Strong coupling asymptotics for Schr\"odinger operators with an interaction supported by an open arc in three dimensions

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 $\lambda_j (H_{\alpha,\Gamma})= \xi_\alpha +\lambda _j(S)+\mathcal{O}(\mathrm{e}^{\pi \alpha })$ as the coupling parameter $\alpha\to\infty$, where $\xi_\alpha = -4\,\mathrm{e}^{2(-2\pi\alpha +\psi(1))}$ and $\lambda _j(S)$ is the $j$-th eigenvalue of the Schr\"odinger operator $S=-\frac{\D^2}{\D s^2 }- \frac14 \gamma^2(s)$ on $L^2(0,L)$ with Dirichlet condition at the interval endpoints in which $\gamma$ is the curvature of $\Gamma$.