Reducibility of One-dimensional Quasi-periodic Schrödinger Equations

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1 Reducibility of One-dimensional Quasi-periodic Schrödinger Equations Jiansheng Geng Department of Mathematics, Nanjing University, Nanjing , P.R.China Zhiyan Zhao Institut de Mathématique de Jussieu-Paris Rive GaucheANR contract), Paris, France Abstract Consider the time-dependent linear Schrödinger equation i q n = ɛq n+1 + q n 1 ) + V x + nω)q n + δ m Z a mn θ + ξt)q m, n Z, where V is a nonconstant real-analytic function on T, ω satisfies a certain Diophantine condition and a mn θ) is real-analytic on T b, b Z +, decaying with m and n. We prove that, if ɛ and δ are sufficiently small, then for a.e. x T and most frequency vectors ξ T b, it can be reduced to an autonomous equation. Moreover, for this non-autonomous system, dynamical localization is maintained in a quasi-periodic time-dependent way. 1 Introduction and Main Results We consider the one-dimensional Schrödinger equation i q n = ɛq n+1 + q n 1 ) + V x + nω)q n + δ m Z a mn θ + ξt)q m, n Z, 1.1) where V is a nonconstant real-analytic function on T = R/Z, ω R is a Diophantine number, and for each m, n Z, a mn : T b R is analytic in a complex neighbourhood Imθ < r 1 of the b torus T b, satisfying sup a mn θ) < e ρ max m, n }, ρ > 0. Imθ <r We are going to prove that, for ɛ and δ small enough, Eq. 1.1) can be reduced to a constant coefficient equationindependent of t) for most value of the parameter ξ, with the corresponding solutions well localized in space all the time. It is stated in the following theorem. 1

2 Theorem 1 There exists a sufficiently small ɛ = ɛ V, ω, r, ρ), such that if 0 < ɛ, δ < ɛ, then for a.e. x T, one can find a Cantor set O ɛ,δ = O ɛ,δ x) T b with LebT b \ O ɛ,δ ) 0 as ɛ, δ 0, such that for each ξ O ɛ,δ and θ T b, the equation 1.1) can be analytically reduced to an autonomous equation. Moreover, given any initial datum q0) satisfying q n 0) < ce ρ n for some constant c > 0, the solution qt) to the equation 1.1) with ξ O ɛ,δ satisfies that for any fixed d > 0, sup n 2d q n t) 2 <. t n Z Remark 1.1 The behaviour of solutions for a dynamical equation in the last statement of Theorem 1 is called dynamical localization. The Equation 1.1) is a perturbation of an autonomous quasi-periodic Schrödinger equation, whose behavior is determined by the spectral property of the operator T : l 2 Z) l 2 Z), defined as T q) n = ɛq n+1 + q n 1 ) + V x + nω)q n. As shown in [12], for ɛ small enough, the spectrum of this operator is pure point for a.e. x T. More precisely, it can be almost block-diagonalized, which is presented by a KAM schemesee Proposition 1 for details). The readers can also refer to [6, 10, 15, 22, 23, 26] for other works on the pure point spectrum and localization of quasi-periodic Schrödinger operators, and see [20, 21] for more about dynamical localization. It is necessary to mention that the KAM theory has been well adapted to Hamiltonian PDE s, especially in the continuous case. Many well-known works have been done for construction of time quasi-periodic solutionse.g., [3, 14, 16, 17]), for reducibility of nonautonomous equationse.g., [1, 13]), and for growth of Sobolev normse.g., [2, 4, 5, 27, 28]). However, the KAM technique is not widely applied to the discrete models, especially the case of the dense normal frequencies. A successful application is the model i q n = ɛq n+1 + q n 1 ) + V x + nα)q n + q n 2 q n, n Z, 1.2) with ɛ sufficiently small, x T, and α Diophantine. It is proved that, if V x) = tan πx [19] or V is a nonconstant real-analytic function on T [18], the equation 1.2) admits a large family of time quasi-periodic solutions q n t)) n Z, which are well localized in space. Corresponding to [19], Zhang-Zhao[30] has proved reducibility of the non-autonomous equation i q n = ɛq n+1 + q n 1 ) + tan πx + nα)q n + v n ξt)q n, n Z, for most frequencies ξ T b, where v n ) is smooth on T b and decaying with n. In the similar sense, the present work corresponds to [18]. Moreover, compared to [18], the gauge invariance of the perturbation is seriously destroyedsee the assumption A5) in [18]), hence we have to state a new KAM theorem and prove it again. Apart from the quasi-periodic discrete models above, some systems with random potentials are also studied by other techniques[7, 8, 9, 29]. 2

