Discrete repulsive oscillator wavefunctions

Abstract

For the study of infinite discrete systems on phase space, the three-dimensional Lorentz algebra and group, and , provide a discrete model of the repulsive oscillator. Its eigenfunctions are found in the principal irreducible representation series, where the compact generator—that we identify with the position operator—has the infinite discrete spectrum of the integers , while the spectrum of energies is a double continuum. The right- and left-moving wavefunctions are given by hypergeometric functions that form a Dirac basis for . Under contraction, the discrete system limits to the well-known quantum repulsive oscillator. Numerical computations of finite approximations raise further questions on the use of Dirac bases for infinite discrete systems.