Reducibility for a fast driven linear Klein-Gordon equation
aa r X i v : . [ m a t h . A P ] J a n Reducibility for a fast driven linear Klein-Gordon equation
L. Franzoi ∗ , A. Maspero † January 25, 2019
Abstract
We prove a reducibility result for a linear Klein-Gordon equation with a quasi-periodicdriving on a compact interval with Dirichlet boundary conditions. No assumptions aremade on the size of the driving, however we require it to be fast oscillating. In particular,provided that the external frequency is sufficiently large and chosen from a Cantor set oflarge measure, the original equation is conjugated to a time independent, diagonal one. Weachieve this result in two steps. First, we perform a preliminary transformation, adapted tofast oscillating systems, which moves the original equation in a perturbative setting. Thenwe show that this new equation can be put to constant coefficients by applying a KAMreducibility scheme, whose convergence requires a new type of Melnikov conditions.
We consider a linear Klein-Gordon equation with quasi-periodic driving(1.1) B tt u ´ B xx u ` m u ` V p ωt, x q u “ , x P r , π s , t P R , with spatial Dirichlet boundary conditions u p t, q “ u p t, π q “ .The potential V : T ν ˆ r , π s Ñ R , is quasi-periodic in time with a frequency vector ω P R ν zt u .The main feature of this driving is that it is not perturbative in size, but we require it to be fastoscillating, namely | ω | " .The goal of our paper is to provide, for any frequency ω belonging to a Cantor set of largemeasure, a reducibility result for the system (1.1). That is, we construct a change of coordinateswhich conjugates equation (1.1) into a diagonal, time independent one.As long as we know, this is the first result of reducibility in an infinite dimensional setting inwhich the perturbation is not assumed to be small in size, but only fast oscillating.The proof is carried out in two steps, combining a preliminary transformation, adapted tofast oscillating systems, with a KAM reducibility scheme which completely removes the timedependence from the equation. In particular we first perform a change of coordinates, following[ADRHH17b], that conjugates (1.1) to an equation with driving of size | ω | ´ , and thus pertur-bative in size. The price to pay is that the new equation might not fit in the standard KAMscheme developed by Kuksin in [Kuk87]. The problem is overcome in our model by exploitingthe pseudodifferential properties of the operators involved, showing that the new perturbationfeatures regularizing properties. ∗ International School for Advanced Studies (SISSA), Via Bonomea 265, 34136, Trieste, Italy
Email: [email protected] † International School for Advanced Studies (SISSA), Via Bonomea 265, 34136, Trieste, Italy
Email: [email protected] balanced
Melnikov conditions(see (1.16)), which allow us to perform a convergent KAM reducibility iteration.To carry out this program, we strongly exploit the fact that the dispersion law of the systemis asymptotically linear in the frequency space; this is used in a direct way to prove the balancedMelnikov conditions, and in an indirect way to prove that the new forcing term generated bythe preliminary transformation is a bounded operator (see also Remark 3.3). This is the mainreason why we consider the Klein-Gordon system. That being said, we suspect a similar resultto be true also for systems with superlinear dispersion law, as the Schrödinger equation, but newideas are needed to overcome the mentioned problems.From a mathematical point of view, our result is part of the attempts to extend classicalFloquet theory and its quasi-periodic generalization to infinite dimensional systems. Whilemany progresses have been made in the last 20 years to prove non perturbative reducibil-ity for finite (and actually low) dimensional systems [Eli01, Kri99, Kri01, Cha11, AFK11],in the infinite dimensional case the only available results nowadays deal with systems whichare small perturbations of a diagonal operator, i.e. of the form D ` ǫV p ωt q , where D is di-agonal, ǫ small and ω in some Cantor set. In this case the literature splits essentially intwo parts: the first one dealing with the case of perturbations which are bounded operators[EK09, GT11, GP16b, GP16a, WL17], while the second one (of more recent interest) with un-bounded ones [BG01, LY10, BBM14, FP15, Bam18, BGMR18].In particular, for the wave and Klein-Gordon equations, the papers [Pös96a, CY00, FHW14,GP16a] are in the first group, while [BBP14, Mon17b] belong to the second one. In any case, allthe previous results require a smallness assumption on the size of the perturbation.In order to deal with perturbations that are periodic in time and fast oscillating, in [ADRH16,ADRHH17a, ADRHH17b] Abanin, De Roeck, Ho and Huveneers developed an adapted normalform that generalizes the classical Magnus expansion [Mag54]. Such a normal form, which fromnow on we call Magnus normal form , allows to extract a time independent Hamiltonian (usuallycalled the effective
Hamiltonian), which approximates well the dynamics up to some finite butvery long times. In [ADRHH17b], the authors apply the Magnus normal form to the study of somequantum many-body systems (spin chains) with a fast periodic driving. Although the Magnusnormal form was developed for periodic systems, we extend it here for quasi-periodic ones andwe use it as a preliminary transformation that moves the problem in a more favourable settingfor starting a KAM reducibility scheme. However, we point out that an important differencebetween [ADRHH17b] and our work lies in the fact that, while in [ADRHH17b] all the involvedoperators are bounded, on the contrary our principal operator is an unbounded one.In case of systems of the form H ` V p t q , where the perturbation V p t q is neither small in sizenor fast oscillating, a general reducibility is not known. However, in same cases it is possibleto find some results of "almost reducibility"; that is, the original Hamiltonian is conjugated toone of the form H ` Z p t q ` R p t q , where Z p t q commutes with H , while R p t q is an arbitrarysmoothing operator, see e.g. [BGMR17]. This normal form ensures upper bounds on the speedof transfer of energy from low to high frequencies; e.g. it implies that the Sobolev norms of eachsolution grows at most as t ǫ when t Ñ 8 , for any arbitrary small ǫ ą . This procedure (or aclose variant of it), has been applied also in [Del10, MR17, Mon17a].There are also examples in [Bou99, Del14, Mas18] where the authors engineer periodic drivingsaimed to transfer energy from low to high frequencies and leading to unbounded growth ofSobolev norms (see also Remark 1.7 below).Finally, we want to mention also the papers [BB08, CG17], where KAM techniques are appliedto construct quasi-periodic solutions with | ω | " . In [BB08] this is shown for a nonlinear waveequation with Dirichlet boundary conditions, however reducibility is not obtained. In [CG17],2AM techniques are applied to a many-body system with fast driving; the authors construct aperiodic orbit with large frequency and prove its asymptotic stability.Before closing this introduction, we mention that periodically driven systems have also a greatinterest in physics, both theoretically and experimentally. Indeed such systems often exhibit arich and surprising behaviour, like the Kapitza pendulum [Kap51], where the fast periodic drivingstabilizes the otherwise unstable equilibrium point in which the pendulum is upside-down. Morerecently, a lot of attention has been dedicated to fast periodically driven many-body systems[JMC15, GD14, KBRD10, JMD ` The potential driving V p ωt, x q is treated as a smooth function V : T ν ˆ r , π s Q p θ, x q ÞÑ V p θ, x q P R , ν ě , which satisfies two conditions:( V1 ) The even extension in x of V p θ, x q on the torus T » r´ π, π s , which we still denote by V , issmooth in both variables and it extends analytically in θ in a proper complex neighbourhoodof T ν of width ρ ą . In particular, for any ℓ P N , there is a constant C ℓ,ρ ą such that ˇˇ B ℓx V p θ, x q ˇˇ ď C ℓ,ρ @ x P T , | Im θ | ď ρ ; ( V2 ) ş T ν V p θ, x q d θ “ for any x P r , π s .To state precisely our main result, equation (1.1) has to be rewritten as a Hamiltonian system.We introduce the new variables(1.2) ϕ : “ B { u ` i B ´ { B t u , ϕ : “ B { u ´ i B ´ { B t u , where(1.3) B : “ a ´ ∆ ` m ; note that the operator B is invertible also when m “ , since we consider Dirichlet boundaryconditions. In the new variables equation (1.1) is equivalent to(1.4) i B t ϕ p t q “ Bϕ p t q ` B ´ { V p ωt q B ´ { p ϕ p t q ` ϕ p t qq . Taking (1.4) coupled with its complex conjugate, we obtain the following system(1.5) i B t ϕ p t q “ H p t q ϕ p t q , H p t q : “ ˆ B ´ B ˙ ` B ´ { V p ωt, x q B ´ { ˆ ´ ´ ˙ , where, abusing notation, we denoted ϕ p t q ” ˆ ϕ p t q ϕ p t q ˙ the vector with the components ϕ, ϕ . Thephase space for (1.5) is H r ˆ H r , where, for r ě ,(1.6) H r : “ ϕ p x q “ ÿ m P N ϕ m sin p mx q , x P r , π s ˇˇˇˇˇ k ϕ k H r : “ ÿ m P N x m y r | ϕ m | ă 8 + . x m y : “ p ` | m | q , which will be kept throughout all the article.We define the ν -dimensional annulus of size M ą by R M : “ B M p qz B M p q Ă R ν ; here we denoted by B M p q the ball of center zero and radius M in the Euclidean topology of R ν . Theorem 1.1.
Consider the system (1.5) and assume ( V1 ) and ( V2 ). Fix arbitrary r, m ě and α P p , q . Fix also an arbitrary γ ˚ ą sufficiently small.Then there exist M ˚ ą , C ą and, for any M ě M ˚ , a subset Ω α “ Ω α p M , γ ˚ q in R M , fulfilling (1.7) meas p R M z Ω α q meas p R M q ď Cγ ˚ , such that the following holds true. For any frequency vector ω P Ω α , there exists an operator T p ωt ; ω q , bounded in L p H r ˆ H r q , quasi-periodic in time and analytic in a shrunk neighbourhoodof T ν of width ρ { , such that the change of coordinates ϕ “ T p ωt ; ω q ψ conjugates (1.5) to thediagonal time-independent system (1.8) i ψ p t q “ H ,α ψ p t q , H ,α : “ ˆ D ,α ´ D ,α ˙ , D ,α “ diag λ j p ω q ˇˇ j P N ( . The transformation T p ωt ; ω q is close to the identity, in the sense that there exists C r ą independent of M such that (1.9) kT p ωt ; ω q ´ k L p H r ˆ H r q ď C r M ´ α . The new eigenvalues p λ j p ω qq j P N are real, Lipschitz in ω , and admit the following asymptoticsfor j P N : (1.10) λ j p ω q ” λ j p ω, α q “ λ j ` ε j p ω, α q , ε j p ω, α q „ O ˆ M j α ˙ , where λ j “ a j ` m are the eigenvalues of the operator B .Remark . In particular, back to the original coordinates, equation (1.1) is reduced to(1.11) B tt u ` p D ,α q u “ . Remark . The parameter α , which one chooses and fixes in the real interval p , q , influ-ences the asymptotic expansion of the final eigenvalues, as one can read from (1.10). Also theconstruction of the set of the admissible frequency vectors heavily depends on this parameter. Remark . We believe that the assumptions of Theorem 1.1 can be weakened, for exampleasking only Sobolev regularity for V p θ, x q , dropping ( V2 ) or using periodic boundary conditions;these issues will be addressed elsewhere. Remark . In Theorem 1.1 we can take also m “ ; this is due to the fact that, with Dirichletboundary conditions, the unperturbed eigenvalues λ j are simple, integers and their correctionsare small (see (1.10)). This implies that it is enough to move the frequency vector ω for avoidingresonances. 4et us denote by U ω p t, τ q the propagator generated by (1.5) such that U ω p τ, τ q “ , @ τ P R .An immediate consequence of Theorem 1.1 is that we have a Floquet decomposition:(1.12) U ω p t, τ q “ T p ωt ; ω q ˚ ˝ e ´ i p t ´ τ q H ,α ˝ T p ωτ ; ω q . Another consequence of (1.12) is that, for any r ě , the norm kU ω p t, q ϕ k H r ˆ H r is boundeduniformly in time: Corollary 1.6.
