Growth of Sobolev norms for abstract linear Schrödinger Equations
Dario Bambusi, Benoit Grébert, Alberto Maspero, Didier Robert
aa r X i v : . [ m a t h . A P ] J u l Growth of Sobolev norms for abstract linearSchr¨odinger equations
D. Bambusi ∗ , B. Gr´ebert † , A. Maspero ‡ , D. Robert § November 6, 2018
Abstract
We prove an abstract theorem giving a h t i ǫ bound ( ∀ ǫ >
0) on thegrowth of the Sobolev norms in linear Schr¨odinger equations of theform i ˙ ψ = H ψ + V ( t ) ψ when the time t → ∞ . The abstract theo-rem is applied to several cases, including the cases where (i) H is theLaplace operator on a Zoll manifold and V ( t ) a pseudodifferential op-erator of order smaller than 2; (ii) H is the (resonant or nonresonant)Harmonic oscillator in R d and V ( t ) a pseudodifferential operator oforder smaller than H depending in a quasiperiodic way on time. Theproof is obtained by first conjugating the system to some normal formin which the perturbation is a smoothing operator and then applyingthe results of [MR17]. In this paper we study growth of Sobolev norms for solutions of the abstractlinear Schr¨odinger equationi ∂ t ψ = H ψ + V ( t ) ψ , (1.1)in a scale of Hilbert spaces H r ; here V ( t ) is a time dependent operator and H a time independent linear operator. We will prove some abstract results ∗ Dipartimento di Matematica Federigo Enriques, Universit`a degli Studi di Milano, ViaSaldini 50, I-20133 Milano, Italy
Email: [email protected] † Laboratoire de Math´ematiques Jean Leray, Universit´e de Nantes, 2 rue de laHoussini`ere BP 92208, 44322 Nantes Cedex 3, France
Email: [email protected] ‡ International School for Advanced Studies (SISSA), Via Bonomea 265, 34136, Trieste,Italy
Email: [email protected] § Laboratoire de Math´ematiques Jean Leray, Universit´e de Nantes, 2 rue de laHoussini`ere BP 92208, 44322 Nantes Cedex 3, France
Email: [email protected] r ≥ ǫ >
0, the H r norm of the solution growsin time at most as h t i ǫ as t → ∞ , where h t i := √ t . The main novelty ofour results is that they allow (1) to weaken the standard gap assumptionson the spectrum of H , in particular to deal with some cases where the gapsare dense in R , and (2) to deal with perturbations which are of any orderstrictly smaller than that of H (see below for a precise definition).The main applications are to the case where(i) H is either the Laplace operator on a Zoll manifold (e.g. the spheres)or an anharmonic oscillator in R , while V is an operator dependingarbitrarily on time and having order strictly smaller than H ;(ii) H is the (possibly nonresonant) multidimensional Harmonic oscillatorand V ( t ) is an operator which depends on time in a quasiperiodic way and has order strictly smaller than H .Further applications will be presented in the paper.We emphasize in particular the results (ii) which, as far as we know arethe first controlling growth of Sobolev norms in higher dimensional systemswithout any gap condition.The proof is based on the combination of the ideas of [Bam17a, Bam17b,BGMR17] (which in turn are a developments of the ideas of [BBM14], seealso [PT01, IPT05]) and the results of [MR17]; precisely, for any positive N ,we construct a (finite) sequence of unitary time dependent transformationsconjugating H + V ( t ) to a Hamiltonian of the form H + Z ( N ) ( t ) + V ( N ) ( t ) , (1.2)where [ H ; Z ( N ) ] = 0 and V ( N ) is a smoothing operator of order N , namelyan operator belonging to L ( H s ; H s + N ) for any s (linear bounded operatorsfrom H s to H s + N ). Then we apply Theorem 1.5 of [MR17] to (1.2) gettingthe h t i ǫ bound on the growth of Sobolev norms.We think that a further point of interest of our paper is that the con-jugation to a system of the form (1.2) is here developed in an abstractcontext, instead then in the framework of classes of pseudodifferential oper-ators adapted to the situation under study; this is the main reason why weget an abstract theory directly applicable to many different contexts.The main point is that we introduce an abstract graded algebra of oper-ators whose properties mimic the properties of pseudodifferential operators.The use of this framework is made possible by the technique we developto solve the homological equations met in the construction of the conjuga-tion of H to (1.2). Indeed, we recall that in previous papers the smoothingtheorem, namely the result conjugating the original system to (1.2) was ob-tained by quantizing the procedure of classical normal form. Here instead,we work directly at the quantum level, in particular solving at this level thetwo homological equations that we find (see eqs. (3.17) and (3.24) below).2t is worth to add a few words on the way we solve the homologicalequations. When dealing with systems related to the applications (i), weassume that H = f ( K ) where f is a superlinear function and K is anoperator s.t. spec( K ) ⊂ N + λ , λ > . (1.3)In this case we solve the homological equation essentially by averaging overthe flow e − i tK of K . In turn this is made possible by the use of a commu-tator expansion lemma proved in [DG97]. When dealing with the d dimen-sional harmonic oscillators instead, we take H = d X j =0 ν j K j , with K j commuting linear operators, each one fulfilling the property (1.3)(think of K j = − ∂ x j + x j ) and ν j >
0; then we consider operators of theform e i τ · K A e − i τ · K (where of course τ · K := τ K + ... + τ d K d ), remark that they are quasiperi-odic in the “angles” τ , and use a Fourier expansion in τ in order to solvethe homological equation.The study of growth of Sobolev norms and the related results on thenature of the spectrum of the Floquet operator has a long history: we recallthe results by [How89, How92, Joy94] showing that the Floquet spectrumof systems with growing gaps and bounded perturbations is pure point, aresult which implies boundedness of the expectation value of the energy.The first h t i ǫ estimates on the expectation value of the energy for system ofthe form (1.1) was obtained by Nenciu in [Nen97] for the case of increasinggaps and bounded perturbations (see also [BJ98, Joy96] for similar results),and by Duclos, Lev and Sˇtov´ıˇcek [DLS08] in case of shrinking gaps. Inthe case of increasing gaps, such results were improved recently by twoof us (see [MR17]) who obtained the h t i ǫ growth of Sobolev norms alsoin the case of unbounded perturbations depending arbitrarily on time, forexample in the case where H = − ∂ x + x k , the result of [MR17] allowsto deal with perturbations growing at infinity as | x | m with m < k − m < k . The result of[MR17] also applies to perturbations of the free Schr¨odiger equation on Zollmanifolds with perturbations of order strictly smaller than 1. Here we dealwith perturbations of order strictly smaller than 2. A study of perturbationsof maximal order has been done independently by Montalto [Mon17] whogot a control of the growth of Sobolev norms for the Schr¨odinger equationon T with H = a ( t, x ) |− ∂ xx | M + V ( t ) with M > / a a smooth positivefunction and V a pseudodifferential operator of order smaller than M .3inally we recall that in [MR17] logarithmic estimates for the growth ofSobolev norms were also obtained in the case of perturbations dependinganalytically on time. Here we do not attack the problem of getting logarith-mic estimates, but we think that our technique would also allow to get suchestimates.A remarkable further result was obtained by Bourgain [Bou99] whoobtained a logarithmic bound on the growth of Sobolev norms for theSchr¨odinger equation on T d ( d = 1 ,
2) in the case of an analytic pertur-bation depending quasiperiodically on time. Such a result is based on theuse of a Lemma on the clustering of resonant sites (in a suitable space timelattice) which does not seem to extend to different geometries. The re-sult of Bourgain was extended by Wang [Wan08] to deal with Schr¨odingerequations on T perturbed by a potential analytic in time (but otherwisedepending arbitrary on time) and greatly simplified by Delort [Del10] whoused it in an abstract framework which allows to deal with the case of T d (any d ≥
1) and also with the case of Zoll manifolds, obtaining a growthbounded by h t i ǫ (see also [FZ12] for analytic potentials on T d ). We alsomention the reducibility result by [EK09] dealing with small quasiperiodicperturbations of the free Schr¨odinger equation on T d ; for such a system, theauthors prove that growth of Sobolev norms cannot happen, provided thefrequency of the quasiperiodic solution is chosen in a nonresonant set. Atpresent our method does not allow to deal with the Schr¨odinger equationon T d for d ≥ R d with d >
1, a couple of reducibil-ity results are known, namely [GP16] in which the authors study smallbounded perturbations of the completely resonant
Harmonic oscillator, and[BGMR17] in which we studied small polynomial perturbations of the reso-nant or nonresonant Harmonic oscillator.As far as we know no results are known on growth of Sobolev norms forperturbations of the harmonic oscillator: H := − ∆ + d X j =1 ν j x j , (1.4)with nonresonant frequencies ν j . This is due to the fact that the differencesbetween two of its eigenvalues { λ a } a ∈ N d , namely λ a − λ b = ν · ( a − b )are dense on the real axis and this prevents the use of any previous tech-nique. As anticipated above here we obtain the h t i ǫ growth for the caseof perturbation of order strictly smaller than the order of the Harmonicoscillator. 4 cknowledgments. During the preparation of this work, we were sup-ported by ANR -15-CE40-0001-02 “BEKAM” of the Agence Nationale dela Recherche. A. Maspero is also partially supported by PRIN 2015 “Varia-tional methods, with applications to problems in mathematical physics andgeometry”.
