Boundary Value Methods for the reconstruction of Sturm-Liouville potentials

In this paper we present numerical procedures for solving the two inverse Sturm–Liouville problems known in the literature as the two-spectra and the half inverse problems. The method proposed looks for a continuous approximation of the unknown potential belonging to a suitable function space of finite dimension. In order to compute such an approximation a sequence of direct problems has to be solved. This is done by applying one of the Boundary Value Methods, generalizing the classical Numerov scheme, recently introduced by the authors. Numerical results confirming the effectiveness of the approach proposed are also reported.