A Proof of Completeness for the Eigenfunctions of the Multi-species Liouville Operator and its Green's Function

In this report, we determine the spectrum and the eigenfunctions of the free-streaming oscillations of electrons and mobile ions about inhomogeneous, collisionless, electrostatic, space-periodic and non-periodic plasma equilibria. The eigenfunctions are given in the Fourier transformed velocity space. We show that the spectrum has a continuous as well as a discrete part, both extending over the whole real axis. The eigenfunctions of the continuous spectrum pertain to particles which are unrestricted in their motion. Those belonging to the discrete spectrum pertain to particles which are either constrained by boundary conditions or trapped in their equilibrium potential wells. We show that, near the boundaries of these wells, the space distribution of the eigenfunctions is algebraically singular. All of the eigenfunctions have two discrete finite degeneracies and two continuous infinite degeneracies. A further discrete, infinite degeneracy appears for space-periodic equilibria. We prove that the eigenfunctions are mutually orthogonal and that they form a complete set. Examples are presented of the eigenfunctions of both electron and ion oscillations in solitary and in double layer equilibria and, for space-periodic equilibria, their Bloch form is also worked out.

In questo lavoro determiniamo lo spettro e le autofunzioni delle oscillazioni "free-streaming" degli elettroni e degli ioni mobili che si muovono intorno ad un equilibro inomogeneo in un plasma non collisionale che presenti o meno di periodicità spaziale.