Normalization in Banach scale Lie algebras via mould calculus and applications

We study a perturbative scheme for normalization problems involving resonances of the unperturbed situation, and therefore the necessity of a non-trivial normal form, in the general framework of Banach scale Lie algebras (this notion is defined in the article). This situation covers the case of classical and quantum normal forms in a unified way which allows a direct comparison. In particular we prove a precise estimate for the difference between quantum and classical normal forms, proven to be of order of the square of the Planck constant. Our method uses mould calculus (recalled in the article) and properties of the solution of a universal mould equation studied in a preceding paper.


Introduction
Perturbation theory is a fascinating subject which appears to have been fundamental for the birth of dynamical systems through Poincaré and quantum mechanics in the Göttingen school.
It is also of fundamental importance for the large computation in physics and chemistry, leading to a panel of different algorithms for computing perturbation series. Each such a method (e.g. generating functions for dynamical systems, functional analysis (expansion of the Neumann series) in quantum mechanics) is very well adapted to emblematic situations (small divisors and KAM theory in classical dynamics, Kato method and existence of dynamics in quantum mechanics), but each methodology seems to be strictly tied to the different underlying paradigms.
In the present article we will present in a unified way new results concerning the use of mould theory for (classical) Birkhoff normal forms (namely in presence of Hamiltonian resonances) and for quantum perturbation theory, this last topics having never met, to our knowledge, mould calculus.
As a by-product we also obtain a precise estimate of the difference between quantum normal forms and the classical ones corresponding to the underlying classical situation, Theorem D. Note 1 that this estimate is of order of the square of the Planck constant and involves only the size of the perturbation.
Mould calculus was introduced and developed by JeanÉcalle ( [E81], [E93]) in the 80-90's in order to give powerful tools for handling problems in local dynamics, typically the normalization of vector fields or diffeomorphisms at a fixed point.
Beside the two topics already mentioned (classical and quantum normal forms), the large difference of paradigm between them has led us to formulate mould calculus in a kind of abstract operational setting able to include both classical and quantum dynamics, and probably many other situations.
This formulation leads to mould resolutions of general perturbation problems, that is problems where a perturbation is added to a bare problem already explicitly solved.
To put it in a nutshell, one of the key ideas of mould calculus can be phrased by saying that mould expansions are done on non-universal -namely related to the perturbation involved in the problem to be solved -objects (comould), with universal -namely dependent only on the unperturbed, solved problem -coefficients (mould). This is quite unfamiliar for people using standard perturbative tools (e.g. Taylor expansions) where universality is more placed on "active" objects. This might explain the poor penetration of the beautiful theory of moulds in other fields than local dynamics.
Thus, in the present article, we want to consider a general formalism that would include the following cases: ‚ the construction of the Birkhoff form for perturbations of integrable Hamiltonian systems hpI, ϕq " h 0 pIq`V pI, ϕq, I P R d , ϕ P T d , ‚ the unitary conjugation to a quantum Birkhoff form B " BpH 1 , . . . , H d q for perturbations of quantum "bare" operators H " H 1`¨¨¨`Hd , rH k , H ℓ s " 0 (e.g. H k "´1 2h 2 B 2 x k`1 2 ω 2 k x 2 k on L 2 pR, dx k q or H k "´ihω k B x k on L 2 pT, dx k q).
Let us notice that the following two situations have also been already considered via mould theory in our companion article [P16]: ‚ the formal linearization, or at least the formal normalization, of a vector field X " Though these four situations are quite different and belong to different paradigms, we would like to emphasize that mould theory can provide a general formulation handling all of them.
Let us present this general framework. It consists of ‚ a Lie algebra L, which is a Banach scale Lie algebra as defined in Section 2, or a filtered Lie algebra (the latter case has been treated in [P16]), ‚ assumptions on L insuring the existence of an exponential map defined on L, ‚ an element B of L, ‚ elements Y, Z of L to be determined so that rX 0 , Zs " 0 and e ad Y pX 0`B q " X 0`Z , (1.1) for an "unperturbed" X 0 P L.
