Reducibility of finitely differentiable quasi-periodic cocycles and its spectral applications

TL;DR

This paper establishes the generic Cantor spectrum for finitely smooth quasi-periodic Schrödinger operators and demonstrates pure point spectrum for certain multi-frequency long-range operators, based on reducibility of cocycles.

Contribution

It proves reducibility conditions for finitely differentiable quasi-periodic cocycles and applies these to spectral properties of related operators, extending previous results to finitely smooth cases.

Findings

Generic Cantor spectrum for finitely smooth quasi-periodic Schrödinger operators.

Pure point spectrum for multi-frequency long-range operators.

Reducibility of $SL(2, )$ cocycles under certain conditions.

Abstract

In this paper, we prove the generic version of Cantor spectrum for quasi-periodic Schr\"{o}dinger operators with finitely smooth and small potentials, and we also show pure point spectrum for a class of multi-frequency $C^k$ long-range operators on $\ell^2(\Z^d)$. These results are based on reducibility properties of finitely differentiable quasi-periodic $SL(2,\R)$ cocycles. More precisely, we prove that if the base frequency is Diophantine, then a $C^k$ $SL(2,\R)$-valued cocycle is reducible if it is close to a constant cocycle, sufficiently smooth and the rotation number of it is Diophantine or rational with respect to the frequency.