On the essential spectrum of transport operators on L1-spaces

Abstract

In a recent article by the first author [J. Math. Phys. 35, 6199–6212 (1994)] the essential spectrum of transport operator was analyzed in Lp-spaces for p∈(1,+∞). The purpose of the present work is to extend this analysis to the case of L1-spaces. After establishing preliminary results we define the notion of the weak spectrum which we characterize by means of Fredholm operators. We show, in particular, that in L1-spaces the weak spectrum is nothing else but the essential spectrum. Using the same techniques as in the above-mentioned work, we prove the stability of the essential spectrum of a one-dimensional transport operator with general boundary conditions where an abstract boundary operator relates the incoming and the outgoing fluxes. Sufficient conditions are given in terms of boundary and collision operators, assuring the stability of the essential spectrum. We show also that our results remain valid for neutron transport operators in arbitrary dimension.