Let us present now briefly mould calculus.
Mould theory relies drastically on the notion of homogeneity, more precisely on the decomposition of the perturbation into homogeneous pieces. In the general setting we suppose that the starting point is an element X of a Lie algebra of the form where B is a "perturbation" of X 0 , for which everything is supposed fully known.
The problem to solve consists in finding a Lie algebra automorphism Θ such that, at any approximation of size any power of B for a certain norm ¨ , where Z is a normal form, namely a 0´homogeneous element is a sense we will explain now.
We define an alphabet Λ Ă C of letters λ through the decomposition To the alphabet Λ we can associate the set of words If λ " λ 1 . . . λ r , then we use the notation rpλq¨" r, with the convention r " 0 for the empty word λ " I.
We can now define the Lie comould as the mapping with the convention B rIs " 0 and we call mould any mapping (in this article we use only complex-valued moulds, but [P16] considers more generally k-valued moulds, where k is the field of scalars of L, an arbitrary field of characteristic zero). To a mould M ‚ , we associate an element of L defined by Returning ot our problem of solving equation (1.2), the key idea will be to process a "mould ansatz", that is looking to a solution of (1.2) of the form In (1.9) one has (1.10) Constructing solutions of (1.9) process in a way familiar to any perturbative setting: first we note that, precisely because B is perturbation of X 0 , Θ must be close to the identity and Z to zero. This entails that S I " 1 and F I " 0 from which it follows that Moreover willing Z to be 0-homogeneous is fulfilled by imposing F ‚ to be resonant, i.e. that ∇F ‚ " 0. Putting all these properties together leads to the fact that F ‚ and the non-resonant part of S ‚ can be determined by induction on the length of letters. What is not determined because it disappears from the equation is the resonant part of S ‚ , since it is "killed" by ∇.
We showed in [P16] that this ambiguity is removed -leading to uniqueness of the solution -by fixing a gauge generator, namely an arbitrary mould A ‚ , resonant and alternal. More precisely, for any gauge A ‚ , (1.9) has a unique solution pS ‚ , F ‚ q. Moreover it happens that S ‚ " e G ‚ (where e has to be understood as the exponential in the algebra of moulds, that is e G ‚ " 8 ř k"0 G ‚ˆk k! wherê is defined in (1.10)) and G ‚ and F ‚ are alternal, a notion we define now.
The notion of alternality has to do with the shuffling two words a and b, which is the set of words λ obtained by interdigitating the letters of a and those of b while preserving their internal order in a or b. The number of different ways a word λ can be obtained out of a and b is denoted by sh`a , b λ˘. Saying that F ‚ is alternal is nothing but saying that for all non-empty words a, b, ř λPΛ sh`a , b λ˘F λ " 0.
To be more precise, in [P16] was proven the following "existence-uniqueness" result for the mould equation. Let us define an operator ∇ 1 : M ‚ Þ Ñ ∇ 1 M ‚ by the formula for an arbitrary mould M ‚ (recall that rpλq denotes the length of λ), and denote by M ‚ 0 the resonant part of the mould, defined by M λ 0¨" 1 tΣpλq"0u M λ for all λ P Λ (where Σpλq is by defined in (1.10)).
Proposition 1.1. Let k be a field of characteristic zero and Λ a subset of k. For any resonant alternal mould A ‚ , there exists a unique pair pF ‚ , G ‚ q of alternal moulds such that (1.13) The proof of Proposition 1.1 is constructive in the sense that we obtain the following simple algorithm to compute the values of F ‚ and S ‚¨" e G ‚ on any word λ by induction on its length rpλq: introducing an auxiliary alternal mould N ‚ , one must take S I " 1, F I " N I " 0 and, for rpλq ě 1, (1.14) where we have used the notation 'λ¨" λ 2¨¨¨λr for λ " λ 1 λ 2¨¨¨λr and the symbol ÿ˚i ndicates summation over non-trivial decompositions (i.e. a, b ‰ I in the above sums); the mould F ‚ thus inductively defined is alternal and then defines the alternal mould G ‚ which solves (1.12)-(1.13).
