2000/05/31 by N. V. Ustinov, Marek Czachor, S. B. Leble +4
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Algebra over a field #Applied mathematics #Class (philosophy) #Computer science #Darboux integral #Differential Equations and Boundary Problems #Geometry #Mathematical analysis #Mathematics #Nonlinear Waves and Solitons #Nonlinear system #Physics #Property (philosophy) #Pure mathematics #Quantum mechanics #Scattering #Type (biology) #Von Neumann architecture #nlin.SI #quant-ph
paper · pdf · doi:10.1016/s0375-9601(01)00013-5
published as Phys. Lett. A 279 (2001) 333 · revtex, 5 pages, 2 eps figures, submitted to Phys.Lett.A infinite-dimensional example is added
arxiv created 2000/11/23 · openalex publication_date 2001/02/01 · arxiv updated 2009/11/30 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
A Darboux-type method of solving the nonlinear von Neumann equation i 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)=ρq-2ρq-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.