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Lattice Oscillator Model on Noncommutative Space: Eigenvalues Problem for the Perturbation Theory

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Abstract

Harmonic oscillator in noncommutative two-dimensional lattice is investigated. Using the properties of non-differential calculus and its applications to quantum mechanics, we provide the eigenvalues and eigenfunctions of the corresponding Hamiltonian. First, we consider the case of ordinary quantum mechanics, and we point out the thermodynamic properties of the model. Then we consider the same question when both coordinates and momenta are noncommutative.

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References

  1. P. Jizba, H. Kleinert, F. Scardigli, J. Phys. Conf. Ser. 306, 012026 (2011). https://doi.org/10.1088/1742-6596/306/1/012026

    Article  Google Scholar 

  2. P. Jizba, H. Kleinert, F. Scardigli, AIP Conf. Proc. 1446, 181 (2012). https://doi.org/10.1063/1.4727995

    Article  ADS  Google Scholar 

  3. P. Jizba, H. Kleinert, F. Scardigli, Phys. Rev. D. 81, 084030 (2010). https://doi.org/10.1103/PhysRevD.81.084030. arXiv:0912.2253 [hep-th]

    Article  ADS  Google Scholar 

  4. J.A. Wheeler, K.W. Ford. (New York, W W Norton and Co Inc., 2018)

  5. K.G. Wilson, Phys. Rev, D. 10, 2445 (1974). https://doi.org/10.1103/PhysRevD.10.2445

    Google Scholar 

  6. A. Amador, J.S. Hoye, K. Olaussen, arXiv:1610.05284 [hep-th]

  7. M.R. Douglas, N.A. Nekrasov, Rev. Mod. Phys. 73, 977 (2001). https://doi.org/10.1103/RevModPhys.73.977 [hep-th/0106048]

    Article  ADS  Google Scholar 

  8. N. Seiberg, E. Witten, JHEP. 9909, 032 (1999). https://doi.org/10.1088/1126-6708/1999/09/032 [hep-th/9908142]

    Article  ADS  Google Scholar 

  9. R.J. Szabo, . Phys. Rept. 378, 207 (2003). https://doi.org/10.1016/S0370-1573(03)00059-0 [hep-th/0109162]

    Article  ADS  Google Scholar 

  10. S. Hellerman, M. Van Raamsdonk, JHEP. 0110, 039 (2001). https://doi.org/10.1088/1126-6708/2001/10/039 [hep-th/0103179]

    Article  ADS  Google Scholar 

  11. P.A. Horvathy, hep-th/0307175

  12. O.F. Dayi, A. Jellal, J. Math. Phys. 43, 4592 (2002). https://doi.org/10.1063/1.1504484 [hep-th/0111267]

    Article  ADS  MathSciNet  Google Scholar 

  13. F.G. Scholtz, B. Chakraborty, S. Gangopadhyay, J. Govaerts, J. Phys. A. 38, 9849 (2005). https://doi.org/10.1088/0305-4470/38/45/008 [cond-mat/0509331 [cond-mat.mes-hall]]

    Article  ADS  MathSciNet  Google Scholar 

  14. S. Doplicher, K. Fredenhagen, J.E. Roberts, Commun. Math. Phys. 172, 187 (1995). [hep-th/0303037]

    Article  ADS  Google Scholar 

  15. D.J. Gross, P.F. Mende, Nucl. Phys. B. 303, 407 (1988). https://doi.org/10.1016/0550-3213(88)90390-2

    Article  ADS  Google Scholar 

  16. E.M.F. Curado, M.A. Rego-Monteiro, H.N. Nazareno, . Phys. Rev. A. 64, 012105 (2001). https://doi.org/10.1103/PhysRevA.64.012105 [hep-th/0012244]

