  In this paper we implement the Darboux transformation, as well as an analogue
of Crum's theorem, for a discrete version of Schr\"odinger equation. The
technique is based on the use of first order operators intertwining two
difference operators of second order. This method, which has been applied
successfully for differential cases, leads also to interesting non trivial
results in the discrete case. The technique allows us to construct the
solutions for a wide class of difference Schr\"odinger equations. The exact
solutions for some special potentials are also found explicitly.
