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We propose a group-theoretical approach to the generalized oscillator algebraAk recently investigated in J. Phys. A: Math. Theor. 43 (2010) 115303. The casek > or 0 corresponds to the noncompact group SU(1,1) (as for the harmonicoscillator and the Poeschl-Teller systems) while the case k < 0 is described bythe compact group SU(2) (as for the Morse system). We construct the phaseoperators and the corresponding temporally stable phase eigenstates for Ak inthis group-theoretical context. The SU(2) case is exploited for derivingfamilies of mutually unbiased bases used in quantum information. Along thisvein, we examine some characteristics of a quadratic discrete Fourier transformin connection with generalized quadratic Gauss sums and generalized Hadamardmatrices.

symmetry

SU(2) and SU(1,1) Approaches to Phase Operators and Temporally Stable  Phase States: Applications to Mutually Unbiased Bases and Discrete Fourier  Transforms