Self-adjoint A Delta Os with vanishing reflection

We review our work concerning ordinary linear second-order analytic differe nce operators (A Delta Os) that admit reflectionless eigenfunctions. This o perator class is far more extensive than the reflectionless Schrodinger and Jacobi operators corresponding to KdV and Toda lattice solitons. A subclas s of reflectionless A Delta Os, which generalizes the latter class of diffe rential and discrete difference operators, is shown to correspond to the so liton solutions of a nonlocal Toda-type evolution equation. Further restric tions give rise to A Delta Os with isometric eigenfunction transformations, which can be used to associate self-adjoint operators on L-2(R, dx) with t he A Delta Os.