Spectral analysis of a complex Schr\"odinger operator in the semiclassical limit

TL;DR

This paper studies the spectral properties of a complex Schrödinger operator in the semi-classical limit, providing detailed asymptotic expansions of eigenvalues in one dimension and spectral bounds in two dimensions.

Contribution

It offers the first complete asymptotic expansion of eigenvalues for the operator in one dimension and characterizes the spectrum's left margin in two dimensions.

Findings

Complete eigenvalue asymptotic expansion in 1D

Identification of the spectrum's left margin in 2D

Analysis under smooth potential with no critical points

Abstract

We consider the Dirichlet realization of the operator $-h^2\Delta+iV$ in the semi-classical limit $h\to0$, where $V$ is a smooth real potential with no critical points. For a one dimensional setting, we obtain the complete asymptotic expansion, in powers of $h$, of each eigenvalue. In two dimensions we obtain the left margin of the spectrum, under some additional conditions.