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JOURNALS // Nanosystems: Physics, Chemistry, Mathematics // Archive

Nanosystems: Physics, Chemistry, Mathematics, 2016 Volume 7, Issue 2, Pages 290–302 (Mi nano202)

This article is cited in 7 papers

INVITED SPEAKERS

Boundary triples for Schrödinger operators with singular interactions on hypersurfaces

J. Behrndta, M. Langerb, V. Lotoreichikc

a Institut für Numerische Mathematik, Technische Universität Graz, Steyrergasse 30, 8010 Graz, Austria
b Department of Mathematics and Statistics, University of Strathclyde, 26 Richmond Street, Glasgow G1 1XH, United Kingdom
c Department of Theoretical Physics, Nuclear Physics Institute CAS, 250 68 Řež near Prague, Czech Republic

Abstract: The self-adjoint Schrödinger operator $A_{\delta,\alpha}$ with a $\delta$-interaction of constant strength $\alpha$ supported on a compact smooth hypersurface $\mathcal{C}$ is viewed as a self-adjoint extension of a natural underlying symmetric operator $S$ in $L^2(\mathbb{R}^n)$. The aim of this note is to construct a boundary triple for $S^*$ and a self-adjoint parameter $\Theta_{\delta,\alpha}$ in the boundary space $L^2(\mathcal{C})$ such that $A_{\delta,\alpha}$ corresponds to the boundary condition induced by $\Theta_{\delta,\alpha}$. 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 $A_{\delta,\alpha}$ in terms of the Weyl function and $\Theta_{\delta,\alpha}$.

Keywords: Boundary triple, Weyl function, Schrödinger operator, singular potential, $\delta$-interaction, hypersurface.

PACS: 02.30.Tb, 03.65.Db

Received: 22.01.2016

Language: English

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



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