Use of partial knowledge of the potential in the phase problem of inverse scattering (Q1331432)

An algorithm for the numerical determination of the potential \(V(x)\) in a one-dimensional Schrödinger equation \(\Phi'' + (k^ 2 - V(x)) \Phi = 0\), \(- \infty < x < \infty\), is proposed using as data \(r(k) = | R(k) |^ 2\), the amplitude of the complex reflection coefficient, and partial knowledge of the potential, \(\{V(x) : 0 < x < \varepsilon\}\). The recovery of the phase involves the solution of a nonlinear optimization problem. Numerical examples showing effective reconstruction of the potential are presented.