Reducibility Of Finitely Differentiable Quasi-periodic Linear Cocycles And Its Applications

In this paper,we are dedicated to studying the spectral and spectrum-related prop-erties of the one-dimensional quasi-periodic Schrodinger operator HV,?,? with finitely smooth potential:(HV,?,?x)n=xn+1+xn-1+V(?+n?)xn,n?Z,(0.2)where ? ? Td denotes the initial phase,? ? Rd(rational independent)is called the frequency and V ? Ck(Td,R)is called the potential.As we know,Schrodinger operator HV,?,? is closely related to Schrodinger cocycle(a,A),where (?),since the solution of HV?,?X= Ex satisfies (?).Therefore,we can use reducibility method to analyze the dynamics of the Ck quasi-periodic linear cocycle(?,A)? Td × Ck(Td,SL(2,R))and then study the spectral properties of the operator(0.2).In the first chapter,we introduce the finitely differentiable quasi-periodic linear cocycle in detail and give a brief introduction of spectral theory along with some basic concepts.In the second chapter,we prove the reducibility properties of finitely differentiable quasi-periodic linear cocycles.In almost reducible part,we first prove an analytic KAM theorem(perturbative KAM)and then use it and analytic approximation to obtain our quantitative Ck almost reducibility theorem.In reducible part,we use the quantitative Ck almost reducibility theorem and two positive measure reducibility lemmas to obtain our quantitative Ck reducibility theorem.In the third chapter,we study the spectral and spectrum-related properties by reducibility theorems established in Chapter two.Firstly,as spectrum-related ap-plications of the quantitative Ck almost reducibility theorem,we prove the 1/2-Holder continuity of the Lyapunov exponent of finitely differentiable quasi-periodic cocycles in some class.Besides,we also prove the 1/2-Holder continuity of the IDS of quasi-periodic Schrodinger operator HV,?,? with small Ck potential by Thouless formula.On the other hand,as spectral applications(spectral types and the structure of the spectrum)of the Ck reducibility theorem,we show pure point spectrum for a class of multi-frequency Ck long-range operators on l2(Zd),In addition,we prove the generic version of Can-tor spectrum for quasi-periodic Schrodinger operators with finitely smooth and small potentials.It is worth pointing out that there are few Ck reducibility results.Our quanti-tative Ck almost reducibility and reducibility theorems extend the analytic result of Eliasson[31].In the process of proving them,we first obtain an analytic KAM theorem and we can get Ch,h'? strong almost reducibility with any fixed h'<h by iteration.This is much stronger than Eliasson's theorem as he can only obtain C? weak almost reducibility,i.e.the analytic radius tends to zero in the end.It is very important because weak almost reducibility has few spectral applications but strong one can be widely applied in studying spectrum.Let us focus on the spectrum-related applications of our Ck almost reducibility theorem.Compared with the striking counter example:discontinuity of Lyapunov exponent of Ck or even smooth cocycles by Wang-You[59],our paper gives a positive result disjoint with them(In almost reducible region,Lyapunov exponent is zero.Since LE is a subharmonic function,it is upper semi-continuous.On the other side,LE ? 0 implies that LE is lower semi-continuous at zero point,thus LE is always continuous in our setting).Moreover,our result of IDS's 1/2-Holder continuity is the promotion of the analytic theorem of Avila-Jitomirskaya[9]and we can deal with multiple frequencies but[9]can only obtain one-frequency result.At last,on the spectral applications of our Ck reducibility theorem,our result of pure point spectrum of Ck quasi-periodic Schrodinger operator's dual operator LV,?,?with multiple frequencies,is the extension of analytic result proved by Avila-You-Zhou[14]and Jitomirskaya-Kachkovskiy[42].Compared with Bourgain-Goldstein[24],the advantage of our method is that we can fix the frequency.Besides,Cantor spectrum of Ck quasi-periodic Schrodinger operator with small potential extends the analytic re-sult of Eliasson[31]and Puig[52].Moreover,notice that their results are in C? topology and we can generalize them to standard Ch'? topology.