Eigenvalues of the tetradiagonal Toeplitz matrices with diagonals 1, 0, 0, 1

Abstract

We perform a thorough analysis of the eigenvalues of tetradiagonal Toeplitz matrices of large order $n$ generated by the Laurent polynomial $a(t)=t^2+t^{-1}$. The spectra of these matrices are invariant under $2\pi/3$-rotation. They are contained in three segments of the complex plane and asymptotically fill these segments as $n$ tends to infinity. We apply Widom's formula for the determinants and transform the characteristic equation into a convenient form that can be solved by the fixed point iteration method. After that, we compute the asymptotic distribution of the eigenvalues. The main results are asymptotic formulas for the eigenvalues, both close to the origin and far from the origin. The main results are verified by numerical tests for moderate values of $n$.