Finite difference methods for half inverse Sturm-Liouville problems

We investigate finite difference solution of the Hochstadt-Lieberman problem for a Sturm-Liouville operator defined on (0, π): given the value of the potential q on (c, π), where 0 < c < π, use eigenvalues to estimate q on (0, c). Our methods use an asymptotic correction technique of Paine, de Hoog and Anderssen, and its extension to Numerov’s method for various boundary conditions. In the classical case c = π/2, Numerov’s method is found to be particularly effective. Since eigenvalue data is scarce in applications, we also examine stability problems associated with the use of the extra information on q when c < π/2, and give some suggestions for further research.