Spectral asymptotics for $\delta'$ interaction supported by a infinite curve

Abstract

We consider a generalized Schr\"odinger operator in $L^2(\mathbb R^2)$ describing an attractive $\delta'$ interaction in a strong coupling limit. $\delta'$ interaction is characterized by a coupling parameter $\beta$ and it is supported by a $C^4$-smooth infinite asymptotically straight curve $\Gamma$ without self-intersections. It is shown that in the strong coupling limit, $\beta\to 0_+$, the eigenvalues for a non-straight curve behave as $-\frac{4}{\beta^2} +\mu_j+\mathcal O(\beta|\ln\beta|)$, where $\mu_j$ is the $j$-th eigenvalue of the Schr\"odinger operator on $L^2(\mathbb R)$ with the potential $-\frac14 \gamma^2$ where $\gamma$ is the signed curvature of $\Gamma$.