Eigenvalue asymptotics for the Schrödinger operator with a $\delta$-interaction on a punctured surface

Abstract

Given $n\ge 2$, we put $r=\min \{i\in \mathbb{N};i>n/2\}$. Let $\Sigma$ be acompact, $C^r$-smooth surface in ${\mathbb{R}}^n$ which contains the origin. Let further $\{S_{\epsilon }{\}}_{0\le \epsilon <\eta }$ be a family of measurable subsets of $\Sigma$ such that ${\sup }_{x\in S_{\epsilon }}|x|=\mathcal{O}(\epsilon )$ as $\epsilon \to 0$. We derive an asymptotic expansion for the discrete spectrum of the Schr{ö}dinger operator $-\Delta -\beta \delta (\cdot -\Sigma \setminus S_{\epsilon })$ in $L^2({\mathbb{R}}^n)$, where $\beta$ is a positive constant, as $\epsilon \to 0$. An analogous result is given also for geometrically induced bound states due to a $\delta$ interaction supported by an infinite planar curve.

LaTeX 2e, 10 pages; an error in the proof corrected

References in corpus (6)

• Geometrically induced spectrum in curved leaky wires
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• Curvature-induced bound states for a $\delta$ interaction supported by a curve in ${\mathbb{R}}^3$
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• Band gap of the Schroedinger operator with a strong delta-interaction on a periodic curve
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Cited by in corpus (7)

• Schrödinger operators with delta and delta'-potentials supported on hypersurfaces
• Self-adjoint elliptic operators with boundary conditions on not closed hypersurfaces
• Scattering by local deformations of a straight leaky wire
• Schroedinger operators with singular interactions: a model of tunneling resonances
• Hiatus perturbation for a singular Schrödinger operator with an interaction supported by a curve in \mathbb{R}^3
• Generalized eigenfunctions and spectral theory for strongly local Dirichlet forms
• Rank One Perturbations Supported by Hybrid Geometries and Their Deformations