We start with a Hilbert space H and a reference operator K , which weassume to be selfadjoint and positive, namely such that h ψ ; K ψ i ≥ c K k ψ k , ∀ ψ ∈ D ( K / ) , c K > , and define as usual a scale of Hilbert spaces by H r = D ( K r ) (the domainof the operator K r ) if r ≥
0, and H r = ( H − r ) ′ (the dual space) if r < H −∞ = S r ∈ R H r and H + ∞ = T r ∈ R H r . We endow H r with the natural norm k ψ k r := k ( K ) r ψ k , where k · k is the norm of H ≡ H . Notice that for any m ∈ R , H + ∞ is a dense linear subspace of H m (this is a consequence of the spectral decomposition of K ).We introduce now a graded algebra A of operators which mimic somefundamental properties of different classes of pseudo-differential operators.For m ∈ R let A m be a linear subspace of T s ∈ R L ( H s , H s − m ) and define A := S m ∈ R A m . We notice that the space T s ∈ R L ( H s , H s − m ) is a Fr´echetspace equipped with the semi-norms: k A k m,s := k A k L ( H s , H s − m ) .One of our aims is to control the smoothing properties of the operatorsin the scale {H r } r ∈ R . If A ∈ A m then A is more and more smoothing if m → −∞ and the opposite as m → + ∞ . We will say that A is of order m if A ∈ A m . Definition 2.1.
We say that S ∈ L ( H + ∞ , H −∞ ) is N -smoothing if ∀ κ ∈ R ,it can be extended to an operator in L ( H κ , H κ + N ) . When this is true forevery N ≥ , we say that S is a smoothing operator. The first set of assumptions concerns the properties of A m : Assumption I: (i) For each m ∈ R , K m ∈ A m ; in particular K is an operator of orderone.(ii) For each m ∈ R , A m is a Fr´echet space for a family of semi-norms { ℘ mj } j ≥ such that the embedding A m ֒ → T s ∈ R L ( H s , H s − m ) is con-tinuous.If m ′ ≤ m then A m ′ ⊆ A m with a continuous embedding.5iii) A is a graded algebra, i.e ∀ m, n ∈ R : if A ∈ A m and B ∈ A n then AB ∈ A m + n and the map ( A, B ) AB is continuous from A m × A n into A m + n .(iv) A is a graded Lie-algebra : if A ∈ A m and B ∈ A n then the com-mutator [ A, B ] ∈ A m + n − and the map ( A, B ) [ A, B ] is continuousfrom A m × A n into A m + n − .(v) A is closed under perturbation by smoothing operators in the followingsense: let A be a linear map: H + ∞ → H −∞ . If there exists m ∈ R such that for every N > A = A ( N ) + S ( N ) ,with A ( N ) ∈ A m and S ( N ) is N -smoothing, then A ∈ A m .(vi) If A ∈ A m then also the adjoint operator A ∗ ∈ A m . The dualityhere is defined by the scalar product h· , ·i of H = H . The adjoint A ∗ is defined by h u, Av i = h A ∗ u, v i for u, v ∈ H ∞ and extended bycontinuity.It is well known that classes of pseudo-differential operators satisfy theseproperties, provided one chooses for K a suitable operator of the right order(see e.g. [H¨or85]).In [Gui85] V. Guillemin has introduced abstract pseudo-differential algebras,called generalized Weyl algebras. For his purpose [Gui85] needs differentproperties than ours, but obviously there is an overlap with our presentation. Remark 2.2.
One has that ∀ A ∈ A m , ∀ B ∈ A n ∀ m, s ∃ N s.t. k A k m,s ≤ C ℘ mN ( A ) , (2.1) ∀ m, n, j ∃ N s.t. ℘ m + nj ( AB ) ≤ C ℘ mN ( A ) ℘ nN ( B ) , (2.2) ∀ m, n, j ∃ N s.t. ℘ m + n − j ([ A, B ]) ≤ C ℘ mN ( A ) ℘ nN ( B ) , (2.3) for some positive constants C ( s, m ) , C ( m, n, j ) , C ( m, n, j ) . For Ω ⊂ R d and F a Fr´echet space, we will denote by C mb (Ω , F ) thespace of C m maps f : Ω ∋ x f ( x ) ∈ F , such that, for every seminorm k · k j of F one hassup x ∈ Ω k ∂ αx f ( x ) k j < + ∞ , ∀ α ∈ N d : | α | ≤ m . (2.4)If (2.4) is true ∀ m , we say f ∈ C ∞ b (Ω , F ).The next property needed is the following Egorov property, also wellknown for pseudo-differential operators. This property will impose the choice of the semi-norms { ℘ mj } j ≥ . We will see in theexamples that the natural choice ( k · k m,s ) s ≥ has to be refined. ssumption II: For any A ∈ A m and τ ∈ R , the map τ A ( τ ) :=e i τK A e − i τK ∈ C b ( R , A m ). Remark 2.3.
From Assumption II one has that, for any B ∈ A n , for any ℓ ∈ N , ad ℓA ( s ) ( B ) ∈ C b (] − T, T [ , A n +( m − ℓ ) , ∀ T > . Here ad A ( B ) :=i[ A, B ] . Remark that
Assumption II is a quantum property for the time evolu-tion of observables. Practically it follows from the time evolution of classicalobservables (Hamilton equation) if some classes of symbols are preservedunder the classical flows. Indeed one might replace
Assumption II by aweaker one (see Appendix B).
Now we state our spectral assumption on K : Assumption A : K has an entire discrete spectrum such thatspec( K ) ⊆ N + λ (2.5)for some λ > H is a function of K . To state it precisely we need the followingdefinition Definition 2.4.
