On the reducibility of a class of almost-periodic linear Hamiltonian systems and its application in Schr¨odinger equation

: In the present paper, we focus on the reducibility of an almost-periodic linear Hamiltonian system


Introduction
In this paper, we will focus on the reducibility of an almost-periodic linear Hamiltonian system where A is a symmetric 2d ×2d constant matrix with possible multiple proper-values, Q(t) is an almostperiodic analytic symmetric matrix with respect to t, J = 0 I d −I d 0 , where I d is an identity matrix of order d, and ε is a sufficiently small parameter.
Let A(t) be a quasi-periodic matrix of order d, and the differential equation is known as reducible if there exists a nonsingular quasi-periodic (q-p) Lyapunov-Perron (L-P) change of variables X = ϕ(t)Y, where ϕ(t) and ϕ −1 (t) are quasi-periodic and bounded, which transforms (1.2) where B is a constant matrix.Over recent years, the reducibility of differential systems has been studied widely by a lot of researchers [1][2][3][4][5][6][7][8][9][10][11][12].The earliest result in this field is the well known Floquet Theory, which states that every periodic differential equation (1.2) can be reduced to a constant coefficient differential equation (1.3) by means of a periodic change of variables with the same period as A(t).However, the result is no longer always true for quasi-periodic systems.A counterexample was provided by Palmer [2].
For example, the quasi-periodic linear systems which come from the quasi-periodic Schrödinger operators, which are defined on L 2 (R) as where θ ∈ T n is known as phase, and q : T n → R is known as the potential.It is notable that the spectrum of L does not depend on the phase when ω is rationally independent, yet it is closely related to the dynamics of Schrödinger equation or, on the other hand, the dynamics of the linear differential systems where Dinaburg and Sinai [10] showed that linear system (1.6) is reducible for most E > E * (q, α, τ), which are sufficiently large, if ω is fixed and fulfills the non-resonance condition |⟨k, ω⟩| ≥ α |k| τ , k ∈ Z r \{0}, where α > 0, τ > 0. The result of [10] was generalized by Rüssmann [7], in which ω satisfied the Brjuno condition.
Eliasson [11] showed the full measure reducibility result for quasi-periodic linear Schrödinger equations.Specifically, he showed that (1.6) is reducible for almost all E > E * (q, ω) in the Lebesgue measure sense, where ω is the Diophantine vector which is fixed.
Jorba and Simó [1] considered the differential equations where A is a constant matrix of order d with d distinct proper-values.They showed that under the nonresonant conditions and non-degeneracy conditions, there exists a non-empty Cantor subset E, such that for ε ∈ E, the system (1.8) is reducible.
Xu [3] considered the case that A has multiple eigenvalues and showed the system (1.8) is reducible for ε ∈ E.
Recently, Xue and Zhao [9] considered the linear q-p Hamiltonian system where A is a constant matrix with possible multiple proper-values, and Q(t) is an analytic matrix with respect to t and with frequencies ω = (ω 1 , ω 2 , . . ., ω r ).Under some nonresonant conditions, using KAM iterations and for most sufficiently small parameters ε they proved that the system (1.9) is reducible by means of a quasi-periodic symplectic change of variables with the same basic frequencies as Q(t).
Rather than the reducibility of a q-p system to a constant coefficient system, Xu and You [5] investigated the reducibility of the following almost-periodic linear differential equations: where A is a constant matrix with distinct proper-values, and Q(t) is an almost periodic analytic matrix of order d with frequencies ω = (ω 1 , ω 2 , . ..).Under some small divisor conditions, using KAM iterations and the "spatial structure" of almost periodic functions, they proved that for most sufficiently small ε, Eq (1.10) is reducible.
Inspired by [5,8], in this paper, we extend the results of [9] to almost-periodic Hamiltonian systems instead of quasi-periodic Hamiltonian systems.Here the related LP change of variables should not only be almost-periodic but also be symplectic.
To state our problem, we should present some notations and definitions.A function f (t) is said to be a quasi-periodic function with essential frequencies ω = (ω 1 , ω 2 , . . ., ω d ), if f (t) = F(θ 1 , θ 2 , . . ., θ d ), where F is 2π periodic in all its arguments, and θ i = ω i t for i = 1, 2, . . ., d. f (t) will be known as an analytic q-p in a strip of width ϱ if F is analytical on D ϱ = {θ||ℑθ l | ≤ ϱ, l = 1, 2, . . ., n}.For the present case, we denote the norm of f (t) as where f m (t) (m = 1, 2, 3, . ..) are all quasi-periodic.Definition 1.1.Let A(t) = (a l j (t)) be a quasi-periodic d × d matrix.If every a l j (t) is analytic in D ϱ , then we call A(t) analytic on D ϱ .The norm of A(t) is defined as If A is a constant matrix, the norm of A is defined as: In [5], we have noticed that "spatial structure" and "approximation function" are valuable tools to study the almost-periodic systems.To overcome the difficulties from infinite frequency which generate the small divisors problems, we require much stronger norms.So, let's introduce these notations from [6,7].Definition 1.2.[6] Suppose that N is the natural number set, τ is the set of a few subsets of N.Then, ∪ Λ∈τ Λ = N, and a weight function Consider k ∈ Z N .Indicate k as the support set, and, is defined as

