KAM theory for the Hamiltonian derivative wave ... - UT Mathematics

KAM theory for the Hamiltonian derivative wave equation Massimiliano Berti, Luca Biasco, Michela Procesi Abstract: We prove an infinite dimensional KAM theorem which implies the existence of Cantor families of small-amplitude, reducible, elliptic, analytic, invariant tori of Hamiltonian derivative wave equations. 2000AMS subject classification: 37K55, 35L05. Th´ eorie KAM pour l’´ equation des ondes hamiltonienne avec d´ eriv´ ees R´ esum´ e: Nous prouvons un th´eor`eme KAM en dimension infinie, qui implique l’existence de familles de Cantor de tores invariants de petite amplitude, r´eductibles, elliptiques et analytiques, pour les ´equations des ondes hamiltoniennes avec d´eriv´ees.

1

Introduction

In the last years many progresses have been done concerning KAM theory for nonlinear Hamiltonian PDEs. The first existence results were given by Kuksin [18] and Wayne [29] for semilinear wave (NLW) and Schr¨ odinger equations (NLS) in one space dimension (1d) under Dirichlet boundary conditions, see [24]-[25] and [21] for further developments. The approach of these papers consists in generating iteratively a sequence of symplectic changes of variables which bring the Hamiltonian into a constant coefficients (=reducible) normal form with an elliptic (=linearly stable) invariant torus at the origin. Such a torus is filled by quasi-periodic solutions with zero Lyapunov exponents. This procedure requires to solve, at each step, constant-coefficients linear “homological equations” by imposing the “second order Melnikov” non-resonance conditions. Unfortunately these (infinitely many) conditions are violated already for periodic boundary conditions. In this case, existence of quasi-periodic solutions for semilinear 1d-NLW and NLS equations, was first proved by Bourgain [3] by extending the Newton approach introduced by Craig-Wayne [9] for periodic solutions. Its main advantage is to require only the “first order Melnikov” non-resonance conditions (the minimal assumptions) for solving the homological equations. Actually, developing this perspective, Bourgain was able to prove in [4], [6] also the existence of quasi-periodic solutions for NLW and NLS (with Fourier multipliers) in higher space dimensions, see also the recent extensions in [1], [28]. The main drawback of this approach is that the homological equations are linear PDEs with non-constant coefficients. Translated in the KAM language this implies a non-reducible normal form around the torus and then a lack of informations about the stability of the quasi-periodic solutions. Later on, existence of reducible elliptic tori was proved by Chierchia-You [7] for semilinear 1dNLW, and, more recently, by Eliasson-Kuksin [12] for NLS (with Fourier multipliers) in any space dimension, see also Procesi-Xu [27], Geng-Xu-You [14]. An important problem concerns the study of PDEs where the nonlinearity involves derivatives. A comprehension of this situation is of major importance since most of the models coming from Physics are of this kind. In this direction KAM theory has been extended to deal with KdV equations by Kuksin [19]-[20], Kappeler-P¨ oschel [17], and, for the 1d-derivative NLS (DNLS) and Benjiamin-Ono equations, by LiuYuan [22]. The key idea of these results is again to provide only a non-reducible normal form around 1

the torus. However, in this cases, the homological equations with non-constant coefficients are only scalar (not an infinite system as in the Craig-Wayne-Bourgain approach). We remark that the KAM proof is more delicate for DNLS and Benjiamin-Ono, because these equations are less “dispersive” than KdV, i.e. the eigenvalues of the principal part of the differential operator grow only quadratically at infinity, and not cubically as for KdV. As a consequence of this difficulty, the quasi-periodic solutions in [19], [17] are analytic, in [22], only C ∞ . Actually, for the applicability of these KAM schemes, the more dispersive the equation is, the more derivatives in the nonlinearity can be supported. The limit case of the derivative nonlinear wave equation (DNLW) -which is not dispersive at all- is excluded by these approaches. In the paper [3] (which proves the existence of quasi-periodic solutions for semilinear 1d-NLS and NLW), Bourgain claims, in the last remark, that his analysis works also for the Hamiltonian “derivation” wave equation  d2 1/2 F (x, y) , ytt − yxx + g(x)y = − 2 dx see also [5], page 81. Unfortunately no details are given. However, Bourgain [5] provided a detailed proof of the existence of periodic solutions for the non-Hamiltonian equation ytt − yxx + my + yt2 = 0 ,

m 6= 0 .

These kind of problems have been then reconsidered by Craig in [8] for more general Hamiltonian derivative wave equations like ytt − yxx + g(x)y + f (x, Dβ y) = 0 ,

x ∈ T,

p −∂xx + g(x). The where g(x) ≥ 0 and D is the first order pseudo-differential operator D := perturbative analysis of Craig-Wayne [9] for the search of periodic solutions works when β < 1. The main reason is that the wave equation vector field gains one derivative and then the nonlinear term f (Dβ u) has a strictly weaker effect on the dynamics for β < 1. The case β = 1 is left as an open problem. Actually, in this case, the small divisors problem for periodic solutions has the same level of difficulty of quasi-periodic solutions with 2 frequencies. The goal of this paper is to extend KAM theory to deal with the Hamiltonian derivative wave equation p ytt − yxx + my + f (Dy) = 0 , m > 0 , D := −∂xx + m , x ∈ T , (1.1) with real analytic nonlinearities (see Remark 7.1) X f (s) = as3 + fk sk ,

a 6= 0 .

(1.2)

k≥5

We write equation (1.1) as the infinite dimensional Hamiltonian system ut = −i∂u¯ H , with Hamiltonian

u ¯t = i∂u H ,

Z

u + u ¯ H(u, u ¯) := u ¯Du + F √ dx , 2 T in the complex unknown 1 u := √ (Dy + iyt ) , 2

Setting u =

X

Z

s

F (s) :=

1 u ¯ := √ (Dy − iyt ) , 2

f,

(1.3)

0

i :=

√

−1 .

uj eijx (similarly for u ¯), we obtain the Hamiltonian in infinitely many coordinates

j∈Z

H=

X j∈Z

Z λj uj u ¯j +

 1 X  (uj eijx + u F √ ¯j e−ijx ) dx 2 j∈Z T 2

(1.4)

where λj :=

p

j2 + m

(1.5)

are the eigenvalues of the diagonal operator D. Note thatZ the nonlinearity in (1.1) is x-independent implying, for (1.3), the conservation of the momentum −i

u ¯∂x u dx. This symmetry allows to simplify T

somehow the KAM proof (a similar idea was used by Geng-You [13]). X For every choice of the tangential sites I := {j1 , . . . , jn } ⊂ Z, n ≥ 2, the integrable Hamiltonian λ j uj u ¯j has the invariant tori {uj u ¯j = ξj , for j ∈ I , uj = u ¯j = 0 for j 6∈ I} parametrized by the j∈Z

actions ξ = (ξj )j∈I ∈ Rn . The next KAM result states the existence of nearby invariant tori for the complete Hamiltonian H in (1.4). Theorem 1.1. The equation (1.1)-(1.2) admits Cantor families of small-amplitude, analytic, quasiperiodic solutions with zero Lyapunov exponents and whose linearized equation is reducible to constant coefficients. Such Cantor families have asymptotically full measure at the origin in the set of parameters. The proof of Theorem 1.1 is based on the abstract KAM Theorem 4.1, which provides a reducible normal form (see (4.12)) around the elliptic invariant torus, and on the measure estimates Theorem 4.2. The key point in proving Theorem 4.2 is the asymptotic bound (4.15) on the perturbed normal frequencies Ω∞ (ξ) after the KAM iteration. This allows to prove that the second order Melnikov non-resonance conditions (4.11) are fulfilled for an asymptotically full measure set of parameters (see (4.19)). The estimate (4.15), in turn, is achieved by exploiting the quasi-T¨ oplitz property of the perturbation. This notion has been introduced by Procesi-Xu [27] in the context of NLS in higher space dimensions and it is similar, in spirit, to the T¨oplitz-Lipschitz property in Eliasson-Kuksin [12]. The precise formulation of quasi-T¨ oplitz functions, adapted to the DNLW setting, is given in Definition 3.4 below. Let us roughly explain the main ideas and techniques for proving Theorems 4.1, 4.2. These theorems concern, as usual, a parameter dependent family of analytic Hamiltonians of the form H = ω(ξ) · y + Ω(ξ) · z z¯ + P (x, y, z, z¯; ξ)

(1.6)

where (x, y) ∈ Tn × Rn , z, z¯ are infinitely many variables, ω(ξ) ∈ Rn , Ω(ξ) ∈ R∞ and ξ ∈ Rn . The frequencies Ωj (ξ) are close to the unperturbed frequencies λj in (1.5). As well known, the main difficulty of the KAM iteration which provides a reducible KAM normal form like (4.12) is to fulfill, at each iterative step, the second order Melnikov non-resonance conditions. Actually, following the formulation of the KAM theorem given in [2], it is sufficient to verify ∞ |ω ∞ (ξ) · k + Ω∞ i (ξ) − Ωj (ξ)| ≥

γ , 1 + |k|τ

γ > 0,

(1.7)

only for the “final” frequencies ω ∞ (ξ) and Ω∞ (ξ), see (4.11), and not along the inductive iteration. The application of the usual KAM theory (see e.g. [18], [24]-[25]), to the DNLW equation provides only the asymptotic decay estimate Ω∞ j (ξ) = j + O(1)

for j → +∞ .

(1.8)

Such a bound is not enough: the set of parameters ξ satisfying (1.7) could be empty. Note that for the semilinear NLW equation (see e.g. [24]) the frequencies decay asymptotically faster, namely like Ω∞ j (ξ) = j + O(1/j). The key idea for verifying the second order Melnikov non-resonance conditions (1.7) for DNLW is to prove the higher order asymptotic decay estimate (see (4.15), (4.2)) Ω∞ j (ξ) = j + a+ (ξ) +

m γ 2/3 + O( ) 2j j 3

for j ≥ O(γ −1/3 )

(1.9)

where a+ (ξ) is a constant independent of j (an analogous expansion holds for j → −∞ with a possibly different limit constant a− (ξ)). In this way infinitely many conditions in (1.7) are verified by imposing only first order Melnikov conditions like |ω ∞ (ξ) · k + h| ≥ 2γ 2/3 /|k|τ , h ∈ Z. Indeed, for i > j > O(|k|τ γ −1/3 ), we get ∞ |ω ∞ (ξ) · k + Ω∞ i (ξ) − Ωj (ξ)| =

≥

|ω ∞ (ξ) · k + i − j +

m(i − j) + O(γ 2/3 /j)| 2ij

2γ 2/3 |k|−τ − O(|k|/j 2 ) − O(γ 2/3 /j) ≥ γ 2/3 |k|−τ

noting that i − j is integer and |i − j| = O(|k|) (otherwise no small divisors occur). We refer to section 6 for the precise arguments, see in particular Lemma 6.2. The asymptotic decay (4.15) for the perturbed frequencies Ω∞ (ξ) is achieved thanks to the “quasiT¨ oplitz” property of the perturbation (Definition 3.4). Let us roughly explain this notion. The new 0 0 normal frequencies after each KAM step are Ω+ j = Ωj +Pj where the corrections Pj are the coefficients of the quadratic form Z X P 0 z z¯ := Pj0 zj z¯j , Pj0 := (∂z2j z¯j P )(x, 0, 0, 0; ξ) dx . Tn

j

We say that a quadratic form P 0 is quasi-T¨oplitz if it has the form P0 = T + R where T is a T¨ oplitz matrix (i.e. constant on the diagonals) and R is a “small” remainder satisfying Rjj = O(1/j) (see Lemma 5.2). Then (1.9) follows with a := Tjj which is independent of j. Since the quadratic perturbation P 0 along the KAM iteration does not depend only on the quadratic perturbation at the previous steps, we need to extend the notion of quasi-T¨oplitz to general (non-quadratic) analytic functions. The preservation of the quasi-T¨ oplitz property of the perturbations P at each KAM step (with just slightly modified parameters) holds in view of the following key facts: 1. the Poisson bracket of two quasi-T¨ oplitz functions is quasi-T¨oplitz (Proposition 3.1), 2. the hamiltonian flow generated by a quasi-T¨oplitz function preserves the quasi-T¨oplitz property (Proposition 3.2), 3. the solution of the homological equation with a quasi-T¨oplitz perturbation is quasi-T¨oplitz (Proposition 5.1). We note that, in [12], the analogous properties 1 (and therefore 2) for T¨opliz-Lipschitz functions is proved only when one of them is quadratic. The definition of quasi-T¨ oplitz functions heavily relies on properties of projections. However, for an analytic function in infinitely many variables, such projections may not be well defined unless the Taylor-Fourier series (see (2.28)) is absolutely convergent. For such reason, instead of the sup-norm, we use the majorant norm (see (2.12), (2.54)), for which the bounds (2.14) and (2.55) on projections hold (see also Remark 2.4). We underline that the majorant norm of a vector field introduced in (2.54) is very different from the weighted norm introduced by P¨ oschel in [23]-Appendix C, which works only in finite dimension, see comments in [23] after Lemma C.2 and Remark 2.3. As far as we know this majorant norm of vector fields is new. In Section 2 we show its properties, in particular the key estimate of the majorant norm of the commutator of two vector fields (see Lemma 2.15). Before concluding this introduction we also mention the recent KAM theorem of Greb´ert-Thomann [16] for the quantum harmonic oscillator with semilinear nonlinearity. Also here the eigenvalues grow 4

to infinity only linearly. We quote the normal form results of Delort-Szeftel [10], Delort [11], for quasi-linear wave equations, where only finitely many steps of normal form can be performed. Finally we mention also the recent work by G´erard-Grellier [15] on Birkhoff normal form for a degenerate “half-wave” equation. The paper is organized as follows: • In section 2 we define the majorant norm of formal power series of scalar functions (Definition 2.2) and vector fields (Definition 2.6) and we investigate the relations with the notion of analiticity, see Lemmata 2.1, 2.2, 2.3, 2.11 and Corollary 2.1. Then we prove Lemma 2.15 on commutators. • In section 3 we define the T¨ oplitz (Definition 3.3) and Quasi-T¨oplitz functions (Definition 3.4). Then we prove that this class of functions is closed under Poisson brackets (Proposition 3.1) and composition with the Hamiltonian flow (Proposition 3.2). • In section 4 we state the abstract KAM Theorem 4.1. The first part of Theorem 4.1 follows by the KAM Theorem 5.1 in [2]. The main novelty is part II, in particular the asymptotic estimate (4.15) of the normal frequencies. • In section 5 we prove the abstract KAM Theorem 4.1. We first perform (as in Theorem 5.1 in [2]) a first normal form step, which makes Theorem 4.1 suitable for the direct application to the wave equation. In Proposition 5.1 we prove that the solution of the homological equation with a quasi-T¨oplitz perturbation is quasi-T¨ oplitz. Then the main results of the KAM step concerns the asymptotic estimates of the perturbed frequencies (section 5.2.3) and the T¨oplitz estimates of the new perturbation (section 5.2.4). • In section 6 we prove Theorem 4.2: the second order Melnikov non-resonance conditions are fulfilled for a set of parameters with large measure, see (4.19). We use the conservation of momentum to avoid the presence of double eigenvalues. • In section 7 we finally apply the abstract KAM Theorem 4.1 to the DNLW equation (1.1)-(1.2), proving Theorem 1.1. We first verify that the Hamiltonian (1.4) is quasi-T¨oplitz (Lemma 7.1), as well as the Birkhoff normal form Hamiltonian (7.8) of Proposition 7.1. The main technical difficulties concern the proof in Lemma 7.4 that the generating function (7.17) of the Birkhoff symplectic transformation is also quasi-T¨oplitz (and the small divisors Lemma 7.2). In section 7.2 we prove that the perturbation, obtained after the introduction of the action-angle variables, is still quasi-T¨ oplitz (Proposition 7.2). Finally in section 7.3 we prove Theorem 1.1 applying Theorems 4.1 and 4.2.

