Eigenvalues of tridiagonal Hermitian Toeplitz matrices with perturbations in the off-diagonal corners
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arXiv
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| Natura: | Preprint |
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2020
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| author | Grudsky, Sergei M. Maximenko, Egor A. Soto-González, Alejandro |
| author_facet | Grudsky, Sergei M. Maximenko, Egor A. Soto-González, Alejandro |
| contents | In this paper we study the eigenvalues of Hermitian Toeplitz matrices with the entries $2,-1,0,\ldots,0,-α$ in the first column. Notice that the generating symbol depends on the order $n$ of the matrix. If $|α|\le 1$, then the eigenvalues belong to $[0,4]$ and are asymptotically distributed as the function $g(x)=4\sin^2(x/2)$ on $[0,π]$. The situation changes drastically when $|α|>1$ and $n$ tends to infinity. Then the two extreme eigenvalues (the minimal and the maximal one) lay out of $[0,4]$ and converge rapidly to certain limits determined by the value of $α$, whilst all others belong to $[0,4]$ and are asymptotically distributed as $g$. In all cases, we transform the characteristic equation to a form convenient to solve by numerical methods, and derive asymptotic formulas for the eigenvalues. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2009_01401 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Eigenvalues of tridiagonal Hermitian Toeplitz matrices with perturbations in the off-diagonal corners Grudsky, Sergei M. Maximenko, Egor A. Soto-González, Alejandro Functional Analysis Spectral Theory (2010): 15B05, 15A18, 41A60, 65F15, 47A55 In this paper we study the eigenvalues of Hermitian Toeplitz matrices with the entries $2,-1,0,\ldots,0,-α$ in the first column. Notice that the generating symbol depends on the order $n$ of the matrix. If $|α|\le 1$, then the eigenvalues belong to $[0,4]$ and are asymptotically distributed as the function $g(x)=4\sin^2(x/2)$ on $[0,π]$. The situation changes drastically when $|α|>1$ and $n$ tends to infinity. Then the two extreme eigenvalues (the minimal and the maximal one) lay out of $[0,4]$ and converge rapidly to certain limits determined by the value of $α$, whilst all others belong to $[0,4]$ and are asymptotically distributed as $g$. In all cases, we transform the characteristic equation to a form convenient to solve by numerical methods, and derive asymptotic formulas for the eigenvalues. |
| title | Eigenvalues of tridiagonal Hermitian Toeplitz matrices with perturbations in the off-diagonal corners |
| topic | Functional Analysis Spectral Theory (2010): 15B05, 15A18, 41A60, 65F15, 47A55 |
| url | https://arxiv.org/abs/2009.01401 |