Stationary states in nonlocal type dynamics of composite systems

Abstract

We consider a model for nonlocal type dynamics of composite quantum systems. It is based on the equation − i ħ K ̇ = K H + H ˆ K + β K f ( K ∗ K ) , describing the time evolution of an operator variable K . Here H and H ˆ are fixed self-adjoint and possibly unbounded operators (subsystem Hamiltonians), z → f ( z ) is an analytic function, assuming real values for a real argument, and β is a real parameter. This article focuses on the problem of characterization of stationary solutions, i.e. solutions that assume the special form K ( t ) = e i ν t / ħ K 0 with K 0 satisfying K 0 H + H ˆ K 0 + β K 0 f ( K 0 ∗ K 0 ) = ν K 0 . The main result is a characterization of stationary solutions subject to certain technical assumptions. In particular, we assume that the Hamiltonians have pure-point spectrum. In addition, the solutions are a priori assumed to be compact operators.