Chain models and the spectra of tridiagonal k-Toeplitz matrices

Abstract

Chain models can be represented by a tridiagonal matrix with periodic entries along its diagonals. Eigenmodes of open chains are represented by spectra of such tridiagonal $k$-Toeplitz matrices, where $k$ represents length of the repeated unit, allowing for a maximum of $k$ distinct types of elements in the chain. We present an analysis that allows for generality in $k$ and values in $\mathbb{C}$ representing elements of the chain, including non-Hermitian systems. Numerical results of spectra of some special $k$-Toeplitz matrices are presented as a motivation. This is followed by analysis of a general tridiagonal $k$-Toeplitz matrix of increasing dimensions, beginning with 3-term recurrence relations between their characteristic polynomials involving a $k^{th}$ order coefficient polynomial, with the variables and coefficients in $\mathbb{C}$. The existence of limiting zeros for these polynomials and their convergence are established, and the conditioned $k^{th}$ order coefficient polynomial is shown to provide a continuous support for the limiting spectra representing modes of the chain. This analysis also includes the at most $2k$ eigenvalues outside this continuous set. It is shown that this continuous support can as well be derived using Widom's conditional theorems (and its recent extensions) for the existence of limiting spectra for block-Toeplitz operators, except in special cases. Numerical examples are used to graphically demonstrate theorems. As an addendum, we derive expressions for $O(k)$ computation of the determinant of tridiagonal $k$-Toeplitz matrices of any dimension