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.
Similar content being viewed by others
References
C M Bender and S Boettcherzx, J. Phys. A: Math. Gen. 31, L273 (1998)
A Khare and B P Mandal, Phys. Lett. A 272, 53 (2000)
F Cannata, M Ioffe, R Roychoudhury, and P Roy, Phys. Lett. A 281, 305 (2001)
B Bagchi, S Mallik, C Quesne, and R Roychoudhury, Phys. Lett. A 289, 34 (2001)
A Khare and U Sukhatme, J. Math. Phys. 46, 082106 (2005)
C M Bender and M Monou, J. Phys. A: Math. Theor. 38, 2179 (2005)
Z Ahmed, C M Bender, and M V Berry, J. Phys. A: Math. Theor. 38, L627 (2005)
B Bagchi, C Quesne, and R Roychoudhury, J. Phys. A: Math. Theor. 41, 022001 (2008)
A Khare and B P Mandal, Pramana – J. Phys. 73, 387 (2009)
A Fring, Phys. Lett. A 379, 873 (2015)
A V Turbiner, Commun. Math. Phys. 118, 467 (1988)
M A Shifman, Int. J. Mod. Phys. A 4, 2897 (1989)
M A Shifman, Int. J. Mod. Phys. A 4, 3305 (1989)
N Kamran and P J Olver, J. Math. Anal. Appl. 145, 342 (1990)
A G Ushveridze, Quasi-exactly solvable models in quantum mechanics (Institute of Physics Publishing, Bristol, 1994)
C M Bender and S Boettcher, Phys. Rev. Lett. 80, 5243 (1998)
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)
C E Rüter, K G Makris, R El-Ganainy, D N Christodoulides, M Segev, and D Kip, Nature Phys. 6, 192 (2010)
J Schindler, A Li, M C Zheng, F M Ellis, and T Kottos, Phys. Rev. A 84, 040101 (2011)
A Regensburger, C Bersch, M A Miri, G Onishchukov, D N Christodoulides, and U Peschel, Nature 488, 167 (2012)
A Regensburger, M A Miri, C Bersch, J Näger, G Onishchukov, D Christodoulides , and U Peschel, Phys. Rev. Lett. 110, 223902 (2013)
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)
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)
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)
S Kar and R R Parwani, Europhys. Lett. 80, 30004 (2007)
R Koley and S Kar, Phys. Lett. A 363, 369 (2007)
H -T Cho and C -L Ho, J. Phys. A 41, 172002 (2008)
H -T Cho and C -L Ho, J. Phys. A 41, 255308 (2008)
L J El-Jaick and B D B Figueiredo, J. Math. Phys. 50, 123511 (2009)
A Dutt, T Nath, S Kar, and R Parwani, Eur. Phys. J. Plus 127, 28 (2012)
J von Neumann and E Wigner, Phys. Z 30, 465 (1929)
F H Stillinger and D R Herrick, Phys. Rev. A 11, 446 (1975)
F Capasso, C Sirtori, J Faist, D L Sivco, S N G Chu, and A Y Cho, Nature 358, 565 (1992)
Y Plotnik, O Peleg, F Dreisow, M Heinrich, S Nolte, A Szameit, and M Segev, Phys. Rev. Lett. 107, 183901 (2011)
B Zhen, C W Hsu, L Lu, A D Stone, and M Soljac̆ić, Phys. Rev. Lett. 113, 257401 (2014)
A Gonzalez-Lopez, N Kamran, and P J Olver, Commun. Math. Phys. 153, 117 (1993)
Ronveaux A (Ed.), Heun’s differential equations (Oxford University Press, Oxford, 1995)
S Y Slavyanov and W Lay, Special functions: A unified theory based on singularities (Oxford University Press, Oxford, 2000)
C -L Ho and H -T Cho, J. Phys. A: Math. Theor. 40, 1325 (2007)
C -L Ho, Ann. Phys. 323, 2241 (2008)
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.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12043-015-1139-9