Chapter 21 Stationary States of Non-Hamiltonian Systems

Abstract

Classical non-Hamiltonian and dissipative systems can have regular or strange attractors. The regular attractors of a non-Hamiltonian system can be considered as a set of (stationary) states for Hamiltonian system that corresponds to the non-Hamiltonian system. Also, the regular “quantum” attractors can be considered as the stationary states of non-Hamiltonian quantum systems. The existence of stationary states for non-Hamiltonian quantum systems is an interesting fact. This chapter describes the stationary pure states of some non-Hamiltonian quantum systems. In the pure stationary states, these non-Hamiltonian systems look like Hamiltonian quantum systems. The quantum analog of dynamical bifurcations are described in the chaspter that are used for classical dynamical systems. Non-Hamiltonian quantum systems with pure stationary states of linear harmonic oscillator are suggested. The stationary states are derived for the Lindblad equation. The suggested approach allows using the theory of bifurcate ions for a wide class of quantum non-Hamiltonian systems. The example of bifurcation of pure stationary states for non-Hamiltonian quantum systems is also described in the chapter.