Self-Adjoint A$\Delta$Os with Vanishing Reflection

We review our work concerning ordinary linear second-order analytic difference operators (A$\Delta$Os) that admit reflectionless eigenfunctions. This operator class is far more extensive than the reflectionless Schrödinger and Jacobi operators corresponding to KdV and Toda lattice solitons. A subclass of reflectionless A$\Delta$Os, which generalizes the latter class of differential and discrete difference operators, is shown to correspond to the soliton solutions of a nonlocal Toda-type evolution equation. Further restrictions give rise to A$\Delta$Os with isometric eigenfunction transformations, which can be used to associate self-adjoint operators on $L^2(\mathbb R,dx)$ with the A$\Delta$Os.