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Abstract

In this paper we prove an abstract KAM theorem for infinite dimensional Hamiltonians systems. This result extends previous works of S.B. Kuksin and J. Pöschel and uses recent techniques of H. Eliasson and S.B. Kuksin. As an application we show that some 1D nonlinear Schrödinger equations with harmonic potential admits many quasi-periodic solutions. In a second application we prove the reducibility of the 1D Schrödinger equations with the harmonic potential and a quasi periodic in time potential.

Nonlinear Schrödinger equation, harmonic potential, KAM theory, Hamiltonian systems, reducibility.

KAM for the quantum harmonic oscillator] KAM for the quantum harmonic oscillator

\subjclass

37K55, 35B15, 35Q55. 1

1 Introduction

Let so that for all . We consider the (complex) Hilbert space defined by the norm

We define the symplectic phase space as

(1.1)

equipped with the canonic symplectic structure:

For we introduce the following Hamiltonian in normal form

(1.2)

where is an external parameter.
In [10], (see also [11] and a slightly generalised version in [16]) S.B. Kuksin has shown the persistence of dimensional tori for the perturbed Hamiltonians with general conditions on the frequencies and perturbation which essentially are the following : Firstly the frequencies satisfy some Melnikov conditions and the external frequencies have to be well separated in the sense that there exists so that roughly speaking (see Assumption 2 below)

(1.3)

Denote by the phase space given by the weight where and . Secondly, the perturbation is real analytic and the corresponding Hamiltonian vector field is so that

(1.4)

where is the constant which appears in (1.3). For instance, the Schrödinger and the wave equation on with Dirichlet boundary conditions satisfy the previous conditions, see respectively the KAM results of Kuksin-Pöschel [13] and Pöschel [18]. Indeed the result in [13] is stronger because there is no external parameter in the equation.
Now, if we consider the nonlinear harmonic oscillator

(1.5)

with real and bounded potential , we have , hence but the Hamiltonian perturbation which is here

(1.6)

does not satisfy the strict smoothing condition (1.4) (see Section 6 for more details).

The aim of this paper is to prove a KAM theorem (Theorem 2.2) in the case and in (1.3) and (1.4). To compensate the lack of smoothing effect of we need some additional conditions (see Assumption 4) on the decay of the derivatives (in the spirit of the so-called Töplitz-Lipschitz condition used by Eliasson & Kuksin in [6]) which will be satisfied by the perturbation (1.6). The general strategy is explained with more details in Section 2.3.
Notice that S.B. Kuksin has already considered in [11] the harmonic oscillator with a smoothing nonlinearity of type where is a fixed smooth function.

We present two applications of our abstract result concerning the harmonic oscillator . Let and denote by the space with . The operator has eigenfunctions (the Hermite functions) which satisfy and form a Hilbertian basis of . Let be a typical element of . Then if and only if . Indeed is a Sobolev space based on and we can check that

In this context, we are able to apply our KAM result to (1.5) and we obtain (see Theorem 6.3 for a more precise statement)

{theo}

Let be an integer. For typical potential and for small enough, the nonlinear Schrödinger equation

(1.7)

has many quasi-periodic solutions in .

Here the notion of “typical potential” is vague. This means that there exists rather a large class of perturbations of the harmonic oscillator so that the result of Theorem 1 holds true (unfortunately our result does not cover the case ). Since the definition of this class is technical, we postpone it to Section 6.

The physical motivation for considering equation (1.7) (for ) comes from the Gross-Pitaevski equation used in the study of Bose-Einstein condensation (see [15]). The harmonic potential arises from a Taylor expansion near the bottom of general smooth well. In our work, we have to add a small linear perturbation to the harmonic potential in order to avoid resonances (see the non resonance condition (2.3) below).

The generalisation of such a result in a multidimensional setting is not evident for a spectral reason: the spectrum of the linear part is no more well separated. We could expect to adapt the tools introduced in [6] but the arithmetic properties of the corresponding spectra are not the same: in [6] the free frequencies are for all , while in our case they are for all . Nevertheless we mention that it is still possible to obtain a Birkhoff normal form for (1.5) as recently proved in [9].
A consequence of Theorem 1 is the existence of periodic solutions to (1.7). There are other approaches to construct periodic solutions of this equation. For instance, the gain of compacity yielded by the confining potential allows the use of variational methods. We develop this point of view in the appendix.

