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.