Non-accretive Schr\"odinger operators and exponential decay of their eigenfunctions

TL;DR

This paper studies non-self-adjoint electromagnetic Schr"odinger operators with complex potentials, establishing conditions under which eigenfunctions decay exponentially, extending understanding of spectral properties in non-Hermitian quantum mechanics.

Contribution

It introduces a Dirichlet realization for these operators and proves exponential decay of eigenfunctions under new sufficient conditions.

Findings

Eigenfunctions exhibit Agmon-type exponential decay.

The Dirichlet realization has a non-empty resolvent set.

Conditions on the electromagnetic potential ensure spectral properties.

Abstract

We consider non-self-adjoint electromagnetic Schr\"odinger operators on arbitrary open sets with complex scalar potentials whose real part is not necessarily bounded from below. Under a suitable sufficient condition on the electromagnetic potential, we introduce a Dirichlet realisation as a closed densely defined operator with non-empty resolvent set and show that the eigenfunctions corresponding to discrete eigenvalues satisfy an Agmon-type exponential decay.