Integrability in the mesoscopic dynamics

Abstract

The Mesoscopic Mechanics (MeM), which has been introduced in a previous paper, is relevant to the electron gas confined to two spatial dimensions. It predicts a special way of collective response of correlated electrons to the external magnetic field. The dynamic variable of this theory is a finite-dimensional operator, which is required to satisfy the mesoscopic Schr\"{o}dinger equation (cf. text). In this article, we describe general solutions of the mesoscopic Schr\"{o}dinger equation. Our approach is specific to the problem at hand. It relies on the unique structure of the equation and makes no reference to any other techniques, with the exception of the geometry of unitary groups. In conclusion, a surprising fact comes to light. Namely, the mesoscopic dynamics "filters" through the (microscopic) Schr\"odinger dynamics as the latter turns out to be a clearly separable part, in fact an autonomous factor, of the evolution. This is a desirable result also from the physical standpoint.