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Supersymmetries in Schrödinger–Pauli Equations and in Schrödinger Equations with Position Dependent Mass

  • Anatoly G. NikitinEmail author
Chapter
Part of the CRM Series in Mathematical Physics book series (CRM)

Abstract

The contemporary results concerning supersymmetries in generalized Schrödinger equations are presented. Namely, position dependent mass Schrödinger equations are discussed as well as the equations with matrix potentials. An extended number of realistic quantum mechanical problems admitting extended supersymmetries are described.

Keywords

Position dependent mass Schrödinger–Pauli equations Extended supersymmetries Matrix potentials Integrable systems 

References

  1. 1.
    Y.A. Gol’fand, E.P. Lichtman, Sov. Phys. JETP Lett. 13, 452 (1971); D.V. Volkov, V.P. Akulov, Phys. Lett. B 46, 109 (1973); J. Wess, B. Zumino, Nucl. Phys. B 70, 39 (1974)Google Scholar
  2. 2.
    J. Lipkin, Phys. Lett. 9, 203 (1964); J. Schwinger, Phys. Rev. 152, 1219 (1966); G.L. Stavraki, in High Energy Physics and the Theory of Elementary Particles, Naukova Dumka, Kiev (1966), p. 296 (in Russian); Preprint ITP 67-21, Kiev, 1967; H. Migazawa, Progr. Theor. Phys. 36, 1266 (1968), Phys. Rev. 170, 1586 (1968); M. Flato; P. Hillon, Phys. Rev. D 1, 1667 (1970); A. Neveu, J.M. Schwartz, Nucl. Phys B 31, 86 (1971); J.L. Gervais, B. Sakita, Nucl. Phys. B 34, 633 (1971); A. Joseph, Nuovo Cimento A 8, 217 (1972); Y. Aharonov, A. Casher, L. Susskind, Phys. Lett. B 35, 512 (1974)Google Scholar
  3. 3.
    V.A. Kostelecky, D.K. Campbell, Phys. D 15, 3 (1985)CrossRefGoogle Scholar
  4. 4.
    M.B. Green, J.H. Schwartz, E. Witten, Superstring theory, in 2 vols (Cambridge University Press, Cambridge, 1987); M. Kaku, Strings, Conformal Field Theory and Topology (Springer, New York, 1989)Google Scholar
  5. 5.
    N. Seiberg, E. Witten, Nucl. Phys. B 426, 19 (1994); N. Seiberg, Phys. Rev. D 49, 6857 (1994)Google Scholar
  6. 6.
    L.E.Gendenshtein, I.V. Krive, Usp. Fiz. Nauk 146, 553 (1985)CrossRefGoogle Scholar
  7. 7.
    F. Cooper, A. Khare, U. Sukhatme, Phys. Rep. 211, 268 (1995)Google Scholar
  8. 8.
    S.V. Sukhumar, J. Phys. A 18, L697 (1985)ADSCrossRefGoogle Scholar
  9. 9.
    F. Ravndal, Phys. Rev. D 21, 2461 (1980); A. Khare, J. Maharana, Nucl. Phys. B 244, 409 (1984)Google Scholar
  10. 10.
    E. Witten, Nucl. Phys. B 185, 513 (1981); 202, 253 (1982)Google Scholar
  11. 11.
    L. Gendenshtein, JETP Lett. 38, 356 (1983)ADSGoogle Scholar
  12. 12.
    G.A. Natanzon, Vestnik Leningrad Univ. 10, 22 (1971); Teor. Mat. Fiz. 38, 146 (1979)Google Scholar
  13. 13.
    A.A. Andrianov, M.V. Ioffe, J. Phys. A 45, 503001 (2012)MathSciNetCrossRefGoogle Scholar
  14. 14.
    T. Tanaka, Nucl. Phys. B 662, 413 (2003)ADSCrossRefGoogle Scholar
  15. 15.
    F. Correa, V. Jakubsky, M.S. Plyushchay, J. Phys. A: Math. Theor. 41, 485303 (2008)CrossRefGoogle Scholar
  16. 16.
    C. Quesne, SIGMA 3, 067 (2007)Google Scholar
  17. 17.
    V.Y. Novokshenov, SIGMA 14, 106 (2018)Google Scholar
  18. 18.
    V.A. Rubakov, V.P. Spiridonov, Mod. Phys. Lett. A 3, 1337 (1988)ADSCrossRefGoogle Scholar
  19. 19.
    J. Beckers, N. Debergh, Nucl. Phys. B 340, 767 (1990)ADSCrossRefGoogle Scholar
  20. 20.
    J, Beckers, N. Debergh, A.G. Nikitin, Mod. Phys. Let A 8, 435 (1993)Google Scholar
  21. 21.
    J. Beckers, N. Debergh, A.G. Nikitin, J. Math. Phys. 33, 152 (1992)ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    G.P. Pron’ko, Y.G. Stroganov, Sov. Phys. JETP 45, 1075 (1977)ADSGoogle Scholar
  23. 23.
    E. Ferraro, N. Messina, A.G. Nikitin, Phys. Rev. A 81, 042108 (2010)ADSCrossRefGoogle Scholar
  24. 24.
    F. Cooper, A. Khare, U. Sukhatme, Phys. Rep. 251, 267 (1995)ADSMathSciNetCrossRefGoogle Scholar
