Loi de Weyl presque sure pour un systeme differentiel en dimension 1

TL;DR

This paper proves a Weyl law almost surely for eigenvalues of general differential operators on the circle with small random perturbations, in both semiclassical and high energy limits, indicating predictable eigenvalue distribution.

Contribution

It establishes almost sure Weyl laws for eigenvalues of perturbed differential operators on the circle in both semiclassical and high energy regimes, extending classical spectral results.

Findings

Eigenvalues inside a subdomain follow a Weyl law with high probability

Large eigenvalues obey a Weyl law almost surely

Results hold in both semiclassical and high energy limits

Abstract

We consider quite general differential operators on the circle with a small random lower order perturbation. We embrace two points a view, the semiclassical and the high energy limits. We show (a) in the semiclassical limit, that the eigenvalues inside a subdomain of the pseudospectrum are distributed according to a Weyl law with a probability close to 1, (b) that the large eigenvalues obey a Weyl law almost surely.