Eigenvalue estimates for the Coulombic one-particle density matrix and the kinetic energy density matrix

TL;DR

This paper derives explicit eigenvalue decay bounds for the one-particle density and kinetic energy matrices in atomic systems, improving previous results with more elementary proofs and considering cases with enhanced regularity at coalescence points.

Contribution

It provides new explicit bounds on eigenvalue decay rates for density matrices, including cases with eigenfunction regularity at coalescence points, using more elementary methods.

Findings

Eigenvalue bounds: rac{1}{k^{8/3}} and rac{1}{k^{2}} for general eigenfunctions.

Faster decay bounds: rac{1}{k^{10/3}} and rac{1}{k^{8/3}} for eigenfunctions vanishing at coalescence points.

Bounds depend explicitly on the eigenfunction and are supported by derivative estimates.

Abstract

Consider a bound state (an eigenfunction) $\psi$ of an atom with $N$ electrons. We study the spectra of the one-particle density matrix $\gamma$ and of the one-particle kinetic energy density matrix $\tau$ associated with $\psi$. The paper contains two results. First, we obtain the bounds $\lambda_k(\gamma)\le C_1 k^{-8/3}$ and $\lambda_k(\tau)\le C_2 k^{-2}$ with some positive constants $C_1, C_2$ that depend explicitly on the eigenfunction $\psi$. 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 $\psi$ vanishes at the particle coalescence points. In particular, this is true for totally antisymmetric $\psi$. In this case the eigenfunction $\psi$ exhibits enhanced regularity at the coalescence points which leads to the faster decay of the eigenvalues: $\lambda_k(\gamma)\le C_3 k^{-10/3}$ and $\lambda_k(\tau)\le C_4 k^{-8/3}$. The proofs rely on the estimates for the derivatives of the eigenfunction $\psi$ 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. \O. S\o rensen.