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Journal of Functional Analysis 271 (2016) 1377–1433

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Journal of Functional Analysis

www.elsevier.com/locate/jfa

Quantum singular complete integrability ✩

Thierry Paul a,∗, Laurent Stolovitch b

a CMLS, Ecole Polytechnique, CNRS, Université Paris-Saclay, 91128 Palaiseau Cedex, Franceb CNRS and Laboratoire J.-A. Dieudonné, Université de Nice, Sophia Antipolis Parc Valrose, 06108 Nice Cedex 02, France

a r t i c l e i n f o a b s t r a c t

Article history:Received 6 October 2015Accepted 28 April 2016Available online 8 June 2016Communicated by B. Schlein

Keywords:Integrable systemsQuantum mechanicsSemiclassical analysisQuantum Birkhoff normal formSmall divisorsK.A.M. method

We consider some perturbations of a family of pairwise com-muting linear quantum Hamiltonians on the torus with pos-sibly dense pure point spectra. We prove that their Rayleigh–Schrödinger perturbation series converge near each unper-turbed eigenvalue under the form of a convergent quantum Birkhoff normal form. Moreover the family is jointly diago-nalized by a common unitary operator explicitly constructed by a Newton type algorithm. This leads to the fact that the spectra of the family remain pure point. The results are uni-form in the Planck constant near � = 0. The unperturbed frequencies satisfy a small divisors condition and we explicitly estimate how this condition can be released when the family tends to the unperturbed one. In the case where the number of operators is equal to the number of degrees of freedom – i.e. full integrability – our construction provides convergent normal forms for general perturbations of linear systems.

© 2016 Elsevier Inc. All rights reserved.

✩ Research of L. Stolovitch was supported by ANR grant “ANR-10-BLAN 0102” for the project DynPDE.* Corresponding author.

E-mail addresses: [email protected] (T. Paul), [email protected](L. Stolovitch).

http://dx.doi.org/10.1016/j.jfa.2016.04.0290022-1236/© 2016 Elsevier Inc. All rights reserved.

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1378 T. Paul, L. Stolovitch / Journal of Functional Analysis 271 (2016) 1377–1433

1. Introduction

Perturbation theory belongs to the history of quantum mechanics, and even to its pre-history, as it was used before the works of Heisenberg and Schrödinger in 1925/1926. The goal at that time was to understand what should be the Bohr–Sommerfeld quan-tum conditions for systems nearly integrable [3], by quantizing the perturbation series provided by celestial mechanics [19]. After (or rather during its establishment) the functional analysis point of view was settled for quantum mechanics, the “modern” perturbation theory took place, mostly by using the Neumann expansion of the per-turbed resolvent, providing efficient and rigorous ways of establishing the validity of the Rayleigh–Schrödinger expansion and leading to great success of this method, in particu-lar the convergence under a simple argument of size of the perturbation in the topology of operators on Hilbert spaces [16], and Borel summability for (some) unbounded per-turbations [13,24]. On the other hand, by relying on the comparison between the size of the perturbation and the distance between consecutive unperturbed eigenvalues, the method has two drawbacks: it remains local in the spectrum in the (usual in dimen-sion larger than one) case of spectra accumulating at infinity and is even inefficient in the case of dense point unperturbed spectra which can be the case in the present article.

In the present article, we consider some commuting families of operators on L2(Td)close to a commuting family of unperturbed Hamiltonians whose spectra are pure point and might be dense for all values of �. As already emphasized, standard (Neumann series expansion) perturbation theory does not apply in this context. Nevertheless, we prove that, under some asumptions, the pure point property is preserved and moreover, we show that the perturbed spectra are analytic functions of the unperturbed ones. All these results are obtained using a method inspired by classical local dynamics, namely the analysis of quantum Birkhoff normal forms (QBNF). Let us first recall some known fact of (classical) Birkhoff normal forms.

In the framework of (classical) local dynamics, Rüssmann proved in [23] (see also [4] and [11]) the remarkable result which says that, when the Birkhoff normal form (BNF), at any order, depends only on the unperturbed Hamiltonian, then it converges provided that the small divisors of the unperturbed Hamiltonian do not accumulate the origin too fast (we refer to [1] for an introduction to this subject). This leads to the integrability of the perturbed system. On the other hand, Vey proved two theorems about the holomorphic normalization of families of l− 1 (resp. l) of commuting germs of holomorphic vector fields, volume preserving (resp. Hamiltonian) in a neighborhood of the origin of Cl (resp. C2l) (and vanishing at the origin) with diagonal and independent 1-jets [28,29].

These results were extended by one of us in [25,26] (see also [27]), in the framework of general local dynamics of a families of 1 � m � l commuting germs of holomorphic vector fields near a fixed point. It is proved that under an assumption on the formal (Poincaré) normal form of the family and under a generalized Brjuno type condition of

T. Paul, L. Stolovitch / Journal of Functional Analysis 271 (2016) 1377–1433 1379

the family of linear parts, there exists an holomorphic transformation of the family to a normal form. This fills up therefore the gap between Rüssman–Brjuno and the complete integrability of Vey. In these directions, we should also mention works by H. Ito [15]and N.-T. Zung [30] in the analytic case and H. Eliasson [9] in the smooth case, and Kuksin–Perelman [17] for a specific infinite dimensional version.

In [12] one of us (the other) gave with S. Graffi a quantum version of the Rüssmann theorem in the framework of perturbation theory of the quantization of linear vector fields on the torus Tl. Moreover, in this setting, it is possible to read on the original perturbation if the “quantum Rüssmann condition” is satisfied and the results are uni-form in the Planck constant belonging to [0, 1]. The method lies in the framework of Lie method perturbation theory initiated in classical mechanics in [8,14] and uses the quantum setting established in [2].

The goal of the present paper is to provide a full spectral resolution for certain fam-ilies of commuting quantum Hamiltonians, not treatable by standard methods due to possible spectral accumulation, through the convergence of quantum normal Birkhoff forms and underlying unitary transformations. These families generalize the quantum version of Rüsmman theorem treated in [12], to the quantum version of “singular com-plete integrability” treated in [25]. The methods use the quantum version of the Lie perturbative algorithm together with a Newton type scheme in order to overcome the difficulty created by small divisors.

Let m � l ∈ N∗. For ω = (ωi)i=1,...,m with ωi = (ωji )j=1,...,l ∈ Rl, let us denote by

Lω = (Lωi)i=1,...,m, the operator valued vector of components

Lωi= −i�ωi.∇x = −i�

l∑j=1

ωji

∂

∂xj, i = 1, . . . ,m

on L2(Tl).We define the operator valued vector H = (Hi)i=1,...,m by

H = Lω + V, (1.1)

where V is a bounded operator valued vector on L2(Tl) whose action is defined after a function

V : (x, ξ, �) ∈ T ∗Tl × [0, 1] �→ V(x, ξ, �) ∈ Rm

by the formula (Weyl quantization)

(V f)(x) =∫

Rl×Rl

V((x + y)/2, ξ, �)eiξ(x−y)

� f(y) dydξ

(2π�)l , (1.2)

where in the integral f(·) and V((x + ·)/2, ξ, �) are extended to Rl by periodicity (see Section 5.1 for details). We make the following assumptions.


1.1. Main assumptions

(A1) The family of frequencies vectors ω fulfills the generalized Brjuno condition

∞∑l=1

logM2k

2k < +∞ where MM := min1�i�m

max0�=|q|�M

|〈ωi, q〉|−1. (1.3)

Here, q = (q1, . . . , ql) ∈ Zl and |q| := |q1| + · · · + |ql|. We will sometimes impose ωto fulfill a strongest collective Diophantine condition: there exist γ > 0, τ � l such that

∀q ∈ Zl, q = 0, min1�i�m

|〈ωi, q〉|−1 � γ|q|τ . (1.4)

Remark: usually, 1MM

is denoted by ωM in the literature [4,25].

(A2) V takes the form, for some V ′ : (Ξ, x, �) ∈ Rm × Tl × [0, 1] �→ V ′(Ξ, x, �) ∈ Rm, analytic in (Ξ, x) and kth times differentiable in �,

V(x, ξ, �) = V ′(ω1.ξ, . . . , ωm.ξ, x, �), (1.5)

(A3) The family H satisfies

[Hi, Hj ] = 0, 1 � i, j � m, 0 � � � 1. (1.6)

Moreover we will suppose that the vectors ωj, j = 1, . . . , m are independent over R and we define

ω :=m∑j=1

|ωj | =m∑j=1

(l∑

i=1(ωi

j)2)1/2

(1.7)

Let us define for ρ > 0, k ∈ {0} ∪N and V ′ : (Ξ, x, �) ∈ Rm×Tl×[0, 1] �→ V ′(Ξ, x, �) ∈Rm

‖V ′‖ρ,ω,k =m∑j=1

k∑r=0

‖∂r�

V ′j‖L1

ρ,ω,r(Rm×Zl)⊗L∞([0,1]) and

‖∇V ′‖ρ,ω,k = maxi=1,...,l

m∑j=1

k∑r=0

‖∂r�∂Ξj

V ′i‖ρ,ω,k,

where · denotes the Fourier transform on S(Rm×Tl) and L1ρ,ω,k(Rm×Zl) is the L1 space

equipped with the weighted norm ∑

q∈Zl

∫Rm |f(p, q)|(1 + |ω · p| + |q|) r

2 eρ(ω|p|+|q|)dp (seeSection 4).


Let us remark that ‖V ′‖ρ,ω,k < ∞ implies that V ′ is analytic in a complex strip x < ρ, ξ < ρω and k-times differentiable in � ∈ [0, 1].

We will denote V ′(Ξ) := 1(2π)l

∫Tl V ′(Ξ, x)dx.

Our assumptions are shown to be non-empty in Remark 5 and the relevance of as-sumption (A2) is discussed in Remark 6, both at the end of Section 2 below.

Our main result reads (see Theorems 29, 30 and 39 for more precise and explicit statements):

Theorem 1. Let k ∈ N ∪ {0} and ρ > 0 be fixed. Let H satisfy the Main Assumption above and ‖V ′‖ρ,ω,k, ‖∇V ′‖ρ,ω,k be small enough.

Then there exists a family of vector-valued functions B�∞(·), ∂j

�B�∞(·) being holomor-

phic in {| zi| < ρ2 , i = 1 . . .m} uniformly with respect to � ∈ [0, 1] and 0 � j � k,

such that the family H is jointly unitary conjugated to B�∞(Lω)1 and therefore the

spectrum of each Hi is pure point and equals the set {(B�∞)i(ω · n), n ∈ Zl} where

ω · n = (〈ωi, n〉)i=1...m.

Note that the use of the generalized Brjuno condition requires the intermediate re-sult Theorem 29 involving an extra condition on ω removed by a scaling argument in Theorem 30, as explained in Section 8.

Our results being uniform in � we get as a partial by-product of the preceding result the following global version of [25]:

Theorem 2. Let ρ > 0 be fixed. Let H be a family of m � l Poisson commuting classical Hamiltonians (Hi)i=1...m on T ∗Tl of the form H = H0 + V, H0(x, ξ) = ω.ξ, ω and Vsatisfying assumption (A1) and V on the form V(x, ξ) = V ′(ω1.ξ, . . . , ωm.ξ, x). Let finally ‖V ′‖ρ,ω, ‖∇V ′‖ρ,ω,0 be small enough (here we consider V ′ as a function constant in �).

Then H is (globally) symplectomorphically and holomorphically conjugated to B0∞(H0).

Once again let us mention that our results are much more explicit, precise and com-plete (in particular concerning radii of convergence and unitary/symplectic conjugations) as expressed in Theorems 29, 30 and 39 and Corollary 35.

Moreover it appears in the proofs that the statement in Theorem 1, as well as in Theorems 29, 30 and 39 and Corollary 35, is valid for fixed value of the Planck constant �under the Main Assumption lowed down by restricting (1.6) to � fixed. More precisely under the Main Assumption with (A3) restricted to, e.g., � = 1, the Theorem 1 is still valid by setting in the statement k = 0 and � = 1. Let us mention also that, as in the original formulations in [23,22,24,25], one easily sees that condition (A2) can be replaced by the fact that the quantum Birkhoff normal form (see section 2 below for the precise definition) at each order is a function of (L1, . . . , Lm) only.

1 Here B�

∞(Lω) is defined through the spectral theorem, that is all the components (B�

∞)i(Lω), i =1, . . . , m are diagonal on the common basis of eigenvectors of the Lωj

, namely {en, Lωjen = 〈ωj , n〉en, n ∈

Zl}, and

[(B�

∞)i(Lω)]en = (B�

∞)i(ω · n)en where ω · n = (〈ωi, n〉)i=1,...,m.


Let us emphasize the two extreme cases, that is m = l and m = 1.

Corollary 1 (Quantum Vey theorem). Assume that the ωj ∈ Rl, j = 1, . . . , l, are inde-pendent over R. Assume that the Hi = Lωi

+Vi, i = 1, . . . , l are pairwise commuting. Let the perturbation Vi be the quantization of any small enough analytic function Vi. Then the family H is jointly unitary conjugated to B�

∞(Lω) as defined in Theorem 1.

We emphasize that this last result do not require neither a small divisors condition nor a condition on the perturbation, see Section 10. This correspond to full quantum integrability. Quantum integrability is a huge subject – see the seminal articles [5,6] to quote only two. The difference that provides our construction is the fact that our results gives convergent result even at � = 1 is the case of perturbations of linear systems.

Corollary 2 (Consolidated Graffi–Paul theorem). Assume that ω ∈ Rl satisfies Brjuno condition (m = 1). Assume that H = Lω + V , where the perturbation V is small enough and V(ξ, x) = V ′(ω.ξ, x). Then H is unitary conjugated to B�

∞(Lω) as defined in Theo-rem 1.

The main difference between this last result and the main result of [12] is the small divisors condition used (a Siegel type condition with constraints).

Let us finally mention as a by-product of our result, a kind of inverse result obtained thanks to the fact that we carefully took care of the precise estimates and constants all a long the proofs. This result is motivated by the remark that, though a small divisors condition is necessary to obtain the perturbed integrability (and Brjuno condition is sufficient), such a condition should disappear when the perturbation vanishes, as the Hamiltonian H0 is always integrable, whatever the frequencies ω are. Our last result quantifies this remark.

Let us define, for ω satisfying (1.4) and α < 2 log 2,

Bα(γ, τ) = 2 log[2τγ( τ

eα)τ]

(note that Bα(γ, τ) → ∞ as γ and/or τ → ∞).The next theorem shows that, in the Diophantine case, the small divisors condition

can be released as Bα(γ, τ) diverges logarithmically as the perturbation vanishes.

Theorem 3. Let k ∈ N ∪ {0} and ρ > 0 be fixed. Let ω and V satisfy (A1) (Diophantine case), (A2) and (A3), and let 0 < ω− � ω � ω+ < ∞ and ‖V ′‖ρ,ω+,k, ‖∇V ′‖ρ,ω+,k be small enough (depending only on k).

Then there exist a constant Cω− such that the conclusions of Theorems 1 hold as soon as, for some α < ρ/2, α < 2 log 2,

Bα(γ, τ) < 13 log

(1

‖V ‖ρ,ω ,k

)+ Cω− .

+


See Corollary 40 for details and the remark after on the case of the Brjuno condition. Let us remark that an equivalent result for Theorem 2 is straightforwardly obtainable.

Let us finish this section by mentioning three comments and remarks concerning our results.

