Solution of the radiative transfer equation in combination with rayleigh and isotropic scattering

A theory is constructed for solving half-space, boundary-value problems for the Chandrasekhar equations, describing the propagation of polarized light, for a combination of Rayleigh and isotropic scattering, with an arbitrary probability of photon survival in an elementary act of scattering. A theorem on resolving a solution into eigenvectors of the discrete and continuous spectra is proven. The proof comes down to solving a vector, Riemann—Hilbert, boundary-value problem with a matrix coefficient, the diagonalizing matrix of which has eight branching points in the complex plane. Isolation of the analytical branch of the diagonalizing matrix enables one to reduce the Riemann—Hilbert problem to two scalar problems based on a [0, 1] cut and two vector problems based on an auxiliary cut. The solution of the Riemann—Hilbert problem is given in the class of meromorphic vectors. The conditions of solvability enable one to uniquely determine the unknown expansion coefficients and free parameters of the solution of the boundary-value problem.