On the Reducibility of Quasiperiodic Linear Hamiltonian Systems and Its Applications in Schrödinger Equation

In this paper, we consider the reducibility of the quasiperiodic linear Hamiltonian system _ x = ðA + εQðtÞÞ, where A is a constant matrix with possible multiple eigenvalues, QðtÞ is analytic quasiperiodic with respect to t, and ε is a small parameter. Under some nonresonant conditions, it is proved that, for most sufficiently small ε, the Hamiltonian system can be reduced to a constant coefficient Hamiltonian system by means of a quasiperiodic symplectic change of variables with the same basic frequencies as QðtÞ. Applications to the Schrödinger equation are also given.


Introduction
In this paper, we are concerned with the reducibility of the quasiperiodic linear Hamiltonian system where A is a constant matrix with possible multiple eigenvalues, QðtÞ is analytic quasiperiodic with respect to t, and ε is a small parameter. Firstly, let us recall the definition of the reducibility for quasiperiodic linear systems. Let AðtÞ be an n × n quasiperiodic matrix, the differential equation is called reducible, if there exists a nonsingular quasiperiodic change of variables where φðtÞ and φ −1 ðtÞ are quasiperiodic and bounded, which changes (2) into where B is a constant matrix. The well-known Floquet theorem states that every periodic differential equation (2) can be reduced to a constant coefficient differential equation (4) by means of a periodic change of variables with the same period as AðtÞ. However, this is not true for the quasiperiodic linear system; one can see [1] for more details. In 1981, Johnson and Sell [2] proved that the quasiperiodic linear system (2) is reducible if the quasiperiodic coefficient matrix AðtÞ satisfies the "full spectrum" condition.
A typical example of quasiperiodic linear systems comes from the (continuous time) quasiperiodic Schrödinger operators, which are defined on L 2 ðRÞ as where q : T n ⟶ R is called the potential and θ ∈ T n is called the phase. It is well known that the spectrum of L does not depend on the phase when ω is rationally independent, but it is closely related to the dynamics of the Schrödinger equations or equivalently the dynamics of the linear systems where Dinaburg and Sinai [3] proved that linear systems (7) are reducible for most E > E * ðq, α, τÞ, which are sufficiently large, if ω is fixed and satisfies the Diophantine condition where α, τ are positive constants. The result was generalized by Rüssmann [4] for ω satisfying the Bruno condition. Eliasson [5] proved a full measure reducibility result for quasiperiodic linear Schrödinger equations. More precisely, Eliasson proved that (7) is reducible for almost all E > E * ðq, ωÞ in Lebesgue measure sense, where ω is a fixed Diophantine vector.
In the case that n = 2, a stronger reducibility result, called a nonperturbative reducibility, is available. The nonperturbative reducibility means that the smallness of the perturbation does not depend on the Diophantine constant α. Hou and You [6] proved, besides other results, the nonperturbative reducibility for (7).
The reducibility of quasiperiodic linear systems with coefficients in glðm, RÞ was considered by Jorba and Simó [7]. Suppose that A is a constant matrix with different eigenvalues, they proved that if the eigenvalues of A and the frequencies of Q satisfy some nonresonant conditions, then there exists sufficiently small ε 0 > 0 and a nonempty Cantor set E ⊂ ð0, ε 0 Þ, such that for any jεj ∈ E, system (1) is reducible. Moreover, the relative measure of the set ð0, ε 0 Þ \ E in ð0, ε 0 Þ is exponentially small in ε 0 . Junxiang [8] obtained the similar result for the multiple eigenvalue case. Later, many authors [7][8][9][10] paid attention to the reducibility of the quasiperiodic linear system (1), which is close to a constant coefficient linear system.
In 1996, Jorba and Simó [10] extended the conclusion of the linear system to the nonlinear system Suppose that A has n different nonzero eigenvalues, they proved that under some nonresonant conditions and nondegeneracy conditions, there exists a nonempty Cantor set E ⊂ ð0, ε 0 Þ, such that for all jεj ∈ E, system (10) is reducible. Later, Wang and Xu [11] further investigated the nonlinear quasiperiodic system where A is a real 2 × 2 constant matrix, and f ðt, 0, εÞ = OðεÞ, ∂ x f ðt, 0, εÞ = OðεÞ as ε ⟶ 0. They proved without any nondegeneracy condition, one of two results holds: (1) system (11) is reducible to _ y = By + OðyÞ for all ε ∈ ð0, ε 0 Þ; (2) there exists a nonempty Cantor set E ⊂ ð0, ε 0 Þ, such that system (11) is reducible to _ y = By + Oðy 2 Þ for all jεj ∈ E. In [12], Her and You considered one-parameter family of quasiperiodic linear system where A ∈ C ω ðΛ, glðm, ℂÞÞðC ω ðΛ, glðm, ℂÞÞ be the set of m × m matrices AðλÞ depending analytically on parameter λ in a closed interval (Λ ⊂ ℝ), and g is analytic and small. They proved that under some nonresonant conditions and nondegeneracy conditions, there exists an open and dense set A in C ω ðΛ, glðm, ℂÞÞ, such that for each A ∈ A, system (12) is reducible for almost all λ ∈ Λ.
Instead of a total reduction to a constant coefficient linear system, Jorba et al. [13] investigated the effective reducibility of the following quasiperiodic system: where A is a constant matrix with different eigenvalues. They proved that under nonresonant conditions, by a quasiperiodic transformation, system (13) is reducible to a quasiperiodic system where R * is exponentially small in ε. Li and Xu [14] obtained the similar result for Hamiltonian systems. Later, Xue and Zhao [15] extended the result to the case of multiple eigenvalues.

