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, $\hat{I}{“}^{(K)}$, 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 $\hat{I}{“}^{(K)}$ by $\hat{I}{»}_n(\hat{I}{“}^{(K)})$ for $n=1,2,\dots$, and obtain the bounds $\hat{I}{»}_n(\hat{I}{“}^{(K)})\le C{n}^{-\hat{I}{\pm }_K}$ for all $n$, where $\hat{I}{\pm }_K=1+7/(3L)$ and $L=\min \{K,N-K\}$.