Canonical discretization I. Discrete faces of (an)harmonic oscillator

A certain notion of canonical equivalence in quantum mechanics is proposed. It is used to relate quantal systems with discrete ones. Discrete systems canonically equivalent to the celebrated harmonic oscillator as well as the quartic and the quasiexactly-solvable anharmonic oscillators are found. Th ey can be viewed as a translation-covariant discretization of the (an)harmo nic oscillator preserving isospectrality. The notion of the q-deformation o f the canonical equivalence leading to a dilatation-covariant discretizatio n preserving polynomiality of eigenfunctions is also presented.