On the near periodicity of eigenvalues of Toeplitz matrices

TL;DR

This paper investigates the near periodic behavior of eigenvalues of finite Toeplitz matrix truncations, proving a conjecture about their asymptotic periodicity in a specific case with piecewise constant symbols.

Contribution

It provides a rigorous proof of the near periodicity of eigenvalues for the square of a Toeplitz matrix with a piecewise constant symbol, extending previous numerical conjectures.

Findings

Eigenvalues of $A_N^2$ exhibit asymptotic periodicity in $N$.

The periodicity relates to the discontinuities of the symbol.

The result confirms conjectures in a specific case with piecewise constant symbols.

Abstract

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 $1$.