Finally we proved in [P16], Propositions 3.8 and 3.9, the following result, crucial for the link between the mould equation and the original problem (1.2).
This result shows that (1.2) is solved by Z " F ‚ B r ‚ s and Θ " e ad Y with Y " G ‚ B r ‚ s , where F ‚ and G ‚ solve (1.12) (see [P16] for the details).
The goal of the present article is twofold: first we want to show how we can solve perturbatively the normal form problem (1.1) in the general setting of an X 0 -extended Banach scale Lie algebra -Theorem A -and second we want to show applications to the aforementioned dynamical problems -Theorems B and C. As a by-product we give also a quantitative estimate concerning the difference between classical and quantum normal forms -Theorem D.
The different situations in dynamics which can be realized as an X 0 -extended Banach scale Lie algebra are displayed in the next table.
The paper is organized as follows. The first part is devoted to the result valid in any X 0 -extended Banach scale Lie algebra whose definition is given in Section 2 and in Section 3 we state the general result of the article, proven in Section 4. The second part is devoted to applying the main result to classical dynamical situations, Section 5, the quantum ones, Section 6, and semiclassical approximation, Section 7.
Appendix A gives the minimal setting in semiclassical analysis necessary to the present paper. The three other appendices provide and prove technical lemmas used in different parts of the article.
Let us finally mention that the present article is self-contained (it uses only Theorem B of [P16], rephrased in Proposition 1.1 of the present article) and all the constants are explicit.
Normalization in X 0 -extended Banach scale Lie algebras 2. X 0 -extended Banach scale Lie algebras Let`L, r¨,¨s˘be a Lie algebra over k " R or C. We say that we have an "X 0 -extended Banach scale Lie algebra " if: (1) L contains a family pB ρ , ¨ ρ q ρPR˚o f Banach spaces over k such that there exists a constant γ ą 0 such that (3) L contains an element X 0 (which does not necessary belong to any of the B ρ 's) and there exists a function χ : R˚Ñ R˚such that Let us denote the adjoint representation of L by ad, i.e. for each Y P L, ad Y is the Lie algebra derivation defined by ad Y X " rY, Xs for all X P L. One can check (see Corollary B.2) that, for an X 0 -extended Moreover, e adY X 0¨" ř kě0 1 k! pad Y q k X 0 too is well-defined and e adY X 0´X0 P B ρ 1 for each ρ 1 as above.

The general result
Notation 3.1. Let Λ be a nonempty subset of k. For a word λ " λ 1¨¨¨λr P Λ of length r ě 1 and a subset σ of t1, . . . , ru, we set For τ P R˚, we define a function β τ : Λ Ñ R`by the formula Theorem A. Let L be an X 0 -extended Banach scale Lie algebra and let ρ ą 0 and B P B ρ . Suppose that there exist a subset Λ of k and a decomposition (ii) for all r P N˚, there exist η r ą 0 and τ r ě 1, such that ÿ λ"λ1,...,λrPΛ B λ1 ρ . . . B λr ρ e ηrβτ r pλq :" ǫ r ă 8.
Remark 3.2. If in Theorem A we take B " Bpǫq depending on a perturbation parameter ǫ so that B λ pǫq ρ ď C λ |ǫ| for each λ P Λ, with non-negative constants C λ , then condition (3.4) factorises and ǫ r " Opǫ r q. In this case, the final estimates reduces to E N ρ 1 " Opǫ N`1 q.
Remark 3.3. Below, in Sections 5 and 6, we will take Λ of the form Λ " t i k¨ω | k P Z d u for a given ω P R d . We shall see that, if there exist α ą 0 and τ ě 1 such that the Diophantine condition @k P Z d , k¨ω " 0 or |k¨ω| ě α|k|´τ (3.6) holds, then one can find X 0 -extended Banach scale Lie algebras such that any B P B ρ has a decomposition satisfying (3.4) provided τ r " τ and η r ď ρ α 1{τ 2 r . Moreover, ǫ r " Op B r ρ q and E N ρ 1 " Op B N`1 ρ q in (3.5) in this case.