First of all, as mentioned earlier, no hypothesis on the minimal distance between two consecutive unperturbed eigenvalues is required in our article. More, the spectra of our unperturbed operators Lωi

might be dense for all value of � (actually in the Diophantine case for m = 1, l > 1 they are) so the Neumann series expansion is not possible. For m > 1 the non-degeneracy of the unperturbed eigenvalues is not even ensured by the arithmetical property of ω because it relies on the minimum over i � m of the inverse of the small denominators of the vector ωi. In fact, for a resonant ωj the operator Hj

will have an eigenvalue with infinite degeneracy, so the projection of the perturbation Vj

on the corresponding and infinite dimensional eigenspace, which leads to the first order perturbation correction to the unperturbed eigenvalue, might have continuous spectrum. Nevertheless our results show that the perturbed spectra are analytic functions of the spectra of the Lωi

’s.Secondly, because of the fact that non-degeneracy of some of the unperturbed spectra

is not even guaranteed by our assumptions, the standard argument on existence of a common eigenbasis of commuting operators with simple spectra cannot be used here. This existence is a by-product of our results.

Finally let us mention that, as it was the case in [12], though our hypothesis on the perturbations are restrictive, our results, compared with the usual construction of quasi-modes [22,7,18,20,21], have the property of being global in the spectra (full di-agonalization), and exact (no smoothing or O(�∞) remainder), together of course with sharing the property of being uniform in the Planck constant.

Let us point out that this paper has been written in order to be self-contained

1.2. Notations

Function valued vectors in Rn will be denoted in general in calligraphic style, and operator valued vectors by capital letters, e.g. V = (Vl)l=1,...,m or V = (Vl)l=1,...,m.

For i, j ∈ Zn we will denote by ·ij or ·,ij when · has already an index, the matrix element of an (vector) operator in the basis {ej , ej(x) = eij.x/(2π) l

2 , θ ∈ Tl}, namely

Vij = (Vl,ij)l=1,...,m = ((ei, Vlej)L2(Tn))l=1,...,m,

and by V the diagonal part of V :

V ij = Viiδij ,

together with


V = (2π)−l

∫Tl

Vdx.

We will denote by | · | the Euclidean norm on Rm (or Cm), |Z|2 =m∑i=1

|Zi|2, and by

‖ · ‖L2(Tl)→L2(Tl) the operator norm on the Hilbert space L2(Tl).Finally for ω = (ωi)i=1,...,m, ωi ∈ Rl, and ξ ∈ Rl, p ∈ Rm, q ∈ Zl we will denote

ω · ξ = (〈ωi, ξ〉Rl)i=1,...,m ∈ Rm, (1.8)

p.ω =(

m∑i=1

piωji

)j=1,...,l

∈ Rl (1.9)

and

p.ω.q =m∑i=1

l∑j=1

piωji qj = 〈p · ω, q〉Rl . (1.10)

2. Strategy of the proofs

The general idea for proving Theorem 1 will be to construct a Newton-type iteration procedure consisting in constructing a family of unitary operators Ur such that (norms will be defined later)

U−1r (B�

r (Lω) + Vr)Ur = B�

r+1(Lω) + Vr+1, (2.1)

with ‖Vr+1‖r+1 � Dr+1‖Vr‖2r and B�

0 (Lω) = Lω, V0 = V .Ur will be chosen of the form

Ur = eiWr� , Wr self-adjoint. (2.2)

It is easy to realize that (2.2) implies (2.1) if Wr satisfies the (approximate) cohomological equation

1i�

[B�

r (Lω),Wr] + Vr = Dr+1(Lω) + O(‖Vr‖2r), (2.3)

or equivalently

1i�

[B�

r (Lω),Wr] + V cor = Dr+1(Lω) + O(‖Vr‖2

r), (2.4)

for any V cor such that ‖V co

r − Vr‖r = O(‖Vr‖2r).

We will solve for each r the equation (2.4) where V cor will be obtained by a suitable

“cut-off” in order to have to solve (2.4) with only small denominators of finite order (see Brjuno condition (1.3)).


In fact we will see in Section 3 that we can find a (scalar) solution of the (vector) equation (2.4) satisfying

1i�

[B�

r (Lω),Wr] + V cor = B�

r+1(Lω) + Rr, (2.5)

where ‖Rr‖k+1 = O(‖Vr‖2r). To do this we will remark that since the components of

B�r (Lω) + Vr commute with each other (since the ones of Lω + V do) we have that

[(B�

r (Lω))l, (Vr)l′ ] − [(B�

r (Lω))l′ , (Vr)l] = [(Vr)l′ , (Vr)l] = O(V 2r ) (2.6)

which is an almost compatibility condition (see Section 3 for details).Summarizing, the solution Wr of (2.4) will provide a unitary operator Ur such that

(2.1) will hold with B�r+1 = B�

r + Dr+1 and Vr+1 being the sum of three terms:

• V 1r+1 = U−1

r (B�r (Lω) + Vr)Ur − (B�

r (Lω) + Vr) − 1i� [B�

r (Lω), Wr]• V 2

r+1 = Vr − V cor

• V 3r+1 = Rr.

The choice of the family of norms ‖ · ‖r will be made in order to have that

‖Vr+1‖r+1 = ‖V 1r+1 + V 2

r+1 + V 3r+1‖r+1 � Dr+1‖Vr‖2

r

with Dr satisfying

R∏r=1

D2R−r

r � C2R

.

Hence, we will have

‖VR+1‖R+1 � (C‖V0‖0)2R

,

so that ‖VR+1‖R+1 → 0 as R → ∞ if ‖V0 = V ‖0 < C−1 and ‖ · ‖∞ exists.

Remark 4 (Propagation of assumptions (A2)–(A3)). It is clear (and it will be explicit in the body of the proof of the main theorem) that Condition (A2) will be satisfied by the solution of equations (2.3), (2.4) as soon as Vr and V co

r do. This last condition can be easily seen to be propagated from the decomposition Vr+1 = V 1

r+1 + V 2r+1 + V 3

r+1 given

before by considering that Ur = eiWr� by (2.2) and Wr satisfying (A2). (A3) is obviously

propagated by (2.1).

Remark 5 (Non-emptiness of the hypothesis). Consider a family of operators of the form Lω + B�(Lω) for B� : Rm → Rm with ‖B�‖ρ,ω,k < +∞. Then for each bounded self-adjoint operator W whose Weyl symbol W satisfies (A2) and ‖W‖ρ′,ω,k < +∞ for some


ρ′ > ρ, consider the family eiW� (Lω+B�(Lω))e−iW

� := Lω+V := (Hi)i=1,...,m. Obviously the family (Hi)i=1,...,m satisfies (A3). By the same argument as the one in Remark 4 one sees easily that the Weyl symbol V of V satisfies (A2) for some V ′. Finally estimates (5.11)and (5.12) in Proposition 16 below show that the expansion ei

W� (Lω + B�(Lω))e−iW

� =Lω + B�(Lω) + [Lω + B�(Lω), iW

�] + 1

2 [[Lω + B�(Lω), iW�

], iW�

] + . . . is actually conver-gent. This implies that ‖V‖ρ,ω,k is bounded. Therefore the family Lω +V satisfies all the assumptions of Section 1.

Remark 6 (Relevance of assumption (A2)). Let us recall some classical facts from dy-namical systems. Let H0 =

n∑i=1

λi(x2i + y2

i ) be a quadratic Hamiltonian on R2n. Any

analytic higher order perturbation H = H0 + higher order terms is formally conjugate to a formal Birkhoff normal form H(x2

1 + y21 , . . . , x

2n + y2

n). Rüssman–Brjuno’s theorem asserts that, if (∗) H = F (H0) (i.e. H is a function of that peculiar linear combination n∑

i=1λi(x2

i + y2i ) and contains no other terms), for some formal power series F of one

variable and if a “small divisors” condition is satisfied, then the transformation to the Birkhoff normal form is analytic in a neighborhood of the origin. Condition (∗) is known as Brjuno’s condition A (cf. [4]). It is a sharp condition for the analyticity of the trans-formation to Birkhoff normal form in the following sense: if a normal form NF doesn’t satisfy it, then it is possible to perturb H in such way that the analytic perturbation H still has NF as normal form and the transformation from H to NF is a divergent power series. In our quantum version, we only focus on the sufficiency of the analogue condition. The linear combination

∑j ω

jξj in our article plays the rôle of “quantum analogue” of

∑i λi(x2

i + y2i ).

3. The cohomological equation: the formal construction

In this section we want to show how it is possible to construct the solution of the equation

1i�

[B�(Lω),W ] + V = D(Lω) + O(V 2), (3.1)

where we denote by Lω, ω = (ωi)i=1,...,m, ωi ∈ Rl, the operator valued vector of compo-nents (with a slight abuse of notation) Lωi

= −i�ωi.∇x, i = 1, . . . , m on L2(Tl) and Vis a “cut-off”ed.

Vij = 0 for |i− j| > M.

We will present the strategy only in the case of the Brjuno condition, the Diophantine case being very close.

Let us recall also that equation (3.1) is in fact a system of m equations and that it might seem surprising at the first glance that the same W solves (3.1) for all = 1, . . . , m.


3.1. First order

At the first order the cohomological equation is

[Lω�,W ]i�

+ V� = D�(Lω), l = 1, . . . ,m (3.2)

solved on the eigenbasis of any Lω�by D�(Lω) = diag(V�) and

Wij = − (V� −D�)ijiω� · (i− j) . (3.3)

Indeed, since Lωlis selfadjoint, we have

〈ej , [Lωl,W ]ei〉 = 〈ej , Lωl

Wei −WLωlei〉 = 〈Lωl

ej ,Wei〉 − 〈ej ,WLωlei〉

= iωl.(j − i)〈ej ,Wei〉

In (3.3) we will picked up, for every ij such that |i − j| � M , an index = i−j which minimize the quantity

|〈ω�q , q〉|−1 := min1�i�m

|〈ωi, q〉|−1 � MM . (3.4)

We define W by

Wij = −(V�i−j

)ijiω�i−j

· (i− j) , i− j = 0 (3.5)

Since [H�, H�′ ] = 0, then we have that [L�′V�] + [V�′ , L�] = −[V�, V�′ ]. Therefore, evalu-ating the operators on ej and taking the scalar product with ei, leads to

ω�′ · (i− j)(V�)ij = ω� · (i− j)(V�′)ij − ([V�, V�′ ])ij (3.6)

that is

(V�)ijω� · (i− j) = (V�′)ij

ω�′ · (i− j) − ([V�, V�′ ])ijω� · (i− j)ω�′ · (i− j)

(note that when ω�′ · (i − j) = 0 on has (V�′)ij = −([V�i−j,V�′ ])ij

ω�i−j·(i−j) ).

Let us remark that, though [V�, V�′ ] is quadratic in V , it has the same cut-off property as V , namely ([V�, V�′ ])ij = 0 if |i − j| > M as seen clearly by (3.6).

This means that W defined by (3.5) satisfies

[Lω,W ] + V = D(Lω) + V ,
i�


where

(V�)ij =([V�, V�i−j

])iji�ω�i−j

· (i− j) . (3.7)

Note that this construction is different from the one used in [25].

3.2. Higher orders

The cohomological equation at order r will follow the same way, at the exception that Lω has to be replaced by B�

r (Lω).The corresponding cohomological equation is therefore of the form

[B�r (Lω),Wr]

i�+ Vr = O((Vr)2), (3.8)

equivalent to

B�r (�ω · i) − B�

r (�ω · j)i�

(Wr)ij + (Vr)ij = O((Vr)2). (3.9)

Lemma 7. For B�r close enough to the identity there exists a m ×m matrix Ar(i, j) such

that

B�r (�ω · i) − B�

r (�ω · j)i�

= (I + Ar(i, j))ω.(i− j), (3.10)

where I is the m ×m identity matrix and ω.(i − j) = (ωl.(i − j))l=1...m. Moreover

‖Ar(i, j)‖Cm→Cm � ‖∇(B�

r − B�

0 )‖(Cm→Cm)⊗L∞(Rm)

� maxj=1,...,m

m∑i=1

‖∇j(B�

r − B�

0 )i‖L∞(Rm). (3.11)

Proof. We have

B�r (�ω · i) − B�

r (�ω · j)i�

= ω.(i− j) +1∫

0

∂t[(B�

r − B�

0 )(t�ω.i + (1 − t)�ω.j)] dt�

= ω.(i− j) +1∫

0

[∇(B�

r − B�

0 )(t�ω.i + (1 − t)�ω.j)]· [ω.(i− j)]dt

so Ar(i, j) =∫ 10 ∇(B�

r − B�0 )(t�ω.i + (1 − t)�ω.j)dt and the first part of (3.11) follows.

The second part is a standard estimate of the operator norm. �


Plugging (3.10) in (3.9) we get that W must solve

ω.(i− j)Wij = (I + Ar(i, j))−1 [−(Vr)ij + O((Vr)2)], (3.12)

and we are reduced to the first order case with Vr → V r where

V rij := (I + Ar(i, j))−1 (Vr)ij . (3.13)

3.3. Toward estimating

We will first have to estimate V r: this will be done out of its matrix coefficients given by (3.13) by the method developed in Section 5.1. We will estimate (I + Ar(i, j))−1

V r

in section 6 by using the formula (I + Ar(i, j))−1 =∞∑k=0

(−Ar(i, j))k and a bound of

the norm of (−Ar(i, j))kV r of the form |C|k times the norm of V r leading to a bound of (I + Ar(i, j))−1

V r of the form 11−|C| times the norm of V r, by summation of the

geometric series ∞∑k=0

Ck, if |C| < 1.

We will then have to estimate W defined through

Wij = −(V r

�i−j)ij

iω�i−j· (i− j) , i− j = 0 (3.14)

with again (V r�i−j

)ij = 0 for |i − j| > M . We get

|Wij | � MM |(V�i−j)ij |,

and we will get an estimate of W , ‖W‖ � MM‖V r‖, for a norm ‖ · ‖ to be specified later.

Finally we will have to estimate

(V rl )ij =

([V rl , V

r�i−j

])iji�ω�i−j

· (i− j) . (3.15)

We will get immediately ‖V rl ‖ � MM‖P‖, Pij =

([V rl ,V r

�i−j])ij

i� and the estimate of the commutator will be done by the method developed in Section 5.

In the two next sections we will define the norms and the Weyl quantization procedure used in order to precise the results of this section.

4. Norms

Let m, l be positive integers. For F ∈ C∞(Rm×Tl× [0, 1]; C) we will use the following normalization for the Fourier transform.


Definition 8 (Fourier transforms). Let p ∈ Rm and q ∈ Zl

F(p, x, �) = 1(2π)m

∫Rm

F(ξ, x, �)e−i〈p,ξ〉 dξ (4.1)

F(ξ, q; �) = 1(2π)l

∫Tl

F(ξ, x; �)e−i〈q,x〉 dx (4.2)

F(p, q, �) = 1(2π)m+l

∫Rm×Tl

F(ξ, x, �)e−i〈p,ξ〉−i〈q,x〉dξdx (4.3)

= 1(2π)m

∫Rm

F(ξ, q, �)e−i〈p,ξ〉dξ (4.4)

= 1(2π)l

∫Tl

F(p, x, �)e−i〈q,x〉dx (4.5)

Note that

F(ξ, x, �) =∫Rm

F(p, x, �)ei〈p,ξ〉 dp (4.6)

=∑q∈Zl

F(ξ, q; �)ei〈q,x〉 (4.7)

=∑q∈Zl

∫Rm

F(p, q, �)ei〈p,ξ〉+i〈q,x〉 dp (4.8)

Set now for k ∈ N ∪ {0} and p · ω = (∑

j=1,...,mpj .ω

ij)i=1...l:

μk(p, q) := (1 + |p · ω|2 + |q|2) k2 (4.9)

(note that μr(p − p′, q − q′) � 2 k2 μr(p, q)μr(p′, q′) because |x − x′|2 � 2(|x|2 + |x′|2) and

that |p · ω| → ∞ as |p| → ∞ because the vectors (ωi)1=1...l are independent over R).

