Trace formulae for Schrödinger operators with singular interactions

Abstract

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