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In this paper, we will are concerned with existence of quasi-periodic solutions for a two-dimensional $ (2D) $ quasi-periodically forced beam equation
$ utt+Δ2u+εϕ(t)(u+u3)=0,x∈T2,t∈R $
|
(1.1) |
with periodic boundary conditions
$ u(t,x1,x2)=u(t,x1+2π,x2)=u(t,x1,x2+2π) $
|
(1.2) |
where $ \varepsilon $ is a small positive parameter, $ \phi(t) $ is a real analytic quasi-periodic function in $ t $ with frequency vector $ \omega = (\omega_1, \omega_2 \ldots, \omega_m) \subset[\varrho, 2\varrho]^m $ for some constant $ \varrho > 0. $ Such quasi-periodic functions can be written in the form
$ \phi(t) = \varphi(\omega_1 t, \ldots, \omega_m t), $ |
where $ \omega_1, \ldots, \omega_m $ are rationally independent real numbers, the "basic frequencies" of $ \phi $, and $ \varphi $ is a continuous function of period $ 2\pi $ in all arguments, called the hull of $ \phi $. Thus $ \phi $ admits a Fourier series expansion
$ \phi(t) = \sum\limits_{k\in{\mathbb{Z}}^m}\varphi_k e^{{\rm i}k\cdot\omega t}, $ |
where $ k\cdot\omega = \sum_{\hat{j} = 1}^m k_{\hat{j}}\cdot\omega_{\hat{j}} $. We think of this equation as an infinite dimensional Hamiltonian system and we study it through an infinite-dimensional KAM theory. The KAM method is a composite of Birkhoff normal form and KAM iterative techniques, and the pioneering works were given by Wayne [25], Kuksin [15] and Pöschel [19]. Over the last years the method has been well developed in one dimensional Hamiltonian PDEs. However, it is difficult to apply to higher dimensional Hamiltonian PDEs. Actually, it is difficult to draw a nice result because of complicated small divisor conditions and measure estimates between the corresponding eigenvalues when the space dimension is greater than one. In [11,12] the authors obtained quasi-periodic solutions for higher dimensional Hamiltonian PDEs by means of an infinite dimensional KAM theory, where Geng and You proved that the higher dimensional nonlinear beam equations and nonlocal Schrödinger equations possess small-amplitude linearly-stable quasi-periodic solutions. In this aspect, Eliasson-Kuksin[9], C.Procesi and M.Procesi[20], Eliasson-Grebert-Kuksin [5] made the breakthrough of obtaining quasi-periodic solutions for more interesting higher dimensional Schrödinger equations and beam equations. However, all of the work mentioned above require artificial parameters, and therefore it cannot be used for classical equations with physical background such as the higher dimensional cubic Schrödinger equation and the higher dimensional cubic beam equation. These equations with physical background have many special properties, readers can refer to [4,16,22,23,24] and references therein.
Fortunately, Geng-Xu-You[10], in 2011, used an infinite dimensional KAM theory to study the two dimensional nonlinear cubic Schrödinger equation on $ \mathbb{T}^2 $. The main approach they use is to pick the appropriate tangential frequencies, to make the non-integrable terms in normal form as sparse as possible such that the homological equations in KAM iteration is easy to solve. More recently, by the same approach, Geng and Zhou[13] looked at the two dimensional completely resonant beam equation with cubic nonlinearity
$ utt+Δ2u+u3=0,x∈T2,t∈R. $
|
(1.3) |
All works mentioned above do not conclude the case with forced terms. The present paper study the problem of existence of quasi-periodic solutions of the equation (1.1)+(1.2). Let's look at this problem through the infinite-dimensional KAM theory as developed by Geng-Zhou [13]. So the main step is to convert the equation into a form that the KAM theory for PDE can be applied. This requires reducing the linear part of Hamiltonian system to constant coefficients. A large part of the present paper will be devoted to proving the reducibility of infinite-dimensional linear quasi-periodic systems. In fact, the question of reducibility of infinite-dimensional linear quasi-periodic systems is also interesting itself.
In 1960s, Bogoliubov-Mitropolsky-Samoilenko [3] found that KAM technique can be applied to study reducibility of non-autonomous finite-dimensional linear systems to constant coefficient equations. Subsequently, the technique is well developed for the reducibility of finite-dimensional systems, and we don't want to repeat describing these developments here. Comparing with the finite-dimensional systems, the reducibility results in infinite dimensional Hamiltonian systems are relatively few. Such kind of reducibility result for PDE using KAM technique was first obtained by Bambusi and Graffi [1] for Schrödinger equation on $ \mathbb{R} $. About the reducibility results in one dimensional PDEs and its applications, readers refer to [2,7,17,18,21] and references therein.
Recently there have been some interesting results in the case of systems in higher space dimensions. Eliasson and Kuksin [6] obtained the reducibility for the linear d-dimensional Schrödinger equation
$ \dot{u} = -{\rm i}(\Delta u-\epsilon V(\phi_0+t\omega, x; \omega)u), \quad x\in\mathbb{T}^d. $ |
Grébert and Paturel [14] proved that a linear d-dimensional Schrödinger equation on $ \mathbb{R}^d $ with harmonic potential $ |x|^2 $ and small $ t $-quasiperiodic potential
$ {\rm i}\partial_t u-\Delta u+|x|^2 u+\varepsilon V(t\omega, x)u = 0, \quad x\in\mathbb{R}^d $ |
reduced to an autonomous system for most values of the frequency vector $ \omega\in \mathbb{R}^n $. For recent development for high dimensional wave equations, Eliasson-Grébert-Kuksin [8], in 2014, studied reducibility of linear quasi-periodic wave equation.
However, the reducibility results in higher dimension are still very few. The author Min Zhang of the present paper has studied the two dimensional Schrödinger equations with Quasi-periodic forcing in [27]. However, it would seem that the result cannot be directly applied to our problems because of the difference in the linear part of Hamiltonian systems and the Birkhoff normal forms. As far as we know, the reducibility for the linear part of the beam equation (1.1) is still open. In this paper, by utilizing the measure estimation of infinitely many small divisors, we construct a symplectic change of coordinates which can reduce the linear part of Hamiltonian system to constant coefficients. Subsequently, we construct a symplectic change of coordinates which can transform the Hamiltonian into some Birkhoff normal form depending sparse angle-dependent terms, which can be achieved by choosing the appropriate tangential sites. Lastly, we show that there are many quasi-periodic solutions for the equation (1.1) via KAM theory.
Remark 1.1. Similar to [13], we introduced a special subset of $ \mathbb{Z}^2 $
$ Z2odd={n=(n1,n2),n1∈2Z−1,n2∈2Z}, $
|
(1.4) |
for the small divisor problem could be simplified. Then we define subspace $ \mathcal{U} $ in $ L^2(\mathbb{T}^2) $ as follows
$ \mathcal{U} = \{u = \sum\limits_{j\in\mathbb{Z}^2_{odd}}u_j\phi_j, \quad \phi_j(x) = e^{{\rm i} \lt j, x \gt }\}. $ |
We only prove the existence of quasi-periodic solutions of the equation ${\rm(1.1)}$ in $ \mathcal{U} $.
The following definition quantifies the conditions the tangential sites satisfy. It acquired from Geng-Xu-You[10].
Definition 1.1. A finite set $ S = \{i^*_1 = (\tilde{x}_{1}, \tilde{y}_{1}), \cdots, i^*_n = (\tilde{x}_{n}, \tilde{y}_{n})\}\subset{\mathbb{Z}}^2_{odd} (n\geq 2) $ is called admissible if
$ \rm(i). $ Any three different points of them are not vertices of a rectangle (if $ n > 2 $) or $ n = 2 $.
$ \rm(ii). $ For any $ d\in{\mathbb{Z}}^2_{odd}\setminus S $, there exists at most one triplet $ \{i, j, l\} $ with $ i, j\in S, l\in{\mathbb{Z}}^2_{odd}\setminus S $ such that $ d-l+i-j = 0 $ and $ {|i|}^2-{|j|}^2+{|d|}^2-{|l|}^2 = 0 $. If such triplet exists, we say that $ d, l $ are resonant in the first type and denote all such $ d $ by $ \mathcal{L}_1 $.
$ \rm(iii). $ For any $ d\in{\mathbb{Z}}^2_{odd}\setminus S $, there exists at most one triplet $ \{i, j, l\} $ with $ i, j\in S, l\in{\mathbb{Z}}^2_{odd}\setminus S $ such that $ d+l-i-j = 0 $ and $ {|d|}^2+{|l|}^2-{|i|}^2-{|j|}^2 = 0 $. If such triplet exists, we say that $ d, l $ are resonant in the second type and denote all such $ d $ by $ \mathcal{L}_2 $.
