Spectral properties of Schroedinger operators with a strongly attractive delta interaction supported by a surface
Abstract
Description
We investigate the operator $-Δ-αδ(x-Γ)$ in $L^2(\mathbb{R}^3)$, where $Γ$ 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 $α\to\infty$ which involves a ``two-dimensional'' comparison operator determined by the geometry of the surface $Γ$. 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 $Γ$ decomposes into compact connected components. Finally, we comment on analogous systems of lower dimension and other aspects of the problem.
AMSTeX, 12 pages; to appear in Proceedings of the NSF Summer Research Conference (Mt. Holyoke 2002); AMS "Contemporary Mathematics" Series, Providence, R.I., 2003
AMSTeX, 12 pages; to appear in Proceedings of the NSF Summer Research Conference (Mt. Holyoke 2002); AMS "Contemporary Mathematics" Series, Providence, R.I., 2003