Asymptotics of eigenvalues for Toeplitz matrices with rational symbols that have a minimum of the 4th order

Barrera M, Grudsky SM. Asymptotics of eigenvalues for pentadiagonal symmetric Toeplitz matrices. In: Large truncated Toeplitz matrices, toeplitz operators, and related topics. Operator theory: advances and applications Vol. 259, Birkhauser, Cham.; 2017; p. 51-77. we have considered the problem about asymptotic formulas for all eigenvalues of T-n(a), as n goes to infinity, assuming that a is a specific model symbol with a unique zero of order 4. In this paper, we continue our investigation and we explore the case where a is a more general real-valued rational symbol with a unique zero of order 4. It should be noted that we apply a different method than the one used in Barrera M, Grudsky SM. Asymptotics of eigenvalues for pentadiagonal symmetric Toeplitz matrices. In: Large truncated Toeplitz matrices, Toeplitz operators, and related topics. Operator theory: advances and applications Vol. 259, Birkhauser, Cham.; 2017; p. 51-77. This method coming from works Bogoya JM, Bottcher A, Grudsky SM, et al. Eigenvalues of Hermitian Toeplitz matrices with smooth simple-loop symbols. J Math Anal Appl. 2015;422(2):1308-1334 and Bogoya JM, Bottcher A, Grudsky SM, et al. Eigenvalues of Hermitian Toeplitz matrices generated by simple-loop symbols with relaxed smoothness. In: Large truncated Toeplitz matrices, Toeplitz operators, and related topics. Operator theory: advances and applications Vol. 259, Birkhauser, Cham.; 2017. p. 179- 212, where it is considered the class of all symbols having zeros of second order and one can reduce the problem to asymptotic analysis of a nonlinear equation. As well, we construct uniform asymptotic expansions for all eigenvalues, which allow us to precise the classical results of Widom and Parter for first and very last eigenvalues.