Exact and approximate solutions to Schrödinger’s equation with decatic potentials

• Authors: David Brandon , Nasser Saad
• Languages of publication: EN

Abstracts

EN

The one-dimensional Schrödinger’s equation is analysed with regard to the existence of exact solutions for decatic polynomial potentials. Under certain conditions on the potential’s parameters, we show that the decatic polynomial potential V (x) = ax
10 + bx
8 + cx
6 + dx
4 + ex
2, a > 0 is exactly solvable. By examining the polynomial solutions of certain linear differential equations with polynomial coefficients, the necessary and sufficient conditions for corresponding energy-dependent polynomial solutions are given in detail. It is also shown that these polynomials satisfy a four-term recurrence relation, whose real roots are the exact energy eigenvalues. Further, it is shown that these polynomials generate the eigenfunction solutions of the corresponding Schrödinger equation. Further analysis for arbitrary values of the potential parameters using the asymptotic iteration method is also presented.

Keywords

EN

differential equations   anharmonic oscillators   polynomial potentials   decatic potentials   energy spectrum   exact solutions  

• Publisher: De Gruyter Open
• Journal: Open Physics
• Year: 2013
• Volume: 11
• Issue: 3
• Pages: 279-290

Dates

published

1 - 3 - 2013

online

28 - 3 - 2013

References

• [1] E. G. Kalnins, W. D. Miller, G. S. Pogosyan, J. Math. Phys. 47, 033502 (2006) http://dx.doi.org/10.1063/1.2174237[Crossref]
• [2] N. Kamran, P. J. Olver, J. Math. Anal. Appl. 145, 342 (1990) http://dx.doi.org/10.1016/0022-247X(90)90404-4[Crossref]
• [3] A. Turbiner, J. Math. Phys. 33, 3989 (1992) http://dx.doi.org/10.1063/1.529848[Crossref]
• [4] A. Turbiner, Commun. Math. Phys. 118, 467 (1988) http://dx.doi.org/10.1007/BF01466727[Crossref]
• [5] F. Finkel, A. González-López, N. Kamran, Peter J. Olver, Miguel A. RodrÍguez, Proceedings of IV Workshop on Differential Geometry and its Applications (Santiago de Compostela, Spain, 1995)
• [6] A. González-López, N. Kamran, P. J. Olver, Philos. T. Roy. Soc. A 354, 1165 (1996) http://dx.doi.org/10.1098/rsta.1996.0044[Crossref]
• [7] M. Znojil, D. Yanovich, Proceedings of Institute of Mathematics of NAS of Ukraine 50, 1010 (2004)
• [8] V. S. Adamchik, A. D. Lizarev, UKRMJ 40, 583 (1988) http://dx.doi.org/10.1007/BF01057173[Crossref]
• [9] H. Azad, A. Laradji, M. T. Mustafa, Adv. Difference Equ. 2011, 58 (2011) http://dx.doi.org/10.1186/1687-1847-2011-58
• [10] D. Agboola, Y.-Z. Zhang, J. Math. Phys. 53, 042101 (2012) http://dx.doi.org/10.1063/1.3701833[Crossref]
• [11] D. Agboola, Y.-Z. Zhang, Mod. Phys. Lett. A 27, 1250112 (2012) http://dx.doi.org/10.1142/S021773231250112X[Crossref]
• [12] D. Agboola, Y.-Z. Zhang, Ann. Phys.-New York 327, 2275 (2012) http://dx.doi.org/10.1016/j.aop.2012.07.002[Crossref]
• [13] A. S. de Castro, A. de Souza Dutra, Phys. Lett. A 269, 281 (2000) http://dx.doi.org/10.1016/S0375-9601(00)00277-2[Crossref]
• [14] F. M. Fernández, Phys. Rev. A 44, 333 (1991)
• [15] J. Dziubanski, Colloq. Math. 78, 149 (1998)
