Localization For Ergodic Long-rang Operator

In this thesis,we are dedicated to the following two classes of ergodic long-rangeoperators:1.Quasi-periodic long-range operators on l2(Zd) where α ∈ Td is the frequency,θ ∈ T is the phase,Vk ∈ C is the Fourier coefficient of a real analytic function V:Td→R.2.The one dimensional Anderson model:where ωn ∈R are independent identically distributed random variables with a common Borel probability distributionμ.We will assume S(?)R,the topological support of μ, is compact,and contains at least two points.In the first chapter,we give a brief introduction to basic spectral theory and ergodic theory,then we introduce ergodic long-range operators and give the definition of Lyapunov ecponent.We also give some basic theories of quasi-periodic Schrodinger operators.In the second chapter,we prove exponential dynamical localization in expectation(EDL)for a class of quasi-periodic long-range operators on l2(Zd).We first give a criterion for EDL which can be viewed as a generalization of the criterion for Anderson localization in[14].Secondly,we apply the criterion to quasi-periodic models.For this purpose,we need a good control of eigenfunctions.The control will be given by Aubry duality and the quantitative almost reducibility of the dual cocycles.In the third chapter,we prove EDL for one dimensional Anderson model.We mention that EDL for one dimensional Anderson model has already been proved if one assumes that the distribution is regular.Our result doesn’t need any assumption on the distribution.Our proof is based on the methods developed in[53,58],thus completely different from the previous proofs.The general idea is that detailed estimates of the eigenfunctions impliy EDL.These estimates can be given by detailed analysis of the large deviation set in[58].