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Trends in Supersymmetric Quantum Mechanics

  • David J. Fernández C.Email author
Chapter
Part of the CRM Series in Mathematical Physics book series (CRM)

Abstract

Along the years, supersymmetric quantum mechanics (SUSY QM) has been used for studying solvable quantum potentials. It is the simplest method to build Hamiltonians with prescribed spectra in the spectral design. The key is to pair two Hamiltonians through a finite order differential operator. Some related subjects can be simply analyzed, as the algebras ruling both Hamiltonians and the associated coherent states. The technique has been applied also to periodic potentials, where the spectra consist of allowed and forbidden energy bands. In addition, a link with non-linear second-order differential equations, and the possibility of generating some solutions, can be explored. Recent applications concern the study of Dirac electrons in graphene placed either in electric or magnetic fields, and the analysis of optical systems whose relevant equations are the same as those of SUSY QM. These issues will be reviewed briefly in this paper, trying to identify the most important subjects explored currently in the literature.

Keywords

Supersymmetric quantum mechanics Coherent states Painlevé equations Painlevé transcendents Polynomial Heisenberg algebras Factorization method Exact solutions Spectral design Graphene 

Notes

Acknowledgements

The author acknowledges the financial support of the Spanish MINECO (project MTM2014-57129-C2-1-P) and Junta de Castilla y León (VA057U16).

References

  1. 1.
    E. Witten, Nucl. Phys. B 185, 513–554 (1981)ADSCrossRefGoogle Scholar
  2. 2.
    L. Infeld, T.E. Hull, Rev. Mod. Phys. 23, 21–68 (1951)ADSCrossRefGoogle Scholar
  3. 3.
    W. Miller, Lie Theory and Special Functions (Academic, New York, 1968)zbMATHGoogle Scholar
  4. 4.
    A. Lahiri, P.K. Roy, B. Bagchi, Int. J. Mod. Phys. A 5, 1383–1456 (1990)ADSCrossRefGoogle Scholar
  5. 5.
    B. Roy, P. Roy, R. Roychoudhury, Fortschr. Phys. 39, 211–258 (1991)MathSciNetCrossRefGoogle Scholar
  6. 6.
    O.L. de Lange, R.E. Raab, Operator Methods in Quantum Mechanics (Clarendon Press, Oxford, 1991)Google Scholar
  7. 7.
    F. Cooper, A. Khare, U. Sukhatme, Phys. Rep. 251, 267–385 (1995)ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    G. Junker, Supersymmetric Methods in Quantum and Statistical Mechanics (Springer, Berlin, 1996)zbMATHCrossRefGoogle Scholar
  9. 9.
    B.N. Zakhariev, V.M. Chabanov, Inv. Prob. 13, R47–R79 (1997)ADSCrossRefGoogle Scholar
  10. 10.
    A. Ramos, J.F. Cariñena, in Symmetries in Quantum Mechanics and Quantum Optics, ed. by A. Ballesteros et al. (Universidad de Burgos, Burgos, 1999), pp. 259–269Google Scholar
  11. 11.
    H.C. Rosu, in Symmetries in Quantum Mechanics and Quantum Optics, ed. by A. Ballesteros et al. (Universidad de Burgos, Burgos, 1999), pp. 301–315Google Scholar
  12. 12.
    B. Bagchi, Supersymmetry in Quantum and Classical Mechanics (Chapman & Hall, Boca Raton, 2001)zbMATHGoogle Scholar
  13. 13.
    F. Cooper, A. Khare, U. Sukhatme, Supersymmetry in Quantum Mechanics (World Scientific, Singapore, 2001)zbMATHCrossRefGoogle Scholar
  14. 14.
    R. de Lima Rodrigues (2002). arXiv:hep-th/0205017v6Google Scholar
  15. 15.
    I. Aref’eva, D.J. Fernández, V. Hussin, J. Negro, L.M. Nieto, B.F. Samsonov (eds.) Special issue dedicated to the subject of the International Conference on Progress in Supersymmetric Quantum Mechanics. J. Phys. A: Math. Gen. 37(43) (2004)Google Scholar
  16. 16.
    B. Mielnik, O. Rosas-Ortiz, J. Phys. A: Math. Gen. 37, 10007–10035 (2004)ADSCrossRefGoogle Scholar
  17. 17.
