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

Abstract

A Darboux-type method of solving the nonlinear von Neumann equation $i\dot{\rho }=[H,f(\rho )]$, with functions $f(\rho )$ commuting with $\rho$, 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 $f(\rho )$ 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 $f(\rho )={\rho }^q-2{\rho }^{q-1}$ 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.

revtex, 5 pages, 2 eps figures, submitted to Phys.Lett.A infinite-dimensional example is added

References in corpus (6)

• Nonlinear quantum mechanics implies polynomial-time solution for NP-complete and #P problems
• Darboux-integrable nonlinear Liouville-von Neumann equation
• Complete positivity of nonlinear evolution: A case study
• Microscopic Foundation of Nonextensive Statistics
• Nonlinear von Neumann-type equations: Darboux invariance and spectra
• Von Neumann equations with time-dependent Hamiltonians and supersymmetric quantum mechanics

Cited by in corpus (5)

• Quantum morphogenesis: A variation on Thom's catastrophe theory
• Darboux Covariant Equations of von Neumann Type and their Generalizations
• Abstract DNA-type systems
• Special solutions of nonlinear von Neumann equations
• Covariant forms of Lax one-field operators