On a boundary value method for computing Sturm–Liouville potentials from two spectra

Abstract

Convergence of a boundary value method (BVM) in Aceto et al. [Boundary value methods for the reconstruction of Sturm–Liouville potentials, Appl. Math. Comput. 219 (2012), pp. 2960–2974] for computing Sturm–Liouville potentials from two spectra is discussed. In Aceto et al. (2012), a continuous approximation of the unknown potential belonging to a suitable function space of finite dimension is obtained by forming an associated set of nonlinear equations and solving these with a quasi-Newton approach. In our paper, convergence of the quasi-Newton approach is established and convergence of the estimate of the unknown potential, provided by the exact solution of the nonlinear equation, to the true potential is proved. To further investigate the properties of the BVM in Aceto et al. (2012), some other spaces of functions are introduced. Numerical examples confirm the theoretically predicted convergence properties and show the accuracy and stability of the BVM.