Spectral Properties of Schrödinger Operators with a Strongly Attractive δ Interaction Supported by a Surface

We investigate the operator −∆− αδ(x − Γ) in L(R), 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 → ∞ 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.