Random weighted Sobolev inequalities on $\mathbb{R}^d$ and application to Hermite functions

Aurélien Poiret (1) , Didier Robert (2) , Laurent Thomann (2)

1 LM-Orsay - Laboratoire de Mathématiques d'Orsay
2 LMJL - Laboratoire de Mathématiques Jean Leray

We extend a randomisation method, introduced by Shiffman-Zelditch and developed by Burq-Lebeau on compact manifolds for the Laplace operator, to the case of $\mathbb{R}^d$ with the harmonic oscillator. We construct measures, thanks to probability laws which satisfy the concentration of measure property, 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. As an application, we show that there exists a basis of Hermite functions with good decay properties in $L^{\infty}(\mathbb{R}^d)$, when $d\geq 2$.

Mots clés: Harmonic oscillator; spectral analysis; concentration of measure; Hermite functions

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

Aurélien Poiret, Didier Robert, Laurent Thomann. Random weighted Sobolev inequalities on $\mathbb{R}^d$ and application to Hermite functions. Annales Henri Poincaré, 2015, 16 (2), pp.651--689. ⟨hal-00846154v2⟩