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

Xue, Nina; Zhao, Wencai

doi:https://doi.org/10.1155/2020/6260253

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Volume 2020 | Article ID 6260253 | https://doi.org/10.1155/2020/6260253

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

Nina Xue¹and Wencai Zhao²

Academic Editor: Liguang Wang

Received30 Jan 2020

Accepted03 Apr 2020

Published05 May 2020

Abstract

In this paper, we consider the reducibility of the quasiperiodic linear Hamiltonian system , where is a constant matrix with possible multiple eigenvalues, is analytic quasiperiodic with respect to , and is a small parameter. Under some nonresonant conditions, it is proved that, for most sufficiently small , 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 . Applications to the Schrödinger equation are also given.

1. Introduction

In this paper, we are concerned with the reducibility of the quasiperiodic linear Hamiltonian system where is a constant matrix with possible multiple eigenvalues, is analytic quasiperiodic with respect to , and is a small parameter.

Firstly, let us recall the definition of the reducibility for quasiperiodic linear systems. Let be an quasiperiodic matrix, the differential equation is called reducible, if there exists a nonsingular quasiperiodic change of variables where and are quasiperiodic and bounded, which changes (2) into where 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 . 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 satisfies the “full spectrum” condition.

A typical example of quasiperiodic linear systems comes from the (continuous time) quasiperiodic Schrödinger operators, which are defined on as where is called the potential and is called the phase. It is well known that the spectrum of does not depend on the phase when 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 , which are sufficiently large, if is fixed and satisfies the Diophantine condition where are positive constants. The result was generalized by Rüssmann [4] for 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 in Lebesgue measure sense, where is a fixed Diophantine vector.

In the case that , 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 . Hou and You [6] proved, besides other results, the nonperturbative reducibility for (7).

The reducibility of quasiperiodic linear systems with coefficients in was considered by Jorba and Simó [7]. Suppose that is a constant matrix with different eigenvalues, they proved that if the eigenvalues of and the frequencies of satisfy some nonresonant conditions, then there exists sufficiently small and a nonempty Cantor set , such that for any , system (1) is reducible. Moreover, the relative measure of the set in is exponentially small in . Junxiang [8] obtained the similar result for the multiple eigenvalue case. Later, many authors [7–10] 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 has different nonzero eigenvalues, they proved that under some nonresonant conditions and nondegeneracy conditions, there exists a nonempty Cantor set , such that for all , system (10) is reducible. Later, Wang and Xu [11] further investigated the nonlinear quasiperiodic system where is a real constant matrix, and as . They proved without any nondegeneracy condition, one of two results holds: (1) system (11) is reducible to for all ; (2) there exists a nonempty Cantor set , such that system (11) is reducible to for all .

In [12], Her and You considered one-parameter family of quasiperiodic linear system where be the set of matrices depending analytically on parameter in a closed interval (), and is analytic and small. They proved that under some nonresonant conditions and nondegeneracy conditions, there exists an open and dense set in , such that for each , system (12) is reducible for almost all .

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 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 is exponentially small in . 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.

In this paper, we will study the reducibility of quasiperiodic linear Hamiltonian system (1), where matrix may have multiple eigenvalues. To this end, the following assumptions are made.

Assumption 1. (nonresonant condition). Let all eigenvalues of matrix be be an analytic quasiperiodic function on with the frequencies . Suppose that and satisfy the nonresonant conditions for all , , , where is a small constant and .

Assumption 2. (nondegeneracy condition). Assume that has different eigenvalues with , , , , , where is a positive constant independently of . Here, we denote the average of by , 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 and a nonempty Cantor subset with positive Lebesgue measure, such that for , Hamiltonian system (1) is reducible, i.e., there is an analytic quasiperiodic symplectic transformation , where has same frequencies as , which changes (1) into the Hamiltonian system , where is a constant matrix. Moreover, if is small enough, the relative measure of in is close to 1.

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 to have multiple eigenvalues. Of course, if the eigenvalues of are different, the nondegeneracy condition holds naturally.

As an example, we apply Theorem 3 to the following Schrödinger equation: where is analytic quasiperiodic with the frequencies . Denote the average of by . If and the frequencies of satisfy the Diophantine condition where is a small constant and , then there exists some sufficiently small , equation (17) is reducible and the equilibrium of (17) is stable in the sense of Lyapunov for most sufficiently small . Moreover, all solutions of equation (17) are quasiperiodic with the frequencies for most sufficiently small , where as . Here, we remark that if we rewrite equation (17) into Hamiltonian system (1), we find that which has multiple eigenvalues . 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 [16–23] 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.

