Eigenvalues of Hermitian Toeplitz matrices with Fisher--Hartwig symbols

We investigate the eigenvalues of Hermitian Toeplitz matrices generated by the symmetric Fisher-Hartwig symbol $\widehat{a}(z)=(1-z{)}^{\alpha \mathrm{/}2}(1-z^{-1}{)}^{\alpha \mathrm{/}2}$ for $\alpha \in (0,2)$. 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 $\alpha >1$, we bridge this gap by covering the full range of $\alpha \in (0,2)$. Our approach uses a construction of approximate eigenvectors that has not previously been applied in this setting.