A function f ∈ C ∞ ( R ) will be said to be a classical symbolof order ρ (at + ∞ ) if there exist real numbers { c j } j ≥ s.t. c ≥ and forall k ≥ , all N ≥ , there exists C k,N s.t. (cid:12)(cid:12) d k dx k (cid:0) f ( x ) − X ≤ j ≤ N − c j x ρ − j (cid:1)(cid:12)(cid:12) ≤ C k,N | x ρ − N − k | , ∀ x ≥ . We will denote by S ρ the space of classical symbols of order ρ .We shall say that f is an elliptic classical symbol of order ρ if f is real and c > . We shall write f ∈ S ρ + .We shall say that f is a classical symbol of order −∞ if f ∈ S m ∀ m < .We shall write f ∈ S −∞ . Some standard properties of classical symbols are recalled in AppendixA. We assume that
Assumption B:
There exists an elliptic classical symbol f of order µ > H = f ( K ) . (2.6)7e will prove (see Lemma A.2) that (2.6) implies H ∈ A µ , i.e. H is anoperator of order µ > H ( t ) := H + V ( t ) (see (1.1)). When the solution ψ ( t ) existsglobally in time, we define the Schr¨odinger propagator U ( t, s ), generated by(1.1), such that ψ ( t ) = U ( t, s ) ψ , U ( s, s ) = (2.7)We are ready to state our main result on systems with increasing gaps: Theorem 2.5.
Assume that A is a graded algebra as defined in Section 2.1and that K , H satisfy assumptions A and B. Furthermore assume thatthe perturbation V ( t ) with domain H ∞ is symmetric for every t ∈ R andsatisfies V ∈ C ∞ b ( R , A ρ ) , with ρ < µ . (2.8) Then H ( t ) = H + V ( t ) generates a propagator U ( t, s ) s.t. U ( t, s ) ∈ L ( H r ) ∀ r ∈ R .Moreover for any r > and any ǫ > there exists C r,ǫ > such that kU ( t, s ) ψ k r ≤ C r,ǫ h t − s i ǫ k ψ k r , ∀ t, s ∈ R . (2.9)This result extends a result by Nenciu [Nen97] for bounded perturbations( ρ = 0). Furthermore in [MR17] two of us had already extended Nenciu’s re-sult to unbounded perturbations with the constraint ρ < min( µ − , H is f ( N + λ ) for some smooth function f (see AssumptionsA and B).As a final remark, we note that Theorem 2.5 gives also a proof of theexistence and of some properties of the propagator U ( t, s ), which in theframework of Theorem 2.5 are not obvious. Zoll manifolds.
Recall that a Zoll manifold is a compact Riemannianmanifold (
M, g ) such that all the geodesic curves have the same period T := 2 π . For example the d -dimensional sphere S d is a Zoll manifold.We denote by △ g the positive Laplace-Beltrami operator on M and by H r ( M ) = Dom(1 + △ g ) r/ , r ≥
0, the usual scale of Sobolev spaces. Finallywe denote by S m cl ( M ) the space of classical real valued symbols of order m ∈ R on the cotangent T ∗ ( M ) of M (see H¨ormander [H¨or85] for moredetails). Definition 2.6.
We say that A ∈ A m if it is a pseudodifferential operator(in the sense of H¨ormander [H¨or85]) with symbol of class S m cl ( M ) .
8n this case the operator K is a perturbation of order − p △ g (seeSect. 4.1), and the norms k ψ k r coincide with the standard Sobolev norms. Corollary 2.7 (Zoll manifolds) . Let V ( t ) be a symmetric pseudo-differentialoperator of order ρ < on M such that its symbol v ∈ C ∞ b ( R ; S ρ cl ( M )) . Thenthe propagator U ( t, s ) generated by H ( t ) = △ g + V ( t ) exists and satisfies (2.9) . Anharmonic oscillators on R . The second application concerns one di-mensional quantum anharmonic oscillatorsi ∂ t ψ = H k,l ψ + V ( t ) ψ , x ∈ R , (2.10)where H k,l is the one degree of freedom Hamiltonian H k,l := D lx + ax k , k, l ∈ N , k + l ≥ , a > . (2.11)Here D x := i − ∂ x . It is well known that H k,ℓ is essentially self-adjoint in L ( R ) [HR82b].Define the Sobolev spaces H r := Dom( H k + l kl rk,l ) for r ≥
0. We define nowsuitable operator classes for the perturbation. Denote k ( x, ξ ) := (1 + x k + ξ l ) k + l kl . Definition 2.8.
A function f will be called a symbol of order ρ ∈ R if f ∈ C ∞ ( R x × R ξ ) and ∀ α, β ∈ N , there exists C α,β > s.t. | ∂ αx ∂ βξ f ( x, ξ ) | ≤ C α,β k ( x, ξ ) ρ − kβ + lαk + l . (2.12) We will write f ∈ S ρ an . As usual to a symbol f ∈ S ρ an we associate the operator f ( x, D x ) whichis obtained by standard Weyl quantization (see formula (4.2) below). Definition 2.9.
We say that F ∈ A ρ if it is a pseudodifferential operatorwith symbol of class S ρ an , i.e., if there exist f ∈ S ρ an and S smoothing (in thesense of Definition 2.1) such that F = f ( x, D x ) + S . In this case the seminorms are defined by ℘ ρj ( F ) := X | α | + | β |≤ j C αβ , with C αβ the smallest constants s.t. eq. (2.12) holds. If a symbol f de-pends on additional parameters (e.g. it is time dependent), we ask that theconstants C α,β are uniform w.r.t. such parameters.9 emark 2.10. With this definition of symbols, one has x ∈ S lk + l an , ξ ∈ S kk + l an , x k + ξ l ∈ S klk + l an , k ( x, ξ ) ∈ S . We get the following:
Corollary 2.11 (1-D anharmonic oscillators) . Consider equation (2.10) with the assumption (2.11) . Assume also that V ∈ C ∞ b ( R ; A ρ ) with ρ < klk + l .Then the propagator U ( t, s ) generated by H ( t ) = H k,l + V ( t ) is well definedand satisfies (2.9) . An example of admissible perturbation is V ( t, x, ξ ) = X lα + kβ< kl a α,β ( t ) x α ξ β with a α,β ∈ C ∞ b ( R , R ). In particular if we choose H = − d dx + x , wecan consider unbounded perturbations of the form x g ( t ) and of course also xg ( t ) with g ∈ C ∞ b ( R , R ). Remark 2.12.
Our class of perturbations contains quite general pseudod-ifferential operators, however it is easy to see that multiplication operators(i.e. operators independent of ∂ x ) must be polynomials in x with coefficientswhich are possibly time dependent.In the similar problem of reducibility more general classes of perturba-tions have been treated in [Bam17b]. We did not try to push the result inthat direction. This is probably non trivial in an abstract framework like theone we are using here. Remark 2.13.
We think that our method should also allow to deal withsome perturbations of the same order as the main term. For example itshould be treatable the case where V is a quasihomogeneous polynomial ofmaximal order fulfilling some sign condition (more or less as in Theorem2.12 of [Bam17a]). In order to deal with perturbations of operators of order 1 we have to restrictto the case where the dependence of the perturbation on time is quasiperi-odic.Let A := ∪ m ∈ R A m be a graded Lie algebra satisfying Assumption I with a reference operator K .Let K , K , · · · , K d be d self-adjoint positive operators such that K j ∈ A , ∀ ≤ j ≤ d . Assume the following modified Assumption II: Assumption II ′ : (i) [ K j , K ℓ ] = 0 for any 0 ≤ j, ℓ ≤ d .10ii) Denote K = ( K , · · · , K d ) and for τ ∈ R d , τ · K := X ≤ j ≤ d τ j K j .Then for any A ∈ A m , the map τ A ( τ ) := e i τ · K A e − i τ · K ∈ C ∞ b ( R d ; A m ). Remark 2.14.