Proof. The fact of having evaluations of F ‚ , G ‚ in the form of sums of bounded functions divided by monomials in the variables mentioned in the statement of Lemma 4.1 is a property obviously stable by mould multiplication. Therefore it is enough to prove it for S ‚ in order to get it satisfied for G ‚ . It is easily shown to be true by induction using (1.14) and (1.15) and the fact, easy to prove, that (once again we take A ‚ " 0) The homogeneity property follows also easily from the induction generated by (1.14) and (1.15). 1 .
Finally the fact that the functions C r σ1,...,σr´1 and D r σ1,...,σr are bounded comes from the way of solving (1.14)-(1.15) by induction on the length of the words and the fact that the possibly unbounded constant (at fixed length r of the word) appearing in (1.14)-(1.15) is Σpλq :" r ř i"1 λ i and it appears only in (1.14) with homogeneity´1. 1 We get also the homogeneity by a simple physical dimension reasoning: since the letters are defined by tX0, B λ u " λB λ and the Poisson bracket by tA, Bu " BA Bp

BB Bq´B
A Bq BB Bp , we have that λ must have the dimension of energy action (the dimension of action is the one of pˆq). An evaluation of the comould on a word of length r, tB λr tB λ r´1 t. . . , B λ 1 uu . . . u has the dimension energy r action r´1 . Finally the dimension of the normal form is the one of an energy. Since all the constants in the mould equation (with zero gauge) are universal and therefore have no dimension, we conclude that the dimension of the evaluation of the mould F ‚ on a word of length r is energyâ ction r´1 energy r "´a ction energy¯r´1 " pdimension of λq´p r´1q . In the same way one sees that since one takes the exponential of adY N , adY N must have no dimension and therefore YN must have the dimension of an action and get the desired homogeneity.
where G r is defined in (4.3).
Performing a truncation of e ad Y N as in Corollary B.2, we get, defining e 4.3. End of the proof. Let us go back now to the mould equation:
‚ (4.11)ñ(4.12) with (4.15): since the support of G ‚ N contains only words of length up to N , we can expand the mould in (4.11) up to words of length N , which appears in (4.12), plus the rest.
The rest, whose support contains only words on length between N`1 and N 2 , gives (4.15) by combinatorial coefficients and Corollary 4.3.

Applications to dynamics
Normal forms have a long history since the seminal work by Poincaré in perturbation theory [P1892].
See also [B28] for a more "dynamical systems" presentation. Their use in stability problems for dynamical systems are presented in the textbooks [M56, A78, G83, L88]. More recent results and surveys are present in the articles [P03, Z05, S09].
Bohr-Sommerfeld quantization of normal forms have been used before the birth of quantum mechanics itself (namely the publication of [H25]): see [B25]. More recently quantum normal forms have been used in spectral problems near minima of potentials ( [S92], [B99] and [C08], for inverse problems in, e.g., [I02] and [G10] and perturbations of integrable systems in [G12, P14].
The link between quantum and classical normal forms has been established in [G87] and [D91].
The three Theorems B, C and D below give systematic precise estimates for the construction of normal forms at any order as rephrasing of the general Theorem A. The estimate between quantum and classical normal forms contained in Theorem D is to our knowledge new.

Quantitative classical formal normal forms
We denote the circle by T¨" R{2πZ.
Let d ě 1 be integer. We are interested in two situations: the phase-space P is either T˚R d » R dˆRd and then X 0 px, ξq " 1 2´d or it is T˚T d » T dˆRd and X 0 px, ξq " ω¨ξ.
In both cases, we suppose that ω P R d has components ω j ą 0, and we will denote the variable in P by px, ξq. The symplectic 2-form being ř dξ j^d x j , the Hamiltonian vector field associated with X 0 is Bξj˘i n the first case, and ř ω j B Bxj in the second.
We want to perturb X 0 by a "small" perturbation B and want to show that it is possible, after a symplectic change of coordinates, to put the new Hamiltonian X 0`B into a normal form X 0`Z , tX 0 , Zu " 0, modulo an arbitrarily small error.
