Pavel Exner and Sylwia Kondej
Strong-coupling asymptotic expansion for Schr\"odinger operators
with a singular interaction supported by a curve in $\mathbb{R}^3$
(74K, LaTeX)
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
strong-coupling 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
one-dimensional 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.