An Extension of the Kunz-Souillard Approach to Localization in One Dimension and Applications to Almost-Periodic Schr\"odinger Operators

TL;DR

This paper extends the Kunz-Souillard localization approach to include correlated potentials and applies it to almost periodic Schrödinger operators, demonstrating dense pure point spectra and eigenvalues for various potentials.

Contribution

It generalizes the Kunz-Souillard method to handle correlations and applies it to show dense pure point spectra in limit-periodic and quasi-periodic Schrödinger operators.

Findings

Pure point spectrum is dense among limit-periodic Schrödinger operators.

Small quasi-periodic potentials can have eigenvalues for some phases.

Eigenvalues can occur for any frequency, including Liouville frequencies.

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 Schr\"odinger operators are presented. On the one hand, we show that the Schr\"odinger operators on $l^2(\mathbb{Z})$ with limit-periodic potential that have pure point spectrum form a dense subset in the space of all limit-periodic Schr\"odinger operators on $l^2(\mathbb{Z})$. 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 Schr\"odinger operators on $l^2(\mathbb{Z})$. On the other hand, we show that Schr\"odinger operators on $l^2(\mathbb{Z})$ 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 $\|\cdot\|_\infty$ 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.