} \in \Lambda } \sum _{S_1 \ni x_{p(1)}} \sup _{x_{p(2)} \in \Lambda } \sum _{S_2 \ni x_{p(2)}} e^{(\kappa + \sigma ) (|S_1| + |S_2|)} F_{S_1, S_2}(A, B)\\&= 4 e^{-(\kappa + 1)} \sigma ^{-1} \Vert A\Vert _{\kappa + \sigma , \rho } \Vert B\Vert _{\kappa + \sigma , \rho }\,, \end{aligned}$$

where we have used the fact that $\sup _{x \in \mathbb R} x e^{-\sigma x} \le \frac{1}{e}$. This proves (A.2). Using that $A, B \in \mathcal {O}^\texttt {s} _{\kappa + \sigma , \rho }$ and that $S'\nsubseteq S$, by (2.3) we deduce that

$$ \begin{aligned} {[[A, B]_S, N^{(\alpha )}_{S'}]}&= \hspace{-10pt}\sum _{\begin{array}{c} S_1, S_2: S_1 \cup S_2 = S\\ S_1 \cap S_2 \ne \emptyset \end{array}} \hspace{-10pt} [[A_{S_1}, B_{S_2}], N^{(\alpha )}_{S'}]\\&= -\hspace{-10pt} \sum _{\begin{array}{c} S_1, S_2: S_1 \cup S_2 = S\\ S_1 \cap S_2 \ne \emptyset \end{array}} \hspace{-10pt} [[B_{S_2}, N^{(\alpha )}_{S'}], A_{S_1}] - \hspace{-10pt}\sum _{\begin{array}{c} S_1, S_2: S_1 \cup S_2 = S\\ S_1 \cap S_2 \ne \emptyset \end{array}}\hspace{-10pt} [[N^{(\alpha )}_{S'}, A_{S_1}], B_{S_2}] \,. \end{aligned} $$

Now we note that, if $S' \nsubseteq S,$ $\forall S_1, S_2$ such that $S = S_1 \cup S_2$ then it must be that both $S' \nsubseteq S_1$, and $S' \nsubseteq S_2$. But then, by definition of strongly local operator $[B_{S_2}, N^{(\alpha )}_{S'}] = 0$ and $[N^{(\alpha )}_{S'}, A_{S_1}] = 0$, which proves $[[A, B]_S, N^{(\alpha )}_{S'}] = 0$, namely $[A,B] \in \mathcal {O}^\texttt {s} _{\kappa , \rho }$. $\square$

Proof of Lemma 4.2

We use the expansion $e^A B e^{-A}=\sum _{q\ge 0} \frac{1}{q!} \textrm{Ad}_A^qB$ and we prove that $\textrm{Ad}^q_A B \in \mathcal {O}^\texttt {s} _{\kappa + \sigma , \rho }$ with

$$\begin{aligned} \Vert \textrm{Ad}_A^q B \Vert _{\kappa , \rho } \le \left( \frac{q}{e}\right) ^q \frac{(4 e^{-\kappa })^q}{\sigma ^q} \Vert A\Vert ^q_{\kappa + \sigma , \rho } \Vert B\Vert _{\kappa + \sigma , \rho }\,. \end{aligned}$$

(A.3)

We start with observing that

$$\begin{aligned} \begin{aligned} \textrm{Ad}^q_{A(\varphi )} B(\varphi )&= \sum _{S \in \mathcal {P}_c(\Lambda )} (\textrm{Ad}^q_{A(\varphi )}B(\varphi ))_S\,, \\ (\textrm{Ad}^q_{A(\varphi )} B(\varphi ))_S&= \sum _{\begin{array}{c} S_1,\dots ,S_{q+1} \, s.t. \\ \bigcup _{j=1}^{q+1} S_j = S \\ S_r \cap (\bigcup _{r+1}^{q+1} S_\ell ) \ne \varnothing \\ \forall r=1, \dots , q \end{array}} [{A_{S_1}}(\varphi ),[{A_{S_2}}(\varphi ),[\dots ,[{A_{S_q}}(\varphi ),{B_{S_{q+1}}}(\varphi )] \dots ]\,. \end{aligned} \end{aligned}$$

(A.4)