The second application concerns the reducibility of a linear harmonic oscillator, , on perturbed by a quasi periodic in time potential. Such kind of reducibility result for PDE using KAM machinery was first obtained by Bambusi & Graffi [1] for Schrödinger equation with an potential, being strictly larger than 2 (notice that in that case the exponent in the asymptotic of the frequencies (1.3)). This result was recently extended by Liu and Yuan [14] to include the Duffing oscillator.
Here we follow the more recent approach developed by Eliasson & Kuksin (see [7]) for the Schrödinger equation on the multidimensional torus. Namely we consider the linear equation

where is a small parameter and the frequency vector of forced oscillations is regarded as a parameter in . We assume that the potential is analytic in on for some , and in , and we suppose that there exists and so that for all and

(1.8)

In Section 7 we consider the previous equation as a linear non-autonomous equation in the complex Hilbert space and we prove (see Theorem 7.1 for a more precise statement) {theo} Assume that satisfies (1.8). Then there exists such that for all there exists of positive measure and asymptotically full measure: as , such that for all , the linear Schrödinger equation

(1.9)

reduces, in , to a linear equation with constant coefficients (with respect to the time variable).

In particular, we prove the following result concerning the solutions of (1.9). {coro} Assume that is in with all its derivatives bounded and satisfying (1.8). Let and . Then there exists so that for all and , there exists a unique solution of (1.9) so that . Moreover, is almost-periodic in time and we have the bounds

for some .

{rema}

In the very particular case where satisfies (1.8) and is independent of , the result of Corollary 1 is easy to prove. In that case, the solution of (1.9) reads

where and are the eigenfunctions and the eigenvalues of , and some . The result follows thanks to the asymptotics of when (see Section 6 for similar considerations.)

The previous results show that all solutions to (1.9) remain bounded in time, for a large set of parameters . A natural question is whether we can find a real valued potential , quasi-periodic in time and a solution so that does not remain bounded when . J.-M. Delort [4] has recently shown that this is the case if is replaced by a pseudo differential operator : he proves that there exist smooth solutions so that for all and , , which is the optimal growth. We also refer to the introduction of [4] for a survey on the problem of Sobolev growth for the linear Schrödinger equation.

Another way to understand the result of Theorem 1 is in term of Floquet operator (see [5] and [23] for mathematical considerations, and [8, 21] for the physical meaning). Consider on the Floquet Hamiltonian

(1.10)

then we have {coro} Assume that satisfies (1.8). There exists so that for all and , the spectrum of the Floquet operator is pure point. A similar result, using a different KAM strategy, was obtained by W.M. Wang in [23] in the case where

where is the first Hermite function.

At the end of Section 7 we make explicit computations in the case of a potential which is independent of the space variable. This example shows that one can not avoid to restrict the choice of parameters to a Cantor type set in Theorem 1.

Acknowledgements.

The first author thanks Hakan Eliasson and Serguei Kuksin for helpful suggestions at the principle of this work. Both authors thank Didier Robert for many clarifications in spectral theory.

2 Statement of the abstract result

We give in this section our abstract KAM result.

2.1 The assumptions on the Hamiltonian and its perturbation


Let be a bounded closed set so that , where Meas denote the Lebesgue measure in . The set is the space of the external parameters . Denote by the difference operator in the variable :

For so that only a finite number of coordinates are non zero, we denote by its length, and . We set

The first two assumptions we make, concern the frequencies of the Hamiltonian in normal form (1.2)

Assumption 1 (Nondegeneracy).

Denote by the internal frequencies. We assume that the map is an homeomorphism from to its image which is Lipschitz continuous and its inverse also.
Moreover we assume that for all

(2.1)

and for all

Assumption 2 (Spectral asymptotics).

Set . We assume that there exists so that for all and uniformly on

Moreover we assume that there exists such that the functions

are uniformly Lipschitz on for .

If the previous assumptions are satisfied (and actually without assuming (2.1)), J. Pöschel [16] proves that there exist a finite set and with when , such that for all

(2.2)

for some large depending on and .
Then assuming (2.1), J. Pöschel proves [16, Corollary C and its proof] that the non resonance condition (2.2) remains valid on all , i.e

(2.3)

In the sequel, we will use the distance

and the semi-norm

Finally, we set

where {rema} The proof of (2.3) crucially uses the control of the Lipschitz semi-norm (see [16, Lemma 5]). For this reason in assumptions 3 and 4 below we have to control the Lipschitz version of each semi-norms introduced on or . Recall that the phase space is defined by (1.1), with a weight so that , as in the beginning of the introduction. As in [16], for we define the (complex) neighbourhood of in .

(2.4)

Let . Then for we define

The next assumption concerns the regularity of the vector field associated to . Denote by

Then

Assumption 3 (Regularity).

We assume that there exist so that

Moreover we assume that for all , is analytic in and that for all , and are Lipschitz continuous on .

We then define the norms

and

where and we define the semi-norms

and

where .
In the sequel, we will often work in the complex coordinates

Notice that this is not a canonical change of variables and in the variables the symplectic structure reads

and the Hamiltonian in normal form is

(2.5)

As we mentioned previously we need some decay on the derivatives of . We first introduce the space : Let , we say that if where :
The norm is defined by the conditions 2

The semi-norm is defined by the conditions

The last assumption is then the following

Assumption 4 (Decay).

for some .

{rema}

The control of the second derivative is the most important condition. The other ones are imposed so that we are able to recover the last one after the KAM iteration (see Lemma 3.2). Furthermore the assumptions on the first derivatives are already contained in Assumption 3 as soon as .

2.2 Statement of the abstract KAM Theorem


Recall that . {theo} Suppose that is a family of Hamiltonians of the form (2.5) on the phase space depending on parameters so that Assumptions 1 and 2 are satisfied. Then there exist and so that every perturbation of which satisfies Assumptions 3 and 4 and the smallness condition

for some and , the following holds. There exist

  1. a Cantor set with as ;

  2. a Lipschitz family of real analytic, symplectic coordinate transformations ;

  3. a Lipschitz family of new normal forms

    defined on ;

such that

where is analytic on and globally of order 3 at . That is the Taylor expansion of only contains monomials with .
Moreover each symplectic coordinate transformation is close to the identity

(2.6)

the new frequencies are close to the original ones

(2.7)

and the new frequencies satisfy a non resonance condition

(2.8)

As the consequence, for each the torus is still invariant under the flow of the perturbed Hamiltonian , the flow is linear ( in the new variables) on these tori and furthermore all these tori are linearly stable.

2.3 General strategy


The general strategy is the classical one used for instance in [10, 11, 16]. For convenience of the reader we recall it. Let be a Hamiltonian, where is given by (2.5) and a perturbation which satisfies the assumptions of the previous section. We then consider the second order Taylor approximation of which is

(2.9)

with and we define its mean value by

Recall that in this setting have homogeneity 1, whereas has homogeneity 2.
Let be a function of the form (2.9) and denote by the flow at time associated to the vector field of . We can then define a new Hamiltonian by , and the Hamiltonian structure is preserved, because is a symplectic transformation. The idea of the KAM step is to find, iteratively, an adequate function so that the new error term has a small quadratic part. Namely, thanks to the Taylor formula we can write

In view of the previous equation, we define the new normal form by , where satisfies the so-called homological equation (the unknown are and )

(2.10)

The new normal form has the form (2.5) with new frequencies given by

where

(2.11)

Once the homological equation is solved, we define the new perturbation term by

(2.12)

where in such a way that

Notice that if was initially of size , then and are of size , and the quadratic part of is formally of size . That is, the formal iterative scheme is exponentially convergent.

Without any smoothing effect on the regularity, there is no decreasing property in the correction term added to the external frequencies (2.11). In that case it would be impossible to control the small divisors (see (2.3)) at the next step. In this work the smoothing condition (1.4) on is replaced by Assumption 4 (see also Remark 4.2). The difficulty is to verify the conservation of this assumption at each step.

Plan of the proof of Theorem 2.2

In Section 3 we solve the homological equation and give estimates on the solutions. Then we study precisely the flow map and the composition . In Section 4 we estimate the new error term and the new frequencies after the KAM step, and Section 5 is devoted to the convergence of the KAM method and the proof of Theorem 2.2.

{enonce*}

Notations In this paper , denote constants the value of which may change from line to line. These constants will always be universal, or depend on the fixed quantities .
We denote by the set of the non negative integers, and . For , we denote by its length (if it is finite), and . We define the space . The notation Meas stands for the Lebesgue measure in .

In the sequel, we will state without proof some intermediate results of [16] which still hold under our conditions ; hence the reader should refer to [16] for the details. For the convenience of the reader we decided to remain as close as possible to the notations of J. Pöschel.

3 The linear step

In this section, we solve equation (2.10) and study the Lie transform .
Following [16], (respectively  ) stands either for or (respectively or  ) and stands for .

3.1 The homological equation


The following result shows that it is possible to solve equation (2.10) under the Diophantine condition (2.3). {lemm}[[16]] Assume that the frequencies satisfy, uniformly on , for some the condition (2.3). Then the homological equation (2.10) has a solution , which is normalised by , , and satisfies for all , and

where only depends on and . The space is not stable under the Poisson bracket. Therefore we need to introduce the space endowed with the norm defined by the following conditions.


This definition is motivated by the following result, which can be understood as a smoothing property of the homological equation

{lemm}

Assume that the frequencies satisfy (2.3), uniformly on . Let be given by Lemma 3.1. Assume moreover that , then there exists so that for any , we have , and

(3.1)

and

where only depends on and .

For the proof of this result, we need the classical lemma {lemm} Let be a periodic function and assume that is holomorphic in the domain , and continuous on . Then there exists so that its Fourier coefficients satisfy

Proof of Lemma 3.1.

In [16], the author looks for a solution of (2.10) of the form of (2.9), i.e.

(3.2)

A direct computation then shows that the coefficients in (3.2) are given by

(3.3)

and that we can set .
In the following we will use the notation , where the 1 is at the position, and .
The variables and exactly play the same role, therefore it is enough to study the derivatives in the variable .
In the sequel we write . Then it easy to check that for any and ,

for some and . In the sequel, may vary from line to line, but will remain independent of .
We first prove that
Observe that , then according to Lemma 3.1, there exists so that