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Computing Sturm–Liouville potentials from two spectra

Alan L Andrew

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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.


PACS

02.30.Zz Inverse problems

02.30.Hq Ordinary differential equations

02.60.Lj Ordinary and partial differential equations; boundary value problems

02.70.Bf Finite-difference methods

MSC

65L12 Finite difference methods

65L09 Inverse problems

65L15 Eigenvalue problems

Subjects

Mathematical physics

Computational physics

Dates

Issue 6 (December 2006)

Received 18 July 2006, in final form 22 September 2006

Published 13 October 2006



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