For any B ∈ A n , for any ℓ ∈ N , one has ad ℓA ( s ) ( B ) ∈ C ∞ b ( R d ; A n + ℓ ( m − ) . We also adapt our spectral conditions:
Assumption A ′ : K = ( K , · · · , K d ) has an entire joint spectrum, spec( K ) ⊆ N d + λ for some λ ∈ R d , λ ≥ Assumption B ′ : There exist { ν j } dj =1 , ν j > H = X ≤ j ≤ d ν j K j , (2.13) K = H . (2.14)In order to fix ideas one can think of the case of Harmonic oscillators, inwhich K j = − ∂ j + x j , 1 ≤ j ≤ d . Remark 2.15.
Since the operators K j are positive, the norm k . k r definedusing the operator K is equivalent to the norm defined using the operator K ′ := P dj =1 K j . We consider both the case where ν := ( ν , ..., ν d )is resonant and the case where it is nonresonant. To state the arithmeticalassumptions on ν , we first recall the following well known lemma whosescheme of proof will be recalled in the Appendix C. Lemma 2.16.
There exists ˜ d ≤ d , a vector ˜ ν ∈ R ˜ d with components inde-pendent over the rationals, and vectors v j ∈ Z d , j = 1 , ..., ˜ d such that ν = ˜ d X j =1 ˜ ν j v j . (2.15) Remark 2.17.
For example(i) if ν is nonresonant, then ˜ ν = ν and v j = e j , the standard basis of R d ;(ii) if ν is completely resonant then ˜ d = 1 ; e.g. if ν = (1 , . . . , , then ˜ ν = 1 , v = (1 , . . . , . heorem 2.18. Assume that V ( t ) = W ( ωt ) with W ∈ C ∞ b ( T n , A ρ ) a quasi-periodic operator of order ρ < . Assume furthermore that (˜ ν, ω ) ∈ R ˜ d + n isa Diophantine vector, namely that there exist γ > , and κ ∈ R s.t., (cid:12)(cid:12) ω · k + ˜ ν · ℓ (cid:12)(cid:12) ≥ γ ( | ℓ | + | k | ) κ , = ( k, ℓ ) ∈ Z n + ˜ d . (2.16) Then the propagator U ( t, s ) generated by H ( t ) = ν · K + W ( ωt ) exists andsatisfies (2.9) . Remark 2.19.
The vector ˜ ν is defined up to linear combinations with in-teger coefficients; clearly condition (2.16) does not depend on the choice of ˜ ν . Remark 2.20.
We recall that Diophantine vectors form a subset of R n + ˜ d of full measure if κ > n + ˜ d − . Relativistic Schr¨odinger equation on Zoll manifolds.
We considerthe reduced Dirac equation on a Zoll manifold M with mass µ > ∂ t ψ = p △ g + µ ψ + V ( ωt, x, D x ) ψ , t ∈ R , x ∈ M .
As in the case of the Schr¨odinger equation on Zoll manifolds, A ρ is the classof pseudodifferential operators with symbols in S ρ cl ( M ) (see Definition 2.6).In this case V is assumed to be quasi-periodic in time. Corollary 2.21 (Relativistic Schr¨odinger equation on Zoll manifolds) . As-sume that V ( t ) = W ( ωt ) with W ∈ C ∞ ( T n , A ρ ) with ρ < . Assumefurthermore that the non resonance condition | ω · k + m | ≥ γ | k | κ , ∀ = k ∈ Z n , ∀ m ∈ Z (2.17) holds for some γ > and κ . Then the propagator U ( t, s ) generated by H ( t ) = p △ g + µ + W ( ωt ) exists and satisfies (2.9) . Harmonic oscillator in R d . Consider the quantum Harmonic oscillatori ∂ t ψ = H ν ψ + V ( t ) ψ , x ∈ R d (2.18) H ν := − ∆ + d X j =1 ν j x j , V ( t ) = W ( ωt, x, D x ) . (2.19)Here W is the Weyl quantization of a symbol belonging to the followingclass 12 efinition 2.22. A function f will be called a symbol of order ρ ∈ R if f ∈ C ∞ ( R dx × R dξ ) and ∀ α, β ∈ N d , there exists C α,β > s.t. | ∂ αx ∂ βξ f ( x, ξ ) | ≤ C α,β (1 + | x | + | ξ | ) ρ − | β | + | α | . (2.20) We will write f ∈ S ρ ho . The class (2.20) is the extension to higher dimensions of the class used inthe anharmonic oscillators (see Definition 2.8) and with k = l = 1. Remark 2.23.
With our numerology, the symbol of the harmonic oscillatoris of order 1, | ξ | + P j ν j x j ∈ S , and not of order 2 as typically in theliterature. The classes A m are defined as in Definition 2.9, with symbols in the class S m ho . Corollary 2.24.
Assume that ν is such that ˜ ν fulfills (2.16) , and that W ∈ C ∞ ( T n ; A ρ ) with ρ < . Then the propagator U ( t, s ) of H ( t ) = H ν + W ( ωt ) exists and fulfills (2.9) . Remark that after a trivial rescaling of the spatial variables, H ν = P dj =1 ν j ( − ∂ j + x j ), thus the corollary is a trivial application of Theorem2.18. Remark 2.25.
In the completely resonant case H (1 ,..., = − ∆ + | x | , one has ˜ ν = 1 and the set of the ω ′ s for which (2.16) is fulfilled has fullmeasure provided κ > n . Remark 2.26.
We note that in the resonant case there have been exhib-ited examples of polynomial growths of the Sobolev norms. In particularsee [Del14] and [BGMR17] for periodic in time perturbations; of course insuch examples the frequency ω does not fulfill (2.16) . Finally we recall also[BJLPN], where some some random in time perturbations are considered. As explained in the introduction, the main step of the proof consists inproving a theorem conjugating the original Hamiltonian to a Hamiltonianof the form (1.2); this will be done in Theorem 3.8. Subsequently we willapply Theorem 1.5 of [MR17], which essentially states that, if H ( t ) is suchthat for some N > − H ( t ) , K ] K N ∈ C b ( R , L ( H r )) , (3.1)13hen ∃ C r,N > kU ( t, s ) ψ k r ≤ C r,N h t − s i r N k ψ k r , ∀ t, s ∈ R . (3.2)We come to the algorithm of conjugation of the original Hamiltonianto (1.2). Before discussing it, we need to know the way a Hamiltonian ischanged by a time dependent unitary transformation. This is the contentof the following lemma. Lemma 3.1.
Let H ( t ) be a time dependent self-adjoint operator, and X ( t ) be a selfadjoint family of operators. Assume that ψ ( t ) = e − i X ( t ) ϕ ( t ) then i ˙ ψ = H ( t ) ψ ⇐⇒ i ˙ ϕ = ˜ H ( t ) ϕ (3.3) where ˜ H ( t ) := e i X ( t ) H ( t ) e − i X ( t ) − Z e i sX ( t ) ˙ X ( t ) e − i sX ( t ) d s . (3.4)This is seen by an explicit computation. For example see Lemma 3.2 of[Bam17a].A further important property giving the expansion of an operator of theform e i X ( t ) A e − i X ( t ) in operators of decreasing order is stated in the followinglemma. Lemma 3.2.