$ \rm(iv). $ Any $ d\in{\mathbb{Z}}^2_{odd}\setminus S $ should not be in $ \mathcal{L}_1 $ and $ \mathcal{L}_2 $ at the same time. It means that $ \mathcal{L}_1\cap\mathcal{L}_2 = \emptyset. $
Remark 1.2. We can give an example to show the admissible set $ S $ above is non-empty. For example, for any given positive integer $ n\geq 2 $, the first point $ (\tilde{x}_1, \tilde{y}_1)\in {\mathbb{Z}}^2_{odd} $ is chosen as $ \tilde{x}_{1} > n^2, \tilde{y}_{1} = 2\tilde{x}_{1}^{5^n}, $ and the second one is chosen as $ \tilde{x}_{2} = \tilde{x}_{1}^5, \tilde{y}_{2} = 2\tilde{x}_{2}^{5^n} $, the others are defined inductively by
$ \tilde{x}_{\hat{j}+1} = \tilde{x}_{\hat{j}}^5\prod\limits_{2\leq{\hat m}\leq{\hat j}, 1\leq {\hat l} \lt {\hat m}}\big({(\tilde{x}_{\hat m}-\tilde{x}_{\hat l})}^2+{(\tilde{y}_{\hat m}-\tilde{y}_{\hat l})}^2+1\big), \quad 2\leq {\hat j}\leq n-1, $ |
$ \tilde{y}_{\hat{j}+1} = 2\tilde{x}_{\hat{j}+1}^{5^n}, \quad 2\leq {\hat j}\leq n-1. $ |
The choice of the admissible set is same to that in [13], where the proof of such admissible set is given.
In this paper, we assume that
$ \rm\bf(H) $ $ \phi(t) $ is a real analytic quasi-periodic function in $ t $ with frequency vector $ \omega, $ and $ [\phi]\neq0 $ where $ [\phi] $ denotes the time average of $ \phi $, coinciding with the space average.
The main result of this paper in the following. The proof is based on an infinite dimensional KAM theorem inspired by Geng-Zhou[13].
Theorem 1.1. (Main Theorem) Given $ \varrho $, $ \phi(t) $ as above. Then for arbitrary admissible set $ S\subset{\mathbb{Z}}^2_{odd} $ and for any $ 0 < \gamma < 1, 0 < \rho < 1 $ and $ \gamma' > 0 $ be small enough, there exists $ \varepsilon^*(\rho, \gamma, \gamma') > 0 $ so that for all $ 0 < \varepsilon < \varepsilon^*, $ there exists $ R\subset [\varrho, 2\varrho]^m $ with $ {\rm meas}\, R > (1-\gamma)\varrho^m $ and there exists $ \Sigma_{\gamma'}\subset\Sigma: = R\times [0, 1]^{n} $ with $ {\rm meas}\, (\Sigma\setminus\Sigma_{\gamma'}) = O(\sqrt[4]{\gamma'}) $, so that for $ (\omega, \tilde{\xi}_{i^*_1}, \ldots, \tilde{\xi}_{i^*_n})\in \Sigma_\gamma', $ the beam equation $ \rm(1.1)+(1.2) $ admits a quasi-periodic solution in the following
$ u(t,x)=∑j∈S(1+gj(ωt,ω,ε))√3˜ξj16|j|2π2(ei˜ωjtei<j,x>+e−i˜ωjte−i<j,x>)+O(|˜ξ|3/2), $
|
where $ g_j(\vartheta, \omega, \varepsilon) = \varepsilon^{\rho}g_j^*(\vartheta, \omega, \varepsilon) $ is of period $ 2\pi $ in each component of $ \vartheta $ and for $ j\in S, \vartheta\in\Theta(\sigma_0/2), \omega\in\Omega $, we have $ |g_j^*(\vartheta, \omega, \varepsilon)|\leq C $. And the solution $ u(t, x) $ is quasi-periodic in terms of $ t $ with the frequency vector $ \tilde{\omega} = (\omega, (\tilde{\omega}_j)_{j\in S}) $, and $ \tilde{\omega}_j = \varepsilon^{-4}|j|^2+O(|\tilde{\xi}|)+O(\varepsilon) $.
Let's rewrite the beam equation (1.1) as follows
$ utt+Δ2u+εϕ(t)(u+u3)=0,x∈T2,t∈R. $
|
(2.1) |
Introduce a variable $ v = u_t $, the equation (2.1) is transformed into
$ {ut=v,vt=−Δ2u−εϕ(t)(u+u3). $
|
(2.2) |
Introducing $ q = \frac{1}{\sqrt{2}}({(-\Delta)^{\frac{1}{2}}}u-{\rm i}{(-\Delta)^{-\frac{1}{2}}}v) $ and (2.2) is transformed into
$ −iqt=−Δq+1√2εϕ(t)(−Δ)−12((−Δ)−12(q+ˉq√2)+((−Δ)−12(q+ˉq√2))3). $
|
(2.3) |
The equation can be written as the Hamiltonian equation $ \dot{q} = {\rm i}\frac{\partial H}{\partial {\bar q}} $ and the corresponding Hamiltonian functions is
$ H=∫T2((−Δ)q)ˉqdx+12εϕ(t)∫T2((−Δ)−12(q+ˉq√2))2dx+14εϕ(t)∫T2((−Δ)−12(q+ˉq√2))4dx. $
|
(2.4) |
The eigenvalues and eigenfunctions of the linear operator $ -\Delta $ with the periodic boundary conditions are respectively $ \lambda_j = |j|^2 $ and $ \phi_j(x) = \frac{1}{2\pi}e^{{\rm i} < j, x > } $. Now let's expand $ q $ into a Fourier series
$ q=∑j∈Z2oddqjϕj, $
|
(2.5) |
the coordinates belong to some Hilbert space $ l^{a, s} $ of sequences $ q = (\cdots, q_j, \cdots)_{j\in{\mathbb{Z}}^2_{odd}} $ that has the finite norm
$ \|q\|_{a, s} = \sum\limits_{j\in{\mathbb{Z}}^2_{odd}}|q_j||j|^s e^{|j|a} \quad(a \gt 0, s \gt 0). $ |
The corresponding symplectic structure is $ {\rm i}\sum_{j\in{{\mathbb{Z}}^2_{odd}}} dq_j\wedge d{\bar q}_j $. In the coordinates, the Hamiltonian equation (2.3) can be written as
$ ˙qj=i∂H∂ˉqj,∀j∈Z2odd $
|
(2.6) |
with
$ H = \Lambda+G $ |
where
$ \Lambda = \sum\limits_{j\in{\mathbb{Z}}^2_{odd}}\big(\lambda_j{|q_j|}^2+\frac{\varepsilon}{4\lambda_j} \phi(t)(q_jq_{-j}+2{|q_j|}^2+{\bar{q}}_j{\bar{q}}_{-j})\big) $ |
$ G=164π2εϕ(t)∑i+j+d+l=0i,j,d,l∈Z2odd1√λiλjλdλl(qiqjqdql+ˉqiˉqjˉqdˉql)+332π2εϕ(t)∑i−j+d−l=0i,j,d,l∈Z2odd1√λiλjλdλlqiˉqjqdˉql+116π2εϕ(t)∑i+j+d−l=0i,j,d,l∈Z2odd1√λiλjλdλl(qiqjqdˉql+ˉqiˉqjˉqdql). $
|
Denote $ \varphi(\vartheta) $ be the shell of $ \phi(t) $, we introduce the action-angle variable $ (J, \vartheta)\in{\Bbb{R}^m\times\Bbb{T}^m}, $ then (2.6) can be written as follows
$ \label{2.7} \dot{\vartheta} = \omega, \quad \dot{J} = -\frac{\partial H}{\partial \vartheta}, \quad \dot{q_j} = {\rm i}\frac{\partial H}{\partial \bar{q}_j}, \quad j\in{\mathbb{Z}}^2_{odd} $ |
and the corresponding Hamiltonian function is
$ H=ˉH+εG4, $
|
(2.7) |
where
$ ˉH=<ω,J>+∑j∈Z2odd(λj|qj|2+ε4λjφ(ϑ)(qjq−j+2|qj|2+ˉqjˉq−j)), $
|
(2.8) |
$ G4=164π2∑i+j+d+l=0i,j,d,l∈Z2odd1√λiλjλdλl(G4,0ijdl(ϑ)qiqjqdql+G0,4ijdl(ϑ)ˉqiˉqjˉqdˉql)+332π2∑i−j+d−l=0i,j,d,l∈Z2odd1√λiλjλdλlG2,2ijdl(ϑ)qiˉqjqdˉql+116π2∑i+j+d−l=0i,j,d,l∈Z2odd1√λiλjλdλl(G3,1ijdl(ϑ)qiqjqdˉql+G1,3ijdl(ϑ)ˉqiˉqjˉqdql) $
|
(2.9) |
and
$ G4,0ijdl(ϑ)=G0,4ijdl(ϑ)={φ(ϑ),i+j+d+l=0, 0,i+j+d+l≠0, $
|
(2.10) |
$ G2,2ijdl(ϑ)={φ(ϑ),i−j+d−l=0, 0,i−j+d−l≠0, $
|
(2.11) |
$ G3,1ijdl(ϑ)=G1,3ijdl(ϑ)={φ(ϑ),i+j+d−l=0, 0,i+j+d−l≠0. $
|
(2.12) |
Now We are going to study the reducibility of the Hamiltonian (2.8). To make this reducibility, we introduce the notations and spaces as follows.
