Sandro Graffi
email: graffi@dm.unibo.it

Thierry Paul
email: paul@math.polytechnique.fr

Thierry Paul Convergence

CONVERGENCE OF A QUANTUM NORMAL FORM AND AN EXACT QUANTIZATION FORMULA

d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

1. Introduction 1.1. Quantization formulae. The establishment of a quantization formula (QF) for the eigenvalues of the Schrödinger operators is a classical mathematical problem of quantum mechanics (see e.g. [FM]). To review the notion of QF, consider first a semiclassical pseudodifferential operator H (for this notion, see e.g. [Ro]) acting on L 2 (R l ), l ≥ 1, of order m, self-adjoint with pure-point spectrum, with (Weyl) symbol σ H (ξ, x) ∈ C ∞ (R l × R l ; R).

Definition 1.1. We say that H admits an M -smooth exact QF, M ≥ 2, if there exists a function µ : (A, ) → µ(A, ) ∈ C M (R l × [0, 1]; R) such that:

(1) µ(A, ) admits an asymptotic expansion up to order M in uniformly on compacts with respect to A ∈ R l ;

(2) ∀ ∈]0, 1], there is a sequence n k := (n k 1 , . . . , n k l ) ⊂ Z l such that all eigenvalues λ k ( ) of H admit the representation:

λ k ( ) = µ(n k , ).
(1.1)

Remark 1.2. (Link with the Maslov index) Consider any function f : R l → R l with the property f (A), ∇µ(A, 0) = ∂ µ(A, 0). Then we can rewrite the asymptotic expansion of µ at second order as :

µ(n k , ) = µ(n k + f (n k )) + O( 2 ). (1.2)
When f (m ) = ν, ν ∈ Q l , the Maslov index [Ma] is recovered. Moreover, when

|λ k ( ) -µ(n k , )| = O( M ), → 0, M ≥ 2 (1.3)
then we speak of approximate QF of order M .

Example 1.3. (Bohr-Sommerfeld-Einstein formula). Let σ H fulfill the conditions of the Liouville-Arnold theorem (see e.g. [START_REF] Convergence | Arnold Mathematical Methods of Classical Mechanics[END_REF], §50). Denote A = (A 1 , . . . , A l ) ∈ R l the action variables, and E(A 1 , . . . , A l ) the symbol σ H expressed as a function of the action variables. Then the Bohr-Sommerfeld-Einstein formula (BSE) QF is λ n, = E((n 1 + ν/4) , . . . , (n l + ν/4) ) + O( 2 ) (1.4)

where ν = ν(l) ∈ N ∪ {0} is the Maslov index [Ma]. When H is the Schrödinger operator, and σ H the corresponding classical Hamiltonian, (1.4) yields the approximate eigenvalues, i.e.

the approximate quantum energy levels. In the particular case of a quadratic, positive definite

Hamiltonian, which can always be reduced to the harmonic oscillator with frequencies ω 1 > 0, . . . , ω l > 0, the BSE is an exact quantization formula in the sense of Definition 1.1 with ν = 2, namely:

µ(A, ) = E(A 1 + /2, . . . , A l + /2) = l k=1 ω k (A k + /2)
To our knowledge, if l > 1 the only known examples of exact QF in the sense of Definition 1.1 correspond to classical systems integrable by separation of variables, such that each separated system admits in turn an exact QF, as in the case of the Coulomb potential (for exact QFs for general one-dimensional Schrödinger operators see [Vo]). For general integrable systems, only the approximate BSE formula is valid. Non-integrable systems admit a formal approximate QF, the so-called Einstein-Brillouin-Keller (EBK), recalled below, provided they possess a normal form to all orders.

In this paper we consider a perturbation of a linear Hamiltonian on T * T l = R l × T l , and prove that the corresponding quantized operator can be unitarily conjugated to a function of the differentiation operators via the construction of a quantum normal form which converges uniformly with respect to ∈ [0, 1]. This yields immediately an exact, ∞-smooth QF. The uniformity with respect to yields also an explicit family of classical Hamiltonians admitting a convergent normal form, thus making the system integrable.

1.2. Statement of the results. Consider the Hamiltonian family H ε : R l × T l → R, (ξ, x) → H ε (ξ, x), indexed by ε ∈ R, defined as follows:

H ε (ξ, x) := L ω (ξ) + εV(x, ξ); L ω (ξ) := ω, ξ , ω ∈ R l , V ∈ C ∞ (R l × T l ; R).
(1.5)

Here ξ ∈ R l , x ∈ T l are canonical coordinates on the phase space R l × T l , the 2l-cylinder. L ω (ξ)

generates the linear Hamiltonian flow ξ i → ξ i , x i → x i + ω i t on R l × T l . For l > 1 the dependence of V on ξ makes non-trivial the integrability of the flow of H ε when ε = 0, provided the frequencies ω := (ω 1 , . . . , ω l ) are independent over Q and fulfill a diophantine condition such as (1.25) below.

Under this assumption it is well known that H ε admits a normal form at any order (for this notion, see e.g. [START_REF]Arnold Geometric Methods in the Theory of Ordinary Differential Equations[END_REF], [SM]). Namely, ∀ N ∈ N a canonical bijection C ε,N : R l × T l ↔ R l × T l close to the identity can be constructed in such a way that:

(H ε • C ε,N )(ξ, x) = L ω (ξ) + N k=1 B k (ξ; ω)ε k + ε N +1 R N +1,ε (ξ, x) (1.6)
This makes the flow of H ε (ξ, x) integrable up to an error of order ε N +1 . In turn, C ε,N is the Hamiltonian flow at time 1 generated by

W N ε (ξ, x) := ξ, x + N k=1 W k (ξ, x)ε k , (1.7)
where the functions W k (ξ, x) : R l × T l → R are recursively computed by canonical perturbation theory via the standard Lie transform method of Deprit [De] and Hori [Ho] (see also e.g [Ca]).

To describe the quantum counterpart, let H ε = L ω + εV be the operator in L 2 (T l ) of symbol H ε , with domain D(H ε ) = H 1 (T l ) and action specified as follows:

∀u ∈ D(H ε ), H ε u = L ω u + V u, L ω u = l k=1 ω k D k u, D k u := -i ∂ x k u, (1.8)
and V is the Weyl quantization of V (formula (1.26) below).

Since uniform quantum normal forms (see e.g. [Sj], [BGP], [START_REF] Popov | Invariant Tori, Effective Stability, and Quasimodes with Exponentially Small Error Terms I -Birkhoff Normal Forms[END_REF], [START_REF] Popov | Invariant Tori, Effective Stability, and Quasimodes with Exponentially Small Error Terms II -Quantum Birkhoff Normal Forms[END_REF]) are not so well known as the classical ones, let us recall here their definition. The construction is reviewed in Appendix.

Definition 1.4. [Quantum normal form (QNF)] We say that a family of operators H ε ε-close (in the norm resolvent topology) to H 0 = L ω admits a uniform quantum normal form (QNF) at any order if (i) There exists a sequence of continuous self-adjoint operators W k ( ) in L 2 (T l ), k = 1, . . . and a sequence of functions B k (ξ 1 , . . . , ξ l , ) ∈ C ∞ (R l × [0, 1]; R), such that, defining ∀ N ∈ N the family of unitary operators:

U N,ε ( ) = e iW N,ε ( )/ , W N,ε ( ) = N k=1 W k ( )ε k (1.9)
we have:

U N,ε ( )H ε U * N,ε ( ) = L ω + N k=1 B k (D 1 , . . . , D l , )ε k + ε N +1 R N +1,ε ( ). (1.10) (ii) The operators B k (D, ) : k = 1, 2 . . ., R N +1 are continuous in L 2 (T l ); the corresponding symbols W k , B k , R N +1 (ε) belong to C ∞ (R l × T l × [0, 1]
), and reduce to the classical normal form construction (1.6) and (1.7) as → 0:

B k (ξ; 0) = B k (ξ); W k (ξ, x, 0) = W k (ξ, x), R N +1,ε (x, ξ; 0) = R N +1,ε (x, ξ) (1.11)
(1.10) entails that H ε commutes with H 0 up to an error of order ε N +1 ; hence the following approximate QF formula holds for the eigenvalues of H ε :

λ n,ε ( ) = n, ω + N k=1 B k (n 1 , . . . , n l , )ε k + O(ε N +1 ).
(1.12) Definition 1.5. (Uniformly convergent quantum normal forms) We say that the QNF converges M -smoothly, M > 2l, uniformly with respect to the Planck constant , if there is

ε * > 0 such that ∞ k=1 sup R l ×T l ×[0,1] |α|≤M |D α W k (ξ, x; )ε k | < +∞ (1.13) ∞ k=1 sup R l ×[0,1] |α|≤M |D α B k (ξ, )ε k | < +∞, |ε| < ε * .
(1.14)

Here D α = ∂ α 1 ξ ∂ α 2 x ∂ α 3 , |α| = |α 1 | + |α 2 | + α 3 .
(1.13,1.14) entail that, if |ε| < ε * , we can define the symbols

W ∞ (ξ, x; ε, ) := ξ, x + ∞ k=1 W k (ξ, x; )ε k ∈ C M (R l × T l × [0, ε * ] × [0, 1]; C), (1.15) B ∞ (ξ; ε, ) := L ω (ξ) + ∞ k=1 B k (ξ; )ε k ∈ C M (R l × [0, ε * ] × [0, 1]; C) (1.16) By the Calderon-Vaillancourt theorem (see §3 below) their Weyl quantizations W ∞ (ε, ), B ∞ (ε, )
are continuous operator in L 2 (T l ). Then:

e iW∞(ε, )/ H ε e -iW∞(ε, )/ = B ∞ (D 1 , . . . , D l ; ε, ).
(1.17)

Therefore the uniform convergence of the QNF has the following straightforward consequences:

(A1) The eigenvalues of H ε are given by the exact quantization formula:

λ n ( , ε) = B ∞ (n , , ε), n ∈ Z l , ε ∈ D * := {ε ∈ R | |ε| < ε * } (1.18) (A2)
The classical normal form is convergent, uniformly on compacts with respect to ξ ∈ R l , and therefore if ε ∈ D * the Hamiltonian H ε (ξ, x) is integrable.

Let us now state explicit conditions on V ensuring the uniform convergence of the QNF.

Given

F(t, x) ∈ C ∞ (R × T l ; R), consider its Fourier expansion F(t, x) = q∈Z l
F q (t)e i q,x .

(1.19) and define furthermore

F ω : R l × T l → R; F ω ∈ C ∞ (R l × T l ; R)
in the following way:

F ω (ξ, x) := F(L ω (ξ), x) = q∈Z l
F ω,q (ξ)e i q,x , (1.20)

F ω,q (ξ) := (F q • L ω )(ξ) = 1 (2π) l/2 R F q (p)e -ipLω(ξ) dp = (1.21) = 1 (2π) l/2 R F q (p
)e -i pω,ξ dp, pω := (pω 1 , . . . , pω l ).

(1.22)

Here, as above, L ω (ξ) = ω, ξ .

Given ρ > 0, introduce the weighted norms:

F ω,q (ξ) ρ := R | F q (p)|e ρ|p| | dp (1.23) F ω (x, ξ) ρ := q∈Z l e ρ|q| F ω,q ρ (1.24)
We can now formulate the main result of this paper. Assume:

(H1) There exist γ > 1, τ > l -1 such that the frequencies ω fulfill the diophantine condition

| ω, q | -1 ≤ γ|q| τ , q ∈ Z l , q = 0. (1.25) (H2) V ω is the Weyl quantization of V ω (ξ, x) (see Sect.3 below), that is: V ω f (x) = R q∈Z l
V q (p)e i q,x + p ω,q /2 f (x + pω) dp, f ∈ L 2 (T l ).

(1.26) with V(ξ, x; ) = V( ω, ξ , x) = V ω (ξ, x) for some function V(t; x) : R × T l → R.

(H3)

V ω ρ < +∞, ρ > 1 + 16γτ τ .
Clearly under these conditions the operator family

H ε := L ω + εV ω , D(H ε ) = H 1 (T l ), ε ∈ R, is self-adjoint in L 2 (T l
) and has pure point spectrum. We can then state the main results.

Theorem 1.6. Under conditions (H1-H3), H ε admits a uniformly convergent quantum normal form B ∞,ω (ξ, ε, ) in the sense of Definition 1.5, with radius of convergence no smaller than: .27) If in addition to (H1-H2) we assume, for any fixed r ∈ N:

ε * (γ, τ ) := 1 e 24(3+2τ ) 2 2τ V ρ . ( 1
(H4)

ρ > λ(γ, τ, r) := 1 + 8γτ [(2(r + 1) 2 ] (1.28)
we can sharpen the above result proving smoothness with respect to :

Theorem 1.7. Let conditions (H1-H2-H4) be fulfilled. For r ∈ N define D * r := {ε ∈ C | |ε| < ε * (γ, τ, r)}, where:

ε * (γ, τ, r) := 1 e 24(3+2τ ) (r + 2) 2τ V ρ (1.29) Then → B ∞ (t, ε, ) ∈ C ∞ ([0, 1]; C ω ({t ∈ C | |ℑt| < ρ/2 × D * r (ρ)}); i.e. there exist C r (ε * ) > 0 such that, for ε ∈ D * r : r γ=0 max ∈[0,1] ∂ γ B ∞,ω (ξ; ε, ) ρ/2 ≤ C r , r = 0, 1, . . . (1.30)
In view of Definition 1.1, the following statement is a straightforward consequence of the above Theorems:

Corollary 1.8 (Quantization formula). H ε admits an ∞-smooth quantization formula in the sense of Definition 1.1. That is, ∀ r ∈ N, ∀ |ǫ| < ε * (γ, τ, r)
given by (1.29), the eigenvalues of H ε are expressed by the formula:

λ(n, , ε) = B ∞,ω (n , ε, ) = L ω (n ) + ∞ s=1 B s (L ω (n ), )ε s (1.31) where B ∞,ω (ξ, ε, ) belongs to C r (R l × [0, ε * (•, r)] × [0, 1]
), and admits an asymptotic expansion at order r in , uniformly on compacts with respect to

(ξ, ε) ∈ R l × [0, ε * (•, r)].

Remarks

(i) (1.30) and (1.31) entail also that the Einstein-Brillouin-Keller (EBK) quantization formula:

λ EBK n,ε ( ) := L ω (n ) + ∞ s=1 B s (L ω (n ))ε s = B ∞,ω (n , ε), n ∈ Z l (1.32)
reproduces here Spec(H ε ) up to order .

(ii) Apart the classical Cherry theorem yielding convergence of the Birkhoff normal form for smooth perturbations of the harmonic flow with complex frequencies when l = 2 (see e.g.

[SM], §30; the uniform convergence of the QNF under these conditions is proved in [GV]), no simple convergence criterion seems to be known for the QNF nor for the classical NF as well. (See e.g. [PM], [Zu], [St] for reviews on convergence of normal forms). Assumptions

(1) and (2) of Theorem 1.6 entail Assertion (A2) above. Hence they represent, to our knowledge, a first explicit convergence criterion for the NF.

