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In this paper, we prove that for any $d$-frequency analytic quasiperiodic Schrodinger operator, if the frequency is weak Liouvillean, and the potential is small enough, then the corresponding operator has absolutely continuous spectrum. Moreover, in the case $d=2$, we even establish the existence of ac spectrum under small potential and some super-Liouvillean frequency, and this result is optimal due to a recent counterexample of Avila and Jitomirskaya.
We give sufficient conditions for the presence of the absolutely continuous spectrum of a Schrodinger operator on a regular rooted tree without loops (also called regular Bethe lattice or Cayley tree).
We provide an abstract framework for the study of certain spectral properties of parabolic systems; specifically, we determine under which general conditions to expect the presence of absolutely continuous spectral measures. We use these general cond
By generalising Rudins construction of an aperiodic sequence, we derive new substitution-based structures which have purely absolutely continuous diffraction and mixed dynamical spectrum, with absolutely continuous and pure point parts. We discuss se
We show that a generic quasi-periodic Schrodinger operator in $L^2(mathbb{R})$ has purely singular spectrum. That is, for any minimal translation flow on a finite-dimensional torus, there is a residual set of continuous sampling functions such that f
In this article, we study algebraic dynamical pairs $(f,a)$ parametrized by an irreducible quasi-projective curve $Lambda$ having an absolutely continuous bifurcation measure. We prove that, if $f$ is non-isotrivial and $(f,a)$ is unstable, this is e