Abstract
Given n≥2, we put r=min\(\left\{ {i \in \mathbb{N};i > n/2} \right\}\). Let Σ be a compact, C r-smooth surface in ℝn which contains the origin. Let further \(\left\{ {S_\varepsilon } \right\}_{0 \leqslant \varepsilon < \eta } \) be a family of measurable subsets of Σ such that \(\sup _{x \in S_\varepsilon } |x| = \mathcal{O}(\varepsilon )\) as \(\varepsilon \to {\text{0}}\). We derive an asymptotic expansion for the discrete spectrum of the Schrödinger operator \( - \Delta - \beta \delta \left( { \cdot - \sum \backslash S_\varepsilon } \right)\) in L 2(ℝn), where β is a positive constant, as \(\varepsilon \to {\text{0}}\). An analogous result is given also for geometrically induced bound states due to a δ interaction supported by an infinite planar curve.
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Exner, P., Yoshitomi, K. Eigenvalue Asymptotics for the Schrödinger Operator with a δ-Interaction on a Punctured Surface. Letters in Mathematical Physics 65, 19–26 (2003). https://doi.org/10.1023/A:1027367605285
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DOI: https://doi.org/10.1023/A:1027367605285