Eigenvalues of Toeplitz matrices emerging from finite differences for certain ordinary differential operators

Abstract

We consider Hermitian Toeplitz matrices emerging from finite linear combinations with non-negative coefficients of the differential operators ${(-1)}^k{d}^{2k}/d{x}^{2k}$ over the interval $(0,1)$ after discretizing them on a uniform grid of step size $1/(n+1)$. The collective distribution in the Szegő–Weyl sense of the eigenvalues of these matrices as n goes to infinity can be described by GLT theory. However, we focus on the asymptotic behavior of the individual eigenvalues, on both the inner eigenvalues in the bulk and on the extreme eigenvalues. The difficulty of the problem is that not only the order of the matrices depends on n but also their so-called symbols. Our main results are third order asymptotic formulas for the eigenvalues in the case $k⩽2$. These results reveal some basic phenomena one should expect when considering the problem in full generality.