SOME REMARKS ON THE THEORY OF AN ELECTRON ON BINARY QUASIPERIODIC LATTICES

Abstract

Some remarks on the theory of an electron on binary quasiperiodic lattices are presented herein. We classify all the binary quasiperiodic lattices that are constructed by a class of scaling transformations: B→A, A→AnB, where n is an arbitrary positive integer at each stage of the scaling transformation, into three types by means of the scaling transformation itself. We give a simple approach to generate a Cantor-set type energy spectrum by the generic trace map on an invariant surface, I=x2+y2+z2−2xyz−1. The Cantor-set type energy spectrum is a result of the existence of the noncompact invariant surface of the invariant I>0. The energy bands correspond to the nonescaping points of the generic trace map on the invariant surface, which are identified with the Julia set of the generic trace map. The density of states and the wavefunction at the band center are discussed, and we are able to classify them into three types respectively by means of its exponents.