Table of Contents Author Guidelines Submit a Manuscript
Advances in High Energy Physics
Volume 2019, Article ID 1248393, 13 pages
https://doi.org/10.1155/2019/1248393
Research Article

A Central Potential with a Massive Scalar Field in a Lorentz Symmetry Violation Environment

Departamento de Física e Química, Universidade Federal do Espírito Santo, Av. Fernando Ferrari, 514, Goiabeiras, 29060-900, Vitória, ES, Brazil

Correspondence should be addressed to R. L. L. Vitória; moc.liamtoh@19siul-odracir

Received 16 January 2019; Accepted 2 May 2019; Published 4 June 2019

Academic Editor: Edward Sarkisyan-Grinbaum

Copyright © 2019 R. L. L. Vitória and H. Belich. 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. R. Pohl, F. Nez, L. M. Fernandes et al., “Laser spectroscopy of muonic deuterium,” Science, vol. 353, pp. 669–673, 2016. View at Google Scholar
  2. A. Songaila and L. L. Cowie, “Fine-structure variable?” Nature, vol. 398, no. 6729, pp. 667-668, 1999. View at Google Scholar · View at Scopus
  3. L. L. Cowie and A. Songaila, “The inconstant constant?” Nature, vol. 428, no. 6979, pp. 132-133, 2004. View at Publisher · View at Google Scholar · View at Scopus
  4. V. A. Kostelecký and S. Samuel, “Spontaneous breaking of Lorentz symmetry in string theory,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 39, p. 683, 1989. View at Publisher · View at Google Scholar
  5. D. Colladay and V. A. Kostelecký, “CPT violation and the standard model,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 55, article 6760, 1997. View at Publisher · View at Google Scholar
  6. D. Colladay and V. A. Kostelecký, “Lorentz-violating extension of the standard model,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 58, Article ID 116002, 1998. View at Publisher · View at Google Scholar
  7. D. Colladay and V. Kostelecký, “Cross sections and lorentz violation,” Physics Letters B, vol. 511, no. 2-4, pp. 209–217, 2001. View at Publisher · View at Google Scholar
  8. R. Lehnert, “Threshold analyses and Lorentz violation,” Physical Review D, vol. 68, Article ID 085003, 2003. View at Google Scholar
  9. R. Lehnert, “Dirac theory within the standard-model extension,” Journal of Mathematical Physics, vol. 45, no. 8, pp. 3399–3412, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  10. B. Altschul, “Compton scattering in the presence of Lorentz and CPT violation,” PHysical Review D, vol. 70, Article ID 056005, 2004. View at Google Scholar
  11. G. M. Shore, “Strong equivalence, Lorentz and CPT violation, anti-hydrogen spectroscopy and gamma-ray burst polarimetry,” Nuclear Physics B, vol. B717, pp. 86–118, 2005. View at Publisher · View at Google Scholar
  12. S. Aghababaei and M. Haghighat, “Muon anomalous magnetic moment in the standard model extension,” Physical Review D, vol. 96, Article ID 115028, 2017. View at Google Scholar
  13. R. Bluhm, V. A. Kostelecký, and C. D. Lane, “CPT and Lorentz Tests with Muons,” Physical Review Letters, vol. 84, no. 6, pp. 1098–1101, 2000. View at Publisher · View at Google Scholar
  14. R. Bluhm, V. A. Kostelecký, C. D. Lane, and N. Russell, “Clock-comparison tests of Lorentz and CPT symmetry in space,” Physical Review Letters, vol. 88, no. 9, Article ID 090801, 2002. View at Publisher · View at Google Scholar
  15. S. M. Carrol, G. B. Field, and R. Jackiw, “Limits on a Lorentz- and parity-violating modification of electrodynamics,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 41, no. 4, pp. 1231–1240, 1990. View at Publisher · View at Google Scholar
  16. A. A. Andrianov, D. Espriu, P. Giacconi, and R. Soldati, “Anomalous positron excess from Lorentz-violating QED,” Journal of High Energy Physics, vol. 2009, no. 09, article 057, 2009. View at Google Scholar
