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 En1 = En1 + 1, n1 ≥ 1, at the real value[Formula: see text] of deformation parameter, as well as the occurrence of other degeneracies En1 = En1 + k, for k ≥ 2, at the corresponding values of q which depend on both n1 and k; (ii) the position and momentum operators X and Pcommute on the state|n1> if q is fixed as [Formula: see text], 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 non-simple Lie algebra su(2) ⊕ u(1)whose nontrivial q-deformed commutation relation is: [J+, J-] = 2J0q2J3-1 where [Formula: see text] and [Formula: see text].
Hidden duality and accidental degeneracy in cycloacene and Möbius cycloacene