Let X ∈ A ρ with ρ < be a symmetric operator. Let A ∈ A m with m ∈ R . Then X is selfadjoint and for any M ≥ we have e i τX A e − i τX = M X ℓ =0 τ ℓ ℓ ! ad ℓX ( A ) + R M ( τ, X, A ) , ∀ τ ∈ R , (3.5) where R M ( τ, X, A ) ∈ A m − ( M +1)(1 − ρ ) .In particular ad ℓX ( A ) ∈ A m − ℓ (1 − ρ ) and e i τX A e − i τX ∈ A m , ∀ τ ∈ R . The proof will be given in Sect. 3.2.We describe now the algorithm which will lead to the smoothing Theorem3.8; the proof is slightly different according to the set of assumptions onechooses. We start by discussing it under the assumptions of Theorem 2.5,namely Assumption A and B. Subsequently we will discuss the changesneeded to deal with Theorem 2.18.We look for a change of variables of the form ϕ = e i X ( t ) ψ where X ( t ) ∈A ρ − µ +1 is a self-adjoint operator which, due to the assumption ρ < µ ,has order smaller then one. Then ϕ fulfills the Schr¨odinger equation i ˙ ϕ = H + ( t ) ϕ with H + ( t ) := e i X ( t ) H ( t ) e − i X ( t ) − Z e i sX ( t ) ˙ X ( t ) e − i sX ( t ) d s = H + i[ X ( t ) , H ] + V ( t ) + i[ X ( t ) , V ( t )] −
12 [ X ( t ) , [ X ( t ) , H ]] + · · ·− Z e i sX ( t ) ˙ X ( t ) e − i sX ( t ) d s.
14n view of the properties of the graded algebra we have [ X , V ] ∈ A ρ − µ ,[ X , [ X , H ]] ∈ A ρ − µ (Assumption I (iv)) and e i sX ( t ) ˙ X ( t ) e − i sX ( t ) ∈A ρ − µ +1 (Lemma 3.2), therefore one has H + ( t ) = H + i[ X ( t ) , H ] + V ( t ) + V +1 ( t ) , (3.6)with V +1 ( t ) ∈ C ∞ b ( R , A min( ρ − µ +1 , ρ − µ ) ).Now we look for X ( t ) s.t.i[ H , X ( t )] = V ( t ) − h V ( t ) i , (3.7)where h V ( t ) i is the average over τ of e i τK V ( t ) e − i τK (see (3.18)), which inparticular commutes with K . We will verify in Lemma 3.5 that there exists X s.t. i[ H , X ( t )] − V ( t ) + h V ( t ) i ∈ A ρ − . Therefore using such a X to generate a unitary transformation, we get H + ( t ) := H + h V ( t ) i + V + ( t ) , (3.8)where V + ( t ) ∈ C ∞ b ( R , A ρ − δ ) with δ := min (1 , µ − , µ − ρ ) > . (3.9)Therefore V + ( t ) is a perturbation of order lower than V ( t ). Furthermore h V ( t ) i commutes with K .Iterating this procedure we will establish an ”almost” reducibility resultthat will be stated and proved in Subsect. 3.4.Then, using Theorem 1.5 of [MR17], we immediately get Theorem 2.5.In the case where H ∈ A the procedure has to be slightly modifiedsince in this case X and therefore ˙ X has the same order as V and thus itcannot be considered as a remainder when analyzing H + . In this case onerewrites H + ( t ) = H + i[ X ( t ) , H ] + V ( t )+ i[ X ( t ) , V ( t )] −
12 [ X ( t ) , [ X ( t ) , H ]] + · · ·− ˙ X − Z (cid:16) i s [ X ( t ) , ˙ X ( t )] + .... (cid:17) d s, so that eq. (3.6) is substituted by H + ( t ) = H + i[ X ( t ) , H ] + V ( t ) − ˙ X ( t ) + V + ( t ) , (3.10)with V + ∈ A ρ − δ ∗ , δ ∗ := 1 − ρ > , (3.11)15o again it is more regular than V ( t ). Thus one is led to consider the newhomological equationi[ H , X ( t )] + ˙ X ( t ) = V ( t ) − h V ( t ) i , (3.12)where h V ( t ) i has to commute with K . In order to be able to solve such anequation we restrict to the case of V ( t ) quasiperiodic in t and, as explainedin the introduction, we develop a procedure based on a suitable Fourierexpansion to construct X and h V ( t ) i . The details are given in Lemma 3.7which will ensure that such a homological equation has a smooth solutionand thus the procedure is well defined also in the case of order 1. Lemma 3.3. ( i ) Let X ∈ A be symmetric w.r.t. the scalar product of H .Then X has a unique self-adjoint extension and e − i τX ∈ L ( H r ) ∀ r ≥ and ∀ τ ∈ R . Furthermore e − i τX is an isometry in H . ( ii ) Assume that X ( t ) is a family of symmetric operators in A s.t. sup t ∈ R ℘ j ( X ( t )) < ∞ , ∀ j ≥ . (3.13) Then there exist c r , C r > s.t. c r k ψ k r ≤ k e − i τX ( t ) ψ k r ≤ C r k ψ k r , ∀ t ∈ R , ∀ τ ∈ [0 , . (3.14) Proof. (i) From the properties of the algebra A we have that XK − and[ X, K ] K − are of order 0. Thus by definition these operators belong to L ( H r ) ∀ r ∈ R . Then the result follows from Theorem 1.2 of [MR17].(ii) By item (i), for any t ∈ R and τ ∈ [0 ,
1] the operator e − i τX ( t ) is anisometry in H , therefore k e − i τX ( t ) ψ k r = k e i τX ( t ) K r e − i τX ( t ) ψ k . Then we havee i τX ( t ) K r e − i τX ( t ) ψ = K r ψ + i Z τ e i τ X ( t ) [ X ( t ) , K r ] e − i τ X ( t ) ψ d τ = K r ψ + i Z τ e i τ X ( t ) [ X ( t ) , K r ] K − r K r e − i τ X ( t ) ψ d τ (3.15)By the properties of the algebra A and (3.13) one has that (using (2.1)–(2.3))sup t ∈ R k [ X ( t ) , K r ] K − r k L ( H ) < C r < + ∞ , therefore taking the norm k · k of (3.15) one gets the inequality k e − i τX ( t ) ψ k r ≤ k ψ k r + Z τ C r k e − i τ X ( t ) ψ k r d τ . k e − i τX ( t ) ψ k r ≤ e C r k ψ k r , ∀ t ∈ R , ∀ τ ∈ [ − , . This proves the majoration in (3.14). The minoration follows simply by theidentity ψ = e i τX ( t ) e − i τX ( t ) ψ and the majoration. Proof of Lemma 3.2.
Selfadjointness was proven in the previous lemma. Letus apply to the l.h.s. of (3.5) the Taylor formula at τ = 0. Then we get,with U X ( τ ) := e − i τX and ad X ( A ) := i[ X, A ] U X ( − τ ) A U X ( τ ) (3.16)= M X j =0 τ j j ! ad jX ( A ) + τ M +1 M ! Z (1 − s ) M +1 U X ( − sτ ) ad M +1 X ( A ) U X ( sτ ) d s . Using Assumption I (iv), we have ad jX ( A ) ∈ A m − j (1 − ρ ) . We define theremainder R M ( τ, X, A ) to be the integral term in (3.16), which, using alsoLemma 3.3, belongs to L ( H s , H s − m +( M +1)(1 − ρ ) ), ∀ s ∈ R . Therefore theremainder R M ( τ, X, A ) is N -smoothing provided M + 1 ≥ N + m − ρ . As M can be taken arbitrary large, e i τX A e − i τX fulfills Assumption I (v), thus itbelongs to A m . The first homological equation.
As we have seen in Section 3.1, toprove Theorem 2.5 we need to study an homological equation of the formi[ H , X ] = A − h A i , (3.17)where A ∈ A m and h A i is the average of A along the periodic flow of K : h A i := 12 π Z π A ( τ ) d τ , A ( τ ) = e i τK A e − i τK . (3.18)Notice that the assumption on the spectrum of K (see Assumption A)entails that e πK = e πλ , thus for any A ∈ A one has e πK A e − πK = A ,namely τ A ( τ ) is 2 π periodic. Lemma 3.4.
Let A ∈ A m , m ∈ R . Then h A i ∈ A m and [ K , h A i ] = 0 . (3.19) Proof. h A i ∈ A m is a consequence of Assumption II. Identity (3.19) followsby a direct computation. 17 emma 3.5. (i) Let A ∈ A m , m ∈ R . Then Y = 12 π Z π τ ( A − h A i )( τ ) d τ (3.20) solves the homological equation i[ K , Y ] = A − h A i . (3.21) Further Y ∈ A m and if A is symmetric, so is Y .(ii) Choose R > such that f ′ ( x ) ≥ if x ≥ R and η ∈ C ∞ ( R ) such that η ( x ) = 1 if x ∈ [0 , R ] , η ( x ) = 0 if x ≥ R + 1 . Define X := (1 − η ( K )) (cid:0) f ′ ( K ) (cid:1) − Y , (3.22) with Y as in (3.20) . Then X ∈ A m − µ +1 , is symmetric provided A is sym-metric and solves (3.17) modulo an error term in A m − . More precisely i[ H , X ] = A − h A i + A m − . (3.23)We note for the sequel that if A ∈ A m then X ∈ A m − ( µ − , namely wehave a gain of µ − > Proof.
Assertion (i) is proved by integration by parts using that A ( τ ) is 2 π -periodic.To prove (ii), first remark that by Assumption B and Lemma A.1, f ′ ∈ S µ − ,thus it is different from zero provided x ≥ R is large enough. It followsthat the function x − η ( x ) f ′ ( x ) ∈ S − µ +1 . Therefore, by Lemma A.2, theoperator (1 − η ( K )) ( f ′ ( K )) − ∈ A − µ +1 . Finally since Y ∈ A m , it followsthat X ∈ A m − µ +1 .We show now that X solves (3.23). This is a consequence of the commutatorexpansion Lemma. Indeed fix N ≥
2, then by Lemma A.3 one has[ H , X ] = [ f ( K ) , X ] = f ′ ( K )[ K , X ] + X ≤ j ≤ N j ! f ( j ) ( K )ad jK ( X ) + R N +1 ( f, X )with R N +1 ( f, X ) ∈ A m − µ +1+[ µ ] − N ⊂ A m − .By Lemma A.1 and Assumption I, for any integer j ≥ f ( j ) ( K ) ad jK ( X ) ∈ A m − µ +1+ µ − j ⊂ A m − . Then we geti[ H , X ] = i f ′ ( K )[ K , X ] + A m − = (1 − η ( K )) i[ K , Y ] + A m − = (1 − η ( K )) ( A − h A i ) + A m − , with A m − ∈ A m − . Now put R := − η ( K ) ( A − h A i ). Since x η ( x ) ∈ S −∞ , R is a smoothing operator and thus A m − + R ∈ A m − .18 he second homological equation. We want to solve eq. (3.12). Usingthe quasiperiodicity assumption V ( t ) = W ( ωt ), we look for a quasiperiodicsolution X ( t ) = X ( ωt ) of the equation ω · ∂ θ X ( ωt ) + i[ H , X ( ωt )] = W ( ωt ) − h W ( ωt ) i . (3.24)In order to define precisely h W ( ωt ) i , consider again the vectors v j and thefrequencies ˜ ν j of Lemma 2.16. First remark that, since ν = P ˜ dj =1 ˜ ν j v j , onehas ν · K = P ˜ dj =1 ( K · v j )˜ ν j , so that, defining˜ K j := K · v j , ˜ K := ( ˜ K , ..., ˜ K ˜ d ) , (3.25)one has H ≡ ν · K = ˜ ν · ˜ K , and furthermore, since v j has integer entries, then the joint spectrum of˜ K ≡ ( ˜ K , ..., ˜ K ˜ d ) is s.t. spec( ˜ K ) ⊂ Z ˜ d + ˜ λ , therefore the map R ˜ d ∋ τ A ( τ ) := e i τ · ˜ K Ae − i τ · ˜ K is periodic in each of the τ j ’s. Define now h A i := 1(2 π ) ˜ d Z T ˜ d e i τ · ˜ K A e − i τ · ˜ K d τ . (3.26) Remark 3.6.
Let A ∈ A m , m ∈ R . Then by Assumption II ′ , h A i ∈ A m and [ ˜ K j , h A i ] = 0 , ≤ j ≤ ˜ d ; [ K , h A i ] = 0 . (3.27) Lemma 3.7.
Let A ∈ C ∞ b ( T n , A m ) , m ∈ R . Provided (2.16) holds, thehomological equation (3.24) has a solution X ∈ C ∞ ( T n , A m ) . Furthermoreif A is symmetric then X is symmetric as well.Proof. For A ∈ C ∞ ( T n , A m ), denote A ♯ ( θ, τ ) := e i τ · ˜ K A ( θ )e − i τ · ˜ K . By As-sumption II ′ , A ♯ ∈ C ∞ ( T n + ˜ d , A m ). Since A ♯ is defined on T n + ˜ d , we canexpand it in Fourier series: A ♯ ( θ, τ ) = X ( k,ℓ ) ∈ Z n + ˜ d ˆ A ♯k,ℓ e i( k · θ + ℓ · τ ) , where ˆ A ♯k,ℓ := 1(2 π ) n + ˜ d Z T n + ˜ d A ♯ ( θ, τ )e − i( k · θ + ℓ · τ ) d θ d τ. Notice that A ( θ ) ≡ A ♯ ( θ,
0) = X ( k,ℓ ) ∈ Z n + ˜ d ˆ A ♯k,ℓ e i k · θ . (3.28)Then, instead of solving directly the homological equation (3.24), we solve ω · ∂ θ X ♯ ( θ, τ )+i[ H , X ♯ ( θ, τ )] = ( W − h W i ) ♯ ( θ, τ ) , ∀ θ ∈ T n , ∀ τ ∈ T ˜ d . (3.29)19learly if we find a smooth solution X ♯ ( θ, τ ) of this equation, then X ( θ ) := X ♯ ( θ,
0) solves the original homological equation (3.24). Now remark thati[ H , X ♯ ( θ, τ )] = ˜ d X j =1 ˜ ν j i[ ˜ K j , X ♯ ( θ, τ )] = ˜ d X j =1 ˜ ν j ddǫ (cid:12)(cid:12)(cid:12)(cid:12) ǫ =0 e i ǫ ˜ K j X ♯ ( θ, τ )e − i ǫ ˜ K j = ˜ d X j =1 ˜ ν j ddǫ (cid:12)(cid:12)(cid:12)(cid:12) ǫ =0 X ♯ ( θ, τ + ǫ e j )= X ( k,ℓ ) ∈ Z n + ˜ d ˆ X ♯k,ℓ ddǫ (cid:12)(cid:12)(cid:12)(cid:12) ǫ =0 ˜ d X j =1 ˜ ν j e i( k · θ + ℓ · ( τ + ǫ e j )) = X ( k,ℓ ) ∈ Z n + ˜ d i˜ ν · ℓ ˆ X ♯k,ℓ e i( k · θ + ℓ · τ ) . Therefore, expanding in Fourier series, equation (3.29) is equivalent toi( ω · k + ˜ ν · ℓ ) ˆ X ♯k,ℓ = \ ( W − h W i ) ♯k,ℓ . Hence define ˆ X ♯k,ℓ = − i \ ( W − h W i ) ♯k,ℓ ( ω · k + ˜ ν · ℓ ) , if ω · k + ˜ ν · ℓ = 0 . (3.30)Since W ♯ is in C ∞ ( T n + ˜ d , A m ) we get that for any j, N ≥ C N,j such that ℘ mj (cid:18) \ ( W − h W i ) ♯k,ℓ (cid:19) ≤ C N,j ( | k | + | ℓ | ) − N . So we get easily that if X is defined by X ( θ ) = X ♯ ( θ,
0) and X ♯ has Fouriercoefficients (3.30) with X ♯k, = 0, then X ∈ C ∞ b ( T n , A m ). We state and prove the iterative Lemma which is the main step for the proofof our main results.
