Position-space rescaling and hierarchical lattice models of disordered one-dimensional systems (invited)

The application of exact and approximate position-space renormalization group techniques to the calculation of densities of states for problems with Gaussian generating functions (such as free tight-binding electrons, harmonic vibrations, spin waves, or random walks on Euclidian or ‘‘fractal’’ lattices) is briefly reviewed. It is also shown that for one-dimensional Gaussian theories with disorder, the approximate recursion relations proposed by Gonçalves da Silva and Koiller (GK) are exact for problems formulated on Berker–Ostlund lattices. A generalization of the GK scheme which allows one to calculate the optical zone-center density of states is formulated and then applied to the study of one- and two-mode behavior in mixed diatomic crystals.