On eigenvalue asymptotics for strong delta-interactions supported by surfaces with boundaries

摘要: Let $S\subset\mathbb{R}^3$ be a $C^4$-smooth relatively compact orientable surface with sufficiently regular boundary. For $\beta\in\mathbb{R}_+$, let $E_j(\beta)$ denote the $j$th negative eigenvalue of operator associated quadratic form \[ H^1(\mathbb{R}^3)\ni u\mapsto \iiint_{\mathbb{R}^3} |\nabla u|^2dx -\beta \iint_S |u|^2d\sigma, \] where $\sigma$ is two-dimensional Hausdorff measure on $S$. We show that for each fixed $j$ one has asymptotic expansion E_j(\beta)=-\dfrac{\beta^2}{4}+\mu^D_j+ o(1) \;\text{ as }\; \beta\to+\infty\,, $\mu_j^D$ $-\Delta_S+K-M^2$ $L^2(S)$, in which $K$ and $M$ are Gauss mean curvatures, respectively, $-\Delta_S$ Laplace-Beltrami Dirichlet condition at boundary If, addition, $S$ $C^2$-smooth, then remainder estimate can improved to ${\mathcal O}(\beta^{-1}\log\beta)$.