Arguing as in Lemma A.2, one obtains

$$\begin{aligned} \Vert {\text {Ad}}_A^q(B)\Vert _{\kappa , p}&\le (2 e^{-\kappa })^{q} \sup _{x \in \Lambda } \sum _{\begin{array}{c} S_1 \cup \dots \cup S_{q+1} \ni x \\ S_r \cap (\bigcup _{s={r+1}}^{q+1} S_s) \ne \varnothing \\ \forall r=1,\dots ,q \end{array}} e^{(|S_1| + \dots + |S_{q+1}|)\kappa } F_{S_1, \dots , S_{q+1}} (A, B)\,, \end{aligned}$$

where we have defined

$$ \begin{aligned} F_{S_1, \dots , S_{q+1}} (A, B)&:= \sum _{l_1, \dots , l_{q+1} \in \mathbb {Z}^m} e^{\rho |l_1|} \Big (\prod _{j=1}^{q} \big \Vert (\widehat{A_{S_j}})_{l_j - l_{j+1}}\big \Vert _{\textrm{op}} \Big ) \big \Vert (\widehat{B_{S_{q+1}}})_{l_{q+1}}\big \Vert _{\textrm{op}}\\&\le \Big (\prod _{j = 1}^{q} \sum _{l_j \in \mathbb {Z}^m} e^{\rho |l_j|} \big \Vert (\widehat{A_{S_j}})_{l_j}\big \Vert _{\textrm{op}}\Big ) \sum _{l_{q+1} \in \mathbb {Z}^m} e^{\rho | l_{q+1}|} \big \Vert (\widehat{B_{S_{q+1}}})_{l_{q+1}}\big \Vert _{\textrm{op}}\,. \end{aligned} $$

Then by Lemma A.1 one obtains

$$\begin{aligned} \Vert {\text {Ad}}^q_A B \Vert _{\kappa ,\rho } \le (4 e^{-\kappa })^q \max _{p \in \mathfrak {S}_{q+1}} \sup _{x_{p(1)} \in \Lambda } \sum _{S_1 \ni x_{p(1)}} \dots \sup _{x_{p(q+1)} \in \Lambda } \sum _{S_{q+1} \ni x_{p(q+1)}} e^{\kappa (|S_1| + \dots |S_{q+1}|)} \cdot \\ \cdot (|S_1| + \dots + |S_{q+1}|)^{q} F_{S_1, \dots , S_{q+1}}(A, B)\,, \end{aligned}$$

which by Cauchy estimates implies (A.3). Arguing analogously as in the proof of Lemma A.2, one also proves $\textrm{Ad}_A^q B \in \mathcal {O}^\texttt {s} _{\kappa , \rho }$. Summing over q, (A.3) gives

$$\begin{aligned}&\Big \Vert \sum _{q \ge 1} \frac{1}{q!}{\text {Ad}}^q_A B \Big \Vert _{\kappa ,\rho } \le \sum _{q \ge 1} \frac{1}{q!} \left( \frac{q}{e} \right) ^q \left( \frac{4 e^{-\kappa } \Vert A\Vert _{\kappa + \sigma ,\rho }}{\sigma }\right) ^q \Vert B\Vert _{\kappa + \sigma , \rho }\\&= \frac{4 e^{-\kappa } \Vert A\Vert _{\kappa + \sigma , \rho }}{\sigma } \sum _{q \ge 0} \frac{1}{(q+1)!} \left( \frac{q+1}{e} \right) ^{q+1} \left( \frac{4 e^{-\kappa } \Vert A\Vert _{\kappa + \sigma , \rho }}{\sigma }\right) ^q \Vert B\Vert _{\kappa + \sigma , \rho }\,. \end{aligned}$$

Since, by Stirling formula $q! \ge \left( \frac{q}{e}\right) ^q \sqrt{2\pi q}$, one has $\frac{1}{(q+1)!} \left( \frac{q+1}{e}\right) ^{q+1} \le 1\,,$ and the series converges due to the condition $4 e^{-\kappa } \Vert A\Vert _{\kappa + \sigma , \rho }<\sigma$, this proves (4.6). Equation (4.7) is proved in a similar way. $\square$

1.2 Proof of Lemma 6.2

First, we use $A(\omega t)=\sum _{S \in \mathcal {P}_c(\Lambda )} A_S(\omega t)$ and triangular inequality to have

$$ \Vert [A(\omega t), e^{-\textrm{i}\tau Z} O e^{\textrm{i}\tau Z}] \Vert _{\textrm{op}} \le \sum _{x \in \Lambda } \sum _{\begin{array}{c} S \in \mathcal {P}_c(\Lambda )\\ {x \in S} \end{array}} \Vert [A_S(\omega t), e^{-\textrm{i}\tau Z} O e^{\textrm{i}\tau Z}] \Vert _{\textrm{op}} =: (\star ). $$

