Separability and the 3d Gelfand Levitan equation

 
 
 
  • Abstract
  • Keywords
  • References
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  • Abstract


    The 1D Gelfand-Levitan equation has been well studied with respect to the separability of the spectral measure function. The analytic solu-tion has been shown to be associated with reflectionless potentials. This paper considers the 3D version of this equation to see if an analytic solution can be found for a separable spectral measure function and if it also corresponds to known reflectionless potentials. Though the analytic solution is shown, it does not correspond to reflectionless potentials.

     

     


  • Keywords


    Inverse Scattering; Gelfand-Levitan Equation; Reflection Coefficient; One-Dimensional Scattering.

  • 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] Spectral Measure Function Separability and Reflectionless Potentials,” Eric Kincanon, Applied Mathematics and Computation, Vol 123, No 3, 409-412. https://doi.org/10.1016/S0096-3003(00)00086-2.

      [6] A Method to Construct Reflectionless Potentials,” Eric Kincanon, Vol 165, No 3, 565-569, Applied Mathematics and Computation, (June 2005). https://doi.org/10.1016/j.amc.2004.04.075.

      [7] Error Propagation in the Gel’Fand Levitan Equation,” Eric Kincanon, Vol 182, No 2, 1639-1641, Applied Mathematics and Computation, (November 2006). https://doi.org/10.1016/j.amc.2006.06.001.

      [8] V. Bargmann, On the Connection between Phase Shifts and Scattering Potential. Rev. Mod. Phys. 21 (1949), 488-493. https://doi.org/10.1103/RevModPhys.21.488.

      [9] I. Kay, H. E. Moses, A simple verification of the Gelfand-Levitan equation for the three-dimensional problem, Comm on Pure and Applied Mathematics, 14 (1961) 435-445. https://doi.org/10.1002/cpa.3160140319.


 

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Article ID: 32034
 
DOI: 10.14419/ijamr.v11i1.32034




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