Reflectionless Potentials for Difference Schr\"odinger Equations

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.