We now define $Q_{S_O}$ as the smallest ball that contains $S_O$; we call $r_Q$ its radius. Then we construct $B_{S_O}$ which is the ball of radius $r_{Q}+v\tau$ and the same center as $Q_{S_O}$. Then by Lemma 6.1 we get for any fixed $\tau >0$

$$\begin{aligned} (\star )\le & {} 2\sum _{x \in B_{S_O}} \sum _{\begin{array}{c} S \in \mathcal {P}_c(\Lambda ) \\ \text {s.t. } x \in S \end{array}} \Vert A_S(\omega \tau ) \Vert _{\textrm{op}} \Vert O \Vert _{\textrm{op}} \\{} & {} + \sum _{x \in \Lambda \setminus B_{S_O}} \sum _{\begin{array}{c} S \in \mathcal {P}_c(\Lambda ) \\ \text {s.t. } x \in S \end{array}} \Vert A_S(\omega \tau ) \Vert _{\textrm{op}} \Vert O \Vert _{\textrm{op}} |S_O| e^{- \kappa (d(S,S_O)-v \tau ))} =:(*). \end{aligned}$$

For any $x \in S$:

$$ \begin{aligned} d(S,S_O)-v\tau&\ge d(x,S_O)-v\tau -|S| \ge d(x, Q_{S_O}) - v \tau - |S| = d(x,B_{S_O})-|S| \end{aligned} $$

which means

$$ e^{-\kappa (d(S,S_O)-v\tau )} \le e^{-\kappa d(x,B_{S_O})} e^{\kappa |S|} $$

and then

$$ \begin{aligned} (*)&\le 2\sum _{x \in B_{S_O}} \sum _{\begin{array}{c} S \in \mathcal {P}_c(\Lambda ) \\ \text {s.t. } x \in S \end{array}} \Vert A_S(\omega \tau ) \Vert _{\textrm{op}} \Vert O \Vert _{\textrm{op}}\\&\quad + \sum _{x \in \Lambda {\setminus } B_{S_O}} \sum _{\begin{array}{c} S \in \mathcal {P}_c(\Lambda ) \\ \text {s.t. } x \in S \end{array}} \Vert A_S (\omega \tau ) \Vert _{\textrm{op}} \Vert O \Vert _{\textrm{op}} |S_O| e^{\kappa |S|} e^{-\kappa d(x,B_{S_O})}. \end{aligned} $$

For any $\ell \in \mathbb {N}$, let $C_\ell = \{x \in \Lambda \, | \, \ell < d(x,B_{S_O}) \le \ell +1 \}$. Thus,

$$ \begin{aligned}&\sum _{x \in \Lambda \setminus B_{S_O}} \sum _{\begin{array}{c} S \in \mathcal {P}_c(\Lambda ) \\ \text {s.t. } x \in S \end{array}} \Vert A_S(\omega \tau ) \Vert _{\textrm{op}} \Vert O \Vert _{\textrm{op}} |S_O| e^{\kappa |S|} e^{-\kappa d(x,B_{S_O})} \\&\quad = \sum _{\ell \ge 0} \sum _{x \in C_\ell } \sum _{\begin{array}{c} S \in \mathcal {P}_c(\Lambda ) \\ \text {s.t. } x \in S \end{array}} \Vert A_S(\omega \tau ) \Vert _{\textrm{op}} \Vert O \Vert _{\textrm{op}} |S_O| e^{\kappa |S|} e^{-\kappa (\ell -1)} \\&\quad \le \sum _{\ell \ge 0} |S_O| (|S_O|+v\tau +\ell +1)^d e^{-\kappa (\ell -1)} \sup _{x \in \Lambda } \sum _{\begin{array}{c} S \in \mathcal {P}_c(\Lambda ) \\ \text {s.t. } x \in S \end{array}} \Vert A_S(\omega \tau ) \Vert _{\textrm{op}} \Vert O \Vert _{\textrm{op}} e^{\kappa |S|} \\&\quad \le |S_O| (S_O+v\tau )^d e^{-1} \sum _{\ell \ge 0} \left( 1+\frac{\ell +1}{|S_O|+v\tau } \right) ^d e^{-\kappa \ell } \sup _{x \in \Lambda } \sum _{\begin{array}{c} S \in \mathcal {P}_c(\Lambda ) \\ \text {s.t. } x \in S \end{array}} \Vert A_S(\omega \tau ) \Vert _{\textrm{op}} \Vert O \Vert _{\textrm{op}} e^{\kappa |S|}\,. \end{aligned} $$

