On the existence of eigenvalues of the one particle discrete Schrödinger operators

We study the spectral properties of the Schrödinger-type operator H^=H^0+V^, describing the motion of one quantum particle in the one-dimensional lattice ℤ in an indefinite sign external field v^. We obtain sufficient conditions for the existence of eigenvalues of the operator H^ lying on either below the bottom of the essential spectrum or above its top.