The Permittivity of a Multispecies Ionized Gas to Electrostatic Perturbations: a Fourier Transform, Integral Equation Approach Through Singular Eigenfunction Reconstruction

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 Boltzmann operator (the Liouville operator) [Palumbo and Nocera 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 Boltzmann 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.