2. Some Preliminaries

We first give the definition of quasiperiodic functions.

Definition 4. A function is said to be a quasiperiodic function with a vector of basic frequencies , if , where is periodic in all its arguments and for . Moreover, if is analytic on , we say that is analytic quasiperiodic on .

It is well known that an analytic quasiperiodic function can be expanded as Fourier series with Fourier coefficients defined by

Denote by the norm

Definition 5. An matrix is said to be analytic quasiperiodic on with frequencies , if all are analytic quasiperiodic on with frequencies .

Define the norm of by .

It is easy to see that

If is a constant matrix, write for simplicity. Denote the average of by , 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 6. Let be a function that satisfies , , , . Then,

Lemma 7. Suppose that is an matrix with different nonzero eigenvalues satisfying , , , and is a regular matrix such that . Set , and choose such that If verifies , then the following conclusions hold: (1) has different nonzero eigenvalues (2)There exists a regular matrix such that , which satisfies , where .

The next lemma is used to perform a step of the inductive procedure in the proof of Theorem 3.

Lemma 8. Consider the differential equation where is a constant Hamiltonian matrix with different eigenvalues , is an analytic quasiperiodic Hamiltonian matrix on with frequencies satisfying .
If for all , and , , for , , , where is a positive constant independent of , then equation (27) has a unique analytic quasiperiodic Hamiltonian solution with , where has frequencies and satisfies with and , where the constant depends only on and .

Proof. Choosing such that , making the change of variable and defining , equation (27) becomes Expanding and into Fourier series yields that where and .

By comparing the coefficients of (30), we obtain that

Since is analytic on , is also analytic on . Therefore, we have

Hence, where and . Here and hereafter, we always use the same symbol to denote different constants in estimates. Hence,

Now, we prove that is Hamiltonian. Since and are Hamiltonian, then and , where and are symmetric. Let , if is symmetric, then is Hamiltonian. Below, we prove that is symmetric. Substituting into equation (27) yields and transposing equation (37), we get

It is easy to see that and are solutions of (27); moreover, . Since the solution of (27) with is unique, we have , which implies that is Hamiltonian. Up to now, we have finished the proof of this lemma.

3. Proof of Theorem 3

From the assumptions of Theorem 3, it follows that is a Hamiltonian matrix with different eigenvalues , and , , , , , where is a positive constant independent of . We rewrite Hamiltonian system (1) into where , , and .

Introduce the change of variables , where will be determined later, under this symplectic transformation, Hamiltonian system (39) is changed into the new Hamiltonian system

Expand and into where

Then, system (40) can be rewritten where

We would like to have 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 , thus, by Lemma 8, (46) is solvable for on a smaller domain, that is, there is a unique quasiperiodic Hamiltonian matrix with frequencies on , which satisfies and where .Therefore, by (46), Hamiltonian system (43) becomes where

From Lemma 6, it follows that

Therefore, if is sufficiently small, we have

Now we consider the iteration step. In the ^th step, we consider the Hamiltonian system where has different eigenvalues with Here, we define , .

Let , then system (54) becomes where . We need to solve

If and has different eigenvalues with by Lemma 8, there is a unique quasiperiodic Hamiltonian matrix with frequencies on , which satisfies

Thus, under the symplectic change of variables , system (56) is changed into where

From Lemma 6, it follows that

Therefore, if is sufficiently small, by (60) we have

Now we prove that the iteration is convergent as . When , we choose

At the ^th step, we define

By (64), we have where the constant depends only on . Hence, it follows that

If , then as . From (60), it follows that

Thus, if , then

Since if , it follows from (71) that where and is the regular matrix in Lemma 7 such that

Thus, it follows from Lemma 7 that has different eigenvalues .

Moreover,

In fact,

Moreover, we have

Thus, if , that is, , then by (68), we have

In the same way as above, we have

Let . By (69), the composition of all the changes converges to as . Obviously,

Furthermore, it follows from (71) that is convergent as . Define . Then, under the symplectic change of variables , Hamiltonian system (1) is changed into .

Now we prove that, for most sufficiently small , such symplectic transformation exists. From the above iteration, we need to prove that the nonresonant conditions for all , hold for most sufficiently small .

