  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 $1$.

Spectral Theory