    A. Khare, U. Sukhatme, J. Phys. A: Math. Gen. 37, 10037–10055 (2004)ADSCrossRefGoogle Scholar
  18. 18.
    C.M. Bender, J. Brod, A. Refig, M.E. Reuter, J. Phys. A: Math. Gen. 37, 10139–10165 (2004)ADSCrossRefGoogle Scholar
  19. 19.
    D. Baye, J.M. Sparenberg, J. Phys. A: Math. Gen. 37, 10223–10249 (2004)ADSCrossRefGoogle Scholar
  20. 20.
    A.A. Andrianov, F. Cannata, J. Phys. A: Math. Gen. 37, 10297–10321 (2004)ADSCrossRefGoogle Scholar
  21. 21.
    A. Khare, AIP Conf. Proc. 744, 133–165 (2005)ADSCrossRefGoogle Scholar
  22. 22.
    C.V. Sukumar, AIP Conf. Proc. 744, 166–235 (2005)ADSCrossRefGoogle Scholar
  23. 23.
    D.J. Fernández, N. Fernández-García, AIP Conf. Proc. 744, 236–273 (2005)ADSMathSciNetCrossRefGoogle Scholar
  24. 24.
    S.H. Dong, Factorization Method in Quantum Mechanics Springer, Dordrecht (2007)zbMATHCrossRefGoogle Scholar
  25. 25.
    D.J. Fernández, AIP Conf. Proc. 1287, 3–36 (2010)ADSGoogle Scholar
  26. 26.
    A.A. Andrianov, M.V. Ioffe, J. Phys. A: Math. Theor. 45, 503001 (2012)CrossRefGoogle Scholar
  27. 27.
    D. Baye, J.M. Sparenberg, A.M. Pupasov-Maksimov, B.F. Samsonov, J. Phys. A: Math. Theor. 47, 243001 (2014)ADSCrossRefGoogle Scholar
  28. 28.
    P.A.M. Dirac, The Principles of Quantum Mechanics, 2nd edn. (Clarendon, Oxford, 1935)zbMATHGoogle Scholar
  29. 29.
    E. Schrödinger, Proc. R. Irish Acad. 46A, 9–16 (1940)Google Scholar
  30. 30.
    E. Schrödinger, Proc. R. Irish Acad. 46A, 183–206 (1941)Google Scholar
  31. 31.
    L. Infeld, Phys. Rev. 59, 737–747 (1941)ADSMathSciNetCrossRefGoogle Scholar
  32. 32.
    T.E. Hull, L. Infeld, Phys. Rev. 74, 905–909 (1948)ADSCrossRefGoogle Scholar
  33. 33.
    B. Mielnik, J. Math. Phys. 25, 3387–3389 (1984)Google Scholar
  34. 34.
    D.J. Fernández, Lett. Math. Phys. 8, 337–343 (1984)ADSMathSciNetCrossRefGoogle Scholar
  35. 35.
    A.A. Andrianov, N.V. Borisov, M.V. Ioffe, Phys. Lett. A 105, 19–22 (1984)ADSMathSciNetCrossRefGoogle Scholar
  36. 36.
    A.A. Andrianov, N.V. Borisov, M.V. Ioffe, Theor. Math. Phys. 61, 1078–1088 (1984)CrossRefGoogle Scholar
  37. 37.
    M.M. Nieto, Phys. Lett. B 145, 208–210 (1984)ADSMathSciNetCrossRefGoogle Scholar
  38. 38.
    C.V. Sukumar, J. Phys. A: Math. Gen. 18, 2917–2936 (1985)ADSCrossRefGoogle Scholar
  39. 39.
    C.V. Sukumar, J. Phys. A: Math. Gen. 18, 2937–2955 (1985)ADSCrossRefGoogle Scholar
  40. 40.
    M. Luban, D.L. Pursey, Phys. Rev. D 33, 431–436 (1986)ADSMathSciNetCrossRefGoogle Scholar
  41. 41.
    J. Beckers, V. Hussin, Phys. Lett. A 118, 319–321 (1986)ADSMathSciNetCrossRefGoogle Scholar
  42. 42.
    J. Beckers, D. Dehin, V. Hussin, J. Phys. A: Math. Gen. 20, 1137–1154 (1987)ADSCrossRefGoogle Scholar
  43. 43.