Assumption 2. (nondegeneracy condition). Assume that
Here, we denote the average of QðtÞ by Q, that is, We are in a position to state the main result.
Theorem 3. Suppose that Hamiltonian system (1) satisfies Assumptions 1 and 2. Then there exist some sufficiently small ε 0 > 0 and a nonempty Cantor subset E ε0 ⊂ ð0, ε 0 Þ with positive Lebesgue measure, such that for ε ∈ E ε0 , Hamiltonian system (1) is reducible, i.e., there is an analytic quasiperiodic symplectic transformation x = ψðtÞy, where ψ ðtÞ has same frequencies as QðtÞ, which changes (1) into the Hamiltonian Now, we give some remarks on this result. Firstly, here we deal with the Hamiltonian system and have to find the symplectic transformation, which is different from that in [7,8]. Secondly, we consider the reducibility, other than the effective reducibility in [13,14]. The last but not the least, we can allow matrix A to have multiple eigenvalues. Of course, if the eigenvalues of A are different, the nondegeneracy condition holds naturally.
As an example, we apply Theorem 3 to the following Schrödinger equation: where aðtÞ is analytic quasiperiodic with the frequencies ω = ðω 1 , ⋯, ω r Þ. Denote the average of aðtÞ by a. If a > 0 and the frequencies ω of aðtÞ satisfy the Diophantine condition where α > 0 is a small constant and τ > r − 1, then there exists some sufficiently small ε 0 > 0, equation (17) is reducible and the equilibrium of (17) is stable in the sense of Lyapunov for most sufficiently small ε ∈ ð0, ε 0 Þ. Moreover, all solutions of equation (17) are quasiperiodic with the frequencies Here, we remark that if we rewrite equation (17) into Hamiltonian system (1), we find that which has multiple eigenvalues λ 1 = λ 2 = 0. One can see Section 4 for more details about this example. There are plenty of works about the stability of all kinds of equations, one can refer to [16][17][18][19][20][21][22][23] for a detailed description. In particular, for quasiperiodic equations, in order to determine the type of stability of the equilibria of quasiperiodic Hamiltonian systems, the authors need to assume that the corresponding linearized system is reducible, and some conditions were added to the system after the reducibility. However, as far as we know, the case that the conditions are added to the original system has not been considered in the literature up to now, which we will study in the future.
The paper is organized as follows. In Section 2, we list some basic definitions and results that will be useful in the proof of the main result. In Section 3, we will prove Theorem 3. Equation (17) will be analyzed in Section 4.

Some Preliminaries
We first give the definition of quasiperiodic functions.

Definition 4. A function f is said to be a quasiperiodic function with a vector of basic frequencies
It is well known that an analytic quasiperiodic function f ðtÞ can be expanded as Fourier series with Fourier coefficients defined by Denote by kf k ρ the norm Definition 5. An n × n matrix QðtÞ = ðq ij ðtÞÞ 1≤i,j≤n is said to be analytic quasiperiodic on D ρ with frequencies ω = ðω 1 , ω 2 , ⋯, ω r Þ, if all q ij ðtÞði, j = 1, 2, ⋯, nÞ are analytic quasiperiodic on D ρ with frequencies ω = ðω 1 , ω 2 , ⋯, ω r Þ.
Define the norm of Q by 3

Journal of Function Spaces
It is easy to see that If Q is a constant matrix, write kQk = kQk ρ for simplicity. Denote the average of QðtÞ by Q = ð q ij Þ 1≤i,j≤n , where See [24] for the existence of the limit. Also, we need two lemmas which are provided in this section for the proof of Theorem 3 that were proved in [10].