Theorem 3.8.
Assume that the assumptions of Theorem 2.5 or of Theorem2.18 are satisfied.There exist δ > and a sequence { X j ( t ) } j ≥ of self-adjoint (time-dependent)operators in H with X j ∈ C ∞ b ( R , A ρ − ( µ − − ( j − δ ) , such that ∀ j , the inequal-ities (3.14) are satisfied; for any N ≥ the change of variables ψ = e − i X ( t ) . . . e − i X N ( t ) ϕ (3.31)20 ransforms H + V ( t ) into the Hamiltonian H ( N ) ( t ) := H + Z ( N ) ( t ) + V ( N ) ( t ) (3.32) where Z ( N ) ∈ C ∞ b ( R , A ρ ) commutes with K , i.e. [ Z ( N ) , K ] = 0 , while V ( N ) ∈ C ∞ b ( R , A ρ − Nδ ) . Furthermore, under the assumptions of Theorem2.18, one has [ Z ( N ) ; ˜ K j ] = 0 , ∀ j = 1 , ..., ˜ d . (3.33) Proof.
It is proved by recurrence. Consider first the assumptions of Theorem2.5. Using Lemmas 3.1, 3.2, 3.3, 3.5, 3.7 one gets the theorem for N = 1with Z (1) ( t ) := h V ( t ) i ∈ C ∞ b ( R , A ρ ). By Lemma 3.4, [ Z (1) ( t ) , K ] = 0. Inthis case δ can be taken as in (3.9).The iterative step N → N + 1 is proved following the same lines, justadding the remark that e i X N +1 Z ( N ) e − i X N +1 − Z ( N ) ∈ A ρ − ( µ − − Nδ + ρ − ⊂A ρ − ( N +1) δ .Under the assumptions of Theorem 2.18, the result is proved along thesame lines, with δ as in (3.11). The property (3.33) follows by Remark3.6. By Theorem 3.8, the operator H ( t ) is conjugated to H ( N ) ( t ). So we applyTheorem 1.5 of [MR17] to the Schr¨odinger equation for H ( N ) ( t ). Moreprecisely we have[ H ( N ) ( t ) , K ] = [ V ( N ) ( t ) , K ] ∈ C b ( R , A ρ − Nδ )and thus, by choosing N large enough, (3.2) ensures the result for the prop-agator U N ( t, s ) of H ( N ) ( t ).Now since H ( t ) is conjugated to H ( N ) ( t ), H ( t ) generates a propagator U ( t, s ) in the Hilbert space scale H r unitarily equivalent to the propagator U N ( t, s ). Therefore, using also (3.14), the propagator U ( t, s ) fulfills (2.9),thus yielding the result. In this section we prove Corollary 2.7, Corollary 2.11 and Corollary 2.21.
To begin with we show how to put ourselves in the abstract setup. Sofirst we define the operator K . This will be achieved by exploiting thespectral properties of the operator △ g . Applying Theorem 1 of Colin deVerdi`ere [CdV79], there exists a pseudodifferential operator Q of order − △ g , such that Spec[ p △ g + Q ] ⊆ N + λ with some λ ≥ λ >
0. If not, denoting Π − the projector on the non positiveeigenvalues, we replace Q by Q + C Π − with C > − commutes with △ g and is a smoothing operator. So we define K := p △ g + Q , H := K . (4.1)Now remark that H = △ g + 2 Q p △ g + Q , so we have H = △ g + Q where Q is a pseudo-differential operator of order 0 and therefore H ( t ) = △ g + V ( t ) ≡ H + ˜ V ( t ) , ˜ V ( t ) := V ( t ) − Q and we are in the setup of the abstract Schr¨odinger equation (1.1) with thenew perturbation ˜ V ( t ).Remark that H r := Dom(( K ) r ), r ≥
0, coincides with the classicalSobolev space H r ( M ) and one has the equivalence of norms c r k ψ k H r ( M ) ≤ k ψ k r ≤ C r k ψ k H r ( M ) , ∀ r ∈ R . We define the class A m to be the class of pseudodifferential operators whose(real valued) symbols belong to S m cl ( M ). Clearly K ∈ A (recall that Π − isa smoothing operator). It is classical that Assumptions I and II are fulfilled[H¨or85]. Proof of Corollary 2.7.
Assumption A holds true by construction of K ,Assumption B holds with f ( x ) = x and therefore µ := 2. Since V ( t )is a pseudodifferential operator of order ρ < C ∞ b ( R , S ρ cl ( M )), one verifies easily, using pseudodifferential calculus (in par-ticular estimates (2.1)–(2.3)), that ˜ V ( t ) = V ( t ) − Q ∈ C ∞ b ( R , A ρ ). Hencethe corollary follows from Theorem 2.5. We recall that for a symbol a (in the sense of Definition 2.8) we denote by a ( x, D x ) its Weyl quantization (cid:16) a ( x, D x ) ψ (cid:17) ( x ) := 12 π Z Z y,ξ ∈ R e i( x − y ) ξ a (cid:18) x + y , ξ (cid:19) ψ ( y ) d y d ξ . (4.2)We endow S ρ an (defined in Definition 2.8) with the family of seminorms ℘ ρj ( a ) := X | α | + | β |≤ j sup ( x,ξ ) ∈ R (cid:12)(cid:12)(cid:12) ∂ αx ∂ βξ a ( x, ξ ) (cid:12)(cid:12)(cid:12) [ k ( x, ξ )] ρ − kβ + lαk + l , j ∈ N . (4.3)22he operator K is defined using the spectral properties of the Hamiltonian H k,l defined in (2.11) that were studied in detail in [HR82b]; in that paper anaccurate Bohr-Sommerfeld rule for the the eigenvalues of H k,l was obtainedand the existence of a pseudodifferential operator Q of order − H k + l kl k,l + Q ] ⊆ N + λ ( λ ≥
0) was proven. Note that for our numerology H k + l kl k,l is of order 1 by definition. Therefore we define K := H k + l kl k,l + Q , H := K klk + l . We define A m to be the class of pseudodifferential operator with symbolsin S m an . Notice that by construction A m ⊂ L ( H s , H s − m ) for all s ∈ R . It isclassical that A fulfills Assumptions I and II (see [HR82b, HR82a]).On the other hand Assumptions A and B are fulfilled with µ := klk + l > k + l ≥ H k,ℓ = ( K − Q ) klk + l = K klk + l + Q where Q is a pseudodifferential operator of order klk + l −
2. Therefore H ( t ) = H k,l + V ( t ) ≡ H + ˜ V ( t ) , ˜ V ( t ) := V ( t ) + Q and once again we are in the setup of the abstract Schr¨odinger equation(1.1) with the new perturbation ˜ V ( t ). Proof of Corollary 2.11.
