PreprintCC BY 4.0arXiv
Eigenvalues of Hermitian Toeplitz matrices with Fisher--Hartwig symbols
Abstract
We investigate the eigenvalues of Hermitian Toeplitz matrices generated by the symmetric Fisher-Hartwig symbol â(z) = (1−z)α/2(1−z−1)α/2 for α∈ (0,2). Using Dirichlet-Neumann bracketing of the discrete Laplacian, we establish explicit, non-asymptotic bounds for the individual eigenvalues. As a consequence, we prove that all eigenvalues are simple. We also obtain a two-term approximation for every eigenvalue, with explicit bounds on the remainder, as the size of the matrix tends to infinity. While known results are limited to α> 1, we bridge this gap by covering the full range of α∈ (0,2). Our approach uses a construction of approximate eigenvectors that has not previously been applied in this setting.
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BibTeX
@misc{wszola2026eigenvalues,
title = {{Eigenvalues of Hermitian Toeplitz matrices with Fisher--Hartwig symbols}},
author = {Jacek Wszoła},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.01498},
eprint = {2610.01498},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.01498},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Eigenvalues of Hermitian Toeplitz matrices with Fisher--Hartwig symbols
AU - Jacek Wszoła
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.01498
UR - https://arxiv.org/abs/2610.01498
AB - We investigate the eigenvalues of Hermitian Toeplitz matrices generated by the symmetric Fisher-Hartwig symbol â(z) = (1−z)^α/2(1−z^−1)^α/2 for α∈ (0,2). Using Dirichlet-Neumann bracketing of the discrete Laplacian, we establish explicit, non-asymptotic bounds for the individual eigenvalues. As a consequence, we prove that all eigenvalues are simple. We also obtain a two-term approximation for every eigenvalue, with explicit bounds on the remainder, as the size of the matrix tends to infinity. While known results are limited to α> 1, we bridge this gap by covering the full range of α∈ (0,2). Our approach uses a construction of approximate eigenvectors that has not previously been applied in this setting.
ER -