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


SIGMA 1 (2005), 008, 17 pages      math-ph/0508032      https://doi.org/10.3842/SIGMA.2005.008

Spectra of Observables in the q-Oscillator and q-Analogue of the Fourier Transform

Anatoliy U. Klimyk
Bogolyubov Institute for Theoretical Physics, 14-b Metrologichna Str., 03143 Kyiv, Ukraine

Received August 26, 2005, in final form October 19, 2005; Published online October 21, 2005

Abstract
Spectra of the position and momentum operators of the Biedenharn-Macfarlane q-oscillator (with the main relation aa+ - qa+a = 1) are studied when q > 1. These operators are symmetric but not self-adjoint. They have a one-parameter family of self-adjoint extensions. These extensions are derived explicitly. Their spectra and eigenfunctions are given. Spectra of different extensions do not intersect. The results show that the creation and annihilation operators a+ and a of the q-oscillator for q > 1 cannot determine a physical system without further more precise definition. In order to determine a physical system we have to choose appropriate self-adjoint extensions of the position and momentum operators.

Key words: Biedenharn-Macfarlane q-oscillator; position operator; momentum operator; spectra; continuous q-1-Hermite polynomials; Fourier transform.

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References

  1. Biedenharn L.C., The quantum group SUq(2) and a q-analogue of the boson operators, J. Phys. A: Math. Gen., 1989, V.22, L873-L879.
  2. Macfarlane A.J., On q-analogues of the quantum harmonic oscillator and the quantum group SUq(2), J. Phys. A: Math. Gen., 1989, V.22, 4581-4588.
  3. Damaskinsky E.V., Kulish P.P., Deformed oscillators and their applications, Zap. Nauch. Sem. LOMI, 1991, V.189, 37-74.
  4. Klimyk A., Schmüdgen K., Quantum groups and their representations, Berlin, Springer, 1997.
  5. Klimyk A.U., Schempp W., Classical and quantum Heisenberg groups, their representations and applications, Acta Appl. Math., 1986, V.45, 143-194.
  6. Burban I.M., Klimyk A.U., Representations of the quantum algebra Uq(su1,1), J. Phys. A: Math. Gen., 1993, V.26, 2139-2151.
  7. Burban I.M., Klimyk A.U., On spectral properties of q-oscillator operators, Lett. Math. Phys., 1993, V.29, 13-18.
  8. Chung W.-S., Klimyk A.U., On position and momentum operators in the q-oscillator, J. Math. Phys., 1996, V.37, 917-932.
  9. Borzov V.V., Damaskinsky E.V., Kulish P.P., On position operator spectral measure for deformed oscillator in the case of indetermine Hamburger moment problem, Rev. Math. Phys., 2001, V.12, 691-710.
  10. Askey R., Continuous q-Hermite polynomials when q > 1, in q-Series and Partitions, Editor D. Stanton, Berlin, Springer, 1998, 151-158.
  11. Klimyk A.U., On position and momentum operators in the q-oscillator, J. Phys. A: Math. Gen., 2005, V.38, 4447-4458.
  12. Gasper G., Rahman M., Basic hypergeometric functions, Cambridge, Cambridge University Press, 1990.
  13. Berezanskii Yu.M., Expansions in eigenfunctions of selfadjoint operators, Providence, R.I., American Mathematical Society, 1969.
  14. Shohat J., Tamarkin J.D., The problem of moments, Providence, R.I., American Mathematical Society, 1943.
  15. Simon B., The classical moment problem as a self-adjoint finite difference operator, Adv. Math., 1998, V.137, 82-203.
  16. Akhiezer N.I., The classical moment problem, New York, NY, Hafner, 1965.
  17. Akhiezer N.I., Glazman I.M., The theory of linear operators in Hilbert spaces, New York, NY, Ungar, 1961.
  18. Ciccoli N., Koelink E., Koornwinder T.H., q-Laguerre polynomials and big q-Bessel functions and their orthogonality relations, Methods of Applied Analysis, 1999, V.6, 109-127.
  19. Ismail M.E.R., Masson D.R., q-Hermite polynomials, biorthogonal functions, and q-beta integrals, Trans. Amer. Math. Soc., 1994, V.346, 63-116.
  20. Koekoek R., Swarttouw R.F., The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue, Delft University of Technology Report 98-17; available from ftp.tudelft.nl.
  21. Santilli R.M., Imbedding of Lie algebras in nonassociative structures, Nuovo Cim. A, 1967, V.51, 570-576.
  22. Askey R., Atakishiyev N.M., Suslov S.K., An analog of the Fourier transformation for a q-harmonic oscillator, in Symmetries in Science VI, Editor B. Gruber, New York, Plenum Press, 1993, 57-64.


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