Regularity of solutions for the Coulomb multi-particle Schroedinger equation, and the one-particle density matrix

Regularity properties for multi-particle systems have been the focus of mathematical research since 1950’s. Such results determine spectral properties of the one-particle density matrix which is the key objects in the quantum-mechanical approximation schemes. In this talk I will give a short survey of recent regularity results with emphasis on sharp bounds for the eigenfunctions, and show how these bounds lead to the asymptotic formula for the eigenvalues of the one-particle density matrix.