Let M ě M ˚ and ω P Ω α . For any r ě one has (1.13) c r k ϕ k H r ˆ H r ď kU ω p t, q ϕ k H r ˆ H r ď C r k ϕ k H r ˆ H r , @ t P R , @ ϕ P H r ˆ H r , for some c r ą , C r ą .More precisely, there exists a constant c r ą s.t. if the initial data ϕ P H r ˆ H r then ˆ ´ c r M ´ α ˙ k ϕ k H r ˆ H r ď kU ω p t, q ϕ k H r ˆ H r ď ˆ ` c r M ´ α ˙ k ϕ k H r ˆ H r , @ t P R . Remark . Corollary 1.6 shows that, if the frequency ω is chosen in the Cantor set Ω α , nophenomenon of growth of Sobolev norms can happen. On the contrary, if ω is chosen resonant,one can construct drivings which provoke norm explosion with exponential rate, see [Bou99] (seealso [Mas18] for other examples). Remark . For nonlinear PDEs, the property that all solutions have uniformly bounded Sobolevnorms is typical connected to integrability. For example, the 1 dimensional defocusing NLS, theKdV and Toda chain exhibit this property (see e.g. [Mas18, BM16, KMMT16]).
Our proof splits into three different parts, which we now summarize.
The Magnus normal form.
In Section 3 we perform a preliminary transformation, adaptedto fast oscillating systems, which moves the non-perturbative equation (1.5) into a pertubativeone where the size of the transformed quasi-periodic potential is as small as large is the moduleof the frequency vector. Sketchily, we perform a change of coordinates which conjugates(1.14) " H p t q “ H ` W p ωt q ”size p W q „ ù " r H p t q “ H ` V p ωt ; ω q ”size p V q „ | ω | ´ ” . This change of coordinates, called below Magnus normal form, is an extension to quasi-periodicsystems of the one performed in [ADRHH17b]. Note that H is the same on both sides of (1.14)provided ş T ν W p θ q d θ “ , which is fulfilled in our case thanks to Assumption (V2) .As we already mentioned, the price to pay is that, in principle, it is not clear that the newperturbation is sufficiently regularizing to fit in a standard KAM scheme (see Remark 3.3 for amore detailed discussion).Here it is essential to employ pseudodifferential calculus, thanks to which we control the order(as a pseudodifferential operator) of the new perturbation, and prove that it is actually enoughregular for the KAM iteration. This is true because the principal term of the new perturbationis a commutator with H (see equation (3.20)), and one can exploit the smoothing properties ofthe commutator of pseudodifferential operators.5 alanced Melnikov conditions. After the Magnus normal form, we perform a KAM re-ducibility scheme in order to remove the time dependence on the coefficients of the equation. Asusual one needs second order Melnikov conditions on the unperturbed eigenvalues λ j “ a j ` m .One might impose that for some γ, τ ą ,(1.15) | ω ¨ k ` λ j ´ λ l | ě γ x k y τ x j ´ l y| ω | , @p k, j, l q P Z ν ˆ N ˆ N , p k, j, l q ‰ p , j, j q ; such conditions are violated for a set of frequencies of relative measure bounded by Cγ , where C is a constant independent of | ω | .These Melnikov conditions are useless in our context; indeed recall that, after the Magnus normalform, the new perturbation has size „ | ω | ´ while the small denominators in (1.15) have size „ | ω | ; so the two of them compensate each others, and the KAM step cannot reduce in size.To overcome the problem, rather than (1.15), we impose new balanced Melnikov conditions, inwhich we balance the loss in size (in the denominator) and gain in regularity (in the numerator)in (1.15). More precisely, we show that for any α P r , s one can impose(1.16) | ω ¨ k ` λ j ´ λ l | ě γ x k y τ x j ´ l y α | ω | α , @p k, j, l q P Z ν ˆ N ˆ N , p k, j, l q ‰ p , j, j q for a set of ω ’s in R M of large relative measure. This is proved in Section 4. By choosing ă α ă ,the l.h.s. of (1.16) is larger than the corresponding one in (1.15), and the KAM transformationreduces in size. However note that the choice of α will influence the regularizing effect given by x j ˘ l y α in the r.h.s. of (1.16); ultimately, this modifies the asymptotic expansion of the finaleigenvalues, as one can see in (1.10). The KAM reducibility.
At this point we perform a KAM reducibility scheme; this step isnowadays quite standard and we only sketch the proofs.
Acknowledgments.
We thanks Dario Bambusi, Massimiliano Berti, Roberto Feola, MatteoGallone and Vieri Mastropietro for many stimulating discussions. We were partially supported byPrin-2015KB9WPT and Progetto GNAMPA - INdAM 2018 “Moti stabili ed instabili in equazionidi tipo Schrödinger”.
Given a set Ω Ă R ν and a Fréchet space F , the latter endowed with a system of seminorms t k ¨ k n | n P N u , we define for a function f : Ω Q ω ÞÑ f p ω q P F the quantities(2.1) | f | n, Ω : “ sup ω P Ω k f p ω q k n , | f | Lip n, Ω : “ sup ω ,ω P Ω ω ‰ ω k f p ω q ´ f p ω q k n | ω ´ ω | . Given w P R ` , we denote by Lip w p Ω , F q the space of functions from Ω into F such that(2.2) k f k Lip p w q n, Ω : “ | f | n, Ω ` w | f | Lip n, Ω ă 8 . remark that the conditions | ω ¨ k ` λ j ˘ λ l | ě γ x k y τ x j ˘ l y are violated on a set of relative measure „ γ | ω | ,which is as large as the size of the frequency vector. .1 Pseudodifferential operators The main tool for the construction of the Magnus transform in Section 3 is the calculus withpseudodifferential operators acting on the scale of the standard Sobolev spaces on the torus T : “ R { π Z , which is defined for any r P R as(2.3) H r p T q : “ ϕ p x q “ ÿ j P Z ϕ j e i jx , x P T ˇˇˇˇˇ k ϕ k H r p T q : “ ÿ j P Z x j y r | ϕ j | ă 8 + . For a function f : T ˆ Z Ñ R , define the difference operator △ f p x, j q : “ f p x, j ` q ´ f p x, j q andlet ∆ β “ ∆ ˝ ... ˝ ∆ be the composition β times of ∆ . Then, we have the following: Definition 2.1.
We say that a function f : T ˆ Z Ñ R is a symbol of order m P R if for any j P Z the map x ÞÑ f p x, j q is smooth and, furthermore, for any α, β P N , there exists C α,β ą such that ˇˇ B αx △ β f p x, j q ˇˇ ď C α,β x j y m ´ β , @ x P T . If this is the case, we write f P S m .We endow S m with the family of seminorms ℘ mℓ p f q : “ ÿ α ` β ď ℓ sup p x,j qP T ˆ Z x j y ´ m ` β ˇˇ B αx △ β f p x, j q ˇˇ , ℓ P N . Analytic families of pseudodifferential operators.
We will consider in our discussion alsosymbols depending real analytically on the variable θ P T ν . To define them, we need to introducethe complex neighbourhood of the torus T νρ : “ t a ` i b P C ν | a P T ν , | b | ď ρ u . Definition 2.2.
Given m P R and ρ ą , a function f : T ν ˆ T ˆ Z Ñ R , p θ, x, j q ÞÑ f p θ, x, j q ,is called a symbol of class S mρ if for any j P N it is smooth in x , it extends analytically in θ in T νρ and, furthermore, for every α, β P N there exists C α,β ą such that ˇˇ B αx △ β f p θ, x, j q ˇˇ ď C α,β x j y m ´ β @ x P T , @ θ P C ν , | Im θ | ď ρ . For such a function we write f P S mρ .We endow the class S mρ with the family of seminorms ℘ m,ρℓ p f q : “ sup | Im θ |ď ρ ÿ α ` β ď ℓ sup p x,j qP T ˆ Z x j y ´ m ` β ˇˇ B αx △ β f p θ, x, j q ˇˇ , ℓ P N . We associate to a symbol f P S mρ the operator f p θ, x, D x q by standard quantization(2.4) ψ p x q “ ÿ j P Z ψ j e i jx ÞÑ p f p θ, x, D x q ψ q p x q : “ ÿ j P Z f p θ, x, j q ψ j e i jx ; here D x “ D : “ i ´ B x is the Hörmander derivative. Definition 2.3.
We say that F P A mρ if it is a pseudodifferential operator with symbol of class S mρ , i.e. if there exists a symbol f P S mρ such that F “ f p θ, x, D x q .If F does not depend on θ , we simply write F P A m .7 emark . For any σ P R , the operator x D y σ ” p ´ B xx q σ is in A σ .As usual we give to A mρ a Fréchet structure by endowing it with the seminorms of the symbols.Finally we define the class of pseudodifferential operators depending on a Lipschitz way on anexternal parameter. Definition 2.5.
We denote by
Lip w p Ω , A mρ q the space of pseudodifferential operators whosesymbols belong to Lip w p Ω , S mρ q and by ´ ℘ n,ρj p¨q Lip p w q Ω ¯ j P N the corresponding seminorms. Remark . Let F P Lip w p Ω , A mρ q and G P Lip w p Ω , A nρ q . Then the symbolic calculus impliesthat F G P Lip w p Ω , A m ` nρ q and r F, G s P
Lip w p Ω , A m ` n ´ ρ q , with the quantitative bounds @ j D N s.t. ℘ m ` n,ρj p F G q Lip p w q Ω ď C ℘ m,ρN p F q Lip p w q Ω ℘ n,ρN p G q Lip p w q Ω , @ j D N s.t. ℘ m ` n ´ ,ρj pr F, G sq Lip p w q Ω ď C ℘ m,ρN p F q Lip p w q Ω ℘ n,ρN p G q Lip p w q Ω . Parity preserving operators.
The space H of (1.6) is naturally identified with the subspaceof H p T q ” L p T q of odd functions. Therefore it makes sense to work with pseudodifferentialoperators preserving the parity. Before describing them, we recall the orthogonal decompositionof the periodic L -functions on T : L p T q “ L even p T q ‘ L odd p T q where, for u p x q “ ř j P Z u j e i jx P L p T q , we have for any j P Z ,(2.5) u P L even p T q ô u ´ j “ u j and u P L odd p T q ô u ´ j “ ´ u j . Definition 2.7.
We denote by P S mρ the class of symbols f P S mρ satisfying the property(2.6) f p θ, x, j q “ f p θ, ´ x, ´ j q @ θ P T ν , x P T , j P Z . We denote by PA mρ the subset of A mρ of parity preserving operators, that is, those operators A P A mρ such that A p L even q Ď L even and A p L odd q Ď L odd . Lemma 2.8.
