Discrete reflectionless potentials, quantum algebras, and \(q\)-orthogonal polynomials
Using Darboux transformations for the lattice Schrödinger equation, we construct two families of discrete reflectionless potentials with \(j\) and \(2j\) bound states (solitons). These potentials are related to the exceptional Askey-Wilson polynomials. In the continuous limit they are reduced to the well-known solvable potential \(u(x) = - j(j+1)/ \cosh^ 2 x\). The limit \(j \to \infty\) for the lattice systems may be defined in two different ways. In the first case, discrete spectrum grows exponentially fast, and the continuous spectrum band, \(-2 \leq \lambda \leq 2\), is preserved. In the second case, the limiting potentials are related to the \(q^{-1}\)-Hermite polynomials. Their spectra are purely discrete and consist of one or two geometric series accumulating near the zero. These series are generated by the \(q\)-Weyl algebra and its ``square root'' version. In contrast to the continuous reflectionless potentials with quantum algebraic symmetries, the derived discrete potentials are expressed in terms of the elementary functions.
scientific article; zbMATH DE number 727781