On the magnetic Dirichlet to Neumann operator on the exterior of the disk -- diamagnetism, weak-magnetic field limit and flux effects
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| Format: | Preprint |
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2025
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| _version_ | 1866912292261593088 |
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| author | Bernard, Helffer Francois, Nicoleau |
| author_facet | Bernard, Helffer Francois, Nicoleau |
| contents | In this paper, we analyze the magnetic Dirichlet-to-Neumann operator (D-to-N map) $\check Λ(b,ν)$ on the exterior of the disk with respect to a magnetic potential $A_{b, ν}=A^b + A_ν$ where, for $b\in \mathbb R$ and $ν\in \mathbb R$, $A^b (x,y)= b\, (-y, x)$ and $A_ν(x,y)$ is the Aharonov-Bohm potential centered at the origin of flux $2πν$. First, we show that the limit of $\check Λ(b,ν)$ as $b\rightarrow 0$ is equal to the D-to-N map $\widehat Λ(ν)$ on the interior of the disk associated with the potential $A_ν(x,y)$. Secondly, we study the ground state energy of the D-to-N map $\check Λ(b,ν)$ and show that the strong diamagnetism property holds. Finally we slightly extend to the exterior case the asymptotic results obtained in the interior case for general domains. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_14008 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the magnetic Dirichlet to Neumann operator on the exterior of the disk -- diamagnetism, weak-magnetic field limit and flux effects Bernard, Helffer Francois, Nicoleau Analysis of PDEs Mathematical Physics Spectral Theory In this paper, we analyze the magnetic Dirichlet-to-Neumann operator (D-to-N map) $\check Λ(b,ν)$ on the exterior of the disk with respect to a magnetic potential $A_{b, ν}=A^b + A_ν$ where, for $b\in \mathbb R$ and $ν\in \mathbb R$, $A^b (x,y)= b\, (-y, x)$ and $A_ν(x,y)$ is the Aharonov-Bohm potential centered at the origin of flux $2πν$. First, we show that the limit of $\check Λ(b,ν)$ as $b\rightarrow 0$ is equal to the D-to-N map $\widehat Λ(ν)$ on the interior of the disk associated with the potential $A_ν(x,y)$. Secondly, we study the ground state energy of the D-to-N map $\check Λ(b,ν)$ and show that the strong diamagnetism property holds. Finally we slightly extend to the exterior case the asymptotic results obtained in the interior case for general domains. |
| title | On the magnetic Dirichlet to Neumann operator on the exterior of the disk -- diamagnetism, weak-magnetic field limit and flux effects |
| topic | Analysis of PDEs Mathematical Physics Spectral Theory |
| url | https://arxiv.org/abs/2503.14008 |