Localization by pseudorandom potentials in one dimension

Localization in one dimension in the presence of a pseudorandom potential is investigated. The localization length of the tight-binding model ${\mathrm{V}}_{\mathrm{n}}$ ${\mathrm{u}}_{\mathrm{n}}$+${\mathrm{u}}_{\mathrm{n}+1\mathrm{}}$+${\mathrm{u}}_{\mathrm{n}\mathrm{-}1}$=${\mathrm{Eu}}_{\mathrm{n}}$ with ${\mathrm{V}}_{\mathrm{n}}$=λ cosπα‖n${\mathrm{\Vert }}^{\nu }$ is calculated numerically and in perturbation theory for λ≪1, for generic values of α and ν. The similarity between the potential ${\mathrm{V}}_{\mathrm{n}}$ and random potentials increases with ν. It is found that for ν≥2 all the states are localized and the localization length is equal to that of the corresponding random model while for 0<ν≤1 there are extended states. The intermediate regime 1<ν<2 is discussed as well.