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

Abstract

Abstract-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]. DOI: 10.1134/S0012266115050092The present paper continues the paper [1] and the corresponding numbering of sections, formulas, theorems, corollaries, and remarks. ALGORITHMS FOR FINDING DISCRETE SPECTRA, EIGENFUNCTIONS, AND NORMALIZING CONSTANTS FOR REDUCED SCALAR CHARACTERISTIC EQUATIONS OF THE TRTThe following assertion can be proved with the use of systems (4), (8), and (9), Theorems 3-6, Corollary 1, Remark 3, and constructions used in [2,3] in the analysis and statement of properties of infinite continued fractions.