Anderson Localization for Schrödinger Operator With Denjoy‐Carleman Potentials

ABSTRACT It is concerned with the Anderson localization for the discrete one‐dimensional quasi‐periodic Schrödinger operator with some particular Denjoy‐Carleman potentials. First, it is proved that when the disorder parameter is sufficiently large and the frequency belongs to the strong Diophantine class (with a measure‐zero set excluded), the operator satisfies Anderson localization. This demonstrates the exponential localization of eigenstates under these conditions, a hallmark of Anderson localization in disordered systems. Second, for all energy values , both the Lyapunov exponent and the integrated density of states (IDS) of the Schrödinger operator are shown to be positive. Additionally, these quantities satisfy a specific modulus of continuity, which provides important insights into the regularity and spectral properties of the operator.