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This paper examines and extends a method, recently proposed by the author, for recovering from eigenvalues a symmetric potential of a Sturm-Liouville operator with Dirichlet boundary conditions. It uses Numerov's method and an extension by Andrew and Paine of an asymptotic correction technique of Paine, de Hoog and Anderssen. The method is extended to deal with natural boundary conditions and its convergence properties are investigated. Numerical results show that the method can extract more information from a given set of data than a related earlier method which uses a second-order discretization of the differential equation. Non-symmetric problems are also considered.

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Numerical results demonstrate the usefulness of the correction even for low values of k.

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This paper introduces and examines some new finite difference methods for computing the (generally nonsymmetric) potential of a Sturm–Liouville operator from its first m Dirichlet eigenvalues and its first m or m + 1 Dirichlet–Neumann eigenvalues. The methods use an asymptotic correction technique of Paine, de Hoog and Anderssen, and its extension to Numerov's method by Andrew and Paine. Numerical results suggest that Numerov's method has even greater advantages over related second-order methods for this problem than those recently reported for problems with symmetric potential.