Strong coupling asymptotics for a singular Schroedinger operator with an interaction supported by an open arc

TL;DR

This paper analyzes the asymptotic behavior of the negative eigenvalues of a singular Schrödinger operator with a delta interaction supported on a smooth open arc in the plane, in the strong coupling limit.

Contribution

It provides the first detailed asymptotic expansion of eigenvalues for such operators with curved support in the strong-coupling regime.

Findings

Eigenvalues behave as -β^2/4 plus a curvature-dependent correction

Asymptotic expansion includes a logarithmic term in β

Eigenvalues relate to a 1D Schrödinger operator with curvature potential

Abstract

We consider a singular Schr\"odinger operator in $L^2(\mathbb{R}^2)$ written formally as $-\Delta - \beta\delta(x-\gamma)$ where $\gamma$ is a $C^4$ smooth open arc in $\mathbb{R}^2$ of length $L$ with regular ends. It is shown that the $j$th negative eigenvalue of this operator behaves in the strong-coupling limit, $\beta\to +\infty$, asymptotically as \[ E_j(\beta)=-\frac{\beta^2}{4} +\mu_j +\mathcal{O}\Big(\dfrac{\log\beta}{\beta}\Big), \] where $\mu_j$ is the $j$th Dirichlet eigenvalue of the operator \[ -\frac{d^2}{ds^2} -\frac{\kappa(s)^2}{4}\, \] on $L^2(0,L)$ with $\kappa(s)$ being the signed curvature of $\gamma$ at the point $s\in(0,L)$.