Eigenvalue asymptotic expansion of large tetradiagonal Toeplitz matrices: cusp case

Abstract In a paper from 2021, Albrecht Böttcher, Juanita Gasca, Sergei M. Grudsky, and Anatoli V. Kozak gave a precise and complete description of all types of the limit Schmidt–Spitzer sets for tetradiagonal Toeplitz matrices. In this paper, we consider one of these possible cases, when the limit set consists of two analytic arcs that join at one point forming a cusp. For this family of Toeplitz matrices, we provide asymptotic formulas for every eigenvalue as the order of the matrix tends to infinity. Our analysis provides a theoretical understanding of the structural behavior of the eigenvalues, while the obtained formulas enable high-precision calculation of the eigenvalues.