Skip to main content
SpringerLink
Log in

Lagrangian Manifolds Related to the Asymptotics of Hermite Polynomials

  • Published:
Mathematical Notes Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  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)].

    Article  MathSciNet  MATH  Google Scholar 

  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)].

    MATH  Google Scholar 

  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. G. Szegö, Orthogonal Polynomials (Amer. Math. Soc., Providence, RI, 1959; Fizmatgiz, Moscow, 1962).

    MATH  Google Scholar 

  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).

    MATH  Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

  7. V. P. Maslov, Operational Methods (Nauka, Moscow, 1973; Mir, Moscow, 1976).

    MATH  Google Scholar 

  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. V. P. Maslov, Perturbation Theory and Asymptotic Methods (Izd. Moskov. Univ., Moscow, 1965) [in Russian].

    Google Scholar 

  10. V. P. Maslov and M. V. Fedoryuk, Semi-Classical Approximation in Quantum Mechanics (Nauka, Moscow, 1976; D. Reidel, Dordrecht–Boston–London, 1981).

    Book  MATH  Google Scholar 

  11. J. Heading, An Introduction to Phase-Integral Methods (Wiley, New York, 1962; Mir, Moscow, 1965).

    MATH  Google Scholar 

  12. S. Yu. Slavyanov, Asymptotics of Solutions of the One-Dimensional Schrödinger Equation (Izd. Leningrad. Gos. Univ., Leningrad, 1991) [in Russian].

    Google Scholar 

  13. M. V. Fedoryuk, Asymptotics: Integrals and Series (Nauka, Moscow, 1987) [in Russian].

    MATH  Google Scholar 

  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)].

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Ishlinsky Institute for Problems in Mechanics RAS, Moscow, 119526, Russia

    S. Yu. Dobrokhotov & A. V. Tsvetkova

  2. Moscow Institute of Physics and Technology (State University), Dolgoprudny, Moscow Oblast, 141701, Russia

    S. Yu. Dobrokhotov & A. V. Tsvetkova

Authors
  1. S. Yu. Dobrokhotov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. V. Tsvetkova
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. Yu. Dobrokhotov.

Additional information

Original Russian Text © S. Yu. Dobrokhotov, A. V. Tsvetkova, 2018, published in Matematicheskie Zametki, 2018, Vol. 104, No. 6, pp. 835–850.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dobrokhotov, S.Y., Tsvetkova, A.V. Lagrangian Manifolds Related to the Asymptotics of Hermite Polynomials. Math Notes 104, 810–822 (2018). https://doi.org/10.1134/S0001434618110263

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0001434618110263

Keywords

Search

Navigation