Random weighted Sobolev inequalities and application to quantum ergodicity

Didier Robert (1) , Laurent Thomann (1)

1 LMJL - Laboratoire de Mathématiques Jean Leray

This paper is a continuation of Poiret-Robert-Thomann (2013) where we studied a randomisation method based on the Laplacian with harmonic potential. Here we extend our previous results to the case of any polynomial and confining potential $V$ on $\mathbb{R}^d$. We construct measures, under concentration type assumptions, on the support of which we prove optimal weighted Sobolev estimates on $\mathbb{R}^d$. This construction relies on accurate estimates on the spectral function in a non-compact configuration space. Then we prove random quantum ergodicity results without specific assumption on the classical dynamics. Finally, we prove that almost all basis of Hermite functions is quantum uniquely ergodic.

Mots clés: concentration of measure; quantum unique ergodicity; spectral analysis; quantum ergodicity; Schrödinger operator with confining potential

Domaines: Equations aux dérivées partielles [math.AP]; Physique mathématique [math-ph]; Théorie spectrale [math.SP]; Mathématiques [math]

Didier Robert, Laurent Thomann. Random weighted Sobolev inequalities and application to quantum ergodicity. Communications in Mathematical Physics, 2015, 335 (3), pp.1181--1209. ⟨hal-00919443v3⟩