Title: Solutions of generic bilinear master equations for a quantum oscillator -- positive and factorized conditions on stationary states,
Abstract: We obtain the solutions of the generic bilinear master equation for a quantum
oscillator with constant coefficients in the Gaussian form. The
well-behavedness and positive semidefiniteness of the stationary states could
be characterized by a three-dimensional Minkowski vector. By requiring the
stationary states to satisfy a factorized condition, we obtain a generic class
of master equations that includes the well-known ones and their
generalizations, some of which are completely positive. A further subset of the
master equations with the Gibbs states as stationary states is also obtained.
For master equations with not completely positive generators, an analysis on
the stationary states suggests conditions on the coefficients of the master
equations that generate positive evolution for a given initial state.