Abstract
The problem of the Zeemann-Stark effect for the hydrogen atom in electromagnetic fields is considered using the irreducible representations of the Karasev-Novikova algebra with quadratic commutation relations. An asymptotics of the series of eigenvalues and the asymptotic eigenfunctions are obtained near the upper boundaries of resonance spectral clusters which are formed near the energy levels of an unperturbed hydrogen atom.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
M. Karasev and E. Novikova, “Algebras with Polynomial Commutation Relations for a Quantum Particle in Electric and Magnetic Fields,” Quantum Algebras and Poisson Geometry in Mathematical Physics, Amer. Math. Soc. Trans. Ser. 2, Amer. Math. Soc., Providence, RI 216, 19–135 (2005).
M. V. Karasev and E. M. Novikova, “Algebra with Polynomial Commutation Relations for the Zeeman-Stark Effect in the Hydrogen Atom,” Teor. Mat. Fiz. 142(3), 530–555 (2005); Theor. and Math. Phys. 142 (3), 447–469 (2005).
M. V. Karasev and E. M. Novikova, “Representation of Exact and Semiclassical Eigenfunctions Via Coherent States. The Hydrogen Atom in a Magnetic Field,” Teor. Mat. Fiz. 108(3), 339–387 (1996); Theor. and Math. Phys. 108 (3), 1119–1159 (1996).
M. V. Karasev, “Birkhoff Resonances and Quantum Ray Method,” Proc. Intern. Seminar Days on Diffraction 2004, St.Petersburg, 114–126 (2004).
A. V. Pereskokov, “Asymptotics of the Spectrum and Quantum Averages of a Perturbed Resonant Oscillator near the Boundaries of Spectral Clusters,” Izv. Ross. Akad. Nauk Ser. Mat. 77(1), 165–210 (2013); Izv. Math. 77 (1), 163–210 (2013).
A. V. Pereskokov, “Asymptotics of the Spectrum of the Hydrogen Atom in a Magnetic Field near the Lower Boundaries of Spectral Clusters,” Trudy Moskov. Mat. Obshch. 73(2), 277–325 (2012); Trans. Moscow Math. Soc. 73, 221–262 (2012).
A. V. Pereskokov, “New Type of Semiclassical Asymptotics of Eigenstates near the Boundaries of Spectral Clusters for Schrödinger-Type Operators,” Proc. Intern. Seminar Days on Diffraction 2016, St.Petersburg, 323–326 (2016).
A. S. Migaeva and A. V. Pereskokov, “Asymptotics of the Spectrum of the Hydrogen Atom in Electromagnetic Field near the Lower Boundaries of Spectral Clusters,” Contemporary Problems of Mathematics and Mechanics, Proc. Intern. Conference Dedicated to the 80th Anniversary of Academician V. A. Sadovnichy, MAKS Press, Moscow, 103–105 (2019).
S. Yu. Slavyanov and V. Lai, Special Functions: Unified Theory Based on Analysis of Singularities (Nevskii Dialekt, St.Petersburg, 2002).
M. Karasev and E. Novikova, “Non-Lie Permutation Relations, Coherent States, and Quantum Embedding,” Coherent Transform, Quantization, and Poisson Geometry, Amer. Math. Soc. Trans. Ser. 2, Amer. Math. Soc., Providence, RI 187, 1–202 (1998).
H. Bateman and A. Erdélyi, Higher Transcendental Functions. Bessel Functions, Parabolic Cylinder Functions, Orthogonal Polynomials (Nauka, Moscow, 1974).
M. V. Fedoryuk, Asymptotic Methods for Linear Ordinary Differential Equations (Nauka, Moscow, 1983).
V. P. Maslov and M. V. Fedoryuk, Semiclassical Approximation for Equations of Quantum Mechanics (Nauka, Moscow, 1976).
Acknowledgment
The research is supported by a grant from the Russian Science Foundation (Project No. 19-11-00033). The author thanks E. M. Novikova for useful discussions of the results of the paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
To the memory of Mikhail Karasev
Rights and permissions
About this article
Cite this article
Pereskokov, A.V. On the Asymptotics of the Spectrum of the Hydrogen Atom in Orthogonal Electric and Magnetic Fields Near the Upper Boundaries of Spectral Clusters. Russ. J. Math. Phys. 26, 391–400 (2019). https://doi.org/10.1134/S1061920819030130
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1061920819030130