  In the previous two parts of this series of papers, we introduced and studied
a large class of analytic difference operators admitting reflectionless
eigenfunctions, focusing on algebraic and function-theoretic features in the
first part, and on connections with solitons in the second one. In this third
part we study our difference operators from a quantum mechanical viewpoint. We
show in particular that for an arbitrary difference operator $A$ from a certain
subclass, the reflectionless $A$-eigenfunctions can be used to construct an
unbounded self-adjoint reflectionless operator $\hat{A}$ on $L^2({\mathbb
R},dx$ whose action on a suitable core coincides with that of $A$.
