Eigenvalues of one family of tridiagonal skew-self-adjoint Toeplitz matrices with complex perturbations on the corner
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arXiv
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| Format: | Preprint |
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2026
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| author | Bernardin, C. Grudsky, S. M. Maximenko, E. A. Soto-González, A. |
| author_facet | Bernardin, C. Grudsky, S. M. Maximenko, E. A. Soto-González, A. |
| contents | In this paper, we study the eigenvalues of the matrices $T_n(a)+γE_{n,1,1}$ where $T_n(a)$ is the Toeplitz matrix with generating symbol $a(t)=t-t^{-1}$, $E_{n,1,1}$ is the $n\times n$ matrix whose upper left component is $1$ and the other components are zero, and $γ$ is a fixed complex number such that $0<|γ|<1$. As $n\to\infty$, the eigenvalues of these matrices are asymptotically distributed as the function $2 i \sin(x)$, $x\in[0,2π]$. Our main result is an asymptotic formula for every eigenvalue with a residue of the order $O(1/n^3)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_05669 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Eigenvalues of one family of tridiagonal skew-self-adjoint Toeplitz matrices with complex perturbations on the corner Bernardin, C. Grudsky, S. M. Maximenko, E. A. Soto-González, A. Spectral Theory 15B05, 15A18, 41A60, 47B36, 65F15 In this paper, we study the eigenvalues of the matrices $T_n(a)+γE_{n,1,1}$ where $T_n(a)$ is the Toeplitz matrix with generating symbol $a(t)=t-t^{-1}$, $E_{n,1,1}$ is the $n\times n$ matrix whose upper left component is $1$ and the other components are zero, and $γ$ is a fixed complex number such that $0<|γ|<1$. As $n\to\infty$, the eigenvalues of these matrices are asymptotically distributed as the function $2 i \sin(x)$, $x\in[0,2π]$. Our main result is an asymptotic formula for every eigenvalue with a residue of the order $O(1/n^3)$. |
| title | Eigenvalues of one family of tridiagonal skew-self-adjoint Toeplitz matrices with complex perturbations on the corner |
| topic | Spectral Theory 15B05, 15A18, 41A60, 47B36, 65F15 |
| url | https://arxiv.org/abs/2605.05669 |