Trace formulae for Schr��dinger operators with singular interactions

Let $��\subset\mathbb{R}^d$ be a $C^\infty$-smooth closed compact hypersurface, which splits the Euclidean space $\mathbb{R}^d$ into two domains $��_\pm$. In this note self-adjoint Schr��dinger operators with $��$ and $��'$-interactions supported on $��$ are studied. For large enough $m\in\mathbb{N}$ the difference of $m$th powers of resolvents of such a Schr��dinger 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 $L^2(��)$.