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Scattering by Obstacles

  • Michael E. Taylor
Part of the Applied Mathematical Sciences book series (AMS, volume 116)

Abstract

In this chapter we study the phenomenon of scattering by a compact obstacle in Euclidean space ℝ3. We restrict attention to the three-dimensional case, though a similar analysis can be given for obstacles in n whenever n is odd. The Huygens principle plays an important role in part of the analysis, and for that part the situation for n even is a little more complicated, though a theory exists there also.

Keywords

Inverse Problem Dirichlet Boundary Condition Pseudodifferential Operator Wave Operator Scattering Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [Ag]
    S. Agmon, Lectures on Elliptic Boundary Problems, Van Nostrand, New York, 1964.Google Scholar
  2. [ACK]
    T. Angell, D. Colton, and A. Kirsch, The three dimensional inverse scattering problem for acoustic waves, J. Diff. Eq. 46(1982), 46–58.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [AKR]
    T. Angell, R. Kleinman, and G. Roach, An inverse transmission problem for the Heimholte equation, Inverse Problems 3(1987), 149–180.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [Ber]
    A.M. Berthier, Spectral Theory and Wave Operators for the Schrodinger Equation, Pitman, Boston, 1982.Google Scholar
  5. [Bs]
    V. Buslaev, Scattered plane waves, spectral asymptotics, and trace formulas in exterior problems, Dokl Akad. Nauk. SSSR 197(1971), 1067–1070.MathSciNetGoogle Scholar
  6. [Co]
    D. Colton, The inverse scattering problem for time-harmonic accoustic waves, SIAM Review 26(1984), 323–350.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [Co2]
    D. Colton, Partial Differential Equations, an Introduction, Random House, New York, 1988.zbMATHGoogle Scholar
  8. [CK]
    D. Colton and R. Kress, Integral Equation Methods in Scattering Theory, Wiley, New York, 1983.zbMATHGoogle Scholar
  9. [CK2]
    D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Springer-Verlag, New York, 1992.zbMATHGoogle Scholar
  10. [CH]
    R. Courant and D. Hilbert, Methods of Mathematical Physics II, J. Wiley, New York, 1966.Google Scholar
  11. [Ei]
    D. Eidus, On the principle of limiting absorption, Mat. Sb. 57(1962), 13–44.MathSciNetGoogle Scholar
  12. [Erd]
    A. Erdélyi, Asymptotic Expansions, Dover, New York, 1956.zbMATHGoogle Scholar
  13. [HeR]
    J. Helton and J. Ralston, The first variation of the scattering matrix, J. Diff. Equations 21(1976), 378–394.MathSciNetzbMATHCrossRefGoogle Scholar
  14. [HiP]
    E. Hille and R. Phillips, Functional Analysis and Semigroups, Colloq. Publ. Vol. 31, AMS, Providence, R.I., 1957.zbMATHGoogle Scholar
  15. [Ho]
    L. Hörmander, The Analysis of Linear Partial Differential Operators, Vols. 3 and 4, Springer-Verlag, New York, 1985.Google Scholar
  16. [Isa]
    V. Isakov, Uniqueness ans stability in multi-dimensional inverse problems, Inverse Problems 9(1993), 579–621.MathSciNetzbMATHCrossRefGoogle Scholar
  17. [JeK]
    A. Jensen and T. Kato, Asymptotic behavior of the scattering phase for exterior domains, Comm. in PDE 3(1978), 1165–1195.MathSciNetzbMATHCrossRefGoogle Scholar
  18. [JM]
    D. Jones and X. Mao, The inverse problem in hard acoustic scattering, Inverse Problems 5(1989), 731–748.MathSciNetzbMATHCrossRefGoogle Scholar
  19. [Kt]
    T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, New York, 1966.zbMATHGoogle Scholar
  20. [Kir]
    A. Kirsch, The domain derivative and two applications in inverse scattering theory, Inverse Problems 9(1993), 81–96.MathSciNetzbMATHCrossRefGoogle Scholar
