Quantum Particle in One-Dimensional Potentials with Incommensurate Periods

Abstract

The energy spectrum and wave functions for a particle in periodic potentials with incommensurate periods are obtained analytically and reduced to specific phase trajectories. The spectrum is of the devil's-stairs type. States may be extended and localized, separated by mobility edges. These results are applicable to incommensurate linear chain structures (such as those in ${\mathrm{Hg}}_{3-\delta }\mathrm{As}{\mathrm{F}}_6$) and to the fine structure of de Haas-van Alphen oscillations.