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

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