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29 March 2024
 
  » arxiv » 2009.01401

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Eigenvalues of tridiagonal Hermitian Toeplitz matrices with perturbations in the off-diagonal corners
Sergei M. Grudsky ; Egor A. Maximenko ; Alejandro Soto-González ;
Date 3 Sep 2020
AbstractIn this paper we study the eigenvalues of Hermitian Toeplitz matrices with the entries $2,-1,0,ldots,0,-alpha$ in the first column. Notice that the generating symbol depends on the order $n$ of the matrix. If $|alpha|le 1$, then the eigenvalues belong to $[0,4]$ and are asymptotically distributed as the function $g(x)=4sin^2(x/2)$ on $[0,pi]$. The situation changes drastically when $|alpha|>1$ 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 $alpha$, 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.
Source arXiv, 2009.01401
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