Two-group transport equation with a separable kernel

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Abstract
A two-group transport equation with a separable kernel is studied in the constant cross-section limit. Eigenfunctions and eigenvalues are derived, and full-range orthogonality proved (in an appendix, normalization integrals are derived). Then, certain completeness theorems are proved, namely: 1) If the transport equation reduces to a Hilbert problem involving continuous coefficients (and this covers all known cases) the eigenfunctions are complete on the full range. 2) If, in addition, the dispersion matrix R(z) is even, the eigenfunctions are also complete on the half range (this occurs, for example, if C(μ′, μ) = C(−μ′, −μ), which is a physically meaningful case.) 3) If the transport equation reduces to a Hilbert problem with discontinuous coefficients, then nothing can be said a priori about full-range completeness, but a procedure is developed, using certain theorems of Vekua, for determining the sign of an index ρ; the sign of ρ determines the existence of a full range expansion. 4) If the transformation matrix of the Hilbert problem obeys the Lipschitz condition, then full range completeness implies half range completeness, and vice versa. (This stringent condition is not necessary in the case of continuous coefficients). Throughout the work, certain reasonable assumptions are made, for example that the functions being expanded obey an extended Hölder condition. In an appendix a scalar singular integral equation is obtained which may be solved by analytical or numerical methods as necessary. This equation, which has quite similar form for the full- and half-range cases, contains a Fredholm term which is, in general, non-degenerate. It is curious that while no techniques are known for solving such equations, a solution can be obtained, at least in the full-range case, from orthogonality. Finally, in another appendix, a classification of types of transport problems is given in terms of the so-called “g-matrix”.