Asymptotics of eigenvalues and eigenvectors of Toeplitz matrices

A Toeplitz matrix is one in which the matrix elements are constant along diagonals. The Fisher-Hartwig matrices are much-studied singular matrices in the Toeplitz family. The matrices are defined for all orders, N. They are parameterized by two constants, alpha and beta. Their spectrum of eigenvalues has a simple asymptotic form in the limit as N goes to infinity. Here we study the structure of their eigenvalues and eigenvectors in this limiting case. We specialize to the case with real alpha and beta and 0 < alpha < vertical bar beta vertical bar < 1, where the behavior is particularly simple. The eigenvalues are labeled by an index l which varies from 0 to N - 1. An asymptotic analysis using Wiener-Hopf methods indicates that for large N, the jth component of the lth eigenvector varies roughly in the fashion ln psi(l)(j) approximate to ip(l)j + O(1/N). The lth wavevector, p(l), varies as p(l) = 2 pi l/N + i(2 alpha + 1) In N/N + O(1/N) (I) for negative values of beta and values of l/(N - 1) not too close to zero or one. Correspondingly the lth eigenvalue is given by epsilon(l) = a(exp(-ip(l))) + o(1/N) (II) where a is the Fourier transform (also called the symbol) of the Toeplitz matrix. Note that p(l) has a small positive imaginary part. For values of j/N not too close to zero or one, this imaginary part acts to produce an eigenfunction which decays exponentially as j/N increases. Thus, the eigenfunction appears similar to that of a bound state, attached to a wall at j = 0. Near j = 0 this decay is modified by a set of bumps, probably not universal in character. For j/N above 0.6 the eigenfunction begins to oscillate in magnitude and shows deviations from the exponential behavior. The case of 0 < alpha < beta < 1 need not be studied separately. It can be obtained from the previous one by a 'conjugacy' transformation which takes psi(j) into psi(N-j-1). This 'conjugacy' produces interesting orthonormality relations for the eigenfunctions.