    N.A. Alves, E. Drigo-Filho, J. Phys. A: Math. Gen. 21, 3215–3225 (1988)ADSCrossRefGoogle Scholar
  44. 44.
    A. Arai, Lett. Math. Phys. 19, 217–227 (1990)ADSMathSciNetCrossRefGoogle Scholar
  45. 45.
    R. Adhikari, R. Dutt, Y.P. Varshni, J. Math. Phys. 32, 447–456 (1991)Google Scholar
  46. 46.
    J. Beckers, N. Debergh, A.G. Nikitin, J. Math. Phys. 33, 152–160 (1992)Google Scholar
  47. 47.
    J. Casahorrán, Physica A 217, 429–439 (1995)ADSCrossRefGoogle Scholar
  48. 48.
    H.C. Fu, R. Sasaki, J. Phys. A: Math. Gen. 29, 2129–2138 (1996)Google Scholar
  49. 49.
    G. Junker, P. Roy, Phys. Lett. A 232, 155–161 (1997)ADSMathSciNetCrossRefGoogle Scholar
  50. 50.
    N. Aizawa, H.T. Sato, Prog. Theor. Phys. 98, 707–718 (1997)ADSCrossRefGoogle Scholar
  51. 51.
    E. Drigo-Filho, J.R. Ruggiero, Phys. Rev. E 56, 4486–4488 (1997)ADSCrossRefGoogle Scholar
  52. 52.
    L.J. Boya, H.C. Rosu, A.J. Segui-Santonja, F.J. Vila, Nuovo Cim. B 113, 409–414 (1998)ADSGoogle Scholar
  53. 53.
    I.F. Márquez, J. Negro, L.M. Nieto, J. Phys. A: Math. Gen. 31, 4115–4125 (1998)ADSCrossRefGoogle Scholar
  54. 54.
    J.F. Cariñena, G. Marmo, A.M. Perelomov, M.F. Rañada, Int. J. Mod. Phys. A 13, 4913–4929 (1998)ADSCrossRefGoogle Scholar
  55. 55.
    C. Quesne, N. Vansteenkiste, Phys. Lett. A 240, 21–28 (1998)ADSMathSciNetCrossRefGoogle Scholar
  56. 56.
    C. Quesne, N. Vansteenkiste, Helv. Phys. Acta 72, 71–92 (1999)MathSciNetGoogle Scholar
  57. 57.
    V.M. Eleonsky, V.G. Korolev, J. Math. Phys. 40, 1977–1992 (1999)Google Scholar
  58. 58.
    J.F. Cariñena, A. Ramos, Mod. Phys. Lett. A 15, 1079–1088 (2000)ADSCrossRefGoogle Scholar
  59. 59.
    B. Bagchi, S. Mallik, C. Quesne, Int. J. Mod. Phys. A 16, 2859–2872 (2001)ADSCrossRefGoogle Scholar
  60. 60.
    C. Quesne, Mod. Phys. Lett. A 18, 515–526 (2003)ADSCrossRefGoogle Scholar
  61. 61.
    H.C. Rosu, J.M. Morán-Mirabal, O. Cornejo, Phys. Lett. A 310, 353–356 (2003)ADSMathSciNetCrossRefGoogle Scholar
  62. 62.
    M.V. Ioffe, S. Kuru, J. Negro, L.M. Nieto, J. Phys. A: Math. Theor. 39, 6987–7001 (2006)ADSCrossRefGoogle Scholar
  63. 63.
    S. Kuru, J. Negro, Ann. Phys. 323, 413–431 (2008)ADSCrossRefGoogle Scholar
  64. 64.
    C. Quesne, J. Phys. A: Math. Theor. 41, 244002 (2008)ADSCrossRefGoogle Scholar
  65. 65.
    A.A. Andrianov, M.V. Ioffe, V.P. Spiridonov, Phys. Lett. A 174, 273–279 (1993)ADSMathSciNetCrossRefGoogle Scholar
  66. 66.
    A.A. Andrianov, M.V. Ioffe, F. Cannata, J.P. Dedonder, Int. J. Mod. Phys. A 10, 2683–2701 (1995)ADSCrossRefGoogle Scholar
  67. 67.
