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arXiv · 1512.06551

Trace formulae for Schrödinger operators with singular interactions

Jussi Behrndt·Matthias Langer·Vladimir Lotoreichik
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Abstract

Let $Σ\subset\mathbb{R}^d$ be a $C^\infty$-smooth closed compact hypersurface, which splits the Euclidean space $\mathbb{R}^d$ into two domains $Ω_\pm$. In this note self-adjoint Schrödinger operators with $δ$ and $δ'$-interactions supported on $Σ$ 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 $L^2(Σ)$.

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Jussi Behrndt, Matthias Langer, Vladimir Lotoreichik. 2015-12-21. Trace formulae for Schrödinger operators with singular interactions. https://doi.org/10.4171/175-1%2F6

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