**Abstract:**

Based on the recent work for compact potentials, we
develop the spectral theory for the one-dimensional discrete
Schrodinger operator
H φ = (-Δ + V)φ = -(φ_{n+1} + φ_{n-1}
- 2 φ_{n}) + V_{n} φ_{n}.

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
| e^{i t H} P_{a.c.}(H) |_{l2σ ->
l2-σ} < t^{-3/2}

for any fixed σ > 5/2 and any t > 0, where P_{a.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
| e^{i t H} P_{a.c.}(H) |_{l1 ->
linf} < t^{-1/3}

These estimates are sharp for the discrete Schrodinger operators
even for V = 0.