    Article  ADS  Google Scholar 

  17. V.E. Tarasov, J. Phys. A. 47, 355204 (2014). https://doi.org/10.1088/1751-8113/47/35/355204

    Article  MathSciNet  Google Scholar 

  18. T.G. Dedovich, M.V. Tokarev, Phys. Part. Nucl. Lett. 13 (2), 169 (2016). https://doi.org/10.1134/S1547477116020060

    Article  Google Scholar 

  19. J. Th‘urigen, arXiv:1510.08706 [gr-qc]

  20. J.H. He, Int. J. Theor. Phys. 53(11), 3698 (2014). https://doi.org/10.1007/s10773-014-2123-8

    Article  Google Scholar 

  21. M.N. Chernodub, S. Ouvry, Phys. Rev. E. 92(4), 042102 (2015). https://doi.org/10.1103/PhysRevE.92.042102. arXiv:1504.02269 [cond-mat.stat-mech]

    Article  ADS  MathSciNet  Google Scholar 

  22. J. Jurkiewicz, J. Wosiek, Nucl. Phys. B. 135, 416 (1978). https://doi.org/10.1016/0550-3213(78)90346-2

    Article  ADS  Google Scholar 

  23. S.D. Drell, M. Weinstein, S. Yankielowicz, Phys. Rev. D. 16, 1769 (1977). https://doi.org/10.1103/PhysRevD.16.1769

    Article  ADS  Google Scholar 

  24. C. Bastos, A.E. Bernardini, O. Bertolami, N. Costa Dias, J. Nuno Prata, Phys. Rev. D. 93(10), 104055 (2016). arXiv:1512.03792 [quant-ph]

    Article  ADS  Google Scholar 

  25. J. Jurkiewicz, J. Wosiek, Nucl. Phys. B. 145, 445 (1978). https://doi.org/10.1016/0550-3213(78)90095-0

    Article  ADS  Google Scholar 

  26. J. Zak, Phys. Rev. B. 21, 3345 (1980). https://doi.org/10.1103/PhysRevB.21.3345

    Article  ADS  Google Scholar 

  27. M.C. Gutzwiller, Annals Phys. 133, 304 (1981). https://doi.org/10.1016/0003-4916(81)90253-0

    Article  ADS  Google Scholar 

  28. H.S. Snyder, Phys. Rev. 71, 38 (1947). https://doi.org/10.1103/PhysRev.71.38

    Article  ADS  Google Scholar 

  29. C.N. Yang, Phys. Rev. 72, 874 (1947). https://doi.org/10.1103/PhysRev.72.874

    Article  ADS  Google Scholar 

  30. C.P. Sun, H.C. Fu, J. Phys. A. 22, L983 (1989). https://doi.org/10.1088/0305-4470/22/21/001

    Article  Google Scholar 

  31. E.G. Floratos, Nucl. Phys. Proc. Suppl. 22A, 144 (1991). https://doi.org/10.1016/0920-5632(91)90361-H

    Article  ADS  Google Scholar 

  32. M. Chaichian, D. Ellinas, P. Kulish, Phys. Rev. Lett. 65, 980 (1990). https://doi.org/10.1103/PhysRevLett.65.980

    Article  ADS  MathSciNet  Google Scholar 

  33. E. Celeghini, T.D. Palev, M. Tarlini, Mod. Phys. Lett. B. 5, 187 (1991). https://doi.org/10.1142/S021798499100023X

    Article  ADS  Google Scholar 

  34. W.B. Schmidke, J. Wess, B. Zumino, Z. Phys. C. 52, 471 (1991). https://doi.org/10.1007/BF01559443

    Article  ADS  MathSciNet  Google Scholar 

  35. D. Ellinas, Phys. Rev. A. 45, 3358 (1992). https://doi.org/10.1103/PhysRevA.45.3358

    Article  ADS  MathSciNet  Google Scholar 

  36. D.V. Boulatov, Int. J. Mod, Phys. A. 8, 3139 (1993). https://doi.org/10.1142/S0217751X93001259 [hep-th/9210032]

    Google Scholar 

  37. M. Valiente, J. Phys. A. 44, 465303 (2011). https://doi.org/10.1088/1751-8113/44/46/465303

    Article  ADS  Google Scholar 

  38. D. Mehta, A. Sternbeck, L. von Smekal, A.G. Williams, PoS QCD -TNT09, 025. arXiv:0912.0450 [hep-lat] (2009)

  39. D. Mehta, M. Kastner, Annals Phys. 326, 1425 (2011). https://doi.org/10.1016/j.aop.2010.12.016. arXiv:1010.5335 [cond-mat.stat-mech]

