For $x\in \mathbb{R}^\mathbb{Z}$, let the discrete Laplacian be defined as \begin{align*} (\Delta x)_k = 2x_k-x_{k+1}-x_{k-1}. \end{align*}

I am looking for good references about its spectrum (or eigen-structure), compactness, and properties of the semigroup it generates.

The spaces to consider include the usual $l^p(\mathbb{Z})$, for my purpose, it is also interesting to work on weighted space $l^p_\rho(\mathbb{Z})$ in which, provided a summable weight $\rho$ with $\sum_k \rho_k <\infty$, the norm is given by $$\|x\|^p_{p,\rho}= \sum_{k\in\mathbb{Z}} |x_k|^p\rho_k.$$