Coulomb-oscillator duality in spaces of constant curvature

Journal of Mathematical Physics

Volumn 41, Issue 5, 2000, Pages 2629-2657

  Kalnins, E G ^a   Miller Jr , W ^b   Pogosyan, G S ^c  

^a UNIVERSITY OF WAIKATO   (New Zealand)

^b UNIVERSITY OF MINNESOTA   (United States)

^c JOINT INSTITUTE FOR NUCLEAR RESEARCH   (Russian Federation)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0034347345     PISSN: 00222488     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.533263     Document Type: Article

References (43)

1. T. Levi-Civita, "Sur la resolution qualitative di probleme rstreint des trois corps," Opere mathematiche: memorie e note, pubblicate a cura dell'Accademia nazionale dei Lincei, Vol. 2, 411, Bologna, N. Zanichelli (1954).
2. P. Kustaanheimo and E. Steifel, "Perturbation theory of Kepler motion based on spinor regularisation," J. Reine Angew. Math. 218, 204 (1965).
3. C. P. Boyer, E. G. Kalnins, and W. Miller, Jr., "Stäckel-equivalent integrable Hamiltonian systems," SIAM (Soc. Ind. Appl. Math.) J. Math. Anal. 17, 778 (1986).
4. W. Thirring, A Course in Mathematical Physics. 1. Classical Dynamical Systems, translated by E. Harrell (Springer, New York, 1978), see Sec. 4.2.
5. M. Kibler and T. Negadi, "Connection between the hydrogen atom and the harmonic oscillator," Croat. Chem. Acta 57, 1509 (1984).
6. W. Thirring, A Course in Mathematical Physics. 3. Quantum Mechanics of Atoms and Molecules, translated by E. Harrell (Springer, New York, 1981), see Sec. 4.1.
7. S. Flügge, Practical Quantum Mechanics (Springer, Berlin, 1971), Vol. 1.
8. A. Cisneros and H. V. McIntosh, "Symmetry of the two-dimensional hydrogen atom," J. Math. Phys. 10, 277 (1969).
9. L. G. Mardoyan, G. S. Pogosyan, A. N. Sissakian, and V. M. Ter-Antonyan, "Hidden symmetry, separation of variables and interbasis expansions in two-dimensional hydrogen atom," J. Phys. A 18, 455 (1984).
10. T. Iwai, "Quantization of the conformal Kepler problem and its application to the hydrogen atom," J. Math. Phys. 23, 1093 (1982); "The four-dimensional conformal Kepler problem reduces to the three-dimensional Kepler problem with a centrifugal potential and the Dirac's monopole field. Classical theory," 27, 1523 (1986).
11. T. Iwai, "Quantization of the conformal Kepler problem and its application to the hydrogen atom," J. Math. Phys. 23, 1093 (1982); "The four-dimensional conformal Kepler problem reduces to the three-dimensional Kepler problem with a centrifugal potential and the Dirac's monopole field. Classical theory," 27, 1523 (1986).
12. V. M. Ter-Antonyan and A. Nersessian, "Quantum oscillator and a bound system of two dyons," Mod. Phys. Lett. A 10, 2633 (1995).
13. D. Boccaletti and G. Pucacco, Theory of orbits. I. Integrable Systems and Non-pertubative Methods (Springer, New York, 1996).
14. L. S. Davtyan, L. G. Mardoyan, G. S. Pogosyan, A. N. Sissakian, and V. M. Ter-Antonyan, "Generalized KS transformation: From five-dimensional hydrogen atom to eight-dimensional oscillator," J. Phys. A 20, 6121 (1987).
15. D. Lambert and M. Kibler, "An algebraic and geometric approach to non-bijective quadratic transformation," J. Phys. A 21, 307 (1988).
16. M. H. Hassan and M. Kibler, "On Hurwitz transformations," Report LYCEN 9110, Lyon, 1991.
17. L. Van Hoang, T. J. Viloria, and L. anh Thu, "On the hydrogen-like atom in five dimensional space," J. Phys. A 24, 3021 (1991).
18. T. Iwai and T. Sunako, "The quantized SU(2) Kepler problem and its symmetry group for negative energies," J. Geom. Phys. 20, 250 (1996).
19. L. G. Mardoyan, A. N. Sissakian, and V. M. Ter-Antonyan, "8D Oscillator as a hidden SU(2)-monopole," Phys. At. Nuclei 61, 1746 (1998).
20. M. V. Pletyukhov and E. A. Tolkachev, "8D Oscillator and 5D Kepler problem: The case of nontrivial constraints," J. Math. Phys. 40, 93 (1999).
21. E. Schrödinger, "A method of determining quantum mechanical eigenvalues and eigenfunctions," Proc. R. Iri. Acad. 46, 9 (1941); "Further studies on solving eigenvalue problems by factorization" 46, 183 (1941); "The factorization of the hypergeometric equation" 47, 53 (1941).
