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NANOSYSTEMS: PHYSICS, CHEMISTRY, MATHEMATICS, 2016, 7 (2), P. 290–302

Boundary triples for Schrödinger operators with singular interactions on hypersurfaces

J. Behrndt – Institut für Numerische Mathematik, Technische Universität Graz, Steyrergasse 30, 8010 Graz, Austria; behrndt@tugraz.at
M. Langer – Department of Mathematics and Statistics, University of Strathclyde, 26 Richmond Street, Glasgow G1 1XH, United Kingdom; m.langer@strath.ac.uk
V. Lotoreichik – Department of Theoretical Physics, Nuclear Physics Institute CAS, 250 68 Rêz near Prague, Czech Republic; lotoreichik@ujf.cas.cz

The self-adjoint Schrödinger operator A_δ,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 Θ_δ,α in the boundary space L²(C) such that A_δ,α corresponds to the boundary condition induced by Θ_δ,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 A_δ,α in terms of the Weyl function and Θ_δ,α.

Keywords: Boundary triple, Weyl function, Schrödinger operator, singular potential, interaction, hypersurface.

PACS 02.30.Tb, 03.65.Db

DOI 10.17586/2220-8054-2016-7-2-290-302

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