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

Abstract

We consider a singular Schrödinger 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 \[ E_j(β)=-\frac{β^2}{4} +μ_j +\mathcal{O}\Big(\dfrac{\logβ}β\Big), \] where ${\mu }_j$ is the $j$th Dirichlet eigenvalue of the operator \[ -\frac{d^2}{ds^2} -\frac{κ(s)^2}{4}\, \] on $L^2(0,L)$ with $\kappa (s)$ being the signed curvature of $\gamma$ at the point $s\in (0,L)$.

19 pages

References in corpus (3)

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