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Seeing the Invisible and Maxwell’s Equations

Chapter
Part of the Trends in Mathematics book series (TM)

Abstract

In this chapter we study inverse scattering for Dirac operators with scalar, vector and quaternionic potentials. For that we consider factorizations of the Helmholtz equation and related fundamental solutions; the standard Green’s function and Faddeev’s Green function. This chapter is motivated by optical coherence tomography.

Keywords

Optical coherence tomography Dirac operator inverse scattering Faddeev’s Green function. 

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References

  1. [1]
    M.J. Ablowitz and A.P. Clarkson. Solitons, Nonlinear Evolution Equations and Inverse Scattering, volume 149 of London Mathematical Society Lecture Note Series. Cambridge University Press, 1991.Google Scholar
  2. [2]
    R. Beals and R.R. Coifman. Multidimensional inverse scattering and nonlinear partial differential equations. In F. Trèves, editor, Pseudodifferential Operators and Applications, volume 43 of Proceedings of Symposia in Pure Mathematics, pages 45–70. American Mathematical Society, 1984.Google Scholar
  3. [3]
    S. Bernstein. Factorization of the Schrödinger operator. In W. Sprößig, editor, Proceedings of the Symposium Analytical and Numerical Methods in Quaternionic and Clifford analysis, pages 1–6, 1996.Google Scholar
  4. [4]
    S. Bernstein. Lippmann–Schwinger’s integral equation for quaternionic Dirac operators. unpublished, available at http://euklid.bauing.uni-weimar.de/ikm2003/ papers/46/M_46.pdf, 2003.
  5. [5]
    D. Colton and R. Kress. Inverse Acoustic and Electromagnetic Scattering Theory, volume 93 of Applied Mathematical Sciences. Springer-Verlag, Berlin, Heidelberg, New York, 1992.Google Scholar
  6. [6]
    L.D. Faddeev. Increasing solutions of the Schrödinger equation. Doklady Akademii Nauk SSSR, 165:514–517, 1965.MathSciNetGoogle Scholar
  7. [7]
    A.F. Fercher. Optical coherence tomography. Journal of Biomedical Optics, 1(2):153– 173, 1996.CrossRefGoogle Scholar
  8. [8]
    A.F. Fercher, W. Drexler, C.K. Hitzenberger, and T. Lasser. Optical coherence tomography – principles and applications. Reports on Progress in Physics, 66:239–303, 2003.CrossRefGoogle Scholar
  9. [9]
    A.S. Fokas. A unified transform method for solving linear and certain nonlinear PDEs. Proceedings of the Royal Society A, Mathematical, Physical & Engineering Sciences, 453:1411–1443, 1997.MathSciNetMATHCrossRefGoogle Scholar
  10. [10]
    K. Gürlebeck and W. Sprößig. Quaternionic and Clifford Calculus for Physicists and Engineers. Wiley, Aug. 1997.Google Scholar
  11. [11]
    K. Gürlebeck. Hypercomplex factorization of the Helmholtz equation. Zeitschrift für Analysis und ihre Anwendungen, 5(2):125–131, 1986.MATHGoogle Scholar
  12. [12]
    K. Imaeda. A new formulation of classical electrodynamics. Nuovo Cimento, 32(1):138–162, 1976.MathSciNetCrossRefGoogle Scholar
  13. [13]
    H. Isozaki. Inverse scattering theory for Dirac operators. Annales de l‘I.H.P., section A, 66(2):237–270, 1997.Google Scholar
  14. [14]
    V.V. Kravchenko. On a new approach for solving Dirac equations with some potentials and Maxwell’s system in inhomogeneous media. In Operator theory: Advances and Applications, volume 121 of Operator Theory: Advances and Applications, pages 278–306. Birkhäuser Verlag, 2001.Google Scholar
  15. [15]
    V.V. Kravchenko. Quaternionic Reformulation of Maxwell’s Equations for Inhomogeneous Media and New Solutions. Zeitschrift für Analysis und ihre Anwendungen, 21(1):21–26, 2002.MathSciNetMATHGoogle Scholar
