Electronic states on a fractal: Inverse-iteration method

Abstract

The quantum dynamics of electrons on the Koch fractal is investigated within the framework of a tight-binding approximation. The invariant sets of an exact real-space renormalization procedure play a crucial role for the mainly analytical methods employed. Inverse iteration of the critical fixed points is the key to the construction of dense spectral subsets. This approach is confirmed by numerical determination of periodic orbits of the decimation map. As a corollary, a classification scheme for the quantum states of the system is proposed. This scheme is directly related to the 4-adic representation of the rotation number of the wave function.