Strong coupling asymptotics for δ-interactions supported by curves with cusps

Abstract

Let Γ⊂R2 be a simple closed curve which is smooth except at the origin, at which it has a power cusp and coincides with the curve |x2|=x1p for some p>1. We study the eigenvalues of the Schrödinger operator Hα with the attractive δ-potential of strength α>0 supported by Γ, which is defined by its quadratic formH1(R2)∋u↦∬R2|∇u|2dx−α∫Γu2ds, where ds stands for the one-dimensional Hausdorff measure on Γ. It is shown that if n∈N is fixed and α is large, then the well-defined nth eigenvalue En(Hα) of Hα behaves asEn(Hα)=−α2+22p+2Enα6p+2+O(α6p+2−η), where the constants En>0 are the eigenvalues of an explicitly given one-dimensional Schrödinger operator determined by the cusp, and η>0. Both main and secondary terms in this asymptotic expansion are different from what was observed previously for the cases when Γ is smooth or piecewise smooth with non-zero angles.