Schrödinger operators with δ and δ′-potentials supported on hypersurfaces

Behrndt, Jussi and Langer, Matthias and Lotoreichik, Vladimir (2013) Schrödinger operators with δ and δ′-potentials supported on hypersurfaces. Annales Henri Poincaré, 14 (2). pp. 385-423. ISSN 1424-0637

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      Abstract

      Self-adjoint Schrödinger operators with δ and δ′-potentials supported on a smooth compact hypersurface are defined explicitly via boundary conditions. The spectral properties of these operators are investigated, regularity results on the functions in their domains are obtained, and analogues of the Birman–Schwinger principle and a variant of Krein’s formula are shown. Furthermore, Schatten–von Neumann type estimates for the differences of the powers of the resolvents of the Schrödinger operators with δ and δ′-potentials, and the Schrödinger operator without a singular interaction are proved. An immediate consequence of these estimates is the existence and completeness of the wave operators of the corresponding scattering systems, as well as the unitary equivalence of the absolutely continuous parts of the singularly perturbed and unperturbed Schrödinger operators. In the proofs of our main theorems we make use of abstract methods from extension theory of symmetric operators, some algebraic considerations and results on elliptic regularity.

      ORCID iDs

      Behrndt, Jussi, Langer, Matthias ORCID logoORCID: https://orcid.org/0000-0001-8813-7914 and Lotoreichik, Vladimir;
      • Item type:Article
        ID code:40524
        Dates:
        Date
        Event
        1 March 2013
        Published
        10 July 2012
        Published Online
        Keywords:mathematical physics, Schrödinger operators, hypersurface, Mathematics, Mathematical Physics, Statistical and Nonlinear Physics, Nuclear and High Energy Physics
        Subjects:Science > Mathematics
        Department:Faculty of Science > Mathematics and Statistics
        Depositing user: Pure Administrator
        Date deposited:24 Jul 2012 04:15
        Last modified:10 Sep 2021 01:42
        Related URLs:
        URI:https://strathprints.strath.ac.uk/id/eprint/40524
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