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 $0\leq k\leq n1$). I can't help thinking that this has been already considered...
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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. 

