A finite quantum oscillator model related to special sets of Racah polynomials

TL;DR

This paper introduces a finite quantum oscillator model derived from special Racah polynomial doubles, providing explicit eigenvalues, wavefunctions, and illustrating their properties through plots.

Contribution

It presents a novel finite oscillator model based on Racah polynomial doubles, with explicit eigenvalues and wavefunctions, expanding the mathematical framework of quantum oscillators.

Findings

Derived a class of two-diagonal matrices with explicit eigenvalues

Constructed a finite quantum oscillator model from Racah doubles

Provided explicit discrete position wavefunctions and their properties

Abstract

In Oste and Van der Jeugt, SIGMA, 12 (2016) we classified all pairs of recurrence relations in which two (dual) Hahn polynomials with different parameters appear. Such pairs are referred to as (dual) Hahn doubles, and the same technique was then applied to obtain all Racah doubles. We now consider a special case concerning the doubles related to Racah polynomials. This gives rise to an interesting class of two-diagonal matrices with closed form expressions for the eigenvalues. Just as it was the case for (dual) Hahn doubles, the resulting two-diagonal matrix can be used to construct a finite oscillator model. We discuss some properties of this oscillator model, give its (discrete) position wavefunctions explicitly, and illustrate their behaviour by means of some plots.