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

Details Eigenvalue asymptotics for the Schrödinger operator with a $δ$-interaction on a punctured surface Eigenvalue asymptotics for the Schrödinger operator with a $δ$-interaction on a punctured surface

2003-03-31

arxiv.org/abs/math-ph/0303072v2

Lett. Math. Phys. 65 (2003), 19-26

Physics

Mathematical Physics

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

Scientific paper

10.1023/A:1027367605285

Given $n\geq 2$, 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{\"o}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.

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