Condition de non-résonance pour l’oscillateur harmonique quantique perturbé

Abstract

On R d , we consider the differential operator −�+jj xjj +R(x)+M. Here R is in the Schwartz class and M is a bounded operator of L (R). Under a localization hypothesis on the spectrum of −� + jj xjj + R(x), which is automatically satisfied if d = 1, one proves that there are arbitrarily small operator M, i.e. jj M jj � 1, such that for all r � 3, for small Cauchy data of order in high Sobolev norms, the nonlinear Schrodinger equation admits a solution which is bounded by 2 on a time existence interval of length � −r . The case R = 0 has been proved by Grebert-Imekraz-Paturel. Here we do not need any explicit spectral analysis (eigenvalues and eigenfunctions) of −� + jj xjj + R(x). The role of M is making the spectrum of −�+ jj xjj + R(x)+ M nonresonant. We also give a partial answer in the case M = 0 : there are some potentials R such that the spectrum of −@2 x + x2 + R(x) is nonresonant. For this, we use some Chelkak-Kargaev-Korotyaev's results about the inverse spectral problem of harmonic oscillator and construct explicit dual basis of the squared Hermite functions. Sur R d , nous considerons l'operateur differentiel −� + jj xjj + R(x) +