There is a vast literature on Jacobi matrices, I just don't know where to start looking. I'm interested in estimating the largest eigenvalue of the $n\times n$ periodic Jacobi matrix $D+P+P^{-1}$, where $P$ is the matrix of the cyclic permutation of coordinates, and $D$ is the diagonal matrix with diagonal entries $2\cos\frac{2k\pi}{n}$ (where $0\leq k\leq n-1$). I can't help thinking that this has been already considered...



Largest eigenvalue of a periodic Jacobi matrix
It's roughly $2 - O(1/n^2)$. For an appropriate choice of $K$ evaluate $(D + P + P^{-1}) u$ for
$$
 u(k) = \begin{cases} 1 - \frac{|k|}{K} & |k| \leq K \\\ 0 & otherwise.\end{cases}
$$
An application of the uncertainty principle shows that this is optimal up to constants.


The main point why this works is that $2 \cos( 2 \pi \frac{k}{n})$ is almost constant in $k$. If you would instead take for some $\ell$ coprime to $n$ the potential $2 \cos(2 \pi \frac{\ell k}{n})$. The question is much harder.


