%0 Journal Article
%T The Ehrenfest system and the rest point spectrum for a Hartree-type Equation
%A V. V. Belov
%A M. F. Kondratieva
%A A. Yu. Trifonov
%J Mathematics
%D 2005
%I arXiv
%X Following Ehrenfest's approach, the problem of quantum-classical correspondence can be treated in the class of trajectory-coherent functions that approximate as $\h\to 0$ a quantum-mechanical state. This idea leads to a family of systems of ordinary differential equations, called Ehrenfest M-systems (M=0,1,2,...), formally equivalent to the semiclassical approximation for the linear Schroedinger 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 Ehrenfest systems, without solving the quantum equation: the semiclassical asymptotics for the spectrum 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.
%U http://arxiv.org/abs/math-ph/0510096v2