  17. J. Alfaro, A. A. Andrianov, M. Cambiaso, P. Giacconi, and R. Soldati, “Bare and induced lorentz and cpt invariance violations in QED,” International Journal of Modern Physics A, vol. 25, no. 16, pp. 3271–3306, 2010. View at Google Scholar
  18. Y. Gomes and P. Malta, “Laboratory-based limits on the Carroll-Field-Jackiw Lorentz-violating electrodynamics,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 94, no. 2, Article ID 025031, 2016. View at Publisher · View at Google Scholar
  19. A. Martn-Ruiz and C. A. Escobar, “Local effects of the quantum vacuum in Lorentz-violating electrodynamics,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 95, no. 3, Article ID 036011, 2017. View at Google Scholar
  20. R. Lehnert and R. Potting, “Vacuum čerenkov radiation,” Physical Review Letters, vol. 93, no. 11, Article ID 110402, 2004. View at Publisher · View at Google Scholar
  21. R. Lehnert and R. Potting, “Publisher's note: čerenkov effect in Lorentz-violating vacua,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 70, no. 12, Article ID 125010, 2004. View at Publisher · View at Google Scholar
  22. C. Kaufhold and F. R. Klinkhamer, “Vacuum Cherenkov radiation and photon triple-splitting in a Lorentz-noninvariant extension of quantum electrodynamics,” Nuclear Physics B, vol. 734, no. 1-2, pp. 1–23, 2006. View at Google Scholar
  23. B. Altschul, “Cerenkov radiation in a Lorentz-violating and birefringent vacuum,” Physical Review D, vol. 75, no. 10, Article ID 105003, 2007. View at Google Scholar
  24. B. Altschul, “Vacuum Čerenkov radiation in lorentz-violating theories without CPT Violation,” Physical Review Lett, vol. 98, Article ID 041603, 2007. View at Google Scholar
  25. C. Kaufhold and F. R. Klinkhamer, “Vacuum Cherenkov radiation in spacelike Maxwell-Chern-Simons theory,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 76, no. 2, Article ID 025024, 2007. View at Publisher · View at Google Scholar
  26. B. Altschul, “Finite duration and energy effects in Lorentz-violating vacuum Cerenkov radiation,” Nuclear Physics B, vol. 796, no. 1-2, pp. 262–273, 2008. View at Google Scholar
  27. C. A. Escobar and M. A. G. Garcia, “Full CPT-even photon sector of the standard model extension at finite temperature,” Physical Review D, vol. 92, Article ID 025034, 2015. View at Google Scholar
  28. A. Martn-Ruiz and C. A. Escobar, “Casimir effect between ponderable media as modeled by the standard model extension,” Physical Review D, vol. 94, no. 7, Article ID 076010, 2016. View at Google Scholar
  29. F. R. Klinkhamer and M. Risse, “Ultrahigh-energy cosmic-ray bounds on nonbirefringent modified Maxwell theory,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 77, no. 1, Article ID 016002, 2008. View at Publisher · View at Google Scholar
  30. F. R. Klinkhamer and M. Risse, “Addendum: Ultrahigh-energy cosmic-ray bounds on nonbirefringent modified Maxwell theory,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 77, no. 11, Article ID 117901, 2008. View at Google Scholar
  31. F. R. Klinkhamer and M. Schreck, “New two-sided bound on the isotropic Lorentz-violating parameter of modified Maxwell theory,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 78, no. 8, Article ID 085026, 2008. View at Google Scholar
  32. A. Moyotl, H. Novales-Sánchez, J. J. Toscano, and E. S. Tututi, “Gauge invariant electromagnetic properties of fermions induced by CPT-violation in the standard model extension,” International Journal of Modern Physics A, vol. 29, no. 8, Article ID 1450039, 2014. View at Google Scholar
