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

Abstract

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

18 pages, to be submitted to Asymptotic Analysis