• [16] C. M. Bender, G. V. Dunne, J. Math. Phys. 37, 6 (1997) http://dx.doi.org/10.1063/1.531373[Crossref]
• [17] C. M. Bender, M. Monou, J. Phys. A: Math. Gen. 38, 2179 (2005) http://dx.doi.org/10.1088/0305-4470/38/10/009[Crossref]
• [18] V. E. Kruglov, Diff. Equat+. 47, 20 (2011) http://dx.doi.org/10.1134/S0012266111010034[Crossref]
• [19] A. Boumenir, Appl. Math. Comput. 215, 2914 (2009) http://dx.doi.org/10.1016/j.amc.2009.09.037[Crossref]
• [20] H. Tezuka, Z. Phys. C Part. Fields 65, 101 (1995) http://dx.doi.org/10.1007/BF01571309[Crossref]
• [21] H. Taseli, J. Comput. Phys. 101, 252 (1992) http://dx.doi.org/10.1016/0021-9991(92)90002-G[Crossref]
• [22] A. de Freitas, P. Martín, E. Castro, J. L. Paz, Phys. Lett. A 362, 371 (2007) http://dx.doi.org/10.1016/j.physleta.2006.10.042[Crossref]
• [23] Shi-Hai Dong, Int. J. Theor. Phys. 41, 89 (2002) http://dx.doi.org/10.1023/A:1013273600656[Crossref]
• [24] Shi-Hai Dong, Found. Phys. Lett. 15, 385 (2002) http://dx.doi.org/10.1023/A:1021220712636[Crossref]
• [25] Shi-Hai Dong, Phys. Scripta 65, 289 (2002) http://dx.doi.org/10.1238/Physica.Regular.065a00289[Crossref]
• [26] Y. Meurice, J. Phys. A: Math. Gen. 35, 8831 (2002) http://dx.doi.org/10.1088/0305-4470/35/41/314[Crossref]
• [27] A. Nanayakkara, I. Dasanayake, Phys. lett. A 294, 158 (2002) http://dx.doi.org/10.1016/S0375-9601(02)00045-2[Crossref]
• [28] H. Ciftci, R. L. Hall, N. Saad, J. Phys. A 36, 11807 (2003) http://dx.doi.org/10.1088/0305-4470/36/47/008[Crossref]
• [29] H. Ciftci, R. L. Hall, N. Saad, E. Dogu, J. Phys. A: Math. Theor. 43, 415206 (2010) http://dx.doi.org/10.1088/1751-8113/43/41/415206[Crossref]
• [30] R. L. Hall, N. Saad, K. D. Sen, J. Math. Phys. 52, 092103 (2011) http://dx.doi.org/10.1063/1.3638444[Crossref]
• [31] R. L. Hall, N. Saad, K. D. Sen, J. Phys. A: Math. Theor. 44, 185307 (2011) http://dx.doi.org/10.1088/1751-8113/44/18/185307[Crossref]
• [32] R. L. Hall, N. Saad, K. D. Sen, J. Math. Phys. 51, 022107 (2010) http://dx.doi.org/10.1063/1.3290740[Crossref]
• [33] A. Eremenko, A. Gabrielov, B. Shapiro, CMFT 8, 513 (2008)
• [34] A. Voros, J. Phys. A: Math. Gen. 32, 5993 (1999) http://dx.doi.org/10.1088/0305-4470/32/32/311[Crossref]
• [35] V. V. Napalkov, A. M. Dilmukhametova, Dokl. Math. 79, 97 (2009) http://dx.doi.org/10.1134/S1064562409010293[Crossref]
• [36] M. Znojil, Phys. Lett. A 116, 207 (1986) http://dx.doi.org/10.1016/0375-9601(86)90132-5[Crossref]
• [37] M. Znojil, Quantum Inversion Theory and Applications: Proceedings of the 109th W.E. Heraeus, Ed. H. V. Geramb, Lecture Notes in Physics, vol. 427, 1993) 458
• [38] M. Znojil, Phys. Lett. A 341, 67 (2005) http://dx.doi.org/10.1016/j.physleta.2005.04.061[Crossref]
• [39] Yao-Zhong Zhang, J. Phys. A: Math. Theor. 45, 065206 (2012) http://dx.doi.org/10.1088/1751-8113/45/6/065206[Crossref]
• [40] A. D. Alhaidari, J. Phys. A: Math. Theor. 40, 6305 (2007) http://dx.doi.org/10.1088/1751-8113/40/24/004[Crossref]
• [41] R. Roychoudhury, P. Roy, M. Znojil, G. Lévai, J. Math. Phys. 42, 1996 (2001) http://dx.doi.org/10.1063/1.1362286[Crossref]
