Eigenvalue asymptotics for the one-particle density matrix

Abstract

The one-particle density matrix $\gamma (x,y)$ for a bound state of an atom or molecule is one of the key objects in the quantum-mechanical approximation schemes. We prove the asymptotic formula ${\lambda }_k\sim (Ak{)}^{-8/3}$, $A\ge 0$, as $k\to \infty$, for the eigenvalues ${\lambda }_k$ of the self-adjoint operator $\mathbf{\Gamma }\ge 0$ with kernel $\gamma (x,y)$.

27 pages, Remarks 2.4., 2.5 added. Subsection 3.3 simplified