For given $ \sigma_0 > 0, \Gamma > 0, 0 < \rho < 1 $, define
$ \sigma_\nu = \sigma_0\left(1-\frac{\sum_{\hat{j} = 1}^{\nu}{\hat{j}}^{-2}}{2\sum_{{\hat{j}} = 1}^{\infty}{\hat{j}}^{-2}}\right), \quad\nu = 1, 2, \ldots $ |
$ \Gamma_\nu = \Gamma\left(1+C\sum\limits_{{\hat{j}} = \nu}^{+\infty}\varepsilon_{\hat{j}}^\rho\right), \quad\nu = 0, 1, \ldots $ |
where $ C $ is a constant. Let
$ \varepsilon_0 = \varepsilon, \quad \varepsilon_{\nu} = \varepsilon^{(1+\rho)^{\nu}}, \quad \nu = 1, 2, \ldots $ |
$ \Theta(\sigma_\nu) = \left\{\vartheta = (\vartheta_1, \ldots, \vartheta_{m})\in \Bbb{C}^{m}/2\pi\Bbb{Z}^{m}:|{\rm Im} \vartheta_{\hat{j}}| \lt \sigma_\nu, {\hat{j}} = 1, 2, \ldots, m\right\}, \nu = 0, 1, 2, \ldots. $ |
and denote
$ Da,sν={(ϑ,J,q,ˉq)∈Cm/2πZm×Cm×la,s×la,s:|Imϑ|<σν,|J|<Γ2ν,‖q‖a,s<Γν,‖ˉq‖a,s<Γν}ν=0,1,2,…, $
|
$ Da,s∞={(ϑ,J,q,ˉq)∈Cm/2πZm×Cm×la,s×la,s:|Imϑ|<σ0/2,|J|<Γ2,‖q‖a,s<Γ,‖ˉq‖a,s<Γ}, $
|
where $ |\cdot| $ stands for the sup-norm of complex vectors and $ l^{a, s} $ stands for complex Hilbert space. For arbitrary four order Whitney smooth function $ F(\omega) $ on closed bounded set $ R^* $, let
$ \|F\|^*_{R^*} = \sup\limits_{\omega\in R^*}\sum\limits_{0\leq \hat{j}\leq 4}|\partial^{\hat{j}}_\omega F|. $ |
Let $ F(\omega) $ is a vector function from $ R^* $ to $ l^{a, s}(or \Bbb{R}^{m_1\times m_2}) $ which is four order whitney smooth on $ R^* $, we denote
$ \|F\|^*_{a, s, R^*} = \|(\|F_i(\omega)\|^*_{R^*})_i\|_{a, s} \quad\left(or \|F\|^*_{R^*} = \max\limits_{1\leq i_1\leq m_1}\sum\limits_{1\leq i_2\leq m_2}(\|F_{i_1i_2}(\omega)\|^*_{R^*})\right). $ |
Given $ \sigma_{D^{a, s}} > 0, \Gamma_{D^{a, s}} > 0, $ we define
$ Da,s={(ϑ,J,q,ˉq)∈Cm/2πZm×Cm×la,s×la,s:|Imϑ|<σDa,s,|J|<Γ2Da,s,‖q‖a,s<ΓDa,s,‖ˉq‖a,s<ΓDa,s}. $
|
If $ \tilde w = (\vartheta, J, q, \bar{q})\in D^{a, s} $, we define the weighted norm for $ \tilde w $ by
$ |\tilde w|_{a, s} = |\vartheta|+\frac{1}{\Gamma_{D^{a, s}}^2}|J|+\frac{1}{\Gamma_{D^{a, s}}}\|q\|_{a, s} +\frac{1}{\Gamma_{D^{a, s}}}\|\bar{q}\|_{a, s}. $ |
Let $ F(\eta; \omega) $ is a function from $ D^{a, s}\times R^* $ to $ l^{a, s}(or \Bbb{R}^{m_1\times m_2}) $ which is four order whitney smooth on $ \omega $, we denote
$ \|F\|^*_{a, s, D^{a, s}\times R^*} = \sup\limits_{\eta\in D^{a, s}}\|F\|^*_{a, s, R^*} \quad\left(or \|F\|^*_{D^{a, s}\times R^*} = \sup\limits_{\eta\in D^{a, s}}\|F\|^*_{R^*}\right). $ |
For given function $ F $, associate a hamiltonian vector field denoted as $ X_F = \{F_J, -F_\vartheta, {\rm i} F_{\bar{q}}, -{\rm i} F_q\} $, we define the weighted norm for $ X_F $ by
$ |XF|∗a,s,Da,s×R∗=‖FJ‖∗Da,s×R∗+1Γ2Da,s‖Fϑ‖∗Da,s×R∗+1ΓDa,s‖Fˉz‖∗a,s,Da,s×R∗+1ΓDa,s‖Fz‖∗a,s,Da,s×R∗. $
|
Assume $ w = (q, \bar{q})\in l^{a, s}\times l^{a, s} $ is a doubly infinite complex sequence, and $ A(\eta; \omega) $ be an operator from $ l^{a, s}\times l^{a, s} $ to $ l^{a, s}\times l^{a, s} $ for $ (\eta; \omega)\in D^{a, s}\times R^*, $ then we denote
$ \|w\|_{a, s} = \|q\|_{a, s}+\|\bar{q}\|_{a, s}, $ |
$ {\|A(\eta;\omega)\|}_{a, s, D^{a, s}\times R^*}^{\diamond} = \sup\limits_{(\eta;\omega)\in D^{a, s}\times R^*}\sup\limits_{w\neq0}\frac{\|A(\eta;\omega)w\|_{a, {s}}}{\|w\|_{a, s}}, $ |
$ \|A(\eta;\omega)\|^{\star}_{a, s, D^{a, s}\times R^*} = \sum\limits_{0\leq {\hat{j}}\leq 4}\|\partial^{\hat{j}}_\omega A\|^{\diamond}_{a, s, D^{a, s}\times R^*}. $ |
Assume $ B(\eta; \omega) $ be an operator from $ D^{a, s} $ to $ D^{a, s} $ for $ (\eta; \omega)\in D^{a, s}\times R^*, $ then we denote
$ |B(\eta;\omega)|^{\diamond}_{a, s, D^{a, s}\times R^*} = \sup\limits_{(\eta;\omega)\in D^{a, s}\times R^*}\sup\limits_{\tilde w\neq0}\frac{|B(\eta;\omega)\tilde w|_{a, {s}}}{|\tilde w|_{a, s}}, $ |
$ |B(\eta;\omega)|^{\star}_{a, s, D^{a, s}\times R^*} = \sum\limits_{0\leq {\hat{j}}\leq 4}|\partial^{\hat{j}}_\omega B|^{\diamond}_{a, s, D^{a, s}\times R^*}. $ |
Reducibility of the autonomous Hamiltonian equation corresponding to the Hamiltonian (2.8) will be proved by an KAM iteration which involves an infinite sequence of change of variables. By utilizing the measure estimation of infinitely many small divisors, we will prove that the composition of these infinite many change of variables converges to a symplectic change of coordinates, which can reduce the Hamiltonian equation corresponding to the Hamiltonian (2.8) to constant coefficients.
At the $ \nu- $step of the iteration, we consider Hamiltonian function of the form
$ Hν=H∗ν+Pν $
|
(3.1) |
where
$ H^\ast_{\nu}: = \lt \omega, J \gt +\sum\limits_{j\in\mathbb{Z}^2_{odd}}{\lambda}_{j, \nu}q_j\bar{q_j}, $ |
$ P_{\nu}: = \varepsilon_{\nu}\sum\limits_{j\in\mathbb{Z}^2_{odd}}[\eta_{j, \nu, 2, 0}(\vartheta, \omega) q_j{q}_{-j}+\eta_{j, \nu, 1, 1}(\vartheta, \omega) q_j\bar{q}_j+\eta_{j, \nu, 0, 2}(\vartheta, \omega) {\bar{q}}_j\bar{q}_{-j}] $ |
where $ \eta_{j, \nu, 2, 0} = \eta_{-j, \nu, 2, 0} $, $ \eta_{j, \nu, 0, 2} = \eta_{-j, \nu, 0, 2} $, $ \eta_{j, \nu, n_1, n_2}(\vartheta, \omega) = \sum_{k\in {\Bbb Z}^{m}}\eta_{j, \nu, k, n_1, n_2}(\omega)e^{{\rm i} < k, \vartheta > } $ when $ n_1, n_2 $ $ \in \mathbb{N}, n_1+n_2 = 2 $,
$ ηj,ν,n1,n2=λ−1jη∗j,ν,n1,n2,‖η∗j,ν,n1,n2‖∗Θ(σν)×Rν≤C,n1,n2∈N,n1+n2=2, $
|
(3.2) |
and
$ λj,0=λj,λj,ν=λj+ν−1∑ˆs=0μj,ν,ˆs, $
|
(3.3) |
with
$ μj,ν,0=ε2λj[ϕ],μj,ν,ˆs=λ−1jεˆsμ∗j,ν,ˆs,‖μ∗j,ν,ˆs‖∗Rν≤C,ˆs=1,2,…,ν. $
|
(3.4) |
We're going to construct a symplectic transformation
$ T_{\nu}: D_{\nu+1}^{a, s}\times R_{\nu+1}\longmapsto D_{\nu}^{a, s}\times R_{\nu} $ |
and
$ Hν+1=Hν∘Tν=H∗ν+1+Pν+1 $
|
(3.5) |
satisfies all the above iterative assumptions (3.1)–(3.4) marked $ \nu+1 $ on $ D_{\nu+1}^{a, s}\times R_{\nu}. $
We assume that there is a constant $ C_{*} $ and a closed set $ R_{\nu} $ satisfies
$ measRν≥ϱm(1−γ3−γ∑νˆi=0(δ(ˆi)+ˆi)−23∑+∞ˆi=0(δ(ˆi)+ˆi)−2) $
|
(3.6) |
and for arbitrary $ k\in\mathbb{Z}^m, j\in\mathbb{Z}^2_{odd}, \omega\in R_{\nu}, $
$ |<k,ω>±(λj,ν+λ−j,ν)|≥ϱC∗(δ(ν)+ν2)(|k|+δ(|k|))m+1, $
|
(3.7) |
where $ \delta(x) = 1 $ as $ x = 0 $ and $ \delta(x) = 0 $ as $ x\neq0. $ We put its proof in the Lemma 4.1 below.