2

Functional setting

Given a finite subset I ⊂ Z (possibly empty), a ≥ 0, p > 1/2, we define the Hilbert space n o X 2 `a,p |zj |2 e2a|j| hji2p < ∞ . I := z = {zj }j∈Z\I , zj ∈ C : kzka,p := j∈Z\I

When I = ∅ we denote `a,p :=

`a,p I .

We consider the direct product a,p E := Cn × Cn × `a,p I × `I

(2.1)

where n is the cardinality of I. We endow the space E with the (s, r)-weighted norm v = (x, y, z, z¯) ∈ E ,

kvkE := kvkE,s,r = 5

|y|1 kzka,p k¯ z ka,p |x|∞ + 2 + + s r r r

(2.2)

where, 0 < s, r < 1, and |x|∞ := max |xh |, |y|1 := h=1,...,n

n X

|yh |. Note that, for all s0 ≤ s, r0 ≤ r,

h=1

kvkE,s0 ,r0 ≤ max{s/s0 , (r/r0 )2 }kvkE,s,r .

(2.3)

We shall also use the notations zj− = z¯j .

zj+ = zj ,

We identify a vector v ∈ E with the sequence {v (j) }j∈J with indices in ( ( {1, . . . , n} if j1 = 1, 2 J := j = (j1 , j2 ), j1 ∈ {1, 2, 3, 4}, j2 ∈ Z\I if j1 = 3, 4

) (2.4)

and components v (1,j2 ) := xj2 , v (2,j2 ) := yj2 (1 ≤ j2 ≤ n), v (3,j2 ) := zj2 , v (4,j2 ) := z¯j2 (j2 ∈ Z \ I) , more compactly v (1,·) := x , v (2,·) := y, , v (3,·) := z, , v (4,·) := z¯ . We denote by {ej }j∈J the orthogonal basis of the Hilbert space E, where ej is the sequence with all zeros, X except the j2 -th entry of its j1 -th components, which is 1. Then every v ∈ E writes v= v (j) ej , v (j) ∈ C. We also define the toroidal domain j∈J

D(s, r) := Tns × D(r) := Tns × Br2 × Br × Br ⊂ E where D(r) := Br2 × Br × Br , n Tns := x ∈ Cn :

o n o max |Im xh | < s , Br2 := y ∈ Cn : |y|1 < r2

h=1,...,n

(2.5)

(2.6)

n and Br ⊂ `a,p I is the open ball of radius r centered at zero. We think T as the n-dimensional torus n n n T := 2πR /Z , namely f : D(s, r) → C means that f is 2π-periodic in each xh -variable, h = 1, . . . , n.

Remark 2.1. If n = 0 then D(s, r) ≡ Br × Br ⊂ `a,p × `a,p .

2.1 2.1.1

Majorant norm Scalar functions

We consider formal power series with infinitely many variables X f (v) = f (x, y, z, z¯) = fk,i,α,β eik·x y i z α z¯β

(2.7)

(k,i,α,β)∈I

with coefficients fk,i,α,β ∈ C and multi-indices in

where

I := Zn × Nn × N(Z\I) × N(Z\I)

(2.8)

n o X N(Z\I) := α := (αj )j∈Z\I ∈ NZ with |α| := αj < +∞ .

(2.9)

j∈Z\I α

β

In (2.7) we use the standard multi-indices notation z α z¯β := Πj∈Z\I zj j z¯j j . We denote the monomials mk,i,α,β (v) = mk,i,α,β (x, y, z, z¯) := eik·x y i z α z¯β . 6

(2.10)

Remark 2.2. If n = 0 the set I reduces to NZ ×NZ and the formal series to f (z, z¯) =

X

fα,β z α z¯β .

(α,β)∈I

We define the “majorant” of f as

M f (v) = M f (x, y, z, z¯) :=

X

|fk,i,α,β |eik·x y i z α z¯β .

(2.11)

(k,i,α,β)∈I

We now discuss the convergence of formal series. Definition 2.1. A series X

ck,i,α,β ,

ck,i,α,β ∈ C ,

(k,i,α,β)∈I

is absolutely convergent if the function I 3 (k, i, α, β) → 7 ck,i,α,β ∈ C is in L1 (I, µ) where µ is the counting measure of I. Then we set Z X ck,i,α,β := ck,i,α,β dµ . I

(k,i,α,β)∈I

By the properties of the Lebesgue integral, given any sequence {Il }l≥0 of finite subsets Il ⊂ I with Il ⊂ Il+1 and ∪l≥0 Il = I, the absolutely convergent series X X X ck,i,α,β := ck,i,α,β = lim ck,i,α,β . k,i,α,β

l→∞

(k,i,α,β)∈I

(k,i,α,β)∈Il

Definition 2.2. (Majorant-norm: scalar functions) The majorant-norm of a formal power series (2.7) is X |fk,i,α,β |e|k|s |y i ||z α ||¯ zβ | (2.12) kf ks,r := sup (y,z,¯ z )∈D(r) k,i,α,β

where |k| := |k|1 := |k1 | + . . . + |kn |. By (2.7) and (2.12) we clearly have kf ks,r = kM f ks,r . For every subset of indices I ⊂ I, we define the projection X (ΠI f )(x, y, z, z¯) := fk,i,α,β eik·x y i z α z¯β

(2.13)

(k,i,α,β)∈I

of the formal power series f in (2.7). Clearly kΠI f ks,r ≤ kf ks,r

(2.14)

ΠI ΠI 0 = ΠI∩I 0 = ΠI 0 ΠI .

(2.15)

and, for any I, I 0 ⊂ I, it results Property (2.14) is one of the main advantages of the majorant-norm with respect to the usual sup-norm |f |s,r :=

|f (v)| .

sup

(2.16)

v∈D(s,r)

We now define useful projectors on the time Fourier indices. Definition 2.3. Given ς = (ς1 , . . . , ςn ) ∈ {+, −}n we define X fς := Πς f := ΠZnς ×Nn ×N(Z\I) ×N(Z\I) f =

fk,i,α,β eik·x y i z α z¯β

(2.17)

k∈Zn ς ,i,α,β

where n Znς := k ∈ Zn

( with

kh ≥ 0 kh < 0

7

if if

ςh = + ςh = −

∀1 ≤ h ≤ n

o .

(2.18)

Then any formal series f can be decomposed as X f=

Πς f

(2.19)

ς∈{+,−}n

and (2.14) implies kΠς f ks,r ≤ kf ks,r . We now investigate the relations between formal power series with finite majorant norm and analytic functions. We recall that a function f : D(s, r) → C is • analytic, if f ∈ C 1 (D(s, r), C), namely the Fr´echet differential D(s, r) 3 v 7→ df (v) ∈ L(E, C) is continuous, • weakly analytic, if ∀v ∈ D(s, r), v 0 ∈ E \ {0}, there exists ε > 0 such that the function {ξ ∈ C , |ξ| < ε } 7→ f (v + ξv 0 ) ∈ C is analytic in the usual sense of one complex variable. A well known result (see e.g. Theorem 1, page 133 of [26]) states that a function f is ⇐⇒

analytic

weakly analytic and locally bounded .

(2.20)

Lemma 2.1. Suppose that the formal power series (2.7) is absolutely convergent for all v ∈ D(s, r). Then f (v) and M f (v), defined in (2.7) and (2.11), are well defined and weakly analytic in D(s, r). If, moreover, the sup-norm |f |s,r < ∞, resp. |M f |s,r < ∞, then f , resp. M f , is analytic in D(s, r). Proof. Since the series (2.7) is absolutely convergent the functions f , M f , and, for all ς ∈ {+, −}n , fς := Πς f , M fς (see (2.17)) are well defined (also the series in (2.17) is absolutely convergent). We now prove that each M fς is weakly analytic, namely ∀v ∈ D(s, r), v 0 ∈ E \ {0}, X M fς (v + ξv 0 ) = |fk,i,α,β |mk,i,α,β (v + ξv 0 ) (2.21) k∈Zn ς ,i,α,β

is analytic in {|ξ| < ε}, for ε small enough (recall the notation (2.10)). Since each ξ 7→ mk,i,α,β (v + ξv 0 ) is entire, the analyticity of M fς (v + ξv 0 ) follows once we prove that the series (2.21) is totally convergent, namely X |fk,i,α,β | sup |mk,i,α,β (v + ξv 0 )| < +∞ . (2.22) k∈Zn ς ,i,α,β

|ξ| µ0 , we have Bs,r (N, θ0 , µ0 ) ⊂ Bs,r (N, θ, µ) . Remark 3.1. The projection ΠN,θ,µ can be written in the form ΠI , see (2.13), for a suitable I ⊂ I. The representation in (3.11) is not unique. It becomes unique if we impose the “symmetric” conditions 0

0

σ,σ σ ,σ fm,n = fn,m .

Note that the coefficients in (3.12)-(3.13) satisfy (3.14).

22

(3.14)

3.1

T¨ oplitz functions

Let N ≥ N0 . Definition 3.3. (T¨ oplitz) A function f ∈ Bs,r (N, θ, µ) is (N, θ, µ)-T¨oplitz if the coefficients in (3.11) have the form 0

0

σ,σ fm,n = f σ,σ s(m), σm + σ 0 n




0

for some f σ,σ (ς, h) ∈ Ls,r (N, µ, h) ,

(3.15)

with s(m) := sign(m), ς = +, − and h ∈ Z. We denote by Ts,r := Ts,r (N, θ, µ) ⊂ Bs,r (N, θ, µ) the space of the (N, θ, µ)-T¨ oplitz functions. For parameters N 0 ≥ N , θ0 ≥ θ, µ0 ≤ µ, r0 ≤ r, s0 ≤ s we have Ts,r (N, θ, µ) ⊆ Ts0 ,r0 (N 0 , θ0 , µ0 ) .

(3.16)

Lemma 3.2. Consider f, g ∈ Ts,r (N, θ, µ) and p ∈ Ls,r (N, µ1 , 0) with 1 < µ, µ1 < 6. For all 0 < s0 < s , 0 < r0 < r and θ0 ≥ θ, µ0 ≤ µ one has ΠN,θ0 ,µ0 {f, p}L , ΠN,θ0 ,µ0 {f, p}x,y ∈ Ts0 ,r0 (N, θ0 , µ0 ) .

(3.17)

µN L + κN b < (θ0 − θ)N

(3.18)

ΠN,θ0 ,µ0 {f, g}H ∈ Ts0 ,r0 (N, θ0 , µ0 ) .

(3.19)

If moreover then 0

σ,σ Proof. Write f ∈ Ts,r (N, θ, µ) as in (3.11) where fm,n satisfy (3.15) and (3.14), namely 0

0

0

σ,σ σ ,σ fm,n = fn,m = f σ,σ (s(m), σm + σ 0 n) ∈ Ls,r (N, µ, σm + σ 0 n) ,

(3.20)

similarly for g. 0 σ Proof of (3.17). Since the variables zm , znσ , |m|, |n| > θN , are high momentum, 0

0

0

σ,σ σ σ σ,σ σ σ {fm,n zm zn , p}L = {fm,n , p}L zm zn 0

0

0

σ,σ σ,σ and {fm,n , p}L does not depend on wH by (3.9) (recall that fm,n , p ∈ Ls,r (N, µ)). The coefficient 0 σ σ L of zm zn in ΠN,θ0 ,µ0 {f, p} is 0

(3.20)

0

σ,σ L σ,σ ΠL = ΠL (s(m), σm + σ 0 n) , p}L ∈ Ls0 ,r0 (N, µ0 , σm + σ 0 n) N,µ0 {fm,n , p} N,µ0 {f

using Lemma 3.1 (recall that p has zero momentum). The proof that ΠN,θ0 ,µ0 {f, p}x,y ∈ Ts0 ,r0 (N, θ0 , µ0 ) is analogous. Proof of (3.19). A direct computation, using (3.4), gives X 0 σ σ0 {f, g}H = pσ,σ m,n zm zn |m|,|n|>θN, σ,σ 0 =±

with

0

pσ,σ m,n = 2i

X

  σ,σ1 −σ1 ,σ 0 σ 0 ,σ1 −σ1 ,σ σ1 fm,l gl,n + fn,l gl,m .

|l|>θN , σ1 =±

23

(3.21)

0

By (3.9) the coefficient pσ,σ m,n does not depend on wH . Therefore X σ,σ 0 σ σ 0 σ,σ 0 σ,σ 0 ΠN,θ0 ,µ0 {f, g}H = qm,n zm zn with qm,n := ΠL N,µ0 pm,n

(3.22)

|m|,|n|>θ 0 N, σ,σ 0 =± 0

σ,σ (recall (3.7)). It results qm,n ∈ Ls0 ,r0 (N, µ0 , σm + σ 0 n) by (3.22), (3.21), and Lemma 3.1 since, i.e., σ,σ1 fm,l ∈ Ls,r (N, µ, σm + σ1 l)

0

−σ1 ,σ and gl,n ∈ Ls,r (N, µ, −σ1 l + σ 0 n) .

