Advertisement

WKB and Turning Point Theory for Second-order Difference Equations

  • Jeffrey S. Geronimo
  • Oscar Bruno
  • Walter Van Assche
Part of the Operator Theory: Advances and Applications book series (OT, volume 154)

Abstract

A turning point method for difference equations is developed. This method is coupled with the LG-WKB method via matching to provide approximate solutions to the initial value problem. The techniques developed are used to provide strong asymptotics for Hermite polynomials.

Keywords

Turning point difference equations WKB orthogonal polynomials hermite polynomials. 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [B]
    P.A. Braun, WKB method for three-term recurrence relations and quasienergies of an anharmonic oscillator, Theoret. and Math. Phys. 37 (1978) 1070–1081.CrossRefGoogle Scholar
  2. [CC]
    O. Costin, R. Costin, Rigorous WKB method for finite-order linear recurrence relations with smooth coefficients, SIAM J. Math. Anal. 27 (1996) 110–134.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [DM]
    P. Deift, K. T.-R. McLaughlin, A Continuum Limit of the Toda Lattice, Memoirs of AMS #624 (1998) Providence, RI.Google Scholar
  4. [DMo]
    R.B. Dingle, G.J. Morgan, WKB methods for difference equations I, Appl. Sci. Res. 18 (1967) 221–237.MathSciNetzbMATHCrossRefGoogle Scholar
  5. [DKMVZ]
    P. Deift, T. Kriecherbauer, K. T.-R. McLaughlin, S. Venakides, X. Zhou, Strong Asymptotics of Orthogonal Polynomials with respect to exponential weights, Comm. Pure. Appl. Math. 52 (1999) 1491–1552.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [G]
    J.S. Geronimo, WKB and Turning point Theory for second order difference equations: External Fields and Strong Asymptotics for Orthogonal Polynomials, In prep.Google Scholar
  7. [GS]
    J.S. Geronimo, D. Smith, WKB (Liouville-Green) Analysis of Second Order Difference Equations and Applications, JAT 69 (1992) 269–301.MathSciNetzbMATHGoogle Scholar
  8. [L]
    R.E. Langer On the asymptotic solutions of ordinary differential equations, with an application to the Bessel function of large order, Trans. Amer. Math. Soc. 33 (1931) 23–64.MathSciNetCrossRefGoogle Scholar
  9. [MV]
    M. Maejina, W. Van Assche, Probabilistic proofs of asymptotic formulas for some classical polynomials, Math. Proc. Cambridge Philos. Soc. 97 (1985) 499–510.MathSciNetCrossRefGoogle Scholar
  10. [ND]
    P. Nevai, J.S. Dehisa, On asymptotic properties of zeros of orthogonal polynomi-als, SIAM J. Math. Anal. 10 (1979) 1184–1192.MathSciNetzbMATHCrossRefGoogle Scholar
  11. [O]
    F.W.J. Olver, Asymptotics and Special Functions (1974) Academic Press London.Google Scholar
  12. [SG]
    K. Schulten, R.G. Gordon, Semiclassical analysis of 3-y and 6-y coefficients for quantum mechanical of angular momentum, JMP 16 (1975) 1971–1988.MathSciNetCrossRefGoogle Scholar
  13. [S]
    G. Szegö, Orthogonal Polynomials, AMS Colloq. Pub. Vol. 23, 4th ed, Providence, RI (1978).Google Scholar
  14. [V]
    W. Van Assche, Asymptotics for Orthogonal Polynomial, Lectures Notes in Mathematics 1265 Springer-Verlag (1967) London.Google Scholar
  15. [VG]
    W. Van Assche, J.S. Geronimo, Asymptotics for orthogonal polynomials with regularly varying recurrence coefficients, Rocky Mtn. J. Math. 19 (1989) 39–49.zbMATHCrossRefGoogle Scholar
  16. [WW]
    Z. Wang, R. Wong, Uniform asymptotic expansion of Jυ (υa) via a difference equation, Numer. Math. 91 (2002) 147–193.MathSciNetzbMATHCrossRefGoogle Scholar
  17. [W]
    G.N. Watson, A treatise on the theory of Bessel functions (2nd ed), Cambridge Univ. Press (1944) London.zbMATHGoogle Scholar

Copyright information

© Springer Basel AG 2004

Authors and Affiliations

  • Jeffrey S. Geronimo
    • 1
  • Oscar Bruno
    • 2
  • Walter Van Assche
    • 3
  1. 1.Georgia Institute of TechnologySchool of MathematicsAtlantaUSA
  2. 2.California Institute of TechnologyApplied & Computational MathematicsPasadenaUSA
  3. 3.Department WiskundeCelestijnenlaan 200BLeuvenBelgium

Personalised recommendations