Almost sure Weyl asymptotics for non-self-adjoint elliptic operators on compact manifolds

Abstract

International audienc

Spectral statistics of non-selfadjoint operators subject to small random perturbations

“…Similar results have also been obtained by Christiansen and Zworski [5], Bordeaux-Montrieux [2] and BordeauxMontrieux and Sjöstrand [3]. .…”

Lecture notes : Spectral properties of non-self-adjoint operators

“…That spectral instability is not only a nuisance, but can be at the origin of nice and previously unexpected results, will hopefully be clear from the second and main part of these lecture notes, where we shall describe some results about Weyl asymptotics of the distribution of eigenvalues of elliptic operators with small random perturbations. These results have been obtained in recent works by M. Hager [36,37,38], Hager and the author [39], W. Bordeaux Montrieux [10], the author [97,98] and Bordeaux Montriex and the author [11].…”

“…The purpose of this section is to describe the work of Bordeaux Montrieux and the author [11] on the almost sure Weyl asymptotics of the large eigenvalues of elliptic operators on compact manifolds. For simplicity, we treat only the scalar case and the random perturbation is a potential.…”

Probabilistic Weyl Laws for Quantized Tori

“…1 and 2. However, as stressed in [3,13], and [17], the results on Weyl laws for small random perturbations have in themselves nothing to do with spectral instability. For normal operators they do not produce new results compared to the standard semiclassical Weyl laws: the distribution of eigenvalues is not affected by small perturbations and satisfies a Weyl law to start with.…”