Enclosing solutions of an inverse Sturm-Liouville problem with finite data

This paper is concerned with the reconstruction of an unknown potentialq(x) in the Sturm-Liouville problem with Dirichlet boundary conditions, when only a finite number of eigenvalues are known. The problem is transformed into a system of nonlinear equations. A solution of this system is enclosed in an interval vector by an interval Newton's method. From the interval vector, an interval functionq](x) is constructed that encloses a potentialq(x) corresponding to the prescribed eigenvalues. To make this numerical existence proof rigorous, of course, all discretization and rounding errors have to be taken into account in the computation.