Joachim Rehberg (Weierstrass Institute, Berlin): A mathematical analysis of the Schrödinger-Poisson system

Abstract

The Schrödinger Poisson system is a semiclassical model which describes nature in quite dfferent areas of physics: quantum optics, semiconductor theory, dark matter...

Being analyzed in the ninetees by Francis Nier under strong assumptions, we now present an analysis which needs an absolute minimum of suppositions. In particular, the deep but demanding technique of the Heffer-Sjöstrand calculus is entirely avoided, and for the required Birman-Solomyak theorem an elementary proof is presented. So we end up with a theorem on existence and uniqueness of the Schrödinger-Poisson system where the solution is, additionally, obtained as the fixed point of a contraction sheme. The latter makes it highly accessible by computer methods.