Bloch Eigenfunctions of the Inhomogeneous Liouville Operator in the Fourier Transformed Velocity Space

In this report, we give a detailed derivation of the eigenvalues and of the corresponding eigenfunctions of the inomogeneous Liouville operator governing the free streaming oscillations of ions and electrons about an equilibrium background characterised by a periodic distribution of the electric potential along the space coordinate x. These eigenfunctions are worked out in the Fourier transformed velocity space, where they are well behaved. It is shown that the spectrum of the Liouville operator has a continuous as well as a discrete part both extending over the whole real axis. The eigenfunctions of the continuous spectrum pertain to those particles which are unrestricted in their motion. They are shown to be mutually orthogonal and to have two finite discrete degeneracies and an infinite continuous degeneracy. This latter corresponds to positive values of a continuously varying degeneracy parameter gamma. It is further shown that the eigenfunctions of the continuous spectrum are also eigenfunctions under discrete translations x mapsto x+ lambda, where lambda is the period of the equilibrium background, and that they are thus amenable to Bloch representation. The eigenfunctions of the discrete spectrum pertain to those particles whose motion is restricted by periodic boundary conditions or which are trapped in their equilibrium potential wells. The latter are first worked out in single-well form whereby they are defined in a single potential well and they are shown to necessarily vanish outside. They are mutually orthogonal, share the discrete degeneracies of the eigenfunctions of the continuous spectrum, and have a continuous degeneracy corresponding to negative values of a continuous degeneracy parameter gamma. They also have a further infinite discrete degeneracy. This latter degeneracy is removed by collating the single well eigenfunctions into suitable N-well functions, N being an integer. It is shown that these latter are eigenfunctions of the Liouville operator and also under the modular translation operation x mapsto [x+lambda] mod N lambda. Their Bloch representation is thus derived. It is further shown that the discrete spectrum becomes dense as the continuous degeneracy parameter approaches 0, beyond which value the eigenfunctions of the trapped particles are transformed into the eigenfunctions of the periodically bound particles.