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

Darboux transformations and second order difference equations