Liouville's equation: III. Symmetries of the linearized equation

Let the linearized Liouville-Poisson equation be iδf/δt=Af,f=f(q,p),p= momentum coordinate. A on f's is not a Hermitian operator. However, an eigenvalue equation Afw=wfw, with real w's and non orthogonal eigenfunctions can be set up. For spherically symmetric potentials A and A2 have 0(3) symmetry. There exists an angular momentum operator, Ji, which commutes with A. This classifies the eigenfunctions into classes specified by a pair of eigennumbers (j,m) belonging to (J2,Jz). This in turn enables one to separate the dependence of the eigenfunctions on the direction angles of (q,p) and reduce the six dimensional phase space problem into a two dimensional one in terms of the magnitudes (q,p). (author). 19 refs