H φ = (-Δ + V)φ = -(φn+1 + φn-1 - 2 φn) + Vn φn.We show that under appropriate decay conditions on the general potential (and a non-resonance condition at the spectral edges), the spectrum of H consists of finitely many eigenvalues of finite multiplicities and the essential (absolutely continuous) spectrum, while the resolvent satisfies the limiting absorption principle and the Puiseux expansions near the edges. These properties imply the dispersive estimates
| ei t H Pa.c.(H) |l2σ -> l2-σ < t-3/2for any fixed σ > 5/2 and any t > 0, where Pa.c.(H) denotes the spectral projection to the absolutely continuous spectrum of H. In addition, based on the scattering theory for the discrete Jost solutions and the previous results, we find new dispersive estimates
| ei t H Pa.c.(H) |l1 -> linf < t-1/3These estimates are sharp for the discrete Schrodinger operators even for V = 0.