On the approximation of the Dirac operator coupled with confining Lorentz scalar $δ$-shell interactions
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866911836280979456 |
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| author | Zreik, Mahdi |
| author_facet | Zreik, Mahdi |
| contents | Let $Ω_+\subset\mathbb{R}^{3}$ be a fixed bounded domain with boundary $Σ= \partialΩ_{+}$. We consider $\mathcal{U}^\varepsilon$ a tubular neighborhood of the surface $Σ$ with a thickness parameter $\varepsilon>0$, and we define the perturbed Dirac operator $\mathfrak{D}^{\varepsilon}_{M}=D_m +Mβ\mathbb{1}_{\mathcal{U}^{\varepsilon}},$ with $D_m$ the free Dirac operator, $M>0$, and $\mathbb{1}_{\mathcal{U }^{\varepsilon}}$ the characteristic function of $\mathcal{U}^{\varepsilon}$. Then, in the norm resolvent sense, the Dirac operator $\mathfrak{D}^{\varepsilon}_M$ converges to the Dirac operator coupled with Lorentz scalar $δ$-shell interactions as $\varepsilon = M^{-1}$ tends to $0$, with a convergence rate of $\mathcal{O}(M^{-1})$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_07784 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the approximation of the Dirac operator coupled with confining Lorentz scalar $δ$-shell interactions Zreik, Mahdi Spectral Theory 81Q10, 81V05, 35P15, 58C40 Let $Ω_+\subset\mathbb{R}^{3}$ be a fixed bounded domain with boundary $Σ= \partialΩ_{+}$. We consider $\mathcal{U}^\varepsilon$ a tubular neighborhood of the surface $Σ$ with a thickness parameter $\varepsilon>0$, and we define the perturbed Dirac operator $\mathfrak{D}^{\varepsilon}_{M}=D_m +Mβ\mathbb{1}_{\mathcal{U}^{\varepsilon}},$ with $D_m$ the free Dirac operator, $M>0$, and $\mathbb{1}_{\mathcal{U }^{\varepsilon}}$ the characteristic function of $\mathcal{U}^{\varepsilon}$. Then, in the norm resolvent sense, the Dirac operator $\mathfrak{D}^{\varepsilon}_M$ converges to the Dirac operator coupled with Lorentz scalar $δ$-shell interactions as $\varepsilon = M^{-1}$ tends to $0$, with a convergence rate of $\mathcal{O}(M^{-1})$. |
| title | On the approximation of the Dirac operator coupled with confining Lorentz scalar $δ$-shell interactions |
| topic | Spectral Theory 81Q10, 81V05, 35P15, 58C40 |
| url | https://arxiv.org/abs/2404.07784 |