Next we will construct a parameter set $ R_{\nu+1}\subset R_{\nu} $ and a symplectic coordinate transformation $ T_{\nu} $ so that the transformed Hamiltonian $ H_{\nu+1} = H^*_{\nu+1}+P_{\nu+1} $ satisfies the above iteration assumptions with new parameters $ \varepsilon_{\nu+1}, \sigma_{\nu+1}, \Gamma_{\nu+1} $ and with $ \omega\in R_{\nu+1} $.
Let $ X_{{\Psi}_\nu} $ be the Hamiltonian vector field for a Hamiltonian $ {\Psi}_\nu: $
$ {\Psi}_\nu = \varepsilon_\nu \Upsilon_\nu = \varepsilon_\nu\sum\limits_{j\in{\mathbb{Z}}^2_{odd}} [\varpi_{j, \nu, 2, 0}(\vartheta;\omega)q_j{q}_{-j}+\varpi_{j, \nu, 1, 1}(\vartheta;\omega)q_j\bar{q}_j +\varpi_{j, \nu, 0, 2}(\vartheta;\omega){\bar q}_j\bar{q}_{-j}] $ |
where
$ \varpi_{j, \nu, 2, 0}(\vartheta;\omega) = \varpi_{-j, \nu, 2, 0}(\vartheta;\omega), \quad \varpi_{j, \nu, 0, 2}(\vartheta;\omega) = \varpi_{-j, \nu, 0, 2}(\vartheta;\omega), $ |
$ ϖj,ν,n1,n2(ϑ;ω)=∑k∈Zmϖj,ν,k,n1,n2(ω)ei<k,ϑ>,n1,n2∈N,n1+n2=2 $
|
(3.8) |
and $ [\varpi_{j, \nu, 1, 1}] = 0. $ Let $ X_{{\Psi}_\nu}^t $ be its time-t map.
Let $ T_{\nu} = X_{{\Psi}_\nu}^1 $ where $ X_{{\Psi}_\nu}^1 $ denote the time-one map of the Hamiltonian vector field $ X_{{\Psi}_\nu}, $ then the system (3.1)$ (\nu) $ is transformed into the form (3.1)$ (\nu+1) $ and satisfies (3.2)$ (\nu+1) $, (3.3)$ (\nu+1) $ and (3.4)$ (\nu+1) $. More precisely, the new Hamiltonian $ H_{\nu+1} $ can be written as follows by second order Taylor formula
$ Hν+1:=Hν∘X1Ψν=H∗ν+Pν+{H∗ν,Ψν}+εν∫10(1−t){{H∗ν,Ψν},Υν}∘XtΨνdt+εν∫10{Pν,Υν}∘XtΨνdt. $
|
(3.9) |
The Hamiltonian $ \Psi_\nu $ is satisfies the homological equation
$ P_{\nu}+\{H^*_{\nu}, {\Psi}_\nu\} = \varepsilon_\nu\sum\limits_{j\in{\mathbb{Z}}^2_{odd}}[\eta_{j, \nu, 1, 1}]q_j\bar{q}_j, $ |
which is equivalent to
$ {−<ω,∂ϑϖj,ν,1,1(ϑ;ω)>+ηj,ν,1,1(ϑ;ω)=[ηj,ν,1,1],i(λj,ν+λ−j,ν)ϖj,ν,0,2(ϑ;ω)−<ω,∂ϑϖj,ν,0,2(ϑ;ω)>+ηj,ν,0,2(ϑ;ω)=0,−i(λj,ν+λ−j,ν)ϖj,ν,2,0(ϑ;ω)−<ω,∂ϑϖj,ν,2,0(ϑ;ω)>+ηj,ν,2,0(ϑ;ω)=0. $
|
(3.10) |
Let's inserting (3.8) into (3.10)
$ \left\{ i<k,ω>ϖj,ν,k,1,1(ω)=ηj,ν,k,1,1(ω),k≠0,i(<k,ω>+λj,ν+λ−j,ν)ϖj,ν,k,2,0(ω)=ηj,ν,k,2,0(ω),i(<k,ω>−λj,ν−λ−j,ν)ϖj,ν,k,0,2(ω)=ηj,ν,k,0,2(ω). \right. $
|
Thus
$ {ϖj,ν,1,1(ϑ;ω)=∑0≠k∈Zmηj,ν,k,1,1(ω)i<k,ω>ei<k,ϑ>,ϖj,ν,2,0(ϑ;ω)=∑k∈Zmηj,ν,k,2,0(ω)i(<k,ω>+λj,ν+λ−j,ν)ei<k,ϑ>,ϖj,ν,0,2(ϑ;ω)=∑k∈Zmηj,ν,k,0,2(ω)i(<k,ω>−λj,ν−λ−j,ν)ei<k,ϑ>. $
|
(3.11) |
Now we're going to estimate $ {\Psi}_\nu $ and $ X_{{\Psi}_\nu}^1 $. By Cauchy's estimate and $ (3.2)(\nu) $
$ |ηj,ν,k,n1,n2|≤‖ηj,ν,n1,n2‖∗Θ(σν)×Rνe−|k|σν≤Cλ−1je−|k|σν,n1,n2∈N,n1+n2=2 $
|
(3.12) |
and
$ |∂ˆiωηj,ν,k,n1,n2|≤‖ηj,ν,n1,n2‖∗Θ(σν)×Rνe−|k|σν≤Cλ−1je−|k|σν,ˆi=1,2,3,4 $
|
(3.13) |
can be obtained. By $ \omega\in R_{\nu} $ and $ (3.7)({\nu}), $
$ \sup\limits_{(\vartheta;\omega)\in \Theta(\sigma_{\nu+1})\times R_{\nu}}|\varpi_{j, \nu, 1, 1}|\leq CC_*\lambda_j^{-1}\varrho^{-1}\sum\limits_{0\neq k\in {\Bbb Z}^{m}}|k|^{m+1}e^{-\sigma_\nu |k|}e^{\sigma_{\nu+1} |k|} $ |
and
$ \sup\limits_{(\vartheta;\omega)\in \Theta(\sigma_{\nu+1})\times R_{\nu}}|\varpi_{j, \nu, n_1, n_2}|\leq CC_*\lambda_j^{-1}\varrho^{-1}(\delta(\nu)+\nu^2)(1+\sum\limits_{0\neq k\in {\Bbb Z}^{m}}|k|^{m+1}e^{-\sigma_\nu |k|}e^{\sigma_{\nu+1} |k|}) $ |
for $ n_1 = 0, n_2 = 2 $ or $ n_1 = 2, n_2 = 0 $. According to Lemma 3.3 in [26], for $ (\vartheta; \omega)\in \Theta(\sigma_{\nu+1})\times R_{\nu} $,
$ |ϖj,ν,1,1|,|ϖj,ν,2,0|,|ϖj,ν,0,2|≤CC∗λ−1jϱ−1(ν+1)4m+4≤Cλ−1j(ν+1)12m+28, $
|
(3.14) |
where $ C: = CC_*\varrho^{-1}. $ Moreover, in view of $ (3.3)({\nu}) $ and $ (3.4)({\nu}) $,
$ |∂ˆiωλj,ν|≤Cελ−1j,ˆi=1,2,3,4. $
|
(3.15) |
Similarly
$ |∂ˆiωϖj,ν,n1,n2|≤Cλ−1j(ν+1)12m+28,ˆi=1,2,3,4,n1,n2∈N,n1+n2=2. $
|
(3.16) |
By (3.14) and (3.16), we have
$ ‖ϖj,ν,n1,n2‖∗Θ(σν+1)×Rν≤Cλ−1j(ν+1)12m+28. $
|
(3.17) |
Similar to the above discussion, the following estimates can be obtained
$ ‖∂ϑϖj,ν,n1,n2‖∗Θ(σν+1)×Rν≤Cλ−1j(ν+1)12m+30, $
|
(3.18) |
$ ‖∂ϑϑϖj,ν,n1,n2‖∗Θ(σν+1)×Rν≤Cλ−1j(ν+1)12m+32. $
|
(3.19) |
Now let's estimate the flow $ X^t_{{\Psi}_\nu}, $ denote
$ Mj,ν(ϑ;ω)=(ϖj,ν,2,0+ϖ−j,ν,2,0ϖ−j,ν,1,1ϖj,ν,1,1ϖj,ν,0,2+ϖ−j,ν,0,2),J2=i(01−10). $
|
By (3.17)–(3.19),
