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 Mat. Zametki, 2018, Volume 104, Issue 6, Pages 835–850 (Mi mz12093)

Lagrangian Manifolds Related to the Asymptotics of Hermite Polynomials

S. Yu. Dobrokhotovab, A. V. Tsvetkovaab

a Ishlinsky Institute for Problems in Mechanics of the Russian Academy of Sciences, Moscow
b Moscow Institute of Physics and Technology (State University), Dolgoprudny, Moscow region

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.

 Funding Agency Grant Number Russian Science Foundation 16-11-10282 This work was supported by the Russian Science Foundation under grant 16-11-10282.

Author to whom correspondence should be addressed

DOI: https://doi.org/10.4213/mzm12093

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English version:
Mathematical Notes, 2018, 104:6, 810–822

Bibliographic databases:

UDC: 517.928

Citation: S. Yu. Dobrokhotov, A. V. Tsvetkova, “Lagrangian Manifolds Related to the Asymptotics of Hermite Polynomials”, Mat. Zametki, 104:6 (2018), 835–850; Math. Notes, 104:6 (2018), 810–822

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/mz12093
• https://doi.org/10.4213/mzm12093
• http://mi.mathnet.ru/eng/mz/v104/i6/p835

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. S. Yu. Dobrokhotov, V. E. Nazaikinskii, “Efficient formulas for the Maslov canonical operator near a simple caustic”, Russ. J. Math. Phys., 25:4 (2018), 545–552
2. A. Fedotov, F. Klopp, “Difference equations, uniform quasiclassical asymptotics and Airy functions”, 2018 Days on Diffraction (DD), International Conference on Days on Diffraction (DD) (June 04–08, 2018, St, Petersburg, Russia), IEEE, 2018, 98–101
3. A. I. Klevin, “Asymptotic eigenfunctions of the “bouncing ball” type for the two-dimensional Schrödinger operator with a symmetric potential”, Theoret. and Math. Phys., 199:3 (2019), 849–863
4. A. Yu. Anikin, S. Yu. Dobrokhotov, A. V. Tsvetkova, “Airy function and transition between the semiclassical and harmonic oscillator approximations for one-dimensional bound states”, Theoret. and Math. Phys., 204:2 (2020), 984–992
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