The result will be expressed in Theorem B below, which will follow from Theorem A. We first have to show how our situation enters in the framework of the first part of this article.

Let
We define the (symplectic) Fourier transform G Ź of a function G P L 1 pP, dxdξq by G Ź pq, pq " 1 p2πq d ż P Gpx, ξqe´i ppx´qξq dxdξ for pq, pq P P ľ . (5.1) Let dµ denote either the Lebesgue measure dqdp on R dˆRd or the product of the Lebesgue measure dq by the sum of Dirac masses on Z d Ă R d (counting measure).

(5.2)
By a slight abuse of notation, from now on, we will denote dµpq, pq by dqdp in both cases.
Let us write X 0 " X 0,1`¨¨¨`X0,d with, for each j " 1, . . . , d, Since the X 0,j 's Poisson-commute and since, for each j, all the solutions of the Hamiltonian vector field tX 0,j ,¨u are 2π ωj -periodic, we get an action of T d on P by defining Φ t 0¨" expˆ" Given k P Z d and an integrable function G, we now define GpΦ t 0 px, ξqqe´i kt dt for px, ξq P P.
Lemma 5.1. For any real-analytic G, one has ř kPZ d G pkq " G (pointwise convergence on P) and, for each Proof. For each px, ξq P P, the function t P R d Þ Ñ G˝Φ t 0 px, ξq is analytic and 2π-periodic in each t j ; for each k P Z d , its kth Fourier coefficient is G pkq px, ξq. The first statement thus follows from the fact that G˝Φ t 0 px, ξq is the sum of its Fourier series.
For each j, tX 0,j , G pkq u is the kth Fourier coefficient of the function tX 0,j , G˝Φ t 0 u, and this function coincides with ω j B Btj`G˝Φ t 0˘, hence tX 0,j , G pkq u " ik j ω j G pkq , and the second statement follows.
For ρ ą 0 we will denote by J ρ the space of all integrable functions G whose associated family of functions G pkq have a Fourier transform whose modulus is integrable with respect to dqdp weighted by e ρp|q|`|p|q , the family of integrals obtained that way being itself summable with the weight e ρ|k| .
Namely, 2 we set J ρ¨" t G P L 1 pP, dxdξq | G ρ ă 8 u, with Note that each function in J ρ is real analytic and has a bounded holomorphic extension to the complex strip px, ξq P C dˆCd | |ℑm x 1 |, . . . , |ℑm x d |, |ℑm ξ 1 |, . . . , |ℑm ξ d | ă ρ ( . J ρ is obviously a Banach space satisfying }G} ρ 1 ď }G} ρ whenever ρ 1 ă ρ. In the case P " R dˆRd let us denote by X  (5.6) In the case P " T˚T d we get easily 2 Note that in the case of T˚T d , G pkq is nothing but the Fourier coefficient of Gp¨, ξq times e´i k¨. Therefore in this case G pkq Ź pq, pq " δ k,p G pkq Ź pq, pq and so G ρ " ş |G Ź pq, pq|e ρp|q|`2|p|q dqdp. We present nevertheless the two cases (T˚T d and T˚R d ) in a unified way.
Let L be the space of real analytic functions on P. The two lemmas of Appendix C show that pL, r¨,¨sq endowed with J ρ " B ρ , 0 ă ρ ă 8 and with r¨,¨s " t¨,¨u, γ " 1 and χpρq " 1 eρ , is an X 0 -extended Banach scale Lie algebra.
Let us remark now that the homogeneous components of a perturbation B are easily deduced from the family pB pkq q kPZ d . Indeed, let us define Λ " t i k¨ω | k P Z d u (5.8) as in Remark 3.3. In view of Lemma 5.1, we see that, for each λ P Λ, and B " ř λPΛ B λ . Moreover, since (5.10) In particular, ř kPZ d B pkq ρ e η|k| is convergent for each η ď ρ.
Let us assume that the Diophantine condition (3.6) is satisfied.
