Localization for quasiperiodic Schrodinger operators with multivariable Gevrey potential functions

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.

42 pages, 3 figures

References in corpus (2)

• On the spectrum of skew-shift Schrödinger operators
• Multiscale Analysis for Ergodic Schrödinger operators and positivity of Lyapunov exponents

Cited by in corpus (7)

• Dynamics and spectral theory of quasi-periodic Schrödinger-type operators
• Continuity of the Lyapunov exponents for quasiperiodic cocycles
• Quantitative inductive estimates for Green's functions of non-self-adjoint matrices
• Large deviation estimates and Hölder regularity of the Lyapunov exponents for quasi-periodic Schrödinger cocycles
• Spectral theory of the multi-frequency quasi-periodic operator with a Gevrey type perturbation
• Anosov-Katok constructions for quasi-periodic $\mathrm{SL}(2,R)$ cocycles
• Holder Continuity of Absolutely Continuous Spectral Measure for Multi-frequency Schrodinger Operators