najoNanosystems: Physics, Chemistry, MathematicsНаносистемы: физика, химия, математика2220-80542305-7971Университет ИТМО10.17586/2220-8054-2016-7-2-290-302najo-827Research ArticleINVITED SPEAKERSINVITED SPEAKERSBoundary triples for Schrӧdinger operators with singular interactions on hypersurfacesBoundary triples for Schrӧdinger operators with singular interactions on hypersurfacesBehrndtJ.BehrndtJ.

Steyrergasse 30, 8010 Graz

Steyrergasse 30, 8010 Graz

behrndt@tugraz.atLangerM.LangerM.

26 Richmond Street, Glasgow G1 1XH

26 Richmond Street, Glasgow G1 1XH

m.langer@strath.ac.ukLotoreichikV.LotoreichikV.

250 68 ˇRez near Prague

250 68 ˇRez near Prague

lotoreichik@ujf.cas.czInstitut fur Numerische Mathematik, Technische Universitat GrazInstitut fur Numerische Mathematik, Technische Universitat GrazDepartment of Mathematics and Statistics, University of StrathclydeDepartment of Mathematics and Statistics, University of StrathclydeDepartment of Theoretical Physics, Nuclear Physics Institute CASDepartment of Theoretical Physics, Nuclear Physics Institute CAS20161308202572Special Issue290302Copyright © Behrndt J., Langer M., Lotoreichik V., 20252025Behrndt J., Langer M., Lotoreichik V.Behrndt J., Langer M., Lotoreichik V.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://nanojournal.ifmo.ru/jour/article/view/827

The self-adjoint Schr¨odinger 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.

The self-adjoint Schr¨odinger 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.

Boundary tripleWeyl functionSchrӧdinger operatorsingular potentialδ-interactionhypersurfaceBoundary tripleWeyl functionSchrӧdinger operatorsingular potentialδ-interactionhypersurfaceJ. Behrndt and V. Lotoreichik gratefully acknowledge financial support by the Austrian Science Fund (FWF): Project P 25162-N26. J. Behrndt also wishes to thank Professor Igor Popov for the pleasant and fruitful research stay at the ITMO University in St. Petersburg in September 2015. V. Lotoreichik was also supported by the Czech Science Foundation (GA CR) under the project 14-06818S.J. Behrndt and V. Lotoreichik gratefully acknowledge financial support by the Austrian Science Fund (FWF): Project P 25162-N26. J. Behrndt also wishes to thank Professor Igor Popov for the pleasant and fruitful research stay at the ITMO University in St. Petersburg in September 2015. V. 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The authors declare that there are no conflicts of interest present.