 03110 Pavel Exner and Sylwia Kondej
 Strongcoupling asymptotic expansion for Schr\"odinger operators
with a singular interaction supported by a curve in $\mathbb{R}^3$
(74K, LaTeX)
Mar 13, 03

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Abstract. We investigate a class of generalized
Schr\"{o}dinger operators in $L^2(\mathbb{R}^3)$ with a singular
interaction supported by a smooth curve $\Gamma$. We find a
strongcoupling asymptotic expansion of the discrete spectrum in
case when $\Gamma$ is a loop or an infinite bent curve which is
asymptotically straight. It is given in terms of an auxiliary
onedimensional Schr\"{o}dinger operator with a potential
determined by the curvature of $\Gamma$. In the same way we obtain
an asymptotics of spectral bands for a periodic curve. In
particular, the spectrum is shown to have open gaps in this case
if $\Gamma$ is not a straight line and the singular interaction is
strong enough.
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