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Extended supersymmetry and hidden symmetries in one-dimensional matrix quantum mechanics

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Abstract

We study properties of nonlinear supersymmetry algebras realized in the one-dimensional quantum mechanics of matrix systems. Supercharges of these algebras are differential operators of a finite order in derivatives. In special cases, there exist independent supercharges realizing an (extended) supersymmetry of the same super-Hamiltonian. The extended supersymmetry generates hidden symmetries of the super-Hamiltonian. Such symmetries have been found in models with (2×2)-matrix potentials.

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References

  1. A. Lahiri, P. K. Roy, and B. Bagchi, Internat. J. Mod. Phys. A, 5, 1383–1456 (1990).

    Article  ADS  MathSciNet  Google Scholar 

  2. F. Cooper, A. Khare, and U. Sukhatme, Phys. Rep., 251, 267–385 (1995).

    Article  ADS  MathSciNet  Google Scholar 

  3. G. Junker, Supersymmetric Methods in Quantum and Statistical Physics, Springer, Berlin (1996).

    Book  MATH  Google Scholar 

  4. A. A. Andrianov and M. V. Ioffe, J. Phys. A, 45, 503001 (2012); arXiv:1207.6799v2 [hep-th] (2012).

    Article  MathSciNet  Google Scholar 

  5. A. A. Andrianov, N. V. Borisov, and M. V. Ioffe, Theor. Math. Phys., 61, 1078–1088 (1984).

    Article  Google Scholar 

  6. A. A. Andrianov, M. V. Ioffe, and V. P. Spiridonov, Phys. Lett. A, 174, 273–279 (1993); arXiv:hep-th/9303005v1 (1993).

    Article  ADS  MathSciNet  Google Scholar 

  7. A. A. Andrianov, F. Cannata, J.-P. Dedonder, and M. V. Ioffe, Internat. J. Mod. Phys. A, 10, 2683–2702 (1995); arXiv:hep-th/9404061v1 (1994).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  8. A. A. Andrianov and A. V. Sokolov, Nucl. Phys. B, 660, 25–50 (2003); arXiv:hep-th/0301062v4 (2003).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  9. M. V. Ioffe and D. N. Nishnianidze, Phys. Lett. A, 327, 425–432 (2004); arXiv:hep-th/0404078v1 (2004).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  10. A. A. Andrianov and F. Cannata, Math. Gen., 37, 10297–10321 (2004); arXiv:hep-th/0407077v2 (2004).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  11. A. A. Andrianov and A. V. Sokolov, J. Math. Sci., 143, 2707–2722 (2007); arXiv:0710.5738v1 [quant-ph] (2007).

    Article  MathSciNet  Google Scholar 

  12. A. A. Andrianov, F. Cannata, and A. V. Sokolov, Nucl. Phys. B, 773, 107–136 (2007); arXiv:math-ph/0610024v3 (2006).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  13. A. V. Sokolov, Nucl. Phys. B, 773, 137–171 (2007); arXiv:math-ph/0610022v2 (2006).

    Article  ADS  MATH  Google Scholar 

  14. A. V. Sokolov, J. Math. Sci., 151, 2924 (2008); arXiv:0903.2835v1 [math-ph] (2009).

    Article  MathSciNet  MATH  Google Scholar 

  15. A. V. Sokolov, J. Math. Sci., 168, 881–900 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  16. Yu. A. Kuperin, D. M. Latypov, and S. P. Merkur’ev, Phys. Atom. Nucl., 55, 1141–1142 (1992).

    Google Scholar 

  17. A. A. Andrianov and M. V. Ioffe, Phys. Lett. B, 205, 507–510 (1988).

    Article  ADS  MathSciNet  Google Scholar 

  18. R. D. Amado, F. Cannata, and J.-P. Dedonder, Phys. Rev. A, 38, 3797–3800 (1988).

    Article  ADS  Google Scholar 

  19. R. D. Amado, F. Cannata, and J.-P. Dedonder, Internat. J. Mod. Phys. A, 5, 3401–3415 (1990).

    Article  ADS  Google Scholar 

  20. A. A. Andrianov and M. V. Ioffe, Phys. Lett. B, 255, 543–549 (1991).

    Article  ADS  MathSciNet  Google Scholar 

  21. A. A. Andrianov, M. V. Ioffe, V. P. Spiridonov, and L. Vinet, Phys. Lett. B, 272, 297–304 (1991).

    Article  ADS  MathSciNet  Google Scholar 

  22. F. Cannata and M. V. Ioffe, Phys. Lett. B, 278, 399–402 (1992).

    Article  ADS  Google Scholar 

  23. F. Cannata and M. V. Ioffe, J. Phys. A: Math. Gen., 26, L89–L92 (1993).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  24. T. Fukui, Phys. Lett. A, 178, 1–6 (1993).