  33. M. Schreck, “Analysis of the consistency of parity-odd nonbirefringent modified Maxwell theory,” Physical Review D, vol. 86, no. 6, Article ID 065038, 2012. View at Google Scholar
  34. B. Agostini, F. Barone, F. Barone, P. Gaete, and J. Helayël-Neto, “Consequences of vacuum polarization on electromagnetic waves in a Lorentz-symmetry breaking scenario,” Physics Letters B, vol. 708, no. 1-2, pp. 212–215, 2012. View at Publisher · View at Google Scholar
  35. L. C. Brito, H. G. Fargnoli, and A. P. Baêta Scarpelli, “Aspects of quantum corrections in a Lorentz-violating extension of the Abelian Higgs model,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 87, no. 12, Article ID 125023, 2013. View at Publisher · View at Google Scholar
  36. T. Mariz, J. R. Nascimento, E. Passos, R. F. Ribeiro, F. A. Brito, and J. High Energy, “A remark on Lorentz violation at finite temperature,” Journal of High Energy Physics, vol. 2005, no. 10, article 019, 2005. View at Google Scholar
  37. J. R. Nascimento, E. Passos, A. Yu. Petrov, and F. A. Brito, “Lorentz-CPT violation, radiative corrections and finite temperature,” Journal of High Energy Physics, vol. 2007, no. 6, article 016, 2007. View at Publisher · View at Google Scholar
  38. A. P. Scarpelli, M. Sampaio, M. C. Nemes, and B. Hiller, “Gauge invariance and the CPT and Lorentz violating induced Chern–Simons-like term in extended QED,” The European Physical Journal C, vol. 56, no. 4, pp. 571–578, 2008. View at Publisher · View at Google Scholar
  39. F. A. Brito, J. R. Nascimento, E. Passos, and A. Y. Petrov, “The ambiguity-free four-dimensional Lorentz-breaking Chern-Simons action,” Physics Letters. B. Particle Physics, Nuclear Physics and Cosmology, vol. 664, no. 1-2, pp. 112–115, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  40. F. A. Brito, L. S. Grigorio, M. S. Guimaraes, E. Passos, and C. Wotzasek, “Induced Chern-Simons-like action in Lorentz-violating massless QED,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 78, no. 12, Article ID 125023, 2008. View at Publisher · View at Google Scholar
  41. O. M. Del Cima, J. M. Fonseca, D. H. Franco, and O. Piguet, “Lorentz and CPT violation in QED revisited: a missing analysis,” Physics Letters B, vol. 688, no. 2-3, pp. 258–262, 2010. View at Publisher · View at Google Scholar
  42. V. A. Kostelecký and M. Mewes, “Electrodynamics with Lorentz-violating operators of arbitrary dimension,” Physical Review D, vol. 80, no. 1, Article ID 015020, 2009. View at Google Scholar
  43. M. Mewes, “Optical-cavity tests of higher-order Lorentz violation,” Physical Review D, vol. 85, no. 11, Article ID 116012, 2012. View at Google Scholar
  44. M. Schreck, “Quantum field theoretic properties of Lorentz-violating operators of nonrenormalizable dimension in the photon sector,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 89, no. 10, Article ID 105019, 17 pages, 2014. View at Publisher · View at Google Scholar
  45. V. A. Kostelecký and M. Mewes, “Fermions with Lorentz-violating operators of arbitrary dimension,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 88, no. 9, Article ID 096006, 2013. View at Publisher · View at Google Scholar
  46. M. Schreck, “Quantum field theoretic properties of Lorentz-violating operators of nonrenormalizable dimension in the fermion sector,” Physical Review D, vol. 90, no. 8, Article ID 085025, 2014. View at Google Scholar
  47. R. C. Myers and M. Pospelov, “Ultraviolet modifications of dispersion relations in effective field theory,” Physical Review Letters, vol. 90, no. 21, Article ID 211601, 4 pages, 2003. View at Publisher · View at Google Scholar
