On Asymptotics of Eigenvalues of Seven-Diagonal Toeplitz Matrices

Asymptotic formulas are derived that admit a uniform estimate of the remainder for Toeplitz matrices of size n as n -> infinity in the case when their symbol a(t) has the form a(t) = (t - 2a(0) + t(-1/3)). This result is a generalization of the result of Stukopin et al. (2021), who obtained similar asymptotic formulas for a seven-diagonal Toeplitz matrix with a similar symbol in the case a(0) = 1. The resulting formulas are of high computational efficiency and generalize the classical results of Parter and Widom on asymptotics of extreme eigenvalues.