Remarks on the Calculation of Eigenvalues and Generalized Eigenfunctions Using a Finite Dimensional Approximation for the Inscattering Part of the Boltzmann Collision Operator

Abstract

From several points of view it is of advantage to know the properties of the collision operators in kinetic equations, e.g. in the well known Boltzmann equation, in particular for the purpose of solving eigenvalue problems. With regard to elastic, exciting and deexciting processes some attemps were recently made to investigate such operators in the Boltzmann equation describing the behaviour of electrons in weakly ionized plasmas. In the following we will prove that in the case of a finite dimensional inscattering operator the eigenvalues and the corresponding eigenfunctions can be represented explicitly. Finite dimensional operators were used successfully in special models to approximate the inscattering operators; they possess the property of compactness and are well suitable for analytical or numerical calculations. The representation has been obtained by solving an adequate linear equation system. The generalized eigenfunctions correspond to the normal solutions used by Case in the neutron transport theory. The regularization of the singular integrals which are necessary to obtain this solution will be given in detail. Further a velocity dependence of the collision frequency which need not be monotonous in the considered case and the dependence on the direction could be included.