General statement of the inverse scattering problem as applied to the Schrodinger equation with nonlocal potentials is proposed for a family of phase-equivalent nonlocal potentials. The main result is that the problem of reconstructing both the family of phase-equivalent wavefunctions and the corresponding family of phase-equivalent potentials is reduced to solving some regular (quadratically nonlinear) integral equation. It is shown that if the S-matrix Sl(k) is rational in k, then for a dense set of nonlocal potentials the main equation turns into a system of second-order algebraic equations, and the corresponding potentials become separable. To illustrate this approach in more detail the author gives an exhaustive description of the family of transparent potentials.
We develop a consistent approach to an inverse scattering problem for the Schrodinger equation with nonlocal potentials. The main result presented in this paper is that for the two-body scattering data, given the problem of reconstructing both the family of phase equivalent two-body wavefunctions and the corresponding family of phase equivalent half-off-shell t-matrices, is reduced to solving a regular integral equation. This equation may be regarded as a generalization of the Gel’fand-Levitan equation.
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