Since V ( t ) is a pseudodifferential operator of order ρ < klk + l whose symbol and its time-derivatives have uniformly (in time)bounded seminorms, one verifies that ˜ V ( t ) = V ( t )+ Q ∈ C ∞ b ( R , A ρ ). Hencethe corollary follows from Theorem 2.5. The proof of Corollary 2.21 is along the lines developed in Subsection 4.1.Let us remark that the operator p △ g + µ − p △ g is of order −
1. Hence,defining K as in (4.1), one has again p △ g + µ = K + Q with Q of order −
1. Therefore H ( t ) = p △ g + µ + V ( ωt, x, D x ) = K + ˜ V ( ωt ) Actually [HR82b] proves that Q has a symbol which is quasi-homogeneous of degree − k − l . Here a symbol f ( x, ξ ) is quasi-homogeneous of degree m if f ( λ l x, λ k ξ ) = λ m f ( x, ξ ) , ∀ λ > , ∀ ( x, ξ ) ∈ R \ { } . It is classical [HR82b, HR82a] that if f is quasi-homogeneous of degree m , then it is asymbol in the class S m/ ( k + l )an . V ( ωt ) ∈ C ∞ ( T n , A ρ ).This time we verify Assumptions II ′ , A ′ and B ′ with d = 1 and K = K = H . Concerning the nonresonance condition just remark that in thiscase we have that ν has only one component given by 1.Thus Theorem 2.18 immediately yields Corollary 2.21. A Technical lemmas on classical symbols
We begin with the following lemma whose proof is completely standard (andwe skip it)
Lemma A.1. (i) If f ∈ S a , g ∈ S b then f g ∈ S a + b .(ii) If f ∈ S a , then f ( j ) ∈ S a − j .(iii) If x η ( x ) is a smooth cut-off function on R , then η ∈ S −∞ .(iv) The function f ( x ) = x a , a > , is a classical elliptic symbol in S a + . Lemma A.2. If g ∈ S µ , µ ∈ R , then g ( K ) ∈ A µ .Proof. By definition g ( x ) = P ≤ j ≤ N − c j x µ − j + R ( x ), | R ( x ) | ≤ C N | x µ − N | for | x | ≥
1. Then g ( K ) = P ≤ j ≤ N − c j K µ − j + R ( K ), where R ( K ) isdefined by functional calculus as R ( K ) := R ∞ R ( λ )d E K ( λ ), d E K ( λ ) beingthe spectral resolution of K . By Assumption I, P ≤ j ≤ N − c j K µ − j ∈ A µ while the operator R ( K ) is N -smoothing (in the sense of Definition 2.1).Since N can be taken arbitrarily large, g ( K ) fulfills Assumption I (v),therefore it belongs to A µ . The other properties are easily verified usingsuch decomposition.Finally, we recall a commutator expansion lemma following from [DG97,Lemma C.3.1]: Lemma A.3.
Let f ∈ S ρ + and W ∈ A m . Then for all N ≥ [ ρ ] we have [ f ( K ) , W ] = X ≤ j ≤ N j ! f ( j ) ( K )ad jK W + R N +1 ( f, K , W ) , where R N +1 ( f, K , W ) ∈ A [ ρ ]+ m − N .Moreover if W depends on time t with uniform estimates in A m then it isalso true for R N +1 ( f, K , W ) .Proof. Apply [DG97, Lemma C.3.1] to the bounded operator B = K − m W .24 An abstract proof of Egorov Theorem
In order to check Assumption II, we introduce the following condition
Assumption II-CL:
For every m ∈ R and every A ∈ A m there existsΦ ( t ) ( A ) ∈ C b ( R t , A m ) and R ( A, t ) ∈ C b ( R t , A m − ) such that Φ (0) ( A ) = A and ddt Φ ( t ) ( A ) = i − [Φ ( t ) ( A ) , K ] + R ( A, t ) (B.1)In applications in a pseudodifferential operator setting, we have A = Op ( a ), a is the symbol of A and one can choose Φ ( t ) ( A ) = Op ( a ◦ φ t ) where φ t isthe classical flow of the symbol of K . Then one has to verify that a ◦ φ t belongs to the same symbol class as a (see for example [Tay91]). Theorem B.1 (Abstract Egorov Theorem) . If Assumption I and Assump-tion II-CL are satisfied then Assumption II holds true.Proof.
We follow [Rob87] (p. 202-207). Let U ( t ) = e − i tK . Compute ddτ (cid:16) U ( τ − t )Φ ( τ ) ( A ) U ( t − τ ) (cid:17) = U ( τ − t ) (cid:18) i[Φ ( τ ) ( A ) , K ] + ddτ Φ ( τ ) ( A ) (cid:19) U ( t − τ ) . So using (B.1) and integrate in τ between 0 and t we get U ( − t ) AU ( t ) = Φ ( t ) ( A ) + Z t U ( τ − t ) R ( A, τ ) U ( t − τ )d τ. (B.2)Now we iterate from this formula. In the following step we apply this formulafor every τ to A new = R ( A, τ ). So we get U ( − t ) AU ( t ) = A ( t ) + A ( t )+ Z t Z t − τ U ( τ + τ − t ) R ( R ( A, τ ) , τ − τ ) U ( t − τ − τ )d τ d τ . where A ( t ) = Φ ( t ) ( A ), A ( t ) = R t Φ ( t − τ ) ( R ( A, τ ))d τ ∈ A m − and R ( R ( A, τ ) , τ − τ ) ∈ A m − .At the step N we get easily by induction: U ( − t ) AU ( t ) = A ( t ) + A ( t ) + · · · + A N ( t )+ Z t Z t − τ · · · Z t − τ −···− τ N d τ d τ · · · d τ N U ( τ + τ + · · · + τ N − t ) R ( N ) ( A, τ , τ , · · · , τ N ) U ( t − τ − τ − · · · − τ N ) , where A j ∈ C b ( R , A m − j ) and R ( N ) ( A, τ , · · · , τ N ) ∈ C b ( R N +1 , A m − N − ).Now we remark that the remainder term is as smoothing as we want by tak-ing N large enough, so the algebra being stable by smoothing perturbationswe get Assumption II. 25 Proof of Lemma 2.16
We reproduce here the proof given in the lecture notes by Giorgilli [Gio](in particular the technical results are contained in Appendix A). A generalpresentation containing also the results that we use here can be found in[Sie89].We start by stating without proof a simple Lemma.
Lemma C.1.
Let e , ..., e d and e ′ , ..., e ′ d be two basis of Z d ; then the matrix M = ( M ij ) s.t. e ′ i = P j M ij e j is unimodular with integer entries. Then one has the following corollary.
Corollary C.2.
A collection of vectors e j ∈ Z d , j = 1 , ..., d , is a basis of Z d if and only if the determinant of the matrix having e j as rows is 1. The corollary immediately follows from Lemma C.1 and the remark thatsuch a property holds for the canonical basis of Z d .Define now the resonance modulus M ν of ν by M ν := n k ∈ Z d : ν · k = 0 o . This is a discrete subgroup of R d which satisfiesspan( M ν ) ∩ Z d = M ν . (C.1)Let 0 ≤ r ≤ d − M ν . It is well known that anydiscrete subgroup of R d admits a basis. Let e , ..., e r , be a basis of M ν , andremark that the vectors e j have integer components. Then the followingresult holds . Lemma C.3.
There exist ˜ d := d − r vectors u , ..., u ˜ d with integer entries,such that e , ..., e r , u , ..., u ˜ d form a basis of Z d . Then one obtains immediately the following
Corollary C.4.
Let M be the matrix with rows given by the vectors e j andthe vectors u j ; define ˇ ν := M ν , then one has ˇ ν i = 0 , ∀ i = 1 , ..., r , while ˜ ν i := ˇ ν r + i , i = 1 , ..., ˜ d are independent over the rationals.Proof of Lemma 2.16. Consider the matrix M − : since M is unimodularwith integer entries, the same is true for M − , and one has ν = M − ˇ ν ;however, since the first r components of ˇ ν vanish, such an expression reducesto a linear combination of vectors with integer entries, the coefficients of thecombination being ˜ ν , ..., ˜ ν ˜ d . this can be found as Theorem 31 in [Sie89], or as Lemma A.6 in [Gio] eferences [Arn89] V. Arnold. Mathematical methods of classical mechanics.
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