Approximation of Schrödinger operators with δ‐interactions supported on hypersurfaces. Issue 8 (22nd September 2016)

• Record Type: Journal Article
• Authors: Behrndt, Jussi; Exner, Pavel; Holzmann, Markus; Lotoreichik, Vladimir
• Is Part Of: Mathematische Nachrichten. Volume 290:Issue 8/9(2017)
• Page Start: 1215
• Page End: 1248
• Publication Date: 2016-09-22
• Subjects: Schrödinger operators -- δ‐interactions supported on hypersurfaces -- approximation by scaled regular potentials -- norm resolvent convergence -- spectral convergence -- 35J10 -- 81Q10; Mathematics -- Periodicals; 510.5
• DOI: 10.1002/mana.201500498
• Languages: English
• ISSNs: 0025-584X

Abstract

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