Stationary states in nonlocal type dynamics of composite systems

We consider a model for nonlocal type dynamics of composite quantum systems. It is based on the equation −iħK̇=KH+HˆK+βKf(K∗K), describing the time evolution of an operator variable KK. Here HH and Hˆ are fixed self-adjoint and possibly unbounded operators (subsystem Hamiltonians), z→f(z)z→f(z) is an analytic function, assuming real values for a real argument, and ββ 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)=eiνt/ħK0 with K0K0 satisfying K0H+HˆK0+βK0f(K0∗K0)=νK0. 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.