Separability and the 3d Gelfand Levitan equation

  • Authors

    • Eric Kincanon Gonzaga University
  • Inverse Scattering, Gelfand-Levitan Equation, Reflection Coefficient, One-Dimensional Scattering.
  • 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.



  • References

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      [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.

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      [5] Spectral Measure Function Separability and Reflectionless Potentials,†Eric Kincanon, Applied Mathematics and Computation, Vol 123, No 3, 409-412.

      [6] A Method to Construct Reflectionless Potentials,†Eric Kincanon, Vol 165, No 3, 565-569, Applied Mathematics and Computation, (June 2005).

      [7] Error Propagation in the Gel’Fand Levitan Equation,†Eric Kincanon, Vol 182, No 2, 1639-1641, Applied Mathematics and Computation, (November 2006).

      [8] V. Bargmann, On the Connection between Phase Shifts and Scattering Potential. Rev. Mod. Phys. 21 (1949), 488-493.

      [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.

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  • How to Cite

    Kincanon, E. (2022). Separability and the 3d Gelfand Levitan equation. International Journal of Applied Mathematical Research, 11(1), 11-13.