Definition 9 (Norms I). For ρ > 0, F ∈ C∞(Rm×Tl×[0, 1]; C) we introduce the weighted norms

‖F‖†ρ = ‖F‖†ρ,ω := max�∈[0,1]

∫Rm

∑q∈Zl

| F(p, q, �)| eρ(ω|p|+|q|) dp. (4.10)

‖F‖†ρ,ω,k = ‖F‖†ρ,ω,k := max�∈[0,1]

k∑j=0

∫Rm

∑q∈Zl

μk−j(p, q)∂j�| F(p, q, �)| eρ(ω|p|+|q|) dp.

(4.11)

Note that ω is given by (1.7) and ‖ · ‖†ρ;0 = ‖ · ‖†σ.


Definition 10 (Norms II). Let Oω be the set of functions F : Rl × Tl × [0, 1] → C such that F(ξ, x; �) = F ′(ω · ξ, x, �) for some F ′ : Rm × Tl × [0, 1] → C. Define, for F ∈ Oω:

‖F‖ρ,ω,k := ‖F ′‖†ρ,ω,k. (4.12)

We will also need the following definition for F ∈ Oω:

‖F‖�ρ,ω,k :=k∑

j=0

∫Rm

∑q∈Zl

μk−j(p, q)∂j�| F ′(p, q, �)| eρ(ω|p|+|q|) dp. (4.13)

Let us note that, obviously, ‖ · ‖�ρ,ω,k � ‖ · ‖ρ,ω,k.

We will need an extension of the previous definition to the vector case. Consider now F ∈ C∞(Rm×Tl×[0, 1]; Cm) and G ∈ C∞(Rm×[0, 1]; Cm). The definition of the Fourier transform is defined as usual, component by component.

Definition 11 (Norms III). Let F = (Fi)i=1,...,m ∈ C∞(Rm × Tl × [0, 1]; Cm). We define

(1) ‖F‖†ρ,ω,k =m∑i=1

‖Fi‖†ρ,ω,k (4.14)

(2) Let

Omω =

{F = (Fi)i=1,...,m : Rm × Tl × [0, 1] → Cm/ Fi ∈ Oω, i = 1, . . . ,m

}(4.15)

Let F ∈ Omω . We define:

‖F‖ρ,ω,k =m∑i=1

‖Fi‖ρ,ω,k (4.16)

Let

Om×mω =

{F = (Fij)i,j=1,...,m : Rm × Tl × [0, 1] → Cm/ Fij ∈ Oω, i, j = 1, . . . ,m

}(4.17)

Let F ∈ Om×mω . We define:

‖F‖ρ,ω,k = supi=1,...,m

∑j=1,...,m

‖Fij‖ρ,ω,k. (4.18)


(3) Finally we denote F the Weyl quantization of F recalled in Section 5 and

‖F‖ρ,ω,k = ‖F‖ρ,ω,k (4.19)

J †k (ρ, ω) = {F | ‖F‖†ρ,ω,k < ∞}, (4.20)

J†k(ρ, ω) = {F | F ∈ J †

k (ρ, ω)}, (4.21)

Jk(ρ, ω) = {F ∈ Oω | ‖F‖ρ,ω,k < ∞}, (4.22)

Jk(ρ, ω) = {F | F ∈ Jk(ρ, ω)}. (4.23)

J �

k (ρ, ω) = {F ∈ Oω | ‖F‖�ρ,ω,k < ∞}, (4.24)

J�

k (ρ, ω) = {F | F ∈ J �

k (ρ, ω)}. (4.25)

Jmk (ρ, ω) = {F ∈ Om

ω | ‖F‖ρ,ω,k < ∞}, (4.26)

Jmk (ρ, ω) = {F | F ∈ Jm

k (ρ, ω)}. (4.27)

Jm×mk (ρ, ω) = {F ∈ Om×m

ω | ‖F‖ρ,ω,k < ∞}, (4.28)

Jm×mk (ρ, ω) = {F | F ∈ Jm×m

k (ρ, ω)} (4.29)

and J@(ρ, ω) = J@k=0(ρ, ω), J@(ρ, ω) = J@

k=0(ρ, ω) ∀@ ∈ {†, m, m ×m}.When there will be no confusion we will forget about the subscript ω in the label of

the norms and also denote by J@k (ρ) = J@

k (ρ, ω).

5. Weyl quantization and first estimates

We express the definitions and results of this section in case of scalar operators and symbols. The extension to the vector case is trivial component by component. The reader only interested by explicit expression can skip the beginning of the next paragraph and go directly to Definition 5.4.

5.1. Weyl quantization, matrix elements and first estimates

In this section we recall briefly the definition of the Weyl quantization of T ∗Tl. The reader is referred to [12] for more details (see also e.g. [10]).

Let us recall that the Heisenberg group over T ∗Tl × R, denoted by Hl(Rl × Zl ×R), is (the subgroup of the standard Heisenberg group Hl(Rl × Rl × R)) topologically equivalent to Rl × Zl × R with group law (u, t) · (v, s) = (u + v, t + s + 1

2Ω(u, v)). Here u := (p, q), p ∈ Rl, q ∈ Zl, t ∈ R and Ω(u, v) is the canonical 2-form on Rl × Zl: Ω(u, v) := 〈u1, v2〉 − 〈v1, u2〉.

The unitary representations of Hl(Rl ×Zl ×R) in L2(Tl) are defined for any � = 0 as follows

(U�(p, q, t)f)(x) := ei�t+i〈q,x〉+�〈p.q〉/2f(x + �p) (5.1)


Consider now a family of smooth phase–space functions indexed by �, A(ξ, x, �) :Rl × Tl × [0, 1] → C, written under its Fourier representation

A(ξ, x, �) =∫Rl

∑q∈Zl

A(p, q; �)ei(〈p.ξ〉+〈q,x〉) dp (5.2)

Definition 12 (Weyl quantization I). By analogy with the usual Weyl quantization on T ∗Rl[10], the (Weyl) quantization of A is the operator A(�) defined as

A(�) := (2π)l∫Rl

∑q∈Zl

A(p, q; �)U�(p, q, 0) dp (5.3)

(note that the factor (2π)l in (5.3) is due to the (convenient for us) normalization of the Fourier transform in Definition 8).

It is a straightforward computation to show that, considering f ∈ L2(Tl) andV((x + ·)/2) as periodic functions on Rl, we get the equivalent definition

Definition 13 (Weyl quantization II).

(A(�)f)(x) :=∫

Rlξ×Rl

y

A((x + y)/2, ξ, �)eiξ(x−y)

� f(y) dξdy

(2π�)l (5.4)

Remark 14. The expression (13) is exactly the same as the definition of Weyl quantization on T ∗Rl except the fact that f is periodic. Note that A(�)f is periodic thanks to the fact that A(x, ξ, �) is periodic:∫

A((x + 2π + y)/2, ξ)eiξ(x+2π−y)

� f(y)dξdy�l

=∫

A((x + 2π + y + 2π)/2, ξ)eiξ(x−y)

� f(y + 2π)dξdy�l

=∫

A((x + y)/2 + 2π, ξ)eiξ(x−y)

� f(y)dξdy�l

=∫

A((x + y)/2, ξ)eiξ(x−y)

� f(y)dξdy�l

= (A(�))f(x).

The first results concerning this definition are contained in the following proposition.

Proposition 15. Let A(�) be defined by the expression (5.4). Then:

(1) ∀ρ > 0, ∀ k � 0 we have:

‖A(�)‖B(L2(Tl)) � ‖A‖ρ,k (5.5)


and, if A(ξ, x, �) = A′(ω · ξ, x; �)

‖A(�)‖B(L2(Tl)) � ‖A′‖ρ,k. (5.6)

(2) Let, for n ∈ Zl, en(x) = einx

(2π)l . Then for all m, n in Zl,

〈em, A(�)en〉L2(Tl) = A((m + n)�/2,m− n, �) (5.7)

(3) Reciprocally, let A(�) be an operator whose matrix elements satisfy (5.7) for some A belonging to J@, @ ∈ {†, m, m ×m}. Then A(�) is the Weyl quantization of A.

Proof. (5.7) is obtained by a simple computation. It also implies that

‖A(�)em‖2L2(Rl) =

∑q∈Zl

|A(�(m + q)/2,m− q, �)|2 � supξ∈Rl

∑q∈Zl

|A(ξ, q, �)|2.

So that

‖A(�)∑Zl

cmem‖2L2(Rl) �

∑Zl

|cm|2 supξ∈Rl

∑q∈Zl

|A(ξ, q, �)|2

� (∑Zl

|cm|2)

⎛⎝∑q∈Zl

supξ∈Rl

|A(ξ, q, �)|

⎞⎠2

.

And therefore, since by (4.6)–(4.7)–(4.8) A(ξ, q, �) =∫Rl

A(p, q, �)ei<ξ,p>dp so that

|A(ξ, q, �)| �∫Rl | A(p, q, �)|dp,

‖A(�)‖B(L2(Tl)) �∑q∈Zl

supξ∈Rl

|A(ξ, q, �)| �∫Rl

∑q∈Zl

| A(p, q, �)|dp � ‖A‖ρ,k, ∀ρ > 0, k � 0.

(5.8)

In the case A(ξ, x, �) = A′(ω · ξ, x; �) we get, ∀ρ > 0, k � 0:

‖A(�)‖B(L2(Tl)) �∑q∈Zl

supξ∈Rl

|A(ξ, q, �)| =∑q∈Zl

supY ∈Rm

|A′(Y, q, �)|

�∫Rm

∑q∈Zl

|A′(p, q, �)|dp � ‖A′‖ρ,k.

(3) is obvious. �


5.2. Fundamental estimates

This section contains the fundamental estimates which will be the blocks of the esti-mates needed in the proofs of our main results. These primary estimates are contained in the following proposition. We shall omit to write the subscript ω in the norms.

Proposition 16. We have:

(1) For F, G ∈ J1k (ρ), FG ∈ J1

k (ρ) and fulfills the estimate

‖FG‖ρ,k � (k + 1)8k‖F‖ρ,k · ‖G‖ρ,k (5.9)

(2) There exists a positive constant C ′ such that for F ∈ Jmk (ρ) and for G ∈ J1

k (ρ), we have, ∀δ1 > 0, δ � 0, ρ > δ + δ1,∥∥∥∥ [F,G]

i�

∥∥∥∥ρ−δ−δ1,k

� 2(k + 1)8k

e2δ1(δ + δ1)‖F‖ρ,k‖G‖ρ−δ,k, (5.10)

1d!‖[G, . . . [G︸︷︷︸

d times

, F ] · · · ]/(i�)d‖ρ−δ,k � 12π

(2(1 + k)8k

δ2

)d

‖F‖ρ,k‖G‖dρ,k, (5.11)

and ∥∥∥∥ [Lω, G]i�

∥∥∥∥ρ−δ,k

� ω

eδ‖G‖ρ,k (5.12)

(3) For F , G ∈ J 1k (ρ), FG ∈ J 1

k (ρ) and

‖FG‖ρ,k � (k + 1)4k‖F‖ρ,k · ‖G‖ρ,k. (5.13)

(4) Let V = (Vl)l=1...m ∈ Jmk (ρ) and let W be defined by 〈em, Wen〉 =

〈em,Vlm−nen〉

ω�m−n·(m−n)

where, ∀m, n ∈ Z, the index lm−n is such that |〈ω�m−n, m −n〉|−1 := min

1�i�m|〈ωi, m −

n〉|−1. Then

‖W‖ρ−d,k � γτ τ

(ed)τ ‖V ‖ρ,k (5.14)

in the Diophantine case and (obviously) when |〈em, Vlm−nen〉| = 0 for |m −n| > M ,

‖W‖ρ,k � MM‖V ‖ρ,k (5.15)

in the case of the Brjuno condition (MM defined by (1.3)).


(5) Let finally V = (Vl)l=1,...,m ∈ Jmk (ρ) and let P be defined by (Pl)ij = ([Vl,V�i−j

])iji�

for any choice of (i, j) → i−j. Then P = (Pl)l=1,...,m ∈ Jmk (ρ − δ), ∀δ1 � 0, δ > 0,

ρ > δ + δ1 and

‖P‖ρ−δ−δ1,k � 2(k + 1)8k

e2δ1(δ + δ1)‖V ‖ρ,k‖V ‖ρ−δ,k (5.16)

(6) Moreover let F : ξ ∈ Rm �→ F(ξ) ∈ Rm be in Jmk (ρ). Let us define ∇F the matrix

((∇F)ij)i,j=1,...,m with

(∇F)ij := ∂ξiFj . (5.17)

Then, for all δ > 0, ∇F ∈ Jm×mk (ρ − δ) and

‖∇F‖ρ−δ,k � 1eδ

‖F‖ρ,k. (5.18)

Let us remark that, as the proof will show, Proposition 16 remains valid when the norm ‖ · ‖ρ,k is replaced by the norm ‖ · ‖�ρ,k.

Proof. Items (1) and (2) are simple extension to the multidimensional case of the cor-responding results for m = 1 proven in [12]. For sake of completeness we give here an alternative proof in the case m = 1. The proof will use the three elementary inequalities,

μk(p + p′, q + q′) � 2 k2 μk(p, q)μk(p′, q′) (5.19)

|(p.ω.q′ − p′ω.q)/2|k � μk(p, q)μk(p′, q′) (5.20)∣∣∣∣∂k�

sin x�

�

∣∣∣∣ � |x|k+1 (5.21)

|p · ω · q| � ω maxj=1,...,m

|pj ||q| � ω|p||q| (5.22)

where we have used the notation (1.10) and the definition (1.7).(In order to prove (5.19), (5.20), (5.21) and (5.22) just use |X + X ′|2 � 2(|X|2 +

|X ′|2) for all X, X ′ ∈ R2l, |(p.ω.q′ − p′ω.q)/2|2 � (|p · ω|2 + |q|2)(|p · ω′|2 + |q′|2), sin x�

�=

x∫0

cos (s�)ds and |p ·ω · q| �m∑j=1

|pj || l∑

i=1ωijqi| =

m∑j=1

|pj ||ωj · q| �m∑j=1

|pj ||ωj ||q| by

Cauchy–Schwarz, respectively.)We start with (5.9). Since F, G ∈ J1(ρ) we know that there exist two functions F ′, G′

such that the symbols of F, G are F(ξ, x) = F ′(ω.ξ, x), G(ξ, x) = G′(ω.ξ, x). By (5.7) we have that


(FG)mn =∑q′∈Zl

Fmq′Gq′n

=∑q′∈Zl

F(m + q′

2 �,m− q′)G(q′ + n

2 �, q′ − n

)

=∑q′∈Zl

F(m + n + q′

2 �,m− n− q′)G(q′ + 2n

2 �, q′)

=∑q′∈Zl

F ′(ω · m + n + q′

2 �,m− n− q′)G′

(ω · q

′ + 2n2 �, q′

)

=∑q′∈Zl

F ′(ω · m + n + q′

2 �,m− n− q′)G′

(ω · q

′ + m + n− (m− n)2 �, q′

).

