On the spectrum of the linear transport operator

Source abstract

In this paper, spectral properties of the time-independent linear transport operator A are studied. This operator is defined in its natural Banach space L 1(D × V), where D is the bounded space domain and V is the velocity domain. The collision operator K accounts for elastic and inelastic slowing down, fission, and low energy elastic and inelastic scattering. The various cross sections in K and the total cross section are piecewise continuous functions of position and speed. The two cases ν0>0 and ν0=0 are treated, where ν0 is the minimum neutron speed. For ν0=0, it is shown that σ(A) consists of a full half-plane plus, in an adjoining strip, point eigenvalues and curves. For ν0>0, σ(A) consists just of point eigenvalues and curves in a certain half-space. In both cases, the curves are due to purely elastic ``Bragg'' scattering and are absent if this scattering does not occur. Finally the spectral differences between the two cases ν0>0 and ν0=0 are discussed briefly, and it is proved that A is the infinitesimal generator of a strongly continuous semigroup of operators.