Scattering for Schrödinger operators with potentials concentrated near a subspace
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866912222692769792 |
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| author | Black, Adam Malinovitch, Tal |
| author_facet | Black, Adam Malinovitch, Tal |
| contents | We study the scattering properties of Schrödinger operators with bounded potentials concentrated near a subspace of $\mathbb{R}^d$. For such operators, we show the existence of scattering states and characterize their orthogonal complement as a set of surface states, which consists of states that are confined to the subspace (such as pure point states) and states that escape it at a sublinear rate, in a suitable sense. We provide examples of surface states for different systems including those that propagate along the subspace and those that escape the subspace arbitrarily slowly. Our proof uses a novel interpretation of the Enss method in order to obtain a dynamical characterisation of the orthogonal complement of the scattering states. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2204_00712 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Scattering for Schrödinger operators with potentials concentrated near a subspace Black, Adam Malinovitch, Tal Mathematical Physics Analysis of PDEs Spectral Theory We study the scattering properties of Schrödinger operators with bounded potentials concentrated near a subspace of $\mathbb{R}^d$. For such operators, we show the existence of scattering states and characterize their orthogonal complement as a set of surface states, which consists of states that are confined to the subspace (such as pure point states) and states that escape it at a sublinear rate, in a suitable sense. We provide examples of surface states for different systems including those that propagate along the subspace and those that escape the subspace arbitrarily slowly. Our proof uses a novel interpretation of the Enss method in order to obtain a dynamical characterisation of the orthogonal complement of the scattering states. |
| title | Scattering for Schrödinger operators with potentials concentrated near a subspace |
| topic | Mathematical Physics Analysis of PDEs Spectral Theory |
| url | https://arxiv.org/abs/2204.00712 |