Solvable Discrete Quantum Mechanics: q-Orthogonal Polynomials with |q|=1 and Quantum Dilogarithm

TL;DR

This paper introduces new solvable discrete quantum mechanical systems featuring q-orthogonal polynomials with |q|=1, whose orthogonality weights involve quantum dilogarithm functions, extending classical gamma functions.

Contribution

It constructs novel q-orthogonal polynomials with |q|=1 as eigenfunctions of discrete quantum systems, linking them to quantum dilogarithms and Askey-Wilson polynomials.

Findings

Orthogonal spaces have finite dimensions.

Polynomials expressed via Askey-Wilson polynomials and limits.

Orthogonality weights involve quantum dilogarithm functions.

Abstract

Several kinds of q-orthogonal polynomials with |q|=1 are constructed as the main parts of the eigenfunctions of new solvable discrete quantum mechanical systems. Their orthogonality weight functions consist of quantum dilogarithm functions, which are a natural extension of the Euler gamma functions and the q-gamma functions (q-shifted factorials). The dimensions of the orthogonal spaces are finite. These q-orthogonal polynomials are expressed in terms of the Askey-Wilson polynomials and their certain limit forms.