Pavel Exner and Sylwia Kondej
Strong coupling asymptotics for Schr\"odinger operators with an interaction supported by an open arc in three dimensions
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ABSTRACT. We consider Schr\"odinger operators with a strongly
attractive singular interaction supported by a finite curve
$\Gamma$ of lenghth $L$ in $\R^3$. We show that if $\Gamma$ is
$C^4$-smooth and has regular endpoints, the $j$-th eigenvalue of
such an operator has the asymptotic expansion $\lambda_j
(H_{lpha,\Gamma})= \xi_lpha +\lambda
_j(S)+\mathcal{O}(\mathrm{e}^{\pi lpha })$ as the coupling
parameter $lpha o\infty$, where $\xi_lpha =
-4\,\mathrm{e}^{2(-2\pilpha +\psi(1))}$ and $\lambda _j(S)$ is
the $j$-th eigenvalue of the Schr\"odinger operator
$S=-rac{\D^2}{\D s^2 }- rac14 \gamma^2(s)$ on $L^2(0,L)$ with
Dirichlet condition at the interval endpoints in which $\gamma$ is
the curvature of $\Gamma$.