Trace formulae for Schrödinger operators with singular interactions
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2015
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| _version_ | 1866913390647050240 |
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| author | Behrndt, Jussi Langer, Matthias Lotoreichik, Vladimir |
| author_facet | Behrndt, Jussi Langer, Matthias Lotoreichik, Vladimir |
| contents | 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 $L^2(Σ)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1512_06551 |
| institution | arXiv |
| publishDate | 2015 |
| record_format | arxiv |
| spellingShingle | Trace formulae for Schrödinger operators with singular interactions Behrndt, Jussi Langer, Matthias Lotoreichik, Vladimir Spectral Theory Mathematical Physics Analysis of PDEs 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 $L^2(Σ)$. |
| title | Trace formulae for Schrödinger operators with singular interactions |
| topic | Spectral Theory Mathematical Physics Analysis of PDEs |
| url | https://arxiv.org/abs/1512.06551 |