Spectral properties of Schroedinger operators with a strongly attractive delta interaction supported by a surface

Abstract

We investigate the operator $-\Delta -\alpha \delta (x-\Gamma)$ in $L^2(\mathbb{R}^3)$, where $\Gamma$ is a smooth surface which is either compact or periodic and satisfies suitable regularity requirements. We find an asymptotic expansion for the lower part of the spectrum as $\alpha\to\infty$ which involves a ``two-dimensional'' comparison operator determined by the geometry of the surface $\Gamma$. In the compact case the asymptotics concerns negative eigenvalues, in the periodic case Floquet eigenvalues. We also give a bandwidth estimate in the case when a periodic $\Gamma$ decomposes into compact connected components. Finally, we comment on analogous systems of lower dimension and other aspects of the problem.