On the spectrum of neutron transport equations with reflecting boundary conditions

This dissertation is devoted to investigating the time dependent neutron transport equations with reflecting boundary conditions. Two typical geometries---slab geometry and spherical geometry---are considered in the setting of Lp including L1. Some aspects of the spectral properties of the transport operator A and the C0 semigroup T( t) generated by A are studied. It is shown under fairly general assumptions that the accumulation points of Pas(A) := sigma(A) ∩ {lcub}lambda : Relambda > -lambda*), if they exist, could only appear on the line Relambda = -lambda* (where lambda* is the essential infimum of the total collision frequency), and the spectrum of T(t) outside the disk {lcub}lambda : |lambda| ≤ exp(-lambda*t){rcub} consists of isolated eigenvalues of T(t) with finite algebraic multiplicity, and the accumulation points of sigma(T(t)) ∩ {lcub}lambda : |lambda| > exp(-lambda*t)), if they exist, could only appear on the circle {lcub}lambda : |lambda| = exp(-lambda* t){rcub}. Consequently, the asymptotic behavior of the time dependent solution is obtained.