  We present a theory of Anderson localization on a one-dimensional lattice
with translation-invariant hopping. We find by analytical calculation, the
localization length for arbitrary finite-range hopping in the single
propagating channel regime. Then by examining the convergence of the
localization length, in the limit of infinite hopping range, we revisit the
problem of localization criteria in this model and investigate the conditions
under which it can be violated. Our results reveal possibilities of having
delocalized states by tuning the long-range hopping.
