Revisiting the Weak Coupling Phenomenon for Two-Dimensional Schr\"odinger Operators

TL;DR

This paper extends Simon's classical results on negative eigenvalues of 2D Schrödinger operators to more general potentials, exploring the existence and multiplicity of weakly coupled eigenvalues.

Contribution

It broadens the class of potentials for which the existence of negative eigenvalues in 2D Schrödinger operators is established, including more singular and slowly decaying potentials.

Findings

Negative eigenvalues exist under broader potential conditions.

Uniqueness of weakly coupled eigenvalues is lost with more general potentials.

Results generalize Simon's original findings from 1976.

Abstract

We study the existence of negative eigenvalues for two-dimensional Schr\"odinger operators with real-valued potentials in the weak coupling regime. In his pioneering paper [Simon 1976] from half a century ago, Simon was the first to describe the unique negative eigenvalue emerging from the threshold of the essential spectrum of one- and two-dimensional Schr\"odinger operators. The aim of this paper is to extend Simon's results in two dimensions to a broader class of potentials, allowing for both stronger singularities and slower decay at infinity, at the cost of losing uniqueness of weakly coupled eigenvalues.