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

Abstract

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 quasi-exactly-solvable anharmonic oscillators are found. They can be viewed as a translation-covariant discretization of the (an)harmonic oscillator preserving isospectrality. The notion of the $q-$deformation of the canonical equivalence leading to a dilatation-covariant discretization preserving polynomiality of eigenfunctions is also presented.

29 pages, LaTeX

Cited by in corpus (5)

• One-Dimensional Quasi-Exactly Solvable Schrödinger Equations
• Umbral Calculus, Difference Equations and the Discrete Schroedinger Equation
• Canonical Commutation Relation Preserving Maps
• The quantum $H_4$ integrable system
• Quantum Many--Body Problems and Perturbation Theory