3 The remaining part of this paper is organized as follows. In Section 2, we shall present a KAM theorem, which can be directly applied to the equation 1.1). This KAM theorem will be proved in Section 3, by giving the details for one step of the KAM iteration and verifying the convergence. Section 4, which is regarded as an appendix, will present some necessary properties of infinite-dimensional matrices and Hamiltonians. 2 A KAM Theorem 2.1 Statement of KAM theorem To state a KAM theorem, which can be adapted to the equation 1.1), we start with recalling some necessary definitions of notations, which are related to the Hamiltonian vector field and the Poisson bracket. Some basic properties about this mechanism have been studied in previous workse.g., [18, 19]). Given d, ρ > 0, let l 1 d,ρ Z) be the space of complex valued sequences q = q n) n Z, with the norm q d,ρ := q n 1 + n 2 ) d 2 e ρ n <. n Z For r, s > 0, let D d,ρ r, s) be the complex b-dimensional neighborhood of T b 0} 0} 0} in T b R b l 1 d,ρ Z) l1 d,ρ Z), i.e., D d,ρ r, s) := θ, I, q, q) : Imθ < r, I < s 2, q d,ρ = q d,ρ < s}, where is the l 1 -norm of b dimensional complex vectors. For a real-analytic function F θ, I, q, q; ξ) on D = D d,ρ r, s), CW 1 i.e., C1 in the sense of Whitney) parametrized by ξ O, a closed region in T b, it can be expanded into the Taylor-Fourier series with respect to θ, I, q, q: F θ, I, q, q; ξ) = F klαβ ξ)e i k,θ I l q α q β, where I l := b j=1 I l j j, qα q β = α n,β n) 0,0) k Z b, l N b α, β qn αn q n βn for α := α n ) n Z, β := β n ) n Z, α n, β n N, with finite non-vanishing components. With ξ F klαβ := b j=1 ξj F klαβ and Fklαβ O := sup ξ O F klαβ + ξ F klαβ ), let F θ, I, q, q) O := k,l,α,β F klαβ O e k Imθ I l q α q β. Define the weighted norm of F as F D, O := sup F θ, I, q, q) O. For the Hamiltonian vector field X F = I F, θ F, i qn F ) n Z, i qn F ) n Z ) on D O, its norm is defined by X F D, O := sup D 1 s D qn F θ, I, q, q) O + qn F θ, I, q, q) O ) 1 + n 2 ) d 2 e n ρ n Z b + Ij F D, O + 1 b s j=1 2 θj F D, O. 2.1) j=1 3

4 Obviously, this norm depends on d, ρ, r, s. Given two real-analytic functions F and G on D, let, } denote the Poisson bracket of such functions, i.e., F, G} = I F, θ G θ F, I G + i n Z qn F qn G qn F qn G). In the remaining part of this paper, all constants labeled with c, c 0, c 1, are positive and independent of the iteration step. We consider the Hamiltonian H = ξ, I + T q, q + P θ, q, q; ξ), real-analytic on some suitable domain of θ, I, q, q), real-analytic parametrized by x T, and CW 1 parametrized by ξ O. We assume that T is the symmetric matrix defined by the quasi-periodic Schödinger operator with pure point spectrum. More precisely, T mn x) = V x + mω), m = n ɛ, m n = ±1 0, otherwise with V and ω as in the equation 1.1), and ɛ smaller than some critical value ɛ 0, which will be given in Proposition 1 later. P can be expanded as P θ, q, q; ξ) = k Z b P k ξ) q, q e i k,θ, with for some positive 0 < ε 1 and r, ρ > 0. P k mn O εe ρ max m, n } e r k 2.2) Remark 2.1 Compared to [18], the gauge invariance of the perturbation is seriously destroyedsee the assumption A5) in [18]), hence we have to state a new KAM theorem and prove it again. For ρ 0 := ρ 2, r 0 := r 2 and any d, s > 0, H is real-analytic on the domain D 0 := D d,ρ0 r 0, s), CW 1 parametrized by ξ O 0 := O. The decay estimate 2.2) implies that there exists a numerical constant c 0 such that X P D0, O 0 c 0 ρ 1 + r 1 )ε =: ε 0, in view of the definition given in 2.1). This provides us the necessary smallness condition for the perturbation argument. Theorem 2 There exists a positive ε, only depends on V, ω, r and ρ, such that if X P D0, O 0 < ε 0 ε, then for a.e. x T, there exists a Cantor set O ε0 = O ε0 x) O 0 with LebO 0 \ O ε0 ) 0 as ε 0 0, such that the following holds. There exists an analytic symplectic diffeomorphism Ψ : D d,0 r 0 /2, s) D 0, which is C 1 W parametrized by ξ O ε 0, satisfying Ψ Id Dd,0 r 0 /2, s), O ε0 0 1 as ε 0 0, such that H Ψ = ξ, I + Ωξ)q, q, independent of θ. 1 Here the norm Dd,0 r 0 /2, s), O ε0 is the operator norm. 4

