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

Pavel Exner
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Abstract

We investigate the operator $-Δ-αδ(x-Γ)$ in $L^2(\mathbb{R}^3)$, where $Γ$ 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 $α\to\infty$ which involves a ``two-dimensional'' comparison operator determined by the geometry of the surface $Γ$. 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 $Γ$ decomposes into compact connected components. Finally, we comment on analogous systems of lower dimension and other aspects of the problem.

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Pavel Exner. 2003-01-15. Spectral properties of Schroedinger operators with a strongly attractive delta interaction supported by a surface. https://arxiv.org/abs/math-ph/0301021

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