On solutions of nonlinear von Neumann hierarchies

Abstract:
We investigate the initial-value problem of the non-linear von Neumann hierarchy. We consider many-particle quantum systems with various statistics (Maxwell-Boltzmann, Fermi, Bose). For the general form of the interaction potential we construct a solution in terms of an expansion over particle clusters whose evolution is described by the corresponding-order cumulant of evolution operators of a system of finitely many particles. For the initial data from the space of trace-class operators the existence of a strong solution of the Cauchy problem for these hierarchies is proved.

Joint work with V.O. Shtyk (Institute of Mathematics of NAS of Ukraine, Kyiv, Ukraine).