Hence the (N, θ0 , µ0 )-bilinear function ΠN,θ0 ,µ0 {f, g}H in (3.22) is written in the form (3.11). It remains to prove that it is (N, θ0 , µ0 )-T¨ oplitz, namely that for all |m|, |n| > θ0 N , σ, σ 0 = ±,
0 0 σ,σ 0 qm,n = q σ,σ s(m), σm + σ 0 n for some q σ,σ (ς, h) ∈ Ls,r (N, µ0 , h) . (3.23) Let us consider in (3.21)-(3.22) the term (with m, n, σ, σ 0 , σ1 fixed) X σ,σ −σ ,σ0 ΠL fm,l 1 gl,n 1 N,µ0

(3.24)

|l|>θN

(the other is analogous). Since f, g ∈ Ts,r (N, θ, µ) we have
σ,σ1 fm,l = f σ,σ1 s(m), σm + σ1 l ∈ Ls,r (N, µ, σm + σ1 l)
0 −σ1 ,σ 0 gl,n = g −σ1 ,σ s(l), −σ1 l + σ 0 n ∈ Ls,r (N, µ, −σ1 l + σ 0 n) .

(3.25) (3.26)

0

σ,σ1 −σ1 ,σ By (3.10), (3.25), (3.26), if the coefficients fm,l , gl,n are not zero then

|σm + σ1 l| , | − σ1 l + σ 0 n| < µN L + κN b .

(3.27)

By (3.27), (3.1), we get cN > |σm + σ1 l| = |σσ1 s(m)|m| + s(l)|l||, which implies, since |m| > θ0 N > N (see (3.22)), that the sign s(l) = −σσ1 s(m) . (3.28) Moreover (3.27)

(3.18)

|l| ≥ |m| − |σm + σ1 l| > θ0 N − µN L − κN b > θN . This shows that the restriction |l| > θN in the sum (3.24) is automatically met. Then ΠL N,µ0

X

σ,σ1 −σ1 ,σ fm,l gl,n

0

(3.26)

=

ΠL N,µ0

X

=

ΠL N,µ0

X

ΠL N,µ0

X

|l|>θN



0 f σ,σ1 s(m), σm + σ1 l g −σ1 ,σ s(l), −σ1 l + σ 0 n

l∈Z



0 f σ,σ1 s(m), j g −σ1 ,σ s(l), σm + σ 0 n − j

j∈Z (3.28)

=



0 f σ,σ1 s(m), j g −σ1 ,σ − σσ1 s(m), σm + σ 0 n − j

j∈Z

depends only on s(m) and σm + σ 0 n, i.e. (3.23).

3.2

Quasi-T¨ oplitz functions

Given f ∈ Hs,r and f˜ ∈ Ts,r (N, θ, µ) we set fˆ := N (ΠN,θ,µ f − f˜) .

(3.29)

All the functions f ∈ Hs,r below possibly depend on parameters ξ ∈ O, see (2.86). For simplicity we shall often omit this dependence and denote k ks,r,O = k ks,r . 24

null Definition 3.4. (Quasi-T¨ oplitz) A function f ∈ Hs,r is called (N0 , θ, µ)-quasi-T¨ oplitz if the quasiT¨ oplitz semi-norm h  i kf kTs,r := kf kTs,r,N0 ,θ,µ := sup inf max{kXf ks,r , kXf˜ks,r , kXfˆks,r } (3.30) N ≥N0

f˜∈Ts,r (N,θ,µ)

is finite. We define n o null QTs,r := QTs,r (N0 , θ, µ) := f ∈ Hs,r : kf kTs,r,N0 ,θ,µ < ∞ . In other words, a function f is (N0 , θ, µ)-quasi-T¨oplitz with semi-norm kf kTs,r if, for all N ≥ N0 , ∀ε > 0, there is f˜ ∈ Ts,r (N, θ, µ) such that ΠN,θ,µ f = f˜ + N −1 fˆ and kXf ks,r , kXf˜ks,r , kXfˆks,r ≤ kf kTs,r + ε .

(3.31)

We call f˜ ∈ Ts,r (N, θ, µ) a “T¨ oplitz approximation” of f and fˆ the “T¨ oplitz-defect”. Note that, by Definition 3.3 and (3.29) ΠN,θ,µ f˜ = f˜ , ΠN,θ,µ fˆ = fˆ . By the definition (3.30) we get kXf ks,r ≤ kf kTs,r

(3.32)

and we complete (2.88) noting that quasi-T¨ oplitz

M − regular

=⇒

=⇒

regular

⇐=

λ − regular .

(3.33)

Clearly, if f is (N0 , θ, µ)-T¨ oplitz then f is (N0 , θ, µ)-quasi-T¨oplitz and kf kTs,r,N0 ,θ,µ = kXf ks,r .

(3.34)

Then we have the following inclusions null Ts,r ⊂ QTs,r , Bs,r ⊂ Hs,r ⊂ Hs,r .

Note that neither Bs,r ⊆ QTs,r nor Bs,r ⊇ QTs,r . Lemma 3.3. For parameters N1 ≥ N0 , µ1 ≤ µ, θ1 ≥ θ, r1 ≤ r, s1 ≤ s, we have QTs,r (N0 , θ, µ) ⊂ QTs1 ,r1 (N1 , θ1 , µ1 ) and kf kTs1 ,r1 ,N1 ,θ1 ,µ1 ≤ max{s/s1 , (r/r1 )2 }kf kTs,r,N0 ,θ,µ . Proof. By (3.31), for all N ≥ N1 ≥ N0 (since θ1 ≥ θ, µ1 ≤ µ) ΠN,θ1 ,µ1 f = ΠN,θ1 ,µ1 ΠN,θ,µ f = ΠN,θ1 ,µ1 f˜ + N −1 ΠN,θ1 ,µ1 fˆ . The function ΠN,θ1 ,µ1 f˜ ∈ Ts1 ,r1 (N, θ1 , µ1 ) and (2.79)

kXΠN,θ

f˜ks1 ,r1 1 ,µ1

(3.31)

≤ kXf˜ks1 ,r1 (2.79)

kXΠN,θ

ˆks1 ,r1

f 1 ,µ1

≤ kf kTs1 ,r1 + ε ,

(3.31)

≤ kXfˆks1 ,r1

≤ kf kTs1 ,r1 + ε .

Hence, ∀N ≥ N1 , inf

f˜∈Ts1 ,r1 (N,θ1 ,µ1 )



 max{kXf ks1 ,r1 , kXf˜ks1 ,r1 , kXfˆks1 ,r1 } ≤ kf kTs1 ,r1 + ε ,

25

(3.35)

applying (2.3) we have (3.35), because ε > 0 is arbitrary. For f ∈ Hs,r we define its homogeneous component of degree l ∈ N, X f (l) := Π(l) f := fk,i,α,β eik·x y i z α z¯β ,

(3.36)

k∈Zn , 2|i|+|α|+|β|=l

and the projections fK := Π|k|≤K f :=

X

fk,i,α,β eik·x y i z α z¯β ,

Π>K f := f − Π|k|≤K f .

(3.37)

|k|≤K,i,α,β

We also set ≤2 fK := Π|k|≤K f ≤2 ,

f ≤2 := f (0) + f (1) + f (2) .

(3.38)

The above projectors Π(l) , Π|k|≤K , Π>K have the form ΠI , see (2.13), for suitable subsets I ⊂ I. Lemma 3.4. (Projections) Let f ∈ QTs,r (N0 , θ, µ). Then, for all l ∈ N, K ∈ N, kΠ(l) f kTs,r,N0 ,θ,µ ≤ kf kTs,r,N0 ,θ,µ

(3.39)

≤2 T kf ≤2 kTs,r,N0 ,θ,µ , kf − fK ks,r,N0 ,θ,µ ≤ kf kTs,r,N0 ,θ,µ

(3.40)

kΠ|k|≤K f kTs,r,N0 ,θ,µ ≤ kf kTs,r,N0 ,θ,µ

(3.41)

kΠk=0 Π|α|=|β|=1 Π(2) f kTr,N0 ,θ,µ ≤ kΠ(2) f kTs,r,N0 ,θ,µ

(3.42)

0

and, ∀ 0 < s < s, 0

kΠ>K f kTs0 ,r,N0 ,θ,µ ≤ e−K(s−s )

s kf kTs,r,N0 ,θ,µ . s0

(3.43)

Proof. We first note that by (2.15) (recall also Remark 3.1) we have Π(l) ΠN,θ,µ g = ΠN,θ,µ Π(l) g ,

∀ g ∈ Hs,r .

(3.44)

Then, applying Π(l) in (3.31), we deduce that, ∀N ≥ N0 , ∀ε > 0, there is f˜ ∈ Ts,r (N, θ, µ) such that Π(l) ΠN,θ,µ f = ΠN,θ,µ Π(l) f = Π(l) f˜ + N −1 Π(l) fˆ

(3.45)

kXΠ(l) f ks,r , kXΠ(l) f˜ks,r , kXΠ(l) fˆks,r ≤ kf kTs,r + ε .

(3.46)

and, by (2.79), (3.31),

We claim that Π(l) f˜ ∈ Ts,r (N, θ, µ), ∀l ≥ 0. Hence (3.45)-(3.46) imply Π(l) f ∈ QTs,r (N0 , θ, µ) and kΠ(l) f kTs,r ≤ kf kTs,r + ε , i.e. (3.39). Let us prove our claim. For l = 0, 1 the projection Π(l) f˜ = 0 because f˜ ∈ Ts,r (N, θ, µ) σ,σ 0 is bilinear. For l ≥ 2, write f˜ in the form (3.11) with coefficients f˜m,n satisfying (3.15). Then also (l) ˜ g := Π f has the form (3.11) with coefficients 0

0

σ,σ σ,σ gm,n = Π(l−2) f˜m,n

which satisfy (3.15) noting that Π(l) Ls,r (N, µ, h) ⊂ Ls,r (N, µ, h). Hence g ∈ Ts,r (N, θ, µ), ∀l ≥ 0, proving the claim. The proof of (3.40), (3.41), (3.42), and (3.43) are similar (use also (2.56)).

26

Lemma 3.5. Assume that, ∀N ≥ N∗ , we have the decomposition G = G0N + G00N

with

kG0N kTs,r,N,θ,µ ≤ K1 , N kXΠN,θ,µ G00N ks,r ≤ K2 .

(3.47)

Then kGkTs,r,N∗ ,θ,µ ≤ max{kXG ks,r , K1 + K2 }. Proof. By assumption, ∀N ≥ N∗ , we have kG0N kTs,r,N,θ,µ ≤ K1 . Then, ∀ε > 0, there exist ˜ 0N ∈ Ts,r (N, θ, µ), G ˆ 0N , such that G ˜ 0N + N −1 G ˆ 0N ΠN,θ,µ G0N = G

and kXG˜ 0 ks,r , kXGˆ 0 ks,r ≤ K1 + ε . N

(3.48)

N

Therefore, ∀N ≥ N∗ , ˜ N + N −1 G ˆN , ΠN,θ,µ G = G

˜ N := G ˜ 0N , G ˆ N := G ˆ 0N + N ΠN,θ,µ G00N G

˜ N ∈ Ts,r (N, θ, µ) and where G (3.48)

kXG˜ N ks,r = kXG˜ 0 ks,r ≤ K1 + ε,

(3.49)

N

(3.48),(3.47)

kXGˆ N ks,r ≤ kXGˆ 0 ks,r + N kXΠN,θ,µ G00N ks,r

≤

N

K1 + ε + K2 .

(3.50)

Then G ∈ QTs,r,N∗ ,θ,µ and kGkTs,r,N∗ ,θ,µ

≤

 sup max kXG ks,r , kXG˜ N ks,r , kXGˆ N ks,r

N ≥N∗ (3.49),(3.50)

≤

max{kXG ks,r , K1 + K2 + ε} .

Since ε > 0 is arbitrary the lemma follows. The Poisson bracket of two quasi-T¨ oplitz functions is quasi-T¨oplitz. Proposition 3.1. (Poisson bracket) Assume that f (1) , f (2) ∈ QTs,r (N0 , θ, µ) and N1 ≥ N0 , µ1 ≤ µ, θ1 ≥ θ, s/2 ≤ s1 < s, r/2 ≤ r1 < r satisfy b s−s1 2

κN1b−L < µ − µ1 , µ1 N1L−1 + κN1b−1 < θ1 − θ, 2N1 e−N1

< 1, b(s − s1 )N1b > 2 .

(3.51)

Then {f (1) , f (2) } ∈ QTs1 ,r1 (N1 , θ1 , µ1 ) and k{f (1) , f (2) }kTs1 ,r1 ,N1 ,θ1 ,µ1 ≤ C(n)δ −1 kf (1) kTs,r,N0 ,θ,µ kf (2) kTs,r,N0 ,θ,µ

(3.52)

where C(n) ≥ 1 and

n r1 o s1 . δ := min 1 − , 1 − s r The proof is based on the following splitting Lemma for the Poisson brackets.

(3.53)

Lemma 3.6. (Splitting lemma) Let f (1) , f (2) ∈ QTs,r (N0 , θ, µ) and (3.51) hold. Then, for all N ≥ N1 , ΠN,θ1 ,µ1 {f (1) , f (2) } = n oH n oL n oL (2) L (1) (2) ΠN,θ1 ,µ1 ΠN,θ,µ f (1) , ΠN,θ,µ f (2) + ΠN,θ,µ f (1) , ΠL f + Π f , Π f N,θ,µ N,2µ N,2µ n ox,y n ox,y (2) (1) + ΠN,θ,µ f (1) , ΠL + ΠL , ΠN,θ,µ f (2) N,µ f N,µ f n o n o + Π|k|≥N b f (1) , f (2) + Π|k| 2 .

(3.80)

Since, by (3.77), µi − µi+1 =

µ − µ0 , j

θi+1 − θi =

θ − θ0 , j

si − si+1 =

s − s0 j

and j < J = ln N (see (3.73)), 0 < b < L < 1 (recall (3.2)), µ0 ≤ µ ≤ 6, the above conditions (3.79)-(3.80) are implied by κN b−L ln N < µ − µ0 , 2N e−N

b

(s−s0 )/2 ln N

(6 + κ)N L−1 ln N < θ0 − θ ,

< 1,

b(s − s0 )N b > 2 ln N .

(3.81)

The last two conditions (3.81) are implied by b(s − s0 )N b > 2 ln2 N and since N ≥ e1/1−b (recall (3.64)). Recollecting we have to verify κN b−L ln N ≤ µ − µ0 , (6 + κ)N L−1 ln N ≤ θ0 − θ , 2N −b ln2 N ≤ b(s − s0 ) .

