 03144 Pavel Exner and Kazushi Yoshitomi
 Eigenvalue asymptotics for the Schr\"odinger operator
with a $\delta$interaction on a punctured surface
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Mar 31, 03

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Abstract. Given $n\geq 2$, we put $r=\min\{\,i\in\mathbb{N};\: i>n/2\,\}$. Let
$\Sigma$ be a compact, $C^{r}$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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