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

N. V. Ustinov·S. B. Leble·M. Czachor·M. Kuna
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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.

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N. V. Ustinov, S. B. Leble, M. Czachor, M. Kuna. 2000-11-23. Darboux-integration of idρ/dt=[H,f(ρ)]. https://doi.org/10.1016/s0375-9601(01)00013-5

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