On random weighted Sobolev inequalities on R^d and applications

Didier Robert (1) , Laurent Thomann (1)

1 LMJL - Laboratoire de Mathématiques Jean Leray

Our aim in this paper is to give an overview of recent results concerning randomization methods applied to Schrödinger equations obtained in [33, 32, 35]. We focus here on properties of the harmonic oscillator and Her-mite functions for simplicity. The general idea is that several well known deterministic results like Sobolev type inequalities are improved in a large extent by introducing some random-ness. This is very useful in particular for N.L.S with supercritical initial data.

Mots clés: Harmonic oscillators; Spectral analysis of operators; Concentration of measure phenomenon; Quantum ergodicity; NLS with random initial states

Domaines: Physique mathématique [math-ph]

Didier Robert, Laurent Thomann. On random weighted Sobolev inequalities on R^d and applications. Spectral Theory and Partial Differential Equations, 640, American Mathematical Society, pp.115--135, 2015, 978-1-4704-0989-0. ⟨10.1090/conm/640/12843⟩. ⟨hal-01620210⟩