Advertisement

Phase Space Factorization and Functional Integral Methods in Direct and Inverse Scattering — Symbol Analysis

  • Louis Fishman
Conference paper
Part of the Woodward Conference book series (WOODWARD)

Abstract

The analysis and fast, accurate numerical computation of the wave equations of classical physics are often quite difficult for rapidly changing, multidimensional environments extending over many wavelengths. This is particularly so for environments characterized by a refractive index field with a compact region of arbitrary (n-dimensional) variability superimposed upon a transversely inhomogeneous ((n-1)-dimensional) background profile. For such environments, the entire domain is in the scattering regime, with the subsequent absence of an “asymptotically free” region. While classical, macroscopic methods have resulted in direct wave field approximations, derivations of approximate wave equations, and discrete numerical approximations, mathematicians studying linear partial differential equations have developed a sophisticated, microscopic phase space analysis centered about the theory of pseudo-differential and Fourier integral operators. In conjunction with the global functional integral techniques pioneered by Wiener (Brownian motion) and Feynman (quantum mechanics), and so successfully applied today in quantum field theory and statistical physics, the n-dimensional classical physics propagators can be both represented explicitly and computed directly. The phase space, or microscopic, methods and path (functional) integral representations provide the appropriate framework to extend homogeneous Fourier methods to inhomogeneous environments, in addition to suggesting the basis for the formulation and solution of corresponding arbitrary-dimensional nonlinear inverse problems.

Keywords

Pseudodifferential Operator Fourier Integral Operator Refractive Index Profile Symbol Analysis Path Integral Representation 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    L. Fishman, J.J. McCoy: J. Math. Phys. 25(2), 285 (1984).MathSciNetADSMATHCrossRefGoogle Scholar
  2. 2.
    L. Fishman, J.J. McCoy: J. Math. Phys. 25(2), 297 (1984).MathSciNetADSMATHCrossRefGoogle Scholar
  3. 3.
    L. Fishman, J.J. McCoy: IEEE Trans. Geosc. Rem. Sens. GE-22(6), 682 (1984).Google Scholar
  4. 4.
    L. Fishman, J.J. McCoy, S.C. Wales: J. Acoust. Soc. Am. 81(5), 1355 (1987).ADSCrossRefGoogle Scholar
  5. 5.
    L. Fishman, S.C. Wales: J. Comp. Appl. Math. 20, 219 (1987).MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    R.P. Gilbert, D.H. Wood: Wave Motion 8,383 (1986).MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    E. Witten: Physics Today 38 (July, 1980).Google Scholar
  8. 8.
    M.E. Taylor: Pseudodifferential Operators (Princeton University Press, Princeton 1981).MATHGoogle Scholar
  9. 9.
    F. Treves: Introduction to Pseudodifferential and Fourier Integral Operators. Volume 2. Fourier Integral Operators (Plenum Press, New York 1980 ).Google Scholar
  10. 10.
    J. Duistermaat: Fourier Integral Operators ( Courant Institute Lecture Notes, New York 1974 ).Google Scholar
  11. 11.
    L. Hörmander: Comm. Pure Appl. Math. 32, 359 (1979).MATHCrossRefGoogle Scholar
  12. 12.
    As yet unpublished work with M. Porter (SACLANT Centre, La Spezia, Italy).Google Scholar
  13. 13.
    A. Bamberger, B. Engquist, L. Halpern, and P. Joly: SIAM J. Appl. Math. 48(1), 129 (1988).MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag New York Inc. 1989

Authors and Affiliations

  • Louis Fishman

There are no affiliations available

Personalised recommendations