Approximate solution of direct and inverse one-dimensional scattering problems on finite intervals for complex-valued potentials

The one-dimensional scattering problem for a complex-valued potential of compact support is investigated, including both the direct and inverse formulations. A method is developed for computing the reflection and transmission coefficients, based on representations of the solutions (and their derivatives) of the Schrödinger equation in the form of Neumann series of Bessel functions. These series converge uniformly with respect to the square root of the spectral parameter on any strip parallel to the real axis, allowing for highly accurate approximations of the reflection and transmission coefficients over wide ranges of the spectral parameter. The inverse one-dimensional scattering problem of recovering a complex-valued potential of compact support is examined in two distinct settings: one where both the reflection and transmission coefficients are provided, and another where only the reflection coefficient is given. A method for approximately solving both inverse scattering problems using Neumann Series of Bessel functions is developed and tested. To determine the value of the unknown potential at a specific point within the interval, two systems of linear algebraic equations must be solved sequentially. Numerical examples demonstrate the accuracy and efficiency of the proposed approach.