Eigenvalue problems for the Boltzmann operator in various formulations

This paper puts into precise mathematical terms the relationship between the integral equation for the scalar density, the integral equation for the angular density, and the integrodifferential equation for the angular density, by relating these problems to the linear monoenergetic Boltzmann equation with an isotropic scattering in plane geometry. In addition to recognizing the two formulations of the equation (integral and integrodifferential), their application to different situations (time-dependent and stationary problems) must be recognized. For the former, eigenvalues in the time-dependent are considered; for the latter, a pseudo-steady state problem is constructed using an appropriate eigenvalue. Only a few aspects of importance to subsequent discussions are brought up: to what extent can integral equations be used instead of integrodifferential equations. How could ''shadowed'' eigenvalues occur. A monoenergetic model is proposed that provides true eigenvalues beyond the continuous part of the spectrum. A model that could simulate the experimental situation reported by Grosshog in connection with pulsed neutron decays in bodies with cavities is constructed