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 Schr\"odinger operator $H_\alpha$ with the attractive $\delta$-potential of strength $\alpha>0$ supported by $\Gamma$, which is defined by its quadratic form \[ H^1(\mathbb{ R}^2)\ni u\mapsto \iint_{ \mathbb{ R}^2} |\nabla u|^2\,\mathrm{ d}x-\alpha\int_\Gamma u^2\, \mathrm{ d}s, \] 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 \[ E_n(H_\alpha)=-\alpha^2 + 2^{ \frac{ 2}{ p+2}} \mathcal{ E}_n \,\alpha^{ \frac{ 6}{ p+2}} + \mathcal{ O}(\alpha^{ \frac{ 6}{ p+2}-\eta}), \] 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~$\Gamma$ is smooth or piecewise smooth with non-zero angles.
Categories: math.SP math-ph math.AP math.MP