Let where we choose such that, for , the above iteration is convergent, and

For , by (71), we have

Hence, where .

If , then and .

Suppose that . By (83), it follows that

Thus,

Let

Then

Thus

Let , then

Therefore, is a nonempty subset of . Thus, for , Hamiltonian system (1) is reducible, i.e., there exists a symplectic transformation , which changes Hamiltonian system (1) into the Hamiltonian system . Thus, Theorem 3 is proved completely.

4. The Applications

As an example, we apply Theorem 3 to the following Schrödinger equation where is an analytic quasiperiodic function on with frequencies . Denote the average of by , and suppose .

Let , then equation (93) can be rewritten in the equivalent form

To apply Theorem 3, we express (94) in the form where

It is easy to see that has multiple eigenvalues ; moreover, has two different eigenvalues , , where stands for the average of the matrix and . It is clear that where we choose , which is a constant independent of . Therefore, Theorem 3 can be applied. It follows from Theorem 3 that the following result holds.

Theorem 9. Assume that is an analytic quasiperiodic function on with frequencies , and . If the frequencies of satisfy the Diophantine condition where is a small constant and .

Then, there exist some sufficiently small and a nonempty Cantor subset with positive Lebesgue measure, such that for , system (95) is reducible. Moreover, if is small enough, the relative measure of in is close to 1.

Remark 10. From Theorem 9, it follows that equation (93) can be changed into a constant coefficient differential equation for most sufficiently small .

Now we want to study the Lyapunov stability of the equilibrium of equation (93), using the results obtained in Section 3. If is periodic in time ( is the period), one famous stability criterion was given by Magnus and Winkler [25] for Hill’s equation

That is, (100) is stable if which can be shown using a Poincaré inequality. Such a stability criterion had been generalized and improved by Zhang and Li in [26], which now is the so-called criterion. Recently, Zhang in [27] had extended such a criterion to the linear planar Hamiltonian system where , are continuous and -periodic functions.

However, for quasiperiodic equation (93), the results above cannot be applied directly. Now, we obtain a result about the stability of the equilibrium of equation (93).

Theorem 11. Under the conditions of Theorem 9, the equilibrium of equation (93) is stable in the sense of Lyapunov for most sufficiently small .

Proof. Theorem 9 tells us that, for most sufficiently small , there exists an analytic quasiperiodic symplectic transformation , where has the same frequencies as , which changes (95) into the Hamiltonian system where is a constant matrix. Moreover, from the proof of Theorem 3 in Section 3, it follows that has two different eigenvalues, satisfying Furthermore, by the proof of Theorem 3, we have Therefore, the two different eigenvalues of are pure imaginary and can be written in the form where can be written in the following form which depends on and only.

Thus, there exists a singular symplectic matrix such that

Let , under this symplectic transformation, system (103) is changed into

Hence, by an analytic quasiperiodic symplectic transformation, equation (93) is changed into

It is easy to see that equation (110) is elliptic. Therefore, the equilibrium of equation (93) is stable in the sense of Lyapunov for most sufficiently small .

For the existence of quasiperiodic solution of equation (93), we have the following result.

Theorem 12. Under the conditions of Theorem 9, all solutions of equation (93) are quasiperiodic with frequencies for most sufficiently small , where is given by (107).

Proof. By Theorem 9, we know that, for most sufficiently small , there exists an analytic quasiperiodic symplectic transformation which has the same frequencies as ; by this transformation, equation (93) is changed into (110). On the other hand, it is easy to see that all solutions of equation (110) are periodic, and the frequency of these solutions is .

Thus, we only need to prove that, for most sufficiently small , the following nonresonant condition holds for all , where is defined in Section 3, that is, , and are the frequencies of .

If , then from the Diophantine condition (99), it follows that (111) holds.

Suppose that . Let , and

It follows from the nondegeneracy condition that

By (107), we have

From the Diophantine condition (99), it follows that

If , then and .

Suppose that , it follows from (113) that

Thus,

Then

Therefore, (111) holds for most sufficiently small .

Thus, all solutions of equation (93) are quasiperiodic with frequencies for most sufficiently small .

Data Availability

There is no additional data in the manuscript, because the main result is theoretical proof.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

Nina Xue was supported by the National Natural Science Foundation of China (11971059) and Doctoral Funding of Weifang University (2019BS02).

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Copyright © 2020 Nina Xue and Wencai Zhao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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