Recursive solution of Liouville's equation

The evolution of a density in phase space may be expressed as the intensity of a time-dependent wave function of coordinates and their momenta, which satisfies the classical Liouville equation and whose power spectrum is stable according to the black body theorem of von Laue. Formulated in this way, the evolution of phase space densities can be obtained from the recursion method which expands their power spectra, with respect to Liouville's equation, in continued fractions, and for which the positivity of the phase space density is preserved in all approximations. To demonstrate this method, we present its application to three simple one-dimensional systems: a harmonic oscillator, an inverted oscillator, and motion under a Gaussian potential.