Mathematical and Physical Meaning of the Crossings of Energy Levels in \({\mathscr {PT}}\)-Symmetric Systems

Conference paper
Part of the Springer Proceedings in Physics book series (SPPHY, volume 184)

Abstract

Unavoided crossings of the energy levels due to a variation of a real parameter are studied. It is found that after the quantum system in question passes through one of its energy-crossing points alias Kato’s exceptional points (EP), its physical interpretation may dramatically change even when the crossing energies themselves do not complexify. The anomalous physical phase-transition mechanism of the change is revealed, attributed to the EP-related mathematics and illustrated via several exactly solvable matrix toy models.

Notes

Acknowledgments

D.B. was supported by RFBR grant no. 14-01-97009-r_povolzhe_a. M.Z. was supported by RV O61389005 and by the GACR grant Nr. 16-22945S.

References

  1. 1.
    C.M. Bender, Rep. Prog. Phys. 70, 947 (2007)ADSCrossRefGoogle Scholar
  2. 2.
    Z. Ahmed, D. Ghosh, J.A. Nathan, G. Parkar, Phys. Lett. A 379, 2424 (2015); M. Znojil. arXiv:1303.4876 (unpublished)
  3. 3.
    D.I. Borisov, Acta Polytech. 54, 93 (2014). arXiv:1401.6316
  4. 4.
    M. Znojil, Phys. Lett. A 259, 220 (1999)ADSMathSciNetCrossRefGoogle Scholar
  5. 5.
    T. Kato, Perturbation Theory for Linear Operators (Springer-Verlag, Berlin, 1966)CrossRefMATHGoogle Scholar
  6. 6.
    M. Znojil, J. Phys. A: Math. Theor. 45, 444036 (2012); Y.N. Joglekar, C. Thompson, D.D. Scott, G. Vemuri, Eur. Phys. J. Appl. Phys. 63, 30001 (2013); D.E. Pelinovsky, P.G. Keverekidis, D.J. Frantzeskakis, Eur. Phys. Lett. 101, 11002 (2013); C.H. Liang, D.D. Scott, Y.N. Joglekar, Phys. Rev. A 89, 030102(R) (2014); D.I. Borisov, F. Ruzicka, M. Znojil, Int. J. Theor. Phys. 54, 4293 (2015)Google Scholar
  7. 7.
    M. Znojil, J. Phys. A: Math. Theor. 40, 4863 (2007); M. Znojil, J. Phys. A: Math. Theor. 40, 13131 (2007)Google Scholar
  8. 8.
    A. Mostafazadeh, J. Phys. A: Math. Gen. 39, 10171 (2006); C.F. de Morison Faria, A. Fring, Czech. J. Phys. 56, 899 (2006); V. Jakubsky, J. Smejkal, Czech. J. Phys. 56 985 (2006); A. Ghatak, B.P. Mandal, Comm. Theor. Phys. 59, 553 (2013)Google Scholar
  9. 9.
    D. Krejčiřík, P. Siegl, J. Železný, Complex Anal. Oper. Theory 8, 255 (2014); D.C. Brody, J. Phys. A: Math. Theor. 49, 10LT03 (2016)Google Scholar
  10. 10.
    D. Krejčiřík, P. Siegl, J. Phys. A: Math. Theor. 43, 485204 (2010); F. Bagarello, M. Znojil, J. Phys. A: Math. Theor. 44, 415305 (2011); D. Borisov, D. Krejcirik, Asympt. Anal. 76, 49 (2012)Google Scholar
  11. 11.
    M. Znojil, SIGMA 5, 001 (2009), arXiv:0901.0700; M. Znojil, Int. J. Theor. Phys. 52, 2038 (2013)
  12. 12.
    F.G. Scholtz, H.B. Geyer, F.J.W. Hahne, Ann. Phys. (NY) 213, 74 (1992)Google Scholar
  13. 13.
    M. Znojil, H.B. Geyer, Fort. d. Physik 61, 111 (2013)ADSCrossRefGoogle Scholar
  14. 14.
    A. Mostafazadeh, Int. J. Geom. Meth. Mod. Phys. 7, 1191 (2010)MathSciNetCrossRefGoogle Scholar
  15. 15.
    M. Znojil, SIGMA 4, 001 (2008). arXiv:0710.4432
  16. 16.
    M. Znojil, in Non-Selfadjoint Operators in Quantum Physics: Mathematical Aspects, pp. 7–58, ed. by F. Bagarello, et al. (eds.), (Wiley, Hoboken, 2015)Google Scholar
  17. 17.
    D. Krejčiřík, H. Bíla, M. Znojil, J. Phys. A 39 10143 (2006); D. Krejčiřík, J. Phys. A: Math. Theor. 41, 244012 (2008); C.M. Bender, K. Besseghir, H.F. Jones, X. Yin, J. Phys. A: Math. Theor. 42, 355301 (2009)Google Scholar
  18. 18.
    M. Znojil, J. Wu, Int. J. Theor. Phys. 52, 2152 (2013)MathSciNetCrossRefGoogle Scholar
  19. 19.
    G. Levai, M. Znojil, Mod. Phys. Lett. A 16, 1973 (2001); A. Sinha, P. Roy, J. Phys. A: Math. Gen. 39, L377 (2006); P. Dorey, C. Dunning, A. Lishman, R. Tateo, J. Phys. A: Math. Theor. 42, 465302 (2009); G. Levai. J. Phys. A: Math. Theor. 45, 444020 (2012)Google Scholar
  20. 20.
    M. Znojil, J. Phys. A: Math. Theor. 41, 244027 (2008)ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Institute of Mathematics CC USC RASUfaRussia
  2. 2.Bashkir State Pedagogical UniversityUfaRussia
  3. 3.University of Hradec KraloveHrader KraloveCzech Republic
  4. 4.Nuclear Physics Institute ASCRŘežCzech Republic

Personalised recommendations