5 It is obvious that the Hamiltonian equation θ = H I, I H = θ, i q n = H q n, i q n = H q n, n Z, is related to Eq. 1.1). So, with ɛ and δ small enough, Theorem 2 can be applied to it. The reducibility statement is obtained directly. Moreover, if q0) satisfies q n 0) < ce ρ n, qt) l 1 d,0 Z) is a b frequency time quasi-periodic solution to 1.1). Hence, sup t n Z n 2d q n t) 2 ) 1 2 < c sup t Therefore, Theorem 1 is a corollary of Theorem 2. n d q n t) < KAM scheme for the quasi-periodic Schrödinger operator Before the analysis on the Hamiltonian H, we first consider its ingredient, the quasiperiodic Schrödinger operator T = T x) : l 2 Z) l 2 Z), defined as n Z T q) n := ɛq n+1 + q n 1 ) + V x + nω)q n, n Z, with V and ω as in the equation 1.1). It is well-known from [12] that if ɛ is sufficiently small, then for a.e. x T, the spectrum of T x) is pure point. Related to the diophantine frequency ω, there exist τ > 1 and γ > 0 such that inf nω k k Z γ, n 0. n τ The non-constant real-analytic potential V is a smooth function in the Gevrey class sup m V x) CL m m!, m 0, x R/Z for some C, L > 0, and satisfying the transversality condition max 0 m s m ϕ V x + ϕ) V x)) ξ > 0, x, ϕ, max 0 m s m x V x + ϕ) V x)) ξ inf ϕ k, x, ϕ, k Z for some ξ, s > 0. Clearly, the case V x) = cos 2πx is included. With any N 0 1, ρ 0 = N0 1, any ε 0 ɛe ρ 0, and M 0 max one can define the following sequences for ν N: 2 s+4 C L s+1 s+1)! ξ }, 2 τ, 8, M ν+1 = M sm ν 3 ν, a ν = 1 τ M 3 sm ν 3 ν, ε ν+1 = ε 1 2 ε aν /2 ν ν, N ν+1 = ε aν ν, ρ ν+1 = ε aν ν, σ ν+1 = 1 3 ρ ν. 2.3) which have already been defined in [18] and originally defined in [12]. 5

6 Proposition 1 There exists a constant ɛ 0 = ɛ 0 C, L, ξ, s, γ, τ) such that if 0 < ɛ < ɛ 0 then the following holds. Fixed x R/Z. There exists a sequence of orthogonal matrices U ν, ν N, with such that U ν T U ν = D ν + Z ν, where U ν I Z ) mn ε e 3 2 σν m n, 2.4) D ν is a symmetric matrix which can be block-diagonalized via an orthogonal matrix Q ν with Q ν ) mn = 0 if m n > N ν. 2.5) More precisely, there is a disjoint decomposition j Λν j = Z such that D ν = Q νd ν Q ν = j D ν Λ ν j 2 with Λ ν j M ν, diamλ ν j M ν N ν, j. 3 Moreover, there exists a full-measure subset X R/Z such that if we fix x X, then for each k Z, there is a ν 0 k) such that Λ ν+1 k) = Λ ν k), ν ν 0 k). Z ν is a symmetric matrix, and Z ν ) mn ε ν e ρν m n. 2.6) Remark 2.2 In this paper, we consider this model for fixed x T. From now on, we shall not report the parameter x explicitly, if this dependence is irrelevant. 3 Proof of KAM Theorem Now we start the KAM iteration for the Hamiltonian H 0 := H = ξ, I + T q, q + P 0, real-analytic on D 0 = D d,ρ0 r 0, s), C 1 W parametrized by ξ O 0, with X P0 D0, O 0 ε 0. Suppose that we have arrived at the ν th step of the KAM iteration, ν = 0, 1, 2,. We consider the real-analytic Hamiltonian H ν = N ν + P ν on D ν := D d, ρν r ν, s) with N ν ξ) = ξ, I + T + W ν ξ))q, q, P ν ξ) = k Z b P k ν ξ)q, q e i k,θ 2 For readers convenience, we represent some notations for infinite-dimensional matrix. Given an infinite-dimensional matrix D, with D mn C the m, n) th entry, for a subset Λ Z, we define Λ := Z\Λ, C Λ := n C Z Dmn, m, n Λ : n i = 0 if i Λ}, D Λ := δ mn, otherwise. Then D Λ : C Λ + C Λ C Λ + C Λ, acts as R Λ C Z D C Z proj C Λ on the first component and as the identity on the second component. When there is no risk for confusion, we will use D Λ also to denote its first component. 3 The disjoint decomposition defines an equivalence relation m n on the integers and, for each n Z, we denote its equivalence class by Λ ν n). 6

