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.Comment: 51 page

Hypergeometric expression for the resolvent of the discrete Laplacian in low dimensions

We present an explicit formula for the resolvent of the discrete Laplacian on the square lattice, and compute its asymptotic expansions around thresholds in low dimensions. As a by-product we obtain a closed formula for the fundamental solution to the discrete Laplacian. For the proofs we express the resolvent in a general dimension in terms of the Appell--Lauricella hypergeometric function of type $C$ outside a disk encircling the spectrum. In low dimensions it reduces to a generalized hypergeometric function, for which certain transformation formulas are available for the desired expansions