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

Jussi Behrndt, Matthias Langer, Vladimir Lotoreichik

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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.
Original languageEnglish
Pages (from-to)385-423
Number of pages39
JournalAnnales Henri Poincaré
Issue number2
Early online date10 Jul 2012
Publication statusPublished - 1 Mar 2013


  • mathematical physics
  • Schrödinger operators
  • hypersurface


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