A q-Oscillator with 'Accidental' Degeneracy of Energy Levels

We study main features of the exotic case of q-deformed oscillators (so-called Tamm-Dancoff cutoff oscillator) and find some special properties: (i) degeneracy of the energy levels E_{n_1} = E_{n_1+1}, n_1\ge 1, at the {\em real value} q=\sqrt{\frac{n_1}{n_1+2}} of deformation parameter, as well as the occurrence of other degeneracies E_{n_1} = E_{n_1+k}, for k \ge 2, at the corresponding values of q which depend on both n_1 and k; (ii) the position and momentum operators X and P {\em commute on the state} |m> if q is fixed as q=\frac{m}{m+1}, that implies unusual uncertainty relation; (iii) two commuting copies of the creation, annihilation, and number operators of this q-oscillator generate the corresponding q-deformation of the {\em non-simple} Lie algebra su(2)\oplus u(1) whose nontrivial q-deformed commutation relation is: [ J_+, J_- ] = 2 J_0 q^{2J_3-1} where J_0\equiv \frac12 (N_1-N_2) and J_3\equiv \frac12 (N_1+N_2).