7 which is C 1 W parametrized by ξ O ν, and satisfies X Pν Dν, O ν ε ν, W ν ) mn Oν pν e σν max m, n }, m, n N ν 0, otherwise, 3.1) P k ν ) mn Oν ε ν e ρν max m, n } e rν k, k Z b, 3.2) with some 0 < p ν 1, 0 < r ν < r 0, and ρ ν, σ ν, ε ν defined as in 2.3). Choose some r ν+1 such that 0 < r ν+1 < r ν, and let J ν := we define the quantities at each KAM sub-step as [ 5 aν 2 ε 2 ν ]. For j = 0, 1,, J ν, ρ j) ν = 1 j 2J ν )ρ ν, r j) ν = r ν jr ν r ν+1 ) J ν, and D ν j) = D d,ρν+1 r ν j), s), ε j) ν = ε j 5 +1 ν. Our goal is to construct a set O ν+1 O ν and a finite sequence of maps Φ j) ν : D j) ν D j 1) ν, j = 1, 2,, J ν, so that the Hamiltonian transformed into the ν + 1) th KAM cycle H ν+1 := H ν Φ 1) ν Φ Jν) ν = N ν+1 + P ν+1 is real-analytic on D ν+1 = D Jν) ν and C 1 W parametrized by ξ O ν+1, with N ν+1 ξ) = ξ, I + T + W ν+1 ξ))q, q, P ν+1 ξ) = k Z b P k ν+1ξ)q, q e i k,θ satisfies 3.1) and 3.2) with new quantities, and X Pν+1 Dν+1, O ν+1 ε Jν) ν ε 1 2 ε aν /2 ν ν = ε ν Construction of O ν+1 In view of 2.4) and 3.1), there exists a constant c 1 > 0 such that U ν W ν U ν ) mn Oν c 1 p ν σ 2 ν e σν max m, n }, by a simple application of Lemma 4.1. Define the truncation Ŵν as Ŵν) mn := U ν W ν U ν ) mn, m, n N ν 0, otherwise. 3.3) It follows that ) Uν W ν U ν Ŵν under the assumption C1) c 1 p ν σ 2 ν e σν ρν)nν ε ν. mn ε ν e ρν max m, n } 3.4) Oν 7

8 Let K ν+1 := N ν+1 M ν + 1)N ν and Λ ν := Λ ν j : Λ ν j [ K ν+1 + N ν ), K ν+1 + N ν ] } [ N ν+1, N ν+1 ]. According to Q ν in Proposition 1, define D Λ ν ν := Dν Λ ν, Wν := Q j νŵνq ν. 3.5) Λ ν j Λν In view of 2.5) and 3.3), we have W ν ) mn 0 if m or n > 2N ν. Since both of D Λ ν ν and W ν are Hermitian, there is an orthogonal matrix O ν such that O ν D ν Λ ν + W ν )O ν = diagµ ν j } j Λ ν, where µ ν j } j Λ ν are eigenvalues of Dν Λ ν + W ν. Due to the block-diagonal structure of D ν Λ ν + W ν, we also have O ν ) mn 0 if m n > 2M ν + 2)N ν. 3.6) Indeed, Dν Λ ν + W ν can be expressed as a product of smaller blocks D Λ ν ν + W ν = D Λ ν + W ν ν ) D Λ ν ν j Λ ν [ 2Nν,2Nν ]= j Λ ν j Λν where Λ ν := Λ ν j : Λν j [ 2N ν, 2N ν ] } with diamλ ν 2M ν + 2)N ν. The diagonalization of D ν Λ ν + Ãν is exactly the diagonalization of D ν Λ ν + W ν ) and D ν Λ ν j. As for the eigenvalues of D Λ ν ν + W ν, it is well-known that µ ν n} n Λ ν CW 1 -smoothly depend on ξ and there exist orthonormal eigenvectors ψn ν corresponding to µ ν n, CW 1 -smoothly depending on ξ see e.g. [11]). In fact, µ ν n = D Λ ν + W ν )ψn, ν ψ n ν and ξj µ ν n = ξj D Λ ν + W ν ))ψ ν n, ψ ν n, j = 1,, b. By the construction of Wν, we have ξj Wν = Q ν ξj Ŵ ν )Q ν, with Ŵν the truncation of U ν W ν ξ)u ν. Since D ν, U ν and Q ν are all independent of ξ, sup ξ µ ν n c sup ξ W ν ) mn cp ν. 3.7) ξ O ν ξ Oν m,n Now we defined the new parameter set O ν+1 O ν as O ν+1 := ξ O ν : k, ξ + µ ν m µ ν ε 1 80 ν n > k τ Nν+1 4, k 0, m, n Λ ν. 3.8) for some τ b. These inequalities are famous small-divisor conditions for controlling the solutions of the linearized equations. Since 3.7) implies combing with τ b, we can get ξ k, ξ µ ν m + µ ν n) k cp ν = O k ), ε 1 80 ν LebO ν \ O ν+1 ) c k τ+1 ε 1 80 ν. 3.9) k 0 8