Remark that L ω (ξ) is also the form taken by harmonic-oscillator Hamiltonian in R 2l ,

P 0 (η, y; ω) := l s=1 ω s (η 2 s + y 2 s ), (η s , y s ) ∈ R 2 , s = 1, . . . , l
if expressed in terms of the action variables ξ s > 0, s = 1, . . . , l, where

ξ s := η 2 s + y 2 s = z s z s , z s := y s + iη s .
Assuming (1.25) and the property

B k (ξ) = (F k • L ω (ξ)) = F k ( l s=1 ω s z s z s ), k = 0, 1, . . . (1.33)
Rüssmann [Ru] (see also [Ga]) proved convergence of the Birkhoff NF if the perturbation V, expressed as a function of (z, z), is in addition holomorphic at the origin in C 2l . No explicit condition on V seems to be known ensuring both (1.33) and the holomorphy. In this case instead we prove that the assumption

V(ξ, x) = V(L ω (ξ), x) entails (1.33), uniformly in ∈ [0, 1]; namely, we construct F s (t; ) : R × [0, 1] → R such that: B s (ξ; ) = F s (L ω (ξ); ) := F ω,s (ξ; ), s = 0, 1, . . . (1.34)
The conditions of Theorem 1.6 cannot however be transported to Rüssmann's case: the map

T (ξ, x) = (η, y) := η i = - √ ξ i sin x i , y i = √ ξ i cos x i , i = 1, . . . , l,
namely, the inverse transformation into action-angle variable, is defined only on R l + × T l and does not preserve the analyticity at the origin. On the other hand, T is an analytic, canonical map between R l + × T l and R 2l \ {0, 0}. Assuming for the sake of simplicity V 0 = 0 the image of H ε under T is:

(H ε • T )(η, y) = l s=1 ω s (η 2 s + y 2 s ) + ε(V • T )(η, y) := P 0 (η, y) + εP 1 (η, y) (1.35)
where

P 1 (η, y) = (V • T )(η, y) = P 1,R (η, y) + P 1,I (η, y), (η, y) ∈ R 2l \ {0, 0}.
(1.36) localization; therefore, and this is the main difference, at each step the coefficients of the homological equation for the operator symbols not only have an additional dependence on but also have to be controlled up to infinity. These difficulties are overcome by exploiting the closeness to the identity of the whole procedure, introducing adapted spaces of symbols i(Section 2), which account also for the properties of differentiability with respect to the Planck constant. The link between quantum and classical settings is provided by a sharp (i.e. without ∞ approximation) Egorov Theorem established in section 4. Estimates for the solution of the quantum homological equation and their recursive properties are obtained in sections 5.1 (Theorem 5.3) and 5.2 (Theorem 5.5) respectively. Recursive estimates are established in Section 6 (Theorem 6.4) and the proof of our main result is completed in section 7. The link with the usual construction of the quantum normal form described in Appendix.

P 1,R (η, y) = 1 2 k∈Z l (ℜV k • H 0 )(η, y) l s=1 η s -iy s η 2 s + y 2 s ks P 1,I (η, y) = 1 2 k∈Z l (ℑV k • H 0 )(η,

Norms and first estimates

Let m, l = 1, 2, . . . . For F ∈ C ∞ (R m × T l × [0, 1]; C), (ξ, x, ) → F(ξ, x; ), and G ∈ C ∞ (R m × [0, 1]; C), (ξ, ) → G(ξ;
), consider the Fourier transforms

G(p; ) = 1 (2π) m/2 R m G(ξ; )e -i p,ξ dx (2.1) F(ξ, q; ) := 1 (2π) m/2
T l F(ξ, x; )e -i q,x dx.

(2.2)

F(ξ, x; ) = q∈Z l
F(ξ, q; )e -i q,x (2.3)

F(p, q; ) = 1 (2π) m/2 R m F(ξ, q; )e -i p,ξ dx (2.4)
It is convenient to rewrite the Fourier representations (2.3, 2.4) under the form a single Lebesgue-Stieltjes integral. Consider the product measure on R m × R l :

dλ(t) := dp dν(s), t := (p, s) ∈ R m × R l ;
(2.5)

dp := m k=1 dp k ; dν(s) := l h=1 q h ≤s h δ(s h -q h ), q h ∈ Z, h = 1, . . . , l (2.6)
Then:

F(ξ, x; ) = R m ×R l
F(p, s; )e i p,ξ +i s,x dλ(p, s) (2.7)

Definition 2.1. For ρ ≥ 0, σ ≥ 0, we introduce the weighted norms

|G| † σ := max ∈[0,1] G(.; ) L 1 (R m ,e σ|p| dp) = max ∈[0,1] R l G(.; ) e σ|p| dp.
(2.8)

|G| † σ,k := max ∈[0,1] k j=0 (1 + |p| 2 ) k-j 2 ∂ j G(.; ) L 1 (R m ,e σ|p| dp) ; |G| † σ;0 := |G| † σ .
(2.9)

Remark 2.2. By noticing that |p| ≤ |p ′ -p| + |p ′ | and that, for x ≥ 0, x j e -δx ≤ 1 e ( j δ ) j , we immediately get the inequalities

|FG| † σ ≤ |F| σ |G| σ , (2.10) |(I -∆ j/2 )F| σ-δ ≤ 1 e j δ j |F| σ , k ≥ 0. (2.11) Set now for k ∈ N ∪ {0}: µ k (t) := (1 + |t| 2 ) k 2 = (1 + |p| 2 + |s| 2 ) k 2 .
(2.12) and note that

µ k (t -t ′ ) ≤ 2 k 2 µ k (t)µ k (t ′ ). (2.13) because |x -x ′ | 2 ≤ 2(|x| 2 + |x ′ | 2 ). Definition 2.3. Consider F(ξ, x; ) ∈ C ∞ (R m × T l × [0, 1]; C), with Fourier expansion F(ξ, x; ) = q∈Z l
F(ξ, q; )e i q,x (2.14)

(1) Set:

F † ρ,k := max ∈[0,1] k γ=0 R m ×R l |µ k-γ (p, s)∂ γ F(p, s; )|e ρ(|s|+|p|) dλ(p, s). (2.15) (2) Let O ω be the set of functions Φ : R l × T l × [0, 1] such that Φ(ξ, x; ) = F(L ω (ξ), x; ) for some F : R × T l × [0, 1] → C. Define, for Φ ∈ O ω : Φ ρ,k := max ∈[0,1] k γ=0 R |µ k-γ (pω, q)∂ γ F(p, s; )|e ρ(|s|+|p| dλ(p, s).
(2.16)

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

J † k (ρ) = {F | F † ρ,k < ∞},
(2.17)

J † k (ρ) = {Op W (F) | F ∈ J † (ρ, k)}, (2.18) J k (ρ) = {F ∈ O ω | F ρ,k < ∞}, (2.19) J k (ρ) = {F | F ρ,k < ∞}, (2.20)
Finally we denote:

L 1 σ (R m ) := L 1 (R m , e σ|p| dp).
Remark 2.4. Note that, if F(ξ, q, ) is independent of q, i.e. F(ξ, q, ) = F(ξ, )δ q,0 , then:

F † ρ,k = |F| † ρ,k ; F ρ,k = |F| ρ,k (2.21)
while in general

F ρ,k ≤ F ρ ′ ,k ′ whenever k ≥ k ′ , ρ ≤ ρ ′ ; (2.22) Remark 2.5. (Regularity properties) Let F ∈ J † k (ρ), k ≥ 0. Then: (1) There exists K(α, ρ, k) such that max ∈[0,1] F(ξ, x; ) C α (R m ×T l ) ≤ K F † ρ,k , α ∈ N (2.23)
and analogous statement for the norm • ρ,k .

(

) Let ρ > 0, k ≥ 0. Then F(ξ, x; ) ∈ C k ([0, 1]; C ω ({|ℑξ| < ρ} × {|ℑx| < ρ}) and sup {|ℑξ|<d}×{|ℑx|<d} ≤ F † ρ,k . (2.24) 2
Analogous statements for F ∈ J k (ρ).

We will show in section 3 that:

Op W (F ) B(L 2 ) ≤ F ρ,k ∀k, ρ > 0. (2.25)
In what follows we will often use the notation F also to denote the function F(L ω (ξ)), because the indication of the belonging to J or J † , respectively, is already sufficient to mark the distinction of the two cases.

Remark 2.6. Without loss of generality we may assume:

|ω| := |ω 1 | + . . . + |ω l | ≤ 1 (2.26)
Indeed, the general case |ω| = α|ω ′ |, |ω ′ | ≤ 1, α > 0 arbitrary reduces to the former one just by the rescaling ε → αε.

3. Weyl quantization, matrix elements, commutator estimates 3.1. Weyl quantization: action and matrix elements. We sum up here the canonical (Weyl) quantization procedure for functions (classical observables) defined on the phase space R l × T l . In the present case it seems more convenient to consider the representation (unique up to unitary equivalences) of the natural Heisenberg group on R l × T l . Of course this procedure yields the same quantization as the standard one via the Brézin-Weil-Zak transform (see e.g. [Fo], §1.10)

and has already been employed in [CdV], [START_REF] Popov | Invariant Tori, Effective Stability, and Quasimodes with Exponentially Small Error Terms I -Birkhoff Normal Forms[END_REF], [START_REF] Popov | Invariant Tori, Effective Stability, and Quasimodes with Exponentially Small Error Terms II -Quantum Birkhoff Normal Forms[END_REF]).

Let H l (R l × R l × R) be the Heisenberg group over R 2l+1 (see e.g. [Fo], Chapt.1). Since the dual space of R l × T l under the Fourier transformation is R l × Z l , the relevant Heisenberg group here

is the subgroup of H l (R l × R l × R), denoted by H l (R l × Z l × R), defined as follows: Definition 3.1. Let u := (p, q), p ∈ R l , q ∈ Z l , and let t ∈ R. Then H l (R l × Z l × R) is the subgroup of H l (R l × R l × R) topologically equivalent to R l × Z l × R with group law (u, t) • (v, s) = (u + v, t + s + 1 2 Ω(u, v)) (3.1)
Here

Ω(u, v) is the canonical 2-form on R l × Z l : Ω(u, v) := u 1 , v 2 -v 1 , u 2 (3.2) H l (R l × Z l × R)
is the Lie group generated via the exponential map from the Heisenberg Lie

algebra HL l (Z l × R l × R) defined as the vector space R l × Z l × R with Lie bracket [(u, t) • (v, s)] = (0, 0, Ω(u, v)) (3.3)
The unitary representations of

H l (R l × Z l × R) in L 2 (T l ) are defined as follows (U (p, q, t)f )(x) := e i t+i q,x + p.q /2 f (x + p) (3.4) ∀ = 0, ∀ (p, q, t) ∈ H l , ∀ f ∈ L 2 (T l
). These representations fulfill the Weyl commutation relations

U (u) * = U (-u), U (u)U (v) = e i Ω(u,v) U (u + v) (3.5)
For any fixed > 0 U defines the Schrödinger representation of the Weyl commutation relations, which also in this case is unique up to unitary equivalences (see e.g. [Fo], §1.10).

Consider now a family of smooth phase-space functions indexed by , A(ξ, x, ) : R l ×T l ×[0, 1] → C, written under its Fourier representation

A(ξ, x, ) = R l q∈Z l A(p, q; )e i( p.ξ + q,x ) dp = R l ×R l A(p, s; )e i( p.ξ + s,x ) dλ(p, s) (3.6)
Definition 3.2. The (Weyl) quantization of A(ξ, x; ) is the operator A( ) definde as

(A( )f )(x) := R l q∈Z l A(p, q; )U (p, q)f (x) dp (3.7) = R l ×R l A(p, s; )U (p, s)f (x) dλ(p, s) f ∈ L 2 (T l )
Remark 3.3. Formula (3.7) can be also be written as

(A( )f )(x) = q∈Z l A(q, )f, (A(q, )f )(x) = R l A(p, q; )U (p, q)f (x) dp (3.8)
From this we compute the action of A( ) on the canonical basis in L 2 (T l ):

e m (x) := (2π) -l/2 e i m,x , x ∈ T l , m ∈ Z l .
Lemma 3.4.

A( )e m (x) =

q∈Z l e i (m+q),x A( (m + q/2), q, ) (3.9)

Proof. By (3.8), it is enough to prove that the action of A(q, ) is

A(q, )e m (x) = e i (m+q),x A( (m + q/2), q, ) (3.10)
Applying Definition 3.2 we can indeed write:

(A(q, )e m )(x) = (2π) -l/2 R l
A(p, q; )e i q,x +i p,q /2 e i m,(x+ p) dp

= (2π) -l/2 e i (m+q),x R l
A(p; q, )e i p,(m+q/2) dp = e i (m+q),x A( (m + q/2), q, ).

. We note for further reference an obvious consequence of (3.10):

A(q, )e m , A(q, )e n L 2 (T l ) = 0, m = n; A(r, )e m , A(q, )e n L 2 (T l ) = 0, r = q. (3.11)
As in the case of the usual Weyl quantization, formula (3.7) makes sense for tempered distributions A(ξ, x; ) [Fo]. Indeed we prove in this context, for the sake of completeness, a simpler, but less general, version of the standard Calderon-Vaillancourt criterion:

Proposition 3.5. Let A( ) by defined by (3.7). Then

A( ) L 2 →L 2 ≤ 2 l+1 l + 2 • π (3l-1)/2 Γ( l+1 2 ) |α|≤2k ∂ k x A(ξ, x; ) L ∞ (R l ×T l ) .
(3.12)

where

k =      l 2 + 1, l even l+1 2 + 1, l odd.
Proof. Consider the Fourier expansion

u(x) = m∈Z l u m e m (x), u ∈ L 2 (T l ).
Since:

A(q, ) u m e m 2 = |A( (m + q/2), q, )| 2 • | u m | 2
by Lemma 3.4 and (3.11) we get:

A( )u 2 ≤ (q,m)∈Z l ×Z l A(q, ) u m e m 2 = (q,m)∈Z l ×Z l |A( (m + q/2), q, )| 2 • | u m | 2 ≤ q∈Z l sup ξ∈R l |A(ξ, q, )| 2 m∈Z l | u m | 2 = q∈Z l sup ξ∈R l |A(ξ, q, )| 2 u 2 ≤ q∈Z l sup ξ∈R l |A(ξ, q, )| 2 u 2 Therefore: A( ) L 2 →L 2 ≤ q∈Z l sup ξ∈R l |A(ξ, q, )|.
Integration by parts entails that, for k ∈ N, and ∀ g ∈ C ∞ (T l ):

T l e i q,x g(x)dx = 1 1 + |q| 2k T l e i q,x (1 + (-△ x ) k )g(x)dx ≤ 1 1 + |q| 2k (2π) l sup T l |α|≤2k |∂ α x g(x)|.
Let us now take:

k =      l 2 + 1, l even l+1 2 + 1, l odd =⇒ 2k -l + 1 = 3, l even 2k -l + 1 = 2, l odd (3.13)
Then 2kl + 1 ≥ 2, and hence:

q∈Z l 1 1 + |q| 2k ≤ 2 R l du 1 • • • du l 1 + u 2k ≤ 2 π (l-1)/2 Γ( l+1 2 ) ∞ 0 ρ l-1 1 + ρ 2k dρ. Now: ∞ 0 ρ l-1 1 + ρ 2k dρ = 1 2k ∞ 0 u l/2k-1 1 + u du ≤ 1 2k 1 0 u l/2k-1 du + ∞ 1 u l/2k-2 du = 1 (4k -l)(2k -l)
This allows us to conclude:

q∈Z l sup ξ |A(ξ, q, )| ≤ (2π) l |α|≤2k ∂ α x A(ξ, x; ) L ∞ (R l ×T l ) • q∈Z l 1 1 + |q| 2k ≤ 2 l+1 • π (3l-1)/2 Γ( l+1 2 ) 1 l + 2 |α|≤2k ∂ k x A(ξ, x; ) L ∞ (R l ×T l ) .
with k given by (3.13). This proves the assertion.

Remark 3.6. Thanks to Lemma 3.4 we immediately see that, when

A(ξ, x, ) = F(L ω (ξ), x; ), A( )f = R q∈Z l F(p, q; )U h (pω, q)f dp (3.14) = R q∈Z l F(p, q; )e i q,x +i p ω,q /2 f (x + pω) dp f ∈ L 2 (T l )
where, again, pω := (pω 1 , . . . , pω l ). Explicitly, (3.10) and (3.9) become:

A( )e m (x) =
q∈Z l e i (m+q),x A( ω, (m + q/2) , q, ) (3.15)

A(q, )e m (x) = e i (m+q),x A( ω, (m + q/2) , q, ) (3.16)

Remark 3.7. If A does not depend on x, then A(ξ, q, ) = 0, q = 0, and (3.9) reduces to the standard (pseudo) differential action

(A( )u)(x) = m∈Z l A(m , ) u m e i m,x = m∈Z l A(-i ∇, ) u m e i m,x (3.17)
because -i ∇e m = m e m . On the other hand, if F does not depend on ξ (3.9) reduces to the standard multiplicative action

(A( )u)(x) = q∈Z l A(q, )e i q,x m∈Z l u m e i m,x = A(x, )u(x) (3.18) Corollary 3.8. Let A( ) : L 2 (T l ) → L 2 (T l
) be defined as in 3.2. Then:

(1) ∀ρ ≥ 0, ∀ k ≥ 0 we have:

A( ) L 2 →L 2 ≤ A † ρ,k (3.19) and, if A(ξ, x, ) = A(L ω (ξ), x; ) A( ) L 2 →L 2 ≤ A ρ,k . (3.20) (2)
e m+s , A(q, )e m = δ q,s A((m + q/2) , q, ) (3.21)

e m+s , A( )e m = A((m + s/2) , s, ) (3.22) and, if A(ξ, x, ) = F(L ω (ξ), x; )
e m+s , F (q, )e m = δ q,s F( ω, (m + q/2) , q, ) = δ q,s F(L ω (m + s/2) , q, ) (3.23)

e m+s , F ( )e m = F( ω, (m + s /2) , s, ) = F(L ω (m + s /2), s, ) (3.24)
Equivalently:

e m , A( )e n = A((m + n) /2, m -n, ) (3.25)
(3) A( ) is an operator of order -∞, namely there exists C(k, s) > 0 such that

A( )u H k (T l ) ≤ C(k, s) u H s (T l ) , (k, s) ∈ R, k ≥ s (3.26)
Proof.

