2024/07/11 by Lingrui Ge, Ge, Lingrui, Svetlana Jitomirskaya +1
Mathematics · #Advanced Harmonic Analysis Research #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Holomorphic and Operator Theory #Mathematical Physics (math-ph) #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2407.08866
openalex publication_date 2024/07/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We show that formal eigenvalue equations of analytic one-frequency Schröd-inger operators admit intrinsic analytic Sp(2k,\C) structures, where k=k(E) is the T-acceleration in global theory. For trigonometric potentials those structures govern the center dynamics of partially hyperbolic dual cocycles; for general analytic potentials they persist, without loss of analyticity, as an intrinsic object even when the dual operator has infinite range and no cocycles exist. For k=1, we also introduce the concept of projectively real cocycles: complex symplectic systems whose projective action is algebraically conjugate, up to a scalar phase, to that of a real \SL(2,\R) cocycle. This allows us to define a rotation pair and establish a rotation--IDS correspondence in the general analytic setting, where standard dynamical methods fail. Using these tools, we solve two spectral arithmetic conjectures: universality of the sharp arithmetic transition in frequency (AAJ) and of the absolute continuity of the integrated density of states for all frequencies, throughout the class of non-critical Type I operators, an open and conjecturally dense set. We also prove universality of sharp 1/2-Hölder continuity of the integrated density of states for Type I operators with Diophantine frequencies, establishing part of You's conjecture. These results also provide the first duality-based spectral framework for general analytic potentials, overcoming the symmetry and finite-range restrictions present in previous work.