Reflectionless Potentials for Difference Schr\"odinger Equations

TL;DR

This paper constructs reflectionless potentials for difference Schrödinger equations with pure imaginary shifts, linking them to $q$-ultraspherical polynomials and proposing a conjecture for hypergeometric connection formulas at $|q|=1$, advancing solvable models in discrete quantum mechanics.

Contribution

It introduces new reflectionless potentials for difference Schrödinger equations and proposes a conjecture for hypergeometric connection formulas at $|q|=1$, connecting special functions with quantum dilogarithms.

Findings

Derived transmission and reflection amplitudes with desirable properties.

Constructed discrete analogues of known continuous potentials.

Supported the conjecture through properties of the amplitudes.

Abstract

As a part of the program `discrete quantum mechanics,' we present general reflectionless potentials for difference Schr\"odinger equations with pure imaginary shifts. By combining contiguous integer wave number reflectionless potentials, we construct the discrete analogues of the $h(h+1)/\cosh^2x$ potential with the integer $h$, which belong to the recently constructed families of solvable dynamics having the $q$-ultraspherical polynomials with $|q|=1$ as the main part of the eigenfunctions. For the general ($h\in\mathbb{R}_{>0}$) scattering theory for these potentials, we need the connection formulas for the basic hypergeometric function ${}_2\phi_1(\genfrac{}{}{0pt}{}{a,b}{c}|q;z)$ with $|q|=1$, which is not known. The connection formulas are expected to contain the quantum dilogarithm functions as the $|q|=1$ counterparts of the $q$-gamma functions. We propose a conjecture of the connection formula of the ${}_2\phi_1$ function with $|q|=1$. Based on the conjecture, we derive the transmission and reflection amplitudes, which have all the desirable properties. They provide a strong support to the conjectured connection formula.