(1) Formulae (3.19) and (3.20) are straighforward consequences of Formula (2.23).

(2) (3.23) immediately yields (3.24). In turn, (3.23) follows at once by (3.10).

(3) The condition A ∈ J (ρ) entails: sup

(ξ; )∈R l ×[0,1] |A(ξ; q, )|e ρ|q| ≤ e ρ|q| max ∈[0,1] A(p; q, ) 1 → 0, |q| → ∞. (3.27) Therefore: A( )u 2 H k ≤ (q,m)∈Z l ×Z l (1 + |q| 2 ) k A((m + q/2) , q, )| 2 • | u m | 2 ≤ q∈Z l sup q,m (1 + |q| 2 ) k |A((m + q/2) , q, )| 2 m∈Z l (1 + |m| 2 ) s | u m | 2 = C(k, s) u 2 H s C(k, s) := q∈Z l sup q,m (1 + |q| 2 ) k |A((m + q/2) , q, )| 2
where 0 < C(k, s) < +∞ by (3.27) above. The Corollary is proved.

3.2. Compositions, Moyal brackets. We first list the main properties which are straightforward consequences of the definition, as in the case of the standard Weyl quantization in R 2l . First introduce the abbreviations

t := (p, s); t ′ = (p ′ , s ′ ); ωt := (pω, s) (3.28) Ω ω (t ′ -t, t ′ ) := (p ′ -p)ω, s ′ -(s ′ -s), p ′ ω = p ′ ω, s -s ′ , pω . (3.29) Given F( ), G( ) ∈ J k (ρ)
, define their twisted convolutions:

( F( ) * G( ))(p, q; ) := R×R l F(t ′ -t; ) G(t ′ ; )e i[ Ωω(t ′ -t,t ′ )/2] dλ(t ′ ) (3.30) (F♯G)(x, ξ, ) := R×R l ( F( ) * G( ))(t, )e i s,x +pLω(ξ) dλ(t) (3.31) C(p, q; ) := 1 R×R l F(t ′ -t, ) G(t ′ , ) sin[ Ω ω (t ′ -t, t ′ )/2] dλ(t ′ ) (3.32) C(x, ξ; ) := R×R l C(p, s; )e ipLω(ξ)+i s,x dλ(t) (3.33)
Once more by the same argument valid for the Weyl quantization in R 2l :

Proposition 3.9. The following composition formulas hold:

F ( )G( ) = R×R l ( F( ) * G( ))(t; )U (ωt) dλ(t). (3.34) [F ( ), G( )] i = R×R l C(t; )U (ωt) dλ(t) (3.35) Remark 3.10. The symbol of the product F ( )G( ) is then (F♯G)(L ω (ξ), x, ) and the symbol of the commutator [F ( ), G( )]/i is C(L ω (ξ), x;
), which is by definition the Moyal bracket of the symbols F, G. From (3.32) we get the asymptotic expansion:

C(p, q; ω; ) = ∞ j=0 (-1) j 2j (2j + 1)! D j (p, q; ω) (3.36) D j (p, q; ω) := R×R l F(t ′ -t, ) G(t ′ , )[Ω ω (t ′ -t, t ′ ) j dλ(t ′ ) (3.37)
whence the asymptotic expansion for the Moyal bracket

{F, G} M (L ω (ξ), x; ) = {F, G}(L ω (ξ), x, ) + (3.38) ∞ |r+j|=0 (-1) |r| |r+j| r!sj [∂ r x ω∂ j L F(L ω (ξ), x)] • [ω∂ j L ∂ r x G(L ω (ξ), x, )] - - ∞ |r+j|=0 (-1) |r| |r+j| r!j! [∂ r x ω∂ j L G(L ω (ξ), x)] • [ω∂ j L ∂ r x F (L ω (ξ), x, )]
Remark that:

{F, G} M (L ω (ξ), x; ) = {F, G}(L ω (ξ), x) + O( ) (3.39) In particular, since L ω (ξ) is linear, we have ∀ F(ξ; x; ) ∈ C ∞ (R l × T l × [0, 1]): {F, L ω (ξ)} M (L ω (ξ), x; ) = {F, L ω (ξ)}(L ω (ξ), x; ) (3.40)
The observables F(ξ, x; ) ∈ J (ρ) enjoy the crucial property of stability under compositions of their dependence on L ω (ξ) (formulae (3.31) and (3.33) above). As in [BGP], we want to estimate the relevant quantum observables uniformly with respect to , i.e. through the weighted norm

(2.16).

Uniform estimates.

The following proposition is the heart of the estimates needed for the convergence of the KAM iteration. The proof will be given in the next (sub)section. Even though we could limit ourselves to symbols in J (ρ), we consider for the sake of generality and further reference also the general case of symbols belonging to J † (ρ).

Proposition 3.11. Let F , G ∈ J † k (ρ), k = 0, 1, . . ., d = d 1 + d 2 .
Let F, G be the corresponding symbols, and 0 < d + d 1 < ρ. Then:

(1 † ) F G ∈ J † k (ρ) and fulfills the estimate F G B(L 2 ) ≤ F♯G † ρ,k ≤ (k + 1)4 k F † ρ,k • G † ρ,k (3.41) (2 † ) [F, G] i ∈ J † k (ρ -d)
and fulfills the estimate

[F, G] i B(L 2 ) ≤ {F, G} M † ρ-d-d 1 ,k ≤ (k + 1)4 k e 2 d 1 (d + d 1 ) F † ρ,k G † ρ-d,k
(3.42)

(3 † ) FG ∈ J † k (ρ), and FG † ρ,k ≤ (k + 1)4 k F † ρ,k • G † ρ,k (3.43) Moreover if F , G ∈ J k (ρ), k = 0, 1, .
. ., and F, G ∈ J k (ρ), then:

(1) F G ∈ J k (ρ) and fulfills the estimate

F G B(L 2 ) ≤ F♯G ρ,k ≤ (k + 1)4 k F ρ,k • G ρ,k (3.44) (2) [F, G] i ∈ J k (ρ -d) and fulfills the estimate [F, G] i B(L 2 ) ≤ {F, G} M ρ-d-d 1 ,k ≤ (k + 1)4 k e 2 d 1 (d + d 1 ) F ρ,k • G ρ-d,k (3.45)
(3) FG ∈ J k (ρ) and

FG ρ,k ≤ (k + 1)4 k F ρ,k • G ρ,k . (3.46)
Remark 3.12. The operators F ( ) with the uniform norm F ρ,k , k = 0, 1, . . . form a Banach subalgebra (without unit) of the algebra of the continuous operators in L 2 (T l ).

Before turning to the proof we state and prove two further useful results.

Corollary 3.13. Let F, G ∈ J k (ρ), and let 0 < d < ρ, r ∈ N. Then:

1 r! {F, {F, . . . , {F, G} M } M . . .} M ρ-d,k ≤ √ 2πr(k + 1)4 k (ed)d r F r ρ,k G ρ,k (3.47)
Proof. We follow the argument of [BGP], Lemma 3.5.

If d = d 1 + d 2 , (3.42) entails: {F, G} M ρ-d,k ≤ C k e 2 dd 1 F ρ,k • G ρ-d 2 ,k , C k := (k + 1)4 k . because G ρ-d,k ≤ G ρ-d 2 ,k and d 1 (d + d 1 ) < d 1 d. Set now d 2 = r -1 r d which yields d 1 = d r .
Then:

{F, G} M ρ-d,k ≤ C k e 2 d d r F ρ,k • G ρ-r-1 r d,k = C k r (ed) 2 F ρ,k • G ρ-r-1 r d,k and
{F, {F, G} M } M ρ-d,k ≤ C k ed d r , k F ρ,k • {F, G} M ρ-r-2 r d,k ≤ ≤ (C k r) 2 (ed) 3 F 2 ρ,k • G ρ-r-1 r d,k
Iterating r times we get:

1 r! {F, {F, • • • , {F, G} M } M , • • • } M ρ-d,k ≤ (C k r) r r! 1 (ed) r+1 F r ρ,k • G ρ-r-1 r d,k .
The Stirling formula and the majorization G ρ-r-1 r d,k ≤ G ρ,k now yield (3.47).

Proposition 3.14. Let F(ξ; x;

) ∈ J k (ρ), ρ > 0, k = 0, 1, . . .. Then {F, L ω } M ∈ J k (ρ -d)
∀ 0 < d < ρ and the following estimates hold:

[F, L ω ]/i ρ-d,k = {F, L ω } M ρ-d,k ≤ 1 d F ρ,k (3.48) [F, [• • • , [F, L ω ] • • • ]/(i ) r ρ-d,k = {F, • • • , {F, L ω } M • • • , } M ρ-d,k (3.49) ≤ 2π(r -1)(k + 1)4 k (ed)d r F r ρ,k
Proof. By (3.40):

{F, L ω } M = {F, L ω } = -ω, ∇ x F(ξ, x; ) = q∈Z l
ω, q e i q,x R F q (p; )e ipLω(ξ) dp and therefore:

{F, L ω } M ρ-d,k ≤ {F, L ω } ρ-d,k ≤ q∈Z l | ω, q |e (ρ-d)|q| F q ρ,k ≤ sup q∈Z l ω, q |e -d|q| q∈Z l e ρ|q| F q ρ,k ≤ 1 d F ρ,k
because |ω| ≤ 1 by Remark 2.6. This proves (3.48). (3.49) is a direct consequence of Corollary 3.13.

3.4. Proof of Proposition 3.11.

3.4.1. Three lemmata. The proof will use the three following Lemmata.

Lemma 3.15. Let p, p ′ ∈ R l , s, s ′ ∈ R l . Define t := (p, s), t ′ := (p ′ , s ′ ). Let Ω ω (•) and µ j (•) be defined by (3.29) and (2.12), respectively. Then:

|Ω ω (t, t ′ )| j ≤ 2 j µ j (t)µ j (t ′ ). (3.50) The proof is straightforward, because |Ω ω (t, t ′ )| ≤ 2|t||t ′ | and |ω| ≤ 1. Lemma 3.16. d m d m sin x/2 ≤ |x| m+1 2 m+1 . (3.51)
.

Proof. Write:

d m d m 1 sin x/2 = d m d m 1 2 x 0 cos t/2 dt = (-) m 2 m+1 x 0 t m cos (m) ( t/2) dt ≤ m 2 m+1 x 0 t m dt. whence d m d m sin x/2 ≤ m 2 m+1 x 0 t m dt = m |x| m+1 2 m+1 (m + 1) ≤ |x| m+1 2 m+1 . Lemma 3.17. Let (F, G) ∈ J † ρ , 0 < d + d 1 < ρ, t = (p, s), t ′ = (p ′ , s ′ ), |t| := |p| + |s|, |t ′ | := |p ′ | + |s ′ |. Then: {F, G} M † ρ-d-d 1 ≤ 1 e 2 d 1 (d + d 1 ) F † ρ G † ρ-d (3.52)
Proof. We have by definition

|{F, G} M † ρ-d-d 1 ≤ 1 R 2l e (ρ-d-d 1 )|t| dλ(t) R 2l |F(t ′ )G(t ′ -t)| • | sin (t ′ -t) ∧ t ′ / | dλ(t ′ ) ≤ R 2l e (ρ-d-d 1 )|t| dλ(t) R 2l |F(t ′ )| • |G(t ′ -t)| • |(t ′ -t)| • |t ′ | dλ(t ′ ) = R 2l e (ρ-d-d 1 )|t| dλ(t) R 2l |F(u + t/2)G(u -t/2)| • |u -t/2| • |u + t/2| dλ(u) = R 2l ×R 2l e (ρ-d-d 1 )(|x|+|y|) |F(x)G(y)| • |x| • |y| dλ(x)dλ(y) ≤ 1 d 1 (d + d 1 ) R 2l |F(x)|e ρ|x| dλ(x) R 2l |G(y)|e (ρ-d)|y| dλ(x) ≤ 1 e 2 d 1 (d + d 1 ) F † ρ G † ρ-d because sup α∈R |α|e -δα = 1 eδ , δ > 0. 3.4.2. Assertion (1 † ). By definition F( )♯G( ) † ρ,k = k γ=0 R 2l ×R 2l |∂ γ [ F(t ′ -t, ) G(t ′ , )e i Ωω(t ′ ,t ′ -t) ]|µ k-γ (t)e ρ|t| dλ(t ′ )dλ(t) whence F( )♯G( ) † ρ,k = k γ=0 γ j=0 γ j R 2l ×R 2l |∂ γ-j [ F(t ′ -t, ) G(t ′ , )]|Ω ω (t ′ -t, t ′ )| j µ k-γ (t)e ρ|t| dλ(t ′ )dλ(t) = k γ=0 γ j=0 γ-j i=0 γ j j i R 2l ×R 2l |∂ γ-j-i F(t ′ -t, )∂ i G(t ′ , )||Ω ω (t ′ -t, t ′ )| j µ k-γ (t)e ρ|t| dλ(t ′ )dλ(t)
By Lemma 3.15 and the inequality

µ k (t ′ -t) ≤ 2 k/2 µ k (t ′ )µ k (t) we get, with t = (p, s) : t ′ = (p ′ , s ′ ) |Ω ω (t ′ -t, t ′ )| j µ k-γ (t) ≤ 2 j µ j (t ′ -t)µ j (t ′ )µ k-γ (t) ≤ 2 j µ j t ′ -t)µ j (t ′ )µ k-γ (t)2 (k-γ)/2 µ k-γ (t ′ -t)µ k-γ (t) ≤ 2 j+(k-γ)/2 µ k-γ+j (t ′ -t)µ k-γ+j (t) Denote now γ -j -i = k -γ ′ , i = k -γ ′′ and remark that j ≤ γ ′ , i ≤ γ -j. Then: 2 j+(k-γ)/2 µ k-γ+j (t ′ -t)µ k-γ+j (t) ≤ 2 k µ γ ′ (t ′ )µ γ ′′ (t)
Since γ j j i ≤ 4 k and the sum over k has (k + 1) terms we get:

F( )♯G( ) † ρ,k ≤ (k + 1)4 k k γ ′ ,γ ′′ =0 R 2l ×R 2l |∂ k-γ ′ F(t ′ -t, )|∂ k-γ ′′ G(t ′ , )|µ γ ′ (t ′ -t)µ γ ′′ (t)e ρ|t| dλ(t ′ )dλ(t)
Now we can repeat the argument of Lemma 3.17 to conclude:

F( )♯G( ) † ρ,k ≤ (k + 1)4 k F † ρ,k • G † ρ,k
which is (3.41). Assertion (3 † ), formula (3.43) is the particular case of (3.41) obtained for Ω ω = 0, and Assertion (3), formula (3.46), is in turn particular case of (3.43) .

