On the Spectrum of the One-Particle Density Matrix

Abstract

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.