  25. 25.
    A.A. Andrianov, N.V. Borisov, M. V. Ioffe, Theor. Math. Phys. 61, 1078 (1984)CrossRefGoogle Scholar
  26. 26.
    M.V. Ioffe, SIGMA 6, 075 (2010)Google Scholar
  27. 27.
    A.A. Andrianov, M.V. Ioffe, Phys. Lett. B 255, 543 (1991); A.A. Andrianov, M.V. Ioffe, V.P. Spiridonov, L. Vinet, Phys. Lett. B 272, 297 (1991)Google Scholar
  28. 28.
    A.A. Andrianov, F. Cannata, M.V. Ioffe, D.N. Nishnianidze, J. Phys. A: Math. Gen. 30, 5037 (1997)ADSCrossRefGoogle Scholar
  29. 29.
    T. Fukui, Phys. Lett. A 178, 1 (1993)ADSMathSciNetCrossRefGoogle Scholar
  30. 30.
    M.V. Ioffe, S. Kuru, J. Negro, L.M. Nieto, J. Phys. A 39, 6987 (2006)ADSMathSciNetCrossRefGoogle Scholar
  31. 31.
    R. de Lima Rodrigues, V.B. Bezerra, A.N. Vaidyac, Phys. Lett. A 287, 45 (2001)ADSCrossRefGoogle Scholar
  32. 32.
    V.M. Tkachuk, P. Roy, Phys. Lett. A 263, 245 (1999); V.M. Tkachuk, P. Roy, J. Phys. A 33, 4159 (2000)Google Scholar
  33. 33.
    A.G. Nikitin, Y. Karadzhov, Matrix superpotentials. J. Phys. A Math. Theor. 44, 305204 (2011)MathSciNetCrossRefGoogle Scholar
  34. 34.
    A.G. Nikitin, Y. Karadzhov, Enhanced classification of matrix superpotentials. J. Phys. A Math. Theor. 44, 445202 (2011)ADSMathSciNetCrossRefGoogle Scholar
  35. 35.
    J. Niederle, A.G. Nikitin, J. Math. Phys. 40, 1280 (1999)ADSMathSciNetCrossRefGoogle Scholar
  36. 36.
    A.G. Nikitin, J. Mod. Phys. A 14, 885 (1999)ADSCrossRefGoogle Scholar
  37. 37.
    Y. Karadzhov, Matrix superpotentials, Thesis, Kiev, Institute of Mathematics, 2015Google Scholar
  38. 38.
    A.G. Nikitin, J. Math. Phys. 53, 122103 (2012)ADSMathSciNetCrossRefGoogle Scholar
  39. 39.
    A.G. Nikitin, Superintegrable and supersymmetric systems of Schrödinger equations, in Proceedings of the Sixth International Workshop on Group Analysis of Differential Equations and Integrable Systems, June 17–21, 2012, Protaras, Cyprus. University of Cyprus, Nikosia, 2013, pp.154–169Google Scholar
  40. 40.
    A.G. Nikitin, J. Phys. A: Math. Theor. 45, 225205 (2012)ADSCrossRefGoogle Scholar
  41. 41.
    A.G. Nikitin, T.M. Zasadko, J. Math. Phys. 56, 042101 (2015)ADSMathSciNetCrossRefGoogle Scholar
  42. 42.
    A.G. Nikitin, T.M. Zasadko, J. Phys. A 49, 365204 (2016)MathSciNetCrossRefGoogle Scholar
  43. 43.
    A.G. Nikitin, J. Math. Phys. 58, 083508 (2017)ADSMathSciNetCrossRefGoogle Scholar
  44. 44.
    A.G. Nikitin, J. Phys. A 48, 335201 (2015)MathSciNetCrossRefGoogle Scholar
  45. 45.
    F.W. Olver , Asymptotics and Special Functions (Academic Press, New York, 1974)zbMATHGoogle Scholar
  46. 46.
    L.E. Gendenshtein, JETP Lett. 39, 234 (1984)Google Scholar
  47. 47.
    V.M. Tkachuk, S.I. Vakarchuk, Phys. Lett. A 228, 141 (1997)ADSMathSciNetCrossRefGoogle Scholar
  48. 48.
    A.G. Nikitin, in Problems of quantum field theory, JINR E2-96-369 (Dubna, 1996), p. 509Google Scholar
  49. 49.
    J. Beckers, N. Debergh, A.G. Nikitin, Int. J. Theor. Phys. 36, 1991 (1997)CrossRefGoogle Scholar
  50. 50.
    T. Tanaka, Mod. Phys. Lett. A 27, 1250051 (2012)ADSCrossRefGoogle Scholar
  51. 51.
    A.V. Sokolov, J. Phys. A 48, 085202 (2015)ADSMathSciNetCrossRefGoogle Scholar
  52. 52.
    A.V. Sokolov, Phys. Lett. A 377, 655 (2013)ADSMathSciNetCrossRefGoogle Scholar
  53. 53.
    A.A. Andrianov, A.V. Sokolov, Phys. Lett. A 379, 279 (2015)MathSciNetCrossRefGoogle Scholar
  54. 54.
    A.A. Andrianov, A.V. Sokolov, Theor. Math. Phys. 186, 2 (2016)CrossRefGoogle Scholar
  55. 55.
    M.V. Ioffe, E.V. Kolevatova, D.N. Nishnianidze, Phys. Lett. A 380, 3349 (2016)ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Institute of MathematicsNational Academy of Sciences of UkraineKyivUkraine

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