Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 3 (2007), 112, 11 pages      arXiv:0710.0841
Contribution to the Proceedings of the Seventh International Conference Symmetry in Nonlinear Mathematical Physics

Deformed Oscillators with Two Double (Pairwise) Degeneracies of Energy Levels

Alexandre M. Gavrilik and Anastasiya P. Rebesh
Bogolyubov Institute for Theoretical Physics, 14-b Metrologichna Str., 03680 Kyiv, Ukraine

Received October 09, 2007, in final form November 13, 2007; Published online November 22, 2007

A scheme is proposed which allows to obtain special q-oscillator models whose characteristic feature consists in possessing two differing pairs of degenerate energy levels. The method uses the model of two-parameter deformed q,p-oscillators and proceeds via appropriately chosen particular relation between p and q. Different versions of quadratic relations p = f(q) are utilized for the case which implies two degenerate pairs E1 = E2 and E3 = E4. On the other hand, using one fixed quadratic relation, we obtain p-oscillators with other cases of two pairs of (pairwise) degenerate energy levels.

Key words: q,p-deformed oscillators; q-oscillators; energy levels degeneracy; energy function.

pdf (277 kb)   ps (179 kb)   tex (93 kb)


  1. Arik M., Coon D.D., Hilbert spaces of analytic functions and generalized coherent states, J. Math. Phys. 17 (1976), 524-527.
  2. Biedenharn L.C., The quantum group SUq(2) and a q-analogue of the boson operators, J. Phys. A: Math. Gen. 22 (1989), L873-878.
  3. Macfarlane A.J., On q-analogues of the quantum harmonic oscillator and the quantum group SUq(2), J. Phys. A: Math. Gen. 22 (1989), 4581-4585.
  4. Odaka K., Kishi T., Kamefuchi S., On quantization of simple harmonic oscillator, J. Phys. A: Math. Gen. 24 (1991), L591-L596.
  5. Chaturvedi S., Srinivasan V., Jagannathan R., Tamm-Dancoff deformation of bosonic oscillator algebras, Modern Phys. Lett. A 8 (1993), 3727-3734.
  6. Chakrabarti A., Jagannathan R., A q,p-oscillator realization of two-parameter quantum algebras, J. Phys. A: Math. Gen. 24 (1991), L711-L718.
  7. Mizrahi S.S., Camargo Lima J.P., Dodonov V.V., Energy spectrum, potential and inertia functions of a generalized f-oscillator, J. Phys. A: Math. Gen. 37 (2004), 3707-3724.
  8. Man'ko V.I., Marmo G., Sudarshan E.C.G., Zaccaria F., f-oscillators and nonlinear coherent states, Phys. Scripta 55 (1997), 528-541, quant-ph/9612006.
  9. Meljanac S., Milekovic M., Pallua S., Unified view of deformed single-mode oscillator algebras, Phys. Lett. B 328 (1994), 55-59, hep-th/9404039.
  10. Landau L.D., Lifshitz E.M., Quantum mechanics (nonrelativistic theory), Fiz.-Mat. Lit., Moscow, 1963 (in Russian).
  11. Kar S., Parwani R.R., Can degenerate bounds states occur in one dimensional quantum mechanics?, Europhys. Lett. 80 (2007), no. 3, 30004, 5 pages, arXiv:0706.1135.
  12. Gavrilik A.M., Rebesh A.P., A q-oscillator with "accidental" degeneracy of energy levels, Modern Phys. Lett. A 22 (2007), 949-960, quant-ph/0612122.
  13. Gavrilik A.M., Rebesh A.P., Occurrence of pairwise energy level degeneracies in q,p-oscillator model, submitted.
  14. Quesne C., Tkachuk V.M., Deformed algebras, position-dependent effective masses and curved spaces: an exactly solvable Coulomb problem, J. Phys. A: Math. Gen. 37 (2004), 4267-4281, math-ph/0403047.
  15. Lorek A., Wess J., Dynamical symmetries in q-deformed quantum mechanics, Z. Phys. C 67 (1995), 671-680, q-alg/9502007.
  16. Gavrilik A.M., Combined analysis of two- and three-particle correlations in the q,p-Bose gas model, SIGMA 2 (2006), 074, 12 pages, hep-ph/0512357.
  17. Anchishkin D.V., Gavrilik A.M., Iorgov N.Z., Two-particle correlations from the q-boson viewpoint, Eur. J. Phys. C. 7 (2000), 229-238, nucl-th/9906034.
  18. Anchishkin D.V., Gavrilik A.M., Iorgov N.Z., q-boson approach to multiparticle correlations, Modern Phys. Lett. A 15 (2000), 1637-1646, hep-ph/0010019.

Previous article   Next article   Contents of Volume 3 (2007)