Boundary triples for Schrödinger operators with singular interactions on hypersurfaces

Abstract

The self-adjoint Schrodinger operator Aδ,α with a δ-interaction of constant strength α supported on a compact smooth hypersurface C is viewed as a self-adjoint extension of a natural underlying symmetric operator S in L2(ℝn). The aim of this note is to construct a boundary triple for S* and a self-adjoint parameter Θδ,α in the boundary space L2(C) such that Aδ,α corresponds to the boundary condition induced by Θδ,α. As a consequence the well-developed theory of boundary triples and their Weyl functions can be applied. This leads, in particular, to a Krein-type resolvent formula and a description of the spectrum of Aδ,α in terms of the Weyl function and Θδ,α.