On the self-adjointness of two-dimensional relativistic shell interactions
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866915514077413376 |
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| author | Benhellal, Badredine Pankrashkin, Konstantin Zreik, Mahdi |
| author_facet | Benhellal, Badredine Pankrashkin, Konstantin Zreik, Mahdi |
| contents | We study the self-adjointness of the two-dimensional Dirac operator coupled with electrostatic and Lorentz scalar shell interactions of constant strength $\varepsilon$ and $μ$ supported on a closed Lipschitz curve. Namely, we present several new explicit ranges of $\varepsilon$ and $μ$ for which there is a unique self-adjoint realization with domain included into $H^{\frac{1}{2}}$. A more precise analysis is carried out for curvilinear polygons, which allows one to take the corner openings into account. Compared to the preceding works on this topic, two new technical ingredients are employed: the explicit use of the Cauchy transform on non-smooth curves and the explicit characterization of the Fredholmness for singular integral operators. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2307_12772 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the self-adjointness of two-dimensional relativistic shell interactions Benhellal, Badredine Pankrashkin, Konstantin Zreik, Mahdi Spectral Theory Mathematical Physics Analysis of PDEs 81Q10, 81V05, 35P15, 58C40 We study the self-adjointness of the two-dimensional Dirac operator coupled with electrostatic and Lorentz scalar shell interactions of constant strength $\varepsilon$ and $μ$ supported on a closed Lipschitz curve. Namely, we present several new explicit ranges of $\varepsilon$ and $μ$ for which there is a unique self-adjoint realization with domain included into $H^{\frac{1}{2}}$. A more precise analysis is carried out for curvilinear polygons, which allows one to take the corner openings into account. Compared to the preceding works on this topic, two new technical ingredients are employed: the explicit use of the Cauchy transform on non-smooth curves and the explicit characterization of the Fredholmness for singular integral operators. |
| title | On the self-adjointness of two-dimensional relativistic shell interactions |
| topic | Spectral Theory Mathematical Physics Analysis of PDEs 81Q10, 81V05, 35P15, 58C40 |
| url | https://arxiv.org/abs/2307.12772 |