Localization for quasiperiodic Schrödinger operators with multivariable Gevrey potential functions

Abstract

We consider an integer lattice quasiperiodic Schrödinger operator. The underlying dynamics is either the skew-shift or the multi-frequency shift by a Diophantine frequency. We assume that the potential function belongs to a Gevrey class on the multidimensional torus. Moreover, we assume that the potential function satises a generic transversality condition, which we show to imply a Łojasiewicz type inequality for smooth functions of several variables. Under these assumptions and for large coupling constant, we prove that the associated Lyapunov exponent is positive for all energies, and continuous as a function of energy, with a certain modulus of continuity. Moreover, in the large coupling constant regime and for an asymptotically large frequency-phase set, we prove that the operator satisfies Anderson localization.