The local density approximation for the Schrödinger energy of Coulomb systems at a prescribed particle density has been widely used in physics and chemistry for a long time, but only recently has the validity of the approximation been proven rigorously in the grand canonical setting by Lewin, Lieb, and Seiringer. In this talk I will briefly present the new developments and discuss an ongoing project to generalize the results to the canonical setting. A central difficulty for this is to construct quantum trial states with a given fixed particle density.

This is joint work with Mathieu Lewin.