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 ℝ 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.
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References
S. Agmon, Lectures on Elliptic Boundary Problems, Van Nostrand, New York, 1964.
T. Angell, D. Colton, and A. Kirsch, The three dimensional inverse scattering problem for acoustic waves, J. Diff. Eq. 46(1982), 46–58.
T. Angell, R. Kleinman, and G. Roach, An inverse transmission problem for the Heimholte equation, Inverse Problems 3(1987), 149–180.
A.M. Berthier, Spectral Theory and Wave Operators for the Schrodinger Equation, Pitman, Boston, 1982.
V. Buslaev, Scattered plane waves, spectral asymptotics, and trace formulas in exterior problems, Dokl Akad. Nauk. SSSR 197(1971), 1067–1070.
D. Colton, The inverse scattering problem for time-harmonic accoustic waves, SIAM Review 26(1984), 323–350.
D. Colton, Partial Differential Equations, an Introduction, Random House, New York, 1988.
D. Colton and R. Kress, Integral Equation Methods in Scattering Theory, Wiley, New York, 1983.
D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Springer-Verlag, New York, 1992.
R. Courant and D. Hilbert, Methods of Mathematical Physics II, J. Wiley, New York, 1966.
D. Eidus, On the principle of limiting absorption, Mat. Sb. 57(1962), 13–44.
A. Erdélyi, Asymptotic Expansions, Dover, New York, 1956.
J. Helton and J. Ralston, The first variation of the scattering matrix, J. Diff. Equations 21(1976), 378–394.
E. Hille and R. Phillips, Functional Analysis and Semigroups, Colloq. Publ. Vol. 31, AMS, Providence, R.I., 1957.
L. Hörmander, The Analysis of Linear Partial Differential Operators, Vols. 3 and 4, Springer-Verlag, New York, 1985.
V. Isakov, Uniqueness ans stability in multi-dimensional inverse problems, Inverse Problems 9(1993), 579–621.
A. Jensen and T. Kato, Asymptotic behavior of the scattering phase for exterior domains, Comm. in PDE 3(1978), 1165–1195.
D. Jones and X. Mao, The inverse problem in hard acoustic scattering, Inverse Problems 5(1989), 731–748.
T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, New York, 1966.
A. Kirsch, The domain derivative and two applications in inverse scattering theory, Inverse Problems 9(1993), 81–96.
A. Kirsch, R. Kress, P. Monk, and A. Zinn, Two methods for solving the inverse scattering problem, Inverse Problems 4(1988), 749–770.
G. Kristensson and C. Vogel, Inverse problems for acoustic waves using the penalised liklihood method, Inverse Problems 2(1984), 461–479.
P. Lax and R. Phillips, Scattering Theory, Academic Press, New York, 1967.
P. Lax and R. Philips, Scattering theory, Rocky Mountain J. Math. 1(1971), 173–223.
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.
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.
P. Lax and R. Phillips, Scattering Theory for Automorphic Functions, Princeton Univ. Press, Princeton, N. J., 1976.
A. Majda and J. Ralston, An analogue of Weyl’s theorem for unbounded domains, Duke Math. J. 45(1978), 513–536.
A. Majda and M. Taylor, Inverse scattering problems for transparent obstacles, electromagnetic waves, and hyperbolic systems, Comm. PDE 2(1977), 395–438.
R. Melrose, Scattering theory and the trace of the wave group, J. Func. Anal. 45(1982), 29–40.
R. Melrose, Polynomial bound on the number of scattering poles, J. Func. Anal. 53(1983), 287–303.
R. Melrose, Weyl asymptotics for the phase in obstacle scattering, Comm. in PDE 11(1988), 1431–1439.
R. Melrose, Geometric Scattering Theory, Cambridge Univ. Press, Cambridge, 1995.
R. Melrose and M. Taylor, Near peak scattering and the corrected Kirchhoff approximation for convex bodies, Adv. in Math. 55(1985), 242–315.
R. Melrose and M. Taylor, Boundary Problems for Wave Equations with Grazing and Gliding Rays. Monograph, in preparation.
K. Miller, Least squares methods for ill-posed problems with a prescribed bound, SIAM J. Math. Anal. 1(1970), 52–74.
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.
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.
C. Morawetz, The limiting amplitude principle, Comm. Pure Appl. Math. 15 (1962), 349–361.
C. Morawetz, Exponential decay of solutions to the wave equation, Comm. Pure Appl. Math. 19(1966), 439–444.
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.
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.
R. G. Newton, Scattering Theory of Waves and Particles, Springer-Verlag, New York, 1966.
H. Nussensweig, High frequency scattering by an impenetrable sphere, Ann. Phys. 34(1965), 23–95.
F. Olver, Asymptotics and Special Functions, Academic Press, New York, 1974.
V. Petkov and G. Popov, Asymptotic behavior of the scattering phase for nontrapping obstacles, Ann. Inst. Fourier (Grenoble) 32(1982), 111–149.
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.
A. Ramm, Scattering by Obstacles, D. Reidel, Dordrecht, 1986.
J. Rauch, Partial Differential Equations, Springer-Verlag, New York, 1991.
J. Rauch and M. Taylor, Potential and scattering theory on wildly perturbed domains, J. Func. Anal. 18(1975), 27–59.
M. Reed and B. Simon, Methods of Mathematical Physics, Academic Press, New York, Vols. 1,2, 1975; Vols. 3,4,1978.
A. Roger, Newton-Kantorovich algorithm applied to an electromagnetic inverse problem, IEEE Trans. Antennas Propagat 29(1981), 232–238.
B. Simon, Quantum Mechanics for Hamiltonians Defined as Quadratic Forms, Princeton Univ. Press, Princeton, N. J., 1971.
B. Simon, Trace Ideals and Their Applications, London Math. Soc. Lecture Notes no. 35, Cambridge Univ. Press, Cambridge, 1979.
I. Stakgold, Boundary Value Problems of Mathematical Physics, Macmillan, New York, 1968.
M. Taylor, Propagation, reflection, and diffraction of singularities of solutions to wave equations, Bull. AMS 84(1978), 589–611.
M. Taylor, Fourier integral operators and harmonic analysis on compact manifolds, Proc. Symp. Pure Math. 35(pt. 2)(1979), 113–136.
M. Taylor, Pseudodifferential Operators, Princeton Univ. Press, Princeton, N. J., 1981.
M. Taylor, Estimates for approximate solutions to acoustic inverse scattering problems, IMA Preprint #1303, 1995.
V. K. Varadan and V. V. Varadan (eds.), Acoustic, Electromagnetic, and Elastic Wave Scattering — Focus on the T matrix Approach, Pergammon, New York, 1980.
G. Watson, Theory of Bessel Functions, Cambridge Univ. Press, Cambridge, 1945.
C. Wilcox, Scattering Theory for the d’Alembert Equation in Exterior Domains, LNM no. 442, Springer-Verlag, New York, 1975.
K. Yosida, Functional Analysis, Springer-Verlag, New York, 1965.
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Taylor, M.E. (1996). Scattering by Obstacles. In: Partial Differential Equations II. Applied Mathematical Sciences, vol 116. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-4187-2_3
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DOI: https://doi.org/10.1007/978-1-4757-4187-2_3
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