9 3.2 Homological equation and its approximate solution From now on, to simplify notations, the subscripts or superscripts) ν of quantities at the ν th step are neglected, and the corresponding quantities at the ν + 1) th step are labeled with +. In addition, we still use the superscript j) to distinguish quantities at various sub-steps. With O + defined as in 3.8), we have Proposition 2 There exist two real-analytic Hamiltonians F = F k q, q e i k,θ, `P = `P k q, q e i k,θ, k Z b \0} k Z b and a Hermitian matrix W, all of which are C 1 W -parametrized by ξ O +, such that Moreover, for ε sufficiently small, N, F } + P = W q, q + `P. 3.10) F k mn O+ ε 4 5 k 2τ+1 e k r e ρ max m, n }, 3.11) `P k mn O+ ε 7 5 k 2τ+1 e k r e ρ1) max m, n }, 3.12) W mn O+ εe ρ max m, n }, m, n N + 0, otherwise. 3.13) Proof of Proposition 2: We accomplish the proof with the following procedures. Approximate linearized equations First of all, we try to construct a Hamiltonian F = k Z b \0} F k q, q e i k,θ, such that N, F } + P = P 0 q, q. 3.14) By a straightforward calculation and simple comparison of coefficients, the equation 3.14) is equivalent to k, ξ I Z T + W ))F k + F k T + W ) = ip k, k ) We can instead consider the equation k, ξ I Z D + Ŵ ) ) ˆF k + ˆF k D + Ŵ ) = i ˆP k, 3.16) where D and Ŵ are defined in the previous subsection, and for k 0, ˆP k mn = U P k U) mn, m, n K + 0, otherwise. 3.17) By 3.2) and 2.4), combining with Lemma 4.1, there exists c 2 > 0 such that U P k U) mn O c 2 σ ρ) 2 εe ρ max m, n } e k r. 3.18) This means U P k U ˆP k ) mn O 1 4 ε 7 5 e ρ 1) max m, n } e k r 3.19) under the assumption that 9

10 C2) c 2 σ ρ) 4 e ρ ρ1) )K ε 2 5. Equation 3.16) provides us with an approximate solution to 3.15), with the error estimated later. Consider the equation k, ξ I Λ D Λ + W )) F k + F k D Λ + W ) = i P k, 3.20) where D Λ, W are defined as in 3.5) via the orthogonal matrix Q, and P k := Q ˆP k Q. Note that Q mn = 0 if m n > N, then by 3.17), we have P k mn 0, if m or n > K + + N. Thus, recalling that Λ := Λ j : Λ j [ K + + N), K + + N] }, solutions of these finite-dimensional equations satisfy F k mn 0, if m or n Λ. Then, in view of the facts k, ξ I Z ± D + W )) F k = k, ξ I Λ ± D Λ + W )) F k, F k D + W ) = F k D Λ + W ), they are also solutions of k, ξ I Z D + W )) F k + F k D + Ã) = i P k, which is equivalent to Equation 3.16) since D can be block-diagonalized by the orthogonal matrix Q. Finally, we can focus on the equation k, ξ µ m + µ n ) ˇF k mn = io P k O) mn, for k 0 and m, n Λ, which is transformed from 3.20) by diagonalizing D Λ + W via the orthogonal matrix O. Obviously, these equations can be solved in O + defined in 3.8). Hence, 3.16) is solved with ˆF k = QO ˇF k O Q. Let F k := U ˆF k U, then we obtain a Hamiltonian F = k 0 F k q, q e i k,θ. It is easy to see that F = F, by noting F k)) = F k. Estimates for coefficients of F By the construction above, one sees that F k mn = i F 0 U mn1 Q n1 n 2 O n2 n 3 O n 3 n 4 Q n 4 n 5 ˆP k n5 n 6 Q n6 n 7 O n7 n 8 O n 8 n 9 Q n 9 n 10 U n 10 n k, ξ µ n3 + µ n8, where the summation notation F 0 denotes n1 Z, n 2 n 1 N, n 3 n 2, n 4 n 3 2M + 2)N, n 5 n 4 N, n 10 Z, n 9 n 10 N, n 8 n 9, n 7 n 8 2M + 2)N, n 6 n 7 N } 10

