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

Veerle Ledoux, Marnix Van Daele UGent and Guido Vanden Berghe UGent (2010) IMA JOURNAL OF NUMERICAL ANALYSIS. 30(3). p.751-776
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.
Please use this url to cite or link to this publication:
author
organization
alternative title
Efficient numerical solution of the one-dimensional Schrodinger eigenvalue problem using Magnus integrators
year
type
journalArticle (original)
publication status
published
subject
keyword
Schrödinger, Sturm-Liouville, eigenvalue, shooting, Modified Magnus, Filon, STURM-LIOUVILLE PROBLEMS, DIFFERENTIAL-EQUATIONS, CP METHODS, PACKAGE
journal title
IMA JOURNAL OF NUMERICAL ANALYSIS
IMA J. Numer. Anal.
volume
30
issue
3
pages
751 - 776
Web of Science type
Article
Web of Science id
000280710400007
JCR category
MATHEMATICS, APPLIED
JCR impact factor
1.425 (2010)
JCR rank
39/235 (2010)
JCR quartile
1 (2010)
ISSN
0272-4979
DOI
10.1093/imanum/drn062
language
English
UGent publication?
yes
classification
A1
copyright statement
I have transferred the copyright for this publication to the publisher
id
1013385
handle
http://hdl.handle.net/1854/LU-1013385
date created
2010-07-16 11:59:50
date last changed
2016-12-19 15:46:13
@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},
}

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.