Asymptotic spectrum of the linear Boltzmann equation with general boundary conditions in finite bodies

The purpose of this paper is the study of the spectral properties of both streaming operator and transport operator with general boundary conditions in multidimensional bounded geometry. We discuss the asymptotic spectrum: existence and nonexistence results of eigenvalues in the half-plane {λ∈C:Reλ>s(TH)} where s(TH) stands for the spectral bound of the streaming operator TH. Next, we discuss the irreducibility of the transport semigroup. In particular, we establish that the transport semigroup is irreducible if the boundary operator is strictly positive. Afterwards, we discuss the strict monotonicity of the leading eigenvalue (when it exists) of the transport operator with respect to different parameters of the equation. Our analysis is based essentially on results from the theory of positive linear operators.