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

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Matematicheskie Zametki, 2018, Volume 104, Issue 6, Pages 835–850
DOI: https://doi.org/10.4213/mzm12093 (Mi mzm12093)

This article is cited in 14 scientific papers (total in 14 papers)

Lagrangian Manifolds Related to the Asymptotics of Hermite Polynomials

S. Yu. Dobrokhotov^ab, A. V. Tsvetkova^ab

^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

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

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.

Received: 14.06.2018

English version:
Mathematical Notes, 2018, Volume 104, Issue 6, Pages 810–822
DOI: https://doi.org/10.1134/S0001434618110263

Bibliographic databases:

Document Type: Article

UDC: 517.928

Language: Russian

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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Linking options:

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https://doi.org/10.4213/mzm12093

https://www.mathnet.ru/eng/mzm/v104/i6/p835

This publication is cited in the following 14 articles:

A. A. Fedotov, “Complex WKB method (one-dimensional linear problems on the complex plane)”, Math Notes, 114:5-6 (2023), 1418
S. Yu. Dobrokhotov, A. V. Tsvetkova, “Global asymptotics for functions of parabolic cylinder and solutions of the Schrödinger equation with a potential in the form of a nonsmooth double well”, Russ. J. Math. Phys., 30:1 (2023), 46
A. Fedotov, E. Shchetka, “Difference equations in the complex plane: quasiclassical asymptotics and Berry phase”, Appl. Anal., 101:1 (2022), 274–296
A. I. Aptekarev, S. Yu. Dobrokhotov, D. N. Tulyakov, A. V. Tsvetkova, “Plancherel–Rotach type asymptotic formulae for multiple orthogonal Hermite polynomials and recurrence relations”, Izv. Math., 86:1 (2022), 32–91
Fedotov A., Klopp F., “Wkb Asymptotics of Meromorphic Solutions to Difference Equations”, Appl. Anal., 100:7 (2021), 1557–1573
S. Yu. Dobrokhotov, D. S. Minenkov, V. E. Nazaikinskii, “Representations of Bessel functions via the Maslov canonical operator”, Theoret. and Math. Phys., 208:2 (2021), 1018–1037
S. Yu. Dobrokhotov, A. V. Tsvetkova, “Asymptotics of multiple orthogonal Hermite polynomials (Z, alpha) determined by a third-order differential equation”, Russ. J. Math. Phys., 28:4 (2021), 439–454
S. Yu. Dobrokhotov, A. V. Tsvetkova, “An approach to finding the asymptotics of polynomials given by recurrence relations”, Russ. J. Math. Phys., 28:2 (2021), 198–223
A. A. Fedotov, “The complex WKB method for a system of two linear difference equations”, St. Petersburg Math. J., 33:2 (2022), 405–425
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
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
A. Fedotov, F. Klopp, “The complex wkb method for difference equations and airy functions”, SIAM J. Math. Anal., 51:6 (2019), 4413–4447
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
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

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

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