The spectrum of a Schr\"odinger operator in a wire-like domain with a purely imaginary degenerate potential in the semiclassical limit

TL;DR

This paper analyzes the spectral properties of a Schrödinger operator with a purely imaginary, degenerate potential in a wire-like domain, providing asymptotic and resolvent estimates in the semiclassical limit.

Contribution

It extends previous spectral analysis results to cases where the potential is constant along the boundary, unlike earlier models with discrete current sets.

Findings

Asymptotic behavior of the spectrum's left margin determined.

Resolvent estimates established on the spectrum's left side.

Extension of spectral results to boundary-constant potentials.

Abstract

Consider a two-dimensional domain shaped like a wire, not necessarily of uniform cross section. Let $V$ denote an electric potential driven by a voltage drop between the conducting surfaces of the wire. We consider the operator ${\mathcal A}_h=-h^2\Delta+iV$ in the semi-classical limit $h\to 0$. We obtain both the asymptotic behaviour of the left margin of the spectrum, as well as resolvent estimates on the left side of this margin. We extend here previous results obtained for potentials for which the set where the current (or $\nabla V$) is normal to the boundary is discrete, in contrast with the present case where $V$ is constant along the conducting surfaces.