Properties of integral transport operators

Abstract

Neutron diffusion in a uniform convex body is considered, and the time-independent distribution sought from the integral version of the Boltzmann equation. After the general properties of the scattering operator [Kcirc] are listed, the integral transport operator Ĝ[Kcirc] is investiated in the space C of bounded continuous functions, in the dual space C^† (containing the measures), as well as in appropriate1 weighted L₂ and L₁ —spaces. For a nonabsorbing medium the norm ‖Ĝ[Kcirc]‖ turns out to be equal to 1 in C-space, but may exceed that bound in the other three spaces. On the other hand, the spectral radius spr(Ĝ[Kcirc]) is shown to be smaller than 1 in the C and L₁—spaces, which assures the convergence of iterative solutions in both topologies. The operator ĜĴĜ[Kcirc], where Ĵ reverses the neutron velocity, is selfadjoint in L₂, which is useful in a variational method.