Separability and the 3d Gelfand Levitan equation


  • 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.




[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.

[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.

[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.