Approximation of Schrödinger operators with delta-interactions supported on hypersurfaces

We show that a Schrödinger operator Aδ,α with a δ-interaction of strength α supported on a bounded or unbounded C2-hypersurface Σ ⊂ Rd, d ≥ 2,, can be approximated in the norm resolvent sense by a family of Hamiltonians with suitably scaled regular potentials. The differential operator Aδ,α with a singular interaction is regarded as a self-adjoint realization of the formal differential expression −∆ − α 〈δ∆, •〉δ∆, where α: Σ → R is an arbitrary bounded measurable function. We discuss also some spectral consequences of this approximation result.