$ \|M_{j, \nu}\|^*_{\Theta(\sigma_{\nu+1})\times R_{\nu}}\leq C\lambda_j^{-1}{(\nu+1)}^{12m+28}, $ |
$ \|\partial_\vartheta M_{j, \nu}\|^*_{\Theta(\sigma_{\nu+1})\times R_{\nu}}\leq C\lambda_j^{-1}{(\nu+1)}^{12m+30}, $ |
$ \|\partial_{\vartheta\vartheta} M_{j, \nu}\|^*_{\Theta(\sigma_{\nu+1})\times R_{\nu}}\leq C\lambda_j^{-1}{(\nu+1)}^{12m+32}. $ |
The vector field $ X_{{\Psi}_\nu} $ is as follows
$ {˙ϑ=0ddt(qjˉq−j)=ενJ2Mj,ν(ϑ;ω)⋅(qjˉq−j),j∈Z2odd˙J=εν∑j∈Z2odd[∂ϑϖj,ν,2,0(ϑ;ω)qjq−j+∂ϑϖj,ν,1,1(ϑ;ω)qjˉqj+∂ϑϖj,ν,0,2(ϑ;ω)ˉqjˉq−j]. $
|
The integral from $ 0 $ to $ t $ of the above equation, we have $ X_{{\Psi}_\nu}^t: $
$ {ϑ=ϑCw(t)=exp(ενJMν(ϑC;ω)t)⋅w(0)J(t)=J(0)+∫t0εν∑j∈Z2odd∂ϑϖj,ν,2,0(ϑC;ω)qj(t)q−j(t)dt+∫t0εν∑j∈Z2odd[∂ϑϖj,ν,1,1(ϑC;ω)qj(t)ˉqj(t)+∂ϑϖj,ν,0,2(ϑC;ω)ˉqj(t)ˉq−j(t)]dt. $
|
(3.20) |
where $ (\vartheta^\mathcal {C}, J(0), w(0)) $ is the initial value,
$ J=i(0˜E∞×∞−˜E∞×∞0), $
|
and $ M_{\nu}(\vartheta; \omega) $ are the corresponding matrices. According to $ \varepsilon_\nu = \varepsilon^{(1+\rho)^\nu} $, then
$ |ε1−ρν(ν+1)12m+32(C∗ϱ−1)5ν|≤C,ν=0,1,… $
|
(3.21) |
as $ \varepsilon < 1, $ where $ C $ is an absolute constant. In view of (3.17), for $ \vartheta\in \Theta(\sigma_{\nu+1}) $,
$ \varepsilon_\nu{\mathcal {J}_2}M_{j, \nu}(\vartheta;\omega ) = \lambda_j^{-1}\varepsilon_\nu{(\nu+1)}^{12m+28}M_{j, \nu}^{*1}(\vartheta;\omega ) = \lambda_j^{-1}\varepsilon_\nu^{\rho}M_{j, \nu}^{*}(\vartheta;\omega), \|M_{j, \nu}^*(\vartheta;\omega)\|^*_{\Theta(\sigma_{\nu+1})\times R_{\nu}}\leq C, $ |
then
$ ‖ενJMν(ϑ;ω)‖⋆a,s,Θ(σν+1)×Rν≤Cερν. $
|
(3.22) |
In view of (3.18),
$ ∂ϑ(ενJ2Mj,ν(ϑ;ω)⋅(qjˉq−j))=ερν⋅∂ϑ(M∗j,ν(ϑ;ω)⋅(qjˉq−j)) $
|
where
$ \left\|\partial_\vartheta \left(M_{j, \nu}^{*}(\vartheta;\omega )\cdot\big(qjˉq−j \big)\right)\right\|^*_{\Theta(\sigma_{\nu+1})\times R_{\nu}}\leq C(|q_j|+|{\bar q}_{-j}|) $
|
then
$ ‖∂ϑ(ενJMν(ϑ;ω)⋅w)‖∗Da,sν+1×Rν≤CερνΓν+1. $
|
(3.23) |
By (3.22) and (3.23),
$ exp(ενJMν(ϑ;ω)t)=Id+g∞ν(ϑ;ω,t) $
|
(3.24) |
and for $ t\in[0, 1] $,
$ ‖g∞ν(ϑ;ω,t)‖⋆a,s,Θ(σν+1)×Rν≤Cερν,‖∂ϑ(g∞ν(ϑ;ω,t)⋅w)‖∗Da,sν+1×Rν≤CερνΓν+1. $
|
(3.25) |
Let's define $ J(t) $ in (3.20) as
$ J(t)=J+gJ,ν(ϑ,w;ω,t). $
|
(3.26) |
By (3.18), (3.25) and (3.21),
$ ‖gJ,ν(ϑ,w;ω,t)‖∗Da,sν+1×Rν≤CερνΓ2ν,t∈[0,1], $
|
(3.27) |
and for any $ w'\in l^{a, s}\times l^{a, s} $,
$ ‖∂w(gJ,ν(ϑ,w;ω,t))⋅w′‖∗Da,sν+1×Rν≤CερνΓν⋅‖w′‖a,s,t∈[0,1]. $
|
(3.28) |
By (3.19), (3.25) and (3.21),
$ ‖∂ϑ(gJ,ν(ϑ,w;ω,t))‖∗Da,sν+1×Rν≤CερνΓ2ν,t∈[0,1]. $
|
(3.29) |
Denote
$ XtΨν=ΠZ+gν(ω,t):Da,sν+1×Rν+1↦Da,sν $
|
(3.30) |
from (3.20), (3.24) and (3.26),
$ {Πϑ∘XtΨν(ϑ,J,w)=ϑ:Da,sν+1×Rν+1↦Θ(σν),Πw∘XtΨν(ϑ,J,w)=(Id+g∞ν(ϑ;ω,t))⋅w:Da,sν+1×Rν+1↦la,s×la,sΠJ∘XtFν(ϑ,J,w)=J+gJ,ν(ϑ,w;ω,t):Da,sν+1×Rν+1↦Cm $
|
(3.31) |
where $ \Pi_\mathcal {Z}, \Pi_\omega $ denote the projectors
$ \Pi_\mathcal {Z}:\mathcal {Z}^{a, s}\times R_{0}\longmapsto\mathcal {Z}^{a, s}, \quad\Pi_\omega:\mathcal {Z}^{a, s}\times R_{0}\longmapsto R_{0}, $ |
and $ \Pi_\vartheta, \Pi_J, \Pi_w $ denote the projectors of $ \mathcal {Z}^{a, s} = \mathbb{C}^m/2\pi\mathbb{Z}^m\times\mathbb{C}^m\times l^{a, s}\times l^{a, s} $ on the first, second and third factor respectively. According to the first equation of (3.25), (3.27) and (3.31),
$ |XtΨν−ΠZ|∗a,s,Da,sν+1×Rν+1≤Cερν. $
|
(3.32) |
By (3.31), we have
$ DXtΨν=(Idm×m00∂ϑ(g∞ν(ϑ;ω,t)w)Id∞×∞+g∞ν(ϑ;ω,t)0∂ϑ(gJ,ν(ϑ,w;ω,t))∂w(gJ,ν(ϑ,w;ω,t))Idm×m) $
|
where $ D $ is the differentiation operator with respect to $ (\vartheta, w, J). $ In view of (3.25), (3.28) and (3.29), for $ \tilde w = (\vartheta', w', J'), (\vartheta, w, J)\in D^{a, s}_{\nu+1}, $
$ |(DXtΨν−Id)˜w|a,s≤Cερν|˜w|a,s. $
|
Thus
$ |DX_{{\Psi}_\nu}^t - Id|^{\diamond}_{a, s, D_{\nu+1}^{a, s}\times R_{\nu+1}} \lt C\varepsilon_\nu^\rho. $ |
Similarly
$ |\partial_\omega^{\hat{i}}(DX_{{\Psi}_\nu}^t - Id)|^{\diamond}_{a, s, D_{\nu+1}^{a, s}\times R_{\nu+1}} \lt C\varepsilon_\nu^\rho, \quad {\hat{i}} = 1, 2, 3, 4 $ |
and
$ |DXtΨν−Id|⋆a,s,Da,sν+1×Rν+1<Cερν. $
|
(3.33) |
Let
$ {\lambda }_{j, \nu+1} = {\lambda }_{j, \nu}+\varepsilon_\nu[\eta_{j, \nu, 1, 1}], $ |
then by $ (3.2)(\nu) $, it is obvious that $ {\lambda }_{j, \nu+1} $ satisfies the conditions $ (3.3)(\nu+1) $ and $ (3.4)(\nu+1) $.
