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| Format: | Preprint |
| Published: |
2024
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| Online Access: | https://arxiv.org/abs/2412.16073 |
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| _version_ | 1866917875614220288 |
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| author | Hearnshaw, Peter |
| author_facet | Hearnshaw, Peter |
| contents | For bound states of atoms and molecules of $N$ electrons we consider the corresponding $K$-particle reduced density matrices, $Γ^{(K)}$, for $1 \le K \le N-1$. Previously, eigenvalue bounds were obtained in the case of $K=1$ and $K=N-1$ by A.V. Sobolev. The purpose of the current work is to obtain bounds in the case of $2 \le K \le N-2$. For such $K$ we label the eigenvalues of the positive, trace class operators $Γ^{(K)}$ by $λ_n(Γ^{(K)})$ for $n=1,2,\dots$, and obtain the bounds $λ_n(Γ^{(K)}) \le Cn^{-α_K}$ for all $n$, where $α_K = 1 + 7/(3L)$ and $L = \min\{K,N-K\}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_16073 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Eigenvalue Bounds for Multi-Particle Reduced Density Matrices of Coulombic Wavefunctions Hearnshaw, Peter Mathematical Physics 35J10 For bound states of atoms and molecules of $N$ electrons we consider the corresponding $K$-particle reduced density matrices, $Γ^{(K)}$, for $1 \le K \le N-1$. Previously, eigenvalue bounds were obtained in the case of $K=1$ and $K=N-1$ by A.V. Sobolev. The purpose of the current work is to obtain bounds in the case of $2 \le K \le N-2$. For such $K$ we label the eigenvalues of the positive, trace class operators $Γ^{(K)}$ by $λ_n(Γ^{(K)})$ for $n=1,2,\dots$, and obtain the bounds $λ_n(Γ^{(K)}) \le Cn^{-α_K}$ for all $n$, where $α_K = 1 + 7/(3L)$ and $L = \min\{K,N-K\}$. |
| title | Eigenvalue Bounds for Multi-Particle Reduced Density Matrices of Coulombic Wavefunctions |
| topic | Mathematical Physics 35J10 |
| url | https://arxiv.org/abs/2412.16073 |