The weight value is denoted by [k], and [k]
Definition 1.3.[7] In the following, the non-resonance conditions are provided for the supposed approximation functions.∆ is called an approximation function, if ), is an increasing function, and fulfills ∆(0) = 1; , where Q Λ (t) are quasi-periodic matrices having frequencies ω Λ = {ω l |l ∈ Λ}, then Q(t) is called an almost-periodic matrix having the spatial structure (τ, [•]) and frequency ω of Q(t), which is the maximum subset of ∪ω Λ in the sense of integer modular.Denote Q = (q l j ) as the average of Q(t) = (q l j (t)), and For ϱ > 0, m > 0, the weighted norm of Q(t) with spatial structure (τ, [.]) is defined as: In our paper, the non-resonant condition is , l j, ∀ 1 ≤ l, j ≤ 2d, and k ∈ Z N \{0}, where α 0 > 0 is the small constant λ 1 , λ 2 , . . ., λ 2d are the proper-values of JA, ω = (ω 1 , ω 2 , . ..) is the frequency of Q(t), and ∆(t) is an approximation function which fulfills So, we are in a position to state our main result.), which depends continuously upon the small parameter ε.Suppose that , where α 0 > 0, and ∆(t) is an approximation function.
Then, there exists some sufficiently small ε * > 0 and a positive measure non-empty Cantor subset E * ⊂ (0, ε * ), s.t. for ε ∈ E * , there is an analytic almost-periodic symplectic change X = ψ(t)Y with the same frequencies and finite spatial structure like Q(t), which changes (1.1) into the Hamiltonian system Ẏ = BY, where B is a constant matrix.Additionally, means ( (0,ε * ) E * ) approaches 1 as ε * goes to 0. Remark 1.1.Here, as we are dealing with the Hamiltonian system, we need to find the symplectic change, which is not the same as that in [1].
Remark 1.2.We allow matrix JA to have multiple eigen-values.Obviously, if the eigen-values of JA are distinct, the non-degeneracy condition holds naturally.
As an example, we apply the Theorem 1.1 to the following Schrödinger equation: where Ja(t) = Ja Λ (t) is an almost-periodic function which is analytic on D ϱ with frequencies ω and has spatial structure (τ, [•]), which is persistently dependent on small parameter ε. a is the average of a(t).If a > 0 and the frequency ω of Ja(t) = Ja Λ (t) fulfills the non-resonance condition where α 0 > 0 is a small constant and ∆(t) is an approximation function, then there exists some sufficiently small ε * > 0, the system (1.11) is reducible, and the equilibrium of (1.11) is stable in the sense of Lyapunov for generally sufficiently small ε ∈ (0, ε * ).In addition, all solutions of Eq (1.11) are quasi-periodic with the frequency Ω = ( √ b, ω 1 , ω 2 , . ..) for generally sufficiently small ε ∈ (0, ε * ), where b = aε + O(ε 2 ) as ε approaches 0. Here, we can see that if we rewrite the system (1.11) into the system (1.1), we have which has various proper-values λ 1 = λ 2 = 0.One can see Section 5 for much more details about this example.This paper is organized as follows: • In Section 2, some Lemmas are given.
• In Section 3, we will prove the first KAM step.
• In Section 4, we will prove the main Theorem 1.1.