(3.82)

Since the function N 7→ N −γ ln N is decreasing for N ≥ e1/γ , we have that (3.82) follows by (3.64)T (3.65). Therefore Proposition 3.1 implies that adi+1 f (g) ∈ Qsi+1 ,ri+1 (N, θi+1 , µi+1 ) and, by (3.52), (3.35), we get i T 0 −1 T T kadi+1 (3.83) f (g)ksi+1 ,ri+1 ,N,θi+1 ,µi+1 ≤ C δi kf ks,r kadf (g)ksi ,ri ,N,θi ,µi where



ri+1 si+1 , 1− δi := min 1 − si ri 32

 ≥

δ j

(3.84)

and δ is defined in (2.65). Then T kadi+1 f (g)ksi+1 ,ri+1 ,N,θi+1 ,µi+1

(3.83),(3.84)

C 0 jδ −1 kf kTs,r,N0 ,θ,µ kadif (g)kTsi ,ri ,N,θi ,µi

≤ (3.78)

(C 0 jδ −1 kf kTs,r )i+1 kgkTs,r

≤ proving (3.78) by induction.

4

An abstract KAM theorem

We consider a family of integrable Hamiltonians N := N (x, y, z, z¯; ξ) := e(ξ) + ω(ξ) · y + Ω(ξ) · z z¯

(4.1)

a,p defined on Tns ×Cn ×`a,p I ×`I , where I is defined in (2.83), the tangential frequencies ω := (ω1 , . . . , ωn ) and the normal frequencies Ω := (Ωj )j∈Z\I depend on n-parameters

ξ ∈ O ⊂ Rn ,

O bounded with positive Lebesgue measure .

For each ξ there is an invariant n-torus T0 = Tn × {0} × {0} × {0} with frequency ω(ξ). In its normal space, the origin (z, z¯) = 0 is an elliptic fixed point with proper frequencies Ω(ξ). The aim is to prove the persistence of a large portion of this family of linearly stable tori under small analytic perturbations H = N + P . (A1) Parameter dependence. The map ω : O → Rn , ξ 7→ ω(ξ), is Lipschiz continuous. With in mind the application to NLW we assume (A2) Frequency asymptotics. We have p Ωj (ξ) = j 2 + m + a(ξ) ∈ R , j ∈ Z \ I ,

(4.2)

for some Lipschiz continuous functions a(ξ) ∈ R. By (A1) and (A2), the Lipschiz semi-norms of the frequency maps satisfy, for some 1 ≤ M1 < ∞, |ω|lip + |Ω|lip ∞ ≤ M1

(4.3)

where the Lipschiz semi-norm is |Ω|lip ∞ :=

sup ξ,η∈O,ξ6=η

|Ω(ξ) − Ω(η)|∞ . |ξ − η|

(4.4)

(A3) Regularity. The perturbation P : D(s, r) × O → C is λ-regular (see Definition 2.8). In order to obtain the asymptotic expansion (4.15) for the perturbed frequencies we also assume ¨ plitz. The perturbation P (preserves momentum and) is quasi-T¨oplitz, see (4.13). (A4) Quasi-To

33

Thanks to the conservation of momentum we restrict to the set of indices n I := (k, l) ∈ Zn × Z∞ , (k, l) 6= (0, 0) , |l| ≤ 2, where

(4.5)

or l = 0 , k · j = 0 , or l = σem , m ∈ Z \ I , k · j + σm = 0 , o or l = σem + σ 0 en , m, n ∈ Z \ I , k · j + σm + σ 0 n = 0 . For η > 0 we define the set of Diophantine vectors n Dη := ω ∈ Rn : |ω · k| ≥

o η , ∀ k ∈ Zn \ {0} . n 1 + |k|

(4.6)

Let P = P00 (x) + P¯ (x, y, z, z¯)

where

P¯ (x, 0, 0, 0) = 0 .

(4.7)

Theorem 4.1. (KAM theorem) I) Suppose that H = N + P satisfies (A1)-(A3). Let γ ∈ (0, 1) be a parameter and λ := γ/M1 . If o n (4.8)  := max γ −2/3 |XP00 |λs,r , γ −1 |XP¯ |λs,r is small enough, then there exist: • (Frequencies) Lipschiz functions ω ∞ : O → Rn , Ω∞ : O → `∞ such that |ω ∞ − ω| + λ|ω ∞ − ω|lip , |Ω∞ − Ω|∞ + λ|Ω∞ − Ω|lip ∞ ≤ Cγ , ∞ lip

(4.9)

|Ω∞ |lip ∞

and |ω | , ≤ 2M1 . • (KAM normal form) A Lipschiz family of analytic symplectic maps Φ : D(s/4, r/4) × O∞ 3 (x∞ , y∞ , w∞ ; ξ) 7→ (x, y, w) ∈ D(s, r)

(4.10)

close to the identity where n O∞ := ξ ∈ O ∩ ω −1 (Dγ 2/3 ) : |ω ∞ (ξ) · k + Ω∞ (ξ) · l| ≥

2γ , (k, l) ∈ I 1 + |k|τ o where I is defined in (4.5) and Dγ 2/3 in (4.6) with η = γ 2/3

(4.11)

such that, H ∞ (·; ξ) := H ◦ Φ(·; ξ) = ω ∞ (ξ) · y∞ + Ω∞ (ξ) · z∞ z¯∞ + P ∞

has

∞ P≤2 = 0.

(4.12)

Then, ∀ξ ∈ O∞ , the map x∞ 7→ Φ(x∞ , 0, 0; ξ) is a real analytic embedding of an elliptic, n-dimensional torus with frequency ω ∞ (ξ) for the system with Hamiltonian H. II) Assume (A4). If, for some 1 < θ, µ < 6, N > 0, n o ε := max γ −2/3 kXP00 ks,r , γ −1 kP¯ kTs,r,N,θ,µ

(4.13)

is small enough, then ∗ • (Asymptotic of frequencies) There exist a∞ ± : O∞ → R where ∗ O∞ :=

n

ξ ∈ O∞ : |ω ∞ (ξ) · k + p| ≥

o 2γ 2/3 , ∀ k ∈ Zn , p ∈ Z , (k, p) 6= (0, 0) τ 1 + |k|

(4.14)

with τ > 1/b (recall (3.2)) and 2/3 sup |Ω∞ ˆ∞ ε j (ξ) − Ωj (ξ) − a s(j) (ξ)| ≤ γ

∗ ξ∈O∞

34

C , |j|

∀|j| ≥ C? γ −1/3 .

(4.15)

Part I) of Theorem 4.1 follows by Theorem 5.1 of [2] because the KAM condition (4.8) implies hypothesis (H3) with d = 1 and µ = 2/3 of Theorem 5.1-[2]. Note that condition (4.8) is weaker than the KAM condition in [24], allowing a direct application to the nonlinear wave equation. There is only a minor p difference in the settings of Theorems 4.1 and Theorem 5.1-[2]. In assumption (A1) the ”tail” Ωj − j 2 + m = a(ξ) is independent of j, unlike in [2] it tends to zero as j → ∞. This difference does not affect the iterative part of the KAM theorem. The decay of the tail was used in [2] (as in [24]) only to prove the measure estimates. The main novelty of Theorem 4.1 is part II). In the next Theorem 4.2 we verify the second order Melnikov non-resonance conditions thanks to 1. the asymptotic decay (4.15) of the perturbed frequencies, 2. the restriction to indices (k, l) ∈ I in (4.11) which is a consequence of the momentum conservation, see (A4). ∗ As in [2], the Cantor set of ”good” parameters O∞ in (4.11) and O∞ in (4.14), are expressed in terms of the final frequencies only (and not inductively as in [24]). This simplifies the measure estimates.

Theorem 4.2. (Measure estimate) Suppose ω(ξ) = ω ¯ + Aξ , ω ¯ ∈ Rn , A ∈ Mat(n × n) ,

Ωj (ξ) =

p j 2 + m + ~a · ξ , a ∈ Rn

(4.16)

and assume the non-degeneracy condition: A invertible

2(A−1 )T ~a ∈ / Zn \ {0} .

and

(4.17)

∗ Then, the Cantor like set O∞ defined in (4.14), with exponent

τ > max{2n + 1, 1/b}

(4.18)

(b is fixed in (3.2)), satisfies ∗ |O \ O∞ | ≤ C(τ )ρn−1 γ 2/3

where

ρ := diam(O) .

(4.19)

Theorem 4.2 is proved in section 6. The asymptotic estimate (4.15) is used for proving the key inclusion (6.11).

5

Proof of the KAM Theorem 4.1

In this section we revisit the KAM scheme of [2] for proving part II of Theorem 4.1.

5.1

First step

We perform a preliminary change of variables in order to improve the smallness conditions. For all ξ ∈ ω −1 (Dγ 2/3 ) ∩ O =: O0

(5.1)

X P00,k eik·x iω(ξ) · k

(5.2)

(see (4.6)) we consider the solution F00 (x) :=

k6=0

of the homological equation − adN F00 + P00 (x) = hP00 i . Note that for any function F00 (x) we have kF00 kTs,r = kXF00 ks,r , see Definition 3.4. 35

(5.3)

We want to apply Proposition 3.2 with s, r, s0 , r0 verified because kF00 kT3s/4,r = kXF00 k3s/4,r

(5.2),(4.6)

≤

3s/4, 3r/4, s/2, r/2. The condition (3.63) is (4.13)

C(s)γ −2/3 kXP00 ks,r ≤ C(s)ε0

and ε0 is sufficiently small. Hence the time–one flow Φ00 := eadF00 : D(s0 , r0 ) × O0 → D(s, r)

with

s0 := s/2 , r0 := r/2 ,

(5.4)

is well defined, analytic, symplectic. Let µ0 < µ, θ0 > θ, N0 > N large enough, so that (3.65) is satisfied with s, r, N0 , θ, µ, s, r, N, θ, µ and s0 , r0 , N00 , θ0 , µ0 s0 , r0 , N0 , θ0 , µ0 . Hence (3.66) implies keadF00 P¯ kTs0 ,r0 ,N0 ,θ0 ,µ0 ≤ 2kP¯ kTs,r,N,θ,µ .

(5.5)

Noting that eadF00 P00 = P00 and eadF00 N = N + adF00 N the new Hamiltonian is H 0 := eadF00 H = eadF00 N + eadF00 P00 + eadF00 P¯

= (5.3)

=

N + adF00 N + P00 + eadF00 P¯

(5.6)

hP00 i + N + eadF00 P¯ =: N0 + P0 .

By (5.5) and (4.13) we have that kP0 kTs0 ,r0 ,N0 ,θ0 ,µ0 < 2γε .

5.2

(5.7)

KAM step

We now consider the generic KAM step for an Hamiltonian ≤2 ≤2 H = N + P = N + PK + (P − PK )

(5.8)

≤2 are defined as in (3.38). where PK

5.2.1

Homological equation

Lemma 5.1. Assume that |Ωj −

p

j 2 + m − as(j) | ≤

γ , |j|

∀ |j| ≥ j∗ ,

(5.9)

for some a+ , a− ∈ R. Let ˜ k,m,n := ω · k + |m| − |n| . ∆

∆k,m,n := ω · k + Ωm − Ωn , √ If |m|, |n| ≥ max{j∗ , m} and s(m) = s(n), then

   1 m |m − n| 1  m2 1 1 ˜ |∆k,m,n − ∆k,m,n | ≤ +γ + + + 3 . (5.10) 2 |n||m| |m| |n| 2 |m|3 |n| √ Proof. For 0 ≤ x ≤ 1 we have | 1 + x − 1 − x/2| ≤ x2 /2. Setting x := m/n2 (which is ≤ 1) and using (5.9), we get 2 Ωn − |n| − m − as(n) ≤ γ + m . |n| 2|n|3 2|n| ˜ k,m,n | = |Ωm − |m| − Ωn + |n|| the estimate An analogous estimates holds for Ωm . Since |∆k,m,n − ∆ (5.10) follows noting that as(m) = as(n) . For a monomial mk,i,α,β := eik·x y i z α z¯β we set ( mk,i,α,β [mk,i,α,β ] := 0

if

k = 0, α = β otherwise.

(5.11)

The following key proposition proves that the solution of the homological equation is quasi-T¨oplitz. 36

Proposition 5.1. (Homological equation) I) Let K ∈ N. For all ξ ∈ O such that |ω(ξ) · k + Ω(ξ) · l| ≥

γ , ∀(k, l) ∈ I (see (4.5)), |k| ≤ K , hkiτ

(5.12)

(h)

null then ∀PK ∈ Hs,r , h = 0, 1, 2 (see (3.36), (3.37)), the homological equations (h)

(h)

(h)

− adN FK + PK = [PK ] ,

h = 0, 1, 2 ,

(h)

(5.13)

(h)

null have a unique solution of the same form FK ∈ Hs,r with [FK ] = 0 and

kXF (h) ks,r < γ −1 K τ kXP (h) ks,r . K

(0)

(1)

(5.14)

K

(2)

≤2 In particular FK := FK + FK + FK solves ≤2 ≤2 ≤2 − adN FK + PK = [PK ].

(5.15)

(h)

II) Assume now that PK ∈ QTs,r (N0 , θ, µ) and Ω(ξ) satisfies (5.9) for all |j| ≥ θN0∗ where n o N0∗ := max N0 , cˆγ −1/3 K τ +1

(5.16)

for a constant cˆ := cˆ(m, κ) ≥ 1. Then, ∀ξ ∈ O such that |ω(ξ) · k + p| ≥

γ 2/3 , ∀|k| ≤ K, p ∈ Z , hkiτ

(5.17)

(h)

we have FK ∈ QTs,r (N0∗ , θ, µ), h = 0, 1, 2, and (h)

(h)

kFK kTs,r,N0∗ ,θ,µ ≤ 4ˆ cγ −1 K 2τ kPK kTs,r,N0 ,θ,µ .

(5.18)

Proof. The solution of the homological equation (5.13) is (h)

FK := −i

X

|k|≤K,(k,i,α,β)6=(0,i,α,α) 2i+|α|+|β|=h

Pk,i,α,β ik·x i α β e y z z¯ , ∆k,i,α,β

∆k,i,α,β := ω(ξ) · k + Ω(ξ) · (α − β) .

