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

TL;DR

This paper compares two recent results on eigenvalue distributions for semi-classical non-self-adjoint operators in two dimensions, highlighting differences in the associated eigenvalue densities.

Contribution

It demonstrates that the eigenvalue densities from the complex Bohr-Sommerfeld rule and Weyl asymptotics generally differ, clarifying their relationship in semi-classical analysis.

Findings

Eigenvalue distributions are governed by two different smooth densities.

The densities from the Bohr-Sommerfeld rule and Weyl asymptotics are generally not the same.

The results connect eigenvalue distribution patterns with different semi-classical approaches.

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 obtained a complex Bohr-Sommerfeld rule, showing that the eigenvalues are situated on a distorted lattice. On the other hand, with M. Hager 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.