, 73:397 | Cite as

Supersymmetric quantum mechanics living on topologically non-trivial Riemann surfaces

  • Miloslav Znojil
  • Vít Jakubský


Supersymmetric quantum mechanics is constructed in a new non-Hermitian representation. Firstly, the map between the partner operators H (±) is chosen antilinear. Secondly, both these components of a super-Hamiltonian \( \mathcal{H} \) are defined along certain topologically non-trivial complex curves r (±)(x) which spread over several Riemann sheets of the wave function. The non-uniqueness of our choice of the map \( \mathcal{T} \) between ‘tobogganic’ partner curves r (+)(x) and r (−)(x) is emphasized.


Supersymmetry Schrödinger equation complexified coordinates 


11.30.Pb 03.65.Fd 93.65.Db 


  1. [1]
    F Cooper, A Khare and U Sukhatme, Phys. Rep. 251, 267 (1995)CrossRefMathSciNetGoogle Scholar
  2. [2]
    M Znojil, Phys. Lett. A342, 36 (2005)MathSciNetADSGoogle Scholar
  3. [3]
    M Znojil, J. Phys. A: Math. Gen. 39, 13325 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  4. [3a]
    M Znojil, Phys. Lett. A372, 3591 (2008)MathSciNetADSGoogle Scholar
  5. [4]
    M Znojil, J. Phys. A: Math. Theor. 41, 215304 (2008)CrossRefMathSciNetADSGoogle Scholar
  6. [5]
    M Znojil, Phys. Lett. A372, 584 (2008)MathSciNetADSGoogle Scholar
  7. [6]
    G J C Wessels, A numerical and analytical investigation into non-Hermitian Hamiltonians (University of Stellenbosch, 2008), Master-degree thesis supervised by H B Geyer and J A C WeidemanGoogle Scholar
  8. [7]
    M Znojil, J. Phys.: Conference Series 128, 012046 (2008)CrossRefADSGoogle Scholar
  9. [8]
    Y A Gelfand and E P Likhtman, JETP Lett. 13, 323 (1971)ADSGoogle Scholar
  10. [8a]
    P Ramond, Phys. Rev. D3, 2415 (1971)MathSciNetADSGoogle Scholar
  11. [8b]
    A Neveu and J Schwarz, Nucl. Phys. B31, 86 (1971)CrossRefADSGoogle Scholar
  12. [9]
    E Witten, Nucl. Phys. B188, 513 (1981)CrossRefADSGoogle Scholar
  13. [10]
    E Witten, Nucl. Phys. B202, 253 (1982)CrossRefMathSciNetADSGoogle Scholar
  14. [11]
    M Znojil, F Cannata, B Bagchi and R Roychoudhury, Phys. Lett. B483, 284 (2000)MathSciNetADSGoogle Scholar
  15. [12]
    V Buslaev and V Grecchi, J. Phys. A: Math. Gen. 26, 5541 (1993)zbMATHCrossRefMathSciNetADSGoogle Scholar
  16. [13]
    C M Bender, D C Brody, J-H Chen, H F Jones, K A Milton and M C Ogilvie, Phys. Rev. D74, 025016 (2006)ADSGoogle Scholar
  17. [14]
    H F Jones and J Mateo, Phys. Rev. D73, 085002 (2006)MathSciNetADSGoogle Scholar
  18. [14a]
    C M Bender and D W Hook, J. Phys. A: Math. Theor. 41, 244005 (2008)CrossRefMathSciNetADSGoogle Scholar
  19. [15]
  20. [16]
    A Sinha and P Roy, Czech. J. Phys. 54, 129 (2004)CrossRefMathSciNetADSGoogle Scholar

Copyright information

© Indian Academy of Sciences 2009

Authors and Affiliations

  1. 1.Nuclear Physics Institute ASCRŘežCzech Republic
  2. 2.Departamento de FísicaUniversidad de Santiago de ChileSantiago 2Chile

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