The divisors ∆k,i,α,β 6= 0, ∀(k, i, α, β) 6= (0, i, α, α), because (k, i, α, β) 6= (0, i, α, α) is equivalent to (k, α − β) ∈ I, and the bounds (5.12) hold. Item I) follows by Lemma 2.18. ≤1 In item II) we notice that the cases h = 0, 1 are trivial since ΠN,θ,µ FK = 0. (2) When h = 2 we first consider the subtlest case when PK contains only the monomials with i = 0, |α| = |β| = 1 (see (3.36)), namely X (2) P := PK = Pk,m,n eik·x zm z¯n , (5.19) |k|≤K,m,n∈Z\I

and, because of the conservation of momentum, the indices k, m, n in (5.19) are restricted to j · k + m − n = 0. (2)

(5.20)

(2)

The unique solution FK of (5.13) with [FK ] = 0 is (2)

F := FK := −i

X

Pk,m,n ik·x e zm z¯n , ∆k,m,n := ω(ξ) · k + Ωm (ξ) − Ωn (ξ) ∆k,m,n

|k|≤K,(k,m,n)6=(0,m,m)

37

(5.21)

Note that by (5.12) and (5.20) we have ∆k,m,n 6= 0 if and only if (k, m, n) 6= (0, m, m). Let us prove (5.18). For all N ≥ N0∗ X

ΠN,θ,µ F = −i

|k|≤K,|m|,|n|>θN

Pk,m,n ik·x e zm z¯n , ∆k,m,n

(5.22)

and note that eik·x is (N, µ)-low momentum since |k| ≤ K < (N0∗ )b ≤ N b by (5.16) and τ > 1/b. By assumption P ∈ QTs,r,N0 ,θ,µ and so, recalling formula (3.45), we may write, ∀N ≥ N0∗ ≥ N0 , ΠN,θ,µ P = P˜ + N −1 Pˆ

X

with P˜ :=

P˜k,m−n eik·x zm z¯n ∈ Ts,r (N, θ, µ)

(5.23)

|k|≤K,|m|,|n|>θN

and kXP ks,r , kXP˜ ks,r , kXPˆ ks,r ≤ 2kPkTs,r .

(5.24)

We now prove that X

F˜ :=

|k|≤K,|m|,|n|>θN

P˜k,m−n ik·x e zm z¯n , ˜ k,m,n ∆

˜ k,m,n := ω(ξ) · k + |m| − |n| , ∆

is a T¨ oplitz approximation of F. Since |m|, |n| > θN ≥ θN0∗ > N0∗ deduce by (5.20) that m, n have the same sign. Then

(5.16)

˜ k,m,n = ω(ξ) · k + |m| − |n| = ω(ξ) · k + s(m)(m − n) , ∆

>

(5.25)

κ K ≥ |j · k| by (3.1), we

s(m) := sign(m) ,

and F˜ in (5.25) is (N, θ, µ)-T¨ oplitz (see (3.15)). Moreover, since |m| − |n| ∈ Z, by (5.17), we get ˜ k,m,n | ≥ γ 2/3 hki−τ , |∆

∀|k| ≤ K, m, n,

(5.26)

and Lemma 2.18 and (5.25) imply kXF˜ ks,r ≤ γ −2/3 K τ kXP˜ ks,r .

(5.27)

The T¨ oplitz defect is N −1 Fˆ

:= (5.22),(5.25)

=

ΠN,θ,µ F − F˜ X

(5.28) P

|k|≤K,|m|,|n|>θN

=

X

h P

=

X

k,m,n

∆k,m,n

|k|≤K,|m|,|n|>θN (5.23)

k,m,n

∆k,m,n

P˜k,m−n  ik·x − e zm z¯n ˜ k,m,n ∆

h

Pk,m,n

−

∆ ˜

Pk,m,n   Pk,m,n − P˜k,m−n i ik·x + e zm z¯n ˜ k,m,n ˜ k,m,n ∆ ∆

− ∆k,m,n  Pˆk,m,n i ik·x + N −1 e zm z¯n . ˜ k,m,n ˜ k,m,n ∆k,m,n ∆ ∆

|k|≤K,|m|,|n|>θN

k,m,n

By (5.10), |m|, |n| ≥ θN ≥ N , and |m − n| ≤ κK (see (5.20)) we get, taking cˆ large enough,   (5.16)  1/3  mκK m2 K cˆγ γ 2/3 2γ cˆ ˜ |∆k,m,n − ∆k,m,n | ≤ + + 3 ≤ +γ ≤ min , . (5.29) 2N 2 N N 4N N 2N 2K τ Hence ˜ k,m,n | − |∆ ˜ k,m,n − ∆k,m,n | |∆k,m,n | ≥ |∆ 38

(5.26),(5.29)

≥

γ 2/3 γ 2/3 γ 2/3 − ≥ . hkiτ 2K τ 2hkiτ

(5.30)

Therefore (5.29), (5.26), (5.30) imply ˜ k,m,n − ∆k,m,n | cˆγ 1/3 2hkiτ hkiτ cˆ 2τ |∆ ≤ ≤ K ˜ k,m,n | 2N γ 2/3 γ 2/3 Nγ |∆k,m,n ||∆ and (5.28), (5.26), and Lemma 2.18, imply (5.24)

kXFˆ ks,r ≤ cˆγ −1 K 2τ kXP ks,r + γ −2/3 K τ kXPˆ ks,r ≤ 4ˆ cγ −1 K 2τ kPkTs,r .

(5.31)

In conclusion (5.14), (5.27), (5.31) prove (5.18) for F. (2) Let us briefly discuss the case when h = 2 and PK contains only the monomials with i = 0, |α| = 2, |β| = 0 or viceversa (see (3.36)). Denoting X (2) P := PK := Pk,m,n eik·x zm zn , (5.32) |k|≤K,m,n∈Z\I

we have X

ΠN,θ,µ F = −i

|k|≤K,|m|,|n|>θN

Pk,m,n eik·x zm zn ω · k + Ωm + Ωn

where |ω · k + Ωm + Ωn | > (|m| + |n|)/2 > θN/2 since |m|, |n| > θN and |k| ≤ K < N b . In this case we may take as T¨ oplitz approximation F˜ = 0. 5.2.2

The new Hamiltonian H +

≤2 Let F = FK be the solution of the homological equation (5.15). If, for s/2 ≤ s+ < s, r/2 ≤ r+ < r, the condition n s+ r+ o kF kTs,r,N0∗ ,θ,µ ≤ c(n) δ+ , δ+ := min 1 − ,1 − (5.33) s r holds (see (3.63)), then Proposition 3.2 (with s0 s+ , r0 r+ , N0 N0∗ defined in (5.16)) implies adF that the Hamiltonian flow e : D(s+ , r+ ) → D(s, r) is well defined, analytic and symplectic. We transform the Hamiltonian H in (5.8), obtaining

H + := eadF H

(2.82)

=

H + adF (H) +

X 1 j ad (H) j! F j≥2

(5.8)

=

≤2 ≤2 N + PK + (P − PK ) + adF N + adF P +

X 1 j ad (H) j! F j≥2

(5.15)

=

N+

≤2 [PK ]

+P −

≤2 PK

X 1 j + adF P + ad (H) := N + + P + j! F j≥2

with new normal form ˆ, N + := N + N ω ˆ i := ∂yi | y=0,z=0 hP i , i = 1, . . . n ,

ˆ := [P ≤2 ] = eˆ + ω ˆ · z¯ N ˆ · y + Ωz K ˆ := (Ω ˆ j )j∈Z\I , Ω ˆ j := [P ]j := ∂ 2 Ω zj z¯j | y=0,z=0 hP i

(5.34)

(the h i denotes the average with respect to the angles x) and new perturbation ≤2 P + := P − PK + adF P ≤2 + adF P ≥3 +

X 1 j ad (H) j! F j≥2

having decomposed P = P ≤2 + P ≥3 with P ≥3 :=

X h≥3

39

P (h) , see (3.36).

(5.35)

5.2.3

The new normal form N +

The next lemma holds uniformly in the parameters ξ. Lemma 5.2. Let P ∈ QTs,r (N0 , θ, µ) with 1 < θ, µ < 6, N0 ≥ 9. Then ˆ ∞ ≤ 2kP (2) kT |ˆ ω |, |Ω| s,r,N0 ,θ,µ and there exist a ˆ± ∈ R satisfying

(5.36)

|ˆ a± | ≤ 2kP (2) kTs,r,N0 ,θ,µ

such that ˆj − a |Ω ˆs(j) | ≤

40 (2) T kP ks,r,N0 ,θ,µ , |j|

∀ |j| ≥ 6(N0 + 1) .

(5.37)

Lemma 5.2 is based on the following elementary Lemma, whose proof is postponed. Lemma 5.3. Suppose that, ∀N ≥ N0 ≥ 9, j ≥ θN , Ωj = aN + bN,j N −1 with aN , bN,j ∈ R , |aN | ≤ c1 , |bN,j | ≤ c1 ,

(5.38)

for some c1 > 0 (indipendent of j). Then there exists a ∈ R, satisfying |a| ≤ c1 , such that |Ωj − a| ≤

20c1 , |j|

∀ |j| ≥ 6(N0 + 1) .

(5.39)

ˆ we set (recall (3.36), (3.42)) proof of Lemma 5.2. The estimate on ω ˆ is trivial. Regarding Ω X (2) P0 := Πk=0 Π|α|=|β|=1 Π(2) P = [P ]j zj z¯j j (2)

since, by the momentum conservation (2.85), all the monomials in P0 [P ]j is defined in (5.34). By Lemma 2.19 (3.30)

(2)

have α = β = ej . Note that

(3.42)

|[P ]j | ≤ kXP (2) kr ≤ kP0 kTr ≤ kP (2) kTs,r .

(5.40)

0

(2)

We now prove (5.37) for j > 0 (the case j < 0 is similar). Since P0 (2) (2) (2) we may write ΠN,θ,µ P0 = P˜0,N + N −1 Pˆ0,N with (2) P˜0,N :=

X

P˜j zj z¯j ∈ Tr (N, θ, µ) ,

(2) Pˆ0,N :=

∈ QTr (N, θ, µ), for all N ≥ N0 ,

X

Pˆj zj z¯j

j>θN

j>θN

and (2)

kXP (2) kr , kXP˜ (2) kr , kXPˆ (2) kr ≤ 2kP0 kTr ≤ 2kP (2) kTs,r . 0

0,N

(5.41)

0,N

For |j| > θN , since all the quadratic forms in (5.41) are diagonal, we have ˆ j = [P ]j = P˜j + N −1 Pˆj := aN,+ + N −1 bN,j Ω (2) where aN,+ := P˜j is independent of j > 0 because P˜0,N ∈ Tr (N, θ, µ) (see (3.15)). Applying Lemma (2) (2) 2.19 to P˜ and Pˆ , we obtain 0,N

0,N

(5.41)

|aN,+ | ≤ kXP˜ (2) ks,r ≤ 2kP (2) kTs,r , 0,N

(5.41)

|bN,j | = |Pˆj | ≤ kXPˆ (2) kr ≤ 2kP (2) kTr . 0,N

Hence the assumptions of Lemma 5.3 are satisfied with c1 = 2kP (2) kTs,r and (5.37) follows. 40

Proof of Lemma 5.3. For all N1 > N ≥ N0 , j ≥ θN1 we get, by (5.38), |aN − aN1 | = |bN1 ,j N1−1 − bN,j N −1 | ≤ 2c1 N −1 . Therefore aN is a Cauchy sequence. Let a :=

(5.42)

lim aN be its limit. Since |aN | ≤ c1 we have |a| ≤ c1 .

N →+∞

Moreover, letting N1 → +∞ in (5.42), we derive |a − aN | ≤ 2c1 N −1 , ∀N ≥ N0 , and, using also (5.38), |Ωj − a| ≤ |Ωj − aN | + |aN − a| ≤ 3c1 N −1 ,

∀ N ≥ N0 , j ≥ 6N .

(5.43)

For all j ≥ 6(N0 + 1) let N := [j/6] (where [·] denotes the integer part). Since N ≥ N0 , j ≥ 6N , (5.43) 3c 3c1 18c1  1  20c1 1 1+ ≤ ≤ ≤ |Ωj − a| ≤ [j/6] (j/6) − 1 j N0 j for all j ≥ 6(N0 + 1). 5.2.4

The new perturbation P +

We introduce, for h = 0, 1, 2, ε(h) := γ −1 kP (h) kTs,r,N0 ,θ,µ ,

ε¯ :=

2 X

Θ := γ −1 kP kTs,r,N0 ,θ,µ

ε(h) ,

(5.44)

h=0

and the corresponding quantities for P + with indices r+ , s+ , N0+ , θ+ , µ+ . Proposition 5.2. (KAM step) Suppose (s, r, N0 , θ, µ), (s+ , r+ , N0+ , θ+ , µ+ ) satisfy s/2 ≤ s+ < s, r/2 ≤ r+ < r, ¯ } (recall (5.16), (3.64)) , N0+ > max{N0∗ , N κ(N0+ )b−L ln N0+ ≤ µ − µ+ ,

2(N0+ )−b ln2 N0+ ≤ b(s − s+ ) ,

(5.45)

(6 + κ)(N0+ )L−1 ln N0+ ≤ θ+ − θ .

(5.46)

Assume that −1 ε¯K τ¯ δ+ ≤ c small enough ,

Θ ≤ 1,

(5.47) θN0∗ .

where τ¯ := 2τ + n + 1 and δ+ is defined in (5.33). Suppose also that (5.9) holds for |j| ≥ ≤2 Then, for all ξ ∈ O satisfying (5.12),(5.17), denoting by F := FK the solution of the homological adF equation (5.15), the Hamiltonian flow e : D(s+ , r+ ) → D(s, r), and the transformed Hamiltonian H + := eadF H = N+ + P+ satisfies (0)

ε+

(1)

ε+

(2)

ε+

−2 2¯ −(s−s+ )K K τ ε¯2 + ε(0) s s−1 l δ+ + e
−2 2¯ −(s−s+ )K l δ+ K τ ε(0) + ε¯2 + ε(1) s s−1 + e
−2 2¯ −(s−s+ )K l δ+ K τ ε(0) + ε(1) + ε¯2 + ε(2) s s−1 + e

Θ+ ≤ Θ(1 +

−2 2¯ Cδ+ K τ ε¯) .

(5.48) (5.49)

The proof of this proposition is split in several lemmas where we analyze each term of P + in (5.35). We note first that (3.41)

≤2 T kPK ks,r,N0 ,θ,µ ≤ kP ≤2 kTs,r,N0 ,θ,µ (0)

(1)

(3.38),(5.44)

≤

γ ε¯ .

