Asymptotics of the eigenvalues of seven-diagonal Toeplitz matrices of a special form

TL;DR

This paper derives uniform asymptotic formulas for all eigenvalues of large 7-diagonal symmetric Toeplitz matrices with specific real-valued generating functions, extending previous simple-loop case analyses.

Contribution

It introduces new asymptotic formulas and nonlinear equations for eigenvalues of complex 7-diagonal Toeplitz matrices with special generating functions.

Findings

Derived uniform asymptotic formulas for eigenvalues.

Established nonlinear equations with complex structure.

Extended analysis beyond simple-loop cases.

Abstract

We find uniform asymptotic formulas for all the eigenvalues of certain 7-diagonal symmetric Toeplitz matrices of large dimension. The entries of the matrices are real and we consider the case where the real-valued generating function such that its first five derivatives at the one endpoint of interval are equal zero. This is not the simple-loop case considered earlier. We obtain nonlinear equations for the eigenvalues. It should be noted that our equations have a more complicated structure than the equations for the simple loop case.