  48. C. M. Reyes, L. F. Urrutia, and J. D. Vergara, “Quantization of the Myers-Pospelov model: The photon sector interacting with standard fermions as a perturbation of QED,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 78, no. 12, Article ID 125011, 2008. View at Publisher · View at Google Scholar
  49. J. Lopez-Sarrion and C. M. Reyes, “Microcausality and quantization of the fermionic Myers–Pospelov model,” The European Physical Journal C, vol. 72, no. 9, article 2150, 2012. View at Publisher · View at Google Scholar
  50. C. Reyes, L. Urrutia, and J. Vergara, “The photon sector in the quantum Myers–Pospelov model: An improved description,” Physics Letters B, vol. 675, no. 3-4, pp. 336–339, 2009. View at Publisher · View at Google Scholar
  51. C. M. Reyes, “Causality and stability for Lorentz-CPT violating electrodynamics with dimension-5 operators,” Physical Review D, vol. 82, Article ID 125036, 26 pages, 2010. View at Publisher · View at Google Scholar
  52. C. M. Reyes, “Unitarity in higher-order Lorentz-invariance violating QED,” Physical Review D, vol. 87, no. 12, Article ID 125028, 7 pages, 2013. View at Publisher · View at Google Scholar
  53. K. Bakke and H. Belich, “Quantum holonomies based on the Lorentz-violating tensor background,” Journal of Physics G: Nuclear and Particle Physics, vol. 40, no. 6, Article ID 065002, 2013. View at Google Scholar
  54. H. Belich, T. Costa-Soares, M. A. Santos, and M. T. D. Orlando, “Lorentz symmetry violation,” Revista Brasileira de Ensino de Física, vol. 29, no. 1, pp. 57–64, 2007. View at Google Scholar
  55. K. Bakke and H. Belich, “Relativistic Landau–He–McKellar–Wilkens quantization and relativistic bound states solutions for a Coulomb-like potential induced by the Lorentz symmetry breaking effects,” Annals of Physics, vol. 333, pp. 272–281, 2013. View at Publisher · View at Google Scholar
  56. R. Casana, M. M. Ferreira, R. V. Maluf, and F. E. P. dos Santos, “Radiative generation of the CPT-even gauge term of the SME from a dimension-five nonminimal coupling term,” Physics Letters, vol. 726, pp. 815–819, 2013. View at Google Scholar
  57. R. Casana, M. M. Ferreira, E. da Hora, and A. B. F. Neves, “Maxwell–Chern–Simons vortices in a CPT-odd Lorentz-violating Higgs electrodynamics,” The European Physical Journal C, vol. 74, article 3064, 2014. View at Google Scholar
  58. R. Casana, M. M. Ferreira, and P. F. E. dos Santos, “Gupta-Bleuler quantization of the anisotropic parity-even and C P T-even electrodynamics of a standard model extension,” Physical Review D, vol. 90, Article ID 105025, 2014. View at Google Scholar
  59. R. Casana, C. F. Farias, and M. M. Ferreira, “Topological self-dual configurations in a Lorentz-violating gauged O(3) sigma model,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 92, Article ID 125024, 2015. View at Publisher · View at Google Scholar
  60. R. Casana, M. M. Ferreira, V. E. Mouchrek-Santos, and E. O. Silva, “Generation of geometrical phases and persistent spin currents in 1-dimensional rings by Lorentz-violating terms,” Physics Letters B, vol. 746, pp. 171–177, 2015. View at Google Scholar
  61. G. Gazzola, H. G. Fargnoli, A. P. Baêta Scarpelli et al., “QED with minimal and nonminimal couplings: on the quantum generation of Lorentz-violating terms in the pure photon sector,” Journal of Physics G: Nuclear and Particle Physics, vol. 39, Article ID 035002, 2012. View at Google Scholar
  62. K. Bakke and H. Belich, “A Landau-type quantization from a Lorentz symmetry violation background with crossed electric and magnetic fields,” Journal of Physics G: Nuclear and Particle Physics, vol. 42, Article ID 095001, 2015. View at Google Scholar
  63. K. Bakke and H. Belich, Spontaneous Lorentz Symmetry Violation and Low Energy Scenarios, LAMBERT Academic Publishing, Saarbrücken, Germany, 2015.
