The Ehrenfest system and the rest point spectrum for a Hartree-type Equation

TL;DR

This paper extends Ehrenfest's approach to nonlinear Hartree-type equations, deriving semiclassical spectral asymptotics from Ehrenfest systems without solving the full quantum problem.

Contribution

It develops a method to obtain quantum spectral properties directly from Ehrenfest systems for nonlinear equations, bypassing the need to solve the quantum equation.

Findings

Semiclassical asymptotics for the spectrum are derived from rest point solutions.

A modified nonlinear superposition principle is established for trajectory-coherent states.

Ehrenfest systems effectively retrieve quantum characteristics without solving the quantum equation.

Abstract

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.