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

Let $\mathrm{\Gamma }\subset {\mathbb{R}}^2$ be a simple closed curve which is smooth except at the origin, at which it has a power cusp and coincides with the curve $\mathrm{\mid }{x}_2\mathrm{\mid }=x_1^p$ for some $p>1$. We study the eigenvalues of the Schr"odinger operator $H_{\alpha }$ with the attractive $\delta$-potential of strength $\alpha >0$ supported by $\mathrm{\Gamma }$, which is defined by its quadratic form $$H^1({\mathbb{R}}^2)\ni u\mapsto {\iint }_{{\mathbb{R}}^2}\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^2\mathrm{d}x-\alpha {\int }_{\mathrm{\Gamma }}{u}^2\mathrm{d}s,$$ where $\mathrm{d}s$ stands for the one-dimensional Hausdorff measure on $\mathrm{\Gamma }$. 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 $$E_n(H_{\alpha })=-{\alpha }^2+2^{\frac{2}{p+2}}{\mathcal{E}}_n{\alpha }^{\frac{6}{p+2}}+\mathcal{O}({\alpha }^{\frac{6}{p+2}-\eta }),$$ where the constants ${\mathcal{E}}_n>0$ are the eigenvalues of an explicitly given one-dimensional Schr"odinger 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~$\mathrm{\Gamma }$ is smooth or piecewise smooth with non-zero angles.