Polishuk D. Properties of solutions of hierarchies of quantum evolution equations for correlation operators

The topic of the Thesis relates to the important area of development of modern mathematical physics, meaning the theory of evolution equations of quantum many-particle systems. The aim of the thesis is to develop the methods of construction and investigation of properties of solutions of the initial value problem for the hierarchy of nonlinear evolution equations that describes the non-equilibrium correlations of quantum statistical systems. For correlation operators that describe the states of finite Bose and Fermi systems the von Neumann hierarchy of evolution equations is formulated. The non-perturbative solution of its Cauchy problem is constructed. The properties of the solution are investigated. It is set that the solution is represented in the form of expansion by the groups of particles, which evolution is governed by the cumulant (semi-invariant) of the appropriate order of groups of operators of the von Neumann equation for the finite-particle systems. For the initial data from the spaces of sequences of symmetric and antisymmetric trace-class operators the theorem on existence of strong and weak solutions of the Cauchy problem of von Neumann hierarchy is proved. The hierarchies of quantum evolution equations for marginal density and marginal correlation operators are rigorously derived. The non-perturbative solution of nonlinear quantum BBGKY hierarchy has been constructed for the first time in the form of expansion by the groups (clusters) of particles which evolutions is governed by the reduced cumulant of the appropriate order of nonlinear groups of operators of von Neumann hierarchy for correlations operators. The properties of the solution are investigated.