Before to state it in the present setting, let us remark that since ǫ r ď B r ρ , one can improve (3.5) (or rather (4.16)). To do so we first remark that in Lemma 4.4, if B ρ ď 1, Therefore for ǫ " inf p1, ρ´ρ 1 8γΓN q we have that the second inequality of (4.16) is satisfied when B ρ ď ǫ. Under the same condition on B ρ we find that Thus we define: 8γΓN˘. (5.12) Finally, Z N and Y N are real functions, and e adY N corresponds to a composition by a symplectic transform.
We get the following rephrasing of Theorem A.
For all N P N˚and 0 ă ρ 1 ă ρ, let D and ǫ be given by (5.12). Then Remark 5.2. In the two geometrical situations present in this section, namely T˚T d and T˚R d , no use is made of the underlying symplectic structure. Therefore it seems to us reasonable to think that our methods apply to the situation of perturbations of Hamiltonian flows on Poisson manifolds. Indeed the method is essentially algebraic, using extensively the derivation ad referring only to the Poisson structure, so we are inclined to believe in the possibility of deriving a mould equation in this situation. The point then will to find a norm not using the Fourier transform (peculiar, say, to the linear or homogeneous spaces situation) but rather, and essentially equivalently, complex extensions of real analytic functions (in the case of real analytic Poisson manifolds). More generally (and more difficult), it would be very interesting to transfer the methods of our paper to the question of the local description of a Poisson manifold around a symplectic leaf through the construction of normal forms as presented in [M04, M12, M14], or even to generalized complex geometry as in [B13]. We thank the referee for mentioning the possible extension of our work to Poisson geometry and pointing out the references quoted in this Remark.

Quantitative quantum formal normal forms
This section constitutes the quantum counterpart of the preceding section.
Let the Hilbert space H be either L 2 pR d q and in this case let X 0 " 1 2 p´h 2 ∆`d ř i"1 ω 2 i x 2 i q or L 2 pT d q and X 0 "´ihω¨∇, corresponding indeed to the quantization of the two situations of Section 5. Let us recall that in both cases X 0 is essentially self-adjoint on H.
Here again we want to perturb X 0 by a "small" perturbation B and want to show that it is possible, after a conjugation by a unitary operator on H, to put the new quantum Hamiltonian X 0`B into a normal form X 0`Z , rX 0 , Zs :" X 0 Z´ZX 0 " 0, modulo an error we want to be as small as we wish.
The result will be expressed in Theorem C below but let us first see how the quantum situations just mentioned enter also in the framework of the first part of this article, though they belong to a very different paradigm than the one of the preceding section.
Let J ρ be the set of all pseudo-differential operators whose Weyl symbols belong to J ρ . We define the norm of an operator belonging to J ρ as the }¨} ρ norm of its symbol and we denote it by the same expression There are different ways of defining Weyl quantization (see Appendix A below for elementary definitions).
In the case P " T˚R d , one of them, actually the historical one exposed in the book by Hermann Weyl [W29] consists in writing again the formula (5.2) for the inverse Fourier transform pq, pqe ippx´qξq dµpq, p, q and replace in the right hand side x and ξ byˆx and´ih∇ respectively, in the case where P " T˚R d . We get the operator B associated to the symbol σ B by the formula The reader can check easily that when σ B " x (resp. ξ) on recover B "ˆx (resp.´ih∇). Moreover, using the Campbell-Hausdorff formula, one gets that This is this formulation that we use in the case where P " T˚T d since px`ihq∇ doesn't make any sense on the torus, so we cannot use e´i ppx`ihq∇q , but e´i px does (remember p is the dual variable of x so is discrete). Therefore we define in both cases

Note that a straightforward computation gives back the usual formula (A.2) or (A.3) of Appendix A:
Bf pxq " But the main interest of this formula for our purpose is the fact that e i pq 2 e´i px e´qh ∇ is unitary, since e i pq 2 e´i px e´qh ∇ ϕpxq " e i pq 2 e´i px ϕpx´hqq, so e i pq 2 e´i px e´qh ∇ L 2 ÑL 2 " 1 3 and therefore Moreover it is straightforward to show that when σ B is real valued, B is a symmetric operator so that, when bounded, is constitutes a symmetric bounded perturbation of X 0 , therefore we just proved the following result.
