Expansions for Eigenfunction and Eigenvalues of large-n Toeplitz Matrices

This note starts from work done by Dai, Geary, and Kadanoff (Hui Dai, Zachary Geary, and Leo P. Kadanoff, H. Dai, Z. Geary and L. P. Kadanoff, Journal of Statistical Mechanics, P05012 (2009)) on exact eigenfunctions for Toeplitz operators. It builds methods for finding convergent expansions for eigenvectors and eigenvalues of large-n Toeplitz matrices, using the infinite-n case as a starting point. One expansion is derived from operator equations having a two-dimensional continuous spectrum of eigenvalues, which include the eigenvalues of the finite-$n$ matrices. Another expansion is derived from the transpose equations, which have no eigenvalues at all. The two expansions work together to give an apparently convergent expansion with an expansion parameter expressed as an inverse power of n. A variational principle is developed which gives an approximate expression for determining eigenvalues. A consistency condition is generated, which gives to lowest order exactly the same condition for the eigenvalue.