Chaotic scaling trajectories and hierarchical lattice models of disordered binary harmonic chains

It is shown that the simple position-space rescaling method developed by Gon {{c}} {c}{}alves da Silva and Koiller to compute the local density of states (DOS) in disordered harmonic chains is exact on certain hierarchical lattices and in the limit of zero impurity concentration. It is demonstrated that their method can be improved even though convergence to the true DOS cannot be achieved simply. Partial information on eigenvectors can be extracted from the scaling trajectories. For an ordered chain, delocalized states can in general be identified with chaotic behavior. Analogies with critical phenomena may be found from a functional-integral formulation of the problem. The relationship to transfer-matrix approaches is also mentioned.