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arXiv · 2604.19284

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

Jussi Behrndt·Petr Siegl·Nicolas Weber
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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.

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Jussi Behrndt, Petr Siegl, Nicolas Weber. 2026-04-21. Revisiting the Weak Coupling Phenomenon for Two-Dimensional Schr\"odinger Operators. https://arxiv.org/abs/2604.19284

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