New Quasi Exactly Solvable Difference Equation

TL;DR

This paper demonstrates the exact solvability of certain discrete quantum mechanics models and introduces a novel quasi exactly solvable difference equation by combining two known dynamics, expanding the understanding of solvable quantum systems.

Contribution

It presents the first demonstration of exact solvability for specific discrete quantum models and constructs a new quasi exactly solvable difference equation by combining these models.

Findings

Exact solvability of Meixner-Pollaczek and continuous Hahn polynomials shown.

Construction of a new quasi exactly solvable difference equation.

Potential implications for quantum mechanics models not previously known.

Abstract

Exact solvability of two typical examples of the discrete quantum mechanics, i.e. the dynamics of the Meixner-Pollaczek and the continuous Hahn polynomials with full parameters, is newly demonstrated both at the Schroedinger and Heisenberg picture levels. A new quasi exactly solvable difference equation is constructed by crossing these two dynamics, that is, the quadratic potential function of the continuous Hahn polynomial is multiplied by the constant phase factor of the Meixner-Pollaczek type. Its ordinary quantum mechanical counterpart, if exists, does not seem to be known.