11 by virtue of the structure of Q and O, i.e, 2.5) and 3.6). Then, by 3.18) and Lemma 4.1, 1 sup Fmnξ) k cε 80 k τ N+)σ 4 ρ) 4 M 4 N 8 e 4M+10)Nρ εe k r e ρ max m, n }. ξ O + Here we have applied the property of the orthogonal matrices Q and O, and used the factor e 4M+10)Nρ to recover the exponential decay. To estimate ξj F k mn, we need to differentiate both sides of 3.20) with respect to ξ j, j = 1, 2,, b. Then we obtain the equation about ξj F k k, ξ I Λ D Λ + W )) ξj F k ) ξj F k ) D Λ + W ) = P k ξ j, which can also be solved by diagonalizing D Λ + Ã via O as above, where We get the formulation P k ξ j := i ξj Rk + F k ξj W ) ξj k, ξ I W )) F k. ξj F k mn = F 1 U mn1 Q n1 n 2 O n2 n 3 O n 3 n 4 P k ξ j ) n4 n 5 O n5 n 6 O n 6 n 7 Q n 7 n 8 U n 8 n k, ξ µ n3 + µ n6, with F 1 denotes n1 Z, n 2 n 1 N, n 3 n 2, n 4 n 3 2M + 2)N, n 8 Z, n 7 n 8 N, n 6 n 7, n 5 n 6 2M + 2)N By the decay property of ˆR k and ξj Â, we have that sup P ξ k j ) mn cε 80 k τ+1 N+)σ 4 ρ) 4 M 4 N 8 e 4M+11)Nρ εe k r e ρ max m, n }. ξ O + Thus there exists c 3 > 0 such that sup Fmn k + ξ Fmn ) k ξ O + 1 c 3 ε 1 40 k 2τ+1 N 8 +)σ ρ) 6 M 8 N 14 e 8M+20)Nρ εe ρ max m, n } e k r ε 4 5 k 2τ+1 e k r e ρ max m, n }, under the assumption C3) c 3 σ ρ) 6 N 8 +M 8 N 14 e 8M+20)Nρ ε Estimate for the error Let W be the truncation of P 0, satisfying }. W mn = P 0 mn, m, n N + 0, otherwise, 11

12 and with Ẁ := W UŴ U, `Z := UZU, let `P = k Z b `P k q, q e i k,θ, satisfying `P 0 := P 0 W, `P k := P k U ˆP k U ) iẁ + `Z)F k + if k Ẁ + `Z), Then we obtain The equation 3.10). By 3.2), we have 3.13) holds and under the assumption C4) e ρ ρ1) )N + ε 2 5. `P 0 mn O+ εe ρ max m, n } ε 7 5 e ρ 1) max m, n }, 3.21) As for the case k 0 in 3.12), by 3.19) and C2), combining with Lemma 4.1, P k U ˆP k U ) = UU P k U ˆP k )U ) 1 mn O mn O 4 ε 7 5 e ρ 1) max m, n } e k r. 3.22) In view of 2.6) and 3.4), Ẁmn O cσ ρ) 2 εe ρ max m, n }, `Z mn cσ ρ) 2 εe ρ m n. Then, by applying Lemma 4.1 again, there exists c 4 > 0 such that F k Ẁ + `Z) ), Ẁ + `Z)F k) O+ mn provided that mn O+ c 4 σ ρ) 2 ρ ρ 1) ) 1 ε 9 5 k 2τ+1 e k r e ρ1) max m, n } 1 4 ε 7 5 k 2τ+1 e k r e ρ1) max m, n }, 3.23) C5) c 4 σ ρ) 2 ρ ρ 1) ) 1 ε Thus, we can obtain the estimate for `P k by putting 3.21) 3.23) together. Let D i := D d,ρ+ r 1) + i 4 r r1) ), i 4s), i = 1, 2, 3, 4. Lemma 4.1 in [18] shows Lemma 3.1 There is a constant c 5 > 0 such that Moreover, if X F D3, O + c 5 r r 1) ) 2τ+b+1) ρ ρ + ) 2 ε 4 5, X `P D3, O + c 5 r r 1) ) 2τ+b+1) ρ 1) ρ + ) 2 ε 7 5. C6) c 5 r r 1) ) 2τ+b+1) ρ 1) ρ + ) 2 ε , we have X F D3, O + ε 3 4 and X `P D3, O + ε 5 4. For the Hamiltonian flow Φ t F associated with F, we have 12