Since the series in $\ell$ is convergent, we define $C(d,\kappa ):= e \sum _{\ell \ge 0} (2+\ell )^d e^{-\kappa \ell }$ and we obtain

$$\begin{aligned}{} & {} \sum _{x \in \Lambda \setminus B_{S_O}} \sum _{\begin{array}{c} S \in \mathcal {P}_c(\Lambda ) \\ \text {s.t. } x \in S \end{array}} \Vert A_S(\omega \tau ) \Vert _{\textrm{op}} \Vert O \Vert _{op} |S_O| e^{\kappa |S|} e^{-\kappa d(x,B_{S_O})} \\{} & {} \quad \le |S_O| (|S_O|+v \tau )^d C(d,\kappa ) \Vert O \Vert _{\textrm{op}} \Vert A(\omega \tau ) \Vert _{\kappa } \,. \end{aligned}$$

Then, for fixed $\tau$, since $C(d,\kappa )\ge 1,$ one has

$$ \begin{aligned} \Vert [A(\omega \tau ), e^{-\textrm{i}\tau Z} O e^{\textrm{i}\tau Z} ] \Vert _{\textrm{op}}&\le 2{(|S_O+v\tau )^d}\Vert O \Vert _{\textrm{op}} \Vert A(\omega \tau ) \Vert _{0} \\&\quad + |S_O| (|S_O|+v \tau )^d C(d,\kappa ) \Vert O \Vert _{\textrm{op}} \Vert A(\omega \tau ) \Vert _{\kappa }\\&\le 3|S_O| (|S_O|+v \tau )^d C(d,\kappa ) \Vert O \Vert _{\textrm{op}} \Vert A(\omega \tau ) \Vert _{\kappa }. \end{aligned} $$

Recalling that $\sup _{\tau \in \mathbb {R}} \Vert A(\omega \tau )\Vert _\kappa \le \Vert A \Vert _{\kappa ,0}$, one has

$$\begin{aligned} \int _0^t \textrm{d}\tau \Vert [A(\omega \tau ),e^{-\textrm{i}\tau Z} O e^{\textrm{i}\tau Z} ] \Vert _{\textrm{op}}\le & {} 3|S_O|\Vert O \Vert _{\textrm{op}} \Vert A \Vert _{\kappa ,0} \frac{C(d,\kappa )}{(d+1) v(d,\kappa ,\Vert Z \Vert _{2 \kappa })} \\{} & {} \left[ (|S_O|+v(d,\kappa ,\Vert Z \Vert _{2 \kappa }) t)^{d+1}-|S_O|^{d+1} \right] , \end{aligned}$$

Through elementary computations one sees that

$$ \frac{(|S_O|+y)^{d+1}-|S_O|^{d+1}}{y} \le 2^{d+1} (d+1) |S_O|^{d+1} \langle y\rangle ^{d+1} \, \qquad \forall y >0. $$

Putting $y=v(Z,\kappa )t$, and recalling that $v(Z,\kappa )=C(d)\kappa ^{-{(d+2)}}e^{\kappa }\Vert Z \Vert _{2 \kappa }$, one gets

$$ \begin{aligned} \int _0^t \textrm{d}\tau \Vert [A,e^{-\textrm{i}\tau Z} O e^{\textrm{i}\tau Z} ] \Vert _{\textrm{op}}&\le 3 \cdot 2^{d+1} |S_0|^{d+2} C(d,\kappa ) \langle tv(\kappa ,Z) \rangle ^{d+1} t \Vert O \Vert _{\textrm{op}} \Vert A \Vert _{\kappa } \\&\le C(d,\kappa ,|S_O|)\langle \Vert Z \Vert _{2\kappa } \rangle ^{d+1} \langle t \rangle ^{d+1} t \Vert O \Vert _{\textrm{op}} \Vert A \Vert _{\kappa } \, , \end{aligned} $$

with $C(|S_O|,d,\kappa )=3\cdot 2^{d+1} |S_O|^{d+2}C(d,\kappa ) \big (C(d) \kappa ^{-{(d+2)}} e^{\kappa }\big )^{d+1}$.