Random Weighted Sobolev Inequalities and Application to Quantum Ergodicity

Abstract

This paper is a continuation of Poiret et al. (Ann Henri Poincaré 16:651–689, 2015), 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 bases of Hermite functions are quantum uniquely ergodic.