Fine Spectral Analysis of Isotropic Partly Elastic Neutron Transport Operators
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Fine Spectral Analysis of Isotropic Partly Elastic Neutron Transport Operators Mustapha Mokhtar-Kharroubi1 · Yahya Mohamed2
Received: 4 November 2016 / Accepted: 21 December 2017 © Springer Science+Business Media B.V., part of Springer Nature 2018
Abstract This paper deals with spectral theory of a new class of neutron transport operators involving collision operators of the form K = Ki + Ke where Ki (resp. Ke ) describes the inelastic (resp. elastic) collisions of neutrons with the host material. We give a fine analysis of their asymptotic point spectra for isotropic space homogeneous cross sections in bounded geometries. We provide a new formalism relying on spectral analysis of some non compact symmetrizable operators arising in transport theory. Additional results on essential spectra are also given. Keywords Neutron transport · Partly elastic · Point spectra · Non compact symmetrizable operator Mathematics Subject Classification 47A75 · 49R05 · 82D75
1 Introduction Neutron transport operators are highly non-self-adjoint and their spectral analysis is an important issue in nuclear reactor theory, e.g. for pulsed neutron experiments or criticality problems; see [1, Chap. 5]. This is a classical mathematical topic which goes back at least to a seminal work by J. Lehner and M. Wing [8, 9] on a model case in slab geometry. The subject is now well understood for general inelastic models; in particular, in bounded geometries, the spectrum of their generators consists of a half plane and isolated eigenvalues
B M. Mokhtar-Kharroubi
[email protected] Y. Mohamed [email protected]
1
Laboratoire de Mathématiques, CNRS-UMR 6623, Université de Bourgogne Franche-Comté, 16 Route de Gray, 25030 Besançon, France
2
Institut National Universitaire Jean-François Champollion, Place de Verdun, 81012 Albi Cedex, France
M. Mokhtar-Kharroubi, Y. Mohamed
with finite algebraic multiplicities, see e.g. the monographs [3, 5, 11] while more recent developments can be found in [14] and references therein. On the other hand, spectral analysis of partly elastic neutron transport equations is a relatively new subject initiated by E.W. Larsen and P.F. Zweifel [4] and has been studied very little since then. Indeed, the presence of an elastic collision operator (besides an inelastic one) induces a loss of compactness which makes the structure of the spectrum of such kinetic models much more complicated; in particular spectral curves occur, see [4]. Significant spectral results on semigroups governing such models were obtained much later by M. Sbihi [19, 20]. The peculiarity of such kinetic equations is that both elastic and inelastic scattering phenomena are taken into account, i.e. the collision operator K consists of two parts of different nature K = Ki + K e where
ki x, v, v ψ x, v dv
Ki : ψ → V
describes the inelastic collisions of neutrons with the host material while ke x, ρ, ω, ω ψ x, ρω dω Ke : ψ → S N−1
describes their elastic collisions. (Actually, K has a third component, the s
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