    V.G.Bagrov, B.F. Samsonov, Theor. Math. Phys. 104, 1051–1060 (1995)CrossRefGoogle Scholar
  68. 68.
    D.J. Fernández, Int. J. Mod. Phys. A 12, 171–176 (1997)ADSCrossRefGoogle Scholar
  69. 69.
    D.J. Fernández, M.L. Glasser, L.M. Nieto, Phys. Lett. A 240, 15–20 (1998)ADSMathSciNetCrossRefGoogle Scholar
  70. 70.
    D.J. Fernández, V. Hussin, B. Mielnik, Phys. Lett. A 244, 309–316 (1998)ADSMathSciNetCrossRefGoogle Scholar
  71. 71.
    D.J. Fernández, H.C. Rosu, Phys. Scripta 64, 177–183 (2001)ADSCrossRefGoogle Scholar
  72. 72.
    J.F. Cariñena, A. Ramos, D.J. Fernández, Ann. Phys. 292, 42–66 (2001)ADSCrossRefGoogle Scholar
  73. 73.
    D.J. Fernández, J. Negro, M.A. del Olmo, Ann. Phys. 252, 386–412 (1996)ADSCrossRefGoogle Scholar
  74. 74.
    D.J. Fernández, E. Salinas-Hernández, Phys. Lett. A 338, 13–18 (2005)ADSMathSciNetCrossRefGoogle Scholar
  75. 75.
    A. Contreras-Astorga, D.J. Fernández, AIP Conf. Proc. 960, 55–60 (2007)ADSCrossRefGoogle Scholar
  76. 76.
    A. Contreras-Astorga, D.J. Fernández, J. Phys. A: Math. Theor. 41, 475303 (2008)ADSCrossRefGoogle Scholar
  77. 77.
    D.J. Fernández, V.S. Morales-Salgado, Ann. Phys. 388, 122–134 (2018)ADSCrossRefGoogle Scholar
  78. 78.
    D.J. Fernández, V. Hussin, L.M. Nieto, J. Phys. A: Math. Gen. 27, 3547–3564 (1994)ADSCrossRefGoogle Scholar
  79. 79.
    D.J. Fernández, L.M. Nieto, O. Rosas-Ortiz, J. Phys. A: Math. Gen. 28, 2693–2708 (1995)ADSCrossRefGoogle Scholar
  80. 80.
    D.J. Fernández, V. Hussin, J. Phys. A: Math. Gen. 32, 3603–3619 (1999)ADSCrossRefGoogle Scholar
  81. 81.
    D. Bermudez, A. Contreras-Astorga, D.J. Fernández, Ann. Phys. 350, 615–634 (2014)ADSCrossRefGoogle Scholar
  82. 82.
    D.J. Fernández, V. Hussin, O. Rosas-Ortiz, J. Phys. A: Math. Theor. 40, 6491–6511 (2007)ADSCrossRefGoogle Scholar
  83. 83.
    D.J. Fernández, V. Hussin, V.S. Morales-Salgado, Eur. Phys. J. Plus 134, 18 (2019)CrossRefGoogle Scholar
  84. 84.
    D.J. Fernández, J. Negro, L.M. Nieto, Phys. Lett. A 324, 139–144 (2004)ADSCrossRefGoogle Scholar
  85. 85.
    J.M. Carballo, D.J. Fernández, J. Negro, L.M. Nieto, J. Phys. A: Math. Gen. 37, 10349–10362 (2004)ADSCrossRefGoogle Scholar
  86. 86.
    J. Mateo, J. Negro, J. Phys. A: Math. Theor. 41, 045204 (2008)ADSCrossRefGoogle Scholar
  87. 87.
    M. Castillo-Celeita, E. Díaz-Bautista, D.J. Fernández, Phys. Scripta 94, 045203 (2019)ADSCrossRefGoogle Scholar
  88. 88.
    D.J. Fernández, R. Muñoz, A. Ramos, Phys. Lett. A 308, 11–16 (2003)ADSMathSciNetCrossRefGoogle Scholar
  89. 89.
    N. Fernández-García, O. Rosas-Ortiz, Ann. Phys. 323, 1397–1414 (2008)ADSCrossRefGoogle Scholar
  90. 90.