22. E. Schrödinger, "A method of determining quantum mechanical eigenvalues and eigenfunctions," Proc. R. Iri. Acad. 46, 9 (1941); "Further studies on solving eigenvalue problems by factorization" 46, 183 (1941); "The factorization of the hypergeometric equation" 47, 53 (1941).
23. E. Schrödinger, "A method of determining quantum mechanical eigenvalues and eigenfunctions," Proc. R. Iri. Acad. 46, 9 (1941); "Further studies on solving eigenvalue problems by factorization" 46, 183 (1941); "The factorization of the hypergeometric equation" 47, 53 (1941).
24. P. W. Higgs, "Dynamical symmetries in a spherical geometry. I," J. Phys. A 12, 309 (1979).
25. H. I. Leemon, "Dynamical symmetries in a spherical geometry. II," J. Phys. A 12, 489 (1979).
26. Yu. A. Kurochkin and V. S. Otchik, "The analog of Runge-Lenz vector and the energy spectrum for the Kepler problem on the three-dimensional sphere," Dokl. Acad. Nauk Belarusi XXIII, 987-990 (1979).
27. A. O. Barut, A. Inomata, and G. Junker, "Path integral treatment of the hydrogen atom in a curved space of constant curvature," J. Phys. A 20, 6271 (1987); "Path integral treatment of the hydrogen atom in a curved space of constant curvature. II. Hyperbolic space" 23, 1179 (1990).
28. A. O. Barut, A. Inomata, and G. Junker, "Path integral treatment of the hydrogen atom in a curved space of constant curvature," J. Phys. A 20, 6271 (1987); "Path integral treatment of the hydrogen atom in a curved space of constant curvature. II. Hyperbolic space" 23, 1179 (1990).
29. L. G. Mardoyan, G. S. Pogosyan, A. N. Sissakian, S. I. Vinitsky, and T. A. Strizh, "A hydrogen atom in the curved space. Expansion over free solution," Sov. J. Nucl. Phys. 56, 61 (1993).
30. D. Bonatos, C. Daskaloyannis, and K. Kokkotas, "Deformed oscillator algebras for two-dimensional quantum superintegrable systems," Phys. Rev. A 50, 3700 (1994).
31. C. Grosche, G. S. Pogosyan, and A. N. Sissakian, "Path integral discussion for Smorodin sky - Winternitz potentials. II. Two-and three-dimensional sphere," Fortschr. Phys. 43, 523 (1995).
32. E. G. Kalnins, W. Miller, Jr., and G. S. Pogosyan, "Superintegrability and associated polynomial solutions. Euclidean space and the sphere in two dimensions," J. Math. Phys. 37, 6439 (1996).
33. E. G. Kalnins, W. Miller, Jr., and G. S. Pogosyan, "Superintegrability on the two dimensional hyperboloid," J. Math. Phys. 38, 5416 (1997).
34. C. Grosche, G. S. Pogosyan, and A. N. Sissakian, "Path integral approach to superintegrable potentials. Two-dimensional hyperboloid," Phys. Part. Nuclei 27, 244 (1996); "Path integral discussion for superintegrable potentials. IV. Three dimensional pseudosphere" 28, 486 (1997).
35. C. Grosche, G. S. Pogosyan, and A. N. Sissakian, "Path integral approach to superintegrable potentials. Two-dimensional hyperboloid," Phys. Part. Nuclei 27, 244 (1996); "Path integral discussion for superintegrable potentials. IV. Three dimensional pseudosphere" 28, 486 (1997).
36. G. S. Pogosyan and A. N. Sissakian, "On the Coulomb problem on three dimensional sphere," Turk. J. Phys. 21, 515 (1997).
37. A. A. Izmest'ev, G. S. Pogosyan, A. N. Sissakian, and P. Winternitz, "Contraction of Lie algebras and separation of variables," J. Phys. A 29, 5940 (1996).
38. A. Erdelyi, W. Magnus, F. Oberhettinger, and F. Tricomi, Tables of Integral Transforms (McGraw-Hill, New York, 1954), Vol II.
39. A. Erdelyi, W. Magnus, F. Oberhettinger, and F. Tricomi, Higher Transdental Functions (McGraw-Hill, New York, 1953), Vols. I and II.
40. L. S. Davtyan, G. S. Pogosyan, A. N. Sissakian, and V. M. Ter-Antonyan, "Two-dimensional hydrogen atom. Expansion of the polar and parabolic bases in the continuous spectrum," Teor. Mat. Fiz. 66, 222 (1986).
41. D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii, Quantum Theory of Angular Momentum (World Scientific, Singapore, 1988).
42. Kh. H. Karayan, L. G. Mardoyan, and V. M. Ter-Antonyan, "The Eulerian bound states:" 5D Coulomb problem," Preprint JINR, E2-94-359, Dubna, 1994.
43. A. Hurwitz, Mathematische Werke, Band II (Birkhäuser, Basel, 1933), 641 pp.