Multigroup caseology in 1D via the Fourier transform

An alternative view of the Fourier transform inversion provides a new solution to the age-old problem of multigroup neutral particle transport equation in one-dimensional plane geometry. Through analytical continuation, the inversion contour shifts from the real line to accommodate pole contributions as discrete matrix eigenfunctions with continuum contributions appear from the branch cut. We recast the solution in terms of continuum singular matrix eigenfunctions to derive a singular eigenfunction expansion. Closure is an immediate consequence of the Fourier transform inversion. Finally, based on knowledge of the matrix weight factors and closure, we construct a singular eigenfunction expansion, incorporating orthogonality. The significance of this work is that a concise and consistent multigroup eigenfunction expansion emerges for anisotropic scattering that is as easily applied as in the one-group case.