Advertisement

Pramana

, Volume 86, Issue 5, pp 965–972 | Cite as

Analytical results for non-Hermitian parity–time-symmetric and Hermitian asymmetric volcano potentials

  • QIONGTAO XIE
  • LINA YAN
  • LINMAO WANG
  • JUN FU
Article

Abstract

We investigate both the non-Hermitian parity–time-(PT-)symmetric and Hermitian asymmetric volcano potentials, and present the analytical solution in terms of the confluent Heun function. Under certain special conditions, the confluent Heun function can be terminated as a polynomial, thereby leading to certain exact analytical results. It is found that the non-Hermitian PT-symmetric volcano potentials support the normalizable and non-normalizable reflectionless states with real energies. The Hermitian asymmetric volcano potentials allow normalizable reflectionless states with complex energies.

Keywords

Volcano potentials Heun confluent equation Heun confluent function reflectionless scattering state. 

PACS Nos

03.65.Ge 03.65.Fd 11.30.Er 

Notes

Acknowledgements

This work was supported by the National Natural Science Foundation of China under Grant No. 11375059 and the Scientific Research Fund of Hunan Provincial Education Department under Grant No. 13A058.

References

  1. [1]
    C M Bender and S Boettcherzx, J. Phys. A: Math. Gen. 31, L273 (1998)ADSCrossRefGoogle Scholar
  2. [2]
    A Khare and B P Mandal, Phys. Lett. A 272, 53 (2000)ADSMathSciNetCrossRefGoogle Scholar
  3. [3]
    F Cannata, M Ioffe, R Roychoudhury, and P Roy, Phys. Lett. A 281, 305 (2001)ADSMathSciNetCrossRefGoogle Scholar
  4. [4]
    B Bagchi, S Mallik, C Quesne, and R Roychoudhury, Phys. Lett. A 289, 34 (2001)ADSMathSciNetCrossRefGoogle Scholar
  5. [5]
    A Khare and U Sukhatme, J. Math. Phys. 46, 082106 (2005)ADSMathSciNetCrossRefGoogle Scholar
  6. [6]
    C M Bender and M Monou, J. Phys. A: Math. Theor. 38, 2179 (2005)ADSMathSciNetCrossRefGoogle Scholar
  7. [7]
    Z Ahmed, C M Bender, and M V Berry, J. Phys. A: Math. Theor. 38, L627 (2005)ADSMathSciNetCrossRefGoogle Scholar
  8. [8]
    B Bagchi, C Quesne, and R Roychoudhury, J. Phys. A: Math. Theor. 41, 022001 (2008)ADSMathSciNetCrossRefGoogle Scholar
  9. [9]
    A Khare and B P Mandal, Pramana – J. Phys. 73, 387 (2009)ADSCrossRefGoogle Scholar
  10. [10]
    A Fring, Phys. Lett. A 379, 873 (2015)MathSciNetCrossRefGoogle Scholar
  11. [11]
    A V Turbiner, Commun. Math. Phys. 118, 467 (1988)ADSMathSciNetCrossRefGoogle Scholar
  12. [12]
    M A Shifman, Int. J. Mod. Phys. A 4, 2897 (1989)ADSMathSciNetCrossRefGoogle Scholar
  13. [13]
    M A Shifman, Int. J. Mod. Phys. A 4, 3305 (1989)ADSMathSciNetCrossRefGoogle Scholar
  14. [14]
    N Kamran and P J Olver, J. Math. Anal. Appl. 145, 342 (1990)MathSciNetCrossRefGoogle Scholar
  15. [15]
    A G Ushveridze, Quasi-exactly solvable models in quantum mechanics (Institute of Physics Publishing, Bristol, 1994)zbMATHGoogle Scholar
  16. [16]
    C M Bender and S Boettcher, Phys. Rev. Lett. 80, 5243 (1998)ADSMathSciNetCrossRefGoogle Scholar
  17. [17]
    A Guo, G J Salamo, D Duchesne, R Morandotti, M Volatier-Ravat, V Aimez, G A Siviloglou, and D N Christodoulides, Phys. Rev. Lett. 103, 093902 (2009)ADSCrossRefGoogle Scholar