(5.23)

Calling P the symbol of FG we have that, by (5.7) again, (FG)mn = P(ξ, q) with ξ = m+n

2 � and q = m − n. Therefore

P(ξ, q) =∑q′∈Zl

F ′(ω.ξ + ω · q

′

2 �, q − q′)G′

(ω.ξ + ω · q

′ − q

2 �, q′), (5.24)

so we see that P(ξ, ·) depends only on ω.ξ: P(ξ, x) = P ′(ω.ξ, x). Moreover, since by (4.3)P ′(p, ·) = 1(2π)m

∫Rm P ′(Ξ, ·)e−i〈Ξ,p〉dΞ we get easily by simple changes of integration

variables and the fact that the Fourier transform of a product is a convolution,

P ′(p, q) =∫Rm

∑q′∈Zl

(F ′(p− p′, q − q′)ei �

2 (p−p′).ω.q′)( G′(p′, q′)ei �

2 p′.ω.(q′−q)

)dp′.

(5.25)

Therefore ‖FG‖ρ,k is equal to the maximum over � ∈ [0, 1] of

k∑γ=0

∫R2m

∑(q,q′)∈Z2l

μk−γ(p, q)∣∣∣∣∂γ

�

[ F ′(p− p′, q − q′)ei�

2 ((p−p′).ω.q′−p′.ω.(q−q′)) G′(p′, q′)]∣∣∣∣

× eρ(ω|p|+|q|)dpdp′. (5.26)

Writing, by (5.20), that∣∣∣∣∂γ�

[ F ′(p− p′, q − q′)ei�

2 ((p−p′).ω.q′−p′.ω.(q−q′)) G′(p′, q′)]∣∣∣∣

�γ∑(

γ

μ

) γ−μ∑(γ − μ

ν

)|∂γ−μ−ν

�

F ′(p− p′, q − q′)||∂ν�e

i�

2 ((p−p′).ω.q′−p′.ω.(q−q′))|
μ=0 ν=0


× |∂μ�

G′(p′, q′)|

�γ∑

μ=0

(γ

μ

) γ−μ∑ν=0

(γ − μ

ν

)|∂γ−μ−ν

�

F ′(p− p′, q − q′)||((p− p′).ω.q′ − p′.ω.(q − q′))/2|ν

× |∂μ�

G′(p′, q′)|

=: P(F ′,G′) (5.27)

�γ∑

μ=0

(γ

μ

) γ−μ∑ν=0

(γ − μ

ν

)μν(p− p′, q − q′)μν(p′, q′)|∂γ−μ−ν

�

F ′(p− p′, q − q′)|

× |∂μ�

G′(p′, q′)|

(changing μ → γ′, ν → ν′ := γ − γ′ − ν)

�γ∑

γ′=0

(γ

γ′

) γ−γ′∑ν′=0

(γ − γ′

ν

)μγ−γ′−ν′(p− p′, q − q′)μγ−γ′−ν′(p′, q′)|∂ν′

�

F ′(p− p′, q − q′)|

× |∂γ′

�

G′(p′, q′)|

(since(m

n

)� 2m, γ � k, γ − γ′ � k)

�k∑

γ′=02k

k∑ν′=0

2kμγ−γ′−ν′(p− p′, q − q′)μγ−γ′−ν′(p′, q′)|∂ν′

�

F ′(p− p′, q − q′)|

× |∂γ′

�

G′(p′, q′)| (5.28)

using (5.19) under the form

μk(p, q) � 2 k2 μk(p− p′, q − q′)μk(p′, q′)

together with the fact that μk(p, q) is increasing in k and μkμk′ = μk+k′ .We find that

muk−γ(p, q)P(F ′,G′)

� 4k2 k2

k∑γ′=0

k∑ν′=0

μk−γ+γ−γ′−ν′(p− p′, q − q′)μk−γ+γ−γ′−ν′(p′, q′)|∂ν′

�

F ′(p− p′, q − q′)|

× |∂γ′

�

G′(p′, q′)|

(replacing 2 k2 by 2k to avoid heavy notations and since k − γ′ − ν′ � k − γ′, k − ν′)

� 8kk∑′

k∑′

μk−ν′(p− p′, q − q′)μk−γ′(p′, q′)|∂ν′

�

F ′(p− p′, q − q′)||∂γ′

�

G′(p′, q′)|. (5.29)
γ =0 ν =0


Note that γ disappeared from (5.29) so the first sum in (5.26) gives a factor (1 + k). We get that (1 + k)−18−k‖FG‖ρ,k is bounded by the maximum over � ∈ [0, 1] (note the change ν′ → γ)

∑q,q′∈Zl

∫R2m

k∑γ,γ′=0

μk−γ(p, q)|∂γ�

F ′(p, q)|μk−γ′(p′, q′)|∂γ′

�

G′(p′, q′)|

× eρ(ω|p|+ω|p′|+|q|+|q′|)dpdp′ (5.30)

which is equal to

‖F ′‖ρ,k‖G′‖ρ,k.

The proof of (5.10) follows the same lines, except that it is easy to see that, in (5.25), ei

�

2 ((p−p′).ω.q′−p′.ω.(q−q′)) has to be replaced by 2 sin(�

2 ((p− p′).ω.q′ − p′.ω.(q − q′))),

since (5.23) becomes([F,G]i�

)mn

=∑q′∈Zl

Fmq′Gq′n −Gmq′Fq′n

i�

= 1i�

∑q′∈Zl

[F(m + n + q′

2 �,m− n− q′)G(m + n + q′ − (m− n)

2 �, q′)

− G(m + n + q′

2 �,m− n− q′)G(m + n + q′ − (m− n)

2 �, q′)]

(5.31)

It generates in (5.26) the replacement of |((p −p′).ω.q′−p′.ω.(q−q′))/2|ν by the term

2|((p− p′).ω.q′ − p′.ω.(q − q′))/2|ν+1 � μν(p− p′, q − q′)μν(p′, q′)(|p · ω · q′ − p′ · ω · q|)

thanks to (5.20), and we get by a discussion verbatim to the same than the one contained in equations (5.27)–(5.30) that

‖[F,G]/i�‖ρ−δ−δ1,k � (1 + k)8k∑

q,q′∈Zl

∫R2m

k∑γ,γ′=0

μk−γ(p, q)μk−γ′(p′, q′)Qdpdp′, (5.32)

where thanks to (5.22),

Q = |∂γ�

F ′(p, q)|(|p · ω · q′| + |p′ · ωq|)|∂γ′

�

G′(p′, q′)|e(ρ−δ−δ1)(ω|p|+|q|+ω|p′|+|q′|)

�[|∂γ

�

F ′(p, q)||∂γ′

�

G′(p′, q′)|eρ(ω|p|+|q|)+(ρ−δ)(ω|p′|+|q′|)]×


× (ω|p||q′| + ω|p′||q|)e−(δ+δ1)(ω|p|+|q|)−δ1(ω|p′|+|q′|))

� 2e2δ1(δ + δ1)

|∂γ�

F ′(p, q)||∂γ′

�

G′(p′, q′)|eρ(ω|p|+|q|)+(ρ−δ)(ω|p′|+|q′|) (5.33)

because (e−x � 1, x � 0 and)

supx∈R+

xe−αx = 1eα

. (5.34)

(5.10) follows immediately from (5.32).The proof of (5.12) follows also the same line and is obtained thanks to the remark

(5.34): indeed since

([Lω,W ]

i�

)mn

= −iω · (m− n)Wmn,

we see, again by (5.7), that the symbol Q(ξ, x) of [Lω, W ]/i� is given through the formula

Q(ξ, q) = ([Lω,W ]/i�)mn for ξ = m + n

2 � and q = m− n.

Therefore Q(ξ, q) = (−iω · q)W(ξ, q), so Q(ξ, x) = Q′(ω.ξ, x) with

Q′(ω.ξ, q) = −iω.q W ′(ω.ξ, q).

We get immediately

Q′(p, q)e(ρ−δ)(ω|p|+|q|) � ω

eδ

W ′(p, q)eρ(ω|p|+|q|)

and (5.12) follows.(5.11) is easily obtained by iteration of (5.10) and the Stirling formula: consider the

finite sequence of numbers δs = d−sd δ. We have δ0 = δ, δd = 0 and δs−1 − δs = δ

d . Let us define G0 := F and Gs+1 := 1

i� [G, Gs], for 0 � s � d − 1. According to (5.10), we have

‖Gs‖ρ−δd−s,k � ck

e2δd−s( δd )

‖G‖ρ,k‖Gs−1‖ρ−δd−s+1,k,

where

ck := 2(k + 1)8k.

Hence, by induction, we obtain


1d!‖Gd‖ρ−δ0,k � cd−1

k

d!e2(d−1)δ0 · · · δd−2( δd )d−1 ‖G‖d−1

ρ,k ‖G1‖ρ−δd−1,k (5.35)

� cdkd!e2dδ0 · · · δd−1δd−1( δ

d)d−1 ‖G‖dρ,k‖F‖ρ,k

� cdkd!e2dd!( δ

d)dδd−1( δd )d−1 ‖G‖dρ,k‖F‖ρ,k

� 12π

(ckd

2

e2δ2

)d 1(d− 1)!d!‖G‖dρ,k‖F‖ρ,k

= 12π

(ckδ2

)d(√

2πddde−d

d!

)2

‖G‖dρ,k‖F‖ρ,k

� 12π

(ckδ2

)d

‖G‖dρ‖F‖ρ,k

since d!√2πde−ddd

� 1. This well know inequality can be seen from Binet’s second expres-sion for the log Γ(z) [31, p. 251]:

log(

n!(ne

)n √2πn

)= 2

∞∫0

arctan(t/n)e2πt − 1 dt � 0.

Finally (5.13) is obtained by noticing that ‖FG‖ρ,k has the same expression as ‖FG‖ρ,k after removing the term ei

�

2 (p.ω.q′−p′.ω.q) in (5.25).To prove (4) it is enough to notice that by (5.7) the symbol of W satisfies W (ξ, q, �) =

V�q (ξ,q,�)ω�q ·q

, so that W (p, q, �) =V 1�q (p,q,�)

ω�q ·qand therefore, for all r ∈ N,

|∂r�

W (p, q, �)| � γ|q|τ sup

l=1,...,m|∂r

�

V l(p, q, �)| � γ|q|τ

m∑l=1

|∂r�

V l(p, q, �)|

out of which we deduce (5.14) by standard arguments (xτe−δx � ( τeδ )τ , x > 0) in the

Diophantine case, and

|∂r�

W (p, q, �)| � MM sup

l=1,...,m|∂r

�

V l(p, q, �)| � MM

m∑l=1

|∂r�

V l(p, q, �)|

from which (5.15) follows.

To prove (5.18) we just notice that ∂ξiFj(p, q, �) = piFj(p, q, �). So

|∂ξiFj(p, q, �)| � |piFj(p, q, �)| � |Fj(p, q, �)||p|.

Therefore |∂r�

∂ξiFj(p, q, �)|e(ρ−δ)|p| � 1 |∂r

�

∂ξiFj(p, q, �)|eρ|p| and (5.18) follows.
eδ


(5) is an easy extension of (5.10). Indeed we find immediately, by (5.7) and the fact that li−j depends only on i − j, that the Fourier transform of the symbol of Pl is P(p, q, �) = X lq (p, q, �) where X lq is the Fourier transform of the symbol of the operator Xlq = [Vl,Vlq ]

i� . Therefore |∂r�

X lq (p, q, �)| � maxl=1,...,m

|∂r�

X l(p, q, �)|, ∀r � 0, q ∈ Zl. So

‖Pl‖ρ−δ,k � maxl′=1,...,m

‖[Vl, Vl′ ]/i�‖ρ−δ,k and

‖P‖ρ−δ,k � maxl,l′=1,...,m

‖[Vl, Vl′ ]/i�‖ρ−δ,k �m∑

l,l′=1

‖[Vl, Vl′ ]/i�‖ρ−δ,k

and we conclude by using (5.10). �6. Fundamental iterative estimates: Brjuno condition case

In all this section the norm subscripts ω and k are omitted.Let us recall from Sections 2 and 3 that we want to find Wr such that

eiWr� (Hr + Vr)e−iWr

� = Hr+1 + Vr+1 (6.1)

where Hr+1 = Hr + hr+1 and Hr = Br(Lω), hr+1 = Vr = Dr(Lω) and, for 0 < δ <

ρ < ∞,

‖hr+1‖ρ = ‖Vr‖ρ � ‖Vr‖ρ, ‖Vr+1‖ρ−δ � Dr‖Vr‖2ρ, (6.2)

and that we look for Wr solving:

1i�

[Hr,Wr] + V co,r = V co,r + V r (6.3)

with

V co,r = Vr − V Mr .

V Mr is given by

V Mrij = (Vr)ij if |i− j| > Mr, V Mr

ij = 0 otherwise. (6.4)

(Note that V co,r = V r.)V r = (V r

l )l=1,...,m is given by

(V rl )ij =

([V rl , V

rl(i−j)])ij

i�ωl(i−j) · (i− j) , V rij := (I + Ar(i, j))−1

V coij , V co,r = Vr − V Mr ,

(6.5)


where Ar(i, j) is the matrix given by Lemma 7, that is:

Br(�ω · i) − Br(�ω · j)i�

= (I + Ar(i, j))ω.(i− j).

Let

Zk = 2(k + 1)8k. (6.6)

Let us denote adW the operator H �→ [W, H]. The l.h.s. of (6.1) is then:

Hr + V r + 1i�

[Hr,Wr] +∞∑j=1

1(−i�)jj! adj

Wr(Vr) +

∞∑j=2

1(−i�)jj! adj

Wr(Hr)

that is

Hr + V co,r + V r + V r − V co,r +∞∑j=1

1(−i�)jj! adj

Wr(Vr) +

∞∑j=2

1(−i�)jj! adj

Wr(Hr)

or

Hr + hr+1 + (Vr − V co,r) + Vr + R1 + R2 (6.7)

Let us set

Vr+1 := (Vr − V co,r) + V r + R1 + R2. (6.8)

We want to estimate Vr+1. We first prove the following proposition.

Proposition 17. Let W be in Jk(ρ) and 0 < δ < ρ. Then

‖[Hr,W ]/i�‖ρ−δ � 1eδ

(ω + Zk‖∇(B�

r − B�

0 )‖ρ)‖W‖ρ (6.9)

and for d � 2,

1d!‖[Hr,W ], . . .︸︷︷︸

d times

]/(i�)d‖ρ−δ � δω

2πZk(1 + Zk‖∇(B�

r − B�

0 )‖ρ)(Zk

δ2r

)d

‖W‖dρ (6.10)

Let now Wr be the (scalar) solution of (6.3). Then, we have

‖Wr‖ρ � MMr

1 − Zk‖D(B�r − B�

0 )‖ρ‖Vr‖ρ, (6.11)

and therefore for d � 2,


1d!‖[Hr,Wr], . . .︸︷︷︸

d times

]/(i�)d‖ρ−δ

� ω1 + Zk‖D(B�

r − B�0 )‖ρ

2πZk/δ

(ZkMMr

/δ2r

1 − Zk‖D(B�r − B�

0 )‖ρ‖Vr‖ρ

)d

(6.12)

(MM is defined in (1.3) and ‖DB‖ρ stands for maxi=1...m

∑j=1...m

‖∇iBj‖ρ).

Proof. We first prove (6.9). Note that the proof is somehow close to the proof of Propo-sition 16, items (1) and (2).