    D. Bermudez, D.J. Fernández, Phys. Lett. A 375, 2974–2978 (2011)ADSMathSciNetCrossRefGoogle Scholar
  91. 91.
    D. Bermudez, SIGMA 8, 069 (2012)Google Scholar
  92. 92.
    D.J. Fernández, J.C. González, Ann. Phys. 359, 213–229 (2015)CrossRefGoogle Scholar
  93. 93.
    B. Mielnik, L.M. Nieto, O. Rosas-Ortiz, Phys. Lett. A 269, 70–78 (2000)ADSMathSciNetCrossRefGoogle Scholar
  94. 94.
    D.J. Fernández, E. Salinas-Hernández, J. Phys. A: Math. Gen. 36, 2537–2543 (2003)ADSCrossRefGoogle Scholar
  95. 95.
    D.J. Fernández, E. Salinas-Hernández, J. Phys. A: Math. Theor. 44, 365302 (2011)ADSCrossRefGoogle Scholar
  96. 96.
    D. Bermudez, D.J. Fernández, N. Fernández-García, Phys. Lett. A 376, 692–696 (2012)ADSMathSciNetCrossRefGoogle Scholar
  97. 97.
    A. Contreras-Astorga, A. Schulze-Halberg, J. Math. Phys. 55, 103506 (2014)ADSMathSciNetCrossRefGoogle Scholar
  98. 98.
    A. Contreras-Astorga, A. Schulze-Halberg, Ann. Phys. 354, 353–364 (2015)ADSCrossRefGoogle Scholar
  99. 99.
    A. Contreras-Astorga, A. Schulze-Halberg, J. Phys. A: Math. Theor. 48, 315202 (2015)CrossRefGoogle Scholar
  100. 100.
    D. Bermudez, Ann. Phys. 364, 35–52 (2016)ADSCrossRefGoogle Scholar
  101. 101.
    A. Contreras-Astorga, A. Schulze-Halberg, J. Phys. A: Math. Theor. 50, 105301 (2017)ADSCrossRefGoogle Scholar
  102. 102.
    D.J. Fernández, J. Negro, L.M. Nieto, Phys. Lett. A 275, 338–349 (2000)ADSMathSciNetCrossRefGoogle Scholar
  103. 103.
    J. Negro, L.M. Nieto, D.J. Fernández, Czech. J. Phys. 50, 1303–1308 (2000)ADSCrossRefGoogle Scholar
  104. 104.
    D.J. Fernández, B. Mielnik, O. Rosas-Ortiz, B.F. Samsonov, Phys. Lett. A 294, 168–174 (2002)ADSMathSciNetCrossRefGoogle Scholar
  105. 105.
    D.J. Fernández, B. Mielnik, O. Rosas-Ortiz, B.F. Samsonov, J. Phys. A: Math. Gen. 35, 4279–4291 (2002)ADSCrossRefGoogle Scholar
  106. 106.
    D.J. Fernández, A. Ganguly, Phys. Lett. A 338, 203–208 (2005)ADSMathSciNetCrossRefGoogle Scholar
  107. 107.
    A. Ganguly, M. Ioffe, L.M. Nieto, J. Phys. A: Math. Gen. 39, 14659–14680 (2006)ADSCrossRefGoogle Scholar
  108. 108.
    D.J. Fernández, A. Ganguly, Ann. Phys. 322, 1143–1161 (2007)ADSCrossRefGoogle Scholar
  109. 109.
    D.J. Fernández, A. Ganguly, Phys. Atom. Nucl. 73, 288–294 (2010)ADSCrossRefGoogle Scholar
  110. 110.
    H. Aoyama, M. Sato, T. Tanaka, Nucl. Phys. B 619, 105–127 (2001)ADSCrossRefGoogle Scholar
  111. 111.
    M. Sato, T. Tanaka, J. Math. Phys. 43, 3484–3510 (2002)Google Scholar
  112. 112.
    T. Tanaka, Nucl. Phys. B 662, 413–446 (2003)ADSCrossRefGoogle Scholar
  113. 113.
    A. González-López, T. Tanaka, Phys. Lett. B 586, 117–124 (2004)ADSMathSciNetCrossRefGoogle Scholar
  114. 114.
    B. Bagchi, T. Tanaka, Ann. Phys. 324, 2438–2451 (2009)ADSCrossRefGoogle Scholar
  115. 115.
