Darboux-integration of idρ/dt=[H,f(ρ)]

Abstract

A Darboux-type method of solving the nonlinear von Neumann equation $i\dot ρ=[H,f(ρ)]$, with functions $f(ρ)$ commuting with $ρ$, is developed. The technique is based on a representation of the nonlinear equation by a compatibility condition for an overdetermined linear system. von Neumann equations with various nonlinearities $f(ρ)$ are found to possess the so-called self-scattering solutions. To illustrate the result we consider the Hamiltonian $H$ of a one-dimensional harmonic oscillator and $f(ρ)=ρ^q-2ρ^{q-1}$ with arbitary real $q$. It is shown that self-scattering solutions possess the same asymptotics for all $q$ and that different nonlinearities may lead to effectively indistinguishable evolutions. The result may have implications for nonextensive statistics and experimental tests of linearity of quantum mechanics.