Since B0(Lω) = Lω, (5.12) reads

‖[H0,Wr]/i�‖ρ−δ � ω

eδ‖Wr‖ρ. (6.13)

Note that ([Hr −H0, Wr]/�)ij = Gr(ω.i�)−Gr(ω.j�)�

Wij where Gr(Y ) = B�r (Y ) − Y, Y ∈

Rm (note that Gr has an explicit dependence in � that we omit to avoid heaviness of notations). Indeed, since each Lωi

is self-adjoint on L2(Tl), B�r (Lω) can be defined by

the spectral theorem. Hence, we have

[B�

r (Lω),W ]ij = (ei, [B�

r (Lω),W ]ej) = (ei,B�

r (Lω)Wej −WB�

r (Lω)ei)

= (B�

r (ω.i�) − B�

r (ω.j�))(ei,Wej).

Using (5.13) we get that

‖[Hr −H0,Wr]/i�‖ρ−δ � ‖Xr‖ρ−δ, (6.14)

where Xr is defined through (Xr)ij = Gr(ω.i�)−Gr(ω.j�)�

(Wr)ij .In order to estimate the norm of Xr we need to express its symbol Xr. This is done

thanks to formula (5.7) and the fact that we know the matrix elements of Xr.Expressing (Xr)ij as a function of ((i + j)�/2, i − j) through i, j = i+j

2 ± i−j2 and

using (5.7) we get that

Xr(ξ, q, �) = Gr(ω.ξ + ω.q�/2) − Gr(ω.ξ − ω.q�/2)�

W ′r(ω.ξ, q, �) := X ′

r(ω.ξ, q, �),

so that, using (remember that we denote p.ω.q =∑

j=1...m

∑i=1...l

pjωijqi)

∫Rm

(Gr(Ξ + ω.q�/2) − Gr(Ξ − ω.q�/2))e−i〈Ξ,p〉dp = 2 sin [p.ω.q�/2]∫Rm

Gr(Ξ)e−i〈Ξ,p〉dp,

X ′r(p, q, �) =

∫Rm

Gri (p− p′) sin [(p− p′).ω.q�/2]

�W ′

r(p′, q, �)dp.


Therefore ‖Xr‖ρ−δ is equal to

m∑i=1

∑q∈Zl

∫R2m

dpdp′|2k∑

γ=1μk−γ(p, q)∂γ

�

[Gri (p− p′) sin [(p− p′).ω.q�/2]

�Wr(p′, q, �)

]

× |e(ρ−δ)(ω|p|+|q|)

�m∑i=1

∑q∈Zl

∫ k∑γ=1

μk−γ(p, q)γ∑

μ=1

γ−μ∑ν=1

(γ

μ

)(γ − μ

ν

)×

× 2|∂γ−μ−ν�

Gri (p− p′)||∂ν

�

sin ((p− p′).ω.q�/2)�

||∂μ�

Wr(p′, q, �)|e(ρ−δ)(ω|p|+|q|)dpdp′

(6.15)

Using now the inequalities (5.21) and (5.22), we get,

∣∣∣∣∣k∑

γ=1μk−γ(p, q)

γ∑μ=1

γ−μ∑ν=1

(γ

μ

)(γ − μ

ν

)∂γ−μ−ν�

Gri (p− p′)∂ν

�

× sin ((p− p′).ω.q�)�

∂μ�

Wr(p′, q, �)∣∣∣∣

� ω maxj=1,...,m

|pj − p′j ||q|k∑

γ=1μk−γ(p, q)

×γ∑

μ=1

γ−μ∑ν=1

(γ

μ

)(γ − μ

ν

)|∂γ−μ−ν

�Gri (p− p′)||(p− p′).ω.q|ν |∂μ

�

Wr(p′, q, �)|.

Therefore we notice (after changing q ↔ q′) that ‖Xr‖ρ−δ is bounded by the maximum over � ∈ [0, 1] of

k∑γ=0

∫R2m

∑(q,q′)∈Z2l

μk−γ(p, q)ω maxj=1...m

|pj − p′j |P(Gri ,Wr)e(ρ−δ)(ω|p|+|q|)dpdp′ (6.17)

where P is defined in (5.27) and

Gri (Ξ, x) = Gr

i (Ξ) so that Gri (p, q) = Gr

i (p)δq=0.

Therefore we can verbatim use the argument contained between formulas (5.27)–(5.29)and we arrive, in analogy with (5.30), to the fact that (1 + k)−18−k‖Xr‖ρ−δ is bounded by the maximum over � ∈ [0, 1] of


∑q,q′∈Zl

∫R2m

ω maxj=1,...,m

|pj ||q′|

×k∑

γ,γ′=0μk−γ(p, q)|∂γ

�

Gri (p, q)|μk−γ′(p′, q′)|∂γ′

�

Wr(p′, q′)|e(ρ−δ)(ω|p|+ω|p′|+|q|+|q′|)dpdp′

=∑q′∈Zl

∫R2m

ω maxj=1,...,m

|pj ||q′|

×k∑

γ,γ′=0μk−γ(p, 0)|∂γ

�Gri (p)|μk−γ′(p′, q′)|∂γ′

�

Wr(p′, q′)|e(ρ−δ)(ω|p|+ω|p′|+|q′|)dpdp′

Since |Gri (p)||pj | = |Gr

i (p)pj | = |∇jGri (p)|, we get that (use ρ − δ � ρ and again

|q′|e−δ|q′| � 1eδ )

‖Xr‖ρ−δ � (1 + k)8kωeδ

‖∇Gr‖ρ‖Wr‖ρ � Zkω

eδ‖∇(B�

r − B�

0 )‖ρ‖Wr‖ρ. (6.18)

Here ‖∇(B�r − B�

0 )‖ρ is understood in the sense of (5.17)–(4.18).(6.9) follows form (6.13) and (6.14)–(6.18).We will prove (6.10) by the same argument as in the proof of Proposition 16. Take

(5.35) with G1 := 1i� [Wr, Hr], Gs+1 = 1

i� [Wr, Gs] and γs = d−sd δ for 1 � s � d − 1,

γ0 = δ, γd−1 = δd .

We get

1d!‖Gd‖ρ−γ0 � Zd−1

k

d!e2(d−1)γ0 · · · γd−2( δd )d−1 ‖Wr‖d−1

ρ ‖G1‖ρ−γd−1

� Zd−1k

d!d!e2(d−1)( δd )2d−2 ‖Wr‖d−1

ρ ‖G1‖ρ−δ/d

� 1 + Zk‖∇(B�r − B�

0 )‖ρd!d!e2d−1( δ

d )2d−1 Zd−1k ‖Wr‖dρ

� δ

2πd2e−1Zk(1 + Zk‖∇(B�

r − B�

0 )‖ρ)(dde−d

√2πd

d!

)2 (Zk

δ2

)d

‖Wr‖dρ

and we get (6.10) by dde−d

√2πd

d! � 1 and d2e−1 � 1 if d � 2, and setting ρ = ρr, γ0 = γr.In order to prove (6.11) we first estimate ‖V r‖ρr

defined by (6.5) where Ar(i, j) is given by (3.10).

Lemma 18. Let V ′ be defined by V ′ij = Ar(i, j)V co

ij . Then

‖V ′‖ρ � Zk‖D(B�

r − B�

0 )‖ρ‖V co‖ρ. (6.19)


Proof. The proof will actually be close to the one of (6.9). Ar(i, j)ω.(i − j) =Gr(ω.j�)−G(ω−i�)

�so

Ar(i, j) =1∫

0

∇Gr((1 − t)ω.j� + tω.i�)dt.

Therefore

V ′ =1∫

0

m∑n=1

∇nGr((1 − t)ω.j� + tω.i�)V con dt

so

‖V ′‖ � sup0�t�1

m∑n=1

‖∇nGr((1 − t)ω.j� + tω.i�)V con ‖.

Let Xrn be defined through

(Xrn)ij = ∇nGr((1− t)ω.j�+ tω.i�)(V co

n )ij = ∇nGr(ω · i + j

2 �− (1− 2t)(i− j)�2 )(V con )ij

By the argument as above, using (5.7), we get that the symbol of Xr satisfies

X rn(ξ, q) = ∇nGr(ω · ξ − (1 − 2t)q�2 )Vco

n (ξ, q) = (X rn)′(ω · ξ, q)

:= ∇nGr(ω · ξ − (1 − 2t)q�2 )(Vcon )′(ω · ξ, q)

since Vcon has the same structure as V so there exists (Vco

n )′ such that Vcon (ξ, x) = (Vco

n )′(ω·ξ, x).

Taking now the Fourier transform of (X rn)′(Ξ, q) with respect to Ξ one gets by

translation–convolution

(X r

n)′(p, q, �) =∫Rm

∇nGr(p− p′)ei(p−p′).ω.q(1−2t)�/2Vcon (p′, q, �)dp′.

So, as before,

|∂γ�

(X r

n)′(p, q, �)|

�∫ γ∑

μ=1

γ−μ∑ν=1

|∂γ−μ−ν�

∇nGr(p− p′)∂ν�e

i(p−p′).ω.q(1−2t)�/2∂μ�

Vcon (p′, q, �)|

(γ

μ

)(γ − μ

ν

)dp′

�∫ γ∑

μ=1

γ−μ∑ν=1

|∂γ−μ−ν�

∇nGr(p− p′)(|p− p′||q|/2)ν∂μ�

Vcon (p′, q, �)|

(γ

μ

)(γ − μ

ν

)dp′

Rm


Following the same lines than in the proof of (6.9) we get that (remember that, by

definition, ‖Xr‖ρ :=m∑

n=1‖Xr

n‖ρ, ‖Xrn‖ρ = ‖(X r

n)′‖ρ by Definitions 11 and 10)

‖Xr‖ρ � Zk

m∑i=1

m∑n=1

‖∇nGri ‖ρ‖V co

n ‖ρ � Zk maxn

‖∇nGr‖ρm∑

n=1‖V co

n ‖ρ � Zk‖∇Gr‖ρ‖V co‖ρ

and the lemma is proved. �Corollary 19. Let V ′′ be defined by V ′′

ij = (1 + Ar(i, j))−1V coij . Then

‖V ′′‖ρ � 11 − Zk‖∇(B�

r − B�0 )‖ρ

‖V co‖ρ. (6.20)

(6.11) is now a consequence of (5.15) and the fact that ‖V co‖ρ � ‖V ‖ρ.(6.12) is obtained by using (6.11) in (6.10). The proposition is proved. �We need finally the following obvious lemma:

Lemma 20. Define

VM (x, ξ) :=∑

|q|�M

Vq(ξ)eiqx. (6.21)

Then

‖VM‖ρ−δ � e−δM‖V‖ρ (6.22)

Corollary 21. Let V M be defined by

V Mij = Vij when |i− j| � M

= 0 when |i− j| < M.

Then

‖V M‖ρ−δ � e−δM‖V ‖ρ. (6.23)

Proof. Just notice that the symbol of V M , VM , satisfies, by (5.7), VM (ξ, q, �) = 0 when |q| � M and apply Lemma 20. �

Let us define, for a decreasing positive sequence (ρr)r�0, ρr+1 = ρr−δr to be specified later,

Gr = ‖D(B�

r −B�

0 )‖ρr= max

i=1,...,m

∑‖∇i(B�

r −B�

0 )j‖ρr. (6.24)

j=1,...,m


We are now in position to derive the following fundamental estimates of the five terms in (6.7):

‖hr+1‖ρr−δr � ‖hr+1‖ρr= ‖V co,r‖ρr

= ‖Vr‖ρr(6.25)

‖Vr − V co,r‖ρr−δr = ‖V Mr‖ρr−δr �(e−Mrδr

‖V r‖ρr

)‖Vr‖2

ρr(6.26)

‖V r‖ρr−δr �Zk

MMr

δ2r

(1 − ZkGr)2‖Vr‖2

ρr(6.27)

‖R1‖ρr−δr �Zk

MMr

δ2r(1−ZkGr)

1 − ZkMMr

δ2r(1−ZkGr)‖V r‖ρr

‖Vr‖2ρr

(6.28)


M2Mr

ω(1+ZkGr)δ3r(1−ZkGr)2

1 − ZkMMr


‖Vr‖2ρr

(6.29)

Indeed:(6.25) is obvious and (6.26) is nothing but Corollary 21.(6.27) is derived by using Proposition 16, item (5) equation (5.16), Lemma 19 and equa-tion (6.5). Note that, as pointed out above, V r is cut-offed as V co,r thanks to (3.6).(6.28) is obtained through the definition R1 =

∑∞j=1

1(−i�)jj!adj

Wr(Vr), the fact that, by

(5.11), ‖ 1(−i�)jj!adj

Wr(Vr)‖ρr−δr � (Zk/δ

2r)j‖Vr‖ρr

‖Wr‖jρrand (6.11), so that

‖R1‖ρr−δr �∞∑j=1

(Zk/δ2r)j‖Vr‖ρr

(MMr

1 − ZkGr‖Vr‖ρr

)j

.

(6.29) is proven by the definition R2 =∞∑j=2

1(−i�)jj!adj

Wr(Hr) and the fact that, by (6.12)

we have that ‖ 1(−i�)jj!adj

Wr(Hr)‖ρr−δr � ω 1+ZkGr

Zk/δr

(ZkMMr/δ

2r

1−ZkGr‖Vr‖ρr

)j

.Collecting all the preceding estimates together with the definition (6.8):

Vr+1 := (Vr − V co,r) + V r + R1 + R2,

we obtain:

Proposition 22. For r ∈ N, we have

‖Vr+1‖ρr−δr � Fr‖Vr‖2ρr

+ e−δrMr‖Vr‖ρr(6.30)

with

Fr = MMrZk

δ2r(1 − ZkGr)2

⎛⎝1 +(1 − ZkGr) + MMr

δrω(1 + ZkGr)

1 − MMr Zk2 ‖Vr‖ρr

⎞⎠ . (6.31)
1−ZkGr δr


7. Fundamental iterative estimates: Diophantine condition case

In all this section the norm subscripts ω and k are omitted.Let 0 < δ < ρ. Let us recall that we want to find Wr such that

eiWr� (Hr + Vr)e−iWr

� = Hr+1 + Vr+1 (7.1)

where Hr+1 = Hr + hr+1 and Hr = B�r (Lω), hr+1 = Vr = Dr(Lω) and

‖hr+1‖ρ = ‖Vr‖ρ � ‖Vr‖ρ, ‖Vr+1‖ρ−δ � Dr‖Vr‖2ρ. (7.2)

In the case where ω satisfies the Diophantine condition (1.4) we look for Wr solving:

1i�

[Hr,Wr] + Vr = Vr + V r (7.3)

with V r = (V rl )l=1,...,m given by

(V rl )ij =

([V rl , V

rl(i−j)])ij

i�ωl(i−j) · (i− j) , V rij := (I + Ar

ε(i, j))−1 (Vr)ij . (7.4)

Here Ar(i, j) is the matrix given by Lemma 7, that is:

B�r (�ω · i) − Br(�ω · j)

i�= (I + Ar(i, j))ω.(i− j). (7.5)

The l.h.s. of (7.1) is:

Hr + Vr + 1i�

[Hr,Wr] +∞∑j=1

1(−i�)jj! adj

Wr(Vr) +

∞∑j=2

1(−i�)jj! adj

Wr(Hr),

that is

Hr + V r + V r +∞∑j=1

1(−i�)jj! adj

Wr(Vr) +

∞∑j=2

1(−i�)jj! adj

Wr(Hr)

or

Hr + hr+1 + V r + R1 + R2. (7.6)

Proposition 23. Let Wr be the (scalar) solution of (7.3). Then, for d � 2, 0 < δ < ρ < ∞,

1d!‖[Hr,Wr], . . .︸︷︷︸

d times

]/(i�)d‖ρ−δ � δω

2πZk(1 + Zk‖∇(B�

r − B�

0 )‖ρ)(Zk

δ2

)d

‖Wr‖dρ−δ (7.7)

‖Wr‖ρ−δ �2τγ( τ

eδ )τ

1 − Zk‖D(B�r − B�

0 )‖ρ‖Vr‖ρ, (7.8)


so

1d!‖[Hr,Wr], . . .︸︷︷︸

d times

]/(i�)d‖ρ−δ

� ω1 + Zk‖D(B�

r − B�0 )‖ρ

2πZk/δ

( 2τγ( τeδ )τ

1 − Zk‖D(B�r − B�

0 )‖ρ‖Vr‖ρ

)d

(7.9)

(Let us recall that MM is defined in (1.3) and ‖DB‖ρ stands for maxj=1...m

∑i=1...m

‖∇iBj‖ρ.)

