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

TL;DR

This paper derives the asymptotic behavior of eigenvalues for a Schrödinger operator with strong delta interactions supported on smooth surfaces with boundaries, revealing the influence of surface curvature and boundary conditions.

Contribution

It provides the first detailed asymptotic expansion of eigenvalues for delta-interactions on surfaces with boundaries, including curvature effects and improved remainder estimates.

Findings

Eigenvalues asymptotically behave as -β^2/4 plus curvature-dependent terms.

The eigenvalue expansion includes a correction term involving the Laplace-Beltrami operator and curvatures.

Remainder estimates are refined for surfaces with smooth 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 \[ H^1(\mathbb{R}^3)\ni u\mapsto \iiint_{\mathbb{R}^3} |\nabla u|^2dx -\beta \iint_S |u|^2d\sigma, \] where $\sigma$ is the two-dimensional Hausdorff measure on $S$. We show that for each fixed $j$ one has the asymptotic expansion \[ E_j(\beta)=-\dfrac{\beta^2}{4}+\mu^D_j+ o(1) \;\text{ as }\; \beta\to+\infty\,, \] 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 ${\mathcal O}(\beta^{-1}\log\beta)$.