Critical electronic wave functions on quasiperiodic lattices: Exact calculation of fractal measures

We examine two wave functions which have recently been found for electrons in quasiperiodic systems, and which can be shown to satisfy an exact self-similarity relation---closely related to the self-similarity of the quasiperiodic lattice itself. The first wave function is for the ground state of an electron on a two-dimensional Penrose lattice, and the second is for the center of the spectrum of the Hamiltonian for an electron on a one-dimensional Fibonacci lattice. We calculate exactly the fractal measure of the singularities of these wave functions, as reflected in the exponent {α}({γ}), defined by J_x≤R{}{Ψ}(x)^γ{~}R^α(γ). .AE