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Advances in High Energy Physics
Volume 2015, Article ID 701042, 9 pages
http://dx.doi.org/10.1155/2015/701042
Research Article

Klein-Gordon Equation with Superintegrable Systems: Kepler-Coulomb, Harmonic Oscillator, and Hyperboloid

Department of Physics, Faculty of Science, Sahand University of Technology, P.O. Box 51335-1996, Tabriz, Iran

Received 27 July 2015; Revised 26 September 2015; Accepted 15 October 2015

Academic Editor: Thomas Rössler

Copyright © 2015 V. Mohammadi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP3.

Linked References

  1. A. Alhaidari, H. Bahlouli, and A. Al-Hasan, “Dirac and Klein–Gordon equations with equal scalar and vector potentials,” Physics Letters A, vol. 349, no. 1–4, pp. 87–97, 2006. View at Publisher · View at Google Scholar
  2. X.-A. Zhang, K. Chen, and Z.-L. Duan, “Bound states of Klein-Gordon equation and Dirac equation for ring-shaped non-spherical oscillator scalar and vector potentials,” Chinese Physics, vol. 14, no. 1, pp. 42–44, 2005. View at Publisher · View at Google Scholar
  3. L.-H. Zhang, X.-P. Li, and C.-S. Jia, “Approximate analytical solutions of the Dirac equation with the generalized Morse potential model in the presence of the spin symmetry and pseudo-spin symmetry,” Physica Scripta, vol. 80, no. 3, Article ID 035003, 2009. View at Publisher · View at Google Scholar
  4. F.-L. Zhang and J.-L. Chen, “Dynamical symmetries of the Klein–Gordon equation,” Journal of Mathematical Physics, vol. 50, no. 3, Article ID 032301, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  5. A. de Souza Dutra and M. Hott, “Dirac equation exact solutions for generalized asymmetrical Hartmann potentials,” Physics Letters A: General, Atomic and Solid State Physics, vol. 356, no. 3, pp. 215–219, 2006. View at Publisher · View at Google Scholar · View at Scopus
  6. J. N. Ginocchio, “U(3) and Pseudo-U(3) symmetry of the relativistic harmonic oscillator,” Physical Review Letters, vol. 95, no. 25, Article ID 252501, 3 pages, 2005. View at Publisher · View at Google Scholar
  7. I. Marquette, “Quadratic algebra approach to relativistic quantum Smorodinsky–Winternitz systems,” Journal of Mathematical Physics, vol. 52, no. 4, Article ID 042301, 2011. View at Publisher · View at Google Scholar · View at Scopus
  8. J. N. Ginocchio, “Relativistic harmonic oscillator with spin symmetry,” Physical Review C—Nuclear Physics, vol. 69, no. 3, pp. 034318–1, 2004. View at Publisher · View at Google Scholar · View at Scopus
  9. S. Aghaei and A. Chenaghlou, “Solution of the Dirac equation with some superintegrable potentials by the quadratic algebra approach,” International Journal of Modern Physics A, vol. 29, no. 6, Article ID 1450028, 2014. View at Publisher · View at Google Scholar · View at Scopus
  10. S. Aghaei and A. Chenaghlou, “Quadratic algebra approach to the Dirac equation with spin and pseudospin symmetry for the 4D harmonic oscillator and U(1) monopole,” Few-Body Systems, vol. 56, no. 1, pp. 53–61, 2015. View at Publisher · View at Google Scholar
  11. F.-L. Zhang, B. Fu, and J.-L. Chen, “Dynamical symmetry of Dirac hydrogen atom with spin symmetry and its connection with Ginocchio's oscillator,” Physical Review A, vol. 78, no. 4, Article ID 040101, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  12. L. Cheng-Shi, “Classification of all single travelling wave solutions to Calogero-Degasperis-Focas equation,” Communications in Theoretical Physics, vol. 48, no. 4, pp. 601–604, 2007. View at Publisher · View at Google Scholar · View at Scopus
  13. X.-Q. Hu, G. Luo, Z.-M. Wu, L.-B. Niu, and Y. Ma, “Solving Dirac equation with new ring-shaped non-spherical harmonic oscillator potential,” Communications in Theoretical Physics, vol. 53, no. 2, pp. 242–246, 2010. View at Publisher · View at Google Scholar
  14. C. Berkdemir and Y.-F. Cheng, “On the exact solutions of the Dirac equation with a novel angle-dependent potential,” Physica Scripta, vol. 79, no. 3, Article ID 035003, 2009. View at Publisher · View at Google Scholar · View at Scopus
  15. Z. Yan and G. Jian-You, “The relativistic bound states for a new ring-shaped harmonic oscillator,” Chinese Physics B, vol. 17, no. 2, pp. 380–384, 2008. View at Publisher · View at Google Scholar · View at Scopus
  16. G.-X. Ju and Z.-Z. Ren, “Supersymmetry and solution of Dirac equation with vector and scalar potentials,” Communications in Theoretical Physics, vol. 49, no. 2, pp. 319–326, 2008. View at Publisher · View at Google Scholar · View at Scopus
  17. T. Jana and P. Roy, “Shape invariance approach to exact solutions of the Klein-Gordon equation,” Physics Letters A: General, Atomic and Solid State Physics, vol. 361, no. 1-2, pp. 55–58, 2007. View at Publisher · View at Google Scholar · View at Scopus
  18. E. Olgar, R. Koç, and H. Tütüncüler, “Bound states of the s-wave equation with equal scalar and vector standard Eckart potential,” Chinese Physics Letters, vol. 23, no. 3, pp. 539–541, 2006. View at Publisher · View at Google Scholar · View at Scopus