Now let's estimate the smaller terms of (3.9). Notice that those terms are polynomials of $ q_j {q}_{-j} $, $ q_j \bar{q}_j $ and $ \bar{q}_j {\bar{q}}_{-j}. $ So we can write it
$ εν∫10(1−t){{H2ν,Ψν},Υν}∘XtΨνdt+εν∫10{Pν,Υν}∘XtΨνdt=ε2ν∑j∈Z2odd[˜ηj,ν+1,2,0(ϑ;ω)qjq−j+˜ηj,ν+1,1,1(ϑ;ω)qjˉqj+˜ηj,ν+1,0,2(ϑ;ω)ˉqjˉq−j], $
|
where from
$ \{H^*_{\nu}, {\Psi}_\nu\} = \varepsilon_\nu\sum\limits_{j\in\mathbb{Z}^2_{odd}}[\eta_{j, \nu, 1, 1}]q_j\bar{q}_j-P_{\nu}, $ |
we know that $ {{\tilde{\eta}}_{j, \nu+1, n_1, n_2}(\vartheta; \omega)} $ is a linear combination of the product of $ \varpi_{j, \nu, n_1, n_2} $ and $ {\eta_{j, \nu, m_1, m_2}} $. By (3.17) and $ (3.2)(\nu) $,
$ \varpi_{j, \nu, n_1, n_2}(\vartheta;\omega) = \lambda_j^{-1}{(\nu+1)}^{12m+28}\varpi^*_{j, \nu, n_1, n_2}(\vartheta;\omega), \quad \|\varpi^*_{j, \nu, n_1, n_2}\|^*_{\Theta(\sigma_{\nu+1})\times R_{\nu}}\leq C $ |
and
$ \eta_{j, \nu, n_1, n_2}(\vartheta;\omega) = \lambda_j^{-1}\eta_{j, \nu, n_1, n_2}^{*}(\vartheta;\omega), \quad \|\eta_{j, \nu, n_1, n_2}^{*}(\vartheta;\omega)\|^*_{\Theta(\sigma_{\nu+1})\times R_{\nu}}\leq C $ |
respectively. Thereby, we have
$ \tilde\eta_{j, \nu+1, n_1, n_2}(\vartheta;\omega) = \lambda_j^{-1}{(\nu+1)}^{12m+28}\tilde\eta^*_{j, \nu+1, n_1, n_2}(\vartheta;\omega), \quad \|\tilde\eta_{j, \nu+1, n_1, n_2}^{*}\|^*_{\Theta(\sigma_{\nu+1})\times R_{\nu}}\leq C. $ |
According to $ \varepsilon_\nu^{1-\rho}{(\nu+1)}^{12m+28}\leq1 $ as $ \varepsilon < 1, $ then
$ \eta_{j, \nu+1, n_1, n_2}: = \varepsilon_\nu^{1-\rho}\tilde{\eta}_{j, \nu+1, n_1, n_2} = \lambda_j^{-1}\eta_{j, \nu+1, n_1, n_2}^{*}, \quad\|\eta_{j, \nu+1, n_1, n_2}^{*}\|^*_{\Theta(\sigma_{\nu+1})\times R_{\nu}}\leq C. $ |
From $ \varepsilon_\nu^{2-(1-\rho)} = \varepsilon_{\nu+1}, $ we have $ (3.1)(\nu+1) $ is defined in $ D_{\nu+1}^{a, s} $ and $ {\lambda}_{j, \nu+1} $ satisfies $ (3.3)(\nu+1), (3.4)(\nu+1) $ and $ {\eta_{j, \nu+1, n_1, n_2}} $ satisfies $ (3.2)(\nu+1) $.
The reducibility of the linear Hamiltonian systems can be summarized as follows.
Theorem 3.1. Given $ \sigma_0 > 0, $ $ 0 < \gamma < 1, 0 < \rho < 1. $ Then there is a $ \varepsilon^*(\gamma) > 0 $ such that for any $ 0 < \varepsilon < \varepsilon^*(\gamma) $, there exists a set $ \underline { R}\subset [\varrho, 2\varrho]^{m}, \varrho > 0 $ with $ \rm meas\underline R\geq(1-\frac{2\gamma}{3})\varrho^m $ and a symplectic transformation $ \Sigma_\infty^0 $ defined on $ D^{a}_\infty\times\underline R $ changes the Hamiltonian $ (\rm {2.8}) $ into
$ \bar{H}\circ \Sigma_\infty^0 = \lt \omega, J \gt +\sum\limits_{j\in\mathbb{Z}^2_{odd}}\mu_j|{q}_j|^2, $ |
where
$ \mu_j = \lambda_{j}+\frac{\varepsilon}{2\lambda_{j}} [\phi]+\frac{1}{\lambda_{j}}\varepsilon^{(1+\rho)}\mu_j^*, \quad \|\mu_j^*\|_{\underline R}^*\leq C, \quad j\in\mathbb{Z}^2_{odd}. $ |
Moreover, there exists a constant $ C > 0 $ such that
$ |\Sigma_\infty^0-id|^*_{a, s, D^{a, s}_\infty\times\underline R}\leq C\varepsilon^{\rho}, $ |
where $ id $ is identity mapping.
Proof. Let $ \eta_{j, 0, 2, 0} = \eta_{j, 0, 0, 2} = \frac{1}{4\lambda_j}\varphi(\vartheta) $, $ \eta_{j, 0, 1, 1} = \frac{1}{2\lambda_j}\varphi(\vartheta) $, we have that $ H_{0} = \bar{ H} $ and $ \eta_{j, 0, n_1, n_2} = \lambda_j^{-1}\eta_{j, 0, n_1, n_2}^{*}, \|\eta_{j, 0, n_1, n_2}^{*}\|^*_{\Theta(\sigma_0)\times R_{0}}\leq C, n_1, n_2\in\mathbb{N}, n_1+n_2 = 2 $ where $ C $ is an absolute constant. i.e., the assumptions (3.1), (3.2), (3.3), (3.4) of the iteration are satisfied when $ \nu = 0 $.
We obtain the following sequences:
$ R_\infty\subset\cdots\subset R_{\nu}\subset\cdots\subset R_1\subset R_{0}\subset {[\varrho, 2\varrho]}^m, $ |
$ D_{0}^{a, s}\supset D_{1}^{a, s}\supset\cdots\supset D_{\nu}^{a, s}\supset\cdots\supset D^{a, s}_\infty. $ |
From (3.30), (3.32) and (3.33), denote
$ Tν=X1Fν=ΠZ+gν(ω,1):Da,sν+1×Rν+1⟼Da,sν $
|
(3.34) |
then
$ |Tν−ΠZ|∗a,s,Da,sν+1×Rν+1≤Cερν,|DTν−Id|⋆a,s,Da,sν+1×Rν+1≤Cερν. $
|
(3.35) |
Similar to [27], it can be seen that the limiting transformation $ T_0\circ T_1\circ\cdots $ converges to a symplectic coordinate transformation $ \Sigma_\infty^0. $ And there exists an absolute constant $ C > 0 $ independent of $ j $ such that
$ |Σ0∞−id|∗a,s,Da,s∞×R_≤Cερ, $
|
(3.36) |
with $ id $ is identity mapping.
In view of the Hamiltonian (2.8) satisfies the conditions $ (3.1)-(3.4), (3.6), (3.7) $ with $ \nu = 0, $ the above iterative procedure can run repeatedly. Thus the transformation $ \Sigma_\infty^0 $ changes the Hamiltonian $ (\rm {2.8}) $ to
$ ˉH∘Σ0∞=<ω,J>+∑j∈Z2oddμj|qj|2, $
|
(3.37) |
with
$ μj=λj+ε2λj[ϕ]+1λjε(1+ρ)μ∗j,‖μ∗j‖∗R_≤C,j∈Z2odd. $
|
(3.38) |
We present the following lemma which has been used in the above iterative procedure. The proof is similar to Lemma 3.1 in [15].
Lemma 3.1. For any given $ k\in\Bbb{Z}^m, j\in\mathbb{Z}^2_{odd}, \hat{l}\in\mathbb{N}, $ denote
$ \mathcal{I}_k^1 = \left\{\omega\in[\varrho, 2\varrho]^m:| \lt k, \omega \gt |\leq\frac{\varrho}{C_*|k|^{m+1}}\right\}, \quad k\neq 0, $ |
$ I2,+k,j,ˆl={ω∈[ϱ,2ϱ]m:|<k,ω>+λj,ˆl+λ−j,ˆl|<ϱC∗(δ(ˆl)+ˆl2)(|k|+δ(|k|))m+1}, $
|
$ I2,−k,j,ˆl={ω∈[ϱ,2ϱ]m:|<k,ω>−λj,ˆl−λ−j,ˆl|<ϱC∗(δ(ˆl)+ˆl2)(|k|+δ(|k|))m+1}, $
|
$ R1=⋃0≠k∈ZmI1k,R2ˆl=⋃j∈Z2odd⋃k∈Zm(I2,+k,j,ˆl⋃I2,−k,j,ˆl) $
|
where $ \delta(x) = 1 $ as $ x = 0 $ and $ \delta(x) = 0 $ as $ x\neq0. $ Then the sets $ R^1, R_{\hat{l}}^2 $ is measurable and
$ measR1≤13γϱm,measR2ˆl≤γ(δ(ˆl)+ˆl)−23+∞∑ˆi=0(δ(ˆi)+ˆi)−2ϱm $
|
(3.39) |
if $ C_*\gg1 $ large enough.
