JMP Nameplate
  • Volume/Page
  • Keyword
  • DOI
  • Citation
  • Advanced
   
 
 
 

Flickr Twitter UniPHY Group iResearch App Facebook

Journal of Mathematical Physics / Volume 27 / Issue 1

J. Math. Phys. 27, 185 (1986); http://dx.doi.org/10.1063/1.527360 (17 pages)

Continued fractions and Rayleigh–Schrödinger perturbation theory at large order

E. R. Vrscay1 and J. Cizek2

1Department of Applied Mathematics, Faculty of Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
2Department of Applied Mathematics, Faculty of Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1 and Department of Chemistry, Faculty of Science, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1 and Guelph–Waterloo Centre for Graduate Work in Chemistry, University of Guelph and University of Waterloo, Waterloo, Ontario, Canada N2L 3G1

(Received 30 May 1985; accepted 10 July 1985)

Concern with the continued fraction representations of divergent Rayleigh–Schrödinger perturbation expansions in quantum mechanics is expressed. The following relation between the large‐order behavior of the continued fraction coefficients cn and the perturbation series coefficients E(n) is shown to exist: If E(n) ∼(−1)n+1Γ( pn+a),  p=0,1,2,..., as n→∞, then cn=O(np) as n→∞. The case  p=1 is studied in detail here, using the problems of the quartic anharmonic oscillator and the hydrogen atom in a linear radial potential as illustrative examples. For  p=1 the asymptotics of the cn are shown to be linked to the infinite field limit E(λ)∼F(0)λα, predicting α and providing convergent estimates of F(0).

KEYWORDS and PACS

Keywords

PERTURBATION THEORY, QUANTUM MECHANICS, RAYLEIGH−SCHROEDINGER FORMULA, SERIES EXPANSION, ANHARMONIC OSCILLATORS, QUANTUM FIELD THEORY, CONTINUED FRACTIONS

PACS

  • 03.65.Fd

    Algebraic methods

  • 02.30.-f

    Function theory, analysis

  • 31.15.-p

    Calculations and mathematical techniques in atomic and molecular physics

  • 02.10.De

    Algebraic structures and number theory

ARTICLE DATA

Digital Object Identifier
http://dx.doi.org/10.1063/1.527360

PUBLICATION DATA

ISSN

0022-2488 (print)  
1089-7658 (online)

For access to fully linked references, you need to log in.
    C. M. Bender and T. T. Wu, Phys. Rev. 184, 1231 (1969).

    C. M. Bender and T. T. Wu, Phys. Rev. Lett. 27, 461 (1971).

    C. M. Bender and T. T. Wu, Phys. Rev. D 7, 1620 (1973).

    J. Cizek and E. R. Vrscay, Phys. Rev. A 30, 1550 (1984).

    E. R. Vrscay, Phys. Rev. A 31, 2054 (1985).

    P. W. Langhoff, C. T. Corcoran, J. S. Sims, F. Weinhold, and R. M. Glover, Phys. Rev. A 14, 1042 (1976).

    E. Eichten, K. Gottfried, T. Kinoshita, K. D. Lane, and T. M. Yan, Phys. Rev. D 17, 3090 (1978).


For access to citing articles, you need to log in.



Close

Google Calendar

ADVERTISEMENT

Your Recent History Recent History Help

Recently Viewed

  1. Continued fractions and Rayleigh–Schrödinger perturbation theory at large order
  2. Simple subgroups of simple Lie groups and nonlinear differential equations with superposition principles
  3. Geometrical properties of the algebraic spinors for R3,1
  4. Fibonacci graphs possessing identical matching polynomials
  5. Ladder and cross terms in second‐order distorted Born approximation
Go to AIP Home
Copyright © American Institute of Physics
close