(5.50)

(2)

Moreover, the solution F = F + F + F of the homological equation (5.15) (for brevity F (h) ≡ (h) ≤2 FK and F ≡ FK ) satisfies, by (5.18) (with N0∗ defined in (5.16)), (3.41), (5.44), kF (h) kTs,r,N0∗ ,θ,µ l K τ¯ ε(h) , h = 0, 1, 2,

kF kTs,r,N0∗ ,θ,µ l K τ¯ ε¯ .

(5.51)

Hence (5.47) and (5.51) imply condition (5.33) and therefore eadF : D(s+ , r+ ) → D(s, r) is well defined. We now estimate the terms of the new perturbation P + in (5.35). 41

Lemma 5.4.

T



adF (P ≤2 )

s+ ,r+ ,N0+ ,θ+ ,µ+

Proof. We have X 1 j ad (H) j! F

=

j≥2

X 1

T

−2 + l δ+ γK 2¯τ ε¯2 . adjF (H) j! s+ ,r+ ,N0+ ,θ+ ,µ+ j≥2

X 1 j X 1 j−1 X 1 j adF (N + P ) = adF (adF N ) + ad (P ) j! j! j! F j≥2

(5.15)

=

j≥2

j≥2

X 1 j−1 ≤2 X 1 j ≤2 adF ([PK ] − PK )+ ad (P ) . j! j! F j≥2

j≥2

By (5.45), (5.46) and (5.33) we can apply Proposition 3.2 with N0 , N00 , s0 , r0 , θ0 , µ0 , δ N0∗ , N0+ , s+ , ∗ r+ , θ+ , µ+ , δ+ . We get (recall N0 ≥ N0 )

T

X 1 2 (3.67),(3.35) 

−1 T l δ kF k kP kTs,r,N0 ,θ,µ adjF (P ) ∗

s,r,N0 ,θ,µ + j! s+ ,r+ ,N0+ ,θ+ ,µ+ j≥2

(5.51),(5.44)

−2 2¯ δ+ K τ ε¯2 γ Θ

l and, similarly,

T

X 1

≤2 ) (P adj−1

K j! F s+ ,r+ ,N0+ ,θ+ ,µ+

X

=

j≥1

j≥2

(3.67)

l (5.51),(5.50)

l

(5.52)

T 1 ≤2 ) adjF (PK (j + 1)! s+ ,r+ ,N0+ ,θ+ ,µ+

≤2 T −1 δ+ kF kTs,r,N0∗ ,θ,µ kPK ks,r,N0 ,θ,µ −1 τ¯ δ+ K γ ε¯2 .

(5.53)

Finally, by Proposition 3.1, applied with N0∗ , N0+ , s+ , r+ , θ+ , µ+ , δ+ ,

N0 , N1 , s1 , r1 , θ1 , µ1 , δ

(5.54)

we get

T

adF (P ≤2 )

(3.52)

s+ ,r+ ,N0+ ,θ+ ,µ+

l (5.51),(5.50)

l

−1 δ+ kF kTs,r,N0∗ ,θ,µ kP ≤2 kTs,r,N0 ,θ,µ −1 τ¯ δ+ K γ ε¯2 .

(5.55)

The bounds (5.52), (5.53), (5.55), and Θ ≤ 1 (see (5.47)), prove the lemma. Lemma 5.5. (5.49) holds. Proof. By Proposition 3.1 (applied with (5.54)) we have

T

−1 l δ+ kF kTs,r,N0∗ ,θ,µ kP ≥3 kTs,r,N0 ,θ,µ

adF (P ≥3 ) + s+ ,r+ ,N0 ,θ+ ,µ+

(5.51),(3.40),(5.44)

l

−1 τ¯ δ+ K γ ε¯ Θ ,

(5.56)

and (5.49) follows by (5.35), (3.40), (3.35), (5.44) (5.56), Lemma 5.4 and ε¯ ≤ 3Θ (which follows by (5.44) and (3.39)). (h)

(0)

We now consider P+ , h = 0, 1, 2. The term adF P ≥3 in (5.35) does not contribute to P+ . On (1) the contrary, its contribution to P+ is {F (0) , P (3) } (5.57) (2)

and to P+ is {F (1) , P (3) } + {F (0) , P (4) } . 42

(5.58)

Lemma 5.6. k{F (0) , P (3) }kTs

+ + ,r+ ,N0 ,θ+ ,µ+

−1 l δ+ γK τ¯ ε(0) Θ and

T

(1) (3)

{F , P } + {F (0) , P (4) }

s+ ,r+ ,N0+ ,θ+ ,µ+

−1 τ¯ l δ+ K γ(ε(0) + ε(1) )Θ .

Proof. By (3.52) (applied with (5.54)), (5.51), (5.44) and (3.39). (h)

(h)

≤2 The contribution of P − PK in (5.35) to P+ , h = 0, 1, 2, is P>K . (h)

−K(s−s+ ) ≤ s s−1 γε(h) + e

Lemma 5.7. kP>K kTs

+ + ,r+ ,N0 ,θ+ ,µ+

Proof. By (3.43) and (5.44). Finally, (5.48) follows by (5.35), Lemmata 5.4, 5.6 (and (5.57)-(5.58)), Lemma 5.7 and Θ ≤ 1.

5.3

KAM iteration (0)

(1)

(2)

∈ (0, 1), i = 0, . . . , ν, satisfy

Lemma 5.8. Suppose that εi , εi , εi (0)

εi+1 (1) εi+1 (2) εi+1 (0)

i

(0)

≤ C∗ Ki ε¯2i + C∗ εi e−K∗ 2
i (0) (1) ≤ C∗ Ki εi + ε¯2i + C∗ εi e−K∗ 2
i (0) (1) (2) ≤ C∗ Ki εi + εi + ε¯2i + C∗ εi e−K∗ 2 ,

(1)

(5.59)

i = 0, . . . , ν − 1 ,

(2)

where ε¯i := εi +εi +εi , for some K, C∗ , K∗ > 1. Then there exist ε¯? , C? > 0, χ ∈ (1, 2) (depending on K, C∗ , K∗ > 0), such that, if ε¯0 ≤ ε¯?

=⇒

i

ε¯i ≤ C? ε¯0 e−K∗ χ , ∀i = 0, . . . , ν .

(5.60)

j

Proof. We first note that ε¯j+1 l Kj ε¯j + ε¯j e−K∗ 2 . Then, applying (5.59) three times, we deduce (0)

εj+3 (1)

εj+3 (2)

εj+3

(0)

j

(1)

j

(2)

j

l

K4j+3 ε¯2j + εj e−K∗ 2

l

K4j+3 ε¯2j + εj e−K∗ 2

l

K4j+3 ε¯2j + εj e−K∗ 2

and, therefore aj := ε¯3j aj+1 ≤ C1 K4j+3 a2j + aj C1 e−K∗ 2

j

(5.61)

for some C1 := C1 (C∗ ) > 1. Claim: There is ε0 > 0, C2 > 1, such that, if a0 ≤ ε0 , then, for all j ∈ N, j

aj ≤ C2 a0 (2C1 )j e−K∗ χ ,

(S)j

χ := 3/2 .

We proceed by induction. The statement (S)0 follows by the assumption a0 ≤ ε0 , for C2 e−K∗ > 1. Now suppose (S)j holds true. Then (S)j+1 follows by (5.61) and aj+1

j

j

≤

C1 K4j+3 C22 a20 (2C1 )2j e−2K∗ χ + C2 a0 (2C1 )j C1 e−K∗ (2


0 large enough, ε0 ∈ (0, 1/2) such that, if Θ0 := γ −1 kP0 kTs0 ,r0 ,N0 ,θ0 ,µ0 ≤ ε0 ,

(5.63)

then (S1)ν ∀0 ≤ i ≤ ν, there exist H i := Ni + Pi : Di × Oi∗ → C with Ni := ei + ω (i) (ξ) · y + Ω(i) (ξ) · z z¯ in (i) (i) normal form, Ω(i) = (Ωj )j∈Z\I fulfills (5.9) for some a± , for all |j| ≥ θi Ni . Above O0∗ := O0 , and, for i > 0, n γ ∗ , ∀(k, l) ∈ I , |k| ≤ Ki−1 , Oi∗ := ξ ∈ Oi−1 : |ω (i−1) (ξ) · k + Ω(i−1) (ξ) · l| ≥ 1 + |k|τ o γ 2/3 |ω (i−1) (ξ) · k + p| ≥ , ∀(k, p) = 6 (0, 0) , |k| ≤ K , p ∈ Z . (5.64) i−1 1 + |k|τ Moreover, ∀ 1 ≤ i ≤ ν, H i = H i−1 ◦ Φi where Φi : Di × Oi∗ → Di−1 is a (Lipschitz) family (in ξ ∈ Oi∗ ) of close-to-the-identity analytic symplectic maps. Define ε¯i :=

2 X

(h)

εi

(h)

, εi

(h)

:= γ −1 kPi kTsi ,ri ,Ni ,θi ,µi , Θi := γ −1 kPi kTsi ,ri ,Ni ,θi ,µi .

(5.65)

h=0 (0)

(1)

(2)

(S2)ν ∀0 ≤ i ≤ ν − 1, the εi , εi , εi s0 K0 /4.

∈ (0, 1) satisfy (5.59) with K = 42¯τ +1 , C∗ = 4K02¯τ , K∗ = i

(S3)ν ∀0 ≤ i ≤ ν, we have ε¯i ≤ C? ε¯0 e−K∗ χ and Θi ≤ 2Θ0 . (S4)ν ∀0 ≤ i ≤ ν and ∀ ξ ∈ Oi∗ , denote (recall (5.34)) ω ˆ (i) := ∇y hPi (ξ)i|y=0,z=¯z=0

and

ˆ (i) (ξ) := ∂ 2 Ω zj z¯j | y=0,z=0 hPi (ξ)i . j

(i)

There exist constants a ˆ± (ξ) ∈ R such that (i) ˆ (i) (ξ)|∞ , |ˆ |ˆ ω (i) (ξ)| , |Ω a± (ξ)| ≤ 2γ ε¯i ,

(i)

(i)

ˆ (ξ) − a |Ω ˆs(j) (ξ)| ≤ 40γ j

ε¯i , |j|

∀|j| ≥ 6(Ni + 1) ,

(5.66)

uniformly in ξ ∈ Oi∗ . Proof. The statement (S1)0 follows by the hypothesis. (S2)0 is empty. (S3)0 is trivial. (S4)0 follows by Lemma 5.2 and (5.44). We then proceed by induction. 44

(S1)ν+1 . We wish to apply the KAM step Proposition 5.2 with N = Nν , P = Pν , N0 = Nν , θ = θν . . . and N0+ = Nν+1 , θ+ = θν+1 , . . . Our definitions in (5.62) (and τ > 1/b) imply that the conditions2 (5.45)-(5.46) are satisfied, for all ν ∈ N, taking K0 > large enough. Moreover, since n sν+1 rν+1 o so that 2−ν−2 ≤ δν+1 ≤ 2−ν−1 , (5.67) δ + = δν+1 := min 1 − ,1 − sν rν and (S3)ν the condition (5.47) is satisfied, for ε¯0 ≤ ε0 small enough, ∀ν ∈ N. Finally, by (S1)ν , ∗ condition (5.9) holds for |j| ≥ θν Nν , and (5.12) and (5.17) hold (by definition) for all ξ ∈ Oν+1 . Hence ∗ adFν ∗ : Dν+1 × Oν+1 → Dν and we Proposition 5.2 applies. For all ξ ∈ Oν+1 the Hamiltonian flow e define ∗ → C, H ν+1 := eadFν H ν = Nν+1 + Pν+1 : Dν+1 × Oν+1 where (recall (5.34)) ˆ (ν) , Ω(ν+1) = Ω(ν) + Ω X (i) (ν) (ν) a ˆ± . By (S3)ν - (S4)ν we = a± + a ˆ± = a(0) +

ω (ν+1) = ω (ν) + ω ˆ (ν) , (ν+1)

ˆ (ν) are defined by (S4)ν . Let a and ω ˆ (ν) , Ω ±

i≤ν

have that, by (4.2), (5.9) holds for Ω(ν+1) and for all |j| > θν+1 Nν+1 > 6(Nν + 1) for ε¯0 ≤ ε0 small enough. (S2)ν+1 follows by (5.48) and (5.62). ν+1

(S3)ν+1 . By (S2)ν we can apply Lemma 5.8 and (5.60) implies ε¯ν+1 ≤ C? ε¯0 e−K∗ χ ε0 small enough, (5.49)

≤

Θν+1

. Moreover, for

  (5.67),(S3)ν −2 Θ0 Πνi=0 1 + Cδi+1 Ki2¯τ ε¯i ≤ 2Θ0 .

(S4)ν+1 follows by Lemma 5.2 and (S3)ν . The fundamental estimate (4.15) follows by (5.66) and the following corollary. Corollary 5.1. For all ξ ∈ ∩ν Oν∗ the X (ν) X ˆ (ν) , a ˆ ∞ := Ω ˆ∞ a ˆ± Ω ± :=

ˆ∞ |Ω ˆ∞ j −a s(j) | l

ˆ ∞ |∞ , |ˆ |Ω a∞ ¯0 ± | l γε

(5.68)

ν≥0

ν≥0

and

satisfy

K0τ +1 ε¯0 γ 2/3 , ∀|j| ≥ 6(N0 + 1) = 6(γ −1/3 cˆK0τ +1 + 1) . |j|

(5.69)

Proof. The bounds in (5.68) follow from (5.66) and (S3)ν . Let us prove (5.69) when j > 0 (the case j < 0 is analogous). For all ∀ν ≥ 0, j ≥ 6(Nν + 1), we have ˆ∞ |Ω ˆ∞ j −a +|

≤

ν X

(n)

ˆ |Ω j

(n)

−a ˆ+ | +

n=0 (5.66)

≤

X

(n)

ˆ |Ω j

(n)

−a ˆ+ |

n>ν

ν X (n) X 40γ X ¯0 γ (n) (5.66),(S3ν ) ε ˆ | + |ˆ ε¯n + |Ω a+ | l +γ ε¯n . j j n=0 j n>ν n>ν

Therefore, ∀ν ≥ 0, 6(Nν + 1) ≤ j < 6(Nν+1 + 1), ˆ∞ |Ω ˆ∞ j −a +|l

X (5.62) ε ε¯0 γ Nν+1 X ¯0 γ γ +γ ε¯n l + γ −1/3 K0τ +1 2ρ(ν+1) ε¯n j j n>ν j j n>ν

and (5.69) follows by (S3)ν . 2

¯ }. For example the first inequality in (5.45) reads Nν+1 ≥ max{Nν , cˆγ −1/3 Kντ +1 , N

45

Proof of Theorem 4.1-II) We apply the iterative Lemma 5.9 to H 0 defined in (5.6). The symplectic transformation Φ in (4.10) is defined by Φ := lim Φ00 ◦ Φ0 ◦ Φ1 ◦ · · · ◦ Φν ν→∞

with Φ00 defined in (5.4). The final frequencies ω ∞ and Ω∞ are X X ˆ∞ = Ω + ˆ (ν) . ω ∞ = lim ω (ν) = ω + ω ˆ (ν) , Ω∞ = lim Ω(ν) = Ω + Ω Ω ν→∞

ν→∞

ν≥0

(5.70)

ν≥0

The KAM iteration procedure that we are using is the same as that of the abstract KAM theorem of [2]. To be more precise in [2] one solves the homological equation (5.13) for all ξ in a larger set where only the Melnikov conditions (5.12) hold (see Proposition 5.1-I), but not (5.17). Clearly, the solutions of the homological equation, the new perturbation P + and the new frequencies ω + , Ω+ in (5.34), coincide with those in [2] on this smaller set of parameters. The procedure is completed by extending ω + , Ω+ to Lipschiz functions in the whole parameter set. By the Kirszbraun theorem (see e.g. [21]) the extended frequencies satisfy the bounds (5.36) for every ξ ∈ O0 . In the set of ξ where (5.12) and (5.17) hold, the extended frequencies satisfy also (5.36)-(5.39). ∗ Lemma 5.10. O∞ ⊂ ∩i Oi∗ (see (4.14) and (5.64)). ∗ Proof. If ξ ∈ O∞ then, for all |k| ≤ Ki , |l| ≤ 2,

|ω (i) (ξ) · k + Ω(i) (ξ) · l|

≥ ≥

|ω ∞ (ξ) · k + Ω∞ (ξ) · l| − |ω ∞ − ω (i) ||k| − 2|Ω∞ − Ω(i) |∞ X X γ 2γ (ν) ˆ (ν) |∞ ≥ − K |ˆ ω | − 2 |Ω i 1 + |k|τ 1 + |k|τ ν>i ν>i

by the definition of Ki in (5.62), (S3)ν and (5.66). The other estimate is analogous. ∗ As a consequence, for ξ ∈ O∞ , Corollary 5.1 holds. Then (4.15) follows by (5.70), (4.2) and (5.69). This concludes the proof of Theorem 4.1.

6

Measure estimates: proof of Theorem 4.2

We have to estimate the measure of [ ∗ O \ O∞ = − (k,l)∈Λ0 ∪Λ1 ∪Λ+ 2 ∪Λ2

[

Rkl (γ)

˜ kp (γ 2/3 ) R

ω −1 (Dγ 2/3 )

n Rkl (γ) := Rτkl (γ) := ξ ∈ O : |ω ∞ (ξ) · k + Ω∞ (ξ) · l| < n ˜ kp (γ 2/3 ) := R ˜ τkp (γ 2/3 ) := ξ ∈ O : |ω ∞ (ξ) · k + p| < R

2γ o 1 + |k|τ 2γ 2/3 o

We first consider the most difficult case Λ− 2 . Setting Rk,i,j (γ) := Rk,ei −ej (γ) we show that [ [ Rk,l (γ) = Rk,i,j (γ) l γ 2/3 ρn−1 (k,i,j)∈ I

46

(6.2)

1 + |k|τ

n o − Λh := (k, l) ∈ I (see (4.5)) , |l| = h , h = 0, 1, 2 , Λ2 = Λ+ 2 ∪ Λ2 , n o n o − Λ+ := (k, l) ∈ Λ , l = ±(e + e ) , Λ := (k, l) ∈ Λ , l = e − e 2 i j 2 i j . 2 2

(k,l)∈Λ− 2

(6.1)

(k,p)∈Zn+1 \{0}

where

and

\

(6.3)

(6.4)

where

n o I := (k, i, j) ∈ Zn × (Z \ I)2 : (k, i, j) 6= (0, i, i) , j · k + i − j = 0 .

(6.5)

Note that the indices in I satisfy ||i| − |j|| ≤ κ |k| and k 6= 0 .

(6.6)

Since the matrix A in (4.16) is invertible, the bound (4.9) implies, for  small enough, that ω ∞ : O → ω ∞ (O) is invertible and |(ω ∞ )−1 |lip ≤ 2kA−1 k .

(6.7)

Lemma 6.1. For (k, i, j) ∈ I, η ∈ (0, 1), we have |Rτk,i,j (η)| l

ηρn−1 . 1 + |k|τ +1

(6.8)

Proof. By (4.9) and (4.16) ∞ ∞ ω ∞ (ξ) · k + Ω∞ i (ξ) − Ωj (ξ) = ω (ξ) · k +

p

i2 + m −

p j 2 + m + rk,i,j (ξ)

where |rk,i,j (ξ)| = O(γ) , |rk,i,j |lip = O() .

(6.9)

∞

We introduce the final frequencies ζ := ω (ξ) as parameters (see (6.7)), and we consider p p fk,i,j (ζ) := ζ · k + i2 + m − j 2 + m + r˜k,i,j (ζ) where also r˜k,i,j := rk,i,j ◦ (ω ∞ )−1 satisfies (6.9). In the direction ζ = sk|k|−1 + w, w · k = 0, the function f˜k,i,j (s) := fk,i,j (sk|k|−1 + w) satisfies (6.9)

f˜k,i,j (s2 ) − f˜k,i,j (s1 ) ≥ (s2 − s1 )(|k| − Cε) ≥ (s2 − s1 )|k|/2 . Since |k| ≥ 1 (recall (6.6)), by Fubini theorem, n ζ ∈ ω ∞ (O) : |fk,i,j (ζ)| ≤

ηρn−1 2η o l . 1 + |k|τ 1 + |k|τ +1

By (6.7) the bound (6.8) follows. We split I = I> ∪ I
:= (k, i, j) ∈ I : min{|i|, |j|} > C] γ −1/3 (1 + |k|τ0 )

(6.10)

where C] > C? in (4.15) for τ0 := n + 1. We set I< := I \ I> . Lemma 6.2. For all (k, i, j) ∈ I> we have 0 0 Rτk,i,j (γ 2/3 ) ⊂ Rτk,i (2γ 2/3 ) 0 ,j0

(6.11)

s(i0 ) = s(i) , s(j0 ) = s(j) , |i0 | − |j0 | = |i| − |j|

(6.12)

h i min{|j0 |, |i0 |} = C] γ −1/3 (1 + |k|τ0 ) .

(6.13)

(see (6.2)), i0 , j0 ∈ Z \ I satisfy

and

47

Proof. Since |j| ≥ γ −1/3 C? , by (4.15) and (4.16) we have the frequency asymptotic  2/3   2 m γ m ∞ ∞ Ωj (ξ) = |j| + +O ε + ~a · ξ + as(j) (ξ) + O . 3 2|j| |j| |j|

(6.14)

0 By (6.6) we have ||i| − |j|| = ||i0 | − |j0 || ≤ C|k|, |k| ≥ 1. If ξ ∈ O \ Rτk,i (2γ 2/3 ), since |i|, |j| ≥ µ0 := 0 ,j0 min{|i0 |, |j0 |} (recall (6.10) and (6.13)), we have

∞ |ω ∞ (ξ) · k + Ω∞ i (ξ) − Ωj (ξ)|

≥

∞ |ω ∞ (ξ) · k + Ω∞ i0 (ξ) − Ωj0 (ξ)| ∞ ∞ ∞ −|Ω∞ i (ξ) − Ωi0 (ξ) − Ωj (ξ) + Ωj0 (ξ)|

(6.14)

≥

4γ 2/3 − ||i| − |i0 | − |j| + |j0 || 1 + |k|τ0 ∞ ∞ ∞ −|a∞ s(i) − as(i0 ) − as(j) + as(j0 ) | −Cε

(6.12)

≥

m ||i| − |j|| m ||i0 | − |j0 || γ 2/3 m2 −C 3 − − µ0 µ0 2 |i| |j| 2 |i0 | |j0 |

γ 2/3 |k| 4γ 2/3 − Cε −C 2 1 + |k|τ0 µ0 µ0

(6.13)

≥

2γ 2/3 1 + |k|τ0

0 taking C] in (6.13) large enough. Therefore ξ ∈ O \ Rτk,i,j (γ 2/3 ) proving (6.11). As a corollary we deduce: [ Rτk,i,j (γ) l γ 2/3 ρn−1 . Lemma 6.3.

(k,i,j)∈I> 0 Proof. Since 0 < γ ≤ 1 and τ ≥ τ0 (see (4.18)), we have (see (6.2)) Rτk,i,j (γ) ⊂ Rτk,i,j (γ 2/3 ). Then Lemma 6.2 and (6.8) imply that, for each p ∈ Z,



[ (k,i,j)∈I> , |i|−|j|=p

γ 2/3 ρn−1 Rτk,i,j (γ) l . 1 + |k|τ0 +1

Therefore

[

Rτk,i,j (γ) l

k,|p|≤C|k|

(k,i,j)∈I>

proving the lemma. [ Lemma 6.4.

X

X γ 2/3 ρn−1 γ 2/3 ρn−1 l τ +1 1 + |k| 0 1 + |k|τ0 k

Rτk,i,j (γ) l γ 2/3 ρn−1 .

(k,i,j)∈I
0, p > 1/2, and the momentum is (see (2.85)) X π(α, β) = j(αj − βj ) . j∈Z

Note that 0 ≤ Gα,β ≤ 4! (recall α! = Πi∈Z αi !) 49

Lemma 7.1. For all R > 0, N0 satisfying (3.1), the Hamiltonian G defined in (7.1) belongs to QTR (N0 , 3/2, 4) and kGkTR,N0 ,3/2,4 = kXG kR l R2 . (7.2) Proof. The Hamiltonian vector field XG := (−i∂u¯ G, i∂u G) has components X α β Gl,σ iσ∂uσl G = iσ ¯ , σ = ±, l ∈ Z, α,β u u |α|+|β|=3,π(α,β)=−σl

where Gl,+ α,β = (αl + 1)Gα+el ,β ,

Gl,− α,β = (βl + 1)Gα,β+el .

Note that 0 ≤ Gl,σ α,β ≤ 5! By Definitions 2.6, 2.8 and (2.2) 1 kXG kR = sup R kuka,p ,k¯uka,p 0, such that, for every m ∈ (0, ∞), |~σ · λ~(m)| ≥

c∗ m >0 (n20 + m)3/2

where

51

n0 := min{hj1 i, hj2 i, hj3 i, hj4 i} .

(7.16)

Proof. In the Appendix. The map Γ := eadF is obtained as the time-1 flow generated by the Hamiltonian F := −

X ~ ·~ σ =0 ,~ σ ·λ~  6=0 and ~ ∈(I / c )4

i u~σ ~σ · λ~ ~

(7.17)

We notice that the condition ~ · ~σ = 0 , ~σ · λ~ 6= 0 is equivalent to requiring that ~ · ~σ = 0 and ~, ~σ satisfy (7.12)-(7.15). By Lemma 7.2 there is a constant c¯ > 0 (depending only on m and I) such that ~ · ~σ = 0 , ~σ · λ~ 6= 0 and ~ ∈ / (I c )4

=⇒

|~σ · λ~| ≥ c¯ > 0 .

(7.18)

We have proved that the moduli of the small divisors in (7.17) are uniformly bounded away from zero. Hence F is well defined and, arguing as in Lemma 7.1, we get kXF kR l R2 .

(7.19)

null Moreover F ∈ HR because in (7.17) the sum is restricted to ~σ · ~ = 0 (see also (7.4)).

Lemma 7.3. F in (7.17) solves the homological equation ˆ {N, F } + G = adF (N ) + G = G + G

(7.20)

ˆ are defined in (7.6). where G, G Proof. We claim that the only ~ ∈ Z4 , ~σ ∈ {±}4 with ~ · ~σ = 0 which do not satisfy (7.12)-(7.15) have the form j1 = j2 , j3 = j4 , σ1 = −σ2 , σ3 = −σ4 (or permutations of the indexes) .

(7.21)

Indeed: X If ~ = 0, σi = 0: the σi are pairwise equal and (7.21) holds. i

If ~ = (0, 0, q, q), q 6= 0, and σ1 = −σ2 : by ~ · ~σ = 0 we have also σ3 = −σ4 and (7.21) holds. If ~ = (p, p, −p, −p), p 6= 0 and σ1 = −σ2 : by ~ · ~σ = 0 we have also σ3 = −σ4 and (7.21) holds. If j1 = j2 , j3 = j4 , j1 , j3 6= 0, j1 6= −j3 : Case 1: j1 6= j3 . Then 0 = ~σ · ~ = (σ1 + σ2 )j1 + (σ3 + σ4 )j3 implies σ1 = −σ2 , σ3 = −σ4 . Case 2: j1 = j3 and so j1 = j2 = j3 = j4 6= 0. Hence 0 = (σ1 + σ2 + σ3 + σ4 )j1 and (7.21) follows. By (7.17) and (7.11) all the monomials in {N, F } cancel the monomials of G in (7.1) except for ˆ (see (7.6)) and those of the form |up |2 |uq |2 , p or q ∈ I, which contribute to G. The those in G expression in (7.6) of G follows by counting the multiplicities. null The Hamiltonian F ∈ HR in (7.17) is quasi-T¨oplitz:

Lemma 7.4. Let R > 0. If N0 := N0 (m, I, L, b) is large enough, then F defined in (7.17) belongs to QTR (N0 , 3/2, 4) and kF kTR,N0 ,3/2,4 l R2 . (7.22) null Proof. We have to show that F ∈ HR verifies Definition 3.4. For all N ≥ N0 , we compute, by (7.17) and Definition 3.2 (in particular (3.12)), the projection X 0 σ,σ 0 ΠN,3/2,4 F = Fm,n (wL )uσm uσn (7.23) |n|,|m|>CN/4 , σ,σ 0 =± ,|σm+σ 0 n| 3N/2, |σm − σn| < 4N L , which implies s(m) = s(n) by (3.1). Then σ|m| − σ|n| = σs(m)m − σs(n)n = s(m)(σm − σn) 53

and (7.32) follows. We have proved that F˜ ∈ TR (N0 , 3/2, 4). The T¨ oplitz defect, defined by (3.29), is X X σ,σ0 0 σ,σ 0 σ,σ 0 Fˆ := Fˆm,n (wL )uσm uσn with Fˆm,n (wL ) := Fˆα,β,m,n uα u ¯β

(7.33)

where the indexes in the two sums have the same restrictions as in (7.23)-(7.25), and 24i N α!β! λα,β + σλm + σλn   1 1 24i − = −N α!β! λα,β + σλm − σλn λα,β + σ|m| − σ|n| 24i N σ(λm − |m| − λn + |n|) = α!β! (λα,β + σλm − σλn )(λα,β + σ|m| − σ|n|)

σ,σ Fˆα,β,m,n

= −

σ,−σ Fˆα,β,m,n

(7.34)

(7.35)

We now proof that the coefficients in (7.34)-(7.35) are bounded by a constant independent of N . The coefficients in (7.34) are bounded because X X √ √ X (|αh | + |βh |) ≤ 4N L + 2 m |λα,β | ≤ λh (|αh | + |βh |) ≤ |h|(|αh | + |βh |) + m h

h

h

by (7.26)-(7.25) (note that λh ≤ |h| +

√

m) and

√ |λα,β + σλm + σλn | ≥ |λm + λn | − |λα,β | ≥ 3N − 4N L − 2 m ≥ 3N/2 for N ≥ N0 large enough. The coefficients in (7.35) are bounded by (7.18), (7.29), and (7.30)

N |λm − |m| − λn + |n|| ≤ N

m 1 1  2 + ≤ m. 2 |m| |m| 3

Hence arguing as in the proof of Lemma 7.1 we get kXFˆ kR l R2 .

