We consider the discrete one-dimensional Schr\"{o}dinger operator H=H0+V,
where (H0x)[n]=−(x[n+1]+x[n−1]−2x[n]) and V is a self-adjoint operator on
ℓ2(Z) with a decay property given by V extending to a compact
operator from ℓ∞,−β(Z) to
ℓ1,β(Z) for some β≥1. We give a complete
description of the solutions to Hx=0, and Hx=4x,
x∈ℓ∞,−β(Z). Using this description we give
asymptotic expansions of the resolvent of H at the two thresholds 0 and
4. One of the main results is a precise correspondence between the solutions
to Hx=0 and the leading coefficients in the asymptotic expansion of the
resolvent around 0. For the resolvent expansion we implement the expansion
scheme of Jensen-Nenciu \cite{JN0, JN1} in the full generality.Comment: 51 page