
Mathematics 2005
The Ehrenfest system and the rest point spectrum for a Hartreetype EquationAbstract: Following Ehrenfest's approach, the problem of quantumclassical correspondence can be treated in the class of trajectorycoherent functions that approximate as $\h\to 0$ a quantummechanical state. This idea leads to a family of systems of ordinary differential equations, called Ehrenfest Msystems (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 Hartreetype 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 trajectorycoherent quantum states.
