Energy-Dependent Neutron Transport Theory in Plane Geometry III. Half-Range Completeness and Half-Space Problems

A set of elementary solutions to the energy-dependent Boltzmann equation, which was derived in the preceding paper, is shown to possess a half-range completeness property that allows the exact solution to energy-dependent half-space problems and the reduction of finite-slab problems to rapidly convergent Fredholm equations. Results follow in analogy with Case's work on the one-velocity transport equation, except that a system of singular integral equations is encountered, which gives rise to the Hilbert problem for matrices. It is shown that the methods of Muskhelishvili and Vekua are applicable to this matrix problem and lead to the consideration of a class of Fredholm equations to obtain the solution. The explicit form of the Fredholm equation for the present problem is derived by extending the analysis of the scalar Hilbert problem to the matrix case. Applications of the completeness proof are made to the albedo and Milne problems for a half space.