    C. Leiva, M.S. Plyushchay, J. High Energy Phys. 0310, 069 (2003)ADSCrossRefGoogle Scholar
  116. 116.
    M. Plyushchay, J. Phys. A: Math. Gen. 37, 10375–10384 (2004)ADSMathSciNetCrossRefGoogle Scholar
  117. 117.
    F. Correa, L.M. Nieto, M.S. Plyushchay, Phys. Lett. B 644, 94–98 (2007)ADSMathSciNetCrossRefGoogle Scholar
  118. 118.
    F. Correa, V. Jakubsky, L.M. Nieto, M.S. Plyushchay, Phys. Rev. Lett. 101, 030403 (2008)ADSMathSciNetCrossRefGoogle Scholar
  119. 119.
    F. Correa, V. Jakubsky, M.S. Plyushchay, J. Phys. A: Math. Theor. 41, 485303 (2008)CrossRefGoogle Scholar
  120. 120.
    S.Y. Dubov, V.M. Eleonsky, N.E. Kulagin, Sov. Phys. JETP 75 446–451 (1992)Google Scholar
  121. 121.
    A.P. Veselov, A.B. Shabat, Funct. Anal. Appl. 27, 81–96 (1993)MathSciNetCrossRefGoogle Scholar
  122. 122.
    V.E. Adler, Physica D 73, 335–351 (1994)ADSMathSciNetCrossRefGoogle Scholar
  123. 123.
    U.P. Sukhatme, C. Rasinariu, A. Khare, Phys. Lett. A 234, 401–409 (1997)ADSMathSciNetCrossRefGoogle Scholar
  124. 124.
    A.A. Andrianov, F. Cannata, M. Ioffe, D. Nishnianidze, Phys. Lett. A 266, 341–349 (2000)ADSMathSciNetCrossRefGoogle Scholar
  125. 125.
    I. Marquette, J. Math. Phys. 50, 012101 (2009)ADSMathSciNetCrossRefGoogle Scholar
  126. 126.
    I. Marquette, J. Math. Phys. 50, 095202 (2009)ADSMathSciNetCrossRefGoogle Scholar
  127. 127.
    D. Bermudez, D.J. Fernández, SIGMA 7, 025 (2011)Google Scholar
  128. 128.
    D. Bermudez, D.J. Fernández, Ann. Phys. 333, 290–306 (2013)ADSCrossRefGoogle Scholar
  129. 129.
    D. Bermudez, D.J. Fernández, AIP Conf. Proc. 1575, 50–88 (2014)ADSCrossRefGoogle Scholar
  130. 130.
    D.J. Fernández, V.S. Morales-Salgado, J. Phys. A: Math. Theor. 47, 035304 (2014)ADSCrossRefGoogle Scholar
  131. 131.
    D.J. Fernández, V.S. Morales-Salgado, J. Phys. A: Math. Theor. 49, 195202 (2016)ADSCrossRefGoogle Scholar
  132. 132.
    D. Bermudez, D.J. Fernández, J. Negro, J. Phys. A: Math. Theor. 49, 335203 (2016)CrossRefGoogle Scholar
  133. 133.
    I. Marquette, C. Quesne, J. Math. Phys. 57, 052101 (2016)ADSMathSciNetCrossRefGoogle Scholar
  134. 134.
    V.Y. Novokshenov, SIGMA 14, 106 (2018)Google Scholar
  135. 135.
    P.A. Clarkson, D. Gómez-Ullate, Y. Grandati, R. Milson (2018). arXiv:1811.09274 [math-ph]Google Scholar
  136. 136.
    J.F. Cariñena, A.M Perelomov, M. Rañada, M. Santander, J. Phys. A: Math. Theor. 41, 085301 (2008)Google Scholar
  137. 137.
    S. Odake, R. Sasaki, Phys. Lett. B 679, 414–417 (2009)ADSMathSciNetCrossRefGoogle Scholar
  138. 138.
    S. Odake, R. Sasaki, Phys. Lett. B 682, 130–136 (2009)ADSMathSciNetCrossRefGoogle Scholar
  139. 139.
    D. Gómez-Ullate, N. Kamran, R. Milson, J. Phys. A: Math. Theor. 43, 434016 (2010)ADSCrossRefGoogle Scholar
  140. 140.
