Eigenvalue estimates for the Coulombic one-particle density matrix and the kinetic energy density matrix
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908413310533632 |
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| author | Sobolev, Alexander V. |
| author_facet | Sobolev, Alexander V. |
| contents | Consider a bound state (an eigenfunction) $ψ$ of an atom with $N$ electrons. We study the spectra of the one-particle density matrix $γ$ and of the one-particle kinetic energy density matrix $τ$ associated with $ψ$. The paper contains two results. First, we obtain the bounds $λ_k(γ)\le C_1 k^{-8/3}$ and $λ_k(τ)\le C_2 k^{-2}$ with some positive constants $C_1, C_2$ that depend explicitly on the eigenfunction $ψ$. The sharpness of these bounds is confirmed by the asymptotic results obtained by the author in earlier papers. The advantage of these bounds over the ones derived by the author previously, is their explicit dependence on the eigenfunction. Moreover, their new proofs are more elementary and direct. The second result is new and it pertains to the case where the eigenfunction $ψ$ vanishes at the particle coalescence points. In particular, this is true for totally antisymmetric $ψ$. In this case the eigenfunction $ψ$ exhibits enhanced regularity at the coalescence points which leads to the faster decay of the eigenvalues: $λ_k(γ)\le C_3 k^{-10/3}$ and $λ_k(τ)\le C_4 k^{-8/3}$.
The proofs rely on the estimates for the derivatives of the eigenfunction $ψ$ that depend explicitly on the distance to the coalescence points. Some of these estimates are borrowed directly from, and some are derived using the methods of a recent paper by S. Fournais and T. Ø. Sørensen. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_16178 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Eigenvalue estimates for the Coulombic one-particle density matrix and the kinetic energy density matrix Sobolev, Alexander V. Spectral Theory Mathematical Physics Primary 35J10, Secondary 47G10, 81Q10 Consider a bound state (an eigenfunction) $ψ$ of an atom with $N$ electrons. We study the spectra of the one-particle density matrix $γ$ and of the one-particle kinetic energy density matrix $τ$ associated with $ψ$. The paper contains two results. First, we obtain the bounds $λ_k(γ)\le C_1 k^{-8/3}$ and $λ_k(τ)\le C_2 k^{-2}$ with some positive constants $C_1, C_2$ that depend explicitly on the eigenfunction $ψ$. The sharpness of these bounds is confirmed by the asymptotic results obtained by the author in earlier papers. The advantage of these bounds over the ones derived by the author previously, is their explicit dependence on the eigenfunction. Moreover, their new proofs are more elementary and direct. The second result is new and it pertains to the case where the eigenfunction $ψ$ vanishes at the particle coalescence points. In particular, this is true for totally antisymmetric $ψ$. In this case the eigenfunction $ψ$ exhibits enhanced regularity at the coalescence points which leads to the faster decay of the eigenvalues: $λ_k(γ)\le C_3 k^{-10/3}$ and $λ_k(τ)\le C_4 k^{-8/3}$. The proofs rely on the estimates for the derivatives of the eigenfunction $ψ$ that depend explicitly on the distance to the coalescence points. Some of these estimates are borrowed directly from, and some are derived using the methods of a recent paper by S. Fournais and T. Ø. Sørensen. |
| title | Eigenvalue estimates for the Coulombic one-particle density matrix and the kinetic energy density matrix |
| topic | Spectral Theory Mathematical Physics Primary 35J10, Secondary 47G10, 81Q10 |
| url | https://arxiv.org/abs/2506.16178 |