Revisiting the Weak Coupling Phenomenon for Two-Dimensional Schr\"odinger Operators
Spectral Theory
2026-04-22 v1 Analysis of PDEs
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.
Cite
@article{arxiv.2604.19284,
title = {Revisiting the Weak Coupling Phenomenon for Two-Dimensional Schr\"odinger Operators},
author = {Jussi Behrndt and Petr Siegl and Nicolas Weber},
journal= {arXiv preprint arXiv:2604.19284},
year = {2026}
}