On multigroup transport theory with a degenerate transfer kernel

Abstract

The multigroup transport operator is studied in plane geometry assuming that the transfer kernel can be represented in a degenerate form. The eigenvalue spectrum is analyzed constructing the pertinent dispersion matrix in a block matrix form. The associated eigensolutions are obtained in terms of generalized functions. The adjoint operator is also considered for the purpose of demonstrating the full-range orthogonality relations. In particular, it is proven that the direct and adjoint eigenvalue spectra are identical. The full-range completeness of the eigensolutions is established under rather general conditions. For the half-range completeness to hold it is additionally required that the scattering kernel is self adjoint and possesses reflection symmetry, i.e., the dispersion matrix is symmetric and even. Finally, the infinite medium Green's function is derived employing the orthogonality relations, and the extrapolation distance for the Milne problem is calculated in terms of the emergent distribution.