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We consider the discrete one-dimensional Schr\"{o}dinger operator $H=H_0+V$,where $(H_0x)[n]=-(x[n+1]+x[n-1]-2x[n])$ and $V$ is a self-adjoint operator on$\ell^2(\mathbb{Z})$ with a decay property given by $V$ extending to a compactoperator from $\ell^{\infty,-\beta}(\mathbb{Z})$ to$\ell^{1,\beta}(\mathbb{Z})$ for some $\beta\geq1$. We give a completedescription of the solutions to $Hx=0$, and $Hx=4x$,$x\in\ell^{\infty,-\beta}(\mathbb{Z})$. Using this description we giveasymptotic expansions of the resolvent of $H$ at the two thresholds $0$ and$4$. One of the main results is a precise correspondence between the solutionsto $Hx=0$ and the leading coefficients in the asymptotic expansion of theresolvent around $0$. For the resolvent expansion we implement the expansionscheme of Jensen-Nenciu \cite{JN0, JN1} in the full generality.

A complete classification of threshold properties for one-dimensional discrete Schr\"{o}dinger operators