Nature of eigenstates on fractal structures

Abstract

The density of states and the nature of the eigenstates of the tight-binding (or any general quadratic) Hamiltonian, on a $d$-dimensional Sierpinski gasket, are investigated. For $d>\mathrm{}1\mathrm{}$, the spectral measure is the superposition of two distinct parts: a pure point measure of relative weight $\frac{d}{\mathrm{}(d+1\mathrm{})\mathrm{}}$, associated with "molecular" localized states, and a pure point measure, with a Cantor set support, associated with "hierarchical" states.