Proof. The proof of Proposition 23 is the same than the one of Proposition 17 done in details in Section 6. The only minor difference is the discussion of the small denominators and is adaptable without pain. We omit the details here. �

Using notation (6.24), Proposition 16, last item (5.10), and Proposition 17 we can derive the following fundamental estimates of the four terms of (7.6)

‖hr+1‖ρr−δr � ‖hr+1‖ρr= ‖Vr‖ρr

‖V r‖ρr−δr �Zk

22+τγ( τeδr

)τ

δ2r

(1 − ZkGr)2‖Vr‖2

ρr


γ( τeδr

)τ

δ2r(1−ZkGr)

1 − Zkγ( τ

eδr)τ


‖Vr‖2ρr


(γ( τeδr

)τ )2ω(1+ZkGr)δ3r(1−ZkGr)2

1 − Zkγ( τ

eδr)τ


‖Vr‖2ρr

Collecting all the preceding results we get:

Proposition 24. For r ∈ N, ‖Vr+1‖ρr−δr � F ′r‖Vr‖2

ρrwith

F ′r =

γ( τeδr

)τZk

δ2r(1 − ZkGr)2

⎛⎝22+τ +(1 − ZkGr) + γ

δr( τeδr

)τω(1 + ZkGr)

1 − γ( τeδr

)τ1−ZkGr

Zk

δ2r‖Vr‖ρr

⎞⎠ . (7.10)

8. Strategy of the KAM iteration

In the case of the Diophantine condition, the strategy consists in finding a sequences δrsuch that, with F ′

r given by (7.10),

∞∑δr = δ < ∞ and

r∏D2r−i

i � R2r

, R > 0. (8.1)
r=1 i=1


Indeed when (8.1) is satisfied and thanks to Proposition 24, the series ∞∑r=1

Vr, and there-

fore ∞∑r=1

hr =∞∑r=1

Vr are easily shown to be convergent in Jk(ρ − δ, ω) for ρ > δ, at the

condition that

‖V ‖ρ,ω,k < R.

This last sum is the quantum Birkhoff normal form B�∞ of the perturbation. Estimates on

the solution of the cohomological equations provide also the existence of a limit unitary operator conjugating the original Hamiltonian to its normal form.

The case of the Brjuno condition follows the same way, except that one has also to find a sequence of numbers Mr so that (6.31) holds. The main difference comes from the extra linear and non-quadratic term in Proposition 22. This difficulty is overcome by deriving out of ‖V ‖ρ,ω,k a sequence of quantities with a quadratic growth as in (7.10). This leads to an extra condition for the convergence of the iteration, condition involving only the arithmetical properties of ω and which can be removed by a scaling argument. These ideas will be implemented in the following section.

9. Proof of the convergence of the KAM iteration

In this section the norm subscripts ω and k might be omitted in the body of the proofs. They are nevertheless reestablished in the main statements.

This section is organized as follows: we first prove the convergence of the KAM it-eration in the Brjuno case with a restriction on ω (Theorem 29), restriction released in Theorem 30 thanks to the scaling argument already mentioned. This proves and pre-cises Theorem 1. We then prove the corresponding classical version (Corollary 35, global Hamiltonian version of the singular integrability of [25]) leading to Theorem 2 precised. We end the section by more refined results under Diophantine condition on ω, Theo-rem 39, leading to the criterion contained in Theorem 3.

9.1. Convergence of the KAM iteration I: constraints on ω

Proposition 25. Let us fix 0 < C < η < 1, ρ > 0 and let us choose

ρ0 = ρ, δr = α2−r, 0 < α � log 2, and Mr = 2r. (9.1)

For E � E0 defined below by (9.13) let us assume:

∞∑r=0

[| logMMr

|2r−1 − 3 log δr

2r + logZkE

2r

]= Ck < ∞ (9.2)

and, for 1 � r � l,


ZkGr < η − C/r, (9.3)

δ3r

δ3r+1

e(δr+1Mr+1−2δrMr) > 2 (9.4)

andMMr

Zk

δ2r

‖Vr‖ρr<

12(1 − η + C/r), (9.5)

together with

ZkG0 = 0, (9.6)

δ30δ31e(δ1M1−2δ0M0) > 2 (9.7)

andMM0Zk

δ20

‖V ‖ρ0 <12 . (9.8)

Then, for r � 0

‖Vr+1‖ρr+1 � (Dk)2r+1

, (9.9)

where

Dk := eCk(‖V ‖ρ + e−αα3

2M21ZkE

). (9.10)

Note that G0 = 0 and that, using (9.3) with r = 1 we get:

1 > η > Zk‖∇V ′‖ρ and C < η − Zk‖∇V ′‖ρ. (9.11)

Therefore we will impose the condition

‖∇V ′‖ρ <η − C

Zk(9.12)

Proof. We first prove the two following lemmas.

Lemma 26. Under the assumptions (6.30), (9.3) and (9.5), and η < 1, we have that, if

E � 3α + (1 + η)ωM1(ω)(1 − η)2M1(ω) =: E0 (9.13)

then

‖Vr+1‖ρr+1 � dr‖Vr‖2ρr

+ e−δrMr‖Vr‖ρrwith dr =

M2Mr

ZkE

δ3r

.


The proof is immediate by noticing that, under Proposition 22, (9.3) and (9.5), (6.31)gives that, for r ∈ N∗,

Fr � MMrZk

δ2r(1 − η + C/r)2

(3 + 2MMr

δrω(1 + η − C/r)

)so, for r ∈ N

Fr � MMrZk

δ2r(1 − η)2

(3 + 2MMr

δrω(1 + η)

).

The case r � 1 is obtained out of the preceding inequality, and the case r = 0 comes from the fact that ZkG0 = 0 � η.

Therefore E must be � 3+ω(1+η)MMr

δr(ω)

(1−η)2MMr

δr(ω)

� 3+ω(1+η)M1(ω)/α(1−η)2M1(ω)/α = 3α+ω(1+η)M1(ω)

(1−η)2M1(ω) since

MMris increasing with Mr and the lemma is proved.

Lemma 27. Let Vr = ‖Vr‖ρr+ e−δrMr

2drwhere dr = M2

MrZkE

δ3r

, Vr satisfy (6.31) and V0 :=V, ρo = ρ. Then

Vr+1 � drV2r . (9.14)

The proof reduces to completing the square in Proposition 22 and noticing that e−2δrMr

4dr− e−δr+1Mr+1

2dr+1> 0 by (9.4), since MMr+1 � MMr

. The lemma has for conse-quence the fact the

Vr+1 �r∏

s=0d2r−s

s V 2r

0 � (eCk V0)2r

. (9.15)

This concludes the proof of Proposition 25 since ‖Vr‖ρr� Vr. �

Proposition 28. Let ‖Vr‖ρr� (Dk)2

r with Dk < e−P and Dk < M , M and P be definedbelow by (9.23) and (9.20). Then (9.3), (9.4) and (9.5) hold.

Note that ∞∑r=0

δr = 2α.

Proof. (9.4): it is trivial to show that (9.4) is satisfied when α � 2 log 2.(9.5): (9.5)–(9.8) are equivalent to

12r logMMr

− log δ2r

2r + logZk

2r + logDk <12r log 1

2(1 − η + C

r) (9.16)


and

logM1 − log δ20 + logZk + logDk < log 1

2 (9.17)

which is implied by

logDk < −∞∑r=0

| logMMr|

2r + infr�0

log δ2r − logZk

2r − Δ (9.18)

< −∞∑r=0

| logMMr|

2r − logZk + 2 logα− 2e− Δ (9.19)

which is implied by

logDk < −∞∑r=0

| logMMr|

2r − logZk − 2e− Δ := −P (9.20)

where

Δ = − infr�1

{12r log 1

2(1 − η + C

r), log 1

2

}< ∞ for η < 1. (9.21)

Note that Δ > 0, P > 0.(9.3): remember that Br+1 = Br + V r+1, and ‖Br+1‖ρr+1 � ‖Br‖ρr+1 + ‖V ′

r+1‖ρr+1 �‖Br‖ρr

+ ‖V ′r+1‖ρr

. So ‖DBr+1‖ρr+1 � ‖DBr‖ρr+1 + ‖DV ′r+1‖ρr+1 .

Moreover one has, by (5.18), ‖DV ′r+1‖ρr+1− δr

2� ‖V′

r+1‖ρr+1δr2 e

� 2‖Vr+1‖ρr+1δre

�

2 (Dk)2r+1

δreout of which we conclude that

ZkGr < η − C/r =⇒ ZkGr+1 < η − C/(r + 1), ∀r � 1, if

2ZkD2r+1

k

α2−re<

C

r− C

r + 1 = C

r(r + 1) (9.22)

which is implied by Dk < M for

M = infr�1

(α2−reC

2Zkr(r + 1)

)2−(r+1)

=(αeC

8Zk

)1/4

< 1 (9.23)

since α � log 2, Zk � 8 and C � 1. �Proposition 28 together with Proposition 25 shows clearly that

(9.12) and[Dk < e−P and Dk < M

]=⇒ ‖Vr‖ρr

� (Dk)2r

(9.24)


where Dk is given by (9.10) i.e. Dk := eCk‖V ‖ρ + eCk e−δ0M0

2d0. Note that since M < 1

so is Dk � M leading to the superquadratic convergence of the sequence (Vr)r=0,.... In order for Dk to satisfy the two conditions of the bracket in the l.h.s. of (9.24) the two terms in Dk will have to both satisfy the two conditions. This remark will be the key of the main theorem below.

Let us denote by ωij, j = 1, . . . , m, i = 1, . . . , l be the ith component of the vector ωj.

Let us remark that

M1(ω) = minj=1,...,m

1min

i=1,...,l|ωi

j |and 1

M1(ω) = maxj=1,...,m

mini=1,...,l

|ωij |. (9.25)

Let us denote

B(ω) :=∞∑r=0

| logM2r |2r . (9.26)

We have that, by (9.2) and (9.20),

Ck(ω) = 2B(ω) − 6 logα + 6 log 2 + 2 log(ZkE) (9.27)

and

P (ω) = B(ω) + logZk + 2e

+ Δ. (9.28)

Theorem 29 (Brjuno case). Let α, ρ, η, and C be strictly positive constants satisfying

α < 2 log 2, ρ > 2α, 0 < C < η < 1. (9.29)

Let us define, for Δ = − infr�1

12r log 1

2 (1 − η + Cr ) and M =

(αeC8Zk

) 14 ,

Rk(ω) = (1 − η)4M1(ω)2

(3α + (1 + η)ωM1(ω))2α6e−2B(ω)

26Z2k

min{e−B(ω)−Δ

21/eZk,M

}. (9.30)

Let us suppose that, in addition to Assumptions (A1), (A2) Brjuno case and (A3), ωsatisfies

3α + (1 + η)ωM1(ω)2eα(1 − η)2M1(ω)3 � α3e−2B(ω)

26Zkmin

{e−B(ω)−Δ

21/eZk,M

}(9.31)

and the perturbation V satisfies

‖V ‖ρ,ω,k < Rk(ω), ‖∇V ′‖ρ,ω,k <η − C

. (9.32)
Zk


Then the QBNF as constructed in section 2 converges in the space J †k (ρ − 2α, ω) to

B�∞ and

‖B�

∞ − B�

0‖ρ−2α,ω,k = O(‖V ‖2ρ,ω,k) as ‖V ‖ρ,ω,k → 0. (9.33)

That is to say that there exists a (scalar) unitary operator U∞ such that the family of operators H = (Hi)i=1...m, Hi = Lωi

+ Vi, satisfies, ∀� ∈ (0, 1],

U−1∞ HU∞ = B�

∞(Lω). (9.34)

U∞ is the limit as r → ∞ of the sequence of operators Ur = eiWr� . . . ei

W0� constructed

in Section 2 and

‖U∞ − Ur‖B(L2(Tl)) �Ar

�= O

(E2r

�

)as r → ∞ for some E < 1,

here Ar is defined by (9.53).Moreover, U∞ − I ∈ J�

0 (ρ − 2α, ω) and

‖U∞ − I‖�ρ−2α,ω,0 = O

(‖V ‖ρ,ω,0

�

)as ‖V ‖ρ,ω,0 → 0, (9.35)

and, for any operator X for which there exists Xk,ρ such that for all W ∈ Jk(ρ, ω),

‖[X,W ]/i�‖ρ−δ,ω,k � Zk

δ2 Xk,ρ‖W‖ρ,ω,k, (9.36)

U−1∞ XU∞ −X ∈ Jk(ρ − 2α− δ, ω) and

‖U−1∞ XU∞ −X‖ρ−2α−δ,ω,k � D

δ2 supρ−2α�ρ′�ρ

Xk,ρ′ = O

(‖V ‖ρ,ω,k


Xk,ρ′

)(9.37)

where D is given by (9.57).

Note that the second condition in (9.32) “touches” only the average V and not the full perturbation V . It can also be replaced for any ρ′ > ρ by ‖V ‖ρ′ � ‖V ‖ρ′ � eρ′−ρ

Zk

since ‖∇V ′‖ρ−δ � ‖V ′‖ρ

eδ for any δ > 0.

9.2. Convergence of the KAM iteration II: general ω

Before we start the proof of Theorem 29, let us show the way of overcoming the condition (9.31).


We first notice that multiplying the family Lω + V by λ > 0 preserves of course integrability. Moreover λ(Lω + V ) = Lλω + λV .

On the other side we see easily that:

B(λω) = B(ω) − 2 log λ, M1(λω)

= λ−1M1(ω) and therefore ωM1 is invariant by scaling. (9.38)

Let us show that, for λ large enough, (9.31) will be satisfied for ωλ := λω. More precisely, let us define

μ = α + 2[(1 − η)α + (1 + η)ωM1(ω)]2eα(1 − η)2M1(ω)3

26Zk

α3e−2B(ω)

ν = e−B(ω)−Δ

21/eZk

we easily see that the following number λ0 is uniquely defined:

λ0 = λ0(ω) := inf {λ > 0 such that Mλ− μ � 0 and νλ3 − μ � 0}

= sup{

μ

M,(μν

) 13}. (9.39)

Elementary algebra leads to

Lemma. ∀ω, ∀λ � λo(ω), (9.31) is satisfied for ωλ := λω.

