Localization for quasiperiodic Schrodinger operators with multivariable Gevrey potential functions

TL;DR

This paper proves Anderson localization and positivity of Lyapunov exponents for quasiperiodic Schrödinger operators with multivariable Gevrey potentials, under large coupling and frequency conditions, extending localization results to more general potentials.

Contribution

It establishes localization and Lyapunov exponent positivity for multivariable Gevrey potentials, introducing a new transversality condition and Lojasiewicz inequality in this context.

Findings

Lyapunov exponent is positive for all energies at large coupling.

Operator exhibits Anderson localization in the large coupling and frequency regime.

Continuity of the Lyapunov exponent as a function of energy with a specific modulus.

Abstract

We consider an integer lattice quasiperiodic Schrodinger 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 multi-dimensional torus. Moreover, we assume that the potential function satisfies a generic transversality condition, which we show to imply a Lojasiewicz 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.