Let
$ R00=[ϱ,2ϱ]m∖R1,R0=R00∖R20,Rˆl+1=Rˆl∖R2ˆl+1,ˆl=0,1,⋯. $
|
(3.40) |
Then we have $ (3.6) $ and $ (3.7) $. Denote
$ R_=∞⋂ˆl=1Rˆl $
|
(3.41) |
then by $ (3.6) $,
$ measR_>(1−2γ3)ϱm. $
|
(3.42) |
In view of the symplectic coordinate transformation $ \Sigma_\infty^0 $ is linear, and (3.36), then
$ q_j\circ \Sigma_\infty^0 = q_j+\lambda_j^{-1}\varepsilon^{\rho}\tilde g_{j, 1, \infty}^*(\vartheta;\omega)q_j +\lambda_j^{-1}\varepsilon^{\rho}\tilde g_{j, 2, \infty}^*(\vartheta;\omega){\bar{q}}_{-j} $ |
where
$ \|\tilde g_{j, \hat{l}, \infty}^*(\vartheta;\omega)\|^*_{\Theta(\sigma_0/2)\times\underline R}\leq C, \quad \hat{l} = 1, 2. $ |
Thus from (3.37), the Hamiltonian (2.8) is transformed into by $ \Sigma_\infty^0 $
$ H00:=ˉH∘Σ0∞=<ω,J>+∑j∈Z2oddμjqjˉqj, $
|
(3.43) |
and the Hamiltonian (2.9) is transformed into
$ ˜G4=G4∘Σ0∞=332π2∑i−j+d−l=0i,j,d,l∈Z2odd1√λiλjλdλl˜G2,2ijdl(ϑ;ω)qiˉqjqdˉql+164π2∑i+j+d+l=0i,j,d,l∈Z2odd1√λiλjλdλl(˜G4,0ijdl(ϑ;ω)qiqjqdql+˜G0,4ijdl(ϑ;ω)ˉqiˉqjˉqdˉql)+116π2∑i+j+d−l=0i,j,d,l∈Z2odd1√λiλjλdλl(˜G3,1ijdl(ϑ;ω)qiqjqdˉql+˜G1,3ijdl(ϑ;ω)ˉqiˉqjˉqdql) $
|
(3.44) |
where
$ ˜Gn1,n2ijdl(ϑ;ω)=Gn1,n2ijdl(ϑ)(1+ερGn1,n2,∗ijdl(ϑ;ω)min(|i|2,|j|2,|d|2,|l|2)),‖Gn1,n2,∗ijdl(ϑ;ω)‖∗Θ(σ0/2)×R_≤C, $
|
(3.45) |
with $ n_1, n_2\in\mathbb{N}, n_1+n_2 = 4, n_1, n_2 = 0, 1, 2, 3, 4 $.
This means that the transformation $ \Sigma_\infty^0 $ changes the Hamiltonian (2.7) into
$ H=H00+ε˜G4. $
|
(3.46) |
The following Lemma gives a regularity result, the proof is similar to [13] and is omitted.
Lemma 3.2. For $ a\geq0 $ and $ s > 0 $, the gradients $ \tilde{G}_q^{4}, \tilde{G}_{\bar q}^{4} $ are real analytic as maps from some neighborhood of origin in $ l^{a, s}\times l^{a, s} $ into $ l^{a, s} $ with $ \|\tilde{G}_q^{4}\|_{a, s} = O(\|q\|^3_{a, s}) $, $ \|\tilde{G}_{\bar q}^{4}\|_{a, s} = O(\|q\|^3_{a, s}) $.
As in [13], Let $ S $ is an admissible set. We define $ \mathbb{Z}_*^2 = \mathbb{Z}^2_{odd}\setminus S $. For simplicity, we define the following three sets:
$ S1={(i,j,d,l)∈(Z2odd)4:i−j+d−l=0,|i|2−|j|2+|d|2−|l|2≠0,#(S∩{i,j,d,l})≥2} $
|
(4.1) |
and
$ S2={(i,j,d,l)∈(Z2odd)4:i+j+d+l=0,|i|2+|j|2+|d|2+|l|2≠0,#(S∩{i,j,d,l})≥2} $
|
(4.2) |
$ S3={(i,j,d,l)∈(Z2odd)4:i+j+d−l=0,|i|2+|j|2+|d|2−|l|2≠0,#(S∩{i,j,d,l})≥2.}. $
|
(4.3) |
Obviously, the set
$ {(i,j,d,l)∈(Z2odd)4:i+j+d+l=0,|i|2+|j|2+|d|2+|l|2=0,} $
|
is empty. Similar to [13], the set
$ {(i,j,d,l)∈(Z2odd)4:i+j+d−l=0,|i|2+|j|2+|d|2−|l|2=0,} $
|
is empty.
For Proposition 4.1, we give the following lemma that will be proved in the "Appendix".
Lemma 4.1. Given $ \varrho > 0, 0 < \gamma < 1, $ and $ C_* $ large enough, $ \varepsilon $ small enough, then there is a subset $ \overline R\subset{[\varrho, 2\varrho]}^m $ with
$ meas¯R>(1−γ3)ϱm $
|
(4.4) |
so that the following statements hold:
$ \rm(i) $ If $ (i, j, d, l)\in S_1 $ or $ i-j+d-l = 0, |i|^2-|j|^2+|d|^2-|l|^2 = 0, \#(S\cap\{i, j, d, l\}) = 2 $ and $ k\neq 0 $, then for any $ \omega\in\overline R, $
$ |μi−μj+μd−μl+<k,ω>|≥ϱC∗(|k|+δ(|k|))m+1,∀k∈Zm; $
|
(4.5) |
$ \rm(ii) $ If $ (i, j, d, l)\in S_2 $, then for any $ \omega\in\overline R, $
$ |μi+μj+μd+μl+<k,ω>|≥ϱC∗(|k|+δ(|k|))m+1,∀k∈Zm; $
|
(4.6) |
$ \rm(iii) $ If $ (i, j, d, l)\in S_3 $, then for any $ \omega\in\overline R, $
$ |μi+μj+μd−μl+<k,ω>|≥ϱC∗(|k|+δ(|k|))m+1,∀k∈Zm; $
|
(4.7) |
where $ \delta(x) = 1 $ as $ x = 0 $ and $ \delta(x) = 0 $ as $ x\neq0. $
Let
$ R = \underline R\cap\overline R, $ |
then
$ {\rm meas} R\geq(1-\gamma)\varrho^m. $ |
Next we transform the Hamiltonian (3.46) into some partial Birkhoff form of order four.
Proposition 4.1. For each admissible set $ S $ there exists a symplectic change of coordinates $ X_F^1 $ that changes the hamiltonian $ H = H_{00}+\varepsilon \tilde{G}^{4} $ with nonlinearity ${\rm(3.44) }$ into
$ H∘X1F=N+A+B+ˉB+P, $
|
(4.8) |
with
$ N=ε−4<ω,J>+ε−4∑j∈SμjIj+ε−4∑j∈Z2∗μj|zj|2+316π2∑i∈S1λ2i[˜G2,2iiii]˜ξiIi+38π2∑i,j∈S,i≠j1λiλj[˜G2,2iijj]˜ξiIj+38π2∑i∈S,j∈Z2∗1λiλj[˜G2,2iijj]˜ξi|zj|2 $
|
(4.9) |
$ A=38π2∑d∈L11√λiλjλdλl[˜G2,2ijdl]√˜ξi˜ξjei(θi−θj)zdˉzl $
|
(4.10) |
$ B=38π2∑d∈L21√λiλjλdλl[˜G2,2dilj]√˜ξi˜ξje−i(θi+θj)zdzl $
|
(4.11) |
$ ˉB=38π2∑d∈L21√λiλjλdλl[˜G2,2idjl]√˜ξi˜ξjei(θi+θj)ˉzdˉzl. $
|
(4.12) |
$ P=O(ε2|I|2+ε2|I|‖z‖2a,s+ε|˜ξ|12‖z‖3a,s+ε2‖z‖4a,s+ε2|˜ξ|3+ε3|˜ξ|52‖z‖a,s+ε4|˜ξ|2‖z‖2a,s+ε5|˜ξ|32‖z‖3a,s). $
|
(4.13) |
Proof. Denote
$ ˜Gn1,n2ijdl(ϑ,ω)=∑k∈ZmGn1,n2ijdl,k(ω)ei<k,ϑ>,n1,n2=0,1,2,3,4,n1+n2=4. $
|
(4.14) |
We find a Hamiltonian
$ F=332π2∑i∈S∑k≠01λ2i⋅G2,2iiii,ki<k,ω>ei<k,ϑ>|qi|4+38π2∑i,j∈S,i≠j∑k≠01λiλj⋅G2,2iijj,ki<k,ω>ei<k,ϑ>|qi|2|qj|2+38π2∑i∈S,j∈Z2∗∑k≠01λiλj⋅G2,2iijj,ki<k,ω>ei<k,ϑ>|qi|2|qj|2+38π2∑d∈L1∑k≠01√λiλjλdλl⋅G2,2ijdl,ki(μi−μj+μd−μl+<k,ω>)ei<k,ϑ>qiˉqjqdˉql+38π2∑d∈L2∑k≠01√λiλjλdλl⋅G2,2dilj,ki(μd+μl−μi−μj+<k,ω>)ei<k,ϑ>ˉqiˉqjqdql+38π2∑d∈L2∑k≠01√λiλjλdλl⋅G2,2idjl,ki(μi−μd+μj−μl+<k,ω>)ei<k,ϑ>qiqjˉqdˉql+38π2∑(i,j,d,l)∈S1∑k∈Zm1√λiλjλdλl⋅G2,2ijdl,ki(μi−μj+μd−μl+<k,ω>)ei<k,ϑ>qiˉqjqdˉql+164π2∑(i,j,d,l)∈S2∑k∈Zm1√λiλjλdλl⋅G4,0ijdl,ki(μi+μj+μd+μl+<k,ω>)ei<k,ϑ>qiqjqdql+164π2∑(i,j,d,l)∈S2∑k∈Zm1√λiλjλdλl⋅G0,4ijdl,ki(−μi−μj−μd−μl+<k,ω>)ei<k,ϑ>ˉqiˉqjˉqdˉql+116π2∑(i,j,d,l)∈S3∑k∈Zm1√λiλjλdλl⋅G3,1ijdl,ki(μi+μj+μd−μl+<k,ω>)ei<k,ϑ>qiqjqdˉql+116π2∑(i,j,d,l)∈S3∑k∈Zm1√λiλjλdλl⋅G1,3ijdl,ki(−μi−μj−μd+μl+<k,ω>)ei<k,ϑ>ˉqiˉqjˉqdql. $
