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.