Strong coupling asymptotics for Schrödinger operators with an interaction supported by an open arc in three dimensions

Abstract

We consider Schrödinger operators with a strongly attractive singular interaction supported by a finite curve $\Gamma$ of lenghth $L$ in ${\mathbb{R}}^3$. 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 Schrödinger operator $S=-\frac{\D^2}{\D s^2 }- \frac14 γ^2(s)$ on $L^2(0,L)$ with Dirichlet condition at the interval endpoints in which $\gamma$ is the curvature of $\Gamma$.

17 pages, no figures

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• Singular Schrödinger operators and Robin billiards. Spectral properties and asymptotic expansions
• A straight waveguide with a wire inducing resonances
• Weakly coupled bound state of 2D Schrödinger operator with potential-measure