Eigenvalue asymptotic expansion of large tetradiagonal Toeplitz matrices: cusp case

Ponente(s): Alejandro Soto González, Dr. Sergei Grudsky

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 limiting Schmidt–Spitzer set for tetradiagonal Toeplitz matrices. In this study, we consider one of these possible cases, when the limiting 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 nfinity. Our analysis provides a theoretical understanding of the structural behavior of the eigenvalues, while the obtained formulas enable high-order precision calculation of the eigenvalues. This is a joint work with Dr. Sergei M. Grudsky and Dr. Anatolii V. Kozak. This research has been supported by SECIHTI (Mexico), project “Ciencia de Frontera” FORDECYT-PRONACES/61517/2020, and by Regional Mathemati-cal Center of the Southern Federal University with the support of the Ministry of Science and Higher Education of Russia, Agreement 075-02-2025-1720.