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

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 compact operator from $\ell^{\infty,-\beta}(\mathbb{Z})$ to $\ell^{1,\beta}(\mathbb{Z})$ for some $\beta\geq1$. We give a complete description of the solutions to $Hx=0$, and $Hx=4x$, $x\in\ell^{\infty,-\beta}(\mathbb{Z})$. 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.