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

Veerle Ledoux (UGent) , Marnix Van Daele (UGent) and Guido Vanden Berghe (UGent)
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Abstract
We discuss a new numerical method, based on a modifiedMagnus integrator, to solve the Sturm-Liouville eigenvalue problem in its Schr¨odinger form. A modified Magnus integrator was already used by Degani & Schiff(2006) to approximate the oscillating solution of a Schr¨odinger problem over the classically allowed region. Here we show that the technique can be succesfully extended to the non-oscillatory 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 to be applied 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.
Keywords
Schrödinger, Sturm-Liouville, eigenvalue, shooting, Modified Magnus, Filon, STURM-LIOUVILLE PROBLEMS, DIFFERENTIAL-EQUATIONS, CP METHODS, PACKAGE

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Chicago
Ledoux, Veerle, Marnix Van Daele, and Guido Vanden Berghe. 2010. “Efficient Numerical Solution of the One-dimensional Schrödinger Eigenvalue Problem Using Magnus Integrators.” Ima Journal of Numerical Analysis 30 (3): 751–776.
APA
Ledoux, Veerle, Van Daele, M., & Vanden Berghe, G. (2010). Efficient numerical solution of the one-dimensional Schrödinger eigenvalue problem using Magnus integrators. IMA JOURNAL OF NUMERICAL ANALYSIS, 30(3), 751–776.
Vancouver
1.
Ledoux V, Van Daele M, Vanden Berghe G. Efficient numerical solution of the one-dimensional Schrödinger eigenvalue problem using Magnus integrators. IMA JOURNAL OF NUMERICAL ANALYSIS. 2010;30(3):751–76.
MLA
Ledoux, Veerle, Marnix Van Daele, and Guido Vanden Berghe. “Efficient Numerical Solution of the One-dimensional Schrödinger Eigenvalue Problem Using Magnus Integrators.” IMA JOURNAL OF NUMERICAL ANALYSIS 30.3 (2010): 751–776. Print.
@article{1013385,
  abstract     = {We discuss a new numerical method, based on a modifiedMagnus integrator, to solve the Sturm-Liouville eigenvalue problem in its Schr{\textasciidieresis}odinger form. A modified Magnus integrator was already used by Degani \& Schiff(2006) to approximate the oscillating solution of a Schr{\textasciidieresis}odinger problem over the classically allowed region. Here we show that the technique can be succesfully extended to the non-oscillatory 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 to be applied 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.},
  author       = {Ledoux, Veerle and Van Daele, Marnix and Vanden Berghe, Guido},
  issn         = {0272-4979},
  journal      = {IMA JOURNAL OF NUMERICAL ANALYSIS},
  keyword      = {Schr{\"o}dinger,Sturm-Liouville,eigenvalue,shooting,Modified Magnus,Filon,STURM-LIOUVILLE PROBLEMS,DIFFERENTIAL-EQUATIONS,CP METHODS,PACKAGE},
  language     = {eng},
  number       = {3},
  pages        = {751--776},
  title        = {Efficient numerical solution of the one-dimensional Schr{\"o}dinger eigenvalue problem using Magnus integrators},
  url          = {http://dx.doi.org/10.1093/imanum/drn062},
  volume       = {30},
  year         = {2010},
}

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