Spectral semi-classical analysis of a complex Schr\"odinger operator in exterior domains

TL;DR

This paper extends spectral analysis of complex Schrödinger operators with imaginary potentials to exterior domains, identifying the spectrum's left margin and conditions for the essential spectrum's absence in the semiclassical limit.

Contribution

It generalizes previous bounded domain results to exterior domains, providing new spectral bounds and conditions for the essential spectrum in semiclassical analysis.

Findings

Determined the left margin of the spectrum in exterior domains.

Proved the essential spectrum is empty under certain conditions.

Extended spectral analysis techniques to unbounded exterior domains.

Abstract

Generalizing previous results obtained for the spectrum of the Dirichlet and Neumann realizations in a bounded domain of a Schr\"odinger operator with a purely imaginary potential $h^2\Delta+iV$ in the semiclassical limit $h\to 0$ we address the same problem in exterior domains. In particular we obtain the left margin of the spectrum, and the emptiness of the essential part of the spectrum under some additional assumptions.