  64. K. Bakke and H. Belich, “On a relativistic scalar particle subject to a Coulomb-type potential given by Lorentz symmetry breaking effects,” Annals of Physics, vol. 360, pp. 596–604, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  65. K. Bakke and H. Belich, “On the harmonic-type and linear-type confinement of a relativistic scalar particle yielded by Lorentz symmetry breaking effects,” Annals of Physics, vol. 373, pp. 115–122, 2016. View at Publisher · View at Google Scholar · View at MathSciNet
  66. R. L. L. Vitória, H. Belich, and K. Bakke, “A relativistic quantum oscillator subject to a Coulomb-type potential induced by effects of the violation of the Lorentz symmetry,” The European Physical Journal Plus, vol. 132, article 25, 2017. View at Google Scholar
  67. R. L. L. Vitória, H. Belich, and K. Bakke, “Coulomb-type interaction under lorentz symmetry breaking effects,” Advances in High Energy Physics, vol. 2017, Article ID 6893084, 5 pages, 2017. View at Publisher · View at Google Scholar
  68. M. Gomes, J. R. Nascimento, A. Petrov et al., “On the aether-like Lorentz-breaking actions,” Physical Review, vol. 81, Article ID 045018, 16 pages, 2010. View at Publisher · View at Google Scholar
  69. M. B. Cruz, E. R. de Mello, and A. Y. Petrov, “Casimir effects in Lorentz-violating scalar field theory,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 96, no. 4, 045019, 12 pages, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  70. G. Arfken and H. J. Weber, Mathematical Methods for Physicists, Elsevier Academic Press, New York, NY, USA, 6th edition, 2005. View at MathSciNet
  71. K. Bakke, “On the rotating effects and the Landau-Aharonov-Casher system subject to a hard-wall confining potential in the cosmic string spacetime,” International Journal of Theoretical Physics, vol. 54, no. 7, pp. 2119–2126, 2015. View at Publisher · View at Google Scholar · View at MathSciNet
  72. K. Bakke, “A geometric approach to confining a Dirac neutral particle in analogous way to a quantum dot,” The European Physical Journal B, vol. 85, article 354, 2012. View at Google Scholar
  73. L. C. N. Santos and C. C. Barros, “Relativistic quantum motion of spin-0 particles under the influence of noninertial effects in the cosmic string spacetime,” The European Physical Journal C, vol. 78, article 13, 2018. View at Google Scholar
  74. L. B. Castro, “Noninertial effects on the quantum dynamics of scalar bosons,” The European Physical Journal C, vol. 76, article 61, 2016. View at Google Scholar
  75. R. L. Vitória and K. Bakke, “Aharonov–Bohm effect for bound states in relativistic scalar particle systems in a spacetime with a spacelike dislocation,” International Journal of Modern Physics D: Gravitation, Astrophysics, Cosmology, vol. 27, no. 2, Article ID 1850005, 2018. View at Publisher · View at Google Scholar · View at MathSciNet
  76. R. L. L. Vitória and K. Bakke, “Rotating effects on the scalar field in the cosmic string spacetime, in the spacetime with space-like dislocation and in the spacetime with a spiral dislocation,” European Physical Journal C, vol. 78, no. 175, 2018. View at Google Scholar
  77. W. Greiner, Relativistic Quantum Mechanics: Wave Equations, Springer, Berlin, Germany, 3rd edition, 2000. View at MathSciNet
  78. M. K. Bahar and F. Yasuk, “Exact solutions of the mass-dependent klein-gordon equation with the vector quark-antiquark interaction and harmonic oscillator potential,” Advances in High Energy Physics, vol. 2013, Article ID 814985, 6 pages, 2013. View at Publisher · View at Google Scholar
  79. E. R. Figueiredo Medeiros and E. R. Bezerra de Mello, “Relativistic quantum dynamics of a charged particle in cosmic string spacetime in the presence of magnetic field and scalar potential,” The European Physical Journal C, vol. 72, article 2051, 2012. View at Publisher · View at Google Scholar
  80. R. L. L. Vitória, C. Furtado, and K. Bakke, “Linear confinement of a scalar particle in a Gödel-type spacetime,” The European Physical Journal C, vol. 78, no. 44, 2018. View at Google Scholar
