Abstract
Following Hejtmanek, we consider neutrons in infinite space obeying a linearized Boltzmann equation describing their interaction with matter in some compact setD. We prove existence of theS-matrix and subcriticality of the dynamics in the (weak-coupling) case where the mean free path is larger than the diameter ofD uniform in the velocity. We prove existence of theS-matrix also for the case whereD is convex and filled with uniformly absorbent material. In an appendix, we present an explicit example where the dynamics is not invertible onL 1+ , the cone of positive elements inL 1.
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References
Bell, I., Glasstone, S.: Nuclear reactor theory. Princeton, New Jersey: Van Nostrand 1970
Chernoff, P.: Note on product formulas for operator semigroups. J. Funct. Anal.2 238–242 (1968)
Cohen, E., Thirring, W.: The Boltzmann equation. Berlin-Heidelberg-New York: Springer 1973
Cook, J.: Convergence to the Mølers wave matrix. J. Math. Phys.36, 82–87 (1957)
Davison, B., Sykes, B.: Neutron transport theory. Oxford: Oxford University Press 1958
Ford, G., Uhlenbeck, G.: Lectures in statistical mechanics, A.M.S. Publications
Hejtmanek, J.: Streutheorie des Linearen Boltzmannoperators, Univ. of Wien Preprint
Hille, E., Phillips, R.S.: Functional analysis and semigroups. A.M.S. Colloq. Publ.31, 1957
Jörgens, K.: An asymptotic expansion in the theory of neutron transport. Comm. Pure Appl. Math.11, 219–242 (1958)
Kato, T.: Perturbation theory for linear operators. Berlin-Heidelberg-New York: Springer 1966
Kato, T.: Wave operators and similarity for non-self adjoint operators. Math. Ann.162, 258–279 (1966)
Lavitz, J., Marchand, J.P.: ed. Mathematical theory of scattering. Amsterdam: North Holland 1974
Lax, P., Phillips, R.: Scattering theory for dissipative hyperbolic systems. J. Funct. Anal.14, 172–235 (1973)
Lehner, J., Wing, G.: On the spectrum of an unsymmetric operator arising in the transport theory of neutrons. Comm Pure Appl. Math8, 217–234 (1955)
Lehner, J., Wing, G.: Solution of the linearized Boltzmann equation for the slab geometry. Duke Math. J.23, 125–142 (1956)
Mihalas, D.: Stellar atmospheres. W.H. Freeman & Co. 1970
Nelson, E.: Analytic vectors. Ann. Math.10, 572–614 (1959)
Reed, M., Simon, B.: Methods of modern mathematical physics, Vol. II Fourier Analysis, Self-Adjointness, New York: Academic Press, in press, expected May, 1975
Reed, M., Simon, B.: Methods of Modern Mathematical Physics, Vol. III, in preparation, expected publication Summer 1976
Trotter, H.: On the product of semigroups of operators. Proc. A.M.S.10, 545–557 (1959)
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Communicated by W. Hunziker
A. Sloan fellow; research partially supported by the U.S. NSF under Grant GP 39048
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Simon, B. Existence of the scattering matrix for the linearized Boltzmann equation. Commun.Math. Phys. 41, 99–108 (1975). https://doi.org/10.1007/BF01608751
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DOI: https://doi.org/10.1007/BF01608751