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IMA Journal of Numerical Analysis Advance Access published online on March 27, 2009
IMA Journal of Numerical Analysis, doi:10.1093/imanum/drn062
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© The author 2009. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Efficient numerical solution of the one-dimensional Schrödinger eigenvalue problem using Magnus integrators

Veerle Ledoux{dagger}, Marnix Van Daele and Guido Vanden Berghe

Vakgroep Toegepaste Wiskunde en Informatica, Ghent University, Krijgslaan 281-S9, B-9000 Gent, Belgium

{dagger} Corresponding author. Email: veerle.ledoux{at}ugent.be

Received on 22 October 2007. Accepted for publication 13 February 2008.


  Abstract

We discuss a new numerical method, based on a modified Magnus integrator, to solve the Sturm–Liouville eigenvalue problem in its Schrödinger form. A modified Magnus integrator has already been used by Degani & Schiff (2006, J. Comput. Appl. Math., 193, 413–436) to approximate the oscillating solution of a Schrödinger problem over the classically allowed region. Here we show that the technique can be successfully extended to the nonoscillatory classically forbidden region. This means that the modified Magnus integrator can be used to propagate the solution over the whole integration interval and is well suited for application in a shooting process to locate the eigenvalues. Such a shooting procedure is formulated and shown to allow the efficient computation of a range of eigenvalues and eigenfunctions.

Key Words: Sturm–Liouville; Schrödinger; eigenvalue; modified Magnus; Filon; shooting


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