Eigenvalue Asymptotics for Confining Magnetic Schrödinger Operators with Complex Potentials

Abstract

This article is devoted to the spectral analysis of the electromagnetic Schrödinger operator on the Euclidean plane. In the semiclassical limit, we derive a pseudo-differential effective operator that allows us to describe the spectrum in various situations and appropriate regions of the complex plane. Not only results of the self-adjoint case are proved (or recovered) in the proposed unifying framework, but also new results are established when the electric potential is complex-valued. In such situations, when the non-self-adjointness comes with its specific issues (lack of a “spectral theorem”, resolvent estimates), the analogue of the “low-lying eigenvalues” of the self-adjoint case are still accurately described and the spectral gaps estimated.