The quantum discrete self-trapping equation in the Hartree approximation

We show how the Hartree approximation (HA) can be used to study the quantum discrete self-trapping (QDST) equation, which - in turn - provides a model for the quantum description of several interesting nonlinear effects such as energy localization, soliton interactions, and chaos. The accuracy of the Hartree approximation is evaluated by comparing results with exact quantum mechanical calculations using the number state method. Since the Hartree method involves solving a classical DST equation, two classes of solutions are of particular interest: (i) Stationary solutions, which approximate certain energy eigenstates, and (ii) Time dependent solutions, which approximate the dynamics of wave packets of energy eigenstates. Both classes of solution are considered for systems with two and three degrees of freedom (the dimer and the trimer), and some comments are made on systems with an arbitrary number of freedoms.