Solution of the Inverse Scattering Problem for the Three-Dimensional Schrödinger Equation Using a Fredholm Integral Equation

It is shown that the inverse scattering problem for the three-dimensional Schrodinger equation with a potential having no spherical symmetry can be solved using a Fredholm integral equation. The integral operator studied here is shown to be compact and self-adjoint with its spectrum in $[ - 1,1]$. The relationship between solutions of this Fredholm equation and of a related Riemann–Hilbert problem is also clarified, and it is shown that the Fredholm integral equation is uniquely solvable if and only if the Riemann–Hilbert problem is uniquely solvable.