Scattering theory for hyperbolic systems with coefficients of Gevrey type

Author:
William L. Goodhue

Journal:
Trans. Amer. Math. Soc. **180** (1973), 337-346

MSC:
Primary 35P25

MathSciNet review:
0415094

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Abstract: Using the techniques developed by P. D. Lax and R. S. Phillips, qualitative results on the location of the poles of the scattering matrix for symmetric, hyperbolic systems are obtained. The restrictions placed on the system are that the coefficient matrices be of Gevrey type and that the bicharacteristic rays tend to infinity.

**[1]**R. Courant and P. D. Lax,*The propagation of discontinuities in wave motion*, Proc. Nat. Acad. Sci. U.S.A.**42**(1956), 872–876. MR**0081420****[2]**Avner Friedman,*Regularity of fundamental solutions of hyperbolic equations*, Arch. Rational Mech. Anal.**11**(1962), 62–96. MR**0144076****[3a]**W. L. Goodhue,*On the location of the poles of the scattering matrix for symmetric hyperbolic equations*, Thesis, New York University, October 1971.**[3b]**William L. Goodhue,*On the regularity of the Riemann function for hyperbolic equations*, Trans. Amer. Math. Soc.**175**(1973), 483–490. MR**0318685**, 10.1090/S0002-9947-1973-0318685-5**[4]**Einar Hille and Ralph S. Phillips,*Functional analysis and semi-groups*, American Mathematical Society Colloquium Publications, vol. 31, American Mathematical Society, Providence, R. I., 1957. rev. ed. MR**0089373****[5a]**Peter D. Lax and Ralph S. Phillips,*Scattering theory*, Pure and Applied Mathematics, Vol. 26, Academic Press, New York-London, 1967. MR**0217440****[5b]***Lecture notes on scattering theory*, Rocky Mountain Regional Summer Math. Seminar, Flagstaff, Arizona, 1969.**[5c]**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****[6]**J.-L. Lions and E. Magenes,*Problèmes aux limites non homogènes et applications. Vol. 3*, Dunod, Paris, 1970 (French). Travaux et Recherches Mathématiques, No. 20. MR**0291887****[7]**D. Ludwig,*Exact and asymptotic solutions of the Cauchy problem*, Report NYO-2545, Atomic Energy Commission Research and Development.

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DOI:
http://dx.doi.org/10.1090/S0002-9947-1973-0415094-5

Keywords:
Scattering matrix poles,
hyperbolic systems,
spectrum of infinitesimal generator

Article copyright:
© Copyright 1973
American Mathematical Society