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

Abstract

Let $\Gamma \subset {\mathbb{R}}^2$ be a simple closed curve which is smooth except at the origin, at which it has a power cusp and coincides with the curve $|x_2|=x_1^p$ for some $p>1$. We study the eigenvalues of the Schrödinger operator $H_{\alpha }$ with the attractive $\delta$-potential of strength $\alpha >0$ supported by $\Gamma$, which is defined by its quadratic form \[ H^1(\mathbb{R}^2)\ni u\mapsto \iint_{\mathbb{R}^2} |\nabla u|^2\,\mathrm{d}x-α\int_Γu^2\, \mathrm{d}s, \] where $\mathrm{d}s$ stands for the one-dimensional Hausdorff measure on $\Gamma$. It is shown that if $n\in \mathbb{N}$ is fixed and $\alpha$ is large, then the well-defined $n$th eigenvalue $E_n(H_{\alpha })$ of $H_{\alpha }$ behaves as \[ E_n(H_α)=-α^2 + 2^{\frac{2}{p+2}} \mathcal{E}_n \,α^{\frac{6}{p+2}} + \mathcal{O}(α^{\frac{6}{p+2}-η}), \] where the constants ${\mathcal{E}}_n>0$ are the eigenvalues of an explicitly given one-dimensional Schrödinger operator determined by the cusp, and $\eta >0$. Both main and secondary terms in this asymptotic expansion are different from what was observed previously for the cases when~$\Gamma$ is smooth or piecewise smooth with non-zero angles.

27 pages