Airy functions and transition between semiclassical and harmonic oscillator approximations for one-dimensional bound states

DOI10.1134/S0040577920080024zbMath1453.81024OpenAlexW3080283770MaRDI QIDQ2214761

S. Yu. Dobrokhotov, A. Yu. Anikin, Anna V. Tsvetkova

Publication date: 10 December 2020

Published in: Theoretical and Mathematical Physics (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1134/s0040577920080024

zbMATH Keywords

asymptoticsSchrödinger operatorsemiclassical approximationeigenfunctionharmonic oscillatorAiry functionbound state

Mathematics Subject Classification ID

Asymptotic behavior of solutions to PDEs (35B40) General topics in linear spectral theory for PDEs (35P05) Selfadjoint operator theory in quantum theory, including spectral analysis (81Q10) Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory (81Q20) Bessel and Airy functions, cylinder functions, ({}_0F_1) (33C10)

Related Items (5)

Global asymptotics for functions of parabolic cylinder and solutions of the Schrödinger equation with a potential in the form of a nonsmooth double well ⋮ On expansions in the exact and asymptotic eigenfunctions of the one-dimensional Schrödinger operator ⋮ Close turning points and the Harper operator ⋮ Complex WKB method (one-dimensional linear problems on the complex plane) ⋮ Semiclassical asymptotics for a difference Schrödinger equation with two coalescent turning points

Cites Work

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• Lagrangian manifolds related to the asymptotics of Hermite polynomials
• Spectral asymptotics via the semiclassical Birkhoff normal form
• Uniform asymptotic solution in the form of an Airy function for semiclassical bound states in one-dimensional and radially symmetric problems
• Non-commutative normal form, spectrum and inverse problem

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