  81. A. L. Cavalcanti de Oliveira and E. R. Bezerra de Mello, “Exact solutions of the Klein-Gordon equation in the presence of a dyon, magnetic flux and scalar potential in the spacetime of gravitational defects,” Classical and Quantum Gravity, vol. 23, no. 17, pp. 5249–5263, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  82. K. Bakke and C. Furtado, “On the Klein–Gordon oscillator subject to a Coulomb-type potential,” Annals of Physics, vol. 355, pp. 48–54, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  83. R. L. L. Vitória, C. Furtado, and K. Bakke, “On a relativistic particle and a relativistic position-dependent mass particle subject to the klein–gordon oscillator and the coulomb potential,” Annals of Physics, vol. 370, no. 128, pp. 128–136, 2016. View at Publisher · View at Google Scholar
  84. G. Soff, B. Müller, J. Rafelski, and W. Greiner, “Solution of the dirac equation for scalar potentials and its implications in atomic physics,” Zeitschrift für Naturforschung A, vol. 28, no. 9, pp. 1389–1396, 1973. View at Publisher · View at Google Scholar
  85. H. Asada and T. Futamase, “Propagation of gravitational waves from slow motion sources in a Coulomb-type potential,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 56, no. 10, pp. R6062–R6066, 1997. View at Publisher · View at Google Scholar · View at Scopus
  86. P. Gribi and E. Sigmund, “Exact solutions for a quasi-one-dimensional Coulomb-type potential,” Physical Review B: Condensed Matter and Materials Physics, vol. 44, no. 8, pp. 3537–3549, 1991. View at Publisher · View at Google Scholar · View at Scopus
  87. F. Gesztesy and B. Thaller, “Born expansions for Coulomb-type interactions,” Journal of Physics A: Mathematical and General, vol. 14, no. 3, pp. 639–657, 1981. View at Publisher · View at Google Scholar
  88. J. A. Reyes and M. del Castillo-Mussot, “1D Schrödinger equations with Coulomb-type potentials,” Journal of Physics A: Mathematical and General, vol. 32, no. 10, pp. 2017–2025, 1999. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  89. A. De Souza Dutra, “Conditionally exactly soluble class of quantum potentials,” Physical Review A: Atomic, Molecular and Optical Physics, vol. 47, no. 4, pp. R2435–R2437, 1993. View at Publisher · View at Google Scholar · View at Scopus
  90. S. M. Ikhdair and M. Hamzavi, “Effects of external fields on a two-dimensional Klein-Gordon particle under pseudo-harmonic oscillator interaction,” Chinese Physics B, vol. 21, p. 110302, 2012. View at Google Scholar
  91. S. M. Ikhdair, B. J. Falaye, and M. Hamzavi, “Nonrelativistic molecular models under external magnetic and AB flux fields,” Annals of Physics, vol. 353, pp. 282–298, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  92. I. I. Guseinov and B. A. Mamedov, “Evaluation of multicenter one-electron integrals of noninteger u screened Coulomb type potentials and their derivatives over noninteger n Slater orbitals,” The Journal of Chemical Physics, vol. 121, no. 4, article 1649, 2004. View at Publisher · View at Google Scholar
  93. I. I. Guseinov, “Unified treatment of multicenter integrals of integer and noninteger u Yukawa-type screened Coulomb type potentials and their derivatives over Slater orbitals,” The Journal of Chemical Physics, vol. 120, no. 20, pp. 9454–9457, 2004. View at Publisher · View at Google Scholar
  94. R. L. L. Vitória and K. Bakke, “Relativistic quantum effects of confining potentials on the Klein-Gordon oscillator,” The European Physical Journal Plus, vol. 131, no. 36, 2016. View at Publisher · View at Google Scholar
  95. R. L. L. Vitória and K. Bakke, “Torsion effects on a relativistic position-dependent mass system,” General Relativity and Gravitation, vol. 48, article 161, 2016. View at Publisher · View at Google Scholar · View at MathSciNet
  96. R. F. Ribeiro and K. Bakke, “On the Majorana fermion subject to a linear confinement,” Annals of Physics, vol. 385, pp. 36–39, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  97. A. Ronveaux, Heun's Differential Equations, Oxford University Press, Oxford, UK, 1995.
