Eigenvalues of even very nice Toeplitz matrices can be unexpectedly erratic

TL;DR

This paper demonstrates that eigenvalues of certain Toeplitz matrices can behave unpredictably, especially when the generating symbol does not meet specific regularity conditions, challenging previous assumptions about their asymptotic behavior.

Contribution

It provides a counter-example showing the necessity of the simple-loop condition for regular eigenvalue asymptotics in Toeplitz matrices.

Findings

Eigenvalues of non-simple-loop Toeplitz matrices lack regular asymptotic expansions.

Counter-example to a conjecture by Ekström, Garoni, and Serra-Capizzano.

Highlights the importance of the simple-loop condition for eigenvalue asymptotics.

Abstract

It was shown in a series of recent publications that the eigenvalues of $n\times n$ Toeplitz matrices generated by so-called simple-loop symbols admit certain regular asymptotic expansions into negative powers of $n+1$. On the other hand, recently two of the authors considered the pentadiagonal Toeplitz matrices generated by the symbol $g(x)=(2\sin(x/2))^4$, which does not satisfy the simple-loop conditions, and derived asymptotic expansions of a more complicated form. We here use these results to show that the eigenvalues of the pentadiagonal Toeplitz matrices do not admit the expected regular asymptotic expansion. This also delivers a counter-example to a conjecture by Ekstr\"{o}m, Garoni, and Serra-Capizzano and reveals that the simple-loop condition is essential for the existence of the regular asymptotic expansion.