  For quasiperiodic Schr\"odinger operators with one-frequency analytic
potentials, from dynamical systems side, it has been proved that the
corresponding quasiperiodic Schr\"odinger cocycle is either rotations reducible
or has positive Lyapunov exponent for all irrational frequency and almost every
energy. From spectral theory side, the "Schr\"odinger conjecture" and the
"Last's intersection spectrum conjecture" have been verified. The proofs of
above results crucially depend on the analyticity of the potentials. People are
curious about if the analyticity is essential for those problems, see open
problems by Fayad-Krikorian and Jitomirskaya-Mar. In this paper, we prove the
above mentioned results for ultra-differentiable potentials.
