Approximate recovery of the Sturm–Liouville problem on a half-line from the Weyl function

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Abstract The inverse problem of the reconstruction of the Sturm–Liouville problem on a half-line from its Weyl function is considered. Given the Weyl function, we obtain the potential in the Schrödinger equation and the boundary condition at the origin. If the boundary condition is known, this problem is equivalent to the inverse scattering problem of the recovery of the potential, which is zero on a half-line, from a given reflection coefficient. We develop a simple and direct method for solving the inverse problem. The method consists of two steps. First, the Jost solution is computed from the Weyl function, by solving a homogeneous Riemann boundary value problem. Second, a system of linear algebraic equations is constructed for the coefficients of series representations of three solutions of the Schrödinger equation. The potential and the boundary condition are then recovered from the first component of the solution vector of the system. A numerical illustration of the functionality of the method is presented.