Self-Adjoint AΔOs with Vanishing Reflection

Abstract

We review our work concerning ordinary linear second-order analytic difference operators (AΔOs) that admit reflectionless eigenfunctions. This operator class is far more extensive than the reflectionless Schrodinger and Jacobi operators corresponding to KdV and Toda lattice solitons. A subclass of reflectionless AΔOs, 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 AΔOs with isometric eigenfunction transformations, which can be used to associate self-adjoint operators on \(L^2 \left( {\mathbb{R},dx} \right)\) with the AΔOs.