We consider a family H ˆ , b ( ? ) = 0 + V > of Schrödinger-type operators on the two dimensional lattice Z 2 where is Laurent-Toeplitz-type convolution operator with given Hopping matrix e and potential taking into account only zero-range one-range interactions, i.e., multiplication by function v such that x for | 1 ? ? R ? { } . Under certain conditions regularity we completely describe discr...