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The exact eigenstates of a one-dimensional tight-binding model with a periodic diagonal potential that can be commensurate or incommensurate with the lattice are found. If the period is incommensurate, all the eigenstates are localized. The localization length and the density of states are identical to those of a related disordered system. The case of a commensurate potential, for which all the states are extended, and the approach to the incommensurate case are also discussed. The solution is achieved by mapping the model into a time-dependent quantum problem in which the potential is periodic in time.
Grempel et al. (Mon,) studied this question.