Periodicity of atomic structure in a Thomas-Fermi mean-field model
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| Format: | Preprint |
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2024
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| _version_ | 1866917830023184384 |
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| author | Bjerg, August Solovej, Jan Philip |
| author_facet | Bjerg, August Solovej, Jan Philip |
| contents | We consider a Thomas-Fermi mean-field model for large neutral atoms. That is, Schrödinger operators $H_Z^{\text{TF}}=-Δ-Φ_Z^{\text{TF}}$ in three-dimensional space, where $Z$ is the nuclear charge of the atom and $Φ_Z^{\text{TF}}$ is a mean-field potential coming from the Thomas-Fermi density functional theory for atoms. For any sequence $Z_n\to\infty$ we prove that the corresponding sequence $H_{Z_n}^{\text{TF}}$ is convergent in the strong resolvent sense if and only if $D_{\text{cl}}Z_n^{1/3}$ is convergent modulo $1$ for a universal constant $D_{\text{cl}}$. This can be interpreted in terms of periodicity of large atoms. We also characterize the possible limiting operators (infinite atoms) as a periodic one-parameter family of self-adjoint extensions of $-Δ-C_\infty\vert\,x\,\vert^{-4}$ for an explicit number $C_\infty$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_19839 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Periodicity of atomic structure in a Thomas-Fermi mean-field model Bjerg, August Solovej, Jan Philip Mathematical Physics 81V45 (Primary), 81Q20 (Secondary) We consider a Thomas-Fermi mean-field model for large neutral atoms. That is, Schrödinger operators $H_Z^{\text{TF}}=-Δ-Φ_Z^{\text{TF}}$ in three-dimensional space, where $Z$ is the nuclear charge of the atom and $Φ_Z^{\text{TF}}$ is a mean-field potential coming from the Thomas-Fermi density functional theory for atoms. For any sequence $Z_n\to\infty$ we prove that the corresponding sequence $H_{Z_n}^{\text{TF}}$ is convergent in the strong resolvent sense if and only if $D_{\text{cl}}Z_n^{1/3}$ is convergent modulo $1$ for a universal constant $D_{\text{cl}}$. This can be interpreted in terms of periodicity of large atoms. We also characterize the possible limiting operators (infinite atoms) as a periodic one-parameter family of self-adjoint extensions of $-Δ-C_\infty\vert\,x\,\vert^{-4}$ for an explicit number $C_\infty$. |
| title | Periodicity of atomic structure in a Thomas-Fermi mean-field model |
| topic | Mathematical Physics 81V45 (Primary), 81Q20 (Secondary) |
| url | https://arxiv.org/abs/2406.19839 |