The Absolutely Continuous Spectrum of Finitely Differentiable Quasi-Periodic Schrödinger Operators

We prove that the quasi-periodic Schrödinger operator with a finitely differentiable potential has purely absolutely continuous spectrum for all phases if frequency is Diophantine and sufficiently small in corresponding $$C^k$$ topology. This extends work of Eliasson [19] Avila–Jitomirskaya [5] from analytic topology to one which much broader, revealing interesting phenomenon oscillation leads both zero Lyapunov exponent whole spectrum. Our result based on refined quantitative $$C^{k,k_0}$$ almost reducibility theorem only requires quite low initial regularity “ $$k>14\tau +2$$ ” $$k_0\le k-2\tau -2$$ conserved end, where $$\tau $$ constant frequency.