ASYMPTOTIC PSEUDOMODES OF TOEPLITZ MATRICES

Questions in probability and statistical physics lead to the problem of finding the eigenvectors associated with the extreme eigenvalues of Toeplitz matrices generated by Fisher-Hartwig symbols. We here simplify the problem and consider pseudomodes instead of eigenvectors. This replacement allows us to treat fairly general symbols, which are far beyond Fisher-Hartwig symbols. Our main result delivers a variety of concrete unit vectors x(n) such that if T-n(a) is the n x n truncation of the infinite Toeplitz matrix generated by a function a is an element of L-1 satisfying mild additional conditions and. is in the range of this function, then parallel to T-n(a)x(n) -lambda x(n)parallel to -> 0.