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For quasiperiodic Schrodinger operators with one-frequency analytic potentials, from dynamical systems side, it has been proved that the corresponding quasiperiodic Schrodinger cocycle is either rotations reducible or has positive Lyapunov exponent for all irrational frequency and almost every energy. From spectral theory side, the Schrodinger conjecture and the Lasts 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.
We construct discontinuous point of the Lyapunov exponent of quasiperiodic Schrodinger cocycles in the Gevrey space $G^{s}$ with $s>2$. In contrast, the Lyapunov exponent has been proved to be continuous in the Gevrey space $G^{s}$ with $s<2$ cite{kl
We give a criterion for exponential dynamical localization in expectation (EDL) for ergodic families of operators acting on $ell^2(Z^d)$. As applications, we prove EDL for a class of quasi-periodic long-range operators on $ell^2(Z^d)$.
Some scales of spaces of ultra-differentiable functions are introduced, having good stability properties with respect to infinitely many derivatives and compositions. They are well-suited for solving non-linear functional equations by means of hard i
We obtain spectral estimates for the iterations of Ruelle operator $L_{f + (a + i b)tau + (c + i d) g}$ with two complex parameters and H{o}lder functions $f,: g$ generalizing the case $Pr(f) =0$ studied in [PeS2]. As an application we prove a sharp
We consider the sum of two large Hermitian matrices $A$ and $B$ with a Haar unitary conjugation bringing them into a general relative position. We prove that the eigenvalue density on the scale slightly above the local eigenvalue spacing is asymptoti