    Article  ADS  Google Scholar 

  40. Y. Li, Phys. Rev. B. 91(19), 195133 (2015). https://doi.org/10.1103/PhysRevB.91.195133. arXiv:1410.6189 [cond-mat.str-el]

    Article  ADS  Google Scholar 

  41. H. Atakis, M.Ö. Oktel, Phys. Rev. A. 88, 033612 (2013). https://doi.org/10.1103/PhysRevA.88.033612

    Article  ADS  Google Scholar 

  42. A. Cucchieri, T. Mendes, Phys. Rev. D. 88, 114501 (2013). https://doi.org/10.1103/PhysRevD.88.114501. arXiv:1308.1283 [hep-lat]

    Article  ADS  Google Scholar 

  43. M. Bhatia, P.N. Swamy, Int. J. Theor. Phys. 50, 1687 (2011). https://doi.org/10.1007/s10773-011-0677-2. arXiv:1011.2544 [quant-ph]

    Article  Google Scholar 

  44. M. Bojowald, A. Kempf, Phys. Rev. D. 86, 085017 (2012). https://doi.org/10.1103/PhysRevD.86.085017. arXiv:1112.0994 [hep-th]

    Article  ADS  Google Scholar 

  45. A. Kempf, J. Math. Phys. 38, 1347 (1997). https://doi.org/10.1063/1.531814 [hep-th/9602085]

    Article  ADS  MathSciNet  Google Scholar 

  46. A. Kempf, Phys. Rev. D. 63, 083514 (2001). https://doi.org/10.1103/PhysRevD.63.083514 [astro-ph/0009209]

    Article  ADS  MathSciNet  Google Scholar 

  47. A. Kempf, IN *Erice, From the Planck length to the Hubble radius* 613-622 [hep-th/9810215] (1998)

  48. A. Kempf, hep-th/9612082

  49. A. Kempf, G. Mangano, Phys. Rev. D. 55, 7909 (1997). https://doi.org/10.1103/PhysRevD.55.7909 [hep-th/9612084]

    Article  ADS  Google Scholar 

  50. A. Kempf, J. Phys. A. 30, 2093 (1997). https://doi.org/10.1088/0305-4470/30/6/030 [hep-th/9604045]

    Article  ADS  MathSciNet  Google Scholar 

  51. C. Bastos, A.E. Bernardini, O. Bertolami, N. Costa Dias, J. Nuno Prata, J. Phys. Conf. Ser. 626 (1), 012050 (2015). https://doi.org/10.1088/1742-6596/626/1/012050. arXiv:1411.2146 [quant-ph]

    Article  Google Scholar 

  52. C. Bastos, A. Bernardini, O. Bertolami, N. Costa Dias, J. Nuno Prata, . Phys. Rev. D. 90 (4), 045023 (2014). https://doi.org/10.1103/PhysRevD.90.045023. arXiv:1406.0740 [quant-ph]

    Article  ADS  Google Scholar 

  53. C. Bastos, A.E. Bernardini, O. Bertolami, N. Costa Dias, J. Nuno Prata, Phys. Rev. A. 89 (4), 042112 (2014). https://doi.org/10.1103/PhysRevA.89.042112. arXiv:1310.4762 [quant-ph]

    Article  ADS  Google Scholar 

  54. J. Gamboa, M. Loewe, F. Mendez, J.C. Rojas, Mod. Phys. Lett. A. 16, 2075 (2001). https://doi.org/10.1142/S0217732301005345 [hep-th/0104224]