• [42] P Koscik, A. Okopinska, J. Phys. A: Math. Gen. 38, 7743 (2005) http://dx.doi.org/10.1088/0305-4470/38/35/008[Crossref]
• [43] R. L. Hall, N. Saad, J. Math. Phys. 38, 4909 (1997) http://dx.doi.org/10.1063/1.531925[Crossref]
• [44] R. Adhikari, R. Dutt, Y. P. Varshni, Phys. Lett. A 141, 1 (1989) http://dx.doi.org/10.1016/0375-9601(89)90433-7[Crossref]
• [45] R. L. Hall, N. Saad, K. D. Sen, H. Ciftci, Phys. Rev. A 80, 032507 (2009) http://dx.doi.org/10.1103/PhysRevA.80.032507[Crossref]
• [46] N. Saad, R. L. Hall, H. Ciftci, J. Phys. A: Math. Gen. 39, 8477 (2006) http://dx.doi.org/10.1088/0305-4470/39/26/014[Crossref]
• [47] N. Saad, R. L. Hall, H. Ciftci, J. Phys. A: Math. Gen. 39, 13445 (2006) http://dx.doi.org/10.1088/0305-4470/39/43/004[Crossref]
• [48] H. J. Weber, Cent. Eur. J. Math. 5, 415 (2007) http://dx.doi.org/10.2478/s11533-007-0004-6[Crossref]
• [49] F. M. Fernández, R. Guardiola, J. Phys. A: Math. Gen. 26, 7169 (1993) http://dx.doi.org/10.1088/0305-4470/26/23/051[Crossref]
• [50] G. P. Flessas, J. Phys. A: Math. Gen. 14, L209 (1981) http://dx.doi.org/10.1088/0305-4470/14/6/001[Crossref]
• [51] E. Magyari, Phys. Lett. A 81, 116 (1981) http://dx.doi.org/10.1016/0375-9601(81)90037-2[Crossref]
• [52] M. Vishwarmittar, Physica A 216, 452 (1995) http://dx.doi.org/10.1016/0378-4371(95)00032-3[Crossref]
• [53] F. Pan, J. R. Klauder, J. P. Draayer, Phys. Lett. A 262, 131 (1999) http://dx.doi.org/10.1016/S0375-9601(99)00651-9[Crossref]
• [54] R. N. Chaudhuri, M. Mondal, Phys. Rev. A 43, 3241 (1991) http://dx.doi.org/10.1103/PhysRevA.43.3241[Crossref]
• [55] M. Bansal, S. Srivastava, Vishwamittar, Phys. Rev. A 44, 8012 (1991) http://dx.doi.org/10.1103/PhysRevA.44.8012[Crossref]
• [56] M. Jafarpour, D. Afshar, J. Phys. A: Math. Gen. 35, 87 (2002) http://dx.doi.org/10.1088/0305-4470/35/1/307[Crossref]
• [57] K. Gustav, R. Colditz, Int. J. Quantum Chem. 21, 1091 (1982) http://dx.doi.org/10.1002/qua.560210613[Crossref]
• [58] E. Z. Liverts, V. B. Mandelzweig, Phys. Scripta 77, 025003 (2008) http://dx.doi.org/10.1088/0031-8949/77/02/025003[Crossref]
• [59] E. J. Weniger, Phys. Rev. Lett. 77, 2859 (1996) http://dx.doi.org/10.1103/PhysRevLett.77.2859[Crossref]
• [60] E. D. Rainville, Am. Math. Mon. 43, 473 (1936) http://dx.doi.org/10.2307/2301168[Crossref]
• [61] A. G. Ushveridze, Quasi-exactly solvable models in quantum mechanics (Institute of Physics Publishing, Bristol, 1994)
• [62] T. Barakat, Phys. Lett. A 344, 411 (2005) http://dx.doi.org/10.1016/j.physleta.2005.06.081[Crossref]
• [63] F. J. Gómez, J. Sesma, J. Phys. A: Math. Gen. 38, 3193 (2005) http://dx.doi.org/10.1088/0305-4470/38/14/009[Crossref]
• [64] Q. D. Katatbeh, R. L. Hall, N. Saad, J. Phys. A: Math. Theor. 40, 13431 (2007) http://dx.doi.org/10.1088/1751-8113/40/44/020[Crossref]
• [65] M. Jafarpour, D. Afshar, Journal of Sciences, Islamic Republic of Iran 18, 75 (2007)
• [66] I. A. Ivanov, J. Phys. A: Math. Gen. 31, 5697 (1998) http://dx.doi.org/10.1088/0305-4470/31/26/009[Crossref]
• [67] B. Champion, R. L. Hall, N. Saad, Int. J. Mod. Phys. A 23, 1405 (2008) http://dx.doi.org/10.1142/S0217751X08039852[Crossref]

Identifiers

DOI

10.2478/s11534-013-0179-3