Let F P A mρ with symbol f P S mρ . Then F P PA mρ if and only if f P P S mρ .Proof. It is easy to check that F p L odd p T qq Ď L odd p T q if and only if the symbol f p x, j q of F fulfills Im rp f p x, j q ´ f p´ x, ´ j qq e i jx s ” . Similarly F p L even p T qq Ď L even p T q if and only if Re rp f p x, j q ´ f p´ x, ´ j qq e i jx s ” . Remark . For all σ P R , the operator x D y σ P PA σ , while, by the assumption ( V1 ), V P PA ρ . Remark . Parity preserving operators are closed under composition and commutators.
Remark . For m “ and σ ą , we define B ´ σ ψ : “ ř j ‰ | j | σ ψ j e i jx for any ψ P L p T q ;clearly B ´ σ P PA ´ σ . Note that BB ´ ψ “ B ´ Bψ “ ψ ´ ψ . However, the restriction B | H of B to the phase space (1.6) is invertible (since the phase space contains only functions with zeroaverage) and B ´ is its inverse. For the KAM reducibility, a second and wider class of operators without a pseudodifferentialstructure is needed on the scale of Hilbert spaces p H r q r P R , as defined as in (1.6). Moreover, let H : “ X r P R H r and H ´8 : “ Y r P R H r . If A is a linear operator, we denote by A ˚ the adjointof A with respect to the scalar product of H , while we denote by A the conjugate operator: Aψ : “ Aψ @ ψ P D p A q . atrix representation of operators. To any linear operator A : H Ñ H ´8 we associateits matrix of coefficients p A nm q m,n P N on the basis p p e n : “ sin p nx qq n P N , defined for m, n P N as A nm ” x A p e m , p e n y H . Remark . If A is a bounded operator, the following implications hold: A “ A ˚ ðñ A nm “ A mn @ m, n P N ; A “ A ˚ ðñ A nm “ A mn @ m, n P N . A useful norm we can put on the space of such operators is in the following:
Definition 2.13.
Given a linear operator A : H Ñ H ´8 and s P R , we say that A has finite s -decay norm provided(2.7) | A | s : “ ˜ ÿ h P N x h y s sup | m ´ n |“ h | A nm | ¸ { ă 8 . One has the following:
Lemma 2.14 (Algebra of the s-decay) . For any s ą there is a constant C s ą such that (2.8) | AB | s ď C s | A | s | B | s . The proof of the Lemma is an easy variant of the one in [BB13] we sketch it in Appendix A.3.
Remark . If A : H Ñ H ´8 has finite s -decay norm with s ą , then for any r P r , s s , A extends to a bounded operator H r Ñ H r . Moreover, by tame estimates, one has the quantitativebound k A k L p H r q ď C r,s | A | s .Next, we consider operators depending analytically on angles θ P T ν . Definition 2.16.
Let A be a θ -depending operator, A : T ν Ñ L p H , H ´8 q . Given s ě and ρ ą , we say that A P M ρ,s if one has(2.9) | A | ρ,s : “ ÿ k P Z ν e ρ | k | ˇˇˇ p A p k q ˇˇˇ s ă 8 , where p A p k q : “ p π q ν ż T ν A p θ q e ´ i k ¨ θ d θ . Remark . If A is a θ -depending bounded operator, the following implications hold: A “ A ˚ ðñ r p A p k qs ˚ “ p A p´ k q @ k P Z ν ðñ p A nm p k q “ p A mn p´ k q @ k P Z ν , @ m, n P N A “ A ˚ ðñ r p A p k qs ˚ “ p A p k q @ k P Z ν ðñ p A nm p k q “ p A mn p k q @ k P Z ν , @ m, n P N If Ω Q ω ÞÑ A p ω q P M ρ,s is a Lipschitz map, we write A P Lip w p Ω , M ρ,s q , provided(2.10) | A | Lip p w q ρ,s, Ω : “ sup ω P Ω | A p ω q| ρ,s ` w sup ω ‰ ω P Ω | A p ω q ´ A p ω q| ρ,s | ω ´ ω | ă 8 . Remark . For any s ą and ρ ą , the spaces M ρ,s and Lip w p Ω , M ρ,s q are closed withrespect to composition, with | AB | ρ,s ď C s | A | ρ,s | B | ρ,s , | AB | Lip p w q ρ,s, Ω ď C s | A | Lip p w q ρ,s, Ω | B | Lip p w q ρ,s, Ω . This follows from Lemma 2.14 and the algebra properties for analytic functions.9 perator matrices.
We are going to meet matrices of operators of the form(2.11) A “ ˆ A d A o ´ A o ´ A d ˙ , where A d and A o are linear operators belonging to the class M ρ,s . Actually, the operator A d on the diagonal will have different decay properties than the element on the anti-diagonal A o .Therefore, we introduce classes of operator matrices in which we keep track of these differences. Definition 2.19.
Given an operator matrix A of the form (2.11), α, β P R , ρ ą , s ě , we saythat A belongs to M ρ,s p α, β q if(2.12) r A d s ˚ “ A d , r A o s ˚ “ A o and one also has x D y α A d , A d x D y α P M ρ,s , (2.13) x D y β A o , A o x D y β P M ρ,s , (2.14) x D y σ A δ x D y ´ σ P M ρ,s , @ σ P t˘ α, ˘ β, u , @ δ P t d, o u . (2.15)We endow M ρ,s p α, β q with the norm(2.16) | A | α,βρ,s : “ ˇˇ x D y α A d ˇˇ ρ,s ` ˇˇ A d x D y α ˇˇ ρ,s ` ˇˇˇ x D y β A o ˇˇˇ ρ,s ` ˇˇˇ A o x D y β ˇˇˇ ρ,s ` ÿ σ Pt˘ α, ˘ β, u δ Pt d,o u ˇˇˇ x D y σ A δ x D y ´ σ ˇˇˇ ρ,s , with the convention that, in case of repetition (when α “ β , α “ or β “ ), the same termsare not summed twice. When A is independent of θ P T ν , we use the norm | A | α,βs , defined as(2.16), but replacing |¨| ρ,s with the s -decay norm |¨| s defined in (2.7).Let us motivate the properties describing the class M ρ,s p α, β q : • Condition (2.12) is equivalent to ask that A is the Hamiltonian vector field of a real valuedquadratic Hamiltonian, see e.g. [Mon17b] for a discussion; • Conditions (2.13) and (2.14) control the decay properties for the coefficient of the coeffi-cients of the matrices associated to A d and A o : indeed the matrix coefficients of x D y α A x D y β are given by „ { x D y α A x D y β nm p k q “ x m y α p A nm p k q x n y β , therefore decay (or growth) properties for the matrix coefficients of the operator A areimplied by the boundedness of the norms | ¨ | ρ,s ; • Condition (2.15) is just for simplifying some computations below.
Remark . Let ă ρ ď ρ , ď s ď s α ě α , β ě β . Then M ρ,s p α, β q Ď M ρ ,s p α , β q withthe quantitative bound | A | α ,β ρ ,s ď | A | α,βρ,s .Finally, if A d p ω q and A o p ω q depend in a Lipschitz way on a parameter ω , we introduce theLipschitz norm(2.17) | A | Lip p w q ρ,s,α,β, Ω : “ sup ω P Ω | A p ω q| α,βρ,s ` w sup ω ‰ ω P Ω | A p ω q ´ A p ω q| α,βρ,s | ω ´ ω | . If such a norm is finite, we write A P Lip w p Ω , M ρ,s p α, β qq .10 mbedding of parity preserving pseudodifferential operators. The introduction of theclasses M ρ,s p α, β q is due to the fact that they are closed with respect the KAM reducibilityscheme, for a proper choice of α and β . In the next lemma we show how parity preservingpseudodifferential operators embed in such classes. Lemma 2.21 (Embedding) . Given α, β, ρ ą , consider F P PA ´ αρ and G P PA ´ βρ . Assumethat F ˚ “ F , G ˚ “ G , (where the adjoint is with respect to the scalar product of H ). Define the operator matrix (2.18) A : “ ˆ F G ´ G ´ F ˙ . Then, for any s ě and ă ρ ă ρ , one has A P M ρ ,s p α, β q . Moreover, there exist C, c ą such that (2.19) | A | α,βρ ,s ď C p ρ ´ ρ q ν ´ ℘ ´ α,ρs ` c p F q ` ℘ ´ β,ρs ` c p G q ¯ . Finally, if F P Lip w p Ω , PA ´ αρ q , G P Lip w p Ω , PA ´ βρ q , one has A P Lip w p Ω , M ρ ,s p α, β qq and (2.19) holds with the corresponding weighted Lipschitz norms. The proof is available in Appendix A.
Commutators and flows.
These classes of matrices enjoy also closure properties under com-mutators and flow generation. We define the adjoint operator(2.20) ad X p V q : “ i r X , V s ; note the multiplication by the imaginary unit in the definition of the adjoint map. Lemma 2.22 (Commutator) . Let α, ρ ą and s ą . Assume V P M ρ,s p α, q and X P M ρ,s p α, α q . Then ad X p V q belongs to M ρ,s p α, α q with the quantitative bound (2.21) ˇˇˇ ad X p V q ˇˇˇ α,αρ,s ď C s | X | α,αρ,s | V | α, ρ,s ; here C s is the algebra constant of (2.7) . Moreover, if V P Lip w p Ω , M ρ,s p α, qq and X P Lip w p Ω , M ρ,s p α, α qq , then ad X p V q P Lip w p Ω , M ρ,s p α, α qq , with (2.22) | ad X p V q| Lip p w q ρ,s,α,α, Ω ď C s | X | Lip p w q ρ,s,α,α, Ω | V | Lip p w q ρ,s,α, , Ω . Also the proof of this lemma is postponed to Appendix A.
Lemma 2.23 (Flow) . Let α, ρ ą , s ą . Assume V P M ρ,s p α, q , X P M ρ,s p α, α q . Then thefollowings hold true:(i) For any r P r , s s and any θ P T ν , the operator e i X p θ q P L p H r q , with the standard operatornorm uniformly bounded in θ ;(ii) The operator e i X V e ´ i X belongs to M ρ,s p α, q , while e i X V e ´ i X ´ V belongs to M ρ,s p α, α q with the quantitative bounds: (2.23) ˇˇ e i X V e ´ i X ˇˇ α, ρ,s ď e C s | X | α,αρ,s | V | α, ρ,s ; ˇˇ e i X V e ´ i X ´ V ˇˇ α,αρ,s ď C s e C s | X | α,αρ,s | X | α,αρ,s | V | α, ρ,s . nalogous assertions hold for V P Lip w p Ω , M ρ,s p α, qq and X P Lip w p Ω , M ρ,s p α, α qq . The proof of this lemma is a standard application of (2.21) and the remark that the operatornorm is controlled by the |¨| α,αρ,s -norm (see also Remark 2.15).
