Diagonalization discrete laplace operator

Question

How do i diagonalize the discrete laplace-operator $\triangle$ on $l^2(\mathbb{Z})$ (defined by e.g. $\triangle e_k = e_{k-1} + e_{k+1}$, with $e_{k}$ being the canoical basisvectors of $l^2(\mathbb{Z})$? I showed that $\triangle$ is linear, bounded and selfadjoint but I can't figure out how exactly to calculate the diagonalization. Do I have to calculate $\mathcal{F} \triangle \mathcal{F}^{-1}$ with $\mathcal{F} : l^2(\mathbb{Z}) \rightarrow L^2([0,2\pi])$, $(\mathcal{F} f)(x) = \sum_{k \in \mathbb{Z}} e^{ikx} f(k)$?