Trace formulae for Schr\"odinger operators with singular interactions

TL;DR

This paper derives trace formulae for Schr"odinger operators with singular delta and delta prime interactions on smooth hypersurfaces, expressing resolvent differences in terms of boundary Neumann-to-Dirichlet maps.

Contribution

It introduces new trace formulae for Schr"odinger operators with singular interactions, linking resolvent differences to boundary operators on the hypersurface.

Findings

Trace class difference of resolvent powers for large m

Trace formulae expressed via Neumann-to-Dirichlet maps

Extension of spectral analysis for 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 Schr\"odinger 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 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 $L^2(\Sigma)$.