Nonlinear von Neumann-type equations: Darboux invariance and spectra

Abstract

Generalized Euler-Arnold-von Neumann density matrix equations can be solved by a binary Darboux transformation given here in a new form: $ρ[1]=e^{P\ln(μ/ν)}ρe^{-P\ln(μ/ν)}$ where $P=P^2$ is explicitly constructed in terms of conjugated Lax pairs, and $μ$, $ν$ are complex. As a result spectra of $ρ$ and $ρ[1]$ are identical. Transformations allowing to shift and rescale spectrum of a solution are introduced, and a class of stationary seed solutions is discussed.