**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).