A. Yu. Anikin, S. Yu. Dobrokhotov, V. E. Nazaikinskii, A. V. Tsvetkova, “Uniform asymptotic solution in the form of an Airy function for semiclassical bound states in one-dimensional and radially symmetric problems”, TMF, 201:3 (2019), 382–414; Theoret. and Math. Phys., 201:3 (2019), 1742

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Teoreticheskaya i Matematicheskaya Fizika, 2019, Volume 201, Number 3, Pages 382–414
DOI: https://doi.org/10.4213/tmf9639 (Mi tmf9639)

This article is cited in 35 scientific papers (total in 35 papers)

Uniform asymptotic solution in the form of an Airy function for semiclassical bound states in one-dimensional and radially symmetric problems

A. Yu. Anikin^ab, S. Yu. Dobrokhotov^ab, V. E. Nazaikinskii^ab, A. V. Tsvetkova^ab

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

Full-text PDF (908 kB) Citations (35)

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DOI: https://doi.org/10.4213/tmf9639

Abstract: We consider stationary scalar and vector problems for differential and pseudodifferential operators leading to the appearance of asymptotic solutions of one-dimensional problems localized in a neighborhood of intervals (“bound states”). Based on the semiclassical approximation and the Maslov canonical operator, we develop a constructive algorithm that allows writing an asymptotic solution globally under certain conditions using an Airy function of complex argument.

Keywords: semiclassical asymptotic solution, bound state, closed trajectory, WKB approximation, canonical operator, Airy function.

Funding agency	Grant number
Russian Science Foundation	16-11-10282
This research is supported by a grant from the Russian Science Foundation (Project No. 16-11-10282).

Received: 09.10.2018
Revised: 03.09.2019

English version:
Theoretical and Mathematical Physics, 2019, Volume 201, Issue 3, Pages 1742–1770
DOI: https://doi.org/10.1134/S0040577919120079

Bibliographic databases:

Document Type: Article

Language: Russian

Citation: A. Yu. Anikin, S. Yu. Dobrokhotov, V. E. Nazaikinskii, A. V. Tsvetkova, “Uniform asymptotic solution in the form of an Airy function for semiclassical bound states in one-dimensional and radially symmetric problems”, TMF, 201:3 (2019), 382–414; Theoret. and Math. Phys., 201:3 (2019), 1742–1770

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This publication is cited in the following 35 articles:

Vladislav Rykhlov, Anatoly Anikin, “High-frequency two-dimensional asymptotic standing coastal trapped waves in nearly integrable case”, Lett Math Phys, 115:1 (2025)
Jian-Kang 建康 Zhan 詹, Sheng-Chun 胜春 Piao 朴, Li-Jia 李佳 Gong 龚, Yang 阳 Dong 董, Yong-Chao 永超 Guo 郭, Guang-Xue 广学 Zheng 郑, “A WKB method based on parabolic cylinder function for very-low-frequency sound propagation in deep ocean”, Chinese Phys. B, 34:3 (2025), 034301
I. A. Bogaevsky, S. Yu. Dobrokhotov, A. A. Tolchennikov, “Arnold Lagrangian singularity in the asymptotics of the solution of a model two-dimensional Helmholtz equation with a localized right-hand side”, Theoret. and Math. Phys., 218:1 (2024), 19–40
S.Yu. Dobrokhotov, D.S. Minenkov, M.M. Votiakova, “Asymptotics of Long Nonlinear Coastal Waves in Basins with Gentle Shores”, Russ. J. Math. Phys., 31:1 (2024), 79
Samir B. Hadid, Rabha W. Ibrahim, “Modal treatment in two dimensions theoretical foundations of VLF-radio wave propagation using the normalized airy functions”, Journal of King Saud University - Science, 36:3 (2024), 103099
Fedor Ozhegov, “Feynman checkers: External electromagnetic field and asymptotic properties”, Rev. Math. Phys., 36:07 (2024)
Anna V. Tsvetkova, “Lagrangian Manifolds in the Theory of Wave Beams and Solutions of the Helmholtz Equation”, Regul. Chaotic Dyn., 29:6 (2024), 866–885
V. V. Rykhlov, “Efficient semiclassical approximation for bound states in graphene in magnetic field with a small trigonal warping correction”, Math. Notes, 116:6 (2024), 1339–1349
A.V. Tsvetkova, “Real Semiclassical Approximation for the Asymptotics of Jacobi Polynomials Given by a Difference Equation”, Russ. J. Math. Phys., 31:4 (2024), 774
A. Yu. Anikin, S. Yu. Dobrokhotov, V. E. Nazaikinskii, M. Rouleux, “Lagrangian manifolds and the construction of asymptotics for (pseudo)differential equations with localized right-hand sides”, Theoret. and Math. Phys., 214:1 (2023), 1–23
S. Yu. Dobrokhotov, S. B. Levin, A. A. Tolchennikov, “Keplerian orbits and global asymptotic solution in the form of an Airy function for the scattering problem on a repulsive Coulomb potential”, Russian Math. Surveys, 78:4 (2023), 788–790
Yu. A. Kordyukov, I. A. Taimanov, “Quasi-Classical Approximation of Monopole Harmonics”, Math. Notes, 114:6 (2023), 1277–1288
S. Yu. Dobrokhotov, V. E. Nazaikinskii, A. V. Tsvetkova, “Nonlinear Effects and Run-up of Coastal Waves Generated by Billiards with Semi-rigid Walls in the Framework of Shallow Water Theory”, Proc. Steklov Inst. Math., 322 (2023), 105–117
D. S. Minenkov, S. A. Sergeev, “Asymptotics of the whispering gallery-type in the eigenproblem for the Laplacian in a domain of revolution diffeomorphic to a solid torus”, Russ. J. Math. Phys., 30:4 (2023), 599
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. V. Tsvetkova, P. S. Petrov, “On uniform asymptotic approximations of whispering gallery modes propagating along curved penetrable interfaces”, Journal of Sound and Vibration, 548 (2023), 117555
V. L. Chernyshev, V. E. Nazaikinskii, A. V. Tsvetkova, “Lattice equations and semiclassical asymptotics”, Russ. J. Math. Phys., 30:2 (2023), 152
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
A. Yu. Anikin, V. V. Rykhlov, “Constructive Semiclassical Asymptotics of Bound States of Graphene in a Constant Magnetic Field with Small Mass”, Math. Notes, 111:2 (2022), 173–192
A. Yu. Anikin, S. Yu. Dobrokhotov, A. A. Shkalikov, “On Expansions in the Exact and Asymptotic Eigenfunctions of the One-Dimensional Schrödinger Operator”, Math. Notes, 112:5 (2022), 623–641

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

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