13 Corollary 1 For ε sufficiently small, we have Φ t F : D 2 D 3, 1 t 1 and moreover, DΦ t F Id D1 < 2ε 3 4. Let F 1), W 1), `P 1) be the corresponding quantities in the homological equation 3.10) respectively, which means that we are in the 1 st sub-step. Then H 1) := H Φ 1 F 1) = N + P ) Φ 1 F 1) = N + W 1) q, q + P 1), with P 1) the same as in some standard argumentse.g., [18, 19]), and, by Lemma 3.1 and Corollary 1, X P 1) D 1), O + ε 6 5 = ε 1). As for the decay estimate of P 1), note that P 1) = `P 1) + P, F 1) } + 1 2! N, F 1) }, F 1) } + 1 2! P, F 1) }, F 1) } n! N, F 1) }, F 1) } + 1 }} n! P, F 1) }, F 1) } +. }} n n Applying Lemma 4.2 to P, F 1) } = k Z b P, F 1) } k q, q e i k, θ, we can find a constant c 6 > 0 such that P, F 1) } k mn O+ c 6 r r 1) ) 2τ+b+1) ρ ρ 1) ) 1 ε 9 5 e r 1) k e ρ1) max m, n } according to 3.2) and 3.11). Since N, F 1) } = P + W 1) q, q + `P 1), with W 1) and `P 1) estimated in Proposition 2, we can apply Lemma 4.2 iteratively, and get Proposition 3 P 1) = k Z b P 1)k ξ)q, q e i k,θ satisfies under the assumption that P 1)k mn O+ ε 1) e r1) k e ρ1) max m, n } C7) c 6 r r 1) ) 2τ+b+1) ρ ρ 1) ) 1 ε 1 5 1, The process above is a sub-step of KAM iteration. 3.3 A succession of symplectic transformations Suppose that we have arrived at the j th sub-step, j = 1,, J, with J = encounter the Hamiltonian j 1 H j 1) = H Φ 1 F Φ 1 1) F = N + W i) q, q + P j 1), j 1) i=1 [ 5 2 ε a 2 ]. We with the superscript 0) labeling quantities before the 1 st sub-step in particular. As demonstrated in Proposition 2, on O +, the following homological equation N, F j) } + P j 1) = W j) q, q + `P j), 3.24) 13

14 can be solved, with F j), W j), `P j) having properties similar to F 1), W 1), `P 1) respectively. Then we obtain It can be summarized that j H j) = H j 1) Φ 1 F = N + W i) q, q + P j). j) i=1 Proposition 4 Consider the Hamiltonian H = N + P. There exist J symplectic transformations Φ j) = Φ 1 F j), j = 1,, J, generated by the corresponding real-analytic Hamiltonians F j) respectively, such that a) F j) = j H j) = H Φ 1) Φ j) = N + W i) q, q + P j). i=1 F j)k ξ)q, q e i k,θ satisfies the equation 3.24) on O +, and k Z b \0} F j)k mn O+ ε 1 4 ε j 1) e rj 1) k e ρj 1) max m, n }. b) W j) satisfies that W j) mn O+ ε j 1) e ρj 1) max m, n }, m, n N + 0, otherwise. 3.25) c) P j) = k Z b P j)k ξ)q, q e i k,θ satisfies and X P j) D j), O + ε j). P j)k mn O+ ε j) e rj) k e ρj) max m, n }, 3.26) Let Φ = Φ 1) Φ J), then Φ : D + O + D O. Let N + = ξ, I + T +W + )q, q, J with W + = W + W j), and P + = P J). From 3.25) and 3.26), we have j=1 W + ) mn O+ p+ e σ + max m, n }, m, n N + 0, otherwise P k +) mn O+ ε + e r + k e ρ + max m, n }, k Z b., with p + = p + ε 1 2, σ + = 1 3ρ. Till now, one step of KAM iterations has been completed, and the next cycle can be started for H +. 14

