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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



An application of functional operator models to dissipative scattering theory

Author: Daniel Alan Bondy
Journal: Trans. Amer. Math. Soc. 223 (1976), 1-43
MSC: Primary 47A40
MathSciNet review: 0428077
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Abstract: An abstract framework for dissipative scattering theory is developed and then applied to two systems previously considered by P. Lax and R. S. Phillips. Results relating the poles and zeroes of the scattering matrix to the spectra of the infinitesimal generators A (which generates the semigroup formed by mapping initial data into solution data at time t) and B (which generates a ``local'' semigroup) are proven. In particular these results are shown to follow from the fact that the characteristic function of A (appropriately defined) and the scattering matrix combine to form the characteristic function of B.

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  • [1] V. M. Adamjan and D. Z. Arov, On a class of scattering operators and characteristic operator-functions of contractions, Dokl. Akad. Nauk SSSR 160 (1965), 9–12 (Russian). MR 0174981
  • [2] V. M. Adamjan and D. Z. Arov, On scattering operators and contraction semigroups in Hilbert space, Dokl. Akad. Nauk SSSR 165 (1965), 9–12 (Russian). MR 0188808
  • [3] V. M. Adamjan and D. Z. Arov, Unitary couplings of semi-unitary operators, Mat. Issled. 1 (1966), no. vyp. 2, 3–64 (Russian). MR 0206711
  • [4] R. G. Douglas and J. William Helton, Inner dilations of analytic matrix functions and Darlington synthesis, Acta Sci. Math. (Szeged) 34 (1973), 61–67. MR 0322538
  • [5] C. Foiaş, On the Lax-Phillips non conservative scattering theory, J. Functional Analysis 19 (1975), 273-301.
  • [6] Peter D. Lax and Ralph S. Phillips, Scattering theory, Pure and Applied Mathematics, Vol. 26, Academic Press, New York-London, 1967. MR 0217440
  • [7] P. D. Lax and R. S. Phillips, The acoustic equation with an indefinite energy form and the Schrödinger equation, J. Functional Analysis 1 (1967), 37–83. MR 0217441
  • [8] P. D. Lax and R. S. Phillips, Scattering theory for dissipative hyperbolic systems, J. Functional Analysis 14 (1973), 172–235. MR 0353016
  • [9] B. Sz.-Nagy and C. Foiaş, Analyse harmonique des opérateurs de l'espace de Hilbert, Masson, Paris; Akad. Kiadó, Budapest, 1967; English rev. transl., North-Holland, Amsterdam; American Elsevier, New York; Akad. Kiadó, Budapest, 1970. MR 37 #778; 43 #947.

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Keywords: Characteristic functions, dissipative scattering theory
Article copyright: © Copyright 1976 American Mathematical Society

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