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

Abstract

Let $\mathit{S}\subset {\mathbb{R}}^3$ be a ${\mathit{C}}^4$ -smooth relatively compact orientable surface with a sufficiently regular boundary. For $\mathit{\beta }\in {\mathbb{R}}_{+}$ , let ${\mathit{E}}_{\mathit{j}}\mathrm{(}\mathit{\beta }\mathrm{)}$ denote the jth negative eigenvalue of the operator associated with the quadratic form $$\begin{array}{c}{\mathit{H}}^1\mathrm{\left(}{\mathbb{R}}^3\mathrm{\right)}\ni \mathit{u}\mapsto {\iiint }_{{\mathbb{R}}^3}|\nabla \mathit{u}{|}^2\mathit{d}\mathit{x}-\mathit{\beta }{\iint }_{\mathit{S}}|\mathit{u}{|}^2\mathit{d}\mathit{\sigma }\mathrm{,}\end{array}$$ where σ is the two-dimensional Hausdorff measure on S. We show that for each fixed j one has the asymptotic expansion $$\begin{array}{c}{\mathit{E}}_{\mathit{j}}\mathrm{(}\mathit{\beta }\mathrm{)}=-{\displaystyle \frac{{\mathit{\beta }}^2}{4}}+{\mathit{\mu }}_{\mathit{j}}^{\mathit{D}}+\mathit{o}\mathrm{(}1\mathrm{)}\text{as }\mathit{\beta }\to +\infty \mathrm{,}\end{array}$$ where ${\mathit{\mu }}_{\mathit{j}}^{\mathit{D}}$ is the jth eigenvalue of the operator $-{\mathrm{\Delta }}_{\mathit{S}}+\mathit{K}-{\mathit{M}}^2$ on ${\mathit{L}}^2\mathrm{(}\mathit{S}\mathrm{)}$ , in which K and M are the Gauss and mean curvatures, respectively, and $-{\mathrm{\Delta }}_{\mathit{S}}$ is the Laplace–Beltrami operator with the Dirichlet condition at the boundary of S. If, in addition, the boundary of S is ${\mathit{C}}^2$ -smooth, then the remainder estimate can be improved to $\mathcal{O}\mathrm{(}{\mathit{\beta }}^{-1}log\mathit{\beta }\mathrm{)}$ .