Invariant Imbedding and Case Eigenfunctions

Abstract

A new approach to the solution of transport problems, based on the ideas introduced into transport theory by Ambarzumian, Chandrasekhar, and Case, is discussed. To simplify the discussion, the restriction to plane geometry and one‐speed isotropic scattering is made. However, the method can be applied in any geometry with any scattering model, so long as a complete set of infinite‐medium eigenfunctions is known. First, the solution for the surface distributions is sought. (In a number of applications this is all that is required.) By using the infinite‐medium eigenfunctions, a system of singular integral equations together with the uniqueness conditions is derived for the surface distributions in a simple and straight‐forward way. This system is the basis of the theory. It can be reduced to a system of Fredholm integral equations or to a system of nonlinear integral equations, suitable for numerical computations. Once the surface distributions are known, the complete solution can be found by quadrature by using the fullrange completeness and orthogonality properties of the infinite‐medium eigenfunctions. The method is compared with the standard methods of invariant imbedding, singular eigenfunctions, and a new procedure recently developed by Case.