    Article  ADS  MathSciNet  Google Scholar 

  25. L. V. Hau, J. A. Golovchenko, and M. M. Burns, Phys. Rev. Lett., 75, 1426–1429 (1995).

    Article  ADS  Google Scholar 

  26. A. A. Andrianov, F. Cannata, D. N. Nishnianidze, and M. V. Ioffe, J. Phys. A, 30, 5037–5050 (1997); arXiv: quant-ph/9707004v1 (1997).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  27. J.-M. Sparenberg and D. Baye, Phys. Rev. Lett., 79, 3802–3805 (1997).

    Article  ADS  Google Scholar 

  28. R. de Lima Rodrigues, P. B. da Silva Filho, and A. N. Vaidya, Phys. Rev. D, 58, 125023 (1998).

    Article  ADS  MathSciNet  Google Scholar 

  29. V. M. Goncharenko and A. P. Veselov, J. Phys. A: Math. Gen., 31, 5315–5326 (1998).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  30. T. K. Das and B. Chakrabarti, J. Phys. A: Math. Gen., 32, 2387–2394 (1999).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  31. V. M. Tkachuk and P. Roy, Phys. Lett. A, 263, 245–249 (1999); arXiv:quant-ph/9905102v1 (1999).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  32. A. L. Sakhnovich, J. Phys. A: Math. Gen., 36, 7789–7802 (2003).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  33. M. V. Ioffe and A. I. Neelov, J. Phys. A: Math. Gen., 36, 2493–2506 (2003); arXiv:hep-th/0302004v1 (2003).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  34. B. F. Samsonov and A. A. Pecheritsin, J. Phys. A: Math. Gen., 37, 239–250 (2004); arXiv:quant-ph/0307145v1 (2003).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  35. F. Cannata, M. V. Ioffe, A. I. Neelov, and D. N. Nishnianidze, J. Phys. A: Math. Gen., 37, 10339–10348 (2004); arXiv:hep-th/0405108v1 (2004).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  36. M. V. Ioffe, Ş. Kuru, J. Negro, and L. M. Nieto, J. Phys. A: Math. Gen., 39, 6987–7001 (2006); arXiv:hep-th/0603005v1 (2006).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  37. E. Ferraro, A. Messina, and A. G. Nikitin, Phys. Rev. A, 81, 042108 (2010); arXiv:0909.5543v3 [quant-ph] (2009).

    Article  ADS  Google Scholar 

  38. A. A. Pecheritsin, A. M. Pupasov, and B. F. Samsonov, J. Phys. A: Math. Theor., 44, 205305 (2011); arXiv: 1102.5255v1 [quant-ph] (2011).

    Article  ADS  MathSciNet  Google Scholar 

  39. A. G. Nikitin and Yu. Karadzhov, J. Phys. A: Math. Theor., 44, 305204 (2011); 44, 445202; arXiv:1107.2525v2 [math-ph] (2011).

    Article  MathSciNet  Google Scholar 

  40. T. Tanaka, Modern Phys. Lett. A, 27, 1250051 (2012); arXiv:1108.0480v2 [math-ph] (2011).

    Article  ADS  MathSciNet  Google Scholar 

  41. A. V. Sokolov, Phys. Lett. A, 377, 655–662 (2013); arXiv:1307.4449v1 [quant-ph] (2013).

    Article  ADS  MathSciNet  Google Scholar 

  42. A. M. Pupasov-Maksimov, “Multichannel generalization of eigen-phase preserving supersymmetric transformations,” arXiv:1301.4199v1 [math-ph] (2013).

    Google Scholar 

  43. F. Correa and V. Jakubský, Phys. Rev. D, 90, 125003 (2014); arXiv:1406.2997v2 [hep-th] (2014).

    Article  ADS  Google Scholar 

  44. T. Tanaka, “Construction of quasi-solvable quantum mechanical matrix models: Lie superalgebra v.s. N-fold supersymmetry,” arXiv:1409.5578v1 [math-ph] (2014).

