# Largest eigenvalue of a periodic Jacobi matrix

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...

-

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.
Don't you mean $4-O(1/n^2)$ ? By the way, let $A_\ell$ be the matrix where, as you suggest, you replace the potential $2\cos\frac{2\pi k}{n}$ by $2\cos\frac{2\pi k\ell}{n}$. Is it true that the maximum of the $\|A_\ell\|$'s is attained for $\ell=1$? –  Alain Valette May 21 '11 at 19:37
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9794032573699951, "perplexity": 93.89809518880388}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-52/segments/1419447556220.137/warc/CC-MAIN-20141224185916-00071-ip-10-231-17-201.ec2.internal.warc.gz"}