Spectral asymptotics of a strong ${\delta }^{\mathrm{\prime }}$ interaction on a planar loop

We consider a generalized Schr"odinger operator in $L^2({\mathbb{R}}^2)$ with an attractive strongly singular interaction of ${\delta }^{\mathrm{\prime }}$ type characterized by the coupling parameter $\beta >0$ and supported by a $C^4$-smooth closed curve $\mathrm{\Gamma }$ of length $L$ without self-intersections. It is shown that in the strong coupling limit, $\beta \to 0_{+}$, the number of eigenvalues behaves as $\frac{2L}{\pi \beta }+\text{OO}(\mathrm{\mid }\ln \beta \mathrm{\mid })$, and furthermore, that the asymptotic behaviour of the $j$-th eigenvalue in the same limit is $-\frac{4}{\beta^2} +\mu_j+\OO(\beta|\ln\beta|)$,where$\mu_j$isthe$j$-th eigenvalue of the Schr"odinger operator on $L^2(0,L)$ with periodic boundary conditions and the potential $-\frac{1}{4}{\gamma }^2$ where $\gamma$ is the signed curvature of $\mathrm{\Gamma }$.