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Spectral representation and scattering theory for the wave equation with two unbounded media

Published online by Cambridge University Press:  24 October 2008

G. F. Roach  and
Bo Zhang
G. F. Roach
Affiliation:
Department of Mathematics, University of Strathclyde, Glasgow G1 1XH, Scotland
Bo Zhang
Affiliation:
Department of Mathematics, University of Strathclyde, Glasgow G1 1XH, Scotland
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Abstract

In this paper, we establish the generalized eigenfunction expansions for wave propagation in inhomogeneous, penetrable media in ℝn(n ≥ 2) with an unbounded interface. We then use them together with the method of stationary phase to prove the existence of the wave operators and to obtain the representations of the wave operators in terms of the generalized Fourier transforms.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 113 , Issue 2 , March 1993 , pp. 423 - 447
Copyright
Copyright © Cambridge Philosophical Society 1993

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References

REFERENCES

[1]Agmon, S.. Spectral properties of Schrödinger operators and scattering theory. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 4 (1975), 151–218.Google Scholar
[2]Agmon, S. and Hormander, L.. Asymptotic properties of solutions of differential equations with simple characteristics. J. Analyse Math. 30 (1976), 1–37.CrossRefGoogle Scholar
[3]Constantin, P.. Scattering for Schrödinger operators in a class of domains with non-compact boundaries. J. Funct. Anal. 44 (1981), 87–119.CrossRefGoogle Scholar
[4]Hormander, L.. The existence of wave operators in scattering theory. Math. Z. 146 (1976), 69–91.CrossRefGoogle Scholar
[5]Ikebe, T.. Spectral representation for the Schrödinger operators with long-range potentials. J. Funct. Anal. 20 (1975), 158–177.CrossRefGoogle Scholar
[6]Il'in, E. M.. Scattering by unbounded obstacles for elliptic operators of second order. Proc. Steklov Inst. Math. 2 (1989), 85–107.Google Scholar
[7]Jager, W.. Ein gewöhnlicher Differentialoperator zweiter Ordnung für Funktionen mit Werten in einem Hilbertraum. Math. Z. 113 (1970), 68–98.CrossRefGoogle Scholar
[8]Kato, T. and Kuroda, S. T.. Theory of simple scattering and eigenfunction expansions. In Functional Analysis and Related Fields (Springer-Verlag, 1970), pp. 99–131.Google Scholar
[9]Kato, T.. Perturbation Theory for Linear Operators (Springer-Verlag, 1984).Google Scholar
[10]Kuroda, S. T.. Scattering theory for differential operators I, II. J.Math. Soc. Japan 25 (1973), 75–104. 224–234.CrossRefGoogle Scholar
[11]Lax, P. D. and Phillips, R. S.. Scattering Theory (Academic Press, 1967).Google Scholar
[12]Leis, R.. Initial Boundary Value Problems in Mathematical Physics (John Wiley, 1986).CrossRefGoogle Scholar
[13]Leis, R. and Roach, G. F.. A transmission problem for the plate equation. Proc. Roy. Soc. Edinburgh Sect. A 99 (1985), 285–312.CrossRefGoogle Scholar
[14]Lyford, W. C.. Asymptotic energy propagation and scattering of waves in waveguides with cylinders. Math. Ann. 219 (1976), 193–212.CrossRefGoogle Scholar
[15]Petkov, V.. Scattering Theory for Hyperbolic Operators (North-Holland, 1989).Google Scholar
[16]Phillips, R. S.. Scattering theory for the wave equation with a short range perturbation. Indiana Univ. Math. J. 31 (1982), 602–639.CrossRefGoogle Scholar
[17]Roach, G. F. and Zhang, Bo. The limiting amplitude principle for the wave propagation problem with two unbounded media..Math. Proc. Cambridge Philos. Soc. 112 (1992), 207–223.CrossRefGoogle Scholar
[18]Saito, Y.. Spectral Representations for Schrodinger Operators with long-range Potentials. Lecture Notes in Math. vol. 727 (Springer-Verlag, 1979).CrossRefGoogle Scholar
[19]Schulenberger, J. R. and Wilcox, C. H.. The limiting absorption principle and spectral theory for steady-state wave propagation in inhomogeneous anisotropic media. Arch. Rational Mech. Anal. 41 (1971), 46–65.CrossRefGoogle Scholar
[20]Shenk, N. A.. Eigenfunction expansions and scattering theory for the wave equation in an exterior region. Arch. Rational Mech. Anal. 21 (1966), 120–150.CrossRefGoogle Scholar
[21]Weder, R.. Spectral and scattering theory in perturbed stratified fluids. J. Math. Pures Appl. (9) 64 (1985), 149–173.Google Scholar
[22]Weder, R.. Spectral and scattering theory in perturbed stratified fluids. II. Transmission problems and exterior domains. J. Differential Equations 64 (1986), 103–131.CrossRefGoogle Scholar
[23]Weder, R.. The limiting absorption principle at thresholds. J. Math. Pures Appl. (9) 67 (1988), 313–338.Google Scholar
[24]Weidmann, J.. Linear Operators in Hilbert Spaces (Springer-Verlag, 1980).CrossRefGoogle Scholar
[25]Wilcox, C. H.. Scattering Theory for the d'Alembert Wave Equation in Exterior Domains. Lecture Notes in Math. vol. 442 (Springer-Verlag, 1975).CrossRefGoogle Scholar