Abstract
We study stability (under perturbations of B in the norm topology) of bounded solutions of the operator differential equation (*) (Tψ(x))′=−(I−B)ψ(x), 0<x<∞, with certain boundary conditions. Here T is a bounded selfadjoint operator with zero kernel, and B is a compact selfadjoint operator with eigenvalues not exceeding 1. An example of such equation is the transport equation which appears in astrophysics and neutron physics. It is proved that a bounded solution of (*) is stable provided it is unique. As a particular case of this stability result a mathematical justification of the cutting of Legendre series (a procedure used by physicists to calculate solutions of the transport equation) is obtained.
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References
[B] Beals, R.: On an abstract treatment of some forward-backward problems of transport and scattering, J. Funct. Anal 34 (1979), 1–20.
[BGK] Bart, H., Gohberg, I., Kaashoek, M.A.: Minimal factorization of matrix and operator functions, Birkhäuser Verlag, Basel, 1979.
[C] Chandrasekhar, S.: Radiative transfer, London, Oxford University Press 1950; New York, Dover Publ., 1960.
[CZ] Case, K.M., Zweifel, P.F.: Linear transport theory, Reading (Mass.), Addison-Wesley, 1967.
[F] Feldman, I.A.: On an approximation method for the equation of radiative energy transfer, Matem. Issled. 10 (3) (1975), 226–230 (Russian).
[GLR1] Gohberg, I., Lancaster, P., Rodman, L.: Matrices and indefinite scalar products, Birkhäuser Verlag, Basel, 1983.
[GLR2] Gohberg, I., Lancaster, P., Rodman, L.: Matrix polynomials, Academic Press, New York, etc., 1982.
[H] Hangelbroek, R.J.: Linear Analysis and solution of neutron transport problems, Transp. Theor. Stat. Phys. 5 (1976), 1–85.
[K] Kato, T.: Perturbation theory for linear operators, Springer Verlag, New York, 1966.
[M1] Van der Mee, C.V.M.: Semigroup and factorization methods in transport theory, Math. Centre Tract No. 146, Amsterdam, 1981.
[M2] Van der Mee, C.V.M.: Spectral analysis of the transport equation, II. Stability and application to the Milne problem, Integral equations and Operator Theory 5 (1982), 573–604.
[RR] Ran, A.C.M., Rodman, L.: Stability of invariant maximal semidefinite subspaces I, to appear in Linear Algebra and its Applications.
[S] Stummel, F.: Discrete convergence of linear operators, II, Math. Zeitschrift 120 (1971), 231–264 (in German).
[So] Sobolev, V.V.: Light scattering in planetary atmospheres, Oxford, Pergamon Press, 1975.
[T] Tricomi, F.G.: Vorlesungen über Orthogonalreihen, Springer-Verlag, Berlin, Göttingen, Heidelberg, 1955 (in German).
[V] Vladimirov, V.S.: Mathematical problems of the one-speed particle transport theory, Trudy Matem. Instituta im. V.A. Steklova, ANSSSR,61 (1961) (in Russian).
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The research of the second author was partially supported by the Fund for Basic Research Administrated by the Israel Academy for Sciences and Humanities. Part of the research was done while the second author was a senior visiting fellow at Vrije Universiteit, Amsterdam.
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Ran, A.C.M., Rodman, L. Stability of solutions of the operator differential equation in transport theory. Integr equ oper theory 8, 75–118 (1985). https://doi.org/10.1007/BF01199983
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DOI: https://doi.org/10.1007/BF01199983