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Mathematical Notes

, Volume 104, Issue 5–6, pp 810–822 | Cite as

Lagrangian Manifolds Related to the Asymptotics of Hermite Polynomials

  • S. Yu. DobrokhotovEmail author
  • A. V. Tsvetkova
Article
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Abstract

We discuss two approaches that can be used to obtain the asymptotics of Hermite polynomials. The first, well-known approach is based on the representation of Hermite polynomials as solutions of a spectral problem for the harmonic oscillator Schrödinger equation. The second approach is based on a reduction of the finite-difference equation for the Hermite polynomials to a pseudodifferential equation. Associated with each of the approaches are Lagrangian manifolds that give the asymptotics of Hermite polynomials via the Maslov canonical operator.

Keywords

Hermite polynomial Lagrangian manifold Maslov canonical operator asymptotics finite-difference equation Schrödinger equation 

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References

  1. 1.
    D. N. Tulyakov, “Plancherel–Rotach type asymptotics for solutions of linear recurrence relations with rational coefficients,” Mat. Sb. 201 (9), 111–158 (2010) [Sb.Math. 201 (9), 1355–1402 (2010)].MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    A. A. Fedotov and E. V. Shchetka, “Complex WKB method for a difference Schrödinger equation with the potential being a trigonometric polynomial,” Algebra Anal. 29 (2), 193–219 (2017) [St. PetersburgMath. J. 29 (2), 363–381 (2018)].zbMATHGoogle Scholar
  3. 3.
    A. Fedotov and F. Klopp, “Difference equations, uniform quasiclassical asymptotics and Airy functions,” in Proc. Int. Conf. “Days on Diffraction 2018” (St. Petersburg, 2018), pp. 47–48.Google Scholar
  4. 4.
    G. Szegö, Orthogonal Polynomials (Amer. Math. Soc., Providence, RI, 1959; Fizmatgiz, Moscow, 1962).zbMATHGoogle Scholar
  5. 5.
    H. Bateman and A. Erdélyi, Higher Transcendental Functions, Vol. 2: Bessel Functions, Parabolic Cylinder Functions, Orthogonal Polynomials (McGraw–Hill,New York–Toronto–London, 1953; Nauka, Moscow, 1974).zbMATHGoogle Scholar
  6. 6.
    P. Deift, T. Kriecherbauer, K. T.-R. McLaughlin, S. Venakides, and X. Zhou, “Strong asymptotics of orthogonal polynomials with respect to exponential weights,” Comm. Pure Appl.Math. 52 (12), 1491–1552 (1999).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    V. P. Maslov, Operational Methods (Nauka, Moscow, 1973; Mir, Moscow, 1976).zbMATHGoogle Scholar
  8. 8.
    V. Maslov, “The characteristics of pseudo-differential operators and difference schemes,” in Actes du Congrè s International des Mathématiciens, Vol. 2 (Gauthier-Villars, Paris, 1971), pp. 755–769.Google Scholar
  9. 9.
    V. P. Maslov, Perturbation Theory and Asymptotic Methods (Izd. Moskov. Univ., Moscow, 1965) [in Russian].Google Scholar
  10. 10.
    V. P. Maslov and M. V. Fedoryuk, Semi-Classical Approximation in Quantum Mechanics (Nauka, Moscow, 1976; D. Reidel, Dordrecht–Boston–London, 1981).CrossRefzbMATHGoogle Scholar
  11. 11.
    J. Heading, An Introduction to Phase-Integral Methods (Wiley, New York, 1962; Mir, Moscow, 1965).zbMATHGoogle Scholar
  12. 12.
    S. Yu. Slavyanov, Asymptotics of Solutions of the One-Dimensional Schrödinger Equation (Izd. Leningrad. Gos. Univ., Leningrad, 1991) [in Russian].Google Scholar
  13. 13.
    M. V. Fedoryuk, Asymptotics: Integrals and Series (Nauka, Moscow, 1987) [in Russian].zbMATHGoogle Scholar
  14. 14.
    S. Yu. Dobrokhotov, D. S. Minenkov, and S. B. Shlosman, “Asymptotics of wave functions of the stationary Scrödinger equation in the Weyl chamber,” Teoret. Mat. Fiz. 197 (2), 269–278 (2018) [Theoret. and Math. Phys. 197 (2), 1626–1634 (2018)].MathSciNetCrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Ishlinsky Institute for Problems in Mechanics RASMoscowRussia
  2. 2.Moscow Institute of Physics and Technology (State University)DolgoprudnyRussia

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