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

Abstract

Let Γ ⊂ R 2 be a simple closed curve which is smooth except at the origin, at which it has a power cusp and coincides with the curve |x 2 | = x p 1 for some p > 1. We study the eigenvalues of the Schrödinger operator H α with the attractive δ-potential of strength α > 0 supported by Γ, which is defined by its quadratic formwhere 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 welldefined nth eigenvalue E n (H α ) of H α behaves as, where the constants E n > 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.with δ being the Dirac distribution and α > 0 being the coupling constant. Such operators describe the motion of particles confined to the graph Γ but allowing for a quantum tunneling between its different