    Article  ADS  Google Scholar 

  55. A. Hatzinikitas, I. Smyrnakis, J. Math. Phys. 43, 113 (2002). https://doi.org/10.1063/1.1416196 [hep-th/0103074]

    Article  ADS  MathSciNet  Google Scholar 

  56. V.P. Nair, A.P. Polychronakos, Phys. Lett. B. 505, 267 (2001). https://doi.org/10.1016/S0370-2693(01)00339-2 [hep- th/0011172]

    Article  ADS  MathSciNet  Google Scholar 

  57. M. Maceda, A. Macias, Phys. Rev. D. 79, 087703 (2009). https://doi.org/10.1103/PhysRevD.79.087703

    Article  ADS  Google Scholar 

  58. A.E. Bernardini, Eur. Phys. J. C. 46, 113 (2006). https://doi.org/10.1140/epjc/s2006-02502-2 [hep-th/0606240]

    Article  ADS  Google Scholar 

  59. C. Bastos, A.E. Bernardini, O. Bertolami, N. Costa Dias, J. Nuno Prata,. Phys. Rev. D. 88 (8), 085013 (2013). https://doi.org/10.1103/PhysRevD.88.085013. arXiv:1305.5792 [quant-ph]

    Article  ADS  Google Scholar 

  60. P.R. Giri, P. Roy, Eur. Phys. J. C. 57, 835 (2008). https://doi.org/10.1140/epjc/s10052-008-0705-4. arXiv:0803.4090 [hep-th]

    Article  ADS  Google Scholar 

  61. S. Dulat, K. Li, Chin. Phys. C. 32, 92 (2008). https://doi.org/10.1088/1674-1137/32/2/003. arXiv:0802.1118 [math-ph]

    Article  ADS  Google Scholar 

  62. A. Halder, S. Gangopadhyay, Int. J. Theor. Phys. 56(6), 1831 (2017). arXiv:1609.06580 [hep-th]

    Article  Google Scholar 

  63. J.F.G. Santos, A.E. Bernardini, C. Bastos, Physica. 438, 340 (2015). https://doi.org/10.1016/j.physa.2015.07.009. arXiv:1411.2941 [quant-ph]

    Article  MathSciNet  Google Scholar 

  64. J.F.G. Santos, A.E. Bernardini, Eur. Phys. J. Plus. 132(6), 260 (2017). https://doi.org/10.1140/epjp/i2017-11538-1. arXiv:1606.05592 [quant-ph]

    Article  Google Scholar 

  65. A.E. Bernardini, O. Bertolami, Phys. Rev. A. 88(1), 012101 (2013). https://doi.org/10.1103/PhysRevA.88.012101. arXiv:1303.0685 [quant-ph]

    Article  ADS  Google Scholar 

  66. I. Jabbari, A. Jahan, Z. Riazi, Turk. J. Phys. 33, 149 (2009). arXiv:1201.0827 [hep-th]

    Google Scholar 

  67. A. Jahan, Braz. J. Phys. 38, 144 (2008). arXiv:1208.0137 [hep-th]

    Article  ADS  Google Scholar 

  68. J.M. Seddon, J.D. Gale. Thermodynamics and statistical mechanics (Royal Society of Chemistry, Cambridge CB4OWF, 2001)

    Google Scholar 

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Acknowledgments

D.O.S research at the Max-Planck Institute is supported by the Alexander von Humboldt foundation. S.L.G thanks the Max-Planck Institute for the invitation and financial support.

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Authors and Affiliations

  1. International Chair in Mathematical Physics and Applications (ICMPA-UNESCO Chair), University of Abomey-Calavi, 072B.P.50, Cotonou, Republic of Benin

    Dine Ousmane Samary, Sêcloka Lazare Guedezounme & Antonin Danvidé Kanfon

  2. Max Planck Institute for Gravitational Physics, Albert Einstein Institute, Am Mühlenberg 1, 14476, Potsdam, Germany

    Dine Ousmane Samary

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Samary, D.O., Guedezounme, S.L. & Kanfon, A.D. Lattice Oscillator Model on Noncommutative Space: Eigenvalues Problem for the Perturbation Theory. Braz J Phys 49, 458–470 (2019). https://doi.org/10.1007/s13538-019-00655-8

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