Then X 0`B is essentially self-adjoint on H.
Let L be the space of hermitian operators on H. The lemma in Appendix D shows that pL, r¨,¨s q q endowed with J ρ " B ρ , 0 ă ρ ă 8 and with r¨,¨s q " 1 ih r¨,¨s, γ " 1 and χpρq " 1 eρ , is an X 0 -extended Banach scale Lie algebra.
The decomposition into λ´homogeneous components of an arbitrary B P J ρ involves the letters of the same alphabet Λ defined by (5.8) as in Section 5. In fact, the homogeneous components of B can be obtained by Weyl quantization of the homogeneous components of the symbol σ B , or directly as where X 0,j "´h 2 B 2 xj`ω 2 j x 2 j in the case of R d and X 0,j "´ihω j B xj for T d , since linear Hamiltonian flows commute with quantization (see Lemma A.2 below). Therefore the hypothesis piq´piiq of Theorem A are satisfied for the same values η r " ρα 1{τ 2´r and τ r " τ (6.4) as in Section 5.
Moreover, if Y N is a self-adjoint operator, then e ad Y N corresponds to conjugation by the unitary transform e 1 ih YN , hence the conclusions of Theorem A for this situation can be rephrased as: For all N P N˚and 0 ă ρ 1 ă ρ, let D and ǫ be given by (5.12). Then (a) the following two expansions converge in B ρ 1 (6.5)

Semiclassical approximation
In this final section we would like to link in a quantitative way the two preceding Section 5 and 6. Since the estimates in Section 6 are uniform in the Planck constant, it is natural to think that the quantum normal form should be "close" to the classical one when the Planck constant is close to zero. Since such a comparison invokes objects of different nature (operators for quantum, functions for classical), it is natural to use the symbol "functor" σ to quantify this link.
Expressing the quantum normal form in its mould-comould expansion, we see that, on one hand, the mould in independent ofh, and (therefore) is the same as the one in the mould-comould expansion of the classical normal form. On the other hand, for any pseudodifferential operator B, the symbol of the commutator (divided by ih) of any two homogeneous components of B tends, by Lemma A.1, to the Poisson bracket of their two symbols, ash Ñ 0. Moreover the symbols of such homogeneous components of B are nothing but the homogeneous parts of the symbol of B by Corollary A.2. Finally, by iteration of Lemma A.1, iteration precisely estimated in Proposition A.3, we see that the symbol of the quantum normal form is, term by term in the mould-comould expansion, graduated by the length of the words, close to the classical normal form, ash Ñ 0.
Our next result expresses quantitatively this fact, improving the results of [G87] and [D91].
For N ě 1, we will denote by Z Q N (resp. Z C N ) the quantum (resp. classical) normal form of X Q 0`B Q (resp. X C 0`B C ) as expressed in Theorem C (resp. Theorem B). Here ihω¨∇ as in Section 6 and X C 0 px, ξq " 1 2´d ř j"1 ξ 2 j`d ř j"1 ω 2 j x 2 j¯o r ω¨ξ as in Section 5. Note that, in both cases, X Q 0 " Op W pX C 0 q. Let us recall that ω satisfies the Diophantine condition (3.6) with parameters α, τ .
N´1qτ´N`2 epρ´ρ 1 q¯N`2 and F N is defined by (4.3).

Note that, when
Proof. By Theorem C paq, we have that Using now the first estimate of Corollary 4.2 with pη r , τ r q given by (6.4), and the definition (3.2), we get ÿ λ1,...,λN PΛ F λ1,...,λN where we have used (5.11) for the last inequality.
Remark 7.1. Theorem D implies σ Z Q N´Z C N ρ 1 ďh 2 C 1 N B 2 ρ p1´ B ρ q´1 with C 1 N " maxtC 2 , . . . , C N u, but this is less precise than the result stated above.
