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

Random weighted Sobolev inequalities and application to quantum ergodicity