To begin with, we recall the Pauli matrices notation. Let us introduce(3.1) σ “ ˆ ˙ , σ “ ˆ ´ ii 0 ˙ , σ “ ˆ ´ ˙ , and, moreover, define σ : “ ˆ ´ ´ ˙ , : “ ˆ ˙ , : “ ˆ ˙ . Using Pauli matrix notation, equation (1.5) reads as(3.2) i ϕ p t q “ H p t q ϕ p t q : “ p H ` W p ωt qq ϕ p t q , H : “ B σ , W p ωt q : “ B ´ { V p ωt q B ´ { σ . Note that, by assumption ( V1 ), one has V P PA ρ (see Remark 2.9); therefore the properties ofthe pseudodifferential calculus and of the associated symbols (see Remarks 2.6 and 2.10) implythat(3.3) B P PA and B ´ { V B ´ { P PA ´ ρ (in case m “ , we use Remark 2.11 to define B ´ { ). The difficulty in treating equation (3.2) isthat it is not perturbative in the size of the potential, so standard KAM techniques do not applydirectly.To deal with this problem, we perform a change of coordinates, adapted to fast oscillatingsystems, which puts (3.2) in a perturbative setting. We refer to this procedure as Magnus normalform. The Magnus normal form is achieved in the following way: the change of coordinates ϕ p t q “ e ´ i X p ωt ; ω q ψ p t q conjugates (3.2) to i B t ψ p t q “ r H p t q ψ p t q , where the Hamiltonian r H p t q isgiven by (see [Bam18, Lemma 3.2]) r H p t q “ e i X p ωt ; ω q H p t q e ´ i X p ωt ; ω q ´ ż e i s X p ωt ; ω q X p ωt ; ω q e ´ i s X p ωt ; ω q d s (3.4) “ H ` i r X , H s ` W ´ X ` i r X , . . . s . (3.5)In (3.5) we wrote, informally, r X , . . . s to remark that all the non written terms are commutatorswith X . Then one chooses X to solve W ´ X “ ; if the frequency ω is large and nonresonant,then X has size | ω | ´ , and the new equation (3.5) is now perturbative in size. The price to payis the appearance of i r X , H s , which is small in size but possibly unbounded as operator. Wecontrol this term by employing pseudodifferential calculus and the properties of the commutators.With this informal introduction, the main result of the section is the following:12 heorem 3.1 (Magnus normal form) . For any ă γ ă , there exist a set Ω Ă R M Ă R ν anda constant c ą (independent of M ), with (3.6) meas p R M z Ω q meas p R M q ď c γ , such that the following holds true. For any ω P Ω and any weight w ą , there exists a timedependent change of coordinates ϕ p t q “ e ´ i X p ωt ; ω q ψ p t q , where X p ωt ; ω q “ X p ωt ; ω q σ , X P Lip w p Ω , PA ´ ρ { q , that conjugates equation (3.2) to (3.7) i ψ p t q “ r H p t q ψ p t q , r H p t q : “ H ` V p ωt ; ω q , where (3.8) V p θ ; ω q “ ˜ V d p θ ; ω q V o p θ ; ω q´ V o p θ ; ω q ´ V d p θ ; ω q ¸ , with r V d s ˚ “ V d , r V o s ˚ “ V o and (3.9) V d P Lip w p Ω , PA ´ ρ { q , V o P Lip w p Ω , PA ρ { q . Furthermore, for any ℓ P N , there exists C ℓ ą such that (3.10) ℘ ´ ,ρ { ℓ p V d q Lip p w q Ω ` ℘ ,ρ { ℓ p V o q Lip p w q Ω ď C ℓ M . Proof.
The proof is splitted into two parts, one for the formal algebraic construction, the otherfor checking that the operators that we have found possess the right pseudodifferential propertieswe are looking for.
Step I).
Expanding (3.4) in commutators we have(3.11) r H p t q “ H ` i r X , H s ´ r X , r X , H ss ` W ´ X ` R , where the remainder R of the expansion is given in integral form by R : “ ż p ´ s q e i s X ad X p H q e ´ i s X d s ` i ż e i s X r X , W s e ´ i s X d s ´ i ż p ´ s q e i s X r X , X s e ´ i s X d s. (3.12)From the properties of the Pauli matrices, we note that σ “ . This means that the terms in(3.12) involving W and X are null, and the remainder is given only by(3.13) R “ ż p ´ s q e i s X ad X p H q e ´ i s X d s. We ask X to solve the homological equation(3.14) “ W ´ X “ ˆ B ´ { V p ωt q B ´ { ´ X p ωt ; ω q ˙ σ . p X p k ; ω q “ ω ¨ k B ´ { p V p k q B ´ { , for k P Z ν zt u , p X p ω q ” where the second of (3.15) is a consequence of ( V2 ). It remains to compute the terms in (3.4)and (3.13) involving H . Using again the structure of the Pauli matrices, we get:(3.16) ad X p H q : “ i r X σ , B σ s “ i XB p ´ σ q ´ i BX p ` σ q “ i r X, B s ´ i r X, B s a σ , where we have denoted by r X, B s a : “ XB ` BX the anticommutator. Similarly one has ad X p H q : “ ´r X σ , r X σ , B σ ss (3.16) “ ´ pr X σ , r X, B s s ´ r X σ , r X, B s a σ sq“ ´pr X, r X, B ss ´ r X, r X, B s a s a q σ “ XBX σ ; (3.17)thus ad X p H q (3.17) “ r X σ , XBX σ s “ . (3.18)This shows that R ” and, imposing (3.15) in (3.4), we obtain(3.19) r H p t q “ H ` V p ωt ; ω q , with V d p θ ; ω q : “ i r X p θ ; ω q , B s ` X p θ ; ω q BX p θ ; ω q ,V o p θ ; ω q : “ ´ i r X p θ ; ω q , B s a ` X p θ ; ω q BX p θ ; ω q . (3.20) Step II).
We show now that
X, V d and V o , defined in (3.15) and (3.20) respectively, are pseudod-ifferential operators in the proper classes, provided ω is sufficiently nonresonant. First consider X . For γ ą and τ ą ν ´ , define the set of Diophantine frequency vectors(3.21) Ω ” Ω p γ , τ q : “ " ω P R M ˇˇˇˇ | ω ¨ k | ě γ x k y τ M @ k P Z ν zt u * . We will prove in Proposition 3.4 below that(3.22) meas p R M z Ω q meas p R M q ď c γ for some constant c ą independent of M and γ . This fixes the set Ω and proves (3.6).We show now that X P Lip w p Ω , PA ´ ρ { q . First note that, by Lemma A.1(i) (in Appendix A) andRemark 2.10, one has B ´ { p V p k q B ´ { P PA ´ (both B and V are independent from ω ) with ℘ ´ ℓ p B ´ { p V p k q B ´ { q ď e ´ ρ | k | ℘ ´ ,ρℓ p B ´ { V B ´ { q ď e ´ ρ | k | C ℓ . Provided ω P Ω , it follows that ℘ ´ ℓ p p X p k ; ¨qq Ω ď „ sup ω P Ω | ω ¨ k | ℘ ´ ℓ p B ´ { p V p k q B ´ { q ď x k y τ γ M e ´ ρ | k | C ℓ .
14o compute the Lipschitz norm, it is convenient to use the notation(3.23) ∆ ω f p ω q “ f p ω ` ∆ ω q ´ f p ω q , with ω , ω ` ∆ ω P Ω , ∆ ω ‰ . In this way one gets ˇˇˇ ∆ ω p X p k ; ω q ˇˇˇ ď | ∆ ω | | ω ¨ k | |p ω ` ∆ ω q ¨ k | ˇˇˇ B ´ { p V p k q B ´ { ˇˇˇ ñ ℘ ´ ℓ p p X p k ; ¨qq LipΩ ď x k y τ p γ M q e ´ ρ | k | C ℓ . As a consequence X p θ ; ω q “ ř k p X p k ; ω q e i k ¨ θ is a pseudodifferential operator in the class Lip w p Ω , PA ´ ρ { q (see Lemma A.1(ii) in Appendix A for details) fulfilling(3.24) ℘ ´ ,ρ { ℓ p X q Lip p w q Ω ď ˆ γ M ` w γ M ˙ C ℓ ρ τ ` ν ď max p , w q M r C ℓ ρ τ ` ν . It follows by Remark 2.10 that V d P Lip w p Ω , PA ´ ρ { q while V o P Lip w p Ω , PA ρ { q with theclaimed estimates (3.10).Finally, V is a real selfadjoint operator, simply because it is a real bounded potential, andtherefore V ˚ “ V “ V . It follows by Remark 2.17 and the explicit expression (3.15) that X ˚ “ X “ X . Using these properties one verifies by a direct computation that r V d s ˚ “ V d and r V o s ˚ “ V o . Estimate (3.24) and the symbolic calculus of Remark 2.10 give (3.10). Remark . Everything works with the more general assumptions V P PA ρ . Remark . Pseudodifferential calculus is used to guarantee that V d has order -1 while V o hasorder 0 (see (3.9)). Without this information it would be problematic to apply the standardKAM iteration of Kuksin [Kuk87], which requires the eigenvalues to have an asymptotic of theform j ` O p j δ q with δ ă . In principle one might circumvent this problem by using the ideasof [BBM14, FP15] to regularize the order of the perturbation. However in our context thissmoothing procedure is tricky, since it produces terms of size | ω | , which are very large andtherefore unacceptable for our purposes. Proposition 3.4.
For γ ą and τ ą ν ´ , the set Ω defined in (3.21) fulfills (3.22) .Proof. For any k P Z ν zt u , define the sets G k : “ ! ω P R M ˇˇˇ | ω ¨ k | ă γ x k y τ M ) . By Lemma 4.2 ˇˇ G k ˇˇ À γ | k | τ ` M ν . Therefore the set G : “ Ť k ‰ G k has measure bounded by | G | ď Cγ M ν , whichproves the claim. As we shall see, in order to perform a converging KAM scheme, we must be able to imposesecond order Melnikov conditions, namely bounds from below of quantities like ω ¨ k ` λ i ˘ λ j ,where the λ j ’s are the eigenvalues of the operator B defined in (1.3). Explicitly,(4.1) λ j : “ a j ` m “ j ` c j p m q j , c j p m q : “ j p a j ` m ´ j q . One can check that ď c j p m q ď m @ j P N . We introduce the notation of the indexes sets:(4.2) I ` : “ Z ν ˆ N ˆ N , I ´ : “ p k, j, l q P I ` ˇˇ p k, j, l q ‰ p , a, a q , a P N ( . Ω as(4.3) m r p Ω q : “ | Ω || R M | ” | Ω | M ν p ν ´ q c ν where | C | is the Lebesgue measure of the set C and c ν is the volume of the unitary ball in R ν .The main result of this section is the following theorem. Theorem 4.1 (Balanced Melnikov conditions) . Fix ď α ď and assume that M ě M : “ min t m , x m y { α u if α P r , s . Then, for ă r γ ď min t γ { , { u and r τ ě ν ` , the set (4.4) U α : “ ω P Ω ˇˇˇˇˇ | ω ¨ k ` λ j ˘ λ l | ě r γ x k y r τ x j ˘ l y α M α @p k, j, l q P I ˘ + is of large relative measure, that is (4.5) m r p Ω z U α q ď C r γ { , where C ą is independent of M and r γ . We will use several times the following standard estimate.
Lemma 4.2.