3.4.3. Assertion(2 † ). By definition:

{F( ), G( )} M † ρ,k = k γ=0 R 2l ×R 2l |∂ γ [ F(t ′ -t, ) G(t ′ , ) sin Ω(t ′ -t, t ′ )/ ]|µ k-γ (t)e ρ|t| dλ(t ′ )dλ(t).
Lemma 3.16 entails:

|∂ j sin Ω(t ′ -t, t ′ )/ | ≤ |Ω(t ′ -t, t ′ )| j+1
and therefore:

{F( ), G( )} M ρ,k ≤ k γ=0 γ j=0 γ j R 2l ×R 2l |∂ γ-j [ F(t ′ -t, ) G(t ′ , )]|Ω ω (t ′ -t, t ′ )| j+1 µ k-γ (t)e ρ(|t| dλ(t ′ )dλ(t) = k γ=0 γ j=0 γ-j i=0 γ j j i R 2l ×R 2l |∂ γ-j-i F(t ′ -t, )∂ i G(t ′ , )||Ω ω (t ′ -t, t ′ )| j+1 µ k-γ (t)e ρ|t| dλ(t ′ )dλ(t)
Let us now absorb a factor |Ω ω (t ′t, t ′ )| j in exactly the same way as above, and recall that

|Ω ω (t ′ -t, t ′ )| ≤ |(t ′ -t)t ′ |.
We end up with the inequality:

{F( ), G( )} M † ρ,k ≤ (k + 1)4 k k γ ′ ,γ ′′ =0 R 2l ×R 2l |∂ k-γ ′ F(t ′ -t, )|∂ k-γ" G(t ′ , )||t ′ -t||t ′ |µ γ ′ (t ′ -t)µ γ ′′ (t ′ )e ρ(|t| dλ(t ′ )dλ(t)
Repeating once again the argument of Lemma 3.17 we finally get:

{F( ), G( )} M † ρ-d-d 1 ,k ≤ (k + 1)4 k e 2 d 1 (d + d 1 ) F † ρ,k • G † ρ-d,k
which is (3.42). Once more, Assertion (2) is a particular case of (3.42) and Assertion (1) a particular case of (3.41). This completes the proof of Proposition 3.10.

A sharper version of the semiclassical Egorov theorem

Let us state and prove in this section a particular variant of the semiclassical Egorov theorem (see e.g. [Ro]) which establishes the relation between the unitary transformation e iεW/i and the canonical transformation φ ε W 0 generated by the flow of the symbol W(ξ, x; )| =0 := W 0 (ξ, x) (principal symbol) of W at time 1. The present version is sharper in the sense that the usual one allows for a O( ∞ ) error term.

Theorem 4.1. Let ρ > 0, k = 0, 1, . . . and let A, W ∈ J † k (ρ) with symbols A, W. Then:

S ε := e i εW (L ω + A)e -i εW = L ω + B
where:

(1) ∀ 0 < d < ρ, B ∈ J † k (ρ -d); (2) B † ρ-d,k ≤ (k + 1)4 k (ed) 2 1 -|ε| W † ρ,k /d -1 A † ρ,k + |ε| W † ρ,k /de
(3) Moreover the symbol B of B is such that:

L ω + B = (L ω + A) • Φ ε W 0 + O( )
where Φ ε W 0 is the Hamiltonian flow of W 0 := W| =0 at time ε. (4) Assertions (1), (2), (3) hold true when (A, B, W

) ∈ J k (ρ) with A † ρ,k , B † ρ,k , W † ρ,k replaced by A ρ,k , B ρ,k , W ρ,k .
Proof. The proof is the same in both cases, since it it is based only on Proposition 3.11. Therefore we limit ourselves to the J k (ρ) case.

By Corollary 3.8, Assertion (3), under the present assumptions H 1 (T l ), the domain of the selfadjoint operator F(L ω ) + A, is left invariant by the unitary operator e i εW . Therefore on H 1 (T l ) we can write the commutator expansion

S ε = L ω + ∞ m=1 (iε) m m m! [W, [W, . . . , [W, L ω ] . . .] + ∞ m=1 (iε) m m m! [W, [W, . . . , [W, A] . . .]
whence the corresponding expansions for the symbols

S(x, ξ; , ε) = L ω (ξ) + ∞ m=1 ε m m! {W, {W, . . . , {W, L ω } . . .} M + ∞ m=1 ε m m! {W, {W, . . . , {W, A} M . . .} M
because {W, L ω } M = {W, L ω } by the linearity of L ω . Now apply Corollaries 3.13 and 3.14. We get, denoting once again

C k = (k + 1)4 k : ∞ m=1 (iε) m m m! [W, [W, . . . , [W, L ω ] . . .] L 2 →L 2 ≤ ∞ m=1 ε m m! {W, {W, . . . , {W, L ω } . . .} M ρ-d,k ≤ ∞ m=1 |ε| m m! {W, {W, . . . , {-i ω, ∇ x W} M . . .} M ρ-d,k ≤ C k ed ∞ m=1 √ 2πm |ε| W ρ,k d m ∞ m=1 (iε) m m m! [W, [W, . . . , [W, A] . . .] L 2 →L 2 ≤ ∞ m=1 ε m m! {W, {W, . . . , {W, A} M . . .} M ρ-d,k ≤ C k ed A ρ,k ∞ m=1 √ 2πm |ε| W ρ,k d m
Now define:

B := ∞ m=1 (iε) m m m! [W, [W, . . . , [W, L ω ] . . .] + ∞ m=1 (iε) m m m! [W, [W, . . . , [W, A] . . .] (4.53)
and remark that ∀ η > 0 we can always find 0

< d ′ < d -η such that √ 2πmd -m ≤ (d ′ ) -m .
Denoting (abuse of notation) d ′ = d we can write:

B ρ-d,k ≤ (k + 1)4 k (ed) 2 [1 -|ε| W ρ,k /d] -1 [ A ρ,k + |ε| W ρ,k /de]
This proves assertions (1) and (2).

By Remark 2.9, we have:

S 0 ε (x, ξ; )| =0 = L ω + B ε (ξ, x; )| =0 = ∞ k=0 (ε) k k! {W 0 , {W, . . . , {W 0 , L + A} . . .} = e εL W 0 (L ω + A)
where L W 0 F = {W, F} denote the Lie derivative with respect to the Hamiltonian flow generated by W 0 . Now, by Taylor's theorem

e εL W 0 (L ω + A) = (L ω + A) • φ ε W 0 (x, ξ)
and this concludes the proof of the Theorem.

Remark 4.2. Let W be a solution of the homological equation (5.1). Then the explicit expression of W 0 clearly is:

W 0 = 1 F ′ (L ω (ξ)) q∈Z ℓ V q (ξ) ω, q e i q,
x and

e εL W 0 (F(L ω ) + εA) = F(L ω ) + εN 0,ε (L ω ) + O(ε 2 ).
Thus W 0 coincides with the expression obtained by first order canonical perturbation theory.

Homological equation: solution and estimate

Let us briefly recall the well known KAM iteration in the quantum context.

The first step consists in looking for an L 2 (T l )-unitary map U 0,ε = e iεW 0 / , W 0 = W * 0 , such that

S 0,ε := U 0,ε (L ω + εV 0 )U * 0,ε = F 1,ε (L ω ) + ε 2 V 1,ε , V 0 := V, F 1,ε (L ω ) = L ω + εN 0 (L ω ).
Expanding to first order near ε = 0 we get that the two unknowns W 0 and N 0 must solve the

equation [L ω , W 0 ] i + V = N 0 V 1,ε
is the second order remainder of the expansion. Iterating the procedure:

U ℓ,ε := e iε 2 ℓ W ℓ / ; S ℓ,ε := U ℓ.ε (F ℓ,ε (L ω ) + ε 2 ℓ V ℓ,ε )U * ℓ,ε == F ℓ+1,ε (L ω ) + ε 2 ℓ+1 V ℓ+1 (ε), [F ℓ,ε (L ω ), W ℓ,ε ] i + V ℓ,ε = N ℓ,ε
With abuse of notation, we denote by F ℓ,ε (L ω , ), N ℓ,ε (L ω , ), V ℓ,ε (L ω , ) the corresponding symbols.

The KAM iteration procedure requires therefore the solution in J k (ρ) of the operator homological equation in the two unknowns W and M (here we have dropped the dependence on ℓ and ε, and changed the notation from N to M to avoid confusion with what follows):

[F(L ω ), W ] i + V = M (L ω ) (5.1)
with the requirement M (L ω ) ∈ J k (ρ); the solution has to be expressed in terms of the corresponding Weyl symbols (L ω , W, V, M) ∈ J k (ρ) in order to obtain estimates uniform with respect to .

Moreover, the remainder has to be estimated in terms of the estimates for W, M .

Equation (5.1), written for the symbols, becomes

{F(L ω (ξ), ), W(x, ξ; )} M + V(x, L ω (ξ); ) = M(L ω (ξ), ) (5.2)
5.1. The homological equation. We will construct and estimate the solution of (5.1), actually solving (5.2) and estimating its solution, under the following assumptions on F:

Condition (1) (u, ) → F(u; ) ∈ C ∞ (R × [0, 1]; R); Condition (2) inf (u, )∈R×[0,1] ∂ u F(u; ) > 0; lim |u|→∞ |F(u, )| |u| = C > 0 uniformly with respect to ∈ [0, 1].
Condition (3) Set:

K F (u, η, ) = η F(u + η, ) -F(u, ) (5.3) Then there is 0 < Λ(F) < +∞ such that sup u∈R,η∈R, ∈[0,1] |K F (u, η, )| < Λ.
(5.4)

The first result deals with the identification of the operators W and M through the determination of their matrix elements and corresponding symbols W and M.

Proposition 5.1. Let V ∈ J(ρ), ρ > 0, and let W and M be the minimal closed operators in L 2 (T n ) generated by the infinite matrices e m , W e m+q = i e m , V e m+q F( ω, m , ) -F( ω, (m + q) , )

, q = 0, e m , W e m = 0 (5.5)

e m , M e m = e m , V e m , e m , M e m+q = 0, q = 0 (5.6)

on the eigenvector basis e m : m ∈ Z l of L ω . Then:

(1) W and M are continuous and solve the homological equation (5.1);

(2) The symbols W(x, ξ; ) and M(ξ, ) have the expression:

M(ξ; ) = V(L ω (ξ); ); W(L ω (ξ), x; ) = q∈Z l ,q =0
W(L ω (ξ), q; )e i q,x (5.7)

W(L ω (ξ), q; ) := i V(L ω (ξ); q; ) F(L ω (ξ); ) -F(L ω (ξ + q), )
, q = 0; W(L ω (ξ); ) = 0.

(5.8)

Here the series in (5.7) is • ρ convergent; V(L ω (ξ); ) is the 0-th coefficients in the Fourier expansion of V(L ω (ξ), x, ).

Proof. Writing the homological equation in the eigenvector basis e m : m ∈ Z l we get

e m , [F(L ω ), W ] i e n + e m , V e n = e m , M (L ω )e n δ m,n
(5.9) which immediately yields (5.5,5.6) setting n = m + q. As far the continuity is concerned, we have:

i F( ω, m , ) -F( ω, (m + q) , ) = ω, q -1 η F( ω, m , ) -F( ω, m + η, )
, η := q, ω .

and therefore, by (5.4) and the diophantine condition:

| e m , W e m+q | ≤ γ|q| τ Λ| e m , V e m+q |.
The assertion now follows by Corollary 3.8, which also entails the • ρ convergence of the series (5.7) because V ∈ J ρ . Finally, again by Corollary 3.8, formulae (3.23), (3.24), we can write e m , W e m+q = W( ω, (m + q/2) , q, ); e m , M e m = M(ω, m , ) = V(L ω (ω, m , 0, ) and this concludes the proof of the Proposition.

The basic example of F is the following one. Let:

• F ℓ (u, ε; ) = u + Φ ℓ (u, ε, ), ℓ = 0, 1, 2, . . . (5.10) • Φ ℓ (ε, ) := εN 0 (u; ε, ) + ε 2 N 1 (u; ε, ) + . . . + ε ℓ N ℓ (u, ε, ), ε j := ε 2 j . (5.11)
where we assume holomorphy of ε → N s (u, ε, ) in the unit disk and the existence of ρ 0 > ρ 1 > . . . > ρ ℓ > 0 such that:

(N s ) max |ε|≤1 |N | ρs < ∞, .
Denote, for ζ ∈ R:

g ℓ (u, ζ; ε, ) := Φ ℓ-1 (u + ζ; ε, ) -Φ ℓ-1 (u; ε, ) ζ (5.12)
Let furthermore: 0 < d ℓ < . . . < d 0 < ρ 0 , 0 < ρ 0 := ρ;

(5.13)

ρ s+1 = ρ s -d s > 0, s = 0, . . . , ℓ -1 δ ℓ := ℓ-1 s=0 d ℓ < ρ (5.14)
and set, for j = 1, 2, . . .:

θ ℓ,k (N , ε) := ℓ-1 s=0 |ε s | |N s | ρs,k ed s , θ ℓ (N , ε) := θ ℓ,0 (N , ε).
(5.15) By Remark 2.4 we have

θ ℓ,k (N , ε) = ℓ-1 s=0 |ε s | N s ρs,k ed s (5.16)
Lemma 5.2. In the above assumptions:

(1)

For any R > 0 the function ζ → g ℓ (u, ζ, ε, ) is holomorphic in {ζ | |ζ| < R | |ℑζ| < ρ},
uniformly on compacts with respect to (u, ε,

) ∈ R × R × [0, 1];
(2) For any n ∈ N ∪ {0}:

sup ζ∈R |[g(u, ζ, ε, )] n | ρ ℓ ≤ [θ ℓ (N , ε)] n
(5.17)

(3) Let:

max |ε|≤L θ ℓ (N , ε) < 1, L > 0.
(5.18) Then:

sup ζ∈R;u∈R |K F (u, ζ, ε, )| ρ ℓ ≤ 1 |ζ| • 1 1 -θ ℓ (N , ε) (5.19) (4) sup ζ∈R |∂ j u g(u, ζ, ε, )| ρ ℓ ≤ θ ℓ,j (N , ε) (5.20) sup ζ∈R |∂ j ζ g(u, ζ, ε, )| ρ ℓ ≤ θ ℓ,j (N , ε) (5.21) sup ζ∈R |∂ j g(u, ζ, ε, )| ρ ℓ ≤ θ ℓ,j (N , ε). (5.22)
Proof. The holomorphy is obvious given the holomorphy of N s (u; ε, ). To prove the estimate (5.17), denoting N s (p, ε, ) the Fourier transform of N s (ξ, ε, ) we write

g ℓ (u, ζ, ε, ) = 1 ζ ℓ-1 s=0 ε s R N ℓ (p, ε, )(e iζp -1)e iup dp = (5.23) 2 ζ ℓ-1 s=0 ε s R N ℓ (p, ε, )e ip(u+ζ)/2 sin ζp/2 dp which entails: sup ζ∈R |g ℓ (u, ζ, ε, )| ρ ℓ = sup ζ∈R R | g ℓ (p, ζ, ε, )|e ρ ℓ |p| dp ≤ max ∈[0,1] ℓ-1 s=0 |ε s | R | N s (p, ε, )p|e (ρs-ds)|p| dp ≤ 1 e ℓ-1 s=0 |ε s | |N s | ρs d s = θ ℓ (N , ε, 1) 0 < d s < ρ s .
Hence Assertion (3) of Proposition 3.11, considered for k = 0, immediately yields (5.17). Finally, if g ℓ is defined by (5.12), then:

K F (u, ζ, ε, ) = 1 ζ 1 1 + g ℓ (u, ζ, ε, )
and the estimate (5.19) follows from (5.17) which makes possible the expansion into the geome-

trical series 1 1 + g ℓ (u, ζ, ε, ) = ∞ n=0 (-1) n g ℓ (u, ζ, ε, ) n (5.24)
convergent in the θ ℓ (N , ε) norm. To see (5.20), remark that (5.23) yields:

∂ j u g ℓ (u, ζ, ε, ) = 2 ζ ℓ-1 s=0 ε s R N ℓ (p, ε, )(ip) j e ip(u+ζ)/2 sin ζp/2 dp.
Therefore:

sup ζ∈R |∂ j u g ℓ (u, ζ, ε, )| ρ ℓ ≤ sup ζ∈R max ∈[0,1] 2 ℓ-1 s=0 |ε s | R | N s (p, ε, )||p| j | sin ζp/2|/ζ|e ρ ℓ |p| dp ≤ sup ζ∈R max ∈[0,1] 2 ℓ-1 s= |ε s | R | N s (p, ε, )||p| j | sin ζp/2|/ζ|e (ρs-ds)|p| dp ≤ sup p∈R [|p| ℓ-1 s=0 |ε s | e -ds|p| ] max ∈[0,1] R |p| j N (p, ε, )e ρs|p| dp ≤ 1 e ℓ-1 s=0 |ε s | |N s | ρs,j d s ≤ θ ℓ,j (N , ε)
(5.21) is proved by exactly the same argument. Finally, to show (5.22) we write:

sup ζ∈R |∂ j g ℓ (u, ζ, ε, )| ρ ℓ ≤ sup ζ∈R max ∈[0,1] 2 ℓ-1 s=0 |ε s | R |∂ j N s (p, ε, )| • | sin ζp/2|/ζ|e ρ ℓ |p| dp ≤ max ∈[0,1] ℓ-1 s=0 |ε s | R |∂ j N (p, ε, )|e (ρs-ds)|p| dp ≤ θ ℓ (N , ε)
This proves the Lemma.

