The Riemann-Hilbert vector boundary-value problem for the scattering of polarized light

A method of solving the Riemann-Hilbert vector boundary-value problem, that arises in the solution of half-space boundary-value problems for the vector equation of radiative transfer and which describes the scattering of polarized light with arbitrary photon survival probability (0 < omega < 1) in an elementary scattering event is developed. The method involves the diagonalization and subsequent factorization of the matrix coefficient of the boundary-value problem. The matrix reducing the coefficient to diagonal form has branch points in the complex plane. This requires the solution of two vector boundary-value problems on cuts joining the branch points, in addition to the basic boundary-value problem, which is given in the ''velocity half-space'' and which reduces to two scalar boundary-value problems. A theorem concerning the expansion of the solution of the boundary-value problem in terms of the eigenvectors of the discrete and continuous spectra is proved.