  19. G. Chen, Z.-D. Chen, and Z.-M. Lou, “Exact bound state solutions of the s-wave Klein–Gordon equation with the generalized Hulthén potential,” Physics Letters A, vol. 331, no. 6, pp. 374–377, 2004. View at Publisher · View at Google Scholar · View at Scopus
  20. G. Chen, Z.-D. Chen, and P.-C. Xuan, “Exactly solvable potentials of the Klein-Gordon equation with the supersymmetry method,” Physics Letters A: General, Atomic and Solid State Physics, vol. 352, no. 4-5, pp. 317–320, 2006. View at Publisher · View at Google Scholar · View at Scopus
  21. P. Alberto, A. S. D. Castro, and M. Malheiro, “Spin and pseudospin symmetries and the equivalent spectra of relativistic spin-1/2 and spin-0 particles,” Physical Review C, vol. 75, no. 4, Article ID 047303, 4 pages, 2007. View at Publisher · View at Google Scholar
  22. C. Daskaloyannis, “Quadratic poisson algebras of two-dimensional classical superintegrable systems and quadratic associative algebras of quantum superintegrable systems,” Journal of Mathematical Physics, vol. 42, no. 3, pp. 1100–1119, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  23. F.-L. Zhang, B. Fu, and J.-L. Chen, “Higgs algebraic symmetry in the two-dimensional Dirac,” Physical Review A, vol. 80, no. 5, Article ID 054102, 2009. View at Publisher · View at Google Scholar
  24. Y. I. Granovskii, A. S. Zhedanov, and I. M. Lutzenko, “Quadratic algebra as a ‘hidden’ symmetry of the Hartmann potential,” Journal of Physics A: Mathematical and General, vol. 24, no. 16, pp. 3887–3894, 1991. View at Publisher · View at Google Scholar · View at Scopus
  25. Y. I. Granovskii, A. S. Zhedanov, and I. M. Lutsenko, “Quadratic algebras and dynamics in curved spaces. I. Oscillator,” Theoretical and Mathematical Physics, vol. 91, no. 2, pp. 474–480, 1992. View at Publisher · View at Google Scholar · View at Scopus
  26. D. Bonatsos, C. Daskaloyannis, and K. Kokkotas, “Quantum-algebraic description of quantum superintegrable systems in two dimensions,” Physical Review A, vol. 48, no. 5, pp. R3407–R3410, 1993. View at Publisher · View at Google Scholar · View at Scopus
  27. D. Bonatsos, C. Daskaloyannis, and K. Kokkotas, “Deformed oscillator algebras for two-dimensional quantum superintegrable systems,” Physical Review A, vol. 50, no. 5, pp. 3700–3709, 1994. View at Publisher · View at Google Scholar · View at Scopus
  28. C. Daskaloyannis and Y. Tanoudis, “Quadratic algebras for three-dimensional superintegrable systems,” Physics of Atomic Nuclei, vol. 73, no. 2, pp. 214–221, 2010. View at Publisher · View at Google Scholar · View at Scopus
  29. C. Quesne, “Quadratic algebra approach to an exactly solvable position-dependent mass Schrödinger equation in two dimensions,” SIGMA, vol. 3, article 67, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  30. I. Marquette, “Construction of classical superintegrable systems with higher order integrals of motion from ladder operators,” Journal of Mathematical Physics, vol. 51, no. 7, Article ID 037006JMP, 2010. View at Publisher · View at Google Scholar · View at Scopus
  31. T. Levi-Civita, “Sur la résolution qualitative du problème restreint des trois corps,” Acta Mathematica, vol. 30, no. 1, pp. 305–327, 1906. View at Publisher · View at Google Scholar · View at Scopus
  32. E. G. Kalnins, W. Miller Jr., and G. S. Pogosyan, “Coulomb-oscillator duality in spaces of constant curvature,” Journal of Mathematical Physics, vol. 41, no. 5, pp. 2629–2657, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  33. B. Fu, F.-L. Zhang, and J. Chen, “Connection between Coulomb and harmonic oscillator potentials in relativistic quantum mechanics,” Physica Scripta, vol. 81, no. 3, Article ID 035001, 2010. View at Publisher · View at Google Scholar
  34. E. G. Kalnins, W. Miller Jr., Y. M. Hakobyan, and G. S. Pogosyan, “Superintegrability on the two-dimensional hyperboloid. II,” Journal of Mathematical Physics, vol. 40, no. 5, pp. 2291–2306, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  35. C. P. Boyer, E. G. Kalnins, and P. Winternitz, “Completely integrable relativistic Hamiltonian systems and separation of variables in Hermitian hyperbolic spaces,” Journal of Mathematical Physics, vol. 24, no. 8, pp. 2022–2034, 1983. View at Publisher · View at Google Scholar
  36. C. Grosche, G. S. Pogosyan, and A. N. Sissakian, “Path-integral approach to superintegrable potentials on the two-dimensional hyperboloid,” Physics of Particles and Nuclei, vol. 27, no. 3, pp. 244–278, 1996. View at Google Scholar · View at Scopus
  37. E. G. Kalnins, W. Miller Jr., and G. S. Pogosyan, “Superintegrability on the two-dimensional hyperboloid,” Journal of Mathematical Physics, vol. 38, no. 10, pp. 5416–5433, 1997. View at Publisher · View at Google Scholar · View at Scopus
  38. W. Miller, Symmetry and Separation of Variables, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, 2012.