  21. [KZ]
    A. Kirsch, R. Kress, P. Monk, and A. Zinn, Two methods for solving the inverse scattering problem, Inverse Problems 4(1988), 749–770.MathSciNetzbMATHCrossRefGoogle Scholar
  22. [KV]
    G. Kristensson and C. Vogel, Inverse problems for acoustic waves using the penalised liklihood method, Inverse Problems 2(1984), 461–479.MathSciNetCrossRefGoogle Scholar
  23. [LP1]
    P. Lax and R. Phillips, Scattering Theory, Academic Press, New York, 1967.zbMATHGoogle Scholar
  24. [LP2]
    P. Lax and R. Philips, Scattering theory, Rocky Mountain J. Math. 1(1971), 173–223.MathSciNetzbMATHCrossRefGoogle Scholar
  25. [LP3]
    P. Lax and R. Phillips, Scattering theory for the acoustic equation in an even number of space dimensions, Indiana Univ. Math. J. 22(1972), 101–134.MathSciNetzbMATHCrossRefGoogle Scholar
  26. [LP4]
    P. Lax and R. Phillips, The time delay operator and a related trace formula, in Topics in Func. Anal., I. Gohberg and M. Kac, eds., Academic Press, New York, 1978, pp. 197–215.Google Scholar
  27. [LP5]
    P. Lax and R. Phillips, Scattering Theory for Automorphic Functions, Princeton Univ. Press, Princeton, N. J., 1976.zbMATHGoogle Scholar
  28. [MjR]
    A. Majda and J. Ralston, An analogue of Weyl’s theorem for unbounded domains, Duke Math. J. 45(1978), 513–536.MathSciNetzbMATHCrossRefGoogle Scholar
  29. [MjT]
    A. Majda and M. Taylor, Inverse scattering problems for transparent obstacles, electromagnetic waves, and hyperbolic systems, Comm. PDE 2(1977), 395–438.MathSciNetzbMATHCrossRefGoogle Scholar
  30. [Mel]
    R. Melrose, Scattering theory and the trace of the wave group, J. Func. Anal. 45(1982), 29–40.MathSciNetzbMATHCrossRefGoogle Scholar
  31. [Me2]
    R. Melrose, Polynomial bound on the number of scattering poles, J. Func. Anal. 53(1983), 287–303.MathSciNetzbMATHCrossRefGoogle Scholar
  32. [Me3]
    R. Melrose, Weyl asymptotics for the phase in obstacle scattering, Comm. in PDE 11(1988), 1431–1439.MathSciNetCrossRefGoogle Scholar
  33. [Me4]
    R. Melrose, Geometric Scattering Theory, Cambridge Univ. Press, Cambridge, 1995.zbMATHGoogle Scholar
  34. [MT1]
    R. Melrose and M. Taylor, Near peak scattering and the corrected Kirchhoff approximation for convex bodies, Adv. in Math. 55(1985), 242–315.MathSciNetzbMATHCrossRefGoogle Scholar
  35. [MT2]
    R. Melrose and M. Taylor, Boundary Problems for Wave Equations with Grazing and Gliding Rays. Monograph, in preparation.Google Scholar
  36. [Mr1]
    K. Miller, Least squares methods for ill-posed problems with a prescribed bound, SIAM J. Math. Anal. 1(1970), 52–74.MathSciNetzbMATHCrossRefGoogle Scholar
  37. [Mr2]
    K. Miller, Efficient numerical methods for backward solutions of parabolic problems with variable coefficients, in Improperly Posed Boundary Value Problems, A. Carasso and A. Stone, eds., Pitman Press, London, 1975.Google Scholar
  38. [MrV]
    K. Miller and G. Viano, On the necessity of nearly-best possible methods for analytic continuation of scattering data, J.Math Phys. 14(1973), 1037–1048.MathSciNetzbMATHCrossRefGoogle Scholar
  39. [Mw]
    C. Morawetz, The limiting amplitude principle, Comm. Pure Appl. Math. 15 (1962), 349–361.MathSciNetzbMATHCrossRefGoogle Scholar
  40. [Mw2]
    C. Morawetz, Exponential decay of solutions to the wave equation, Comm. Pure Appl. Math. 19(1966), 439–444.MathSciNetzbMATHCrossRefGoogle Scholar
  41. [MRS]
    C. Morawetz, J. Ralston, and W. Strauss, Decay of solutions of the wave equation outside nontrapping obstacles, Comm. Pure Appl. Math. 30(1977), 447–508.MathSciNetzbMATHCrossRefGoogle Scholar
