Uniqueness of approximate solutions to the Gelfand Levitan equation

 
 
 
  • Abstract
  • Keywords
  • References
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  • Abstract


    This brief paper considers a potential issue of using iterative solutions for the Gelfand-Levitan equation. Iterative solutions require approx-imation methods and this could lead to a loss of uniqueness of solutions. The calculations in this paper demonstrate that this is not the case and that uniqueness is preserved.

     

     


  • Keywords


    Gelfand-Levitan Equation; Inverse Scattering; Ambiguities; Approximation Methods.

  • References


      [1] I.M. Gelfand, B.M. Levitan, On the determination of a differential equation by its spectral function, Dokl. Akad. Nauk. USSR 77 (1951) 557-560.

      [2] I.M. Gelfand, B.M. Levitan, On the determination of a differential equation by its spectral measure function, Izv. Akad. Nauk. SSR 15 (1951) 309-360.

      [3] K. Chadan, P.C. Sabatier, Inverse Problems in Quantum Scattering Theory, Springer-Verlag, New York, 1977. https://doi.org/10.1007/978-3-662-12125-2.

      [4] R. Jost, W. Kohn, On the relation between phase shift energy levels and the potential, Danske Vid. Selsk. Math. Fys. 27 (1953) 3-19.

      [5] E. Kincanon, An Orthogonal Set Composed from the Functions enx, Applied Mathematics and Computation, 41, (1991) 69-75. https://doi.org/10.1016/0096-3003(91)90107-X.

      [6] E. Kincanon, Approximate solution it the Gelfand-Levitan equation, Applied Mathematics and Computation, 53 (1993) 121-128. https://doi.org/10.1016/0096-3003(93)90097-X.

      [7] I. Kay, H.E. Moses, A Simple Verification of the Gelfand-Levitan Equation for the Three-Dimensional Scattering Problem, Communications on Pure and Applied Mathematics, 24 (1961) 435-445. https://doi.org/10.1002/cpa.3160140319.


 

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Article ID: 30522
 
DOI: 10.14419/ijamr.v9i1.30522




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