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Transactions of the Moscow Mathematical Society

ISSN 1547-738X(online) ISSN 0077-1554(print)



Asymptotics of the spectrum of the hydrogen atom in a magnetic field near the lower boundaries of spectral clusters

Author: A. V. Pereskokov
Translated by: ??
Original publication: Trudy Moskovskogo Matematicheskogo Obshchestva.
Journal: Trans. Moscow Math. Soc. 2012, 221-262
MSC (2010): Primary 81R30; Secondary 34Mxx, 81Q20, 33Cxx
Published electronically: March 21, 2013
MathSciNet review: 3184977
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Abstract: We investigate the second-order Zeeman effect in a magnetic field using irreducible representations of an algebra with the Karasev - Novikova quadratic commutation relations. To each such representation there corresponds a spectral cluster near the energy level of the unperturbed hydrogen atom. Using this model as an example, we describe a general method for constructing asymptotic solutions near the boundaries of spectral clusters based on a new integral representation. We also study the problem of computing quantum averages near the lower boundaries of clusters.

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  • 1. H. Friedrich and D. Wintgen, The hydrogen atom in a uniform magnetic field - an example of chaos, Phys. Rep. 183 (1989), no. 2, 37-79. MR 1024519 (90j:81279)
  • 2. V. S. Lisitsa, New results on the Stark and Zeeman effects in the hydrogen atom, Uspekhi Fiz. Nauk 153 (1987), no. 3, 379-421; English transl., Soviet Phys. Uspekhi 30 (1987), no. 11, 927-951. MR 966185 (89i:81148)
  • 3. J. E. Avron, I. W. Herbst, and B. Simon, Schrödinger operators with magnetic fields. III. Atoms in homogeneous magnetic field, Comm. Math. Phys. 79 (1981), no. 4, 529-572. MR 623966 (85h:81084)
  • 4. E. A Solov'ëv, The hydrogen atom in a weak magnetic field, Zh. Eksp. Teor. Fiz. 82 (1982), no. 6, 1762-1771; English transl., Sov. Phys. JETP 55 (1982), no. 6, 1017-1022
  • 5. D. R. Herrick, Symmetry of the quadratic Zeeman effect for hydrogen, Phys. Rev. A (3) 26 (1982), no. 1, 323-329. MR 668295 (83i:81075)
  • 6. D. Delande and J. C. Gay, Group theory applied to the hydrogen atom in a strong magnetic field. Derivation of the effective diamagnetic Hamiltonian, J. Phys. B 17 (1984), no. 11, L335-L340. MR 748791 (85e:81133)
  • 7. V. V. Belov and J. L. Volkova, Investigation of the Zeeman effect in quasiclassical trajectory-coherent approximation, Preprint no. 35, Tomsk Scientific Centre, Siberian Division, AS USSR, 1991.
  • 8. V. V. Belov, V. M. Olivé, and J. L. Volkova, The Zeeman effect for the ``anisotropic hydrogen atom'' in the complex WKB approximation. I. Quantization of closed orbits for the Pauli operator with spin-orbit interaction, J. Phys. A 28 (1995), no. 20, 5799-5810. MR 1364760 (96m:81053a)
  • 9. M. V. Karasev and E. M. Novikova, Representation of exact and quasiclassical eigenfunctions in terms of coherent states. The hydrogen atom in a magnetic field, Teoret. Mat. Fiz. 108 (1996), no. 3, 339-387; English transl., Theoret. and Math. Phys. 108 (1996), no. 3, 1119-1159. MR 1430079 (98b:81083)
  • 10. M. V. Karasev, Birkhoff resonances and quantum ray method, Proc. Intern. Seminar ``Days of Diffraction- 2004'', St. Petersburg University and Steklov Math. Institute, St. Petersburg, 2004, 114-126.
  • 11. M. V. Karasev, Noncommutative algebras, nano-structures, and quantum dynamics generated by resonances. I, Quantum Algebras and Poisson Geometry in Mathematical Physics, Amer. Math. Soc., Providence, RI, 2005, 1-18; MR 2180511 (2006h:81143). II, Adv. Stud. Contemp. Math. (Kyungshang) 11 (2005), no. 1, 33-56; III, Russ. J. Math. Phys. 13 (2006), no. 2, 131-150. MR 2151283 (2006h:81144); MR 2262820 (2007h:81122)
  • 12. M. V. Karasev and E. M. Novikova, An algebra with quadratic commutation relations for an axially perturbed Coulomb-Dirac field, Teoret. Mat. Fiz. 141 (2004), no. 3, 424-454; English transl., Theoret. and Math. Phys. 141 (2004), no. 3, 1698-1724. MR 2141137 (2006e:81120)
  • 13. M. V. Karasev and E. M. Novikova, An algebra with polynomial commutation relations for the Zeeman effect in the Coulomb-Dirac field, Teoret. Mat. Fiz. 142 (2005), no. 1, 127-147; English transl., Theoret. and Math. Phys. 142 (2005), no. 1, 109-127. MR 2137424 (2006b:81134)
  • 14. M. V. Karasev and E. M. Novikova, An algebra with polynomial commutation relations for the Zeeman-Stark effect in the hydrogen atom, Teoret. Mat. Fiz. 142 (2005), no. 3, 530-555; English transl., Theoret. and Math. Phys. 142 (2005), no. 3, 447-469. MR 2165905 (2006d:81348)
  • 15. A. V. Pereskokov, Asymptotics near the boundaries of spectral clusters, International Conference in memory of I. G. Petrovsky (23rd joint meeting of the MMS and the Petrovsky Seminar), Abstracts of talks, MGU Publishing, Moscow, 2011, 260-261.
