2D Schrödinger operators with singular potentials concentrated near curves
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arXiv
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| Format: | Preprint |
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2020
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| _version_ | 1866910919448068096 |
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| author | Golovaty, Yuriy |
| author_facet | Golovaty, Yuriy |
| contents | We investigate the Schrödinger operators $H_\varepsilon=-Δ+W+V_\varepsilon$ in $\mathbb{R}^2$ with the short-range potentials $V_\varepsilon$ which are localized around a smooth closed curve $γ$. The operators $H_\varepsilon$ can be viewed as an approximation of the heuristic Hamiltonian $H=-Δ+W+a\partial_νδ_γ+bδ_γ$, where $δ_γ$ is Dirac's $δ$-function supported on $γ$ and $\partial_νδ_γ$ is its normal derivative on $γ$. Assuming that the operator $-Δ+W$ has only discrete spectrum, we analyze the asymptotic behaviour of eigenvalues and eigenfunctions of $H_\varepsilon$. The transmission conditions on $γ$ for the eigenfunctions $u^+=αu^-$, $α\, \partial_νu^+-\partial_νu^-=βu^-$, which arise in the limit as $\varepsilon\to 0$, reveal a nontrivial connection between spectral properties of $H_\varepsilon$ and the geometry of $γ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2007_10761 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | 2D Schrödinger operators with singular potentials concentrated near curves Golovaty, Yuriy Spectral Theory Mathematical Physics 35P05, 81Q10, 81Q15 We investigate the Schrödinger operators $H_\varepsilon=-Δ+W+V_\varepsilon$ in $\mathbb{R}^2$ with the short-range potentials $V_\varepsilon$ which are localized around a smooth closed curve $γ$. The operators $H_\varepsilon$ can be viewed as an approximation of the heuristic Hamiltonian $H=-Δ+W+a\partial_νδ_γ+bδ_γ$, where $δ_γ$ is Dirac's $δ$-function supported on $γ$ and $\partial_νδ_γ$ is its normal derivative on $γ$. Assuming that the operator $-Δ+W$ has only discrete spectrum, we analyze the asymptotic behaviour of eigenvalues and eigenfunctions of $H_\varepsilon$. The transmission conditions on $γ$ for the eigenfunctions $u^+=αu^-$, $α\, \partial_νu^+-\partial_νu^-=βu^-$, which arise in the limit as $\varepsilon\to 0$, reveal a nontrivial connection between spectral properties of $H_\varepsilon$ and the geometry of $γ$. |
| title | 2D Schrödinger operators with singular potentials concentrated near curves |
| topic | Spectral Theory Mathematical Physics 35P05, 81Q10, 81Q15 |
| url | https://arxiv.org/abs/2007.10761 |