This article is concerned with eigenvalue problems of the form Au = λTu in a Hilbert space H, where Ais a selfadjoint positive operator generated by a second-order Sturm-Liouville differential expression and T a selfadjoint indefinite multiplicative operator which is one-to-one. Emphasis is on the full-range and partial-range expansionproperties of the eigenfunctions.

An attempt is made to provide a sound basis for the method of singular eigenfunction expansions which has been in vogue in linear transport theory for some decades. The procedure is exemplified by a treatment of the one-dimensional neutron transport equation with a degenerate scattering function. Full-range as well as half-range results are derived. At the end of the paper the implications for a certain matrix factorization problem are given.

The asymptotic behaviour of second-order recurring seequences is studied.

The aim of this paper is to give a functional analytic treatment of the homogeneous and inhomogeneous linear transport equation in the case that the parameter c occurring in that equation equals 1. The larger part of the paper is devoted to the study of a certain operator T−1 A in the space L2(– 1, 1). A peculiarity not arising in the case c < 1 (treated amongst others by Hangelbroek) is that, for c = 1, the operator T−1A has a double eigenvalue 0 and that it is no longer hermitian. The Spectral Theorem is used to diagonalise the operator as far as possible, and full-range and half-range formulae are derived. The results are applied inter alia to give a new treatment of the Milne problem concerning the propagation of light in a stellar atmosphere.

We use the notations and results of Part I of this paper [see 7]. Sections and formulae of that paper will be referred to as section I.4, formula (1.5), etc. In particular, T, A, TN0, K, σ, F and Λ(λ) will have the same meaning as in [7].