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A stationary criticality problem in general \(L_ p\)-space for energy dependent neutron transport in cylindrical geometry. (English) Zbl 0547.45001

The authors study the energy-dependent neutron transport integral equation in a homogeneous cylinder of radius R and infinite height, with isotropic scattering, as an abstract equation \(f=Kf\) in the space \(L_ 1((0,1)\times (E_ m,E_ M))\). By means of techniques based on the theory of positive operators in Banach spaces, it is proved that the eigenvalue problem for the integral operator K admits as a solution a unique a.e. positive eigenfunction to which the leading eigenvalue \(\lambda_ 0\) corresponds. After establishing continuity and strictly increasing monotonicity of \(\lambda_ 0\) in R, the criticality problem is discussed and solved under the assumption of subcriticality for a non- multiplying medium. The formulation of the eigenvalue problem for K is finally extended to any \(L_ p\) space \(1\leq p<\infty\). Recalling that K is a Riesz operator in \(L_ p\), it is proved, as a general result, that the spectrum of K, acting on \(L_ p\), is independent of p.

MSC:

45C05 Eigenvalue problems for integral equations
47B60 Linear operators on ordered spaces
82C70 Transport processes in time-dependent statistical mechanics
82D45 Statistical mechanics of ferroelectrics
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