An extension of the Kunz–Souillard approach to localization in one dimension and applications to almost-periodic Schrödinger operators

Abstract

We generalize the approach to localization in one dimension introduced by Kunz–Souillard, and refined by Delyon–Kunz–Souillard and Simon, in the early 1980's in such a way that certain correlations are allowed. Several applications of this generalized Kunz–Souillard method to almost periodic Schrödinger operators are presented.On the one hand, we show that the Schrödinger operators on ℓ2(Z) with limit-periodic potential that have pure point spectrum form a dense subset in the space of all limit-periodic Schrödinger operators on ℓ2(Z). More generally, for any bounded potential, one can find an arbitrarily small limit-periodic perturbation so that the resulting operator has pure point spectrum. Our result complements the known denseness of absolutely continuous spectrum and the known genericity of singular continuous spectrum in the space of all limit-periodic Schrödinger operators on ℓ2(Z).On the other hand, we show that Schrödinger operators on ℓ2(Z) with arbitrarily small one-frequency quasi-periodic potential may have pure point spectrum for some phases. This was previously known only for one-frequency quasi-periodic potentials with ‖⋅‖∞ norm exceeding 2, namely the super-critical almost Mathieu operator with a typical frequency and phase. Moreover, this phenomenon can occur for any frequency, whereas no previous quasi-periodic potential with Liouville frequency was known that may admit eigenvalues for any phase.