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The action of the quantum mechanical volume operator, introduced inconnection with a symmetric representation of the three-body problem andrecently recognized to play a fundamental role in discretized quantum gravitymodels, can be given as a second order difference equation which, by a complexphase change, we turn into a discrete Schr\"odinger-like equation. Theintroduction of discrete potential-like functions reveals the surprisingcrucial role here of hidden symmetries, first discovered by Regge for thequantum mechanical 6j symbols; insight is provided into the underlyinggeometric features. The spectrum and wavefunctions of the volume operator arediscussed from the viewpoint of the Hamiltonian evolution of an elementary"quantum of space", and a transparent asymptotic picture emerges of thesemiclassical and classical regimes. The definition of coordinates adapted toRegge symmetry is exploited for the construction of a novel set of discreteorthogonal polynomials, characterizing the oscillatory components oftorsion-like modes.

Hamiltonian dynamics of a quantum of space: hidden symmetries and spectrum of the volume operator, and discrete orthogonal polynomials