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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 admitcertain regular asymptotic expansions into negative powers of $n+1$. On theother hand, recently two of the authors considered the pentadiagonal Toeplitzmatrices generated by the symbol $g(x)=(2\sin(x/2))^4$, which does not satisfythe simple-loop conditions, and derived asymptotic expansions of a morecomplicated form. We here use these results to show that the eigenvalues of thepentadiagonal Toeplitz matrices do not admit the expected regular asymptoticexpansion. This also delivers a counter-example to a conjecture by Ekstr\"{o}m,Garoni, and Serra-Capizzano and reveals that the simple-loop condition isessential for the existence of the regular asymptotic expansion.

Eigenvalues of even very nice Toeplitz matrices can be unexpectedly erratic