EXPLICIT GREEN-FUNCTIONS FOR HIERARCHICAL LATTICES

Explicit formulas for one-electron propagators x and y between same and distal corner sites are given for several families of hierarchical lattices. These give x and y - a function of lattice size for a set of energies dense in the Cantor portion of the spectrum of the large-lattice limit. With linear chains as quantum lead wires, the Kubo-Greenwood conductance is expressed succinctly in terms of x and y. Lattice families treated are (1) the Sierpinski simplex family S of Dhar in Euclidean dimension k, (2) the R family introduced previously, also k dimensions, (3) a star-delta dual to the R family, (4) a two-parameter family of k dimensional, q sheeted lattices related to those of de Menezes and de Magalhaes, (5) a dual to the two-parameter family, and (6) the first several members of a family related to the lattices of Hilfer and Blumen. The exact results derive from a formalism for solving the dynamics of the renormalization map explicitly. Special properties that make the renormalization maps solvable are enumerated, and the general solution procedure is reviewed. Power-law scaling of conductance with lattice size is found at special energies for each lattice. In several cases there are uncountably many energies at which conductance becomes independent of lattice size.