Study of the generalized quantum isotonic nonlinear oscillator potential. (English) Zbl 1238.81108

Summary: We study the generalized quantum isotonic oscillator Hamiltonian given by \(H = - d^2/dr^2 + l(l + 1)/r^2 + w^2r^2 + 2g(r^2 - a^2)/(r^2 + a^2)^2,~g > 0\). Two approaches are explored. A method for finding the quasipolynomial solutions is presented, and explicit expressions for these polynomials are given, along with the conditions on the potential parameters. By using the asymptotic iteration method, we show how the eigenvalues of this Hamiltonian for arbitrary values of the parameters \(g, w\), and \(a\) may be found to high accuracy.


81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
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[1] J. F. Cariñena, A. M. Perelomov, M. F. Rañada, and M. Santander, “A quantum exactly solvable nonlinear oscillator related to the isotonic oscillator,” Journal of Physics A, vol. 41, no. 8, Article ID 085301, 2008. · Zbl 1138.81380
[2] J. M. Fellows and R. A. Smith, “Factorization solution of a family of quantum nonlinear oscillators,” Journal of Physics A, vol. 42, no. 33, Article ID 335303, 2009. · Zbl 1177.81034
[3] J. Sesma, “The generalized quantum isotonic oscillator,” Journal of Physics A, vol. 43, no. 18, Article ID 185303, 2010. · Zbl 1188.81072
[4] A. Ronveaux, Ed., Heun’s Differential Equations, Oxford Science Publications, Oxford University Press, New York, NY, USA, 1995. · Zbl 0847.34006
[5] B. Champion, R. L. Hall, and N. Saad, “Asymptotic iteration method for singular potentials,” International Journal of Modern Physics A, vol. 23, no. 9, pp. 1405-1415, 2008. · Zbl 1156.81366
[6] H. Ciftci, R. L. Hall, N. Saad, and E. Dogu, “Physical applications of second-order linear differential equations that admit polynomial solutions,” Journal of Physics A, vol. 43, no. 41, Article ID 415206, 2010. · Zbl 1200.81049
[7] H. Ciftci, R. L. Hall, and N. Saad, “Asymptotic iteration method for eigenvalue problems,” Journal of Physics A, vol. 36, no. 47, pp. 11807-11816, 2003. · Zbl 1070.34113
[8] H. Ciftci, R. L. Hall, and N. Saad, “Construction of exact solutions to eigenvalue problems by the asymptotic iteration method,” Journal of Physics A, vol. 38, no. 5, pp. 1147-1155, 2005. · Zbl 1069.34127
[9] R. L. Hall, N. Saad, and A. B. von Keviczky, “Spiked harmonic oscillators,” Journal of Mathematical Physics, vol. 43, no. 1, pp. 94-112, 2002. · Zbl 1059.81044
[10] N. M. Temme, Special Functions, John Wiley & Sons, New York, NY, USA, 1996. · Zbl 0863.33002
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