Eigenvalue asymptotics for the one-particle density matrix

Abstract

The one-particle density matrix $\mathit{\gamma }(x,y)$ 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 ${\mathit{\lambda }}_k\sim {(Ak)}^{-8∕3}$, $A\ge 0$, as $k\to \mathrm{\infty }$, for the eigenvalues ${\mathit{\lambda }}_k$ of the self-adjoint operator $\mathrm{\Gamma }\ge 0$ with kernel $\mathit{\gamma }(x,y)$.