    S. Odake, R. Sasaki, Phys. Lett. B 702, 164–170 (2011)ADSMathSciNetCrossRefGoogle Scholar
  141. 141.
    C. Quesne, Mod. Phys. Lett. A 26, 1843–1852 (2011)ADSCrossRefGoogle Scholar
  142. 142.
    C. Quesne, Int. J. Mod. Phys. A 26, 5337–5347 (2011)ADSCrossRefGoogle Scholar
  143. 143.
    D. Gómez-Ullate, N. Kamran, R. Milson, J. Math. Ann. App. 387, 410–418 (2012)Google Scholar
  144. 144.
    D. Gómez-Ullate, N. Kamran, R. Milson, Found. Comp. Math. 13, 615–666 (2013)CrossRefGoogle Scholar
  145. 145.
    I. Marquette, C. Quesne, J. Math. Phys. 54, 042102 (2013)ADSMathSciNetCrossRefGoogle Scholar
  146. 146.
    I. Marquette, C. Quesne, J. Math. Phys. 55, 112103 (2014)ADSMathSciNetCrossRefGoogle Scholar
  147. 147.
    D. Gómez-Ullate, Y. Grandati, R. Milson, J. Phys. A: Math. Theor. 47, 015203 (2014)ADSCrossRefGoogle Scholar
  148. 148.
    D. Gómez-Ullate, Y. Grandati, R. Milson, J. Math. Phys. 55, 043510 (2014)ADSMathSciNetCrossRefGoogle Scholar
  149. 149.
    A. Schulze-Halberg, B. Roy, Ann. Phys. 349, 159–170 (2014)ADSCrossRefGoogle Scholar
  150. 150.
    D. Gómez-Ullate, Y. Grandati, R. Milson, J. Phys. A: Math. Theor. 51, 345201 (2018)CrossRefGoogle Scholar
  151. 151.
    S. Kuru, J. Negro, L.M. Nieto, J. Phys.: Condens. Matter 21, 455305 (2009)ADSGoogle Scholar
  152. 152.
    E. Milpas, M. Torres, G. Murguia, J. Phys.: Condens. Matter 23, 245304 (2011)ADSGoogle Scholar
  153. 153.
    V. Jakubsky, M.S. Plyushchay, Phys. Rev. D 85, 045035 (2012)ADSCrossRefGoogle Scholar
  154. 154.
    O. Panella, P. Roy, Phys. Lett. A 376, 2580–2583 (2012)ADSCrossRefGoogle Scholar
  155. 155.
    V. Jakubsky, S. Kuru, J. Negro, S. Tristao, J. Phys.: Condens. Matter 25, 165301 (2013)ADSGoogle Scholar
  156. 156.
    F. Queisser, R. Schuetzhold, Phys. Rev. Lett. 111, 046601 (2013)ADSCrossRefGoogle Scholar
  157. 157.
    B. Midya, D.J. Fernández, J. Phys. A: Math. Theor. 47, 285302 (2014)CrossRefGoogle Scholar
  158. 158.
    S. Kuru, J. Negro, S. Tristao, J. Phys.: Condens. Matter 27, 285501 (2015)Google Scholar
  159. 159.
    A. Schulze-Halberg, P. Roy, J. Phys. A: Math. Theor. 50, 365205 (2017)CrossRefGoogle Scholar
  160. 160.
    Y. Concha, A. Huet, A. Raya, D. Valenzuela, Mat. Res. Express 5, 065607 (2018)ADSCrossRefGoogle Scholar
  161. 161.
    D. Jahani, F. Shahbazi, M.R. Setare, Eur. Phys. J. Plus 133, 328 (2018)CrossRefGoogle Scholar
  162. 162.
    S.M. Chumakov, K.B. Wolf, Phys. Lett. A 193, 51–53 (1994)ADSMathSciNetCrossRefGoogle Scholar
  163. 163.
    M.A. Miri, M. Heinrich, D.N. Christodoulides, Phys. Rev. A 87, 043819 (2013)ADSCrossRefGoogle Scholar
  164. 164.
    A. Zúñiga-Segundo, B.M. Rodríguez-Lara, D.J. Fernández, H.M. Moya-Cessa, Opt. Express 22, 987–994 (2014)ADSCrossRefGoogle Scholar
  165. 165.