  16. [16]
    V.V. Kravchenko. Applied Quaternionic Analysis, volume 28 of Research and Exposition in Mathematics. Heldermann Verlag, 2003.Google Scholar
  17. [17]
    V.V. Kravchenko. On force-free magnetic fields: Quaternionic approach. Mathematical Methods in the Applied Sciences, 28:379–386, 2005.MathSciNetMATHCrossRefGoogle Scholar
  18. [18]
    V.V. Kravchenko and R.P. Castillo. An analogue of the Sommerfeld radiation condition for the Dirac operator. Mathematical Methods in the Applied Sciences, 25:1383– 1394, 2002.MathSciNetMATHCrossRefGoogle Scholar
  19. [19]
    V.V. Kravchenko and M.P. Ramirez. New exact solutions of the massive Dirac equation with electric or scalar potential. Mathematical Methods in the Applied Sciences, 23:769–776, 2000.MathSciNetMATHCrossRefGoogle Scholar
  20. [20]
    V.V. Kravchenko and M.V. Shapiro. On a generalized system of Cauchy-Riemann equations with a quaternionic parameter. Russian Academy of Sciences, Doklady., 47:315–319, 1993.Google Scholar
  21. [21]
    R. Leis. Initial Boundary Value Problems in Mathematical Physics. John Wiley, New York, 1986.Google Scholar
  22. [22]
    X. Li. On the inverse problem for the Dirac operator. Inverse Problems, 23:919–932, 2007.MathSciNetMATHCrossRefGoogle Scholar
  23. [23]
    A. McIntosh and M. Mitrea. Clifford algebras and Maxwell’s equations in Lipschitz domains. Mathematical Methods in the Applied Sciences, 22(18):1599–1999, 1999.MathSciNetMATHCrossRefGoogle Scholar
  24. [24]
    C. Müller. On the behavior of solutions of the differential equation δu = (f(x,u)) in a neighborhood of a point. Communications on Pure and Applied Mathematics, 7:505–515, 1954.MathSciNetMATHCrossRefGoogle Scholar
  25. [25]
    A.I. Nachman and M.J. Ablowitz. A multidimensional inverse scattering method. Studies in Applied Mathematics, 71:243–250, 1984.MathSciNetMATHGoogle Scholar
  26. [26]
    G. Nakamura and T. Tsuchida. Uniqueness for an inverse boundary value problem for Dirac operators. Communications in Partial Differential Equations, 25:7–8:557–577, 2000.Google Scholar
  27. [27]
    E.I. Obolashvili. Partial Differential Equations in Clifford Analysis, volume 96 of Pitman Monographs and Surveys in Pure and Applied Mathematics. Harlow: Addison Wesley Longman Ltd., 1998.Google Scholar
  28. [28]
    K.-E. Peiponen, E.M. Vartiainen, and T. Asakura. Dispersion, Complex Analysis and Optical Spectroscopy. Springer Tracts in Modern Physics. Springer Verlag, Berlin, Heidelberg, 1999.Google Scholar
  29. [29]
    M.H. Protter. Unique continuation principle for elliptic equations. Transactions of the American Mathematical Society, 95:81–90, 1960.MathSciNetMATHCrossRefGoogle Scholar
  30. [30]
    L. Päivärinta. Analytic methods for inverse scattering theory. In Y.K.K. Bingham and E. Somersalo, editors, New Analytic and Geometric Methods in Inverse Problems, pages 165–185. Springer, Berlin, Heidelberg, New York, 2003.Google Scholar
  31. [31]
    J. Radon. Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten. Berichte über die Verhandlungen der Sächsischen Akademie der Wissenschaften (Reports on the proceedings of the Saxony Academy of Science), 69:262–277, 1917.Google Scholar
  32. [32]
    J. Sylvester and G. Uhlmann. A global uniqueness theorem for an inverse boundary value problem. Annals of Mathematics, 125(1):153–169, Jan. 1987.MathSciNetMATHCrossRefGoogle Scholar

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© Springer Basel 2013

Authors and Affiliations

  1. 1.Institute of Applied AnalysisTU Bergakademie FreibergFreibergGermany

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