On the Spectrum of the Transport Operator with Abstract Boundary Conditions in Slab Geometry

The paper is concerned with the spectral properties of the one-dimensional transport operator with general boundary conditions where an abstract boundary operator relates the incoming and the outgoing fluxes. We first give some existence and nonexistence results of eigenvalues in the half-plane λ ∈ Reλ > −λ∗ where −λ∗ is the type of the semigroup generated by the streaming operator. Next, we discuss the irreducibility of the transport semigroup. In particular, we show that the transport semigroup is irreducible if the boundary operator is strictly positive. We end the paper by investigating the strict monotonicity of the leading eigenvalue of the transport operator with respect to different parameters of the equation. Finally, an open problem is indicated