    Google Scholar 

  45. D. Baye, J.-M. Sparenberg, A. M. Pupasov-Maksimov, and B. F. Samsonov, J. Phys. A: Math. Theor., 47, 243001 (2014); arXiv:1401.0439v2 [quant-ph] (2014).

    Article  ADS  MathSciNet  Google Scholar 

  46. A. V. Sokolov, J. Phys. A, 48, 085202 (2015); arXiv:1406.0191v2 [math-ph] (2014).

    Article  ADS  MathSciNet  Google Scholar 

  47. A. A. Andrianov and A. V. Sokolov, Phys. Lett. A, 379, 279–283 (2015); arXiv:1409.6695v2 [math-ph] (2014).

    Article  MathSciNet  Google Scholar 

  48. A. A. Andrianov, M. V. Ioffe, and D. N. Nishnianidze, Theor. Math. Phys., 104, 1129–1140 (1995).

    Article  MathSciNet  MATH  Google Scholar 

  49. M. A. Naimark, Linear Differential Operators, Frederick Ungar, New York (1967).

    MATH  Google Scholar 

  50. A. A. Andrianov and A. V. Sokolov, Proc. Inst. Math. NAS Ukraine, 50, 539–546 (2004).

    MathSciNet  Google Scholar 

  51. F. Correa, V. Jakubský, and M. S. Plyushchay, J. Phys. A: Math. Theor., 41, 485303 (2008); arXiv:0806.1614v3 [hep-th] (2008).

    Article  Google Scholar 

  52. A. A. Andrianov and A. V. Sokolov, SIGMA, 5, 064 (2009); arXiv:0906.0549v2 [hep-th] (2009).

    MathSciNet  Google Scholar 

  53. B. F. Samsonov, Phys. Lett. A, 263, 274–280 (1999); arXiv:quant-ph/9904009v1 (1999).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  54. A. Khare and U. Sukhatme, J. Math. Phys., 40, 5473–5494 (1999); arXiv:quant-ph/9906044v1 (1999).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  55. D. J. Fernández C, J. Negro, and L. M. Nieto, Phys. Lett. A, 275, 338–349 (2000).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  56. D. J. Fernández C, B. Mielnik, O. Rosas-Ortiz, and B. F. Samsonov, J. Phys. A: Math. Gen., 35, 4279–4291 (2002); arXiv:quant-ph/0303051v1 (2003).

    Article  ADS  MATH  Google Scholar 

  57. D. J. Fernández C. R. Muñoz, and A. Ramos, Phys. Lett. A, 308, 11–16 (2003); arXiv:quant-ph/0212026v1 (2002).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  58. D. J. Fernández C and E. Salinas-Hernández, J. Phys. A: Math. Gen., 36, 2537–2543 (2003); arXiv:quant-ph/ 0303123v1 (2003).

    Article  ADS  MATH  Google Scholar 

  59. B. F. Samsonov, Phys. Lett. A, 358, 105–114 (2006); arXiv:quant-ph/0602101v3 (2006).

    Article  ADS  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. St. Petersburg State University, St. Petersburg, Russia

    A. A. Andrianov & A. V. Sokolov

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  1. A. A. Andrianov
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  2. A. V. Sokolov
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Correspondence to A. A. Andrianov.

Additional information

This research was supported by the Russian Foundation for Basic Research (Grant No. 13-01-00136 a and 16-01-00375 a) and St. Petersburg State University (Research Grant No. 11.38.660.2013).

The participation of A. V. Sokolov in PHHQP’15 (15th International Workshop on Pseudo-Hermitian Hamiltonians in Quantum Physics) was supported by St. Petersburg State University (Grant No. 11.41.847.2015).

The participation of A. A. Andrianov in PHHQP’15 (15th International Workshop on Pseudo-Hermitian Hamiltonians in Quantum Physics) was supported by St. Petersburg State University (Grant No. 11.41.775.2015).

Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 186, No. 1, pp. 5–26, January, 2016.

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Andrianov, A.A., Sokolov, A.V. Extended supersymmetry and hidden symmetries in one-dimensional matrix quantum mechanics. Theor Math Phys 186, 2–20 (2016). https://doi.org/10.1134/S0040577916010025

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