Constructive theory of scalar characteristic equations of the theory of radiation transport: II. algorithms for finding solutions and their analytic representations

We present methods for finding discrete spectra and derive analytic expressions for the eigenfunctions of scalar characteristic equations of the theory of radiation transport. We obtain new two-term recursion formulas and analytic representations for solutions of infinite tridiagonal systems of linear algebraic equations. We obtain analytic forms of the resolvents of scalar characteristic equations for phase functions square integrable on the closed interval [−1, 1]. In addition, we derive a general analytic expression for the Green function of a two-dimensional (with respect to the angular variables) integro-differential equation of the radiation transport for the case in which the phase functions satisfy the Hölder condition on the closed interval [−1, 1].