Trace formulae for Schr\"odinger operators with singular interactions

Let $\mathrm{\Sigma }\subset {\mathbb{R}}^d$ be a $C^{\mathrm{\infty }}$-smooth closed compact hypersurface, which splits the Euclidean space ${\mathbb{R}}^d$ into two domains ${\mathrm{\Omega }}_{\pm }$. In this note self-adjoint Schr"odinger operators with $\delta$ and ${\delta }^{\mathrm{\prime }}$-interactions supported on $\mathrm{\Sigma }$ are studied. For large enough $m\in \mathbb{N}$ the difference of $m$th powers of resolvents of such a Schr"odinger 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 $L^2({\mathbb{R}}^d)$ is written in terms of Neumann-to-Dirichlet maps on the boundary space $L^2(\mathrm{\Sigma })$.