The Eigenfunctions of the Multi-species Liouville Operator for Solitary Waves, Phase-space Holes and Double Layers

In this report, we give a detailed derivation of the spectrum and the eigenfunctions of the free streaming oscillations of electrons and massive, mobile ions about three physically relevant types of inhomogeneous equilibria: the bell-shaped solitary wave, the phase-space hole and the double layer. It is shown that the spectrum of the oscillations has a continuous as well as a discrete part both extending over the whole real axis. The eigenfunctions are given in the Fourier transformed velocity space. Those of them belonging to the continuous spectrum pertain to particles which are unrestricted in their motion. Those belonging to the discrete spectrum pertain to particles whose motion is constrained either by boundary conditions or which are trapped in their equilibrium potential wells. We prove that near the walls of these wells the eigenfunctions are singular. All of the eigenfunctions are mutually orthogonal and are twice discretely as well as continuously degenerate.