Eigenvalue Bounds for Multi-Particle Reduced Density Matrices of Coulombic Wavefunctions

Abstract

For bound states of atoms and molecules of $N$ electrons we consider the corresponding $K$-particle reduced density matrices, $\Gamma^{(K)}$, 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 $\Gamma^{(K)}$ by $\lambda_n(\Gamma^{(K)})$ for $n=1,2,\dots$, and obtain the bounds $\lambda_n(\Gamma^{(K)}) \le Cn^{-\alpha_K}$ for all $n$, where $\alpha_K = 1 + 7/(3L)$ and $L = \min\{K,N-K\}$.