Numerov's method for inverse Sturm–Liouville problems

Abstract

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.

Solving inverse Sturm-Liouville problems: theory and practice

“…The idea is that, before solving the MIEVP, we add to each known eigenvalue a correction calculated so that, when q is constant, the MIEVP produces the same constant solution. This eliminates the source of failure of earlier attempts to solve ISLPs by finite difference methods: the asymptotic difference between the continuous and discrete eigenvalues [5,6,7,8,9]. In many important cases [4,8,9], the correction is known in closed form.…”

“…The symmetric problem [5]. If c 1 + c 2 = π , and q is known to satisfy q(x) = q(π − x) for almost all x ∈ (0, π) , then q is uniquely determined in L 2 (0, π) by the eigenvalues of (1-2).…”

“…Of the many available methods [9] for solving inverse Sturm-Liouville problems (ISLPs), we concentrate on those using the C127 technique of 'asymptotic correction' [4] (sometimes called the AAdHP correction [23]). This allows us to use a method such as Numerov's method to replace the ISLP by a more easily solved matrix inverse eigenvalue problem (MIEVP) [5]. The idea is that, before solving the MIEVP, we add to each known eigenvalue a correction calculated so that, when q is constant, the MIEVP produces the same constant solution.…”

“…Numerov's method [5,7,11] has the advantage that the matrices whose eigenvalues are required are tridiagonal, but the disadvantage that the mesh length is fixed by the number of available data points [5,7,8]. Gao et al [14] showed that this limitation could be overcome by using interpolation to refine the mesh.…”

“…These methods are much more computationally expensive than the simple Numerov's method [5,7,8], but are useful when available accuracy is limited by availability of data rather than by computational costs. Comparing results obtained by different methods can further increase the information obtained from limited data [5].…”