Bulg. J. Phys. vol.40 no.2 (2013), pp. 115-120

The u(2)α and su(2)α Hahn Harmonic Oscillators

E.I. Jafarov1, N.I. Stoilova2, J. Van der Jeugt3
1Institute of Physics, Azerbaijan National Academy of Sciences, Javid av. 33, AZ-1143 Baku, Azerbaijan
2Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, 72 Tsarigradsko Chaussee, 1784 Sofia, Bulgaria
3Department of Applied Mathematics and Computer Science, Ghent University, Krijgslaan 281-S9, B-9000 Gent, Belgium
Abstract. New models for the finite one-dimensional harmonic oscillator are proposed based upon the algebras u(2)α and
su(2)α. These algebras are deformations of the Lie algebras u(2) and su(2) extended by a parity operator, with deformation parameter α. Classes of irreducible unitary representations of u(2)α and su(2)α are constructed. It turns out that in these models the spectrum of the position operator can be computed explicitly, and that the corresponding (discrete) wavefunctions can be determined in terms of Hahn polynomials.

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