  98. C. Furtado, “Landau levels in the presence of disclinations,” Physics Letters A, vol. 195, no. 1, pp. 90–94, 1994. View at Publisher · View at Google Scholar
  99. M. Eshghi and H. Mehraban, “Study of a 2D charged particle confined by a magnetic and AB flux fields under the radial scalar power potential,” The European Physical Journal Plus, vol. 132, article 121, 2017. View at Publisher · View at Google Scholar
  100. S. Bruce and P. Minning, “The klein-gordon oscillator,” Il Nuovo Cimento A, vol. 106, no. 5, pp. 711–713, 1993. View at Publisher · View at Google Scholar · View at MathSciNet
  101. V. V. Dvoeglazov, “The Klein-Gordon oscillator,” Il Nuovo Cimento A, vol. 107, pp. 1413–1418, 1994. View at Google Scholar
  102. N. A. Rao and B. A. Kagali, “Energy profile of the one-dimensional Klein–Gordon oscillator,” Physica Scripta, vol. 77, Article ID 015003, 2008. View at Google Scholar
  103. A. Boumali, A. Hafdallah, and A. Toumi, “Comment on 'Energy profile of the one-dimensional Klein–Gordon oscillator',” Physica Scripta, vol. 84, Article ID 037001, 2011. View at Google Scholar
  104. B. Mirza and M. Mohadesi, “The Klein-Gordon and the Dirac oscillators in a noncommutative space,” Communications in Theoretical Physics, vol. 42, no. 5, pp. 664–668, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  105. L. D. Landau and E. M. Lifshitz, Quantum Mechanics: The Nonrelativistic Theory, Pergamon, Oxford, UK, 3rd edition, 1977. View at MathSciNet
  106. Y. Aharonov and D. Bohm, “Significance of electromagnetic potentials in the quantum theory,” Physical Review A: Atomic, Molecular and Optical Physics, vol. 115, no. 3, article 485, 1959. View at Publisher · View at Google Scholar
  107. X.-Q. Song, C.-W. Wang, and C.-S. Jia, “Thermodynamic properties for the sodium dimer,” Chemical Physics Letters, vol. 673, pp. 50–55, 2017. View at Google Scholar
  108. H. Hassanabadi and M. Hosseinpour, “Thermodynamic properties of neutral particle in the presence of topological defects in magnetic cosmic string background,” The European Physical Journal C, vol. 76, article 553, 2016. View at Google Scholar
  109. A. N. Ikot, B. C. Lutfuoglu, M. I. Ngwueke, M. E. Udoh, S. Zare, and H. Hassanabadi, “Klein-Gordon equation particles in exponential-type molecule potentials and their thermodynamic properties in D dimensions,” The European Physical Journal Plus, vol. 131, article 419, 2016. View at Publisher · View at Google Scholar
  110. B. Hamil and M. Merad, “Dirac and Klein-Gordon oscillators on anti-de Sitter space,” The European Physical Journal Plus, vol. 133, pp. 133–174, 2018. View at Google Scholar
  111. B. Khosropour, “Statistical aspects of the Klein–Gordon oscillator in the frame work of GUP,” Indian Journal of Physics, vol. 92, no. 1, pp. 43–47, 2018. View at Google Scholar