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

Abstract

In this paper, we prove the generic version of Cantor spectrum for quasi-periodic Schrödinger 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 ${\ell }^2({\mathbb{Z}}^d)$. These results are based on reducibility properties of finitely differentiable quasi-periodic $SL(2,\mathbb{R})$ cocycles. More precisely, we prove that if the base frequency is Diophantine, then a $C^k$ $SL(2,\mathbb{R})$-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.

29 pages,second version

References in corpus (2)

• Almost reducibility and absolute continuity I
• Asymptotics of spectral gaps of quasi-periodic Schrödinger operators

Cited by in corpus (2)

• Polynomial decay of the gap length for C^k quasi-periodic Schrodinger operators and spectral application
• Arithmetic version of Anderson localization via reducibility