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

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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.

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References

  1. C M Bender and S Boettcherzx, J. Phys. A: Math. Gen. 31, L273 (1998)

    Article  ADS  Google Scholar 

  2. A Khare and B P Mandal, Phys. Lett. A 272, 53 (2000)

    Article  ADS  MathSciNet  Google Scholar 

  3. F Cannata, M Ioffe, R Roychoudhury, and P Roy, Phys. Lett. A 281, 305 (2001)

    Article  ADS  MathSciNet  Google Scholar 

  4. B Bagchi, S Mallik, C Quesne, and R Roychoudhury, Phys. Lett. A 289, 34 (2001)

    Article  ADS  MathSciNet  Google Scholar 

  5. A Khare and U Sukhatme, J. Math. Phys. 46, 082106 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  6. C M Bender and M Monou, J. Phys. A: Math. Theor. 38, 2179 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  7. Z Ahmed, C M Bender, and M V Berry, J. Phys. A: Math. Theor. 38, L627 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  8. B Bagchi, C Quesne, and R Roychoudhury, J. Phys. A: Math. Theor. 41, 022001 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  9. A Khare and B P Mandal, Pramana – J. Phys. 73, 387 (2009)

    Article  ADS  Google Scholar 

  10. A Fring, Phys. Lett. A 379, 873 (2015)

    Article  MathSciNet  Google Scholar 

  11. A V Turbiner, Commun. Math. Phys. 118, 467 (1988)

    Article  ADS  MathSciNet  Google Scholar 

  12. M A Shifman, Int. J. Mod. Phys. A 4, 2897 (1989)

    Article  ADS  MathSciNet  Google Scholar 

  13. M A Shifman, Int. J. Mod. Phys. A 4, 3305 (1989)

    Article  ADS  MathSciNet  Google Scholar 

  14. N Kamran and P J Olver, J. Math. Anal. Appl. 145, 342 (1990)

    Article  MathSciNet  Google Scholar 

  15. A G Ushveridze, Quasi-exactly solvable models in quantum mechanics (Institute of Physics Publishing, Bristol, 1994)

    MATH  Google Scholar 

  16. C M Bender and S Boettcher, Phys. Rev. Lett. 80, 5243 (1998)

    Article  ADS  MathSciNet  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  18. C E Rüter, K G Makris, R El-Ganainy, D N Christodoulides, M Segev, and D Kip, Nature Phys. 6, 192 (2010)

    Article  ADS  Google Scholar 

  19. J Schindler, A Li, M C Zheng, F M Ellis, and T Kottos, Phys. Rev. A 84, 040101 (2011)

    Article  ADS  Google Scholar 

  20. A Regensburger, C Bersch, M A Miri, G Onishchukov, D N Christodoulides, and U Peschel, Nature 488, 167 (2012)

    Article  ADS  Google Scholar 

  21. A Regensburger, M A Miri, C Bersch, J Näger, G Onishchukov, D Christodoulides , and U Peschel, Phys. Rev. Lett. 110, 223902 (2013)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  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)

  25. S Kar and R R Parwani, Europhys. Lett. 80, 30004 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  26. R Koley and S Kar, Phys. Lett. A 363, 369 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  27. H -T Cho and C -L Ho, J. Phys. A 41, 172002 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  28. H -T Cho and C -L Ho, J. Phys. A 41, 255308 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  29. L J El-Jaick and B D B Figueiredo, J. Math. Phys. 50, 123511 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  30. A Dutt, T Nath, S Kar, and R Parwani, Eur. Phys. J. Plus 127, 28 (2012)

    Article  Google Scholar 

  31. J von Neumann and E Wigner, Phys. Z 30, 465 (1929)

    Google Scholar 

  32. F H Stillinger and D R Herrick, Phys. Rev. A 11, 446 (1975)

    Article  ADS  Google Scholar 

  33. F Capasso, C Sirtori, J Faist, D L Sivco, S N G Chu, and A Y Cho, Nature 358, 565 (1992)

    Article  ADS  Google Scholar 

  34. Y Plotnik, O Peleg, F Dreisow, M Heinrich, S Nolte, A Szameit, and M Segev, Phys. Rev. Lett. 107, 183901 (2011)

    Article  ADS  Google Scholar 

  35. B Zhen, C W Hsu, L Lu, A D Stone, and M Soljac̆ić, Phys. Rev. Lett. 113, 257401 (2014)

    Article  ADS  Google Scholar 

  36. A Gonzalez-Lopez, N Kamran, and P J Olver, Commun. Math. Phys. 153, 117 (1993)

    Article  ADS  MathSciNet  Google Scholar 

  37. Ronveaux A (Ed.), Heun’s differential equations (Oxford University Press, Oxford, 1995)

  38. S Y Slavyanov and W Lay, Special functions: A unified theory based on singularities (Oxford University Press, Oxford, 2000)

    MATH  Google Scholar 

  39. C -L Ho and H -T Cho, J. Phys. A: Math. Theor. 40, 1325 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  40. C -L Ho, Ann. Phys. 323, 2241 (2008)

    Article  ADS  Google Scholar 

Download references

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.

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Authors and Affiliations

  1. College of Physics and Electronic Engineering, Hainan Normal University, Haikou, 571158, China

    QIONGTAO XIE, LINA YAN, LINMAO WANG & JUN FU

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  1. QIONGTAO XIE
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  2. LINA YAN
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  3. LINMAO WANG
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  4. JUN FU
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Correspondence to QIONGTAO XIE.

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XIE, Q., YAN, L., WANG, L. et al. Analytical results for non-Hermitian parity–time-symmetric and Hermitian asymmetric volcano potentials. Pramana - J Phys 86, 965–972 (2016). https://doi.org/10.1007/s12043-015-1139-9

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