Pavel Exner and Michal Jex
Spectral asymptotics of a strong $\delta'$ interaction on a planar loop
(174K, pdf)
ABSTRACT. We consider a generalized Schr\"odinger operator in $L^2(\R^2)$ with an attractive strongly singular interaction of $\delta'$ type characterized by the coupling parameter $eta>0$ and supported by a $C^4$-smooth closed curve $\Gamma$ of length $L$ without self-intersections. It is shown that in the strong coupling limit, $eta o 0_+$, the number of eigenvalues behaves as $rac{2L}{\pieta} + \mathcal{O}(|\lneta|)$, and furthermore, that the asymptotic behaviour of the $j$-th eigenvalue in the same limit is $-rac{4}{eta^2} +\mu_j+\mathcal{O}(eta|\lneta|)$, where $\mu_j$ is the $j$-th eigenvalue of the Schr\"odinger operator on $L^2(0,L)$ with periodic boundary conditions and the potential $-rac14 \gamma^2$ where $\gamma$ is the signed curvature of $\Gamma$.