Quantum BGK modes for the Wigner-poisson system

Abstract

Abstract It is shown that an infinite family of solutions to the stationary Wigner equation can be constructed as arbitrary functions of a complete set of commuting observables. These are generalizations of the famous BGK modes of classical Vlasov theory; their existence was first proposed by M. Buchanan. For one specific function corresponding to the canonical distribution, a pseudo-differential equation-the Bloch equation-can be written down for the Wigner function. This equation is known to have a unique solution. For other functions, an equation which the Wigner function obeys remains to be discovered.