Spectral analysis of singular, inhomogeneous, collision-less plasma structures

Several laboratory, geophysical and astrophysical problems deal with the development of oscillations within spatially inhomogeneous plasmas, double layers and BGK states being some of the most common configurations. We analyze this problem in the collision-less plasma regime where the effects of inhomogeneity (continuous spectra), multi-phase (electrons and several species of ions), kinetics (particle reflection or trapping) and spatial asymmetry (leading to discontinuous particle velocity distributions) determine a subtle structure of the spectrum of the oscillations, including up to four countably finite and one continuously infinite degeneracy of the oscillation eigenfunctions. These effects have so far been considered separately and in limiting situations (the so called fluid, kinetic and homogeneous limits). By casting the oscillation problem in the Fourier-transformed velocity space, we show that those effects may be treated simultaneously and that the Green's function of the Vlasov operator may be worked out. This result is useful to assess the ability of inhomogeneous plasma structures to emit and absorb radiation and to remain stable under perturbations.