Trace formulae for Schrödinger operators with singular interactions

Let Σ⊂ℝd be a C∞-smooth closed compact hypersurface, which splits the Euclidean space ℝd into two domains Ω±. In this note self-adjoint Schrödinger operators with δ and δ'-interactions supported on Σ are studied. For large enough m∈ℕ the difference of mth 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) is written in terms of Neumann-to-Dirichlet maps on the boundary space L2(Σ)