Self-adjoint \(A\Delta O_s\) with vanishing reflection (Q1347434)

In the present survey, the author reviews his earlier works on a class of ordinary linear second-order analytic difference operators \((A \Delta Os)\) admitting reflectionless eigenfunctions [J. Math. Phys. 40, No. 3, 1627-1663 (1999; Zbl 0985.34080); Publ. Res. Inst. Math. Sci. 36, No. 6, 707-753 (2000; Zbl 1006.81019); J. Nonlinear Math. Phys. 8, No. 1, 106-138 (2001; Zbl 0973.35180) and ibid. 8, Suppl., 240-248 (2001; Zbl 0977.39011) or to appear in J. Nonlin. Math. Phys.]. This operator class under review is far more extensive than the Schrödinger and Jacobi operators corresponding to KdV and Toda lattice solitons. A subclass of reflectionless \(A\Delta Os\) is shown to correspond to soliton solutions of a nonlocal Toda-type evolution equation. Further restrictions 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 the \(A\Delta Os\).