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 by Avila-Fayad-Krikorian. From spectral theory side, the ``Schr\"odinger conjecture"  has been verified by Avila-Fayad-Krikorian and the ``Last's intersection spectrum conjecture" has been proved by Jitomirskaya-Marx. 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, some open problems in this aspect were raised by  Fayad-Krikorian and Jitomirskaya-Marx. In this paper, we prove the above mentioned results for ultra-differentiable potentials.

Abstract: 

It was found in the 1990s that special linear maps playing a role in the representation theory of the symmetric group share common features with random matrices. We construct a representation-theoretic operator which shares some properties with the Anderson model (or, perhaps, with magnetic random Schroedinger operators), and show that indeed it boasts Lifshitz tails. The proof relies on a close connection between the operator and the infinite board version of the fifteen puzzle.
No background in the representation theory of the symmetric group will be assumed. Based on joint work with Ohad Feldheim.