Spectral properties of Schroedinger operators with a strongly attractive delta interaction supported by a surface

TL;DR

This paper analyzes the spectral behavior of Schrödinger operators with a strongly attractive delta interaction supported on a surface, providing asymptotic expansions and geometric insights for large interaction strength.

Contribution

It introduces asymptotic formulas for the spectrum of such operators, linking spectral properties to the geometry of the supporting surface, including cases of compact and periodic surfaces.

Findings

Asymptotic expansion for the lower spectrum as interaction strength grows

Spectral analysis for both compact and periodic surfaces

Bandwidth estimates for periodic surface decompositions

Abstract

We investigate the operator $-\Delta -\alpha \delta (x-\Gamma)$ in $L^2(\mathbb{R}^3)$, where $\Gamma$ is a smooth surface which is either compact or periodic and satisfies suitable regularity requirements. We find an asymptotic expansion for the lower part of the spectrum as $\alpha\to\infty$ which involves a ``two-dimensional'' comparison operator determined by the geometry of the surface $\Gamma$. In the compact case the asymptotics concerns negative eigenvalues, in the periodic case Floquet eigenvalues. We also give a bandwidth estimate in the case when a periodic $\Gamma$ decomposes into compact connected components. Finally, we comment on analogous systems of lower dimension and other aspects of the problem.