Consider the Laplace operator over a 1D domain with homogeneous Neumann B.C.s. The cell-centered Finite Volumes discretization of this operator on a uniform grid looks as follows, which is a well-known result: \begin{equation} L_n = \frac{1}{h}\begin{bmatrix} -1 & 1\\ 1 & -2 &1\\ &\ddots &\ddots &\ddots\\ & & 1 & -2 & 1\\ & & & 1 & -1 \end{bmatrix}\in\mathbb{C}^{n\times n}. \end{equation}
My question is whether there exists a closed form solution for the eigenvalues of $L_n$. I am aware that closed forms exist for the periodic and homogeneous Dirichlet cases, but I haven't been able to find anything for the homogeneous Neumann case. I also haven't been able to make much progress in terms of a proof, but I have found that given $x\in\mathbb{C}^n$: \begin{equation} x^*L_nx = \frac{1}{h}\sum_{k=1}^{n-1} |x_k - x_{k+1}|^2. \end{equation}
Does anyone have a suggestion of where to look or how to proceed with a proof?
Thanks in advance!
After some suggested reading and a few attempts, the eigenvectors turned out to be: \begin{align} \xi_k = \begin{bmatrix} \cos\left(\frac{\pi k}{n}\left(j+\frac{1}{2}\right)\right) \end{bmatrix}_{\forall j\in\{0\dots n-1\}},\quad \forall k\in\{0\dots n-1\}, \end{align}
with corresponding eigenvalues: \begin{align} \lambda_k = \frac{2}{h}\left(\cos\left(\tfrac{\pi k}{n}\right)-1\right), \quad\forall k\in\{0\dots n-1\}. \end{align}