Dependence of the fundamental time eigenvalue of linear transport operator on the system size and other parameters - An application of the Perron-Frobenius theorem

Many papers have been devoted to the study of the spectral properties of the linear (neutron) transport equation. Most of the theoretical investigations have concentrated on the existence (or otherwise) of a continuous spectrum, point spectrum, a leading/dominant eigenvalue, and a corresponding positive eigenvector. It is shown that the fundamental time eigenvalue of the linear transport operator increases with the size of the system. This follows from the increase in the largest eigenvalue of a non-negative irreducible matrix whenever any matrix element his increased. This result of matrix analysis is generalized to more general Krein-Rutman operators that leave a cone of vectors invariant