A discrete Schrödinger spectral problem and associated evolution equations

A recently proposed discrete version of the Schrodinger spectral problem is considered. The whole hierarchy of differential-difference nonlinear evolution equations associated with this spectral problem is derived. It is shown that a discrete version of the KdV, sine-Gordon and Liouville equations is included and that the so-called 'inverse' class in the hierarchy is local. The whole class of related Darboux and Backlund transformations is also exhibited.