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Uniform Asymptotic Solution in the Form of an Airy Function for Semiclassical Bound States in One-Dimensional and Radially Symmetric Problems

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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.

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Authors and Affiliations

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

    A. Yu. Anikin, S. Yu. Dobrokhotov, V. E. Nazaikinskii & A. V. Tsvetkova

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

    A. Yu. Anikin, S. Yu. Dobrokhotov, V. E. Nazaikinskii & A. V. Tsvetkova

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  1. A. Yu. Anikin
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  2. S. Yu. Dobrokhotov
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  3. V. E. Nazaikinskii
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  4. A. V. Tsvetkova
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Correspondence to A. Yu. Anikin, S. Yu. Dobrokhotov, V. E. Nazaikinskii or A. V. Tsvetkova.

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Conflicts of interest. The authors declare no conflicts of interest.

Additional information

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

Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 201, No. 3, pp. 382–414, December, 2019.

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Anikin, A.Y., Dobrokhotov, S.Y., Nazaikinskii, V.E. et al. Uniform Asymptotic Solution in the Form of an Airy Function for Semiclassical Bound States in One-Dimensional and Radially Symmetric Problems. Theor Math Phys 201, 1742–1770 (2019). https://doi.org/10.1134/S0040577919120079

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  • DOI: https://doi.org/10.1134/S0040577919120079

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