Semiclassical spectrum for a Hartree-type equation corresponding to a rest point of the Hamilton–Ehrenfest system

Abstract

Following Ehrenfest's approach, the problem of quantum–classical correspondence can be treated in the class of trajectory-coherent functions that approximate a quantum-mechanical state as → 0. This idea leads to a family of systems of ordinary differential equations, called Hamilton–Ehrenfest M-systems (M = 0, 1, 2, ...). As noted in the authors' previous works, every M-system is formally equivalent to the semiclassical approximation of order M for the linear Schrodinger equation. In this paper a similar approach is undertaken for a nonlinear Hartree-type equation with a smooth integral kernel. It is demonstrated how quantum characteristics can be retrieved directly from the corresponding Hamilton–Ehrenfest systems, without solving the quantum equation: the semiclassical spectral series are obtained from the rest point solution. One of the key steps is derivation of a modified nonlinear superposition principle valid in the class of trajectory-coherent quantum states.