  18. [18]
    C E Rüter, K G Makris, R El-Ganainy, D N Christodoulides, M Segev, and D Kip, Nature Phys. 6, 192 (2010)ADSCrossRefGoogle Scholar
  19. [19]
    J Schindler, A Li, M C Zheng, F M Ellis, and T Kottos, Phys. Rev. A 84, 040101 (2011)ADSCrossRefGoogle Scholar
  20. [20]
    A Regensburger, C Bersch, M A Miri, G Onishchukov, D N Christodoulides, and U Peschel, Nature 488, 167 (2012)ADSCrossRefGoogle Scholar
  21. [21]
    A Regensburger, M A Miri, C Bersch, J Näger, G Onishchukov, D Christodoulides , and U Peschel, Phys. Rev. Lett. 110, 223902 (2013)ADSCrossRefGoogle Scholar
  22. [22]
    M Brandstetter, M Liertzer, C Deutsch, P Klang, J Schöberl, H E Türeci, G Strasser , K Unterrainer, and S Rotter, Nature Commun. 5, 4034 (2014)ADSCrossRefGoogle Scholar
  23. [23]
    B Peng, S K Ozdemir, S Rotter, H Yilmaz, M Liertzer, F Monifi, C M Bender, F Nori , and L Yang, Science 346, 328 (2014)ADSCrossRefGoogle Scholar
  24. [24]
    B Peng, Sahin Kaya Ozdemir, Fuchuan Lei, Faraz Monifi, Mariagiovanna Gianfreda, Gui Lu Long, Shanhui Fan, Franco Nori, Carl M Bender and Lan Yang, Nature Phys. 10, 394 (2014)Google Scholar
  25. [25]
    S Kar and R R Parwani, Europhys. Lett. 80, 30004 (2007)ADSMathSciNetCrossRefGoogle Scholar
  26. [26]
    R Koley and S Kar, Phys. Lett. A 363, 369 (2007)ADSMathSciNetCrossRefGoogle Scholar
  27. [27]
    H -T Cho and C -L Ho, J. Phys. A 41, 172002 (2008)ADSMathSciNetCrossRefGoogle Scholar
  28. [28]
    H -T Cho and C -L Ho, J. Phys. A 41, 255308 (2008)ADSMathSciNetCrossRefGoogle Scholar
  29. [29]
    L J El-Jaick and B D B Figueiredo, J. Math. Phys. 50, 123511 (2009)ADSMathSciNetCrossRefGoogle Scholar
  30. [30]
    A Dutt, T Nath, S Kar, and R Parwani, Eur. Phys. J. Plus 127, 28 (2012)CrossRefGoogle Scholar
  31. [31]
    J von Neumann and E Wigner, Phys. Z 30, 465 (1929)Google Scholar
  32. [32]
    F H Stillinger and D R Herrick, Phys. Rev. A 11, 446 (1975)ADSCrossRefGoogle Scholar
  33. [33]
    F Capasso, C Sirtori, J Faist, D L Sivco, S N G Chu, and A Y Cho, Nature 358, 565 (1992)ADSCrossRefGoogle Scholar
  34. [34]
    Y Plotnik, O Peleg, F Dreisow, M Heinrich, S Nolte, A Szameit, and M Segev, Phys. Rev. Lett. 107, 183901 (2011)ADSCrossRefGoogle Scholar
  35. [35]
    B Zhen, C W Hsu, L Lu, A D Stone, and M Soljac̆ić, Phys. Rev. Lett. 113, 257401 (2014)ADSCrossRefGoogle Scholar
  36. [36]
    A Gonzalez-Lopez, N Kamran, and P J Olver, Commun. Math. Phys. 153, 117 (1993)ADSMathSciNetCrossRefGoogle Scholar
  37. [37]
    Ronveaux A (Ed.), Heun’s differential equations (Oxford University Press, Oxford, 1995)Google Scholar
  38. [38]
    S Y Slavyanov and W Lay, Special functions: A unified theory based on singularities (Oxford University Press, Oxford, 2000)zbMATHGoogle Scholar
  39. [39]
    C -L Ho and H -T Cho, J. Phys. A: Math. Theor. 40, 1325 (2007)ADSMathSciNetCrossRefGoogle Scholar
  40. [40]
    C -L Ho, Ann. Phys. 323, 2241 (2008)ADSCrossRefGoogle Scholar

Copyright information

© Indian Academy of Sciences 2016

Authors and Affiliations

  1. 1.College of Physics and Electronic EngineeringHainan Normal UniversityHaikouChina

Personalised recommendations