On Fundamental Equations of Spatially Independent Problems in Neutron Thermalization Theory

The linearized Boltzmann equation describing the velocity distribution of neutrons in an infinite monatomic gas is studied in the framework of the theory of Hilbert space. The corresponding eigenvalue problem is also investigated. It is found that the set of all eigenfunctions does not form a complete set. This result is contrary to the anticipation which seems to have been existed widely up to date. The more precise nature of the spectrum is examined by means of the perturbation theory of operators in Hilbert space. Especially the essential spectrum and the absolutely continuous part of the spectrum are determined completely. The asymptotic behavior of the solution of the time-dependent equation is stated as a consequence of our results on the spectrum. Proofs are mathematically regorous and any argument of heuristic or inconclusive character is avoided. Finally a conjecure on the eigenfunction expansion formula is stated.