The Eigenfunctions of the Inhomogeneous Free-streaming Collisionless Boltzmann Operator in the Fourier Transformed Velocity Space

In this report, we give a detailed derivation of the eigenvalues and of the corresponding eigenfunctions of the collisionless Boltzamann equations governing the vibrations of the electron and ion fluids in an electrostatic sheath embedded in an inhomogeneously distributed equilibrium electric field. We show that the sought eigenvalues may continuously take any real value and that they are degenerate: to each eigenvalue sigma correspond infinitely many eigenfunctions, which are labelled by a degeneracy parameter extending continuously over the non negative real axis. These eigenfunctions form an orthonormal and complete set over the whole real domain of the space coordinate. Furthermore, if the profile of the potential has a relative extremum, then a real quantity omega_min exists such that, if sigma>k*omega_min, k=1,2,3,..., then there are k extra eigenfunctions having eigenvalue sigma. For each k, and for l=1,2,3,... k, there are k integer sequences m_l=l,2,3,..., such that each of the above extra eigenfunctions having eigenvalue sigma belongs to a set of infinitely countably many eigenfunctions having non degenerate eigenvalues +-m_(k-l+1)*sigma*k. The eigenfunctions of each of these discrete sets are defined over a set dependent, bounded domain of the space coordinate and they are orthonormal and complete over that domain.