Few-Body Systems

, Volume 55, Issue 12, pp 1223–1232 | Cite as

Bound State Solutions of the Klein-Gordon Equation for the Mathews-Lakshmanan Oscillator

  • Axel Schulze-Halberg
  • Jie Wang


We study a boundary-value problem for the Klein-Gordon equation that is inspired by the well-known Mathews-Lakshmanan oscillator model. By establishing a link to the spheroidal equation, we show that our problem admits an infinite number of discrete energies, together with associated solutions that form an orthogonal set in a weighted L 2-Hilbert space.


Gordon Equation Continue Fraction Expansion Nonrelativistic Quantum Bound State Energy Bound State Solution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abramowitz M., Stegun I.: Handbook of mathematical functions with formulas, graphs, and mathematical tables. Dover Publications, New York (1964)zbMATHGoogle Scholar
  2. 2.
    Arda A., Sever R., Tezcan C.: Approximate analytical solutions of the Klein Gordon equation for the Hulthen potential with the position dependent mass. Phys. Scr. 79, 015006 (2009)ADSCrossRefGoogle Scholar
  3. 3.
    Carinena J.F., Ranada M.F., Santander M.: The quantum free particle on spherical and hyperbolic spaces: A curvature dependent approach. II. J. Math. Phys. 53, 102109 (2012)ADSCrossRefMathSciNetGoogle Scholar
  4. 4.
    Carinena J.F., Ranada M.F., Santander M.: A quantum exactly-solvable nonlinear oscillator with quasi-harmonic behaviour. Ann. Phys. 322, 434–459 (2007)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Carinena J.F., Ranada M.F., Santander M., Senthilvelan M.: A non-linear oscillator with quasi-harmonic behaviour: two- and n-dimensional oscillators. Nonlinearity 17, 1941–1963 (2004)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Carinena J.F., Ranada M.F., Santander M.: One-dimensional model of a quantum non-linear harmonic oscillator. Rep. Math. Phys. 54, 285–293 (2004)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Farrokh M., Shojaei M.R., Rajabi A.A.: Klein–Gordon equation with Hulthen potential and position-dependent mass. Eur. Phys. J. Plus 128, 14 (2013)CrossRefGoogle Scholar
  8. 8.
    Higgs P.W.: Dynamical symmetries in a spherical geometry. J. Phys. A 12, 309–323 (1979)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Ikhdair S.M.: Exact Klein–Gordon equation with spatially dependent masses for unequal scalar-vector Coulomb-like potentials. Eur. Phys. J. A 40, 143–149 (2009)ADSCrossRefGoogle Scholar
  10. 10.
    Jia C.-S., Li X.-P., Zhang L.-H.: Exact solutions of the Klein–Gordon equation with position-dependent mass for mixed vector and scalar kink-like potentials. Few-Body Syst. 52, 11–18 (2012)ADSCrossRefGoogle Scholar
  11. 11.
    Lakshmanan M., Eswaran K.: Quantum dynamics of a solvable nonlinear chiral model. J. Phys. A 8, 1658–1669 (1975)ADSCrossRefGoogle Scholar
  12. 12.
    Mathews P.M., Lakshmanan M.: On a unique nonlinear oscillator. Quart. Appl. Math. 32, 215–218 (1974)zbMATHMathSciNetGoogle Scholar
  13. 13.
    Mathews P.M., Lakshmanan M.: A quantum-mechanically solvable nonpolynomial Lagrangian with velocity-dependent interaction. Nuovo Cimento A 26, 299–316 (1975)ADSCrossRefMathSciNetGoogle Scholar
  14. 14.
    Morse P.M., Feshbach H.: Methods of theoretical physics, vol. 2. Mc Graw-Hill, New York (1953)Google Scholar
  15. 15.
    Midya B., Roy B.: A generalized quantum nonlinear oscillator. J. Phys. A 42, 285301 (2009)CrossRefMathSciNetGoogle Scholar
  16. 16.
    Ronveaux A.: Heun’s differential equations. Oxford University Press, New York (1995)zbMATHGoogle Scholar
  17. 17.
    Schulze-Halberg A., Morris J.R.: Special function solutions of a spectral problem for a nonlinear quantum oscillator. J. Phys. A 45, 305301 (2012)ADSCrossRefMathSciNetGoogle Scholar
  18. 18.
    Slavianov S.Y., Lay W.: Special functions: a unified theory based on singularities. Oxford University Press, New York (2000)Google Scholar
  19. 19.
    Roos, O.von : Position-dependent effective masses in semiconductor theory. Phys. Rev. B 27, 7547–7552 (1983)Google Scholar
  20. 20.
    Roos, O.von , Mavromatis, H.: Position-dependent effective masses in semiconductor theory. II. Phys. Rev. B 31, 2294–2298 (1985)Google Scholar

Copyright information

© Springer-Verlag Wien 2014

Authors and Affiliations

  1. 1.Department of Mathematics and Actuarial Science and Department of PhysicsIndiana University NorthwestGaryUSA
  2. 2.Department of Computer Information SystemsIndiana University NorthwestGaryUSA

Personalised recommendations