By Condition (1) the operator family → F(L ω ; ε, ), defined by the spectral theorem, is self-adjoint in L 2 (T l ); by Condition (2) D(F(L ω )) = H 1 (T l ). Since L ω is a first order operator with symbol L ω , the symbol of F(L ω ; ε, ) is F(L ω (ξ), ε, ). We can now state the main result of this section. Let F ℓ (x, ε, ) be as in Lemma 5.2, which entails the validity of Conditions (1),

(2), (3).

Theorem 5.3.

Let V ℓ ∈ J k (ρ ℓ ), ℓ = 0, 1 . . ., V 1 ≡ V for some ρ ℓ > ρ ℓ+1 > 0, k = 0, 1, . . .. Let V ℓ (L ω (ξ), x; ε, ) ∈ J k (ρ)
be its symbol. Then for any θ ℓ (N , ε) < 1 the homological equation (5.1), rewritten as

[F ℓ (L ω ), W ℓ ] i + V ℓ = N ℓ (L ω , ε) (5.25) {F ℓ (L ω (ξ), ε, ), W ℓ (x, ξ; ε, )} M + V ℓ (x, L ω (ξ); ε, ) = N ℓ (L ω (ξ), ε, ) (5.26) admits a unique solution (W ℓ , N ℓ ) of Weyl symbols W ℓ (L ω (ξ), x; ε, ), N ℓ (L ω (ξ), ε, ) such that (1) W ℓ = W * ℓ ∈ J k (ρ ℓ ), with: W ℓ ρ ℓ+1 ,k = W ρ ℓ+1 ,k ≤ A(ℓ, k, ε) V ℓ ρ ℓ ,k (5.27) A(ℓ, k, ε) = γ τ τ (ed ℓ ) τ 1 + 2 k+1 (k + 1) 2(k+1) k k (eδ ℓ ) k [1 -θ ℓ (N , ε)] k+1 θ k+1 ℓ,k .
(5.28)

(2)

N ℓ = V ℓ ; therefore N ℓ ∈ J k (ρ ℓ ) and N ρ ℓ ,k ≤ V ℓ ρ ℓ ,k .
Proof. The proof of ( 2) is obvious and follows from the definition of the norms • ρ and • ρ,k . The self-adjointess property W = W * is implied by the construction itself, which makes W symmetric and bounded.

Consider W ℓ as defined by (5.7). Under the present assumptions, by Lemma 5.2 we have:

W ℓ (L ω (ξ), q; ε, ) := 1 ω, q i V ℓ (L ω (ξ); q; ε, ) 1 + g ℓ (L ω (ξ); ω, q , ε, ) , q = 0; W ℓ (•, 0; ) = 0.
By the • ρ ℓ -convergence of the series (5.24) we can write

∂ γ W ℓ (L ω (ξ), q; ε, ) = ∞ n=0 (-ε) n ∂ γ W ℓ,n (L ω (ξ), q; ε, ), (5.29) W ℓ,n (L ω (ξ), q; ε, ) = 1 ω, q V ℓ (L ω (ξ); q; ε, )[g ℓ (L ω (ξ); ω, q , ε, )] n (5.30) ∂ γ W ℓ,n (L ω (ξ), q; ε, ) = (5.31) γ j=0 γ j ∂ γ-j V ℓ (L ω (ξ); q; ε, )D j [g ℓ (L ω (ξ); ω, q , ε, )] n
where D denotes the total derivative with respect to . We need the following preliminary result.

Lemma 5.4. Let ζ( ) := ω, q . Then:

(1)

|D j g ℓ (L ω (ξ), ζ( ), ε, )| ρ ℓ ≤ (j + 1)(2|q|) j θ ℓ,j (N , ε) 2 (5.32) (2) |D j [g ℓ (L ω (ξ); ζ( ), ε, )] n | ρ ℓ ≤ 2n j (θ ℓ (N , ε)) n-j [2(j + 1)|q|] j θ ℓ,j (N , ε) 2j .
(5.33)

Proof. The expression of total derivative D g is:

D g(•; ω, q , ε, ) = ( ω, q ∂ ∂ζ + ∂ ∂ ) g ℓ (•; ζ, ε, )| ζ= ω,q
(5.34) By Leibnitz's formula we then have:

D j g ℓ (•; ω, q , ε, ) = j i=0 j i ω, q j-i ∂ j-i g ℓ ∂ζ j-i ∂ i g ℓ ∂ i (5.35)
Apply now (3.46) with k = 0, (5.20) and (5.22). We get:

∂ j-i g ℓ ∂ζ j-i ∂ i g ℓ ∂ i ρ ℓ ≤ (j + 1)2 j θ ℓ,j (N , ε) 2
whence, since |ω| ≤ 1:

D j g ℓ D j ρ ℓ ≤ (j + 1)(2) j |q| j θ ℓ,j (N , ε) 2 (5.36)
This proves Assertion (1). To prove Assertion (2), let us first note that

D j [g ℓ (L ω (ξ); ω, q , ε, )] n = P n,j g ℓ , Dg ℓ D , . . . , D j g ℓ D j .
(5.37)

where P n,j (x 1 , . . . , x j ) is a homogeneous polynomial of degree n with n j terms. Explicitly:

P n,j g ℓ , Dg ℓ D , . . . , D j g ℓ D j = n j=1 g ℓ n-j j k=1 j 1 +...+j k =j D j k g ℓ D j k .
Now (5.32), (5.36) and Proposition 3.11 (3) entail:

|D j [g ℓ (L ω (ξ); ω, q , ε, )] n | ρ ℓ ≤ n j |g| n-j ρ ℓ j k=1 j 1 +...+j k =j 2(j k + 1) (2|q|) j k θ ℓ,j k (N , ε) 2 ≤ 2n j (θ ℓ (N , ε)) n-j [2(j + 1)|q|] j θ ℓ,j (N , ε) 2j .
This concludes the proof of the Lemma.

To conclude the proof of the theorem, we must estimate the • ρ ℓ+1 ,k norm of the derivatives

∂ γ W ℓ,n (L ω (ξ), x; ε, ). Obviously: W ℓ (ξ, x; ε, ) ρ ℓ +1,k ≤ ∞ n=0 W ℓ,n (ξ, x; ε, ) ρ ℓ+1,k .
(5.38)

For n = 0:

W ℓ,0 (ξ, x; ε, ) ρ ℓ+1,k ≤ γ k γ=0 R×R l |∂ γ W ℓ,0 (p, s; •)||s| τ µ k-γ (pω, s) e ρ ℓ+1 (|p|+|s|) dλ(p, s) ≤ γ k γ=0 R×R l |∂ γ V ℓ,0 (p, s; •)||s| τ µ k-γ (pω, s) e ρ ℓ+1 (|p|+|s|) dλ(p, s) ≤ γ τ τ (ed ℓ ) τ V ℓ ρ ℓ,k
where the inequality follows again by the standard majorization

e ρ ℓ+1 (|p|+|s|) = e ρ ℓ (|p|+|s|) e -d ℓ (|p|+|s|) , sup s∈R l [|s| τ e -d ℓ |s| ] ≤ γ τ τ (ed ℓ ) τ
on account of the small denominator estimate (1.25). For n > 0 we can write, on account of (2.5,2.6):

W ℓ,n (ξ, x; •) ρ ℓ+1 ,k = k γ=0 R×R l |∂ γ W ℓ,n (p, s; •)||s| τ µ k-γ (pω, s) e ρ ℓ+1 (|p|+|s|) dλ(p, s) ≤ ≤ γ τ τ (ed ℓ ) τ k γ=0 γ j=0 γ j R l Q(s, •)e ρ ℓ |s| dν(s) where Q(s, •) := R |[∂ γ-j V ℓ (p; s; •)] * [D j g * n ℓ (p; ω, s , •)]µ k-γ (pω, s) e ρ ℓ |p| dp
Here * denotes convolution with respect only to the p variable, and g * i n ℓ (p, ζ, •) denotes the n-th convolution of g ℓ with itself, i.e. the p-Fourier transform of g n ℓ . Now, by Assertion (3) of Proposition (3.11) and the above Lemma:

R l Q(s, •)e ρ ℓ |s| dν(s) = = R×R l |[∂ γ-j V ℓ (p; s; •)] * ξ [D j g * ξ n ℓ (p; ω, s , •)]µ k-γ (pω, s) e ρ ℓ (|p|+|s|) dλ(p, s) ≤ R l R |[∂ γ-j V ℓ (p; s; )] * [D j g * n (p; ω, s , •)]|µ k-γ (pω, s) e ρ ℓ |p| dp e ρ ℓ |s| dν(s) ≤ 2A(j) j θ ℓ (N , ε) n-j R l R |∂ γ-j V ℓ (p; s; •)|µ k-γ (pω, s) e ρ ℓ |p| |s| j e ρ ℓ |s| dpdν(s), with A(j) := 2n(j + 1)θ ℓ,j (N , ε) 2 .
This yields, with δ ℓ defined by (5.13):

W ℓ,n (ξ, x; •) ρ ℓ +1,k ≤ γ τ τ (ed ℓ ) τ k γ=0 R×R l |∂ γ W ℓ,n (p, s; •)µ k-γ (pω, s) e ρ ℓ (|p|+|s|) dλ(p, s) ≤ ≤ γτ τ (k + 1)(2A(k)) k (ed ℓ ) τ θ ℓ (N , ε) n-j k γ=0 R×R l |∂ γ V ℓ (p; s; •)| • µ k-γ (pω, s) e ρ ℓ |p| |s| j e ρ ℓ |s| dλ(p, s) ≤ γτ τ (k + 1)(2A(k)) k (ed ℓ ) τ k k (eδ ℓ ) k θ ℓ (N , ε) n-j k γ=0 R l R |∂ γ V ℓ (p; s; •)|µ k-γ (pω, s)e ρ|p| e ρ|s| dλ(p, s) ≤ γ τ τ (ed ℓ ) τ (k + 1)k k (eδ ℓ ) k 2(2n) k (θ ℓ (N , ε)) n-j (k + 1) k θ 2k ℓ,k V ℓ ρ,k .
Therefore, by (5.38):

W ℓ (ξ; x; ε, ) ρ ℓ+1 ,k ≤ ∞ n=0 W ℓ,n (ξ; x; ε, ) ρ ℓ+1 ,k ≤ ≤ γ τ τ (ed ℓ ) τ V ℓ ρ ℓ ,k 1 + 2 k+1 (k + 1) k+1 k k (eδ ℓ ) k θ 2k ℓ,k ∞ n=1 n k (θ ℓ (N , ε)) n-j ≤ γ τ τ (ed ℓ ) τ V ℓ ρ ℓ ,k 1 + 2 k+1 (k + 1) k+1 k k (eδ ℓ ) k θ 2k-j ℓ,k ∞ n=1 n k (θ ℓ (N , ε)) n ≤ γ τ τ (ed ℓ ) τ V ℓ ρ ℓ ,k 1 + 2 k+1 (k + 1) 2(k+1) k k (eδ ℓ ) k [(1 -θ ℓ (N , ε) k+1 ] θ k+1 ℓ,k
.

because j ≤ k, and

∞ n=1 n k x n ≤ ∞ n=1 (n + 1) • • • (n + k)x n = d k dx k ∞ n=1 x n+k = d k dx k x k+1 1 -x = (k + 1)! k+1 j=0 k + 1 -j j x k+1-j (1 -x) j ≤ 2 k+1 (k + 1)! (1 -x) k+1 .
By the Stirling formula this concludes the proof of the Theorem.

Towards KAM iteration.

Let us now prove the estimate which represents the starting point of the KAM iteration:

Theorem 5.5. Let F ℓ and V ℓ be as in Theorem 5.3, and let W ℓ be the solution of the homological equation (5.1) as constructed and estimated in Theorem 5.3. Let (5.18) hold and let furthermore

|ε| < ε ℓ , ε ℓ := d ℓ W ℓ ρ ℓ+1 ,k 2 -ℓ .
(5.39)

Then we have:

e iε ℓ W ℓ / (F ℓ (L ω ) + ε ℓ V ℓ )e -iε ℓ W ℓ / = (F ℓ + ε ℓ N ℓ )(L ω ) + ε 2 ℓ V ℓ+1,ε (5.40)
where, ∀ 0 < 2d ℓ < ρ ℓ and k = 0, 1, . . .:

V ℓ+1,ε ρ ℓ -2d ℓ ,k ≤ C(ℓ, k, ε) V ℓ 2 ρ ℓ ,k 1 -|ε ℓ |A(ℓ, k, ε) V ρ ℓ ,k /d ℓ (5.41) C(ℓ, k, ε) := (k + 1) 2 4 2k (ed ℓ ) 3 A(ℓ, k.ε) 2 + |ε ℓ | (k + 1)4 k (ed ℓ ) 2 A(ℓ, k.ε) V ℓ ρ ℓ ,k (5.42)
Here A(ℓ, k, ε) is defined by (5.28).

Remark 5.6. We will verify in the next section (Remark 6.26 below) that (5.39) is actually fulfilled

for |ε| < 1/|V| ρ .
Proof. To prove the theorem we need an auxiliary result, namely:

Lemma 5.7. For ℓ = 0, 1, . . . let ρ ℓ > 0, ρ 0 := ρ, A ∈ J k (ρ), W ℓ ∈ J k (ρ ℓ ), k = 0, 1, . . .. Let W * ℓ = W ℓ , and define: A ε ( ) := e iε ℓ W ℓ / Ae -iε ℓ W/ .

(5.43)

Then, for |ε| < (d ′ ℓ / W ρ ℓ+1 ,k ) 2 -ℓ , and ∀ 0 < d ′ ℓ < ρ ℓ , k = 0, 1, . . .: A ε ( ) ρ-d ′ ℓ ,k ≤ (k + 1)4 k ed ′ ℓ A ρ ℓ ,k 1 -|ε ℓ | W ρ ℓ+1 ,k /d ′ ℓ (5.44)
Proof. Since the operators W ℓ and A are bounded, there is ε 0 > 0 such that the commutator expansion for A ε ( ):

A ε ( ) = ∞ m=0 (iε ℓ ) m m m! [W ℓ , [W ℓ , . . . , [W ℓ , A] . . .] is norm convergent for |ε| < ε 0 if ∈]0, 1[ is fixed.
The corresponding expansion for the symbols is

A ε ( ) = ∞ m=0 (ε ℓ ) m m! {W ℓ , {W, . . . , {W ℓ , A} M . . .} M
Now we can apply once again Corollary 3.13. We get, with the same abuse of notation of Theorem 4.1:

1 m! {W ℓ , {W ℓ , . . . , {W ℓ , A} M . . .} M ρ-d ′ ℓ ,k ≤ (k + 1)4 k ed 1 W ℓ ρ ℓ ,k d ′ ℓ m A ρ ℓ ,k (5.45) Therefore A ε ( ) ρ ℓ -d ′ ℓ ,k ≤ (k + 1)4 k ed ′ ℓ A ρ ℓ ,k ∞ m=0 |ε| m [ W ρ ℓ+1 ,k /d ′ ℓ ] m = (k + 1)4 k ed ′ ℓ A ρ ℓ ,k 1 -|ε ℓ | W ρ ℓ+1 ,k /d ′ ℓ
and this concludes the proof.