(7.36)

In conclusion, (7.19), (7.31), (7.36) imply (7.22) (recall (3.30)). Proof of Proposition 7.1 completed. We have eadF H

=

eadF N + eadF G = N + {N, F } +

X1 X1 adiF (N ) + G + adi (G) i! i! F i≥2

(7.20)

=

ˆ+ N +G+G

X i≥1

=

i≥1


X1 i 1 adiF adF (N ) + ad (G) (i + 1)! i! F i≥1

ˆ+K N +G+G

where, using again (7.20), K :=

X i≥1

X1 1 ˆ − G) + adiF (G + G adi G =: K1 + K2 . (i + 1)! i! F

(7.37)

i≥1

Proof of (7.9). We claim that in the expansion of K in (7.37) there are only monomials u~~σ with ˆ contain only monomials of degree four and, for any ~ ∈ Z2d , ~σ ∈ {+, −}2d , d ≥ 3. Indeed F, G, G, G monomial m, adF (m) contains only monomials of degree equal to the deg(m) + 2. The restriction ˆ preserve momentum, i.e. Poisson ~σ · ~ = 0 follows by the Jacobi identity (2.81), since F, G, G, G 54

commute with M . Proof of (7.10). We apply Proposition 3.2 with (no (x, y) variables and) ( ˆ − G for K1 , G+G r R , r0 f F, g R/2 , G for K2 , θ N0 defined in Lemma 7.4 and

3/2 ,

N00

θ0

2,

µ

4,

µ0

δ

1/2 ,

3,

≥ N0 satisfying (3.64) and

κ(N00 )b−L ln N00 ≤ 1 ,

(6 + κ)(N00 )L−1 ln N00 ≤ 1/2 .

(7.38)

Note that (3.65) follows by (7.38). By (7.22), the assumption (3.63) is verified for every 0 < R < R0 , with R0 small enough. Then Proposition 3.2 applies and (7.10) follows by (3.67) (with h 1), (7.2), (7.22) and (7.7).

7.2

Action–angle variables

We introduce action-angle variables on the tangential sites I := {j1 , . . . , jn } (see (7.5)) via the analytic and symplectic map Φ(x, y, z, z¯; ξ) := (u, u ¯) (7.39) defined by ujl :=

p p ξl + yl eixl , u ¯jl := ξl + yl e−ixl , l = 1, . . . , n , uj := zj , u ¯j := z¯j , j ∈ Z \ I .

(7.40)

o n % O% := ξ ∈ Rn : ≤ ξl ≤ % , l = 1, . . . , n . 2

(7.41)

Let

Lemma 7.5. (Domains) Let r, R, ρ > 0 satisfy 16r2 < % ,

% = C∗ R2

with

C∗−1 := 48nκ2p e2(s+aκ) .

(7.42)

Then, for all ξ ∈ O% ∪ O2% , the map Φ( · ; ξ) : D(s, 2r) → D(R/2) := BR/2 × BR/2 ⊂ `a,p × `a,p

(7.43)

is well defined and analytic (D(s, 2r) is defined in (2.5) and κ in (3.1)). (7.42)

Proof. Note first that for (x, y, z, z¯)p ∈ D(s, 2r) we have (see (2.6)) that |yl | < 4r2 < ρ/4 < ξl , ∀ξ ∈ Oρ ∪ O2ρ . Then the map yl 7→ ξl + yl is well defined and analytic. Moreover, for ξl ≤ 2ρ, |jl | ≤ κ, x ∈ Tns , kzka,p < 2r, we get ku(x, y, z, z¯; ξ)k2a,p

(7.39)

=

n X X (ξl + yl )|e2ixl ||jl |2p e2a|jl | + |zj |2 hji2p e2a|j| l=1

≤

j∈Z\I

 (7.42) %  2s 2p 2aκ e κ e + 4r2 < R2 /4 n 2% + 4

proving (7.43) (the bound for u ¯ is the same). Given a function F : D(R/2) → C, the previous Lemma shows that the composite map F ◦ Φ : D(s, 2r) → C. The main result of this section is Proposition 7.2: if F is quasi-T¨oplitz in the variables (u, u ¯) then the composite F ◦ Φ is quasi-T¨oplitz in the variables (x, y, z, z¯) (see Definition 3.4). We write X (1) (1) (2) (2) F = Fα,β mα,β , mα,β := (u(1) )α (¯ u(1) )β (u(2) )α (¯ u(2) )β , (7.44) α,β

55

where u = (u(1) , u(2) ) ,

u(1) := {uj }j∈I , u(2) := {uj }j∈Z\I ,

similarly for u ¯,

and (α, β) = (α(1) + α(2) , β (1) + β (2) ) , (α(1) , β (1) ) := {αj , βj }j∈I , (α(2) , β (2) ) := {αj , βj }j∈Z\I . (7.45) We define

n d HR := F ∈ HR : F =

o Fα,β uα u ¯β .

X

(7.46)

|α(2) +β (2) |≥d

Proposition 7.2. (Quasi–T¨ oplitz) Let N0 , θ, µ, µ0 satisfying (3.1) and b N0

(µ0 − µ)N0L > N0b ,

N0 2− 2κ +1 < 1 .

(7.47)

d with d = 0, 1, then f := F ◦ Φ ∈ QTs,r (N0 , θ, µ) and If F ∈ QTR/2 (N0 , θ, µ0 ) ∩ HR/2

kf kTs,r,N0 ,θ,µ,O% l (8r/R)d−2 kF kTR/2,N0 ,θ,µ0 .

(7.48)

The rest of this section is devoted to the proof of Proposition 7.2. Introducing the action-angle variables (7.40) in (7.44), and using the Taylor expansion     X γ  γ γ γ(γ − 1) . . . (γ − h + 1) (1 + t)γ = th , := 1 , := , h ≥ 1, (7.49) h! h 0 h h≥0

we get f := F ◦ Φ =

(2)

X

fk,i,α(2) ,β (2) eik·x y i z α z¯β

(2)

(7.50)

k,i,α(2) ,β (2)

with Taylor–Fourier coefficients fk,i,α(2) ,β (2) :=

X

Fα,β

α(1) −β (1) =k

n Y

(1) (1) α +β l l 2

ξl

−il

(1)   α(1) l +βl

2

.

il

l=1

(7.51)

We need an upper bound on the binomial coefficients. Lemma 7.6. For |t| < 1/2 we have  k  X h 2 (i) |t| ≤ 2k , ∀k ≥ 0 , h

(ii)

X h≥1

h≥0

 k  |t| 2 ≤ 3k |t| , ∀k ≥ 1 . h h

Proof. By (7.49) and the definition of majorant (see (2.11)) we have  X  1  k  X k X  k  k (2.39) 1 2 h ≺ th 2 th = M (1 + t) 2 ≺ (M (1 + t) 2 )k = t h h h≥0

h≥0

(7.52)

(7.53)

h≥0

 1  because 2 ≤ 1 by (7.49). For |t| < 1/2 the bound (7.53) implies (7.52)-(i). Ne h X h≥1

 k   k  (7.49) X  k  | k − h|  k  (7.52)−(i) X X 2 |t|h 2 ≤ |t| |t|h |t|h 2 2 ≤ k|t| |t|h 2 ≤ k2k |t| = |t| h h+1 h h+1 h h≥0

h≥0

h≥0

which implies (7.52)-(ii) for k ≥ 1.

56

d Lemma 7.7. (M -regularity) If F ∈ HR/2 then f := F ◦ Φ ∈ Hs,2r and

kXf ks,2r,O% ∪O2% l (8r/R)d−2 kXF kR/2 .

(7.54)

Moreover if F preserves momentum then so does F ◦ Φ. Proof. We first bound the majorant norm kf ks,2r,O% ∪O2%

(7.50),(7.46)

:=

sup

ξ∈O% ∪O2% (y,z,¯ z )∈D(2r)

(2)

X

sup

|fk,i,α(2) ,β (2) |e|k|s |y i ||z α ||¯ zβ

(2)

|.

(7.55)

k,i,|α(2) +β (2) |≥d

Fix α(2) , β (2) . Since for all ξ ∈ O% ∪ O2% , y ∈ B(2r)2 , we have |yl /ξl | < 1/2 by (7.42), we have X

e|k|s

X

≤

(7.56)

i

k (7.51)

|fk,i,α(2) ,β (2) ||y|i

X

e

s(|α(1) |+|β (1) |)

|Fα,β |ξ

α(1) +β (1) 2

≤

X

(1)

es(|α

|+|β (1) |)

|Fα,β |ξ

α(1) +β (1) 2

X

n Y

(1)

2αl

(1)

+βl

(7.58)

l=1

α(1) ,β (1)

≤

(7.57)

l=1 il ≥0

α(1) ,β (1) (7.52)

(1)  n X il  α(1) Y yl l +βl 2 ξl il

(1)

es(|α

|+|β (1) |)

|Fα,β |(2%)

|α(1) |+|β (1) | 2

(1)

2|α

|+|β (1) |

α(1) ,β (1)

X

=

(2es

p

(1)

2%)|α

|+|β (1) |

|Fα,β | .

α(1) ,β (1)

Then, substituting in (7.55), kf ks,2r,O% ∪O2%

≤

sup

G(z, z¯)

where

(7.59)

kzka,p ,k¯ z ka,p 1 by (7.42)), G(z, z¯) ≤ (8r/R)d (M F )(u∗ , u ¯∗ ) ≤ (8r/R)d kF kR/2 , ∀ kzka,p , k¯ z ka,p < 2r . Hence by (7.59) kf ks,2r,O% ∪O2% ≤ (8r/R)d kF kR/2 .

(7.62)

This shows that f is M -regular. Similarly we get k∂z f ks,2r,O% ∪O2% ≤ k∂u(2) F kR/2 (8r/R)d−1 , same for ∂z¯ . Moreover, by the chain rule, and (7.62) k∂xi f ks,2r,O% ∪O2%

≤

k∂yi f ks,2r,O% ∪O2%

≤

p (k∂u(1) F kR/2 + k∂u¯(1) F kR/2 ) 2% + %/4es (8r/R)d i

i

(k∂u(1) F kR/2 + k∂u¯(1) F kR/2 ) p i

i

Then (7.54) follows by (7.42) (recalling (2.2)).

57

es %/2 − %/4

(8r/R)d .

(7.63)

(1)

Definition 7.1. For a monomial mα,β := (u(1) )α (¯ u(1) )β p(mα,β ) :=

n X

(1)

(1)

hjl i(αjl + βjl ) ,

(1)

(2)

(u(2) )α (¯ u(2) )β

(2)

(as in (7.44)) we set

hji := max{1, |j|} .

(7.64)

l=1

For any F as in (7.44), K ∈ N, we define the projection X Πp≥K F := Fα,β mα,β , Πp θN ,

j∈Z\I

58

, |α(1) − β (1) | < N b .

(7.71)

(1)

We deduce the contradiction that mα,β = (u(1) )α (¯ u(1) )β b bilinear because (recall that we suppose p(mα,β ) < N ) n X

(1)

(1)

|jl |(αjl + βjl ) +

l=1

X

(1)

(2)

˜(2) σ σ 0 um un

(u(2) )α˜ (¯ u(2) )β

(7.64),(7.71)

(2) (2) |j|(α ˜ j + β˜j )


θN 0

Each coefficient F σ,σ (s(m), σm + σ 0 n) ◦ Φ depends on n, m, σ, σ 0 only through s(m), σm + σ 0 n, σ, σ 0 . 0 Hence, in order to conclude that F ◦ Φ ∈ Ts,2r (N, θ, µ0 ) it remains only to prove that F σ,σ (s(m), σm + 0 σ 0 n) ◦ Φ ∈ Ls,2r (N, µ0 , σm + σ 0 n), see Definition 3.1. Each monomial mα,β of F σ,σ (s(m), σm + σ 0 n) ∈ 0 0 LR/2 (N, µ , σm + σ n) satisfies n X

(αjl + βjl )|jl | +

l=1

X

(αj + βj )|j| < µ0 N L

and

p(mα,β ) < N b

j∈Z\I

by the hypothesis Πp≥N b F = 0. Hence mα,β ◦ Φ (see (7.70)) is (N, µ0 )-low momentum, in particular |α(1) − β (1) | ≤ p(mα,β ) < N b . Proof of Proposition 7.2. Since F ∈ QTR/2 (N0 , θ, µ0 ) (see Definition 3.4), for all N ≥ N0 , there is a T¨ oplitz approximation F˜ ∈ TR/2 (N, θ, µ0 ) of F , namely ΠN,θ,µ0 F = F˜ + N −1 Fˆ

kXF kR/2 , kXF˜ kR/2 , kXFˆ kR/2 < 2kF kTR/2,N0 ,θ,µ0 .

with

(7.72)

In order to prove that f := F ◦ Φ ∈ QTs,r (N0 , θ, µ) we define its candidate T¨oplitz approximation f˜ := ΠN,θ,µ ((Πp