  42. [MTW]
    R. Murch, D. Tan, and D. Wall, Newton-Kantorovich method applied to two-dimensional inverse scattering for an exterior Helmholtz problem, Inverse Problems 4(1988), 1117–1128.MathSciNetzbMATHCrossRefGoogle Scholar
  43. [New]
    R. G. Newton, Scattering Theory of Waves and Particles, Springer-Verlag, New York, 1966.Google Scholar
  44. [Nus]
    H. Nussensweig, High frequency scattering by an impenetrable sphere, Ann. Phys. 34(1965), 23–95.CrossRefGoogle Scholar
  45. [Olv]
    F. Olver, Asymptotics and Special Functions, Academic Press, New York, 1974.Google Scholar
  46. [PP]
    V. Petkov and G. Popov, Asymptotic behavior of the scattering phase for nontrapping obstacles, Ann. Inst. Fourier (Grenoble) 32(1982), 111–149.MathSciNetzbMATHCrossRefGoogle Scholar
  47. [R1]
    J. Ralston, Propagation of singularities and the scattering matrix, in Singularities in Boundary Value Problems, H. Garnir, ed., D. Reidel, Dordrecht, 1981, pp. 169–184.CrossRefGoogle Scholar
  48. [Rm]
    A. Ramm, Scattering by Obstacles, D. Reidel, Dordrecht, 1986.zbMATHCrossRefGoogle Scholar
  49. [Rau]
    J. Rauch, Partial Differential Equations, Springer-Verlag, New York, 1991.zbMATHCrossRefGoogle Scholar
  50. [RT]
    J. Rauch and M. Taylor, Potential and scattering theory on wildly perturbed domains, J. Func. Anal. 18(1975), 27–59.MathSciNetzbMATHCrossRefGoogle Scholar
  51. [RS]
    M. Reed and B. Simon, Methods of Mathematical Physics, Academic Press, New York, Vols. 1,2, 1975; Vols. 3,4,1978.Google Scholar
  52. [Rog]
    A. Roger, Newton-Kantorovich algorithm applied to an electromagnetic inverse problem, IEEE Trans. Antennas Propagat 29(1981), 232–238.MathSciNetzbMATHCrossRefGoogle Scholar
  53. [Si]
    B. Simon, Quantum Mechanics for Hamiltonians Defined as Quadratic Forms, Princeton Univ. Press, Princeton, N. J., 1971.zbMATHGoogle Scholar
  54. [Si2]
    B. Simon, Trace Ideals and Their Applications, London Math. Soc. Lecture Notes no. 35, Cambridge Univ. Press, Cambridge, 1979.zbMATHGoogle Scholar
  55. [Stk]
    I. Stakgold, Boundary Value Problems of Mathematical Physics, Macmillan, New York, 1968.zbMATHGoogle Scholar
  56. [T1]
    M. Taylor, Propagation, reflection, and diffraction of singularities of solutions to wave equations, Bull. AMS 84(1978), 589–611.zbMATHCrossRefGoogle Scholar
  57. [T2]
    M. Taylor, Fourier integral operators and harmonic analysis on compact manifolds, Proc. Symp. Pure Math. 35(pt. 2)(1979), 113–136.Google Scholar
  58. [T3]
    M. Taylor, Pseudodifferential Operators, Princeton Univ. Press, Princeton, N. J., 1981.zbMATHGoogle Scholar
  59. [T4]
    M. Taylor, Estimates for approximate solutions to acoustic inverse scattering problems, IMA Preprint #1303, 1995.Google Scholar
  60. [VV]
    V. K. Varadan and V. V. Varadan (eds.), Acoustic, Electromagnetic, and Elastic Wave ScatteringFocus on the T matrix Approach, Pergammon, New York, 1980.Google Scholar
  61. [Wat]
    G. Watson, Theory of Bessel Functions, Cambridge Univ. Press, Cambridge, 1945.Google Scholar
  62. [Wil]
    C. Wilcox, Scattering Theory for the d’Alembert Equation in Exterior Domains, LNM no. 442, Springer-Verlag, New York, 1975.zbMATHGoogle Scholar
  63. [Yo]
    K. Yosida, Functional Analysis, Springer-Verlag, New York, 1965.zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1996

Authors and Affiliations

  • Michael E. Taylor
    • 1
  1. 1.Department of MathematicsUniversity of North CarolinaChapel HillUSA

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