Advanced search
2 files | 842.24 KB

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

Veerle Ledoux (UGent) and Marnix Van Daele (UGent)
Author
Organization
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.
Keywords
ALGORITHM, COMPUTATION, ORDER, SERIES, PACKAGE, INTEGRATORS, QUANTUM DYNAMICS, EIGENVALUE PROBLEM, OSCILLATION-THEORY, CP method, STURM-LIOUVILLE PROBLEMS, Perturbation, Schrodinger, Eigenvalue problem

Downloads

  • (...).pdf
    • full text
    • |
    • UGent only
    • |
    • PDF
    • |
    • 421.21 KB
  • Binder1.pdf
    • full text
    • |
    • open access
    • |
    • PDF
    • |
    • 421.02 KB

Citation

Please use this url to cite or link to this publication:

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.
@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},
}

Altmetric
View in Altmetric
Web of Science
Times cited: