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 $\delta'$-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 $L^2(\Sigma)$.