W ℓ solves the homological equation (5.1). Then by Theorem 5.3

W ℓ = W * ℓ ∈ J k (ρ ℓ -d ℓ )
, k = 0, 1, . . .; in turn, by Assertion (3) of Corollary 3.8 the unitary operator e iε ℓ W ℓ / leaves H 1 (T l ) invariant. Therefore the unitary image of H ε under e iε ℓ W/ is the real-holomorphic operator family

in L 2 (T l ) ε → S ε := e iε ℓ W ℓ / (F ℓ (L ω ) + ε ℓ V ℓ )e -iε ellW/ , D(S(ε)) = H 1 (T l ) (5.46)
Computing its Taylor expansion at ε ℓ = 0 with second order remainder we obtain:

S ε u = F ℓ (L ω )u + ε ℓ N ℓ (L ω )u + ε 2 ℓ V ℓ+1,ε u, u ∈ H 1 (T l ) (5.47) V ℓ+1,ε = 1 2 ε ℓ 0 (ε ℓ -t)e itW ℓ / [N ℓ , W ℓ ] i + [W ℓ , V ℓ ] i + t [W ℓ , [W ℓ , V ℓ ]] (i ) 2 e -itW ℓ / dt (5.48)
To see this, first remark that S 0 = F(L ω ). Next, we compute, as equalities between continuous operators in L 2 (T l ):

S ′ ε = e iε ℓ W/ ([F ℓ (L ω ), W ℓ ]/i + V ℓ + ε ℓ [V, W ]/i )e -iε ℓ W/ = e iε ℓ W/ (N ℓ + ε ℓ [V ℓ , W ℓ ]/i )e iε ℓ W ℓ / ; S ′ 0 = N ℓ S ′′ ε = e iε ℓ W ℓ / ([N ℓ , W ℓ ]/i + [V ℓ , W ℓ ]/i + ε ℓ [W ℓ , [W ℓ , V ℓ ]]/(i ) 2 )e -iε ℓ W ℓ / ,
and this proves (5.47) by the second order Taylor's formula with remainder:

S ε = S(0) + εS ′ 0 + 1 2 ε ℓ 0 (ε -t)S ′′ (t), dt
The above formulae obviously yield

V l+1,ε ≤ |ε ℓ | 2 max 0≤|t|≤|ε ℓ | S ′′ (t) (5.49) Set now: R ℓ+1,ε := [N ℓ , W ℓ ]/i + [V ℓ , W ℓ ]/i + ε ℓ [W ℓ , [W ℓ , V ℓ ]]/(i ) 2 (5.50) R ℓ+1,ε is a continuous operator in L 2 , corresponding to the symbol R ℓ+1,ε (L ω (ξ), x; ) = {N ℓ , W ℓ } M + {V ℓ , W ℓ } M + ε ℓ {W ℓ , {W ℓ , V ℓ } M } M (5.51)
Let us estimate the three terms individually. By Theorems 5.3 and 3.11 we can write, with A(ℓ, k, ε) given by (5.28):

[N ℓ , W ℓ ]/i ρ ℓ -d ℓ ,k ≤ {N ℓ , W ℓ } M ρ ℓ -d ℓ ,k ≤ (k + 1)4 k (ed ℓ ) 2 W ℓ ρ ℓ+1 ,k N ℓ ρ ℓ ,k ≤ (k + 1)4 k (ed) 2 A(ℓ, k, ε) V ℓ 2 ρ ℓ ,k [V ℓ , W ℓ ]/i ρ ℓ -d ℓ ,k ≤ {V ℓ , W ℓ } M ρ ℓ -d ℓ ,k ≤ (k + 1)4 k (ed ℓ ) 2 V ℓ ρ ℓ ,k W ℓ ρ ℓ+1 ,k ≤ ≤ (k + 1)4 k (ed ℓ ) 2 A(ℓ, k.ε) V ℓ 2 ρ ℓ ,k [W ℓ , [W ℓ , V ℓ ]]/(i ) 2 ρ ℓ -d ℓ ,k ≤ {W ℓ , {W ℓ , V ℓ } M } M ρ ℓ -d ℓ ,k ≤ (k + 1) 2 4 2k (ed ℓ ) 4 W ℓ 2 ρ ℓ+1 ,k V ℓ ρ ℓ ,k ≤ (k + 1) 2 4 2k (ed ℓ ) 4 A(ℓ, k, ε) 2 V ℓ 3 ρ ℓ ,k
We can now apply Lemma 5.7, which yields:

e iε ℓ W ℓ / [N ℓ , W ℓ ]e -iε ℓ W ℓ / /i ρ ℓ -d ℓ -d ′ ℓ ,k ≤ (k + 1) 2 4 2k (ed ℓ ) 2 ed ′ ℓ Ξ(ℓ, k) e iε ℓ W ℓ / [V ℓ , W ℓ ]e -iε ℓ W ℓ / /i ρ ℓ -d ℓ -d ′ ℓ ,k ≤ (k + 1) 2 4 2k (ed ℓ ) 2 ed ′ ℓ Ξ(ℓ, k) e iε ℓ W ℓ / [W ℓ , [W ℓ , V ℓ ]]e -iε ℓ W ℓ / /(i ) 2 ρ ℓ -d ℓ -d ′ ℓ ,k ≤ (k + 1) 3 4 3k (ed ℓ ) 4 ed ′ ℓ Ξ 1 (ℓ, k) where Ξ(ℓ, k) := A(ℓ, k) • V ℓ 2 ρ ℓ ,k 1 -|ε ℓ | W ρ ℓ+1 ,k /d ′ ℓ (5.52) Ξ 1 (ℓ, k) = A(ℓ, k, ε) 2 • V 3 ρ ℓ ,k 1 -|ε ℓ | W ρ ℓ+1 ,k /d ′ ℓ (5.53)
Therefore, summing the three inequalities we get

V ℓ+1,ε ρ ℓ -d ℓ -d ′ ℓ ,k ≤ (k + 1) 2 4 2k (ed ℓ ) 2 ed ′ ℓ A(ℓ, k, ε) × × V ℓ 2 ρ ℓ ,k 1 -|ε ℓ | W ℓ ρ ℓ+1 ,k /d ′ ℓ 2 + |ε ℓ | (k + 1)4 k (ed ℓ ) 2 A(ℓ, k, ε) V ℓ ρ ℓ ,k
If we choose d ′ ℓ = d ℓ this is (5.41) on account of Theorem 5.3. This concludes the proof of Theorem 5.5.

Recursive estimates

Consider the ℓ-th step of the KAM iteration. Summing up the results of the preceding Section we can write:

• S ℓ,ε := e iε ℓ W ℓ / • • • e iε 2 W 1 / e iεW 0 / (F(L ω ) + εV )e -iεW 0 / e -iε 2 W 1 / • • • e -iε ℓ W ℓ / = e iε ℓ W ℓ / (F ℓ,ε (L ω ) + ε 2 ℓ V ℓ,ε )e -iε ℓ W ℓ / = F ℓ+1,ε (L ω ) + ε ℓ+1 V ℓ+1,ε , • F ℓ,ε (L ω ) = F(L ω ) + ℓ-1 k=1 ε k N k (L ω ), [F ℓ (L ω ), W ℓ ]/i + V ℓ,ε = N ℓ (L ω , ε) •V ℓ+1,ε = 1 2 ε ℓ 0 (ε ℓ -t)e itW ℓ / R ℓ+1,t e -itW ℓ / dt • R ℓ+1,ε := [N ℓ , W ℓ ]/ + [W ℓ , V ℓ,ε ]/ + ε ℓ [W ℓ , [W ℓ , V ℓ,ε ]]/ 2
We now proceed to obtain recursive estimates for the above quantities in the • ρ ℓ ,k norm.

Consider (5.41) and denote:

Ψ(ℓ, k) = (k + 1) 2 4 k (ed ℓ ) 3 Π(ℓ, k); Π(ℓ, k) := [2(k + 1) 2 ] k+1 k k e k δ k ℓ (6.1) P (ℓ, k, ε) := θ ℓ,k (N , ε) k+1 [1 -θ ℓ (N , ε)] k+1 (6.2)
where θ ℓ,k (N , ε) is defined by (5.16). (6.1) and (6.2) yield

A(ℓ, k, ε) = γ τ τ (ed ℓ ) τ [1 + Π(ℓ, k)P (ℓ, k, ε)]. (6.3)
Set furthermore:

E(ℓ, k, ε) := Ψ(ℓ, k)B(ℓ, k, ε)[2 + |ε ℓ |eΨ(ℓ, k)A(ℓ, k, ε) V ℓ,ε ρ ℓ ,k ] 1 -|ε ℓ |A(ℓ, k, ε) V ℓ,ε ρ ℓ ,k /d ℓ (6.4)
Then we have: Lemma 6.1. Let:

|ε ℓ |A(ℓ, k, ε) V ℓ,ε ρ ℓ ,k /d ℓ < 1. (6.5) Then: V ℓ+1,ε ρ ℓ+1 ,k ≤ E(ℓ, k, ε) V ℓ,ε 2 ρ ℓ ,k (6.6)
Remark 6.2. The validity of the assumption (6.5) is to be verified in Proposition 6.3 below.

Proof. Since d ℓ < 1, by (5.42), (6.1) and ( 6.3) we can write:

C(ℓ, k, ε) ≤ Ψ(ℓ, k)A(ℓ, k, ε)) [2 + |ε ℓ |eΨ(ℓ, k)A(ℓ, k, ε) V ℓ,ε ρ ℓ ,k ] (6.7)
and therefore, by (5.41):

V ℓ+1,ε ρ ℓ -2d ℓ ,k ≤ C(ℓ, k, ε) V ℓ 2 ρ ℓ ,k 1 -|ε ℓ |A(ℓ, k, ε) V ρ ℓ ,k /d ℓ ≤ Ψ(ℓ, k)A(ℓ, k, ε) [2 + |ε ℓ |eΨ(ℓ, k)A(ℓ, k, ε) V ℓ,ε ρ ℓ ,k ] 1 -|ε ℓ |A(ℓ, k, ε) V ℓ,ε ρ ℓ ,k /d ℓ V ℓ 2 ρ ℓ ,k = E(ℓ, k, ε) V ℓ 2 ρ ℓ ,k .
This yields (6.6) and proves the Lemma. Now recall that the sequence {ρ j } is decreasing. Therefore:

N j,ε ρ ℓ ,k ≤ N j,ε ρ j ,k = V j,ε ρ j ,k ≤ V j,ε ρ j ,k , j = 0, . . . , ℓ -1. (6.8)
At this point we can specify the sequence d ℓ , ℓ = 1, 2, . . ., setting:

d ℓ := ρ (ℓ + 1) 2 , ℓ = 0, 1, 2, . . . (6.9)
Remark that (6.9) yields

d - ∞ ℓ=0 d ℓ = ρ - π 2 6 > ρ 2 .
as well as the following estimate

Π(ℓ, k) ≤ [2(k + 1) 2 ] k+1 e k ρ k (6.10)
We are now in position to discuss the convergence of the recurrence (6.6).

Proposition 6.3. Let:

|ε| < ε * (γ, τ, k) := 1 e 24(3+2τ ) (k + 2) 2τ V ρ,k (6.11) ρ > λ(k) := 1 + 8γτ τ [2(k + 1) 2 ].
(6.12)

Then the following estimate holds:

V ℓ,ε ρ ℓ ,k ≤ e 8(3+2τ ) V 0 ρ,k 2 ℓ = e 8(3+2τ ) V 0 ρ,k 2 ℓ , ℓ = 0, 1, 2, . . . V 0 := V. (6.13)
Proof. We proceed by induction. The assertion is true for ℓ = 0. Now assume inductively:

|ε j | V j,ε ρ j ,k ≤ (k + 2) -2τ (j+1) , 0 ≤ j ≤ ℓ. (6.14)
Out of (6.14) we prove the validity of (6.13) and of (6.5); to complete the induction it will be enough to show that (6.13) implies the validity of (6.14) for j = ℓ + 1.

Let us first estimate θ ℓ (N , ε) as defined by (5.15) assuming the validity of (6.14) . We obtain:

θ ℓ (N , ε) ≤ θ ℓ,k (N , ε) ≤ ℓ-1 s=0 |ε s | V ρs,k /d s = 1 ρ ℓ-1 s=0 (s + 1) 2 (k + 2) -2τ (s+1) = 1 4ρ d 2 dτ 2 ℓ-1 s=0 (k + 2) -2τ (s+1) = 1 4ρ d 2 dτ 2 [(k + 2) -2τ 1 -(k + 2) -2τ ℓ 1 -(k + 2) -2τ ≤ 1 ρ (k + 2) -2 ≤ 1 ρ because τ > l -1 ≥ 1. Now ρ > 1 entails that 1 1 -θ ℓ < ρ ρ -1 . (6.15)
Hence we get, by (6.2) and (5.16), the further (ℓ, ε)-independent estimate:

P (ℓ, k, ε) ≤ ρ k+1 (ρ -1) k+1 (k + 2) 2 ρ -k-1 = 1 (ρ -1)(k + 2) 2 k+1 .
(6.16) whence, by (6.3):

A(ℓ, k, ε) ≤ γ τ τ (ℓ + 1) 2τ (eρ) τ [1 + [2(k + 1) 2 ] k+1 (ρ -1)(k + 2) 2 -(k+1) (eρ 3 ) -k ] ≤ γ τ τ (ℓ + 1) 2τ (eρ) τ [1 + 2 (ρ -1) k+1 (eρ 3 ) -k ].
(6.17)

Upon application of the inductive assumption we get:

|ε ℓ |Ψ ℓ,k A(ℓ, k, ε) V ρ ℓ ,k /d ℓ ≤ 4 k [2(k + 1) 2 ] k+3 e k+3 ρ k+4 (ℓ + 1) 2τ +8 |ε ℓ |A(ℓ, k, ε) V ρ ℓ ,k ≤ γ τ τ (ℓ + 1) 2(τ +4) (eρ) τ [1 + 2 (ρ -1) k+1 (eρ 3 ) -k ] 4 k [2(k + 1) 2 ] k+3 e k+3 ρ k+4 (k + 2) -2(ℓ+1)τ ≤ 2(τ + 4) 2τ ln (k + 2) 2(τ +4) (k + 2) - 4(τ +4) 2τ ln (k+2) 4 k [2(k + 1) 2 ] k+3 e k+3 ρ k+4 γτ τ (eρ) τ [1 + 2 (ρ -1) k+1 (eρ 3 ) -k ] because sup ℓ≥0 (ℓ + 1) 2(τ +4) (k + 2) -2(ℓ+1)τ = 2(τ + 4) 2τ ln (k + 2) 2(τ +4) (k + 2) - 4(τ +4) 2τ ln (k+2) .
Hence:

|ε ℓ |Ψ ℓ,k A(ℓ, k, ε) V ρ ℓ ,k /d ℓ ≤ 1 2e (6.18) provided ρ ≥ λ(k); λ(k) = 1 + 8γτ τ [2(k + 1) 2 ]. (6.19) Since Ψ ℓ,k ≥ 1, if (6.19) holds, (6.18) a fortiori yields |ε ℓ |A(ℓ, k, ε) V ρ ℓ ,k /d ℓ ≤ 1 2 .
Therefore, by (6.4):

E(ℓ, k, ε) ≤ 3Ψ ℓ,k A(ℓ, k, ε) ≤ 6γ τ τ (ℓ + 1) 2τ (eρ) τ Ψ ℓ,k
and (6.6) in turn entails:

V ℓ+1 ρ ℓ +1,k ≤ Φ ℓ,k V ℓ 2 ρ ℓ ,k , Φ ℓ,k := 6γ τ τ (ℓ + 1) 2τ (eρ) τ Ψ ℓ,k .
This last inequality immediately yields

V ℓ+1 ρ ℓ ,k ≤ [ V ρ,k ] 2 ℓ+1 ℓ m=0 Φ 2m ℓ-m,k . (6.20) Now: Φ ℓ,k = 6γ τ τ (ℓ + 1) 2τ (eρ) τ (k + 1) 2 4 2k ed 3 ℓ [2(k + 1) 2 ] k+1 e k+τ d τ ℓ δ k ℓ ≤ γν(k, τ, ρ)(ℓ + 1) 6+4τ ν(k, τ, ρ) := 6 τ τ 4 2k [2(k + 1) 2 ] k+2 e k+τ +1 ρ k+τ +3 ≤ 6 τ τ 4 2k [2(k + 1) 2 ] k+2 e k+τ +1 λ(k) k+τ +3 ≤ ≤ 6 τ τ 4 2k [2(k + 1) 2 ] k+2 e k+τ +1 [8γτ τ 2(k + 1) 2 ] k+τ +3 ≤ 6 2 e k 1 e τ +1 γ k+τ +3 [2(k + 1) 2 ] τ +1 ≤ ≤ 6 γ τ +3 τ τ 2 +2 (2e) τ +1
Therefore γν(k, τ, ρ) ≤ 6 γ τ +2 τ τ 2 +2 (2e) τ +1 < 1 (6.21) because τ > 1 and γ > 1. As a consequence, since Φ j,k ≤ Φ ℓ,k , j = 1, . . ., we get: ℓ+1) ≤ [γν(k, τ, ρ)] ℓ(ℓ+1) (ℓ + 1) (6+4τ )ℓ(ℓ+1) ≤ (ℓ + 1) (6+4τ )ℓ(ℓ+1)

ℓ m=1 Φ 2m ℓ+1-m,k ≤ [Φ ℓ,k ] ℓ(
Now ℓ(ℓ + 1) < 2 ℓ+1 , ∀ ℓ ∈ N . Hence we can write:

(ℓ + 1) (6+4τ )ℓ(ℓ+1) < [e (24+16τ ) ] 2 ℓ+1 .

