Eigenvalue distributions and Weyl laws for semiclassical non-self-adjoint operators in 2 dimensions

Abstract

In this note we compare two recent results about the distribution of eigenvalues for semi-classical pseudodifferential operators in two dimensions. For classes of analytic operators A. Melin and the author [6] obtained a complex Bohr-Sommerfeld rule, showing that the eigenvalues are situated on a distorted lattice. On the other hand, with M. Hager [4] we showed in any dimension that Weyl asymptotics holds with probability close to 1 for small random perturbations of the operator. In both cases the eigenvalues distribute to leading order according two smooth densities and we show here that the two densities are in general different. RésuméDans cette note nous comparons deux résultats récents sur la distribution des valeurs propres d'opérateurs pseudodifférentiels en deux dimensions. Pour des classes d'opérateurs analytiques A. Melin et l'auteur [6] a obtenu une loi de Bohr-Sommerfeld complexe qui montre que les valeurs propres sont situés sur un réseau déformé. D'autre part, avec M. Hager [4] nous avons montré dans toute dimension que l'asymptotique de Weyl est valable avec probabilité proche de 1 pour des petites perturbations aléatoires de l'opérateur. Dans les deux cas, le valeurs propres sont distribuées (à des petites corrections près) selon des densités lisses, et ici nous montrons que ces densités sont en général différentes.

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

“…Indeed, in the analytic case there is the possibility to make an analytic distorsion (for instance by replacing the underlying compact analytic manifold by a small deformation) which will not change the spectrum (by ellipticity and analyticity) but which will replace the given real phase space by a deformation, likely to change the Weyl law. In one and two dimensions analytic distorsions have been used to determine the spectrum (by making the operator more normal) in the two-dimensional semi-classical case this was done by one of the authors, first with A. Melin, and in [8] it was shown that the resulting law is in general different from the Weyl law (naively because a complex Bohr-Sommerfeld law relies on going out in the complex domain while the Weyl law only uses the real cotangent space).…”

“…-There is a constant C > 0 such that (4.7) holds with probability 8) and the α j are independent. Now, using standard functional calculus for R as in [9,10], we see that…”

Diophantine Tori and Weyl Laws for Non-selfadjoint Operators in Dimension Two

“…In the paper [25], it has been shown that for large and stable classes of non-selfadjoint analytic operators in dimension two, the individual eigenvalues can be determined accurately in the semiclassical limit by means of a Bohr-Sommerfeld quantization condition, defined in terms of suitable complex Lagrangian tori close to the real domain. (See also [35] for the formulation of the corresponding Weyl laws.) The work [25] was subsequently continued in a series of papers [12]- [14], [16], [15], all of them concerned with the case of non-selfadjoint perturbations of selfadjoint operators of the form, P ε (x, hD x ) = p w (x, hD x ) + iεq w (x, hD x ), 0 < ε ≪ 1, with the leading symbol p ε (x, ξ) = p(x, ξ) + iεq(x, ξ), (x, ξ) ∈ T * R 2 .…”