BooksecrCollected Volumepp. 129–152

Trace formulae for Schrödinger operators with singular interactions

  • Jussi Behrndt

    TU Graz, Austria
  • Matthias Langer

    University of Strathclyde, Glasgow, UK
  • Vladimir Lotoreichik

    Nuclear Physics Institute, Řež - Prague, Czech Republic
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Abstract

Let ΣRd\Sigma\subset\mathbb R^d be a CC^\infty-smooth closed compact hypersurface, which splits the Euclidean space Rd\mathbb R^d into two domains Ω±\Omega_\pm. In this note self-adjoint Schrödinger operators with δ\delta and δ\delta'-interactions supported on Σ\Sigma are studied. For large enough mNm\in\mathbb N the difference of mmth powers of resolvents of such a Schrödinger operator and the free Laplacian is known to belong to the trace class. We prove trace formulae, in which the trace of the resolvent power difference in L2(Rd)L^2(\mathbb R^d) is written in terms of Neumann-to-Dirichlet maps on the boundary space L2(Σ)L^2(\Sigma).