The following estimate is thus established 3+2τ ) ] 2 ℓ+1 . (6.22)

ℓ m=0 Ψ 2m ℓ-m,k ≤ [e 8(
If we now define: 3+2τ ) , µ ℓ := µ 2 ℓ (6.23) then (6.20) and (6.22) yield:

µ := e 8(
V ℓ+1,ε ℓ+1,k ≤ [µ ℓ V ℓ ρ ℓ ,k ] 2 ≤ [ V ρ,k µ] 2 ℓ+1 (6.24) ε ℓ+1 V ℓ+1,ε ℓ+1,k ≤ [ V ρ ℓ ,k µ ℓ ε ℓ ] 2 ≤ [ V ρ,k µε] 2 ℓ+1 (6.25)
Let us now prove out of (6.24,6.25) that the condition (6.14) preserves its validity also for j = ℓ+1.

We have indeed, by the inductive assumption (6.14) and (6.24):

|ε ℓ+1 |V ℓ+1,ε ℓ+1,k ≤ [ V ρ ℓ ,k µ ℓ ε ℓ ] 2 ≤ (k + 2) -2τ (ℓ+1) ε ℓ (µ ℓ ) 2 V ρ ℓ ,k ≤ (k + 2) -2τ (ℓ+1) εµ 3 V ρ,k 2 ℓ ≤ (k + 2) -2τ (ℓ+2) provided |ε| < 1 µ 3 V ρ,k (k + 2) 2τ = 1 e 24(3+2τ ) V ρ,k (k + 2) 2τ := ε * (γ, τ, k) (6.26)
where the last expression follows from (6.23). This proves (6.11), and concludes the proof of the Proposition.

Theorem 6.4. [Final estimates of W ℓ , N ℓ , V ℓ ] Let V fulfill Assumption (H2-H4). Then the following estimates hold, ∀ℓ ∈ N:

ε ℓ W ℓ,ε ρ ℓ+1 ,k ≤ γ τ e τ (ℓ + 1) 2τ (1 + 8γτ τ [2(k + 1) 2 ]) -τ • (µε V ρ ) 2 ℓ . (6.27)

ε ℓ N ℓ,ε ρ ℓ ,k ≤ ε ℓ V ℓ,ε ρ ℓ ,k ≤ [ V ρ εµ] 2 ℓ .
(6.28) (6.29) Proof. Since V does not depend on , obviously |V ρ,k ≡ V ρ . Then formula (5.27) yields, on account of (6.17), (6.15), (6.19), (6.24), (6.25) and of the obvious inequalities eρ -3 < 1, ρ/(ρ -1) > 1 when ρ > λ(k):

ε ℓ+1 V ℓ+1,ε ρ ℓ+1 ,k ≤ [ V ρ εµ] 2 ℓ+1 .
ε ℓ W ℓ,ε ρ ℓ ,k ≤ γ τ τ (ℓ + 1) 2τ (eρ) τ [1 + 2 (ρ -1) k+1 (eρ 3 ) -k ](µε V ρ ) 2 ℓ ≤ 2γ τ τ (ℓ + 1) 2τ (eρ) τ (µε V ρ ) 2 ℓ ≤ γ τ e τ (ℓ + 1) 2τ (1 + 8γτ τ [2(k + 1) 2 ]) -τ • (µε V ρ ) 2 ℓ .
because of the straightforward inequality [1 + 2 (ρ -1) k+1 (eρ 3 ) -k ] < 1 which in turn follows from γ > 1. This proves (6.27). Moreover, since N ℓ,ε = V ℓ,ε , again by (6.24), (6.25):

ε ℓ N ℓ,ε ρ ℓ ,k = ε ℓ V ℓ,ε ρ ℓ ,k ≤ [ V ρ εµ] 2 ℓ .
The remaining assertion follows once more from (6.25). This concludes the proof of the Theorem. ≤ [K(ℓ + 1) 2(τ +1) ] 2 -ℓ V ρ → V ρ , ℓ → ∞ so that (5.39) is actually fulfilled for |ε| < 1 V ρ .

Corollary 6.6. In the above assumptions set:

U n,ε ( ) := n s=0 e iε n-s W n-s,ε , n = 0, 1, . . . . (6.30)

Then:

(1) U n,ε ( ) is a unitary operator in L 2 (T l ), with U n,ε ( ) * = U n,ε ( ) -1 = n s=0 e -iεsWs,ε

(2) Let:

S n,ε ( ) := U n,ε ( )(L ω + εV )U n,ε ( ) -1 (6.31)

Then: Here the operators W s,ε , N s,ε , V ℓ+1,ε and their symbols W s,ε , N s,ε , V ℓ+1,ε fulfill the above estimates.

S n = D n,ε ( ) + ε n+1 V n+1,ε (
(3) Let ε * be defined as in (6. whenever ρ ≤ ρ ′ , and that ρ ℓ < ρ/2, ∀ ℓ ∈ N, (6.29) yields:

ε n+1 V n+1,ε ρ/2,k ≤ ε n+1 V n+1,ε ρ n+1 ,k ≤ [ V ρ,k µε] 2 n+1 → 0, n → ∞, k fixed.
In the same way, by (6.28):

N n,ε ρ/2,k ≤ N n,ε ρn,k = V n,ε ρn,k ≤ V n,ε ρn,,k ≤ [ V ρ,k µε] 2 n → 0, n → ∞, k fixed. → 0, n → ∞, k fixed.
This concludes the proof of the Corollary.

7. Convergence of the iteration and of the normal form.

Let us first prove the uniform convergence of the unitary transformation sequence as n → ∞.

Recall that ε * (•, k) > ε * (•, k + 1), k = 0, 1, . . ., and recall the abbreviation • ρ,0 := • ρ . Define moreover:

ε * := ε * 0 = ε * (γ, τ, 0). (7.1)

where ε * (γ, τ, 0) is defined by (6.26). Then:

Lemma 7.1. Let be fixed, and |ε| < ε * 0 . Consider the sequence {U n,ε ( )} of unitary operators in L 2 (T l ) defined by (6.30). Then there is a unitary operator U ∞,ε ( ) in L 2 (T l ) such that

lim n→∞ U n,ε ( ) -U ∞,ε ( ) L 2 →L 2 = 0
Proof. Without loss we can take = 1. We have, for p = 1, 2, . . .:

U n+p,ε -U n,ε = ∆ n+p,ε e iεnWn • • • e iεW 1 , ∆ n+p,ε := (e iε n+p W n+p • • • e iε n+1 W n+1 -I) U n+p,ε -U n,ε L 2 →L 2 ≤ 2 ∆ n+p,ε L 2 →L 2
Now we apply the mean value theorem and obtain

e iε ℓ W ℓ,ε = 1 + β ℓ,ε β ℓ,ε := iε ℓ W ℓ,ε ε ℓ 0 e iε ′ ℓ W ℓ,ε dε ′ ℓ ,
whence, by (6.27) in which we make k = 0:

β ℓ,ε ≤ ε ℓ W ℓ,ε ρ ℓ ≤ ε ℓ W ℓ,ε ρ ℓ ,k ≤ γτ τ (ℓ + 1) 2τ (1 + 8γτ τ [2(k + 1) 2 ]) 2-τ 64γ 2 τ 2τ [2(k + 1) 2 ] 4 • (µε V ρ ) 2 ℓ ≤ A ℓ (7.2)
for some A < 1. Now:

∆ n+p,ε = [(1 + β n+p,ε ε n+p )(1 + β n+p-1,ε ε n+p-1 ) • • • (1 + β n+1,ε ε n+1 )] = p j=1 β n+j,ε ε n+j + p j 1 <j 2 =1 β n+j 1 ,ε ε n+j 1 β n+j 2 ,ε ε n+j 2 + p j 1 <j 2 <j 3 =1 β n+j 1 ,ε ε n+j 1 β n+j 2 ,ε ε n+j 2 β n+j 3 ,ε ε n+j 3 + . . . + β n+1,ε • • • β n+p,ε ε n+1 • • • ε n+p

  The corresponding symbols are:S n (ξ, x; ) = D n,ε (L ω (ξ), ) + ε n+1 V n+1,ε (L ω (ξ), x; ) (6.34) D n,ε (L ω (ξ), ) = L ω (ξ) + n s=1 ε s N s,ε (L ω (ξ), ).(6.35)

  11). Remark that ε * (•, k) > ε * (•, k + 1), k = 0, 1, . . .. Then, if |ε| < ε(k, •): lim n→∞ D n,ε (L ω (ξ), ) = D ∞,ε (L ω (ξ), ) (6.36)where in the convergence takes place in the C k ([0, 1]; C ω (ρ/2)) topology, namelylim n→∞ D n,ε (L ω (ξ), ) -D ∞,ε (L ω (ξ), ) ρ/2,k = 0. (6.37)Proof. Since Assertions (1) and (2) are straightforward, we limit ourselves to the simple verifica-tion of Assertion (3). If |ε| < ε * (•, k) then V ρ,k µε < Λ < 1. Recalling that • ρ,,k ≤ • ρ ′ ,k

  y)

	If V fulfills Assumption (H3) of Theorem 1.6, both these series converge uniformly in any compact of R 2l away from the origin and P 1 is holomorphic on R 2l \ {0, 0}. Therefore Theorem 1.6 imme-diately entails a convergence criterion for the Birkhoff normal form generated by perturbations
	holomorphic away from the origin. We state it under the form of a corollary:
	Corollary 1.9. (A convergence criterion for the Birkhoff normal form) Under the assumptions of
	Theorem 1.6 on ω and V, consider on R 2l \ {0, 0} the holomorphic Hamiltonian family P ε (η, y) := P 0 (η, y) + εP 1 (η, y), ε ∈ R, where P 0 and P 1 are defined by (1.35,1.36). Then the Birkhoff normal form of H ε is uniformly convergent on any compact of R 2l \ {0, 0} if |ε| < ε * (γ, τ ).
	1.3. Strategy of the paper. The proof of Theorem 1.6 rests on an implementation in the quan-
	tum context of Rüssmann's argument[Ru] yielding convergence of the KAM iteration when the
	complex variables (z, z) belong to an open neighbourhood of the origin in C 2l . Conditions (1.25,
	1.34) prevent the occurrence of accidental degeneracies among eigenvalues at any step of the quan-
	tum KAM iteration, in the same way as they prevent the formation of resonances at the same
	step in the classical case. However, the global nature of quantum mechanics prevents phase-space
	l s=1	η 2 s + y 2 s η s -iy s	ks

Therefore, by (7.2):

Hence {U n,ε ( )} n∈N is a Cauchy sequence in the operator norm, uniformly with respect to |ε| < ε * 0 , and the Lemma is proved.

We are now in position to prove existence and analyticity of the limit of the KAM iteration, whence the uniform convergence of the QNF.

Proof of Theorems 1.6 and 1.7

The operator family H ε is self-adjoint in L 2 (T l ) with pure point spectrum ∀ ε ∈ R because V is a continuous operator. By Corollary 6.6, the operator sequence {D n,ε ( )} n∈N admits for |ε| < ε * 0 the uniform norm limit

of symbol D ∞, (L ω (ξ)). The series is norm-convergent by (6.28). By Lemma (7.1), D ∞, (L ω , ) is unitarily equivalent to H ε . The operator family ε → D ∞,ε ( ) is holomorphic for |ε| < ε * 0 , uniformly with respect to ∈ [0, 1]. As a consequence, D ∞,ε ( ) admits the norm-convergent expansion:

On the other hand, (6.37) entails that the symbol D ∞,ε (L ω (ξ), ) is a J (ρ/2)-valued holomorphic function of ε, |ε| < ε * 0 , continuous with respect to ∈ [0, 1]. Therefore it admits the expansion

) is the symbol yielding the quantum normal form via Weyl's quantization. Likewise, the symbol W ∞,ε (ξ, x, ) is a J(ρ/2)-valued holomorphic function of ε, |ε| < ε * , continuous with respect to ∈ [0, 1], and admits the expansion:

) is a direct consequence of (6.37) on account of the fact that

Remark indeed that by (6.37) the series (7.3) converges in the

) and the formula (1.31) follows from (7.3) upon Weyl quantization. This concludes the proof of the Theorem.

Appendix A. The quantum normal form

The quantum normal form in the framework of semiclassical analysis has been introduced by Sjöstrand [Sj]. We follow here the presentation of [BGP].

The formal construction Given the operator family

where [B p , L 0 ] = 0, p = 1, . . . , k -1. Recall the formal commutator expansion:

and look for W (ε) under the form of a power series:

becomes:

where

Since V s depends on W 1 , . . . , W s-1 , (A1) and (A3) yield the recursive homological equations:

To solve for S, W s , B s , we can equivalently look for their symbols. The equations (A.2), (A.3), (A.4) become, once written for the symbols:

where

In turn, the recursive homological equations become:

Solution of the homological equation and estimates of the solution

The key remark is that {A, L ω } M = {A, L ω } for any smooth symbol A(ξ; x; ) because L ω is linear in ξ. The homological equation (A.9) becomes therefore

We then have:

Proposition A.1. Let V s (ξ, x; ) ∈ J (ρ s ). Then the equation

given by:

Hence all terms of the quantum normal form and the remainder can be recursively estimated in we actually have, applying without modification the argument of [BGP], Proposition 3.2:

Proposition A.2. Let µ s < 1/2, s = 1, . . . , k. Set:

Then the following estimates hold for the quantum normal form:


A V Kazeykina

R G Novikov

Large time asymptotics for the Grinevich-Zakharov potentials

HAL is

Introduction

We consider the following 2 + 1-dimensional analog of the KdV equation (Novikov-Veselov equation):

∂ t v = 4Re 4∂ 3 z v + ∂ z (vw) -E∂ z w , ∂ z w = -3∂ z v, v = v, E ∈ R, v = v(x, t), w = w(x, t), x = (x 1 , x 2 ) ∈ R 2 , t ∈ R, (1.1)
where

∂ t = ∂ ∂t , ∂ z = 1 2 ∂ ∂x 1 -i ∂ ∂x 2 , ∂ z = 1 2 ∂ ∂x 1 + i ∂ ∂x 2 . (1.2)
We assume that v is sufficiently regular and has sufficient decay as |x| → ∞, w is decaying as |x| → ∞.

(1.3) Equation (1.1) is contained implicitly in the paper of S.V. Manakov [M] as an equation possessing the following representation:

∂(L -E) ∂t = [L -E, A] + B(L -E) (1.4) (Manakov L -A -B triple)
, where L = -∆ + v(x, t), ∆ = 4∂ z ∂ z , A and B are suitable differential operators of the third and zero order respectively. Equation (1.1) was written in an explicit form by S.P. Novikov and A.P.

Veselov in [START_REF] Novikov | Finite-zone, two-dimensional, potential Schrödinger operators. Explicit formula and evolutions equations[END_REF], [START_REF] Novikov | Finite-zone, two-dimensional Schrödinger operators. Potential operators[END_REF], where higher analogs of (1.1) were also constructed. Note that both Kadomtsev-Petviashvili equations can be obtained from (1.1) by considering an appropriate limit E → ±∞ (see [ZS], [START_REF] Grinevich | The scattering transform for the two-dimensional Schrödinger operator with a potential that decreases at infinity at fixed non-zero energy[END_REF]).

