Eigenvalue asymptotics for randomly perturbed non-selfadjoint operators

Abstract

We consider quite general h-pseudodifferential operators on R n with small random perturbations and show that in the limit h → 0 the eigenvalues are distributed according to a Weyl law with a probabality that tends to 1. The first author has previously obtained a similar result in dimension 1. Our class of perturbations is different. RésuméNous considérons des opérateurs h-pseudodifférentiels assez généraux et nous montrons que dans la limite h → 0, les valeurs propres se distribuent selon une loi de Weyl, avec une probabilité qui tend vers 1. Le premier auteur a déjà obtenu un résultat semblable en dimension 1. Notre classe de perturbations est différente.

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

“…In [6] the results were generalized to higher dimension and the boundary of the range of p could be included, but the perturbations where no more multiplicative. In [9,10] further improvements of the method were introduced and the case of multiplicative perturbations was handled in all dimensions.…”

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

“…Due to the ellipticity assumption (1.6) and the growth condition on the weigth m, one can show the following result by an argument based on a compact deformation of Fredholm operators, see [15,17].…”

“…Macroscopic spectral distribution: A probabilistic Weyl law. In a series of works by Hager [16,15,17] and Sjöstrand [25,24], the authors considered randomly perturbed operators P δ of the types given in (1.16) and (1.17). Under more restrictive assumptions on the random variables, than (1.14), they have shown the following result.…”

“…Hager [16,15,17] and Sjöstrand [25,24] give explicit control over both, the error term in the Weyl law, and the error term in the probability estimate. Similar results have also been obtained by Christiansen and Zworski [5], Bordeaux-Montrieux [2] and BordeauxMontrieux and Sjöstrand [3].…”

“…In this section we recall the construction of the Grushin problem. The following has been taken from [35], however, the ideas go back to [15,17]. As reviewed in [30], the central idea is to set up an auxiliary problem of the form…”