Hiatus perturbation for a singular Schrödinger operator with an interaction supported by a curve in \mathbb{R}^3

Abstract

We consider Schrödinger operators in $L^2({\mathbb{R}}^3)$ with a singular interaction supported by a finite curve $\Gamma$. We present a proper definition of the operators and study their properties, in particular, we show that the discrete spectrum can be empty if $\Gamma$ is short enough. If it is not the case, we investigate properties of the eigenvalues in the situation when the curve has a hiatus of length $2\epsilon$. We derive an asymptotic expansion with the leading term which a multiple of $\epsilon \ln \epsilon$.

LaTeX, 29 pages

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• Schrödinger operators with δ- and δ'-interactions on Lipschitz surfaces and chromatic numbers of associated partitions
• Asymptotics of the bound state induced by $\delta$-interaction supported on a weakly deformed plane
• Spectral theory for Schrödinger operators with $\delta$-interactions supported on curves in ${\mathbb{R}}^3$
• On absence of bound states for weakly attractive ${\delta }^{\prime }$-interactions supported on non-closed curves in ${\mathbb{R}}^2$
• Singular interactions supported by embedded curves
• A Klein Gordon Particle Captured by Embedded Curves
• Infinitely many singular interactions on noncompact manifolds