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

TL;DR

This paper derives the asymptotic behavior of eigenvalues for three-dimensional Schrödinger operators with strong attractive interactions supported on smooth finite curves, linking them to a one-dimensional operator involving curvature.

Contribution

It provides a precise asymptotic expansion for eigenvalues of Schrödinger operators with singular interactions supported on smooth curves in three dimensions, connecting spectral properties to geometric curvature.

Findings

Eigenvalues asymptotically behave as specified by the derived expansion.

The leading term involves an exponential function of the coupling parameter.

Eigenvalues relate to a one-dimensional Schrödinger operator with curvature-dependent potential.

Abstract

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 $\Gamma$.