Weak coupling for Schr\"odinger operators with complex potentials

TL;DR

This paper investigates the emergence and properties of discrete eigenvalues near the spectrum threshold for non-self-adjoint Schrödinger operators with complex potentials in weak coupling, extending classical results to non-Hermitian cases.

Contribution

It provides necessary and sufficient conditions for eigenvalue existence, uniqueness, and multiplicity in complex potential Schrödinger operators, extending classical weak coupling theory.

Findings

Derived conditions for eigenvalue existence and absence

Analyzed eigenvalue uniqueness and multiplicity

Extended classical weak coupling results to non-self-adjoint operators

Abstract

We study the discrete eigenvalues emerging from the threshold of the essential spectrum of one or two-dimensional Schr\"odinger operators with complex-valued $ L^p $-potentials in a weak coupling regime. We derive necessary and sufficient conditions on the potential for the existence or absence of discrete eigenvalues in this regime and also analyze their uniqueness and algebraic multiplicity. Our results can be viewed as natural non-self-adjoint extensions of the well-known classical weak coupling phenomenon for self-adjoint Schr\"odinger operators with real-valued potentials going back half a century to Simon's famous paper [Simon 1976].