Since the QBNF of λH is the QBNF of H multiplied by λ we get that the latter will exist and be convergent if λ‖V ‖ρ,λω,k � Rk(λω) and λ‖∇V ′‖ρ,λω,k � λη−C

Zk. We get

Theorem 30. Let α, ρ, η, and C be strictly positive constants satisfying

α < 2 log 2, ρ > 2α, 0 < C < η < 1. (9.40)

Let us define Δ = − infr�1

12r log 1

2 (1 − η + Cr ), M =

(αeC8Zk

) 14 = inf

r�1

(α2−reC

2Zkr(r+1)

)2−(r+1)

and, for λ � λ0(ω) given by (9.39),

Rλ,k(ω) = Rk(λω)λ

= λ(1 − η)4M1(ω)2

(3α + (1 + η)ωM1(ω))2α6e−2B(ω)

26Z2k

min{λ2 e

−B(ω)−Δ

21/eZk,M

}.

(9.41)

Let us assume that the general assumption (A1), (A2) Brjuno case and (A3) hold and


‖V ‖ρ,λω,k < Rλ,k(ω), ‖∇V ′‖ρ,λω,k � η − C

Zk. (9.42)

Then the BNF as constructed in section 2 converges in the space J †k (ρ − 2α, λω) to

B�∞ and

‖B�

∞ − B�

0‖ρ−2α,λω,k = O(‖V ‖2ρ,λω,k). (9.43)

That is to say that there exists a (scalar) unitary operator U∞, U∞− I ∈ Jk(ρ − 2α, λω)such that the family of operators H = (Hi)i=1,...,m, Hi = Lωi

+Vi, satisfies, ∀� ∈ (0, 1],

U−1∞ HU∞ = B�

∞(Lω). (9.44)

U∞ is the limit as r → ∞ of the sequence of operators Ur = eiWr� · · · eiW0

� constructed in Section 2 and


�= O

(E2r

�

)as r → ∞ for some E < 1,

here Ar is defined by (9.53).Moreover, U∞ − I ∈ J�

0 (ρ − 2α, λω) and

‖U∞ − I‖�ρ−2α,λω,0 = O

(‖V ‖ρ,λω,0

�

)as ‖V ‖ρ,λω,0 → 0, (9.45)

and, for any operator X for which there exists Xk,ρ,λ such that for all W ∈ Jk(ρ, λω),

‖[X,W ]/i�‖ρ−δ,λω,k � Zk

δ2 Xk,ρ,λ‖W‖ρ,λω,k, (9.46)

U−1∞ XU∞ −X ∈ Jk(ρ − 2α− δ, λω) and

‖U−1∞ XU∞ −X‖ρ−2α−δ,λω,k � D


Xk,ρ′,λ

= O

(‖V ‖ρ,λω,k


Xk,ρ′,λ

)(9.47)

where D is given by (9.57).

Remark 31. Note that, for λ large enough, Rλ,k(ω) = λRk(ω). Therefore the radius of convergence increases by dilating ω. But this fact is compensated by the fact that the norm in the condition of convergence (9.42) (we take here V = 0), ‖V ′‖ρ,λω,0 < Rλ,k(ω), increases at least as λ when λ is large, as shown by the following lemma. Therefore the optimization on λ of (9.42) remains between bounded values of λ.


Lemma 32. For ω′ � ω, ‖F‖ρ,ω′,k − ‖F‖ρ,ω.k � (ω′ − ω)ρ‖∇F‖ρ,ω,k.

The proof is an immediate consequence of

eρ′X − eρX = eρX(e(ρ′−ρ)X − 1) � eρX(ρ′ − ρ)X, X � 0.

9.3. Proof of Theorem 29

First notice that B�r − B�

r−1 = V r = Vr(·, 0, �) so ‖B�r − B�

0‖ρr�

r∑1‖Vl‖l which is

convergent under (9.31) and (9.32). It remains to show that the sequence of unitary operators Ur := ei

Wr� · · · eiW1

� converges to a unitary operator on L2(Tl). This is done by proving that the sequence Ur is Cauchy (� ∈ (0, 1]). For p > n let us denote

Enp = eiWn+p

� eiWn+p−1

� · · · eiWn+1

� − I, (9.48)

so that Un+p − Un = EnpUn. We have for all r,

eiWr� = I + Tr with Tr = i

Wr

�

1∫0

eitWr� dt. (9.49)

Therefore

�‖Tr‖B(L2(Tl)) � ‖Wr‖B(L2(Tl)) � ‖Wr‖ρr,k. (9.50)

By (6.11) we have also that

‖Wr‖ρr= ‖Wr‖ρr,k � MMr

1 − η + C/r‖Vr‖ρr,k � MMr

1 − η + C/rD2r

k l > 0

‖W0‖ρ = ‖W0‖ρ,k � M1‖V ‖ρ,k (9.51)

Note that, by the Brjuno condition, we have for all r, MMr� eB(ω)2r and, by the

condition on Dk ensuring the convergence of the BNF, Dk < e−B(ω), so that:

A :=∞∑r=1

MMr

1 − η + C/rD2r

k � (eB(ω)Dk)2r

1 − η + C/r< ∞. (9.52)

We also define, for n � 1,

An =∞∑

r=n

MMr

1 − η + C/rD2r

k . (9.53)

Note that An = O(E2n) as n → ∞ for E = eBDk < 1 by (9.24).


By (9.49) we get that

Enp = eiWn+p

� Enp−1 − I + eiWn+p

�

= eiWn+p

� Enp−1 + Tn+p

= eiWn+p

� eiWn+p−2

� Enp−1 + eiWn+p

� Tn+p−1 + Tn+p. (9.54)

By iteration, we find easily that

Enp =p∑

k=2

eiWn+p

� · · · eiWn+p−k+1

� Tn+p−k + eiWn+p

� Tn+p−1 + Tn+p

and, by unitarity of eiWr� and (9.50),

‖Enp‖B(L2(Tl)) �p∑

k=0

‖Tn+k‖B(L2(Tl)) �p∑

k=0

‖Tn+k‖ρn+k�

p∑k=0

‖Wn+k‖ρn+k

�

�∞∑k=0

‖Wn+k‖ρn+k

�� An

�→ 0 as n → ∞ since A < ∞.

So ‖Enp‖B(L2(Tl)) → 0 as n → ∞ and so does Un+p − Un = EnpUn by unitarity of Un, and Un converges to U∞ in the operator topology. Moreover we get as a by-product of the preceding estimate that


�.

Since U∞ is a perturbation of the identity which doesn’t belong to any J(ρ), ρ > 0, there is no hope to estimate ‖U∞‖ρ,ω,k. Nevertheless, and somehow more interesting, we will estimate U∞ − I in the ‖ · ‖ρ−2α,ω,0 topology. In the sequel of this proof we will denote ‖ · ‖ρ := ‖ · ‖ρ,ω,0 and use the fact that ‖ · ‖ρr

� ‖ · ‖ρ−2α, ∀r ∈ N.We first remark that, for r � 0, since ‖ · ‖�ρ � ‖ · ‖ρ,

‖Tr‖�ρ−2α = ‖eiWr� − I‖�ρ−2α � ‖eiWr

� − I‖�ρr�

∞∑j=1

(Z0)j−1‖Wr

�‖jρr

j! = eZ0‖Wr‖ρr

� − 1Z0

We first remark also that

(I + Tr+1)Ur = Ur+1.

Therefore, denoting Pr = Ur − I,

Pr+1 = (I + Tr+1)Pr + Tr+1


so

‖Pr+1‖ρ � ‖Pr‖(Z0‖Tr+1‖ρ + 1) + ‖Tr+1‖ρ = (‖Pr‖ρ + 1Z0

)‖(Z0‖Tr+1‖ρ + 1) − 1Z0

so ‖Pr+1‖ρ + 1Z0

� (‖Pr‖ρ + 1Z0

)‖(Z0‖Tr+1‖ρ + 1) and

‖Pr+1‖�ρ−2α + 1Z0

� (‖P0‖�ρ + 1Z0

)r+1∏j=1

(Z0‖Tj‖�ρj+ 1) � (‖P0‖�ρ + 1

Z0)r+1∏j=1

e‖Wj‖ρj

�

= e

r+1∑j=1

‖Wj‖ρj� (‖P0‖�ρ + 1

Z0)

� eA� (‖P0‖�ρ + 1

Z0).

Therefore

‖U∞ − I‖�ρ−2α = ‖P∞‖�ρ−2α � eA�

(M1‖V ‖ρ

�+ 1

Z 0

)− 1

Z 0

Let us note that, by construction, A = O( D2k

1−η ) and that Dk depends on η through (9.10).

Lemma 33. ∃η = η(‖V ‖ρ) such that

D2k

1 − η= O(‖V ‖ρ) as ‖V ‖ρ → 0.

Proof. By looking at the expression of the radius of convergence which tends to 0 as η → 1 we see that as V → 0 one can take values of η → 1 which makes the second term in the definition of Dk of order ‖V ‖ρ and the ratio Dk

1−η of order ‖V ‖ρ. �By application of the lemma we find that

‖U∞ − I‖�ρ−2α = ‖P∞‖ � eA�

(M1‖V ‖ρ

�+ 1

Z 0

)− 1

Z 0= O

(‖V ‖ρ�

)which gives (9.45).

In order to prove (9.47) we first denote Vr = eiWr� .

We have, actually for any operator X, that VrXV −1r −X =

∞∑j=1

1j!adj

Wr(X).

Let us suppose now that ∃Xρ,k such that for all W ∈ Jk(ρ)

‖[X,W ]/i�‖ρ−δ � Zk

δ2 Xk,ρ‖W‖ρ (9.55)

(e.g. Xk,ρ = ‖X‖k,ρ).


Using (5.11), we get (since 2πl > 1)

‖VrFV −1r ‖ρ−δ � ‖F‖ρ

1 − Zk

δ2 ‖Wr‖ρ, (9.56)

and also (let us recall again that ρr+1 = ρr − δr, ρ0 = ρ)

‖ 1j!�j adj

Wr(X)‖ρr−δr−δ=ρr+1−δ �

(Zk

δ2r

)j−1

‖[X,Wr]/i�‖ρr−δ‖Wl‖j−1ρr−δ

�(Z0

δ2r

)j−1

‖[X,Wr]/i�‖ρr−δ‖Wl‖j−1ρr

,

out of which we get

‖V0XV −10 −X‖ρ1−δ � ‖[X,W0]/i�‖ρ0−δ

1 − Zk

δ20‖W0‖ρ

by V1V0XV −10 V −1

1 − V1XV −11 = V1(V0XV −1

0 −X)V −11 and (9.56)

‖V1V0XV −10 V −1

1 − V1XV −11 ‖ρ2−δ � ‖V0XV −1

0 −X‖ρ1−δ

1 − Zk

δ21‖W1‖ρ1

� ‖[X,W0]/i�‖ρ0−δ

(1 − Zk

δ20‖W0‖ρ)(1 − Zk

δ21‖W1‖ρ1)

and by iteration

‖UrXU−1r − UrV

−10 XV0U

−1r ‖ρr+1−δ � ‖[X,W0]/i�‖ρ0−δ

(1 − Zk

δ20‖W0‖ρ) · · · (1 − Zk

δ2r‖Wr‖ρr

)

and by X → V −10 XV0

‖UrV−10 XV0U

−1r − UrV

−11 V −1

0 XV0V1U−1r ‖ρr+1−δ � ‖[X,W1]/i�‖ρ1−δ

(1 − Zk

δ21‖W1‖ρ) · · · (1 − Zk

δ2r‖Wr‖ρr

)

...

‖VrXV −1r −X‖ρr+1−δ � ‖[X,Wr]/i�‖ρr−δ

1 − Zk

δ2r‖Wr‖ρr

.

So that by summing the telescopic sequence we get

‖UrXU−1r −X‖ρr+1−δ �

r∑s=0

‖[X,Ws]/i�‖ρs−δ

r∏j=s

11 − Zk

δ2j‖Wj‖ρj

�r∑ Xk,ρs

δ2 Zk‖Ws‖ρse−

r∑j=s

log (1−Zkδ2j

‖Wj‖ρj)

s=0


and since (1 − a)(1 + 2a) � 1 if 0 < a � 1/2

‖UrXU−1r −X‖ρr+1−δ �

r∑s=0

Xk,ρs

δ2 Zk‖Ws‖ρse

r∑j=s

log (1+2Zkδ2j

‖Wj‖ρj)

�r∑

s=0

Xk,ρs

δ2 Zk‖Ws‖ρse2

r∑j=s

Zkδ2j

‖Wj‖ρj

�r∑

s=0

Xk,ρs

δ2 Zk‖Ws‖ρse2

∞∑j=0

Zkδ2j

‖Wj‖ρj

�sup

ρ−2α�ρ′�ρXk,ρ′

δ2 Zk

r∑s=0

‖Ws‖ρse2B

�sup


δ2 D

with B = Zk

α2 M1‖V ‖ρ +∞∑j=1

Zk2jMMj

α2(1−η+C/j)D2j

k < ∞ and

D = Zk(M1‖V ‖ρ + A)e2B = O(‖V ‖ρ) (9.57)

by Lemma 33.Therefore we get, by letting r → ∞ so that ρr → ρ − 2α,

‖U−1∞ XU∞ −X‖ρ−2α−δ � D


Xk,ρ′ (9.58)

∞ > ‖U−1∞ XU∞ −X‖ρ−2α−δ = O

(‖V ‖ρδ2 sup


)(9.59)

The theorem is proved.

Remark 34 (Diophantine case). In the Diophantine case one immediately sees that

B(ω) � 2 log [γ2τ ] (9.60)

Moreover one easily sees that Rk(ω) and Rλ,k(ω), together with λ0(ω), are decreasing functions of B(ω). Therefore Rk(ω) � RDio

k (ω) and Rλ,k(ω) � RDioλ,k (ω) where RDio

k (ω)and RDio

λ,k (ω) are obtained by replacing B(ω) by 2 log [γ2τ ] in the r.h.s. of (9.30) and (9.41). It follows that Theorem 29 (resp. Theorem 30) is valid with Rdio

k (ω) instead of Rk(ω) (resp. Rdio

λ,k(ω) instead of Rλ,k(ω)).


9.4. Convergence of the KAM iteration III: the classical limit

Since all the estimates are uniform in �, the methods of the present paper allow to prove the following result

Corollary 35. Let H be a family of m � l classical Hamiltonians (Hi)i=1,...,m on T ∗(Tl)of the form H(x, ξ) = ω ·ξ+V(x, ξ) = H0(ω ·ξ) +V ′(ω ·ξ, x). Then, under the assumptions on ω of Theorem 30 (resp. Theorem 29) and the conditions

{Hi,Hj} = 0 1 � i, j � m

‖V‖ρ < Rλ,0(ω) (resp. < R0(ω))

‖∇V ′‖ρ <η − C

Z0

H is (globally) symplectomorphically and holomorphically conjugated to B0∞(ω.ξ): for all

δ > 0, there exist a symplectomorphism Φ−1∞ such that

H ◦ Φ−1∞ = B0

∞(H0).

Moreover, Φ−1∞ − I ∈ J(ρ − 2α − δ) (in particular Φ−1

∞ is holomorphic) and for any positive δ < ρ,

‖Φ−1∞ − I‖ρ−2α−δ � D

δ‖V‖ρ (9.61)

where D is given by (9.68) below.Finally, for any function X satisfying (9.66), we have

‖X ◦ Φ−1∞ −X‖ρ−2α−δ � D


X 0,ρ′ = O

(‖V ‖ρδ2 sup

ρ−2α�ρ′�ρX 0,ρ′

)(9.62)

where X 0,ρ′ is defined in (9.66).

Proof. Once again the function B�∞ is by construction uniform in � ∈ [0, 1] so it has a

limit B0∞ as � → 0. It is easy to get convinced that the construction of B0

∞ is the same as the one of B�

∞ after the substitution (we use capital letters for operators and calligraphic ones for their symbols at � = 0):

AB −→ A× B[A,B]i�

−→ {A,B}

eiW� −→ eLW

eiW1� ei

W2� −→ eLW1 ◦ eLW2

eiW� Ae−iW

� −→ A ◦ eLW .


