PDEA Seminar: The inverse problem of Schrödinger dynamics

Abstract: Consider the time-dependent Schrödinger equation. If the time-dependent external potential is given, it is relatively easy to solve this equation and find the time-dependent wave function and hence its one-particle density. The inverse problem, of finding the external potential giving rise to some prescribed one-particle density, is immensily more difficult. I will explain how, with a careful rewriting of the problem, this inverse problem can be viewed as a nonlinear PDE with a loss of derivatives and our solution to this problem. If time permits, I will explain how this proves the well-posedness of Kohn--Sham theory, where the density of an interacting system is reproduced exactly in a non-interacting one with an appropriately modified nonlinear external potential, which can, however, be computed without ever solving the interacting problem.

Joint work with Mathieu Lewin and Jakob Oldenburg.

This is a seminar in the PDEs and Applications seminar series.