Reflectionless Analytic Difference Operators III. Hilbert Space Aspects

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  on L(R, dx), whose action on a suitable core coincides with that of A.