In the present article we are focused on a very interesting family of solutions for equation (1.1) for E = E f ix > 0 constructed by P.G. Grinevich and V.E. Zakharov, see [START_REF] Grinevich | Rational solitons of the Veselov-Novikov equation are reflectionless potentials at fixed energy[END_REF], [START_REF] Grinevich | The scattering transform for the two-dimensional Schrödinger operator with a potential that decreases at infinity at fixed non-zero energy[END_REF] (containing also a reference to private communication from V.E. Zakharov). The solutions of this family are given by v(x, t) = -4∂ z ∂ z ln det A, w(x, t) = 12∂ 2 z ln det A,

(1.5)

where

A = (A lm ) is 4N × 4N-matrix, A ll = iE 1/2 2 z - z λ 2 l -3iE 3/2 t λ 2 l - 1 λ 4 l -γ l , A lm = 1 λ l -λ m for l = m,
(1.6)

E 1/2 > 0, z = x 1 + ix 2 , z = x 1 -ix 2 , ∂ z , ∂ z are defined in (1.
2), and λ 1 , . . . , λ 4N , γ 1 , . . . , γ 4N are complex numbers such that

λ j = 0, |λ j | = 1, j = 1, . . . , 4N, λ l = λ m for l = m, λ 2j = -λ 2j-1 , γ 2j-1 -γ 2j = 1 λ 2j-1 , j = 1, . . . , 2N, λ 4j-1 = 1 λ 4j-3 , γ 4j-1 = λ2 4j-3 γ4j-3 , j = 1, . . . , N.
(1.7)

The functions v, w of (1.5)-(1.7) satisfy the Novikov-Veselov equation (1.1) for positive E of (1.6) and have also, in particular, the following properties (see [START_REF] Grinevich | Rational solitons of the Veselov-Novikov equation are reflectionless potentials at fixed energy[END_REF], [START_REF] Grinevich | The scattering transform for the two-dimensional Schrödinger operator with a potential that decreases at infinity at fixed non-zero energy[END_REF]):

v = v, w ∈ C ∞ (R 2 × R), v(x, t), w(x, t) are rational functions of x and t, v(x, t) = O |x| -2 , w(x, t) = O |x| -2 , |x| → ∞, for each t ∈ R;
(1.8) the Schrödinger equation Lψ = Eψ, where L = -∆ + v(x, t), has zero scattering amplitude for fixed E > 0 and t ∈ R.

(1.9)

Because of property (1.9) the potentials v of (1.5)-(1.7) are called transparent potentials.

Note that in many respects the solutions (1.5)-(1.7) of (1.1) are similar to related solutions of the KP1 equation, see [FA], [START_REF] Grinevich | Rational solitons of the Veselov-Novikov equation are reflectionless potentials at fixed energy[END_REF], [START_REF] Grinevich | The scattering transform for the two-dimensional Schrödinger operator with a potential that decreases at infinity at fixed non-zero energy[END_REF].

Note that the potentials v of (1.5)-(1.7) play an important role in the direct and inverse scattering theory for the aforementioned Schrödinger equation Lψ = Eψ at fixed energy E > 0 and in the theory of integrable systems, see [START_REF] Grinevich | The scattering transform for the two-dimensional Schrödinger operator with a potential that decreases at infinity at fixed non-zero energy[END_REF], [JS], [VW], [WY], [FN], [N] and references therein. However, the properties of these potentials were not yet studied sufficiently in the literature.

In order to formulate our results we use the following definition. We say that a solution (v, w) of (1.1) is a travel wave iff

v(x, t) = V (x -ct), w(x, t) = W (x -ct), x ∈ R 2 , t ∈ R,
(1.10) for some functions V , W on R 2 and some velocity c ∈ R 2 . In addition, we

identify c = (c 1 , c 2 ) ∈ R 2 with c = c 1 + ic 2 ∈ C.
The main results of the present note consist of the following:

(1) We show that (v, w) of the form (1.5)-(1.7) is a travel wave iff N = 1. See Lemma 2.1 of Section 2.

(2) We show that there are no travel waves of the form (1.5)-(1.7), N = 1, for c ∈ U E , and that there is an unique (modulo translations) travel wave of the form (1.5)-(1.7), N = 1, for c ∈ C\U E , where c denotes travel wave velocity and U E is defined by formula (2.2). In addition we show that there is one-to-one correspondence between permitted velocities c ∈ C\U E and λ-sets

{λ 1 , λ 2 , λ 3 , λ 4 } of (1.5)-(1.7), N = 1. See Lemma 2.2 of Section 2.
(3) We show that the large time asymptotics for the Grinevich-Zakharov potentials, that is for (v, w) defined by (1.5)-(1.7), is described by a sum of N localized travel waves propagating with different velocities. See Theorem 2.1 of Section 2. For KP1 equation a prototype of this result was given for the first time in [MZBIM].

Main results

The main results of this article consist of Lemmas 2.1, 2.2 and Theorem 2.1 presented below.

Lemma 2.1. Let (v, w) be defined by (1.5)-(1.7). Then (v, w) admits the representation (1.10) (and is a travel wave solution for (1.1)) if and only if N = 1. In addition,

c = 6E λ2 + 1 λ 2 + λ 2 λ 2 (2.1)
where c is the travel wave velocity and λ is any of λ j , j = 1, 2, 3, 4, which, in virtue of (1.7), determines uniquely the λ set {λ 1 , λ 2 , λ 3 , λ 4 } for E > 0.

Lemma 2.1 is proved in Section 3. Let (c) There is a one-to-one correspondence between c ∈ C\U E and the sets

U = {u ∈ C : u = re iϕ , r ≤ |6(2e -iϕ + e 2iϕ )|, ϕ ∈ [0, 2π]}, U E = {u ∈ C : u/E ∈ U}. (2.2) One can see that U 1 = U.
{λ 1 , λ 2 , λ 3 , λ 4 } satisfying (1.7).
The proof of this Lemma is given in Section 3 and is based principally on the following auxiliary lemma. This Lemma is a corollary of Lemma 3.1 from [KN].

Theorem 2.1. Let (v, w) be a solution of (1.1) constructed via (1.5)-(1.7).

Then the asymptotical behavior of (v, w) can be described as follows:

v ∼ N k=1 ν k (ξ k ), w ∼ N k=1 ω k (ξ k ) as t → ∞, (2.3)
where ξ k = zc 4k t and

c l = 6E λ2 l + 1 λ 2 l + λ 2 l λ 2 l
.

(2.4)

The functions ν k , ω k are defined by the formulas

ν k = -4∂ z ∂ z ln det A (k) , ω k = 12∂ 2 z ln det A (k) ,
(2.5)

where matrix A (k) is a 4×4 submatrix of matrix A, defined by formulas (1.6), such that

A (k) = {A lm } 4k l,m=4(k-1)+1 .
(2.6)

Remark. The relation (2.3) is understood in the following sense:

lim t→∞ v = lim t→∞ N k=1 ν k (ξ k ) for fixed ξ = z -ct, (2.7)
where

lim t→∞ ν k (ξ k ) = 0, for fixed ξ = z -ct, c = c 4k , ν k (ξ), for fixed ξ = z -c 4k t.
(2.8)

Theorem 2.1 is proved in Section 3. The scheme of the proof of this theorem follows principally the scheme of the derivation of the large time asymptotics for the multi-soliton solutions of the classic KdV equation (see, for example, [G]).

3 Proofs of Lemmas 2.1, 2.2 and Theorem 2.1

The text of the proofs presented below does not completely follow the order of statements in Section 2 as it was constructed to form a whole logical unit. However, we specify in due course which statement is being proved.

Proof of the sufficiency part of Lemma 2.1

Let us first consider the Grinevich-Zakharov potentials defined by (1.5)-(1.7) with N = 1. Then A is a 4 × 4 matrix and, in virtue of (1.7), the choice of any of λ j , j = 1, 2, 3, 4, uniquely determines the set {λ 1 , λ 2 , λ 3 , λ 4 }. Let us find c j such that A jj = A jj (zc j t). Such value c j is a solution of the following equation

i 2 E 1/2 z -ct - 1 λ 2 j (z -ct) = i 2 E 1/2 z - z λ 2 j -3iE 3/2 t λ 2 j - 1 λ 4 j .
(3.1) If |λ j | = 1, then this equation is uniquely solvable and its solution is given by

c j = 6E λ2 j + 1 λ 2 j + λ 2 j λ2 j . (3.2)
It is easy to see that due to (1.7)

c 1 = c 2 = c 3 = c 4 . Thus A = A(z -ct),
and the representation (1.10) with c defined by (2.1) holds. Thus sufficiency in Lemma 2.1 is proved.

Proof of Lemma 2.2

If c ∈ U E , then, as follows from item (a) of Lemma 2.3, c = c j , defined by (3.2), j = 1, 2, 3, 4 for any λ 1 , λ 2 , λ 3 , λ 4 satisfying the conditions indicated in (1.7) for N = 1. This and the sufficiency part of Lemma 2.1 imply item (a) of Lemma 2.2.

If c ∈ C\U E , then, as follows from item (b) of Lemma 2.3, it determines via (3.2) uniquely the set of λ 1 , λ 2 , λ 3 , λ 4 satisfying the conditions indicated in (1.7) for N = 1. Then the solution (v, w), constructed according to formulas (1.5)-(1.7) with N = 1, constitutes a travel wave solution of equation (1.1) with the given velocity c. In the construction procedure one of the parameters γ j can be chosen arbitrarily and it determines uniquely the whole set {γ 1 , . . . , γ 4 }.

One can see that the transform

z → z + ζ, t → t + τ
turns the potential (v, w) into another Grinevich-Zakharov potential (ṽ, w) with the parameters {λ 1 , . . . , λ 4 , γ1 , . . . , γ4 }, where

γ j -γj = iE 2 ζ - ζ λ 2 j -3iE 3/2 τ λ 2 j - 1 λ 4 j (3.3) for j = 1, 2, 3, 4.
On the other hand, if (ṽ, w) is a Grinevich-Zakharov potential with the set of parameters {λ 1 , . . . , λ 4 , γ1 , . . . , γ4 }, then it can be obtained from (v, w) by a translation, i.e. ṽ(z, t) = v(z + ζ, t + τ ), w(z, t) = w(z + ζ, t + τ ) for appropriate ζ ∈ C and t ∈ R such that (3.3) holds for some j (equations (3.3) are equivalent for j = 1, 2, 3, 4 in virtue of (1.7)). In addition, one can assume, for example, that τ = 0 in this translation.

Thus we have proved that any c ∈ C\U E determines uniquely, modulo translations, the solution of (1.1) of the form (1.5)-(1.7) with N = 1 and the given travel velocity c. This proves the point (b) of Lemma 2.2.

Item (c) of Lemma 2.2 follows immediately from Lemma 2.3. Lemma 2.2 is proved.

Proof of Theorem 2.1

Let us consider a more convenient representation of (v, w) defined by (1.5)-(1.7). For this purpose we first perform the differentiation with respect to z in the right-hand side of formula for v in (1.5):

v = -4∂ z (det A) -1 ∂ z (det A) = -4∂ z (det A) -1 4N i,j=1 ∂A ij ∂ z Âij , (3.4)
where Âij is the (i, j) cofactor of the matrix A. Similarly,

w = 12∂ z (det A) -1 4N i,j=1
∂A ij ∂z Âij .

(3.5)

In matrix A only diagonal elements depend on z, z, thus

v = -2iE 1/2 ∂ z (det A) -1 4N j=1 Âjj , w = -6iE 1/2 ∂ z (det A) -1 4N j=1 1 λ 2 j Âjj .
(3.6) Let us consider the following families V (j) and W (j) of systems of linear algebraic equations for functions ψ (3.9)

(j) k , η (j) k , j, k = 1, . . . 4N: V (j) : 4N k=1 A mk ψ (j) k = -2iE 1/2 δ mj , m = 1, . . . , 4N, (3.7) W (j) : 4N k=1 A mk η (j) k = -6iE 1/2 1 λ 2 j δ mj , m = 1, . . . , 4N
In order to write a system of linear algebraic equations for

∂ψ (j) k ∂z = ψ (j) k z
, we differentiate (3.7) with respect to z:

4N k=1 (A mk ) z ψ (j) k + 4N k=1 A mk ψ (j) k z = 0, m = 1, . . . , 4N,
and thus obtain

4N k=1 A mk ψ (j) k z = iE 1/2 2λ 2 m ψ (j) m , m = 1, . . . , 4N. (3.10)
Now let us note that A jj can be represented in the form

A jj = iE 1/2 2 (z -cj t) - 1 λ 2 j (z -c j t) -γ j ,
where c j is given by formula (2.4). As follows from item (c) of Lemma 2.2 c j = c k iff ⌊(j -1)/4⌋ = ⌊(k -1)/4⌋, where ⌊x⌋ denotes the integer part of x. Now let us fix ξ = zct and find the limits ψ

(j) k t → ∞ ξ fixed , ψ (j) k z t → ∞ ξ fixed
. We note that

A jj = iE 1/2 2 { ξ + (c -cj )t} - 1 λ 2 j {ξ + (c -c j )t} . If c = c j , then A jj = iE 1/2 2 ξ -1 λ 2 j
ξ and is independent of t. Otherwise, |A jj | → ∞ as t → ∞ at fixed ξ. We substitute this into (3.7) and consider the leading term in the Cramer's formula for ψ

(j) k t → ∞ ξ fixed = ψ(j) k (ξ), k, j : c k = c j = c, ψ (j) k t → ∞ ξ fixed = 0, k : c k = c or j : c j = c.
Here ψ(j) k (ξ) denotes some function of ξ independent of t at fixed ξ. Similarly, from (3.10) we obtain that

ψ (j) k z t → ∞ ξ fixed = ψ(j) k (ξ), k, j : c k = c j = c, ψ (j) k z t → ∞ ξ fixed = 0, k : c k = c or j : c j = c,
and, as previously, ψ(j) k (ξ) denotes some function of ξ independent of t at fixed ξ.

In addition, one can see that if there exists k such that c = c 4(k-1)+1 = . . . = c 4k , then

v| t → ∞ ξ fixed = 4k j=4(k-1)+1 ψ(j) j (ξ) = ν k (ξ), (3.11)
where ν k is defined by formula

ν k = -4∂ z ∂ z ln det A (k) , ( 3
.12) matrix A (k) is a 4 × 4 submatrix of matrix A from (1.6), such that A (k) = {A lm } 4k l,m=4(k-1)+1 . Similarly, for the case of function w we have

Proof of the necessity part of Lemma 2.1

From (3.15), taking into account (2.7)-(2.8), one can see that (v, w) can be a travel wave only if N = 1. This completes the proof of Lemma 2.1.

  Lemma 2.2. (a) Let c ∈ U E . Then there is no travel wave solution of (1.1) of the form (1.5)-(1.7) with N = 1 and the given travel wave velocity c. (b) Let c ∈ C\U E . Then there exists unique (modulo translations) solution of (1.1) of the form (1.5)-(1.7) with N = 1 and the given travel wave velocity c.

  Lemma 2.3. (a) Let c ∈ U E . Then equation (2.1) has no solution λ satisfying |λ| = 1. (b) Let c ∈ C\U E , then equation (2.1) has exactly four solutions λ 1 , λ 2 , λ 3 , λ 4 satisfying the conditions indicated in (1.7) for N = 1.

k

  as t → ∞. Thus we obtain

  ψ

  k (ξ), k, j : c k = c j = c, η k : c k = c or j : c j = c,where ηj k (ξ), ηj k (ξ) are some functions of ξ independent of t at fixed ξ. If there exists k such that c = c 4(k-1)+1 = . . . = c 4k , thenw| t → ∞ ξ fixed = 4k j=4(k-1)+1 η(j) j (ξ) = ω k (ξ), (3.13)where ω k is defined by formulaω k = 12∂ 2 z ln det A (k) (3.14)and matrix A(k) is the same as in (3ξ k ), t → ∞.(3.15)Here ξ k = zc 4k t, ν k , ω k are defined by (2.5)-(2.6) and the meaning of the relation (3.15) is specified by (2.7)-(2.8). Theorem 2.1 is proved.