  • 16. A. V. Pereskokov, Asymptotics of the spectrum and quantum averages near the boundaries of spectral clusters for perturbed two-dimensional oscillators, Mat. zam. 92 (2012), no. 4, 583-596; English transl., Math Notes 92 (2012), no. 4, 532-543.
  • 17. V. M. Babich and V. S. Buldyrev, Short-wavelength diffraction theory. Asymptotic methods, Nauka, Moscow, 1972; English transl., Springer Series on Wave Phenomena, 4, Springer-Verlag, Berlin, 1991. MR 1245488 (94f:78004)
  • 18. V. P. Maslov, The complex WKB method for nonlinear equations. Nauka, Moscow, 1977; English transl., Progress in Physics, 16. Birkhäuser Verlag, Basel, 1994. MR 1306505 (95m:58126)
  • 19. M. V. Karasev and V. P. Maslov, Asymptotic and geometric quantization, Uspekhi Mat. Nauk 39 (1984), no. 6(240), 115-173; English transl., Russian Math. Surveys 39 (1985), no. 6, 133-205. MR 771100 (86m:58064)
  • 20. A Weinstein, Asymptotics of eigenvalue clusters for the Laplacian plus a potential, Duke Math. J. 44 (1977), no. 4, 883-892. MR 0482878 (58:2919)
  • 21. M. V. Karasev and E. M. Novikova, Non-Lie permutation relations, coherent states, and quantum embedding, Coherent transform, quantization, and Poisson geometry, 1-202, Amer. Math. Soc. Transl. Ser. 2, 187, Amer. Math. Soc., Providence, RI, 1998. MR 1728668 (2001i:81116)
  • 22. A. Erdélyi, W. Magnus, F. Oberhettinger, and F. Tricomi, Higher transcendental functions, vols. I, II, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1953. MR 0058756 (15:419i)
  • 23. A. M. Perelomov, Generalized coherent states and their applications, Nauka, Moscow, 1987; English transl., Texts and Monographs in Physics, Springer-Verlag, Berlin, 1986. MR 858831 (87m:22035)
  • 24. V. V. Golubev, Lectures on analytic theory of differential equations, Gostekhisdat, Moscow, 1950. (Russian) MR 0014552 (7:301a)
  • 25. S. Yu. Slavyanov and W. Lay, Special functions. A unified theory based on singularities, Nevski Dialekt, St. Petersburg, 2002; English transl., Oxford Mathematical Monographs. Oxford Science Publications. Oxford University Press, Oxford, 2000. MR 1858237 (2004e:33003)
  • 26. P. A. M. Dirac, Quantum electrodynamics, Comm. Dublin Inst. Adv. Stud., Ser. A 1 (1943), 1-36. MR 0013042 (7:100e)
  • 27. M. V. Fedoryuk, Asymptotic methods for linear ordinary differential equations, Nauka, Moscow, 1983. (Russian) MR 732787 (85m:34001)
  • 28. F. Olver, Asymptotics and special functions, A K Peters, Ltd., Wellesley, MA, 1997. MR 1429619 (97i:41001)
  • 29. M. A. Lavrentév and B. V. Shabat, Methods of the theory of functions of a complex variable, Nauka, Moscow, 1987. (Russian) MR 1087298 (91k:30003)
  • 30. I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, Nauka, Moscow, 1975; English transl., Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London-Toronto, Ont., 1980. MR 582453 (81g:33001)
  • 31. J. Jenkins, Univalent functions and conformal mappings, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1958. MR 0096806 (20:3288)
  • 32. M. Abramowitz and I. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, 55, Washington, D.C., 1964. MR 0167642 (29:4914)
  • 33. M. V. Fedoryuk, Asymptotics: integrals and series, Nauka, Moscow, 1987. (Russian) MR 950167 (89j:41045)
  • 34. P. A. Braun, WKB method for three-term recursion relations and quasienergies of an anharmonic oscillator, Teoret. Mat. Fiz. 37 (1978), no. 3, 355-370; English transl., Theoret. and Math. Phys. 37 (1978), no. 3, 1070-1081. MR 524700 (80b:81020)
  • 35. A. B. Vasil'eva, The correspondence between certain properties of the solutions of linear difference systems and systems of ordinary linear differential equations, Trudy Sem. Teor. Differencial. Uravneniís Otklon. Argumentom, Univ. Druzhby Narodov Patrisa Lumumby 5 (1967), 21-44. MR 0227645 (37:3229)
  • 36. V. P. Maslov and M. V. Fedoryuk, Quasiclassical approximation for the equations of quantum mechanics, Nauka, Moscow, 1976. (Russian) MR 0461590 (57:1575)
  • 37. V. P. Maslov, Complex Markov chains and the Feynman path integral for nonlinear equations, Nauka, Moscow, 1976. (Russian) MR 0479126 (57:18574)
  • 38. L. I. Schiff, Quantum mechanics, McGraw-Hill, 1965.

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Additional Information

A. V. Pereskokov
Affiliation: Moscow Power Engineering Institute, Moscow Institute of Electronics and Mathematics of the Higher School of Economics - National Research University

Keywords: Operator averaging method, coherent transformation, WKB approximation, turning point, spectral cluster
Published electronically: March 21, 2013
Additional Notes: The author was partially supported by the RFFI (Project 12-01-00627-a), the Ministry of Education and Science of the RF (Contract 14.V37.21.0864), and the Grant Council of the President of the RF (Project NSh-2033.2012.1).
Article copyright: © Copyright 2013 American Mathematical Society

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