A New Class of Reflectionless Second-order $\mathrm{A}\Delta\mathrm{Os}$ and Its Relation to Nonlocal Solitons

We study an extensive class of second-order analytic difference operators admitting reflectionless eigenfunctions. The eigenvalue equation for our $\mathrm{A}\Delta\mathrm{Os}$ may be viewed as an analytic analog of a discrete spectral problem studied by Shabat. Moreover, the nonlocal soliton evolution equation we associate to the $\mathrm{A}\Delta\mathrm{Os}$ is an analytic version of a discrete equation Boiti and coworkers recently associated to Shabat's problem. We show that our nonlocal solitons $G(x,t)$ are positive for $(x,t) \in \mathbb{R}^2$ and obtain evidence that the corresponding $\mathrm{A}\Delta\mathrm{Os}$ can be reinterpreted as self-adjoint operators on $L^2(\mathbb{R},dx)$. In a suitable scaling limit the KdV solitons and reflectionless Schrodinger operators arise.