The spectrum of the multigroup neutron transport operator for bounded spatial domains

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Abstract
The spectrum of the multigroup neutron transport operator A is studied for bounded spatial regions D which consist of a finite number of material subregions. Our main results provide simple conditions on the material cross sections which guarantee that (1) A possesses eigenvalues in the finite plane; (2) A possesses a ’’leading’’ eigenvalue λ0 which is real, not less than the real part of any other eigenvalue, and to which there corresponds at least one nonnegative eigenfunction ψλ0; and (3) A possesses a ’’dominant’’ eigenvalue λ0 which is real, simple, greater than the real part of any other eigenvalue, and whose eigenfunction ψλ0 satisfies ψλ0⩾0 and ∫ψλ0d2Ω≳0. We give examples to illustrate the results and to show that a leading eigenvalue need not be simple, nor its eigenfunction(s) positive.

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