Discrete Eigenvalues of Case Spectrum with Anisotropic Scattering

Abstract

Discrete eigenvalues of a one-speed linear transport equation with anisotropic scattering are studied. It is shown that there is only one pair of real discrete eigenvalues for linear, quadratic, or triplet scattering for a nonmultiplicative medium. For a multiplicative medium there is one imaginary pair of eigenvalues or at most four eigenvalues. These can form one real and one imaginary pair, two imaginary pairs, or a quartet. The range of parameters for these different situations is derived analytically. These are then supported by numerical results that are tabulated in tables for each type of scattering.

Keywords:

• Discrete eigenvalues
• transport equation
• anisotropic scattering