Fix k P Z ν zt u and let R M Q ω ÞÑ ς p ω q P R be a Lipschitz function fulfilling | ς | Lip R M ď c ă | k | . Define f p ω q “ ω ¨ k ` ς p ω q . Then, for any δ ě , the measure of the set A : “ t ω P R M | | f p ω q| ď δ u satisfies the upper bound (4.6) | A | ď δ | k | ´ c p M q ν ´ . Proof.
Take ω “ ω ` ǫk , with ǫ sufficiently small so that ω P R M .Then | f p ω q ´ f p ω q|| ω ´ ω | ě | k | ´ | ς | Lip R M ą | k | ´ c and the estimate follows by Fubini theorem.In the rest of the section we write a À b , meaning that a ď Cb for some numerical constant C ą independent of the relevant parameters.The result of Theorem 4.1 is carried out in two steps. The first one is the following lemma. Lemma 4.3.
Fix ď α ď . There exist r γ ą and τ ą ν ` α such that the set (4.7) T : “ " ω P Ω ˇˇˇˇ | ω ¨ k ` l | ě r γ x k y τ x l y α M α @p k, l q P Z ν ` zt u * has relative measure m r p Ω z T q ď C r γ , where C ą is independent of M and r γ .Proof. If k “ and l ‰ , the estimate in (4.7) holds. The same is true if k ‰ and l “ .Therefore, let both k and l be different from zero. For | l | ą M | k | , the inequality in (4.7) holdstrue taking r γ ď . Indeed: | ω ¨ k ` l | ě | l | ´ | ω | | k | ě | l | ´ M | k | ě | l | ě | l | α ě r γ x k y τ M α | l | α . Then, consider the case ď | l | ď M | k | (so, only a finite number of l P Z zt u ). For fixed k and l , define the set(4.8) G kl : “ " ω P R M ˇˇˇˇ | ω ¨ k ` l | ď r γ x k y τ | l | α M α * .
16y Lemma 4.2, the measure of each set can be estimated by(4.9) ˇˇ G kl ˇˇ À M ν ´ r γ x k y τ | l | α M α | k | À r γ M ν ´ ´ α | l | α x k y τ ` . Let G : “ Ω X Ť G kl ˇˇ p k, l q P Z ν ` zt u , | l | ď M | k | ( . Then | G | ď ÿ k P Z ν zt u ÿ l P Z zt u| l |ď M | k | ˇˇ G kl ˇˇ (4.9) À r γ M ν ´ ´ α ÿ k ‰ ÿ | l |ď M | k | | l | α x k y τ ` À r γ M ν ´ ´ α ÿ k ‰ x k y τ ` p M | k |q α ` À r γ M ν ÿ k ‰ x k y τ ´ α À r γ M ν (4.10)provided τ ą ν ` α . It follows that the relative measure of G is given by(4.11) m r p G q ď C r γ , where C ą is independent of M and r γ . The thesis follows, since T “ Ω z G . Remark . In case m “ , Lemma 4.3 implies Theorem 4.1.From now on assume that m ą . The second step is the next lemma. Lemma 4.5.
There exist ă r γ ď min t γ , r γ { u and τ ě τ ` ν ` such that the set (4.12) T : “ " ω P T ˇˇˇˇ | ω ¨ k ` λ j ˘ λ l | ě r γ x k y τ x j ˘ l y α M α @p k, j, l q P I ˘ * fulfills m r p T z T q ď C r γ r γ , where C ą is independent of M , r γ , r γ .Proof. Let p k, j, l q P I ˘ . We can rule out some cases for which the inequality in (4.12) is alreadysatisfied when ω P T Ă Ω : • For ˘ “ ` and k “ , we have λ j ` λ l “ j ` l ` c j p m q j ` c l p m q l ě j ` l ě r γ M α x j ` l y α ; • For ˘ “ ´ and k ‰ , j “ l , we have | ω ¨ k | ě γ x k y τ M ; • For ˘ “ ´ and k “ , j ‰ l , and α P p , s , it holds that | λ j ´ λ l | “ ˇˇˇˇż jl x ? x ` m d x ˇˇˇˇ ě x m y | l ´ j | ě r γ M α x j ´ l y α . For α “ the estimate is trivially verified.Therefore, for the rest of this argument, let k ‰ and j ‰ l . Assume first that | j ˘ l | ě M | k | .In this case, one has: | ω ¨ k ` λ j ˘ λ l | ě | j ˘ l | ´ ˇˇˇˇ c j p m q j ˘ c l p m q l ˇˇˇˇ ´ | ω ¨ k | ě | j ˘ l | ´ M | k | ě | j ˘ l | . | j ˘ l | ă M | k | . In the region j ă l assume(4.13) j x j ˘ l y α ě R p k q : “ m M α x k y τ r γ , where r γ and τ are the ones of Lemma 4.3. So, for ω P T , we get | ω ¨ k ` λ j ˘ λ l | ě | ω ¨ k ` l ˘ j | ´ ˇˇˇˇ c j p m q j ˘ c l p m q l ˇˇˇˇ ě r γ x k y τ x j ˘ l y α M α ´ m j p . q ě r γ x k y τ x j ˘ l y α M α . (4.14)Thus, we consider just those j and l with j x j ˘ l y α ă R p k q . The symmetric argument shows thatwe can take those l ă j for which l x j ˘ l y α ă R p k q .Like in the previous proof, consider the set(4.15) G k, ˘ j,l : “ " ω P R M ˇˇˇˇ | ω ¨ k ` λ j ˘ λ l | ă r γ x k y τ x j ˘ l y α M α * defined for those k ‰ and j ‰ l in the regions(4.16) P ˘ : “ t | j ˘ l | ă M | k | u X ´ t j x j ˘ l y α ă R p k q , j ă l u Y t l x j ˘ l y α ă R p k q , l ă j u ¯ . Using Lemma 4.2, the estimate for its Lebesgue measure is(4.17) ˇˇˇ G k, ˘ j,l ˇˇˇ À r γ M ν ´ ´ α x j ˘ l y α | k | τ ` . Define G ˘ : “ T X Ť ! G k, ˘ j,l ˇˇˇ p k, j, l q P P ˘ ) . By symmetry of the summand, we estimate ˇˇ G ´ ˇˇ ď ÿ p k,j,l qP P ´ ˇˇˇ G k, ´ j,l ˇˇˇ (4.17) À r γ M ν ´ ´ α ÿ p k,j,l qP P ´ x j ´ l y α | k | τ ` À r γ M ν ´ ´ α ÿ k ‰ ÿ j ă lj x j ´ l y α ă R p k q ÿ | j ´ l |ă M | k | x j ´ l y α | k | τ ` À r γ M ν ´ ´ α ÿ k ‰ ÿ l ´ j “ : h ą h ă M | k | ÿ j ă R p k qx h y ´ α x h y α | k | τ ` (4.13) À r γ r γ M ν ´ ÿ k ‰ ÿ h ă M | k | | k | τ ` ´ τ À r γ r γ M ν ÿ k ‰ | k | τ ´ τ ď r γ r γ M ν (4.18)provided τ ą τ ` ν . The same computation holds for G ` . We conclude that(4.19) m r p T z T q ď m r p G ´ X G ` q ď C r γ r γ , where C ą is independent of M , r γ , r γ . 18 roof of Theorem 4.1. Take r γ “ r γ { , r γ “ r γ { with some r γ ą sufficiently small so that r γ and r γ fulfill the assumptions of the previous lemmas. Similarly, choose τ “ ν ` and τ “ ν ` . By definition, U α ” T Ă Ω . Since Ω z U α “ p Ω z T q Y p T z T q , we get by Lemma4.3 and Lemma 4.5 that m r p Ω z U α q ď C r γ ` C r γ r γ ď C r γ { , C “ p C ` C q . The new potential V p ωt ; ω q that we have found in Theorem 3.1 is perturbative, in the sense thatthe smallness of its norm is controlled by the size M of the frequency vector ω . Thus, we arenow ready to attack with a KAM reduction scheme, presenting first the algebraic constructionof the single iteration, then quantifying it via the norms and seminorms that we have introducedin Section 2. The complete result for this reduction transformation, together with its iterativelemma, is proved at the end of this section. Actually, for the KAM scheme it is more convenient to work with operators of type M ρ,s . Ofcourse, as we have seen in Section 2, pseudodifferential operators analytic in θ belong to such aclass. Lemma 5.1.
Fix an arbitrary s ą { and put ρ : “ ρ { . Then the operator V p ω q defined in (3.8) belongs to Lip w p Ω , M ρ ,s p , qq with the quantitative bound (5.1) | V | Lip p w q ρ ,s , , , Ω ď C M ; here C ą is independent of M .Proof. It is sufficient to apply the embedding Lemma 2.21 and (3.10).
