DISCRETE REFLECTIONLESS POTENTIALS, QUANTUM ALGEBRAS, AND Q-ORTHOGONAL POLYNOMIALS

Using Darboux transformations for the lattice Schrodinger equation, we construct two families of discrete reflectionless potentials with j a nd 2j bound states (solitons). These potentials are related to the exc eptional Askey-Wilson polynomials. In the continuous limit they are re duced to the well-known solvable potential u(x) = -j(j + 1)/cosh(2) x. The limit j --> infinity for the lattice systems may be defined in tw o different ways. In the first case, discrete spectrum grows exponenti ally fast, and the continuous spectrum band, -2 less than or equal to lambda less than or equal to 2, is preserved. In the second case, the limiting potentials are related to the q(-1)-Hermite polynomials. Thei r spectra are purely discrete and consist of one or two geometric seri es accumulating near the zero. These series are generated by the q-Wey l algebra and its ''square root'' version. In contrast to the continuo us reflectionless potentials with quantum algebraic symmetries, the de rived discrete potentials are expressed in terms of the elementary fun ctions. (C) 1995 Academic Press, Inc.