On the Spectrum of the One-Particle Density Matrix

The one-particle density matrix \gamma(x, y) is one of the key objects in quantum-mechanical approximation schemes. The self-adjoint operator \Gamma with kernel \gamma(x, y) is trace class, but no sharp results on the decay of its eigenvalues were previously known. The note presents the asymptotic formula \lambda_k \sim (Ak)^{-8/3}, A \ge 0, as k\to\infty for the eigenvalues \lambda_k of the operator \Gamma and describes the main ideas of the proof.