Abstract
The unanswered mathematical questions of reactor theory can be divided roughly into two classes. The first class concerns the mathematical theory of the basic transport equations and of the approximations thereto. Thus, the multiplication and criticality equations are characteristic value equations but their operators do not belong to the class for which the characteristic value theory is well established: they are not normal. Much progress was made recently concerning the character and properties of the highest characteristic value and the corresponding characteristic vector but the properties of the lower characteristic values and vectors are not known. In particular, the extent of a continuous spectrum and the completeness of the whole set of characteristic vectors are not established in general. A simple example is given in which the transport operator has a continuous spectrum which has not been recognized to date. The need for a generalization of the characteristic value problem is pointed out and such a generalization is proposed.
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References
L Boltzmann, Vorlesungen über Gastheorie, Leipzig, JA Barth, 1896.
D Hawkins and S Ulam, MDDC 287 (1944), CJ Everett and S Ulam, LADC-533-534: LA-683, LA-690, and LA-707 (1948). F r a very brief summary, see F. de Hoffmann, The science and engineering of nuclear power, New York, Addison-Wesley Press, 1949, vol. 2, pp. 116–119. Also RP Feynman, F de Hoffmann and R Serber, J Nucl. En. vol. 3 (1956) p. 64: and LD Pal, Paper 1710, Second United Nations Conference on the Peaceful Uses of Atomic Energy, 1958 (“Second Geneva Conference ”)
S Borowitz and M Hamermesh, Phys. Rev. vol. 74 (1948) p. 1285.
See in particular CC Grosjean, Nuovo Cim. vol. 3 (1956) p. 1262 and Papers 1691-1692 of the Second Geneva Conference, 1958. See also E. P. Wigner, Phys. Rev. vol. 94 (1954) p. 17.
AM Weinberg and EP Wigner, The physical theory of neutron chain reactors, Chicago, University of Chicago Press, 1958, p. 406ff.
Garrett Birkhoff and RS Varga, Reactor criticality and non-negative matrices, Westinghouse Atomic Power Division Report no. 166 (1957)
G Frobenius, S-Ber. Akad. bliss. Berlin, 1912, p. 456.
GM Wing, J. Math. Mech. vol. 7 (1958) p. 757: J Lehner and GM Wing, Comm. Pure Appl. Math. vol. 8 (1955) p. 217. See also other articles quoted by Wing at the present Symposium. Chapter VII of Linear operators (Part I) by N. Dun-ford and J. T. Schwartz, New York, Interscience Publishers, 1958, reviews most of the known general theorems concerning the spectra of linear, not necessarily normal, operators: it deals mostly with bounded operators. See also K. Friedrichs, Comm. Pure Appl. Math. Vol. 1 (1948) p. 361, and F. Rellich, Proceedings of the International Congress of Mathematics, Cambridge, Massachusetts, vol. 1 (1950) p. 606
EP Wigner, Gruppentheorie, and ihre Anwendung auf die Quantenmechanik der Atomspektren, Braunschweig, Friedr. Vieweg, 1931, Chapter 19
V Bargmann, Ann. of Math. Vol. 48 (1947) p. 568
E Greuling and C Marvin, Graphical method of obtaining critical masses of water-tamped boilers, Los Alamos Report 493. For a general discussion, see reference [5], p. 531ff and R. T. Ackroyd, Paper 36, Second Geneva Conference, 1958 and N. C. Francis, J. C. Stewart, L. S. Bohl, ibid., Paper 627.
B Davison, Neutron transport theory, Oxford, Clarendon Press, 1957, p. 98.
E Inönü, J. Nucl. Sci. Eng. Vol. 5 (1959) p. 248
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© 1992 Springer-Verlag Berlin Heidelberg
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Wigner, E.P. (1992). Mathematical Problems of Nuclear Reactor Theory. In: Weinberg, A.M. (eds) Nuclear Energy. The Collected Works of Eugene Paul Wigner, vol A / 5. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-77425-6_30
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DOI: https://doi.org/10.1007/978-3-642-77425-6_30
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