Consider the system(5.2) i ψ p t q “ H p t q ψ p t q , H p t q : “ A p ω q ` P p ωt ; ω q , where the frequency vector ω varies in some set Ω Ă R ν , M ď | ω | ď M ; the time-independentoperator A p ω q is diagonal, with(5.3) A p ω q “ ˆ A p ω q ´ A p ω q ˙ , A p ω q : “ diag t λ ´ j p ω q | j P N u Ă p , N ; and the quasi-periodic perturbation P p ωt ; ω q has the form(5.4) P p ωt ; ω q “ ˆ P d p ωt ; ω q P o p ωt ; ω q´ P o p ωt ; ω q ´ P d p ωt ; ω q ˙ , P d “ r P d s ˚ , P o “ r P o s ˚ . H p t q through a transformation ψ : “ e ´ i X ` p ωt ; ω q ϕ of the form(5.5) X ` p ωt ; ω q “ ˆ X d p ωt ; ω q X o p ωt ; ω q´ X o p ωt ; ω q ´ X d p ωt ; ω q ˙ , X d “ r X d s ˚ , X o “ r X o s ˚ , so that the transformed Hamiltonian, as in (3.4), is(5.6) H ` p t q : “ e i X ` p ωt ; ω q H p t q e ´ i X ` p ωt ; ω q ´ ż e i s X ` p ωt ; ω q X ` p ωt ; ω q e ´ i s X ` p ωt ; ω q d s . Its expansion in commutators is given by(5.7) H ` p t q “ A ` P ` i r X ` , A s ´ X ` ` R , R : “ e i X ` A e ´ i X ` ´ p A ` i r X ` , A sq ` e i X ` P e ´ i X ` ´ P ´ ˆż e i s X ` X ` e ´ i s X ` d s ´ X ` ˙ . We ask now X ` to solve the "quantum" homological equation:(5.8) i r X ` p θ q , A s ´ ω ¨ B θ X ` p θ q ` Π N P p θ q “ Z where Π N P p θ ; ω q : “ ř | k |ď N p P p k ; ω q e i k ¨ θ is the projector on the frequencies smaller than N , while Z is the diagonal, time independent part of P d :(5.9) Z “ Z p ω q : “ ˆ Z p ω q ´ Z p ω q ˙ , Z “ diag t { p P d q jj p ω q | j P N u . With this choice, the new Hamiltonian becomes H p t q ` “ A ` ` P p ωt q ` with(5.10) A ` “ A ` Z , P ` : “ Π K N P ` R , Π K N P : “ p ´ Π N q P . In order to solve equation (5.8), note that it reads block-wise as(5.11) " i r X d , A s ´ ω ¨ B θ X d ` P d “ Z ´ i r X o , A s a ´ ω ¨ B θ X o ` P o “ . Expanding both with respect to the exponential basis of B (for the space) and in Fourier inangles (for the time), we get the solutions(5.12) { p X d q jl p k ; ω q : “ $&% p ω ¨ k ` λ ´ j p ω q ´ λ ´ l p ω qq { p P d q jl p k ; ω q p k, j, l q P I ´ N otherwise , (5.13) { p X o q jl p k ; ω q : “ $&% p ω ¨ k ` λ ´ j p ω q ` λ ´ l p ω qq { p P o q jl p k ; ω q p k, j, l q P I ` N otherwise , where, following the notation in (4.2), we have defined(5.14) I ˘ N : “ p k, j, l q P I ˘ ˇˇ | k | ď N ( . Remark that A ` “ diag t λ ` j p ω q | j P N u with λ ` j p ω q : “ λ ´ j p ω q ` { p P d q jj p ω q . .3 Estimates for the general step Both for well-posing the solutions (5.12) and (5.13) and ensuring convergence of the norms,second order Melnikov conditions are required to be imposed. In particular, we choose thefrequency vector from the following set(5.15) Ω ` : “ " ω P Ω ˇˇˇˇ ˇˇ ω ¨ k ` λ ´ j p ω q ˘ λ ´ l p ω q ˇˇ ě γ x N y τ x j ˘ l y α M α , @ p k, j, l q P I ˘ N * with γ, τ ą to be fixed later on. Here I ˘ N has been defined in (5.14).The fact that Ω ` is actually a set of large measure, that is m r p Ω z Ω ` q “ O p γ q , will be clear as adirect consequence of Lemma 5.9 of Section 5.4.From now on, we choose as Lipschitz weight w : “ γ { M α and, abusing notation, we denote Lip γ p Ω , F q : “ Lip γ { M α p Ω , F q . Furthermore, we fix once for all s ą { and α P p , q .For V P Lip γ p Ω , M ρ,s p α, qq , we write | V | : “ | V | α, s , | V | ρ : “ | V | α, ρ,s , | V | Lip p γ q ρ, Ω : “ | V | Lip p γ { M α q ρ,s ,α, , Ω ” | V | ρ, Ω ` γ M α | V | Lip ρ, Ω , while, for V P Lip γ p Ω , M ρ,s p α, α qq , we denote ||| V ||| ρ : “ | V | α,αρ,s , ||| V ||| Lip p γ q ρ, Ω : “ | V | Lip p γ { M α q ρ,s ,α,α, Ω ” ||| V ||| ρ, Ω ` γ M α ||| V ||| Lip ρ, Ω . Remark . Note that | V | Lip p γ q ρ , Ω ď k V k Lip p γ q ρ , Ω .Now, we provide the estimate on the generator X ` of the previous transformation. For sakeof simplicity during the forthcoming proof, as short notation we define(5.16) g k, ˘ j,l p ω q : “ ω ¨ k ` λ ´ j p ω q ˘ λ ´ l p ω q for p k, j, l q P I ˘ N . Lemma 5.3.
Assume that:(a) P P Lip γ p Ω , M ρ,s p α, qq , with an arbitrary ρ ą ;(b) There exists ă C ď such that for any j P N , ω, ∆ ω P Ω ` one has (5.17) ˇˇ ∆ ω λ ´ j p ω q ˇˇ ď C | ∆ ω | . Let X ` “ X ` p ωt, ω q be defined by (5.12) and (5.13) . Then X ` P Lip γ p Ω ` , M ρ,s p α, α qq withthe quantitative bound (5.18) ˇˇˇˇˇˇ X ` ˇˇˇˇˇˇ Lip p γ q ρ, Ω ` ď x N y τ ` M α γ | P | Lip p γ q ρ, Ω . Proof.
We start with the seminorm ||| X ` ||| ρ, Ω ` . Fix ω P Ω ` and | k | ď N . Then, when j ‰ l , wehave(5.19) ˇˇˇˇ { p X d q jl p k ; ω q ˇˇˇˇ ď ˇˇˇ g k, ´ j,l p ω q ˇˇˇ ˇˇˇˇ{ p P d q jl p k ; ω q ˇˇˇˇ ď x N y τ M α γ ˇˇˇˇ{ p P d q jl p k ; ω q ˇˇˇˇ x j ´ l y α j, l P N (5.20) ˇˇˇˇ { p X o q jl p k ; ω q ˇˇˇˇ ď x N y τ M α γ ˇˇˇˇ{ p P o q jl p k ; ω q ˇˇˇˇ x j ` l y α . From the assumption p a q , we have that all the terms ˇˇˇ x D y α x P d p k ; ω q ˇˇˇ s , ˇˇˇ x P d p k ; ω q x D y α ˇˇˇ s , ˇˇˇ x D y σ x P δ p k ; ω q x D y ´ σ ˇˇˇ s (with σ “ ˘ α, , δ “ d, o ) are bounded. In order to bound ˇˇˇˇˇˇˇˇˇy X ` p k ; ω q ˇˇˇˇˇˇˇˇˇ ,what we have to prove is that we can control also the terms ˇˇˇ x D y α x X δ p k ; ω q ˇˇˇ s , ˇˇˇ x X δ p k ; ω q x D y α ˇˇˇ s , ˇˇˇ x D y σ x X δ p k ; ω q x D y ´ σ ˇˇˇ s . The seminorms involving the diagonal term X d can be easily handled, since, by (5.19), they areessentially bounded by the same seminorms for P d . The similar bound in (5.20) is enough alsowhen we consider the terms ˇˇˇ x D y σ x X o p k ; ω q x D y ´ σ ˇˇˇ s . Consider now the term x D y α x X o p k ; ω q .Applying again (5.20), we get ˇˇˇˇ´ x D y α x X o p k ; ω q ¯ jl ˇˇˇˇ “ ˇˇˇˇ x l y α { p X o q jl p k ; ω q ˇˇˇˇ ď x N y τ M α γ x l y α x j ` l y α ˇˇˇˇ{ p P o q jl p k ; ω q ˇˇˇˇ ď x N y τ M α γ ˇˇˇˇ{ p P o q jl p k ; ω q ˇˇˇˇ . (5.21)The same bound holds for ˇˇˇˇ´ x X o p k ; ω q x D y α ¯ jl ˇˇˇˇ . We obtain that ˇˇˇˇˇˇ X ` ˇˇˇˇˇˇ ρ, Ω ` ď x N y τ M α γ | P | ρ, Ω . We deal now with the estimates on the Lipschitz seminorm ||| X ` ||| Lip ρ, Ω ` . Using the notation (3.23)we have, for δ “ d, o : ∆ ω { p X δ q jl p k ; ω q “ ´ i ∆ ω p g k, ˘ j,l p ω qq g k, ˘ j,l p ω ` ∆ ω q g k, ˘ j,l p ω q { p P δ q jl p k ; ω q ` i g k, ˘ j,l p ω ` ∆ ω q ∆ ω { p P δ q jl p k ; ω q . (5.22)By the assumption in (5.17), we have that(5.23) ˇˇˇ ∆ ω p g k, ˘ j,l p ω qq ˇˇˇ “ ˇˇ ∆ ω ¨ k ` ∆ ω p λ ´ j ˘ λ ´ l q ˇˇ (5.17) ď | k | | ∆ ω | ` C | ∆ ω | ď x N y | ∆ ω | uniformly for every j, l P N and k P Z ν , | k | ď N . We can thus estimate (5.22) by ˇˇˇˇ ∆ ω { p X δ q jl p k ; ω q ˇˇˇˇ ď x N y τ ` M α | ∆ ω | γ ˇˇˇˇ{ p P δ q jl p k ; ω q ˇˇˇˇ x j ˘ l y α ` x N y τ M α γ ˇˇˇˇ ∆ ω { p P δ q jl p k ; ω q ˇˇˇˇ x j ˘ l y α , (5.24)from which one deduces easily the claimed estimate (5.18).22 emma 5.4. Let P P Lip γ p Ω , M ρ,s p α, qq . Assume (5.17) and, for some fixed C s ą , (5.25) C s x N y τ ` M α γ | P | Lip p γ q ρ, Ω ă . Then P ` “ Π K N P ` R , defined as in (5.10) , belongs to Lip γ p Ω ` , M ρ ` ,s p α, qq for any ρ ` P p , ρ q ,with bounds (5.26) ˇˇ Π K N P ˇˇ Lip p γ q ρ ` , Ω ď e ´p ρ ´ ρ ` q N | P | Lip p γ q ρ, Ω , ||| R ||| Lip p γ q ρ, Ω ` ď C s M α γ x N y τ ` ´ | P | Lip p γ q ρ, Ω ¯ . Proof.
The estimate on Π K N P follows by using that it contains only high frequencies. To estimatethe remainder R , use (5.7),(5.8) to write it as(5.27) R “ ż p ´ s q e i s X ` ad X ` p Z ´ P q e ´ i s X ` d s ` ż e i s X ` ad X ` p P q e ´ i s X ` d s . Then, apply Lemma 2.23 and Lemma 5.3.
Remark . Defining the quantities η : “ M α γ | P | Lip p γ q ρ, Ω , η ` : “ M α γ ˇˇ P ` ˇˇ Lip p γ q ρ ` , Ω ` and choosing N “ ´p ρ ´ ρ ` q ´ ln η , Lemma 5.4 implies that(5.28) η ` ď ´ e ´p ρ ´ ρ ` q N ` x N y τ ` η ¯ η ď ˜ ` p ρ ´ ρ ` q τ ` ˆ ln 1 η ˙ τ ` ¸ η . Once that the general step has been illustrated, we are ready for setting our iterative scheme.The Hamiltonian the iteration starts with is the one that we have found after the Magnus normalform in Section 3:(5.29) H p q p t q “ H p q ` V p q p ωt ; ω q , ˇˇˇ V p q ˇˇˇ Lip p γ q ρ , Ω ď C M , where H p q : “ H and V p q : “ V as in Theorem 3.1. All the iterated objects are constructedfrom the transformation in Sections 5.2, 5.3 by setting for n ě H p n q p t q : “ A p ω q ` P p ωt ; ω q , A : “ H p n q , P : “ V p n q Z p n q : “ Z , X p n q : “ X , R p n q : “ R . Given reals γ, ρ , η ą and a sequence of nested sets t Ω n u n ě , we fix the parameters δ n : “ π p ` n q ρ , ρ n ` : “ ρ n ´ δ n , η n : “ M α γ ˇˇˇ V p n q ˇˇˇ Lip p γ q ρ n , Ω n , N n : “ ´ δ n ln η n Proposition 5.6 (Iterative Lemma) . Fix τ ą . There exists k ” k p τ, δ q ą such that forany ă γ ă r γ , any M ą for which η : “ M α γ ˇˇˇ V p q ˇˇˇ Lip p γ q ρ , Ω ď k e ´ , (5.30) the following items hold true for any n P N : i) Setting Ω as in (3.21) , we have recursively for n ě n ` : “ " ω P Ω n ˇˇˇˇ ˇˇˇ ω ¨ k ` λ p n q j p ω q ˘ λ p n q l p ω q ˇˇˇ ą γ N τn x j ˘ l y α M α , @p k, j, l q P I ˘ N n * ; (ii) For every ω P Ω n , the operator X p n q p ω, ¨q P Lip γ p Ω n , M ρ n ´ ,s p α, α qq and (5.31) ˇˇˇˇˇˇˇˇˇ X p n q ˇˇˇˇˇˇˇˇˇ Lip p γ q ρ n ´ , Ω n ď ? η e p ´ p q n ´ q . The change of coordinates e i X p n q conjugates H p n ´ q to H p n q “ H p n q ` V p n q such that:(iii) The Hamiltonian H p n q p ω q is diagonal and time independent, H p n q p ω q “ diag t λ p n q j p ω qu j P N σ ,and the functions λ p n q j p ω q “ λ p n q j p ω ; M , α q are defined over all Ω , fulfilling (5.32) ˇˇˇ λ p n q j ´ λ p n ´ q j ˇˇˇ LipΩ ď η e ´ p q n ´ ; (iv) The new perturbation V p n q P Lip γ p Ω n , M ρ n ,s p α, qq and η n ” M α γ ˇˇˇ V p n q ˇˇˇ Lip p γ q ρ n , Ω n ď η e ´ p q n . (5.33) Proof.
