Advertisement

Solutions of Tridiagonal Recurrence Relations, Application to Ordinary and Partial Differential Equations

  • Hannes Risken
Part of the Springer Series in Synergetics book series (SSSYN, volume 18)

Abstract

As shown in the next chapter, the Fokker-Planck equation describing the Brownian motion in arbitrary potentials, i.e., the Kramers equation, can be cast into a tridiagonal vector recurrence relation by suitable expansion of the distribution function. In this chapter we shall investigate the solutions of tridiagonal vector recurrence relations. As it turns out, the Laplace transform of these solutions as well as the eigenvalues and eigenfunctions can be obtained in terms of matrix continued fractions. Therefore, the corresponding solutions of the Kramers equation can also be given in terms of matrix continued fractions. This method has the advantage that no detailed balance condition is needed for its application. This matrix continued-fraction method is especially suitable for numerical calculations and for some problems it seems to be the most accurate and fastest method, as will be discussed in other chapters. Besides its advantage for numerical purposes, the matrix continued-fraction solutions are also very useful for analytical evaluations. By a proper Taylor series expansion of the matrix continued fractions we obtain, for instance, in Sect. 10.4 the high-friction limit solutions of the Kramers equation.

Keywords

Recurrence Relation Continue Fraction Schrodinger Equation Bloch Equation Pade Approximants 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 9.1
    O. Perron: Die Lehre von den Kettenbrüchen, Vols. I, II (Teubner, Stuttgart 1977)Google Scholar
  2. 9.2
    H. S. Wall: Analytic Theory of Continued Fractions (Chelsea, Bronx, NY 1973)Google Scholar
  3. 9.3
    W. B. Jones, W. J. Thron: Continued Fractions, Encyclopedia of Mathematics and its Applications, Vol. 11 (Addison-Wesley, Reading, MA 1980)Google Scholar
  4. 9.4
    G. A. Baker, Jr.: Essentials of Padé Approximants (Academic, New York 1975)zbMATHGoogle Scholar
  5. 9.5
    P. Hänggi, F. Rösel, D. Trautmann: Z. Naturforsch. 33a, 402 (1978)ADSGoogle Scholar
  6. 9.6
    W. Götze: Lett. Nuovo Cimento 7, 187 (1973)CrossRefGoogle Scholar
  7. 9.7
    J. Killingbeck: J. Phys. A10, L 99 (1977)ADSGoogle Scholar
  8. 9.8
    G. Haag, P. Hänggi: Z. Physik B34, 411 (1979) and B39, 269 (1980)ADSGoogle Scholar
  9. 9.9
    S. H. Autler, C. H. Townes: Phys. Rev. 100, 703 (1955)ADSCrossRefGoogle Scholar
  10. 9.10
    S. Stenholm, W. E. Lamb: Phys. Rev. 181, 618 (1969)ADSCrossRefGoogle Scholar
  11. 9.11
    S. Stenholm: J. Phys. B5, 878 (1972)ADSGoogle Scholar
  12. 9.12
    S. Graffi, V. Grecchi: Lett. Nuovo Cimento 12, 425 (1975)MathSciNetCrossRefGoogle Scholar
  13. 9.13
    M. Allegrini, E. Arimondo, A. Bambini: Phys. Rev. A15, 718 (1977)ADSGoogle Scholar
  14. 9.14
    H. Risken, H. D. Vollmer: Z. Physik B33, 297 (1979)ADSGoogle Scholar
  15. 9.15
    H. D. Vollmer, H. Risken: Z. Physik B34, 313 (1979)MathSciNetADSGoogle Scholar
  16. 9.16
    H. D. Vollmer, H. Risken: Physica 110A, 106 (1982)MathSciNetADSGoogle Scholar
  17. 9.17
    H. Risken, H. D. Vollmer: Mol. Phys. 46, 555 (1982)ADSCrossRefGoogle Scholar
  18. 9.18
    H. Risken, H. D. Vollmer: Z. Physik B39, 339 (1980)MathSciNetADSGoogle Scholar
  19. 9.19
    H. Risken, H. D. Vollmer, M. Mörsch: Z. Physik B40, 343 (1981)ADSGoogle Scholar
  20. 9.20
    W. Dieterich, T. Geisel, I. Peschel: Z. Physik B29, 5 (1978)ADSGoogle Scholar
  21. 9.21
    H. J. Breymayer, H. Risken, H. D. Vollmer, W. Wonneberger: Appl. Phys. B28, 335 (1982)ADSGoogle Scholar
  22. 9.22
    S. N. Dixit, P. Zoller, P. Lambropoulos: Phys. Rev. A21, 1289 (1980)ADSGoogle Scholar
  23. 9.23
    P. Zoller, G. Alber, R. Salvador: Phys. Rev. A24, 398 (1981)ADSGoogle Scholar
  24. 9.24
    H. Denk, M. Riederle: J. Appr. Theory 35, 355 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 9.25
    H. Meschkowski: Differenzengleichungen, Studia Mathematica Vol. XIV (Vanderhoeck & Ruprecht, Göttingen 1959) Chap. XGoogle Scholar
  26. 9.26
    W. Magnus, F. Oberhettinger, R. P. Soni: Formulas and Theorems for the Special Functions of Mathematical Physics (Springer, New York 1966)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1984

Authors and Affiliations

  • Hannes Risken
    • 1
  1. 1.Abteilung für Theoretische PhysikUniversität UlmUlmFed. Rep. of Germany

Personalised recommendations