KAM for the quantum harmonic oscillator

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. P\"oschel and uses recent techniques of H. Eliasson and S.B. Kuksin. As an application we show that some 1D nonlinear Schr\"odinger equations with harmonic potential admits many quasi-periodic solutions. In a second application we prove the reducibility of the 1D Schr\"odinger equations with the harmonic potential and a quasi periodic in time potential.Comment: 54 pages. To appear in Comm. Math. Phy

Tame majorant analyticity for the Birkhoff map of the defocusing nonlinear Schr\uf6dinger equation on the circle

For the defocusing nonlinear Schr\uf6 dinger equation on the circle, we construct a Birkhoff map \u3a6 which is tame majorant analytic in a neighborhood of the origin. Roughly speaking, majorant analytic means that replacing the coefficients of the Taylor expansion of \u3a6 by their absolute values gives rise to a series (the majorant map) which is uniformly and absolutely convergent, at least in a small neighborhood. Tame majorant analytic means that the majorant map of \u3a6 fulfills tame estimates. The proof is based on a new tame version of the Kuksin-Perelman theorem (2010 Discrete Contin. Dyn. Syst. 1 1-24), which is an infinite dimensional Vey type theorem

Global Birkhoff coordinates for the periodic Toda lattice

In this paper we prove that the periodic Toda lattice admits globally defined Birkhoff coordinates.Comment: 32 page

Estimates on periodic and Dirichlet eigenvalues for the Zakharov-Shabat system

Consider the 2 \Theta 2 first order system due to Zakharov-Shabat, LY := i ` 1 0 0 \Gamma1 ' Y 0 + ` 0 / 1 / 2 0 ' Y = Y with / 1 ; / 2 being complex valued functions of period one in the weighted Sobolev space H w j H w C : Denote by spec(/ 1 ; / 2 ) the set of periodic eigenvalues of L(/ 1 ; / 2 ) with respect to the interval [0; 2] and by spec Dir (/ 1 ; / 2 ) the set of Dirichlet eigenvalues of L(/ 1 ; / 2 ) when considered on the interval [0; 1]. It is well known that spec(/ 1 ; / 2 ) and spec Dir (/ 1 ; / 2 ) are discrete. Theorem Assume that w is a weight such that, for some ffi ? 0, w \Gammaffi (k) = (1 + jkj) \Gammaffi w(k) is a weight as well. Then for any bounded subset B of 1-periodic elements in H w \ThetaH w there exist N 1 and M 1 so that for any jkj N , and (/ 1 ; / 2 ) 2 B , the set spec(/ 1 ; / 2 )"f 2 1 C j j \Gamma kj ! =2g contains exactly one isolated pair of eigenvalues f + k ; \Gamma k g and spec Dir (/ 1 ; / 2 ) " f 2 C j ..

Density of finite gap potentials for the Zakharov-Shabat system

For various spaces of potentials we prove that the set of finite gap potentials of the Zakharov-Shabat system is dense. In particular our result holds for Sobolev spaces and for spaces of analytic potentials of a given type

KAM theorem for the nonlinear Schrödinger equation

We prove the persistence of finite dimensional invariant tori associated with the dfocusing nonlinear Schrödinger equation under small Hamiltonian perturbations. The invariant tori are not necessarily small

Gap estimates of the spectrum of the Zakharov-Shabat system

We prove new gap estimates for the Zakharov-Shabat systems with complex periodic potentials. Our method allows us to characterize in a precise way the decreasing properties of the gap length sequence in terms of the regularity of complex potentials in weighted Sobolev spaces