Computing Sturm–Liouville potentials from two spectra

Abstract

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.

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.…”

“…Section 3 reviews recent results on sufficient conditions for the computed potential to converge to the true potential, q , as the number of data points increases. It also announces a new result on the convergence of Numerov's method [7] for the two spectra problem, and discusses possible extensions.…”

“…U.2 The (closely related [7]) two spectra problem. If both λ i (q, c 1 , c 2 ) and λ i (q, c 1 , c 3 ) are given for all i ∈ N , and sin(c 2 − c 3 ) = 0 , then q is uniquely determined in L 2 (0, π) .…”

“…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.…”

“…Gao et al [14] showed that this limitation could be overcome by using interpolation to refine the mesh. An older approach allowing indefinite mesh refinement seeks a potential of the form n i=1 c i φ i , for some specified φ i [7]. Asymptotic correction has proved useful for this also [1,8,15].…”