Note that the correction is of order 2 in the Planck constant, which means that the classical perturbation theory incorporates the entire Bohr-Sommerfeld quantization, including the Maslov index.

Appendix A. Weyl quantization and all that
Weyl quantization has been defined in Section 6. Defining the unitary operator U pq, pq where pq, pq are the Fourier variables of P " T˚R d or T˚T d by: U pq, pqφpxq " e i pq 2 e ipx e´qh ∇ φpxq " e i pq 2 e ipx φpx´hqq, φ P L 2 pR d or T d q, the Weyl quantization of a function σ V on P is the operator where dpdq is used for dµpq, pq as in Section 5 page 15 and¨Ź is the symplectic Fourier transform defined by (5.1) 4 .
Obviously, as mentioned earlier, U pq, pq L 2 ÑL 2 " 1 and therefore Note that (A.1) makes also sense when σ V is a polynomial on P (polynomial in the variable ξ in the case P " T˚T d ) since U pq, pq as defining an unbounded operator. One check easily that this is the case for X 0 in the two examples of Section 5 and 6.
The following result is the fundamental one concerning the transition quantum-classical and, as presented here, is the only one we really need in the present article.
Lemma A.1. Let V " Op W pσ V q, V 1 " Op W pσ V 1 q with σ V , σ V 1 either belong to J ρ for a certain ρ ą 0 or are polynomials on P. Then where˚h is defined through the Fourier transform by The goal of this appendix is not to give a crash course on pseudo-differential operators, but rather to recall the strict minimum used in the present paper. The reader is referred to [F89] for a general exposition. The reader not familiar with the presentation here can recognize easily the Weyl quantization of a symbol σ V being, e.g. of the Schwartz class. That is to say that, when σ V P SpR 2d q, V defined by (A.1) acts on a function ϕ P L 2 pR d q through the formula and, in the case where σ V P C 8 pT d q b SpR d q, V acts on a function ϕ P L 2 pT d q by the same formula where, in (A.3), it is understood that σ V p¨, ξq and ϕ are extended to R d by periodicity (see [P14] In particular lim hÑ0 σ˚h σ 1 " tσ, σ 1 u. and σ 2˚h σ 1 " tσ 2 , σ 1 u when σ 2 is a quadratic form. Therefore, in this case, Using (6.3) and (A.6) we get the following result.
Corollary A.2. Let X 0 be as in Section 6. Then the homogeneous component B λ , λ " k¨ω, k P Z d of any pseudodifferential operator B is the Weyl quantization of the λ´homogeneous part (with respect to σ X0 ) of σ B , that is Using the operator A of Lemma A.1 and similar arguments that the ones used in the proof of Lemma B.1 we get the following result.
Proof. The proof will be using the methods of the one of Lemma D.1.
We have obviously that Therefore, using a last time the magic tool x β e´η x ď´β eη¯β , β, η, x ě 0, we get that d ÿ k"1 |x k | 3 ź l‰k |x l |e ρ 1 p|q1|`|p1|`¨¨¨`|q d |`|p d |q ďˆd`2 epρ´ρ 1 q˙d`2 e ρp|q1|`|p1|`¨¨¨`|q d |`|p d |q . (A.7) Defining P k " p k´pk´1 and Q k " q k´qk´1 and using in e ρp|q1|`|p1|`¨¨¨`|q d |`|p d |q , we get the result by (A.7) and the change of variables pp k , q k q Ñ pP k , Q k q (note that the covariance property with respect to the flow generated by X 0 is exactly the same as explained in the beginning of the proof of Lemma D.1).
Hence, by induction, we obtain, since δ 0 " δ and G 0 :" Y , The proof of (B.3) follows exactly the same lines.
Proof. We will first prove, in the two cases P " R dˆRd and P " T˚T d the following identity.
The same argument, used this time the weighted sum in k, leads to the next result.

Appendix D. Estimating commutators
It has been proven in [B99] for L 2 pR d q and [G12, P14] for L 2 pT d q the following Lemma.