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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 | Abstract: | In 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 | Services: | Forum | Review | PDF | Favorites |
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