Two-dimensional Schrödinger operators with non-local singular potentials
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915101821370368 |
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| author | Heriban, Lukáš Holzmann, Markus Stelzer-Landauer, Christian Stenzel, Georg Tušek, Matěj |
| author_facet | Heriban, Lukáš Holzmann, Markus Stelzer-Landauer, Christian Stenzel, Georg Tušek, Matěj |
| contents | In this paper we introduce and study a family of self-adjoint realizations of the Laplacian in $L^2(\mathbb{R}^2)$ with a new type of transmission conditions along a closed bi-Lipschitz curve $Σ$. These conditions incorporate jumps in the Dirichlet traces both of the functions in the operator domains and of their Wirtinger derivatives and are non-local. Constructing a convenient generalized boundary triple, they may be parametrized by all compact hermitian operators in $L^2(Σ;\mathbb{C}^2)$. Whereas for all choices of parameters the essential spectrum is stable and equal to $[0, +\infty)$, the discrete spectrum exhibits diverse behaviour. While in many cases it is finite, we will describe also a class of parameters for which the discrete spectrum is infinite and accumulates at $-\infty$. The latter class contains a non-local version of the oblique transmission conditions. Finally, we will connect the current model to its relativistic counterpart studied recently in [L. Heriban, M. Tušek: Non-local relativistic $δ$-shell interactions]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_10448 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Two-dimensional Schrödinger operators with non-local singular potentials Heriban, Lukáš Holzmann, Markus Stelzer-Landauer, Christian Stenzel, Georg Tušek, Matěj Spectral Theory Analysis of PDEs In this paper we introduce and study a family of self-adjoint realizations of the Laplacian in $L^2(\mathbb{R}^2)$ with a new type of transmission conditions along a closed bi-Lipschitz curve $Σ$. These conditions incorporate jumps in the Dirichlet traces both of the functions in the operator domains and of their Wirtinger derivatives and are non-local. Constructing a convenient generalized boundary triple, they may be parametrized by all compact hermitian operators in $L^2(Σ;\mathbb{C}^2)$. Whereas for all choices of parameters the essential spectrum is stable and equal to $[0, +\infty)$, the discrete spectrum exhibits diverse behaviour. While in many cases it is finite, we will describe also a class of parameters for which the discrete spectrum is infinite and accumulates at $-\infty$. The latter class contains a non-local version of the oblique transmission conditions. Finally, we will connect the current model to its relativistic counterpart studied recently in [L. Heriban, M. Tušek: Non-local relativistic $δ$-shell interactions]. |
| title | Two-dimensional Schrödinger operators with non-local singular potentials |
| topic | Spectral Theory Analysis of PDEs |
| url | https://arxiv.org/abs/2410.10448 |