Separability and the 3d Gelfand Levitan equation

Authors

  • Eric Kincanon Gonzaga University

DOI:

https://doi.org/10.14419/ijamr.v11i1.32034

Published:

2022-05-15

Keywords:

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

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.

 

 

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.

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[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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