Time averaging for the strongly confined nonlinear Schrödinger equation, using almost- periodicity

Naoufel Ben Abdallah (1) , François Castella (2) , Florian Méhats (2)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 IRMAR - Institut de Recherche Mathématique de Rennes

We study the limiting behavior of a nonlinear Schrödinger equation describing a 3-dimensional gas that is strongly confined along the vertical, z direction. The confinement induces fast oscillations in time, that need to be averaged out. Since the Hamiltonian in the z direction is merely assumed confining, without any further specification, the associated spectrum is discrete but arbitrary, and the fast oscillations induced by the nonlinear equation entail countably many frequencies that are arbitrarily distributed. For that reason, averaging cannot rely on small denominator estimates or like. To overcome these difficulties, we prove that the fast oscillations are almost-periodic in time, with values in a Sobolev-like space that we completely identify. We then exploit the existence of long-time averages for almost-periodic functions to perform the necessary averaging procedure in our nonlinear problem.

Mots clés: Adiabatic approximation; Sobolev scale associated with a self-adjoint operator; Error estimates; Nonlinear analysis; Two-dimensional electron gas; Almost-periodic functions

Naoufel Ben Abdallah, François Castella, Florian Méhats. Time averaging for the strongly confined nonlinear Schrödinger equation, using almost- periodicity. Journal of Differential Equations, 2008, 245 (1), pp.154-200. ⟨hal-00359082⟩