On the near periodicity of eigenvalues of Toeplitz matrices
Levitin, M., Sobolev, A. and Sobolev, D. (2010) On the near periodicity of eigenvalues of Toeplitz matrices. In: Operator Theory and Its Applications: In Memory of V. B. Lidskii (1924-2008). American Mathematical Society Translations - Series 2, Advances in the Mathematical Sciences (231). American Mathematical Society, Providence, RI, pp. 115-126. ISBN 9780821852729
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Let $A$ be an infinite Toeplitz matrix with a real symbol $f$ defined on $[-\pi, \pi]$. It is well known that the sequence of spectra of finite truncations $A_N$ of $A$ converges to the convex hull of the range of $f$. Recently, Levitin and Shargorodsky, on the basis of some numerical experiments, conjectured, for symbols $f$ with two discontinuities located at rational multiples of $\pi$, that the eigenvalues of $A_N$ located in the gap of $f$ asymptotically exhibit periodicity in $N$, and suggested a formula for the period as a function of the position of discontinuities. In this paper, we quantify and prove the analog of this conjecture for the matrix $A^2$ in a particular case when $f$ is a piecewise constant function taking values $-1$ and $1$.