The boundary-value problem for the equations of radiation transfer of polarized light

The theory of the solution of half-space boundary-value problems for Chandrasekhar's equations describing the scattering of polarized light in the case of a combination of Rayleigh and isotropic scattering with arbitrary photon survival probability in an elementary scattering is constructed. A theorem on the expansion of the solution in terms of eigenvectors of discrete and continuous spectra is proved. The proof reduces to solving the Riemann–Hilbert vector boundary-value problem with a matrix coefficient. The matrix that reduces the coefficient to diagonal form has eight branch points in the complex plain. The definition of an analytical branch of a diagonalizing matrix gives us the opportunity to reduce the Riemann–Hilbert vector boundary-value problem to two scalar boundary-value problems on the major cut $[0,1]$ and two vector boundary value problems on the supplementary cut. The solution of the Riemann–Hilbert boundary-value problem is given in the class of meromorphic vectors. The solvability conditions enable unique determination of the unknown coefficients of the expansion and the free parameters of the solution.