Weyl law for semi-classical resonances with randomly perturbed potentials

Sjöstrand, Johannes

Mémoires de la Société Mathématique de France, no. 136 (2014) , 150 p.

Abstract
Résumé

We consider semi-classical Schrödinger operators with potentials supported in a bounded strictly convex subset $𝒪$ of $ℝ^{n}$ with smooth boundary. Letting $h$ denote the semi-classical parameter, we consider classes of small random perturbations and show that with probability very close to 1, the number of resonances in rectangles $[a, b] - i [0, c h^{\frac{2}{3}} [$ , is equal to the number of eigenvalues in $[a, b]$ of the Dirichlet realization of the unperturbed operator in $𝒪$ up to a small remainder.

On considère des opérateurs de Schrödinger dont les potentiels ont leur supports dans un ensemble strictement convexe à bord lisse $𝒪 ⋐ ℝ^{n}$ . En désignant par $h$ le paramètre semi-classique, nous considérons des classes de petites perturbations aléatoires et montrons qu’avec une probabilité très proche de 1, le nombre de résonances dans des rectangles $[a, b] - i [0, c h^{\frac{2}{3}} [$ est égal (à un petit reste près) au nombre de valeurs propres dans $[a, b]$ de la réalisation de Dirichlet de l’opérateur dans $𝒪$ .

MR Zbl

DOI: 10.24033/msmf.446

Classification: 81U99, 35P20, 35P25
Keywords: Resonance, Weyl law, Random
Mot clés : résonance, loi de Weyl, aléatoire

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Sjöstrand, Johannes. Weyl law for semi-classical resonances with randomly perturbed potentials. Mémoires de la Société Mathématique de France, Serie 2, no. 136 (2014), 150 p. doi : 10.24033/msmf.446. http://numdam.org/item/MSMF_2014_2_136__1_0/

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