We argue by induction. For n “ one requires (5.30). Now, assume that the statementshold true up to a fixed n P N . Define Ω n ` as in item p i q . In order to apply Lemma 5.3 andLemma 5.4, we need to check that the assumptions in (5.17) and (5.25) are verified, respectively.First, note that, by item p iii q ,(5.34) ˇˇˇ λ p n q j ˇˇˇ LipΩ ď n ÿ m “ ˇˇˇ λ p m q j ´ λ p m ´ q j ˇˇˇ LipΩ ` | λ j | LipΩ ď η e ÿ m “ e ´ p q m ´ ď η e, so that (5.17) is satisfied, provided simply η e ď .We prove now that (5.25) is fulfilled. We have x N n y τ ` η n ď ˆ ` n δ ˙ τ ` η n (5.33) ď p η e q e ´ p q n ˆ ` n δ ˙ τ ` ď ¨ ¨ C s as long as η e is sufficiently small (depending only on δ , τ q . Therefore we can apply Lemma5.3 and Lemma 5.4 with P ” V p n q and define X p n ` q P Lip γ p Ω n ` , M ρ n ,s p α, α qq , the neweigenvalues(5.35) λ p n ` q j p ω q : “ λ p n q j p ω q ` { p V d, p n q q jj p ω q @ j P N and the new perturbation V p n ` q . We are left only with the quantitative estimates.We start with item p iv q . By Remark 5.5, one has(5.36) η n ` ď ˜ ` δ τ ` n ˆ ln 1 η n ˙ τ ` ¸ η n ď ˆ ` n δ ˙ τ ` p η e q e ´ p q n . n ` provided again that η e is sufficiently small(depending only on δ , τ q . For item p iii q , it is sufficient to note that(5.37) ˇˇˇ λ p n ` q j ´ λ p n q j ˇˇˇ LipΩ n “ ˇˇˇˇ { p V d, p n q q jj p , ¨q ˇˇˇˇ LipΩ n ď ˇˇˇ V p n q ˇˇˇ Lip ρ n , Ω n ď M α γ ˇˇˇ V p n q ˇˇˇ Lip p γ q ρ n , Ω n (5.33) ď η e ´ p q n . Now, by Kirszbraun theorem, we can extend the functions λ p n q j p ω, M q to all Ω preserving theirLipschitz constant; this proves p iii q . Item p ii q is proved in the same lines, using (5.18) and theinductive assumption; we skip the details.A consequence of the iterative lemma is the following result. Corollary 5.7 (Final eigenvalues) . Fix τ ą r τ (of Theorem 4.1). Assume (5.30) . Then for every ω P Ω and for every j P N , the sequence t λ p n q j p¨ ; M , α qu n ě is a Cauchy sequence. We denote by λ j p ω ; M , α q its limit, which is given by λ j p ω q “ λ j ` ε j p ω q and one has the estimate (5.38) sup j P N ˇˇ j α ε j ˇˇ Lip p γ q Ω ď γ M α η e . Proof.
By (5.35) we have ε j p ω q : “ ř n “ { p V p n q ,d q jj p , ω q . The thesis follows using(5.39) ˇˇˇˇ j α { p V p n q ,d q jj p ω q ˇˇˇˇ ď ˇˇˇ x D y α { V p n q ,d p ω q ˇˇˇ s ď ˇˇˇ V p n q ˇˇˇ Lip p γ q ρ n , Ω n (5.33) ď γ M α η e ´ p q n . Corollary 5.8 (Iterated flow) . Fix an arbitrary r P r , s s ; under the same assumptions ofCorollary 5.7, for any ω P X n Ω n and θ P T n , the sequence of transformations (5.40) W n p θ ; ω q : “ e ´ i X p q p θ ; ω q ˝ ¨ ¨ ¨ ˝ e ´ i X p n q p θ ; ω q is a Cauchy sequence in L p H r ˆ H r q fulfilling kW n p θ ; ω q ´ k L p H r ˆ H r q ď ? η e Σ e ? η e Σ (5.41) where Σ : “ ř q “ e ´ p q q . We denote by W p θ ; ω q its limit in L p H r ˆ H r q .Proof. The convergence of the transformations is a standard argument, while the control of theoperator norm L p H r ˆ H r q follows from Remark 2.15; we skip the details.Since for any j P N the sequence t λ p n q j u n ě converges to a well defined Lipschitz function λ j defined on Ω , we can now impose second order Melnikov conditions only on the final frequencies. Lemma 5.9 (Measure estimates) . Consider the set (5.42) Ω ,α : “ " ω P U α | ˇˇ ω ¨ k ` λ j p ω q ˘ λ l p ω q ˇˇ ě γ x k y τ x j ˘ l y α M α , @p k, j, l q P I ˘ * . Then Ω ,α Ď X n Ω n . Furthermore, taking τ ą ν ` α ` r τα , γ P r , r γ { s and M ě M (defined inTheorem 4.1), there exists a constant C ą , independent of M and γ , such that (5.43) m r p U α z Ω ,α q ď C γ . roof. The proof that Ω ,α Ď X n Ω n is standard, see e.g. Lemma 7.6 of [MP18].To prove the measure estimate, let ω P U α and p k, j, l q P I ˘ . We can rule out the casesas at the beginning of Lemma 4.5 with essentially the same arguments. Thus, we restrict toconsider all p k, j, l q P I ˘ for which k ‰ and j ‰ l . Furthermore, if | j ˘ l | ě M | k | , we getagain that ˇˇ ω ¨ k ` λ j p ω q ˘ λ l p ω q ˇˇ ě | j ˘ l | (recall M ą m ). So, we can work in the regions | j ˘ l | ă M | k | . Now, for j ă l satisfying(5.44) j x j ˘ l y ě ˜ η e x k y r τ c p γ, r γ q ¸ α “ : r R p k q , where c p γ, r γ q : “ r γγ ´ ą (recall that r γ { ą γ ), we have (using also (5.38)) ˇˇ ω ¨ k ` λ j p ω q ˘ λ l p ω q ˇˇ ě | ω ¨ k ` λ j ˘ λ l | ´ ˇˇ ε j p ω q ˇˇ ´ | ε l p ω q|ě r γ x k y r τ x j ˘ l y α M α ´ γ M α η ej α ě γ M α x j ˘ l y α x k y r τ . Therefore, we can further restrict ourselves to consider just those j ă l satisfying j x j ˘ l y ă r R p k q .The symmetric argument leads to work in the sector j ă l under the condition l x l ˘ j y ă r R p k q .Now, define the set(5.45) G k, ˘ j,l : “ " ω P R M ˇˇˇˇ ˇˇ ω ¨ k ` λ j p ω q ˘ λ l p ω q ˇˇ ă γ x k y τ x j ˘ l y α M α * for those k ‰ and j ‰ l in the region(5.46) R ˘ : “ t| j ˘ l | ă M | k |u X ` t j x j ˘ l y ă r R p k q , j ă l u Y t l x j ˘ l y ă r R p k q , l ă j u ˘ ; Recall that f ˘ kjl p ω q : “ ω ¨ k ` λ j p ω q ˘ λ l p ω q are Lipschitz functions on R M . For k ‰ , since | λ l | Lip R M ă | k |{ , by Lemma 4.2 we get ˇˇˇ G k, ˘ j,l ˇˇˇ À M ν ´ α γ x j ˘ l y α | k | τ ` . Define G ˘8 : “ Ť ! G k, ˘ j,l ˇˇˇ p k, j, l q P R ˘ ) X U α . We have ˇˇ G ´8 ˇˇ À γ M ν ´ α ÿ k ‰ ÿ j ă lj x j ´ l yă r R p k q ÿ | j ´ l |ă M | k | x j ´ l y α | k | τ ` À γ M ν ´ α ÿ k ‰ ÿ l ´ j “ : h ą | h |ă M | k | ÿ j ă r R p k qx h y ´ x h y α | k | τ ` À γc p γ, r γ q α M ν ÿ k ‰ | k | τ ` ´ α ´ r τα À γc p γ, r γ q α M ν À γ M ν , taking τ ` ´ α ´ r τα ą ν . The same computation holds for G `8 , and proves (5.43). Theorem 5.10 (KAM reducibility) . Fix α P p , q , s ą { , and τ ą ν ` ` α ` r τα . For any ă γ ă r γ , there exists M ˚ “ M ˚ p m , α, γ, ρ q ą such that for any M ě M ˚ the following holds true.There exist functions t λ j p ω ; M , α qu j P N , defined and Lipschitz in ω in the set R M such that:(i) The set Ω ,α “ Ω ,α p γ, τ, M q Ă R M defined in (5.42) fulfills m r p R M z Ω q ď C p γ ` r γ { ` γ q ,where γ is defined in Theorem 3.1 and r γ in Theorem 4.1. ii) For each ω P Ω ,α there exists a change of coordinates ψ “ W p ωt, ω q φ which conjugatesequation (3.7) to a constant-coefficient diagonal one: (5.47) i φ “ H φ , H “ H p ω, α q “ diag t λ j p ω, α q | j P N u σ . Furthermore for any r P r , s s one has (5.48) kW ´ k L p H r ˆ H r q ď ? η e Σ e ? η e Σ . Proof.