Lemma 8. Consider the differential equation
where Λ is a constant Hamiltonian matrix with n different eigenvalues ν 1 , ⋯, ν n , R is an analytic quasiperiodic Hamiltonian matrix on D ρ with frequencies ω satisfying for all 0 ≠ k ∈ ℤ r , and jν i j ≥ δε, jν i − ν j j ≥ δε, for i ≠ j, 0 ≤ i, j ≤ n, where δ is a positive constant independent of ε, then equation (27) has a unique analytic quasiperiodic Hamiltonian solution PðtÞ with P = 0, where PðtÞ has frequencies ω and satisfies with ν = 3τ + r and 0 < s < ρ, where the constant c depends only on τ and r.
By comparing the coefficients of (30), we obtain that Since R is analytic on D ρ , Y is also analytic on D ρ . Therefore, we have Hence, where ν = 3τ + r and 0 < s < ρ. Here and hereafter, we always use the same symbol c to denote different constants in estimates. Hence, Now, we prove that P is Hamiltonian. Since Λ and R are Hamiltonian, then Λ = JΛ J and R = JR J , where Λ J and R J are symmetric. Let P J = J −1 P, if P J is symmetric, then P is Hamiltonian. Below, we prove that P J is symmetric. 4 Journal of Function Spaces Substituting P = JP I into equation (27) yields and transposing equation (37), we get It is easy to see that JP J and JP T J are solutions of (27); moreover, JP J = JP T J = 0. Since the solution of (27) with P = 0 is unique, we have JP J = JP T J , which implies that P is Hamiltonian. Up to now, we have finished the proof of this lemma.

Proof of Theorem 3
From the assumptions of Theorem 3, it follows that A + ε Q is a Hamiltonian matrix with n different eigenvalues μ 1 , ⋯, μ n , and where A 1 = A + ε Q,QðtÞ = QðtÞ − Q, andQðtÞ = 0. Introduce the change of variables x = e εPðtÞ x 1 , where PðtÞ will be determined later, under this symplectic transformation, Hamiltonian system (39) is changed into the new Hamiltonian system Expand e εP and e −εP into e εP = I + εP + B, where Then, system (40) can be rewritten where We would like to havẽ which is equivalent to By Assumption 2 of Theorem 3, it is easy to see that the inequalities hold. Moreover, if the equalities also hold, where α 0 = α/2, thus, by Lemma 8, (46) is solvable for P on a smaller domain, that is, there is a unique quasiperiodic Hamiltonian matrix PðtÞ with frequencies ω on D ρ−s , which satisfies P = 0 and where s = ð1/2Þρ.Therefore, by (46), Hamiltonian system (43) becomes where From Lemma 6, it follows that

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Therefore, if ε is sufficiently small, we have Now we consider the iteration step. In the m th step, we consider the Hamiltonian system where A m has n different eigenvalues λ m 1 , ⋯, λ m n with Here, we define whereQ m = Q m ðtÞ − Q m . We need to solve If and A m+1 has n different eigenvalues λ m+1 by Lemma 8, there is a unique quasiperiodic Hamiltonian matrix P m ðtÞ with frequencies ω on D ρ m −s m , which satisfies Thus, under the symplectic change of variables x m = e ε 2m P m ðtÞ x m+1 , system (56) is changed into where From Lemma 6, it follows that Therefore, if jεj is sufficiently small, by (60) we have Now we prove that the iteration is convergent as m ⟶ ∞. When m = 1, we choose At the m th step, we define Journal of Function Spaces By (64), we have where the constant c depends only on α, ρ. Hence, it follows that If cF 1 < 1, then cF m ⟶ 0 as m ⟶ ∞. From (60), it follows that Thus, if cF 1 < 1/2, then Since if cF 1 ≤ δε/ð3n − 1Þβ 2 m , it follows from (71) that where β m = max fkS m k, kS −1 m kg and S m is the regular matrix in Lemma 7 such that Thus, it follows from Lemma 7 that A m+1 has n different eigenvalues λ m+1 1 , ⋯, λ m+1 n . Moreover, In fact, Moreover, we have Thus, if cF 1 ≤ min f1/2, ð1/4Þδε, δε/ð3n − 1Þβ 2 m g, that is, 0 < ε ≤ min f1, c/kQk ρ , c/kQk 2 ρ g, then by (68), we have In the same way as above, we have (69), the composition of all the changes e ε 2 m P m converges to ψ as m ⟶ ∞. Obviously, Furthermore, it follows from (71) that A m is convergent as m ⟶ ∞. Define B = lim m→∞ A m . Then, under the symplectic change of variables x = ψðtÞy, m ⟶ ∞ Hamiltonian system (1) is changed into _ y = By. Now we prove that, for most sufficiently small ε, such symplectic transformation exists. From the above iteration, we need to prove that the nonresonant conditions for all 0 ≠ k ∈ ℤ r , 1 ≤ i, j ≤ n, m = 0, 1, 2, ⋯, hold for most sufficiently small ε.
where we choose such that, for ε ∈ ð0, ε 0 Þ, the above iteration is convergent, and

The Applications
As an example, we apply Theorem 3 to the following Schrödinger equation