In this paper, we prove the generic version of Cantor spectrum property 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.

在本文中，我们证明了具有有限平滑和小电势的准周期Schrödinger算子的Cantor频谱特性的通用版本，并且我们还展示了一类多频\（C ^ k \）远程范围的纯点谱\（\ ell ^ 2（{\ mathbb Z} ^ d）\）上的运算符。这些结果基于有限可微拟准周期\（SL（2，{\ mathbb R}）\）联合循环的可约性。更准确地说，我们证明了如果基频是丢虫精，那么如果\（C ^ k \） \（SL（2，{\ mathbb R}）\）值的cocycle近似于恒定cocycle，那么它是可约的，足够平滑，并且其转数就频率而言是丢丢定或合理的。