  The finite q-oscillator is a model that obeys the dynamics of the harmonic
oscillator, with the operators of position, momentum and Hamiltonian being
functions of elements of the q-algebra su_q(2). The spectrum of position in
this discrete system, in a fixed representation j, consists of 2j+1
"sensor"-points x_s=(1/2)[2s]_q, s=-j, -j+1,..., j, and similarly for the
momentum observable. The spectrum of energies is finite and equally spaced, so
the system supports coherent states. The wave functions involve dual q-Kravchuk
polynomials, which are solutions to a finite-difference Schrodinger equation.
Time evolution (times a phase) defines the fractional Fourier-q-Kravchuk
transform. In the classical limit q -> 1 we recover the finite oscillator Lie
algebra, the N=2j -> infinity limit returns the Macfarlane-Biedenharn
q-oscillator and both limits contract the generators to the standard
quantum-mechanical oscillator.