    B. Midya, Phys. Rev. A 89, 032116 (2014)ADSCrossRefGoogle Scholar
  166. 166.
    S. Dehdashti, R. Li, X. Liu, M. Raoofi, H. Chen, Laser Phys. 25, 075201 (2015)ADSCrossRefGoogle Scholar
  167. 167.
    S. Cruz y Cruz, Z. Gress, Ann. Phys. 383, 257–277 (2017)Google Scholar
  168. 168.
    B. Midya, W. Walasik, N.M. Litchinitser, L. Feng, Opt. Lett. 43, 4927 (2018)ADSCrossRefGoogle Scholar
  169. 169.
    A. Contreras-Astorga, V. Jakubsky, Phys. Rev. A 99, 053812 (2019)ADSCrossRefGoogle Scholar
  170. 170.
    A. Schulze-Halberg, Eur. Phys. J. Plus 128, 68 (2013)CrossRefGoogle Scholar
  171. 171.
    Y. Grandati, C. Quesne, SIGMA 11, 061 (2015)Google Scholar
  172. 172.
    J.C. Krapez, Int. J. Heat Mass Transfer 99, 485–503 (2016)CrossRefGoogle Scholar
  173. 173.
    A.R.P. Rau, J. Phys. A: Math. Gen. 37, 10421–10427 (2004)ADSCrossRefGoogle Scholar
  174. 174.
    M.I. Estrada-Delgado, D.J. Fernández, J. Phys.: Conf. Ser. 698, 012027 (2016)Google Scholar
  175. 175.
    J.M. Sixdeniers, K.A. Penson, A.I. Solomon, J. Phys. A: Math. Gen. 32, 7543–7563 (1999)ADSCrossRefGoogle Scholar
  176. 176.
    J.M. Sixdeniers, K.A. Penson, J. Phys. A: Math. Gen. 33, 2907–2916 (2000)ADSCrossRefGoogle Scholar
  177. 177.
    J.R. Klauder, K.A. Penson, J.M. Sixdeniers, Phys. Rev. A 64, 013817 (2001)ADSCrossRefGoogle Scholar
  178. 178.
    M. Castillo-Celeita, D.J. Fernández, in Physical and Mathematical Aspects of Symmetries, ed. by S. Duarte et al. (Springer International Publishing AG, Basel, 2017), pp. 111–117zbMATHCrossRefGoogle Scholar
  179. 179.
    E. Díaz-Bautista, D.J. Fernández, Eur. Phys. J. Plus 132, 499 (2017)CrossRefGoogle Scholar
  180. 180.
    A.A. Andrianov, M.V. Ioffe, F. Cannata, J.P. Dedonder, Int. J. Mod. Phys. A 14, 2675–2688 (1999)ADSCrossRefGoogle Scholar
  181. 181.
    T. Goliński, A. Odzijewicz, J. Comput. Appl. Math. 176, 331–355 (2005)Google Scholar
  182. 182.
    A. Dobrogowska, A. Odzijewicz, J. Comput. Appl. Math. 193, 319–346 (2006)Google Scholar
  183. 183.
    A. Dobrogowska, A. Odzijewicz, J. Phys. A: Math. Theor. 40, 2023–2036 (2007)ADSCrossRefGoogle Scholar
  184. 184.
    A. Dobrogowska, K. Janglajew, J. Differ. Equ. Appl. 13, 1171–1177 (2007)CrossRefGoogle Scholar
  185. 185.
    A. Dobrogowska, G. Jakimowicz, Appl. Math. Lett. 74, 161–166 (2017)MathSciNetCrossRefGoogle Scholar
  186. 186.
    A. Dobrogowska, D.J. Fernández (2018). arXiv:1807.06895 [math.DS]Google Scholar
  187. 187.
    A.F. Nikiforov, S.K. Suslov, V.B. Uvarov, Classical Orthogonal Polynomials of a Discrete Variable. Springer Series in Computational Physics (Springer, Berlin, 1991)Google Scholar
  188. 188.
    R. Alvarez-Nodarse, N.M. Atakishiyev, R.S. Costas-Santos, Electron. Trans. Numer. Anal. 27 34–50 (2007)MathSciNetGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Departamento de FísicaCinvestavCiudad de MéxicoMexico

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