Localization for a class of discrete long-range quasi-periodic operators

In this paper we study discrete quasi-periodic operators with certain long-range hopping and monotonic meromorphic potentials. The hopping amplitude decays with the interparticle distance vertical bar n vertical bar as e(-rlogt(1+vertical bar n vertical bar )) (t > 1, r > 0, n is an element of Z(v)). By employing the KAM method, we prove such operators have pure point spectrum with eigenfunctions {phi(i)}(i)(is an element of Zv) obeying vertical bar phi(i)(n)vertical bar <= 2e(-r/2)log(t)(1+vertical bar n-i vertical bar) for all i is an element of Z(v), n is an element of Z(v).