Having fixed α, s and τ , we can produce the constant k p δ , τ q of the iterative Lemma5.6. Having fixed also ă γ ă r γ , we produce M ˚ ą in such a way that for every M ě M ˚ , theestimate (5.30) is fulfilled. We can now apply the iterative Lemma 5.6, Corollary 5.7 and Lemma5.9 to get the result. The KAM reducibility scheme that we have presented has transformed Equation (3.7) into (5.47),where the asymptotic for the final eigenvalues are given, using Equation (5.38), by(5.49) λ j p ω, α q ´ λ j „ O ˆ η M α j α ˙ (5.29) „ O ˆ M j α ˙ . One can argue that the asymptotic λ j p α q ´ λ j „ O p M ´ j ´ α q is not that satisfying, since thepertubation V p q at the beginning of the KAM scheme belongs to the class M ρ ,s p , q and soits diagonal elements have a smoothing effect of order 1 which could be expected to be preservedin the effective Hamiltonian.Actually, it is possible to modify our reducibility scheme for achieving this result: we explainnow briefly how to do it. After the Magnus normal form, we conjugate system (3.7) through e ´ i Y p ωt q , where(5.50) Y p ωt q : “ ˆ Y o p ωt q´ Y o p ωt q ˙ so that Y o solves the homological equation(5.51) ´ i r Y o p θ q , B s a ` V o p θ q ´ ω ¨ B θ Y o p θ q “ ñ { p Y o q lj p k q : “ { p V o q lj p k q i p ω ¨ k ` λ j ` λ l q @ k, j, l . We ask now the frequency vector ω to belong to U X U (see (4.4)). In this way one gets (in thesame lines of the proof of Lemma 5.3) that Y P Lip γ { M p U , M r ρ ,s p , qq , since we have chosen ω P U , with the bound(5.52) | Y | Lip p γ { M q r ρ ,s , , ď C M γ ˇˇˇ V p q ˇˇˇ Lip p γ { M q ρ ,s , , ď C M γ ˇˇˇ V p q ˇˇˇ Lip p γ { M q ρ ,s , , (5.1) ď r C .
The new perturbation(5.53) Ć V p q p ωt q : “ ˆ V d p ωt q ´ V d p ωt q ˙ ` ż p ´ s q e i s Y p ωt q ad Y p ωt q r V p q p ωt qs e ´ i s Y p ωt q d s belongs to the class Lip γ { M p U , M r ρ ,s p , qq fulfilling estimate (5.1).Thus, one can perform a KAM reducibility scheme as in Section 5.3–5.4, in which one takes α “
27n (5.15), the perturbations appearing in the iterations stay in the class
Lip γ { M p Ă Ω n , M r ρ n ,s p , qq and the new final eigenvalues Ă λ j satisfy the nonresonance condition(5.54) ˇˇˇ ω ¨ k ` Ă λ j ˘ Ă λ l ˇˇˇ ě γ x k y τ , @ p k, j, l q P I ˘ . In particular, we obtain better asymptotics on the final eigenvalues, that is Ă λ j ´ λ j „ O p M ´ j ´ q .The price that we pay for this result is that the preliminary change of coordinate e ´ i Y p ωt q is not atransformation close to identity, as the generator Y p ωt q is just a bounded operator and not smallin size, see (5.52). The main consequence is that the effective dynamics of the original system, asCorollary 1.6 is no more valid. In this case, it is possible to conclude just that the Sobolev normsstay uniformly bounded in time and do not grow, but in general their (almost-)conservation islost. A Technical results
A.1 Properties of pseudodifferential operators
Recall that if F is an operator, we denote by p F p k q its k th Fourier coefficient defined as in (2.9).If F is a pseudodifferential operator with symbol f , so p F p k q is, with symbol given by p f p k, x, j q : “ p π q ν ż T ν f p θ, x, j q e ´ i θ ¨ k d θ. Lemma A.1.
Let ρ ą and µ P R . The following holds true:(i) If F P A µρ , then the operator p F p k q belongs to A µ for any k P Z ν and ℘ µℓ p p F p k qq ď e ´ ρ | k | ℘ µ,ρℓ p F q @ ℓ P N . (ii) Assume to have @ k P Z ν an operator p F p k q P A µ fulfilling (A.1) ℘ µℓ p p F p k qq ď x k y τ e ´ ρ | k | C ℓ @ k P Z ν , @ ℓ P N , for some τ ě , ρ ą and C ℓ ą independent of k . Define the operator F p θ q : “ ř k P Z ν p F p k q e i θ ¨ k . Then, F belongs to A µρ for any ă ρ ă ρ and one has ℘ µ,ρ ℓ p F q ď C ℓ p ρ ´ ρ q τ ` ν @ ℓ P N . On the classes
Lip w p Ω , PA µρ q , these assertions extend naturally without any further loss of ana-lyticity.Proof. (i) By Cauchy estimates, it is well-known the analytic decay for the Fourier coefficientsof the symbol f p θ ; x, j q :(A.2) ˇˇˇ p f p k, x, j q ˇˇˇ ď e ´ ρ | k | sup | Im θ |ď ρ | f p θ, x, j q| . ℘ µℓ p p F p k qq , we get the claim;(ii) It is possible to control the seminorm ℘ µ,ρ ℓ p F q in terms of the ones for the Fourier coefficients:(A.3) ℘ µ,ρ ℓ p F q ď ÿ k P Z ν e ρ | k | ℘ µℓ p p F p k qq (A.1) ď ÿ k P Z ν e p ρ ´ ρ q| k | x k y τ C l ď C l p ρ ´ ρ q τ ` ν . In the next Proposition we essentially prove that pseudodifferential operators as in Definition2.5 have matrices which belong to the classes
Lip w p Ω , M ρ,s q extended from Definition 2.16. Proposition A.2.
Let F P Lip w p Ω , PA µρ q , with ρ ą . For any ă ρ ă ρ and s ą , thematrix of the operator x D y α F x D y β , α ` β ` µ ď , belongs to Lip w p Ω , M ρ ,s q . Moreover for any s ą , @ α ` β ď ´ µ , there exists σ ą such that (A.4) ˇˇˇ x D y α F x D y β ˇˇˇ Lip p w q ρ ,s, Ω ď C p ρ ´ ρ q ν ℘ µ,ρs ` σ p F q Lip p w q Ω . Proof.
Since x D y P PA is clearly independent of parameters, without loss of generality let F belong to PA µρ . We start by proving the result in the case µ “ α “ β “ . Let an arbitrary s ą be fixed. Then p F nm p k q : “ p π q ν ż T ν ˆr ,π s F p θ, x, D x qr sin p mx qs sin p nx q e ´ i k ¨ θ d θ d x “ p π q ν ż T ν ˆr´ π,π s F p θ, x, D x qr sin p mx qs sin p nx q e ´ i k ¨ θ d θ d x “ p π q ν ż T ν ` f p θ, x, m qp e i p m ´ n q x ´ e i p m ` n q x q e ´ i k ¨ θ d θ d x , (A.5)where f P P S mρ is the symbol of F . Consider first the case m ‰ n . Then, integrating by parts r s -times in x , with r s : “ t s ` u ` , and shifting the contour of integration in θ to T ν ´ i ρ sgn p k q (here sgn p k q : “ p sgn p k q , ..., sgn p k ν qq P t´ , u ν ), one gets that for any n, m P N , n ‰ m , k P Z ν ˇˇˇ p F nm p k q ˇˇˇ ď e ´ ρ | k | ˆ | m ` n | r s ` | m ´ n | r s ˙ sup | Im θ |ă ρ p x,m qP T ˆ N ˇˇˇ B r sx f p θ ; x, m q ˇˇˇ ď e ´ ρ | k | | m ´ n | r s ℘ ,ρ r s p f q . If m “ n , in a similar way one proves the bound sup m P N ˇˇˇ p F mm p k q ˇˇˇ ď e ´ ρ | k | ℘ ,ρ p f q . It follows thatfor any ă ρ ă ρ , one has | F | ρ ,s ď C p ρ ´ ρ q ´ ν ℘ ,ρ r s p f q ă 8 , which proves (A.4) in the case α “ β “ µ “ . To treat the general case, it is sufficient to note that, by Remarks 2.6, 2.9 and2.10, the operator x D y α F x D y β P PA ρ , so we have(A.6) ˇˇˇ x D y α F x D y β ˇˇˇ ρ ,s ď C p ρ ´ ρ q ν ℘ ,ρs ` σ px D y α F x D y β q ď C α,β p ρ ´ ρ q ν ℘ µ,ρs ` σ p F q . .2 Proof of Lemma 2.21 (Embedding) The result now follows immediately by applying Proposition A.2 to F P Lip w p Ω , PA ´ αρ q and G P Lip w p Ω , PA ´ βρ q . Indeed, we obtain ˇˇˇ x D y σ F x D y ´ σ ˇˇˇ Lip p w q ρ ,s, Ω , |x D y α F | Lip p w q ρ ,s, Ω , | F x D y α | Lip p w q ρ ,s, Ω ď C p ρ ´ ρ q ν ℘ ´ α,ρs ` σ p F q Lip p w q Ω . The estimates for G are analogous. A.3 Proof of Lemma 2.14
Denote by A e the extension of the operator A on L p T q which coincides with A on L odd p T q ” H and is identically zero on L even p T q . Since A e is parity preserving, one verifies for any m, n P Z that @ A e e i mx , e i nx D L p T q “ x A sin p mx q , sin p nx qy H . Therefore, (2.8) is equivalent to the classicalalgebra property developed on the exponential basis (for instance, see [BB13]); we skip thedetails. A.4 Proof of Lemma 2.22(Commutator)
We start with operators independent of θ P T ν . Let X “ ˆ X d X o ´ X o ´ X d ˙ , V “ ˆ V d V o ´ V o ´ V d ˙ . One has i r X , V s “ i p XV ´ VX q “ ˆ i Z d i Z o ´p i Z o q ´p i Z d q ˙ , where Z d : “ X d V d ´ X o V o ´ V d X d ` V o X o , Z o : “ X d V o ´ X o V d ´ V d X o ` V o X d . Omitting sake of simplicity conjugate operators and labels for diagonal and anti-diagonal ele-ments, by Remark 2.18, the following inequalities hold (here σ “ ˘ α, ): ˇˇˇ x D y σ XV x D y ´ σ ˇˇˇ s ď C s ˇˇˇ x D y σ X x D y ´ σ ˇˇˇ s ˇˇˇ x D y σ V x D y ´ σ ˇˇˇ s ; |x D y α XV | s ď C s |x D y α X | s | V | s ; | XV x D y α | s ď C s | X x D y α | s ˇˇˇ x D y ´ α V x D y α ˇˇˇ s ; (A.7)the same for those terms involving V X . All these norms extend easily to the analytic case.Therefore, by the assumption and from the definition in (2.16), properties 2.13, 2.14 and 2.15are satisfied. It remains to show the symmetries conditions in (2.12). Note that p i Z d q ˚ “ i Z d and p i Z o q ˚ “ i Z o if and only if p Z d q ˚ “ ´ Z d , p Z o q ˚ “ Z o . We check the condition for Z d . Wehave p Z d q ˚ “ p V d q ˚ p X d q ˚ ´ p V o q ˚ p X o q ˚ ´ p X d q ˚ p V d q ˚ ` p X o q ˚ p V o q ˚ “ V d X d ´ V o X o ´ X d V d ` X o V o ˚ “ ´ Z d . (A.8)In the same way one checks that p Z o q ˚ “ Z o . The Lipschitz dependence is easily checked.30 eferences [ADRH16] D. Abanin, W. De Roeck, and F. Huveneers. Theory of many-body localization in peri-odically driven systems. Annals of Physics , 372:1 – 11, 2016.[ADRHH17a] D. Abanin, W. De Roeck, W. Ho, and F. Huveneers. Effective hamiltonians, prethermal-ization, and slow energy absorption in periodically driven many-body systems.
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