4 Ju l 2 00 3 Darboux transformations for 5 - point and 7 - point self - adjoint schemes and

With this paper we begin an investigation of difference schemes that possess Darboux transformations and can be regarded as natural discretizations of elliptic partial differential equations. We construct, in particular, the Darboux transformations for the general self ad-joint schemes with five and seven neighbouring points. We also introduce a distinguished discretization of the two-dimensional stationary Schrödinger equation, described by a 5-point difference scheme involving two potentials, which admits a Darboux transformation.