The
then, at that point, the accompanying conclusions hold: (1) C 1 has 2d distinct non-zero proper-values µ 1 1 , . . ., µ 1 2d ; (2) ∃ the regular matrix B 1 such that B −1 The next lemma is the inductive lemma which is used for the inductive procedure in the proof of Theorem 1.1.
with α 0 > 0 a constant and with the approximation function ∆(t).Consider 0 < ϱ < ϱ, 0 < m < m.Then, ∃ a unique analytic almost-periodic Hamiltonian matrix S (t) with similar finite spatial structure and with similar frequency as Q(t), which gives the solution of Eq (2.1) and fulfills where Γ(ϱ) = sup t≥0 [∆ 3 (t)e −ϱt ], and c > 0 is the constant.Proof: Setting S such that S −1 JAS = D = dia(λ 1 , λ 2 , . . ., λ 2d ), making transformation S (t) = BV(t)B −1 and R(t) = B −1 QB(t), Eq (2.1) becomes Substituting above into VΛ = DV Λ − V Λ D + R Λ and by comparing the coefficients on both sides, we obtain v jl Λ0 = 0; or for k 0, QB is also analytic on D ϱ .So, using Eq (2. 2), we have Thus, Let V = Λ∈τ V Λ .From Definition 1.2, we have Then, by utilizing Lemmas 2.1 and 2.2, we can write To show that S = Λ∈τ S Λ is Hamiltonian, we simply need to make sure that S l = J −1 S is symmetric.Since we have that JA is Hamiltonian and Q = Λ∈τ Q Λ is Hamiltonian, using the definition, A is symmetric, and we can denote Q = JQ l , where Q l is symmetric.Putting S = JS l and Q = JQ l into Eq (2.1), we get (2.3) Taking the transpose on the two sides of Eq (2. 3), we have Multiplying both sides of Eqs (2.3) and (2.4) by J, we get J Ṡ l = (JA)JS l − JS l (JA) + Q, and J Ṡ t l = (JA)JS t l − JS t l (AJ) + Q.This shows that JS l and JS t l are solutions of Eq (2.1).As v l j Λ0 =, 1 ≤ l, j ≤ 2d, we have V = 0, and so S = 0. Thus, J S l = J S t l = 0.As Eq (2.1) has unique solution with S = 0, we get JS l = JS t l ; and this implies that S l = S t l , which shows that S is the Hamiltonian.□

The first KAM step
Choose A 0 = JA, Q 0 (t) = JQ(t).By condition A 3 of Theorem 1.1, (A 0 + εQ 0 ) is the Hamiltonian matrix with 2d distinct proper-values λ 1 l , (1 ≤ l ≤ 2d) with |λ 1 l | ≥ 2ηε, and (0 ≤ l, j ≤ 2d) with |λ 1 l − λ 1 j | ≥ 2ηε, where η > 0 is the constant independent from ε.Thus, Hamiltonian system (1.1) can be rewritten in the form: where Let regular matrix B 1 be such that B −1 Using symplectic change of variables X = e εS (t) X 1 , where S (t) will be found later, the system (3.1) is converted into Then, the Hamiltonian system (3.2) can be rewritten as where We would like to have By the condition A 3 of Theorem 1.1, it is not difficult to see that the inequalities also holds, where α 1 = α 0 4 , then Eq (3.4) can be solved for a unique almost-periodic Hamiltonian matrix S = S Λ on D ϱ−ϱ with similar frequencies and similar spatial structure(τ, [•]) as Q , which fulfills S = 0 and Therefore, by using (3.4), the system (3.3) can be written as where, Consequently, under the symplectic transformation X = e εS (t) X 1 , system (3.1) is converted into system (3.7).

Proof of Theorem 1.1
Now, we consider the iteration step.At the n th step, suppose the Hamiltonian system , where η > 0 is independent from ε.By defining the average of Q n (t) as Q n , the system (4.1) is rewritten as where Presently, by making the symplectic change X n = e ε 2 n S n (t) X n+1 , where S n (t) will be found later, the system (4.2) becomes By series expansion, we can indicate where Then, the system (4.3) can be rewritten as where We would like to have or we have Since and 3, there is a unique almost-periodic matrix S n (t) on D ϱ n −ϱ n+1 having frequencies ω and with finite spatial structure (τ, [•]), which fulfills S n = 0 and Then, the Hamiltonian system (4.4) becomes where, Thus, under the symplectic change X n = e ε 2 n S n (t) X n+1 , system (4.1) is transformed into system (4.7).Let regular matrix B n+1 be such that B −1 n+1 A n+1 B n+1 = diag(λ n+1 1 , . . ., λ n+1 2d ) and Then, from Lemma 2.2, we can suppose β n+1 = 2β n , and so β n = 2 n−1 β 1 .

Conclusions
In this research work, we discussed the reducibility of almost-periodic Hamiltonian systems and proved that the almost-periodic linear Hamiltonian system (1.1) is reduced to a constant coefficients Hamiltonian system by means of an almost-periodic symplectic transformation.The result was proved for sufficiently small parameter ε by using some non-resonant conditions, non-degeneracy conditions and the rapidly convergent method that is KAM iterations.The result was also verified for Schrödinger equation.
and A 1 and Q(t) are the Hamiltonian matrices.
ϱ n .Using the condition A 1 of Theorem 1.1, Eqs (4.6) and(4.11), the composition of all the transformations e ε 2 n S n is convergent to ψ(t) as n → ∞.Moreover, it follows from (4.12) that A n converges always as n → ∞.Define B = lim n→∞ A n .Then, at that point, using symplectic change X = ψ(t)Y, the Hamiltonian system (1.1) is transformed into Ẏ = BY with constant coefficient matrix B.