Riccati equation and the problem of decoherence II: Symmetry and the solution of the Riccati equation

In this paper we revisit the problem of decoherence by applying the block operator matrices analysis. The Riccati algebraic equation associated with the Hamiltonian describing the process of decoherence is studied. We prove that if the environment responsible for decoherence is invariant with respect to the antilinear transformation then the antilinear operator solves the Riccati equation in question. We also argue that this solution leads to neither a linear nor an antilinear operator similarity matrix. Therefore, we cannot use the standard procedure for solving a linear differential equation (e.g., Schrödinger equation). Furthermore, the explicit solution of the Riccati equation is found for the case where the environmental operators commute with each other. We discuss the connection between our results and the standard description of decoherence (one that uses the Kraus representation). We show that the reduced dynamics we obtain does not have the Kraus representation if the initial correlations between the system and its environment are present. However, for any initial state of the system (even when the correlations occur) reduced dynamics can be written in a manageable way.