15 3.4 Iteration lemma and convergence Besides the sequences defined in 2.3), we define, with p 0 = ε 1 2 0, ) p ν+1 = p ν + ε 1 ν+1 2 ν, r ν = r i. Now, we have defined all sequences appearing in the KAM cycles. In the previous work, the assumptions C1) C7) have been verified for these sequencesrefer to the proof of Lemma 5.1 in [18]). So the preceding analysis can be summarized as follows. Lemma 3.2 There exists ε 0 sufficiently small such that the following holds for all ν N. a) H ν = N ν + P ν is real-analytic on D ν, and C 1 W -parametrized by ξ O ν, where N ν = ξ, I + T + W ν ξ))q, q, P ν = k Z b P k ν ξ)q, q e i k,θ, i=2 satisfying X Pν Dν, O ν ε ν, and W ν ) mn Oν pν e σν max m, n }, m, n N ν 0, otherwise P k ν ) mn Oν ε ν e rν k e ρν max m, n }, k Z b., b) For each ν, there is a symplectic transformation Φ ν : D ν+1 D ν with such that H ν+1 = H ν Φ ν. DΦ ν Id Dν+1, O ν+1 ε 1 2 ν, Let O ε0 := ν=0 O ν. In view of 3.9), it is clear that LebO 0 \ O ε0 ) ε Fix x X, with X defined as in Proposition 1. This means that, for each n Z, there is a ν 0 n) such that Λ ν+1 n) = Λ ν n), ν ν 0 n). In this case, the local decay rate for n will not shrink with ν necessarilyρ ν is the global upper bound of the rates for all n Z). Define Ψ ν = Φ 0 Φ 1 Φ ν 1, ν N. An induction argument shows that Ψ ν : D ν+1 D 0, and H 0 Ψ ν = H ν = N ν + P ν. As in standard arguments e.g. [24, 25]), thanks to Corollary 1, it concludes that H ν, N ν, P ν, Ψ ν and W ν converge uniformly on D d,0 1 2 r 0, s) O ε0 to, say, H, N, P, Ψ and W respectively, in which case it is clear that N = ξ, I + T + W )q, q. Since X Pν Dν,Oν ε ν with ε ν 0, it follows that X P Dd,0 1 2 r 0,0), O ε0 = 0. 4 Appendix: Decay Property of Matrices and Hamiltonians Lemma 4.1 Lemma 2.1 in [18]) Given two matrices F = F mn ) m,n Z and G = G mn ) m,n Z. Let K = F G. 15

16 1) If F mn c F e ρ F m n, G mn c G e ρ G m n for some c F, c G, ρ F, ρ G > 0, then we have K mn c K e ρ K m n for any 0 < ρ K < minρ F, ρ G }, with c K = c c F c G minρ F, ρ G } ρ K ) 1. 2) In the cases that: F mn c F e ρ F max m, n }, G mn c G e ρ G m n, F mn c F e ρ F m n, G mn c G e ρ G max m, n }, F mn c F e ρ F max m, n }, G mn c G e ρ G max m, n }, we have K mn c K e ρ K max m, n }. Moreover, if ρ F ρ G, the conclusions above hold with ρ K = minρ F, ρ G }, c K = c c F c G ρ F ρ G 1. It can be adapted to the Hamiltonian: Lemma 4.2 Given two Hamiltonians F = k Z b F k ξ)q, q e i k,θ, G = k Z b G k ξ)q, q e i k,θ, C 1 W parametrized by ξ O, satisfying F k mn O c F k d e r F k e ρ F max m, n }, G k mn O c G k d e r G k e ρ G max m, n } for some d, c F, c G, ρ F, ρ G, r F, r G > 0. Then for K = F, G} = k Z b K k ξ)q, q e i k,θ, we have K k mn O c K e ρ K max m, n } e r K k for any 0 < ρ K < minρ F, ρ G }, 0 < r K < minr F, r G }, with c K = c c F c G minρ F, ρ G } ρ K ) 1 minr F, r G } r K ) b+d). Proof: The matrix element of K k can be formulated as Kmn k = ) F k 1 G k 2 G k 1 F k 2 k 1,k 2 Z b k 1 +k 2 =k ) By Lemma 4.1, we know that F k 1 G k 2 G k 1 F k 2 mn mn. is bounded by O c c F c G minρ F, ρ G } ρ K ) 1 k 1 d k 2 d e r F k 1 e r G k 2 + e r G k 1 e r F k 2 ) e ρ K max m, n }, with the part e r F k 1 e r G k 2 + e r G k 1 e r F k 2 smaller than e minr F, r G } r K ) k 1 + k 2 ) e r K k 1 + k 2 ). After the summation on k 1 and k 2, the proof is finished. 16

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