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On CP, LP and other piecewise perturbation methods for the numerical solution of the Schrödinger equation

Veerle Ledoux and Marnix Van Daele UGent (2011) ZEITSCHRIFT FUR ANGEWANDTE MATHEMATIK UND PHYSIK. 62(6). p.993-1011
abstract
The piecewise perturbation methods (PPM) have proven to be very efficient for the numerical solution of the linear time-independent Schrödinger equation. The underlying idea is to replace the potential function piecewisely by simpler approximations and then to solve the approximating problem. The accuracy is improved by adding some perturbation corrections. Two types of approximating potentials were considered in the literature, that is piecewise constant and piecewise linear functions, giving rise to the so-called CP methods (CPM) and LP methods (LPM). Piecewise polynomials of higher degree have not been used since the approximating problem is not easy to integrate analytically. As suggested by Ixaru (Comput Phys Commun 177:897–907, 2007), this problem can be circumvented using another perturbative approach to construct an expression for the solution of the approximating problem. In this paper, we show that there is, however, no need to consider PPM based on higher-order polynomials, since these methods are equivalent to the CPM. Also, LPM is equivalent to CPM, although it was sometimes suggested in the literature that an LP method is more suited for problems with strongly varying potentials. We advocate that CP schemes can (and should) be used in all cases, since it forms the most straightforward way of devising PPM and there is no advantage in considering other piecewise polynomial perturbation methods.
Please use this url to cite or link to this publication:
author
organization
alternative title
On CP, LP and other piecewise perturbation methods for the numerical solution of the Schrodinger equation
year
type
journalArticle (original)
publication status
published
subject
keyword
ALGORITHM, COMPUTATION, ORDER, SERIES, PACKAGE, INTEGRATORS, QUANTUM DYNAMICS, EIGENVALUE PROBLEM, OSCILLATION-THEORY, CP method, STURM-LIOUVILLE PROBLEMS, Perturbation, Schrodinger, Eigenvalue problem
journal title
ZEITSCHRIFT FUR ANGEWANDTE MATHEMATIK UND PHYSIK
Z. Angew. Math. Phys.
volume
62
issue
6
pages
993 - 1011
Web of Science type
Article
Web of Science id
000297345100003
JCR category
MATHEMATICS, APPLIED
JCR impact factor
0.951 (2011)
JCR rank
81/245 (2011)
JCR quartile
2 (2011)
ISSN
0044-2275
DOI
10.1007/s00033-011-0158-8
language
English
UGent publication?
yes
classification
A1
copyright statement
I have transferred the copyright for this publication to the publisher
id
1948599
handle
http://hdl.handle.net/1854/LU-1948599
date created
2011-11-24 12:33:51
date last changed
2016-12-19 15:42:20
@article{1948599,
  abstract     = {The piecewise perturbation methods (PPM) have proven to be very efficient for the numerical solution of the linear time-independent Schr{\"o}dinger equation. The underlying idea is to replace the potential function piecewisely by simpler approximations and then to solve the approximating problem. The accuracy is improved by adding some perturbation corrections. Two types of approximating potentials were considered in the literature, that is piecewise constant and piecewise linear functions, giving rise to the so-called CP methods (CPM) and LP methods (LPM). Piecewise polynomials of higher degree have not been used since the approximating problem is not easy to integrate analytically. As suggested by Ixaru (Comput Phys Commun 177:897--907, 2007), this problem can be circumvented using another perturbative approach to construct an expression for the solution of the approximating problem. In this paper, we show that there is, however, no need to consider PPM based on higher-order polynomials, since these methods are equivalent to the CPM. Also, LPM is equivalent to CPM, although it was sometimes suggested in the literature that an LP method is more suited for problems with strongly varying potentials. We advocate that CP schemes can (and should) be used in all cases, since it forms the most straightforward way of devising PPM and there is no advantage in considering other piecewise polynomial perturbation methods.},
  author       = {Ledoux, Veerle and Van Daele, Marnix},
  issn         = {0044-2275},
  journal      = {ZEITSCHRIFT FUR ANGEWANDTE MATHEMATIK UND PHYSIK},
  keyword      = {ALGORITHM,COMPUTATION,ORDER,SERIES,PACKAGE,INTEGRATORS,QUANTUM DYNAMICS,EIGENVALUE PROBLEM,OSCILLATION-THEORY,CP method,STURM-LIOUVILLE PROBLEMS,Perturbation,Schrodinger,Eigenvalue problem},
  language     = {eng},
  number       = {6},
  pages        = {993--1011},
  title        = {On CP, LP and other piecewise perturbation methods for the numerical solution of the Schr{\"o}dinger equation},
  url          = {http://dx.doi.org/10.1007/s00033-011-0158-8},
  volume       = {62},
  year         = {2011},
}

Chicago
Ledoux, Veerle, and Marnix Van Daele. 2011. “On CP, LP and Other Piecewise Perturbation Methods for the Numerical Solution of the Schrödinger Equation.” Zeitschrift Fur Angewandte Mathematik Und Physik 62 (6): 993–1011.
APA
Ledoux, Veerle, & Van Daele, M. (2011). On CP, LP and other piecewise perturbation methods for the numerical solution of the Schrödinger equation. ZEITSCHRIFT FUR ANGEWANDTE MATHEMATIK UND PHYSIK, 62(6), 993–1011.
Vancouver
1.
Ledoux V, Van Daele M. On CP, LP and other piecewise perturbation methods for the numerical solution of the Schrödinger equation. ZEITSCHRIFT FUR ANGEWANDTE MATHEMATIK UND PHYSIK. 2011;62(6):993–1011.
MLA
Ledoux, Veerle, and Marnix Van Daele. “On CP, LP and Other Piecewise Perturbation Methods for the Numerical Solution of the Schrödinger Equation.” ZEITSCHRIFT FUR ANGEWANDTE MATHEMATIK UND PHYSIK 62.6 (2011): 993–1011. Print.