Wave Scattering by an Asymmetric Nonmonotonic Double Layer

In this report, we give a detailed derivation of the eigenvalues and of the corresponding eigenfunctions of the collisionless Boltzamann equation governing the vibrations of a multispecies ionized gas. These eigenfunctions are worked out as a superposition of the singular, triply discretely degenerate and doubly continuously degenerate eigenfunctions of the free-streaming Vlasov operator (the Liouville operator) [Palumbo 2014a]. The superposition is carried out in the Fourier transformed velocity space, where the Liouville eigenfunctions are smooth. We prove that, by a judicious superposition of these Liouville eigenfunctions, a peculiar, non degenerate eigenfunction of the Vlasov operator can be worked out, such that its limit value, as the conjugate velocity coordinate tends to infinity, equals the permittivity of the ionized gas. Requiring that this limit vanish, as demanded by Lebesgue's lemma, yields the dispersion relation of electrostatic oscillations.

In questo lavoro determiniamo gli autovalori e le corrispondenti autofunzioni dell'equazione di Boltzmann non collisionale che governa le oscillazioni di un un gas ionizzato composto da più specie di particelle.