We consider a wide class of Schrödinger operators H H V V = 0  describing a particle in an external force field v ˆ in the d -dimensional cubic integer lattice , 3 d d  . We study the existence or absence of bound states of the one-particle Schrödinger operator HV depending on the potential v ˆ and the dimension of the lattice , 3 d d  . We obtain some sufficient conditions for the existence of eigenvalues of the operator HV lying below the essential spectrum.