Here × is the usual function multiplication, {., .} denotes the Poisson bracket and eLW

the Hamiltonian flow at time 1 of Hamiltonian W (Lie exponential).What remains to prove is the convergence of the sequence of flows eLWr · · · eLW1 := Φr

as r → ∞. This is done by the same Cauchy argument than in the proof of Theorem 29.

For Φ : T ∗Tl → T ∗Tl we denote ‖Φ‖ρ =2l∑i=1

‖Φi‖ρ where the Φi’s are the components

of Φ and we define Enp by

Enp = eLWn+p ◦ eLWn+p−1 ◦ · · · ◦ eLWn+1 − IT∗Tl→T∗Tl ,

so that Φn+p − Φn = Enp ◦ Φn.We will need the following

Lemma 36. Let F(z, θ) be analytic in {| z|, | θ| � ρ}. Then,

‖F‖L∞(| z|,| θ|�ρ) � ‖F‖ρ. (9.63)

Proof. As in section 5 write

|F(z, θ)| = |∑q

∫ F(p, q)ei〈p,ξ〉+i〈q,x〉dp| �∑q

∫| F(p, q)|eρ(|p|+|q|)dp = ‖F‖ρ. �

We will denote

‖ · ‖∞ρ = ‖ · ‖L∞(| z|,| θ|�ρ).

Proposition 37. Let Hρ = {(z, θ). | z| � ρ and | θ| � ρ}.Under the assumptions of Theorems 29 and 30, Φr is analytic Hρ → Hρr

.

Remember that ρr = ρ −r−1∑j=0

δr, δr = α2−r.

Proof. Let us first remark that the “rule” [A,B]i� −→ {A, B} is in fact (and of course) a

lemma.

Lemma 38. Let F ∈ Jm(ρ), G ∈ J1(ρ). Then [F,G]i� is the Weyl quantization of a function

σ� on T ∗Tl and

lim�→0

σ� = σ0 = {F ,G}.

The proof is an easy exercise which consists (again) in computing the symbol of [F,G]i�

through its matrix elements using Proposition 15, after expressing these matrix elements out of the ones of F, G, themselves expressed through the symbols F , G of F, G thanks


of formula (5.7). The limit � → 0 leads naturally to the Poisson bracket. Since these techniques have been extensively used through the present article, we omit the details.

Let us, by a slight abuse of notation, define again adW the operator F �→ {W, F}. Being uniform in �, the formula (5.11) taken with k = 0 leads, for any F ∈ Jm(ρr), to

1j!‖adj

Wr(F)‖ρr−δr �

(Z0

δ2r

)j

‖F‖ρr‖Wr‖jρr

Let us denote ϕr = eLWr and adWr(F) := {Wr, F}. Since F ◦ ϕ−1

r =∞∑j=0

1j!adj

Wr(F) we

get

‖F ◦ ϕ−1r ‖ρr+1 � ‖F‖ρr

1 − Z0δ2r‖Wr‖ρr

.

Therefore, under the assumptions of Theorem 30 (resp. Theorem 29), F ◦ϕ−1r is analytic

in Hρr+1 for all F analytic in Hρrand so ϕ−1

r maps analytically Hρr+1 to Hρrand so ϕr

maps analytically Hρrto Hρr+1 . Writing Φr = ϕr ◦ ϕr−1 ◦ · · · ◦ ϕ1 gives the result. �

Let us write now for all r in N,

ϕr = I + Tr,

with, as for (5.18),

‖Tr‖∞ρ−2α � ‖Tr‖∞ρr−δr =2l∑i=1

‖(Tr)i‖∞ρr−δr � ‖∇Wr‖∞ρr−δr := maxi

∑j

‖(∇jWr)i‖∞ρr−δr

�‖Wr‖∞ρr

eδr,

and so

‖∇Tr‖∞ρr−δr �‖∇Wr‖∞ρr−δr/2

eδr/2� 4

‖Wr‖∞ρr

e2δ2r

�‖Wr‖∞ρr

δ2r

.

In analogy with (9.54) we write

Enp = ϕn+p ◦ (Enp−1 + I) − I = ϕn+p ◦ (Enp−1 + I) − ϕn+p + (ϕn+p − I)

so

‖Enp‖∞ρ−2α � ‖∇ϕn+p‖∞ρ−2α‖Enp−1‖∞ρ−2α + ‖Tn+p‖∞ρ−2α

and, by induction,


‖Enp‖∞ρ−2α �p−1∑k=0

‖Tn+k‖∞ρ−2α

p−1∏s=k

(‖∇ϕn+s‖∞ρ−2α) + ‖Tn+p‖∞ρ−2α

�p−1∑k=0


p−1∏s=k

(1 + ‖∇Tn+s‖∞ρ−2α) + ‖Tn+p‖∞ρ−2α

�p∑

k=0


∞∏s=k

(1 + ‖∇Tn+s‖∞ρ−2α)

�p∑

k=0

‖Wn+k‖ρn+k

δn+ke

∞∑s=0

‖Ws‖ρsδ2s

� AneA → 0 as n → ∞.

Here we defined

A :=∞∑j=1

MMl

δ2l (1 − η + C/l)D

2l

k < ∞, (9.64)

An =∞∑j=n

MMl

δl(1 − η + C/l)D2l

k → 0 as n → ∞ since A < ∞ (9.65)

and used (9.51).So Φr converges to Φ∞ in the L∞(Hρ) topology.The proof of (9.62) is exactly the one of (9.37) by using the dictionary expressed above.

Since it is a by-product of the proof of (9.61) we repeat it here. We denote ‖ ·‖ρ := ‖ ·‖ρ,ωand use the fact that ‖ · ‖ρr

� ‖ · ‖ρ−2α−δ, ∀r ∈ N.

We have, actually for any operator X , that X ◦ ϕ−1r −X =

∞∑j=1

1j!adj

Wr(X ).

Taking (9.55) at k = 0 we have

‖{X ,W}‖ρ−δ � Z0

δ2 X 0,ρ‖W‖ρ (9.66)

We have

‖F ◦ ϕ−1r ‖ρ−δ � ‖F‖ρ

1 − Z0δ2 ‖Wr‖ρ

, (9.67)

and also (let us recall again that ρr+1 = ρr − δr, ρ0 = ρ)

‖ 1j!�j adj

Wr(X )‖ρr−δr−δ=ρr+1−δ �

(Z0

δ2r

)j−1

‖{X ,Wr}/i�‖ρr−δ‖Wr‖j−1ρr−δ

�(Z0

2

)j−1

‖{X ,Wr}/i�‖ρr−δ‖Wr‖j−1ρr

,
δr


out of which we get

‖X ◦ ϕ−10 −X‖ρ1−δ � ‖{X ,W0}‖ρ0−δ

1 − Z0δ20‖W0‖ρ

by X ◦ ϕ−10 ◦ ϕ−1

1 −X ◦ ϕ−11 = (X ◦ ϕ−1

0 −X ) ◦ ϕ−11 and (9.67)

‖X ◦ ϕ−10 ◦ ϕ−1

1 −X ◦ ϕ−11 ‖ρ2−δ � ‖X ◦ ϕ−1

0 −X‖ρ1−δ

1 − Z0δ21‖W1‖ρ1

� ‖{X ,W0}‖ρ0−δ

(1 − Z0δ20‖W0‖ρ)(1 − Z0

δ21‖W1‖ρ1)

and by iteration

‖X ◦ Φ−1s −X ◦ ϕ0 ◦ Φ−1

s ‖ρs+1−δ � ‖{X ,W0}‖ρ0−δ

(1 − Z0δ20‖W0‖ρ) · · · (1 − Z0

δ2s‖Ws‖ρs

)

By the same argument we get

‖X ◦ ϕ0 ◦ Φ−1s −X ◦ ϕ0 ◦ ϕ1 ◦ Φ−1

s ‖ρs+1−δ � ‖{X ,W1}‖ρ1−δ

(1 − Z0δ21‖W1‖ρ) · · · (1 − Z0

δ2s‖Ws‖ρs

)...

‖X ◦ ϕ−1r −X‖ρr+1−δ � ‖{X ,Wr}‖ρr−δ

1 − Z0δ2r‖Wr‖ρr

.

so that by summing the telescopic sequence

�r∑

s=0

Xk,ρs

δ2 Z0‖Ws‖ρse−

r∑j=s

log(1−Z0

δ2j‖Wj‖ρj

)

and since (1 − a)(1 + 2a) � 1 if 0 < a � 1/2

�r∑

s=0

Xk,ρs

δ2 Z0‖Ws‖ρse

r∑j=s

log(1+ 2Z0

δ2j‖Wj‖ρj

)�

r∑s=0

Xk,ρs

δ2 Z0‖Ws‖ρse2

r∑j=s

Z0δ2j

‖Wj‖ρj

�r∑

s=0

Xk,ρs

δ2 Z0‖Ws‖ρse2

∞∑j=0

Z0δ2j

‖Wj‖ρj �sup


δ2 Z0

r∑s=0

‖Ws‖ρse2B

�sup


δ2 D

with B = Z0α2M1‖V ‖ρ +

∞∑j=1

Z02jMMj

α2(1−η+C/j)D2j

k < ∞ and

D = Z0(M1‖V ‖ρ + A)e2B = O(‖V ‖ρ) (9.68)

by Lemma 33.


Therefore we get, by letting r → ∞ so that ρr → ρ − 2α,

‖X ◦ Φ−1∞ −X‖ρ−2α−δ � D


X k,ρ′ = O

(‖V ‖ρδ2 sup

ρ−2α�ρ′�ρX k,ρ′

). (9.69)

Let now X ∈ {ξ1, . . . , ξl, x1, . . . , xl}, {X , W1} = ±∂ΞW1 where Ξ is the conjugate quan-tity to X . Therefore ‖{X , W0}‖ρ−δ � ‖∇W0‖ρ−δ � 1

δ‖W0‖ρ. Therefore X k,ρ = 1 and ‖X ◦ Φ−1

r −X‖ρr+1−δ � Dδ2 , ∀X ∈ {ξ1, . . . , ξl, x1, . . . , xl} which means that

‖Φ−1r − I‖ρr+1−δ � D

δ2 (9.70)

In fact we just proved that Φ−1∞ = I + Φ with ‖Φ‖ρ∞−δ=ρ−2α−δ � D

δ2 . Corollary 35 is proved. �9.5. Convergence of the KAM iteration IV: the Diophantine case

Though the Diophantine case is covered by the Theorem 29 (see Remark 34), we can also use directly Proposition 24 in order to low down the hypothesis of the theorem.

In fact Proposition 24 shows that the same proof will be possible by only replacing MMr

(ω) by γ( τeδr

)τ and (9.13) by

E �22+τα + 2[α + (1 + η)ωγ( τ

eα )τ ](1 − η)2γ( τ

eα )τ = E1 (9.71)

Indeed (7.10) is verbatim the same as (6.31) after replacing MMr(ω) by γ( τ

eδr)τ and the

first term in the parenthesis, namely 1, by 22+τ . Therefore the proof will be the same by replacing B(ω) by Bα(γ, τ)

Bα(γ, τ) =∞∑r=0

log (γ( τ

eδr)τ )2−r = 2 log

[γ( τ

eα)τ]

+ 2τ log 2 = 2 log[2τγ( τ

eα)τ]

and of course Ck and P by the corresponding expressions C ′k, P

′.The very last change will concern Dk which now will be Dk = eCk‖V ‖ρ because

the estimate of Proposition 24 reads now directly ‖V r+1‖ρl−δl � F ′r‖V r‖2

ρl: this will

imply that in the Diophantine case there is no condition for ω similar to (9.31), and no condition α < 2 log 2. We get:

Theorem 39 (Diophantine case). Let α, ρ, η, C and E be strictly positive constants satis-fying

ρ > 2α, 0 < C < η < 1. (9.72)


Let us define, for Δ = − infr�1

12r log 1

2 (1 − η + Cr ) and M =

(αeC8Zk

) 14 ,

Rk(ω) =( (1 − η)2γ( τ

eα )τ

22+τα + 2[α + (1 + η)ωγ( τeα )τ ]

)2α6

26Z2k(2τγ( τ

eα )τ )4

× min{

(2τγ( τeα )τ )−2e−Δ

21/eZk,M

}. (9.73)

Then if

‖V ‖ρ,ω,k < Rk(ω), ‖∇V ′‖ρ,ω,k <η − C

Zk, (9.74)

the same conclusions as in Theorem 29 and Corollary 35 hold.

9.6. Bound on the Brjuno constant ensuring integrability

As mentioned in the introduction we can use Theorem 30 to estimate the rate of di-vergence of the Brjuno constant as the system remains integrable while the perturbation is vanishing.

Let us suppose that we let ω vary in a way such that ω remains in a bounded set [ω−, ω+] of (0, +∞). (9.73) tells us that, in order that Theorem 39 holds, ω can be taken as we want as soon as γ and τ satisfy ‖V ‖ρ,ω,k < r.h.s. of (9.73) and ‖∇V‖ρ,ω,k < η−C

Zk.

It is easy to check that, since Bα(γ, τ) := 2 log[2τγ( τ

eα )τ]→ ∞ as γ, τ or both of them

diverge, we have, for Bα(γ, τ) large enough (in order that the min in (9.73) is reached by the first term and that 22+τ < Bα(γ, τ)),

Rk(ω) � 2Ke−3Bα(γ,τ)

with K = (1−η)4α6

(α+2(1+η)ω−)2621/eZ3k. Therefore for ‖∇V ′‖ρ,ω,k < η−C

Zkand ‖V ‖ρ,ω,k small

enough (namely ‖V ‖ρ,ω,k � 2Ke−3B−α where B−

α is the smallest value of Bα(γ, τ) which makes the min in (9.73) reached by the first term and which is larger than 22+τ), we have

Corollary 40. The conclusions of Theorem 39 hold as soon as

Bα(γ, τ) < 13 log

(1

‖V ‖ρ,ω+,k

)+ 1

2 log 2K.

Remark. In the case of the Brjuno condition, Theorem 29, it happens that λ0(ω) ∼C ′e2B(ω) and Rλ0(ω) ∼ C as B(ω) → ∞ for some bounded constants C, C ′. Therefore our condition of convergence takes the form ‖V ‖ρ,C′e2B(ω)ω,k < C. This leads to a sufficient condition on B(ω) depending on the way V → 0. For example it is easy to check that,


if V → 0 as V = εV0, ε → 0 and V0 with a symbol V ′0 whose Fourier transform in ξ is

compactly supported, one gets a condition of the form B(ω) < D log log 1ε +D′ for some

constants D, D′.

10. The case m = l

Lemma 41. Let the vectors ωj, j = 1, . . . , m = l, be independent over R and let Ω be thematrix Ω = (Ωij)i,j=1,...,l with Ωij := ωj

i . Then

(1) any V satisfies (1.5)(2) ∀q ∈ Zl, q = 0, min

1�i�m|〈ωi, q〉|−1 � l/|Ω| (there is no small denominator).

Proof. Ω is invertible by the independence of the ωj’s. This proves (1). Moreover one has immediately that 1 � |q| � l|Ω−1| max

j=1,...,l|〈ωj , q〉|. �

Therefore the main assumption reduces to:

Main Assumptions (Extreme case).

ωj ∈ Rl, j = 1, . . . , l, are independent over R and [Hi, Hj ] = 0, ∀1 � i, j � l. (10.1)

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