On The Discrete Spectra of Schrodinger-Type Operators on one Dimensional Lattices

We consider a family (H) over cap mu = (H) over cap (0) + mu(V) over cap, mu > 0, of Schrodinger-type operators on the one dimensional lattice Z, where (H) over cap (0 )is a Laurent-Toeplitz-type convolution operator with a given Hopping matrix (e) over cap and (V) over cap is a potential taking into account only the zero-range and one-range interactions, i.e., a multiplication operator by a function (v) over cap such that (v) over cap (0) = a, (v) over cap (x) = b for vertical bar x vertical bar = 1 and (v) over cap (x) = 0 for vertical bar x vertical bar >= 2, where a, b is an element of N \ {0}. Under certain conditions on the regularity of (e) over cap we completely describe the discrete spectrum of (H) over cap (mu), lying above the essential spectrum and study the dependence of eigenvalues on parameters mu, a and b.