Localized eigenstates in one-dimensional quasiperiodic and limit-periodic potentials

We study the single-particle eigenstates of ordered, nonperiodic potentials in one dimension, focusing on the scaling of the inverse participation rate (IPR) of the ground state with the overall amplitude of the potential. For quasiperiodic potentials given by the sum of three incommensurate cosine waves, the scaling exponent depends on the incommensurate ratios of wavelengths in the potential. For a limit-periodic potential given by the sum of cosine waves with wavenumbers 2^-nk₀, we find a rich set of states that are technically classified as extended but have large IPR’s. We will discuss the scaling of the IPR with the strength of the potential for the ground state and for the highest energy states in a finite system with periodic boundary conditions.