ON FAITHFUL MATRIX REPRESENTATIONS OF
q-DEFORMED LIE ALGEBRA FOR COUPLED
QUANTIZED OSCILLATORS
L.A-M. Hanna
Dept. of Maths., Faculty of Science, Kuwait Univ.
P.O. Box 5969 Safat 13060
Kuwait City – KUWAIT

Abstract

The polynomially deformed Lie algebra, $\mathfrak{l}_{q}:\left[ K_{0},K_{+}% \right] _{q}=G\left( K_{+}\right) $, $\left[ K_{-},K_{0}\right] _{q}=G\left( K_{-}\right) ,\left[ K_{+},K_{-}\right] _{q}=P\left( K_{0}\right) $, is introduced as a generalized model of the coupled quantized oscillators model, where $G$ and $P$ are real polynomial functions, subject to the physical properties: $K_{0}$ is a real diagonal operator, and $% K_{-}=K_{+}^{\dagger }.$ Matrix representations are discussed and conditions are given for $G$ and $P$ to guarantee the existence of the faithful representations.

Citation details of the article



Journal: International Journal of Applied Mathematics
Journal ISSN (Print): ISSN 1311-1728
Journal ISSN (Electronic): ISSN 1314-8060
Volume: 33
Issue: 6
Year: 2020

DOI: 10.12732/ijam.v33i6.9

Download Section



Download the full text of article from here.

You will need Adobe Acrobat reader. For more information and free download of the reader, please follow this link.

References

  1. [1] V.P. Karassiov, New Lie-algebraic structures in nonlinear problems of quantum optics and laser physics, J. of Soviet Laser Research, 13 (1992), 188-195.
  2. [2] V.P. Karassiov, G-invariant polynomial extensions of Lie algebras in quantum many-body physics, J. Phys. A: Math. Gen., 27 (1994), 153-165.
  3. [3] V.P. Karassiov and A.B. Klimov, An algebraic approach to solving evolution problems in some nonlinear quantum models, Phys. Lett. A, 189 (1994), 43-51.
  4. [4] A.B. Klimov and J.L. Romero, An algebraic solution of Lindblad-type master equation, J. Opt. B: Quantum Semiclass. Opt., 5 (2003), S316- S321.
  5. [5] L. A-M. Hanna, M.E. Khalifa and S.S. Hassan, On representations of Lie algebras for quantized Hamiltonians, Linear Algebra Appl., 266 (1997), 69-79.
  6. [6] L. A-M. Hanna, On the classification of the Lie algebras, Linear Algebra Appl., 370 (2003), 251-256.
  7. [7] L. A-M. Hanna, On matrix representations of deformed Lie algebras for quantized Hamiltonians, Linear Algebra Appl., 434 (2011), 507-513.
  8. [8] L. A-M. Hanna, A deformed Tavis-Cummings model and its matrix representation, JP J. of Algebra, Number Theory and Applications, 35 (2014), 49-65.
  9. [9] L. A-M. Hanna and S.S. Hassan, On matrix representations of deformed Lie algebras, Int. J. Theol. Phys., Group Th. and Nonlinear Opt., 13 (2010), 137-143.
  10. [10] L. Hanna, R. Alharbey, S. Abdalla and S. Hassan, Algebraic method of solution of Schr¨odinger’s equation of a quantum model, WSEAS Trans. on Mathematics, 19 (2020), 421-429.
  11. [11] A. Algin, A comparative study on q-deformed Fermion oscillators, Int. J. Theor. Phys., 50 (2011), 1554-1568.
  12. [12] P. Narayana Swamy, Interpolating statistics and q-deformed oscillator algebras, Intern. J. of Modern Physics, B, 20 (2006), 697-713.
  13. [13] P. Narayana Swamy, q-deformed Fermions, Eur. Phys. J., B, 50 (2006), 291-294.
  14. [14] P. Narayana Swamy, q-deformed Fermions: Algebra, Fock space and thermodynamics, Inten. J. of Modern Physics, B 20 (2006), 2537-2550.
  15. [15] S. Dey and V. Hussin, Noncommutative q-photon-added coherent states, Phys. Rev. A, 93 (2016), Art. 053824.
  16. [16] S. Sargolzaeipor, H. Hassanabadi, W.S. Chung and A.N. Ikot, q-deformed oscillator algebra in fermionic and bosonic limits, Indian Acad. Sci., Pra- mana – J. Phys., 93 (2019), Art. 68.
  17. [17] W.S. Chung and H. Hassanabadi, q-deformed quantum mechanics based on the q-addition, In: Advanced Science News, Fortschr. Phys. # 1800111, Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim (2019).
  18. [18] R.J.C. Spreeuw and J.P. Woerdman, Optical atoms, Progress in Optics, 31 (1993), 263-319.
  19. [19] C.S. Adams, M. Sigel and Mlynek, Atom optics, Physics Reports, 240 (1994), Art. 143.
  20. [20] W. Witschel, Ordered operator expansions by comparison, J. Phys. A, 8 (1975), 143-155.
  21. [21] S. Steinberg, Applications of the Lie algebraic formulas of Baker, Campbell, Hausdorff, and Zassenhaus to the calculation of explicit solutions of partial differential equations, J. Differential Equations, 26 (1997), 404-434.
  22. [22] S. Steinberg, Lie series, Lie transformations, and their applications, In: Lie Methods in Optics (J.S. Mondrag´on and K.B. Wolf, Eds., Le´on, 1985), Lecture Notes in Phys., 250 (1986), 45-103.
  23. [23] G. Dattoli, J.C. Gallardo and A. Torre, An algebraic view to the operatorial ordering and its applications to optics, La Rivisita del Nuovo Cimento, 11 (1988), Art. 1.
  24. [24] G. Dattoli, P. Di Lazzaro, A. Torre, A spinor approach to the propagation in self-focusing fibers. Nuovo Cim. B, 105 (1990), 165-178.
  25. [25] G. Dattoli, M. Richetta, G. Schettini and A. Torre, Lie algebraic methods and solutions of linear partial differential equations, J. Math. Phys., 31 (1990), Art. 2856.
  26. [26] P.W. Higgs, Dynamical symmetries in a spherical geometry, I, J. Phys. A, 12 (1979), 309-323.
  27. [27] M.A. Al-Gwaiz, M.S. Abdalla and S. Deshmuckh, Lie algebraic approach to the coupled-mode oscillator, J. Phys. A, 27 (1994), Art. 1275.
  28. [28] M. Sebawe Abdalla, H. Eleuch, T. Barakat, Exact analytical solutions of the wave function for some q-deformed potentials, Reports on Math. Phys., 71 (2013), 217-229.