|
(4.15) |
Let $ X_{F}^1 $ be the time-1 map of the Hamiltonian vector field of $ \varepsilon F $ and denote variables as follows
$ q_j = \left\{ qj,j∈S,zj,j∈Z2∗, \right. $
|
then it satisfies
$ ˆH=H∘X1F=H00+ε˜G4+ε{H00,F}+ε2{˜G4,F}+ε2∫10(1−t){{H,F},F}∘XtFdt=<ω,J>+∑j∈Sμj|qj|2+∑j∈Z2∗μj|zj|2+3ε32π2∑i∈S1λ2i[˜G2,2iiii]|qi|4+3ε8π2∑i,j∈S,i≠j1λiλj[˜G2,2iijj]|qi|2|qj|2+3ε8π2∑i∈S,j∈Z2∗1λiλj[˜G2,2iijj]|qi|2|qj|2+3ε8π2∑d∈L11√λiλjλdλl[˜G2,2ijdl]qiˉqjqdˉql+3ε8π2∑d∈L21√λiλjλdλl[˜G2,2dilj]ˉqiˉqjqdql+3ε8π2∑d∈L21√λiλjλdλl[˜G2,2idjl]qiqjˉqdˉql+O(ε|q|‖z‖3a,s+ε‖z‖4a,s+ε2|q|6+ε2|q|5‖z‖a,s+ε2|q|4‖z‖2a,s+ε2|q|3‖z‖3a,s). $
|
Now we introduce the parameter vector $ \tilde{\xi} = (\tilde{\xi}_j)_{j\in S} $ and the action-angle variable by setting
$ qj=√Ij+˜ξjeiθj,ˉqj=√Ij+˜ξje−iθj,j∈S. $
|
(4.16) |
From the symplectic transformation (4.16), the Hamiltonian $ \widehat{H} $ is changed into
$ ˆH=<ω,J>+∑j∈SμjIj+∑j∈Z2∗μj|zj|2+3ε16π2∑i∈S1λ2i[˜G2,2iiii]˜ξiIi+3ε8π2∑i,j∈S,i≠j1λiλj[˜G2,2iijj]˜ξiIj+3ε8π2∑i∈S,j∈Z2∗1λiλj[˜G2,2iijj]˜ξi|zj|2+3ε8π2∑d∈L11√λiλjλdλl[˜G2,2ijdl]√˜ξi˜ξjei(θi−θj)zdˉzl+3ε8π2∑d∈L21√λiλjλdλl[˜G2,2dilj]√˜ξi˜ξje−i(θi+θj)zdzl+3ε8π2∑d∈L21√λiλjλdλl[˜G2,2idjl]√˜ξi˜ξjei(θi+θj)ˉzdˉzl+O(ε|I|2+ε|I|‖z‖2a,s+ε|˜ξ|12‖z‖3a,s+ε‖z‖4a,s+ε2|˜ξ|3+ε2|˜ξ|52‖z‖a,s+ε2|˜ξ|2‖z‖2a,s+ε2|˜ξ|32‖z‖3a,s) $
|
Through scaling variables
$ \tilde{\xi}\rightarrow\varepsilon^{3}\tilde{\xi}, \quad J\rightarrow\varepsilon^{5}J , \quad I\rightarrow\varepsilon^{5}I , \quad \vartheta\rightarrow\varepsilon^{4}\vartheta, \quad \theta\rightarrow\theta, \quad z\rightarrow\varepsilon^{\frac{5}{2}}z , \quad \bar{z}\rightarrow\varepsilon^{\frac{5}{2}}\bar{z}, $ |
and scaling time $ t\rightarrow \varepsilon^{9}t $, the rescaled Hamiltonian can be obtained
$ H=ε−9ˆH(ε3˜ξ;ε9J,ε5I,ϑ,θ,ε52z,ε52ˉz). $
|
Then $ H $ satisfies the equation (4.8)–(4.13).
Now let's give the estimates of the perturbation $ P. $ For this purpose, we need to introduce the notations which are taken from [13]. Let $ l^{a, s} $ is now the Hilbert space of all complex sequence $ w = {(\ldots, w_j, \ldots)}_{j\in \mathbb{Z}^2_*} $ with
$ \|w\|_{a, s} = \sum\limits_{j\in \mathbb{Z}^2_*}|w_j|e^{a|j|}\cdot|j|^s \lt \infty, \quad a \gt 0, s \gt 0. $ |
Let $ x = \vartheta\oplus\theta $ with $ \theta = (\theta_j)_{j\in S}, y = J\oplus I, $ $ z = (z_j)_{j\in \mathbb{Z}^2_*} $ and $ \zeta = \omega\oplus (\tilde{\xi}_j)_{j\in S}, $ and let's introduce the phase space
$ {\mathcal P}^{a, s} = \widehat{{\Bbb T}}^{m+n}\times {\Bbb C}^{m+n}\times l^{a, s}\times l^{a, s}\ni (x, y, z, \bar{z}) $ |
where $ \widehat{{\Bbb T}}^{m+n} $ is the complexiation of the usual $ (m+n) $-torus $ {\Bbb T}^{m+n}. $ Let
$ D_{a, s}(s', r): = \{(x, y, z, \bar{z})\in {\mathcal P}^{a, s}: |{\rm Im}x| \lt s', |y| \lt r^2, \|z\|_{a, s}+\|\bar{z}\|_{a, s} \lt r\}, $ |
and
$ |W|_r = |x|+\frac{1}{r^2}|y|+\frac{1}{r}\|z\|_{a, s}+\frac{1}{r}\|\bar{z}\|_{a, s} $ |
for $ W = (x, y, z, \bar{z})\in {\mathcal P}^{a, s} $. Set $ \alpha\equiv(\ldots, \alpha_j, \ldots)_{j\in \mathbb{Z}^2_*} $, $ \beta\equiv(\ldots, \beta_j, \ldots)_{j\in \mathbb{Z}^2_*} $, $ \alpha_j $ and $ \beta_j\in \mathbb{N} $ with finitely many nonzero components of positive integers. The product $ z^\alpha {\bar z}^\beta $ denotes $ \prod_jz_j^{\alpha_j} {\bar z}_j^{\beta_j} $. Let
$ P(x, y, z, \bar z) = \sum\limits_{\alpha, \beta}P_{\alpha\beta}(x, y)z^\alpha{\bar z}^\beta, $ |
where $ P_{\alpha\beta} = \sum_{k, b}P_{kb\alpha\beta} y^b e^{{\rm i} < k, x > } $ are $ C_{W}^4 $ functions in parameter $ \zeta $ in the sense of Whitney. Let
$ \|P\|_{D_{a, s}(s', r), \underline{\Sigma}}\equiv\sup\limits_{\|z\|_{a, s} \lt r, \|\bar z\|_{a, s} \lt r}\sum\limits_{\alpha, \beta}\|P_{\alpha\beta}\||z^\alpha||\bar z^\beta|, $ |
where, if $ P_{\alpha, \beta} = \sum_{k\in \mathbb{Z}^{m+n}, b\in \mathbb{N}^{m+n}}P_{kb\alpha\beta}(\zeta) y^b e^{{\rm i} < k, x > }, $ $ P_{\alpha\beta} $ is short for
$ \|P_{\alpha\beta}\|\equiv\sum\limits_{k, b}|P_{kb\alpha\beta}|_{\underline{\Sigma}}r^{2|b|}e^{|k|s'}, \quad |P_{kb\alpha\beta}|_{\underline{\Sigma}}\equiv\sup\limits_{\zeta\in\underline{\Sigma}}\sum\limits_{0\leq s\leq 4}|\partial_\zeta^s P_{kb\alpha\beta}| $ |
the derivatives with respect to $ \zeta $ are in the sense of Whitney. Denote by $ X_P $ the vector field corresponding the Hamiltonian $ P $ with respect to the symplectic structure $ dx\wedge dy+{\rm i}dz\wedge d\bar{z}, $ namely,
$ X_P = (\partial_yP, -\partial_xP, {\rm i}\nabla_{\bar{z}}P, -{\rm i}\nabla_zP). $ |
Its weighted norm is defined by
$ ‖XP‖Da,s(s′,r),Σ_≡‖Py‖Da,s(s′,r),Σ_+1r2‖Px‖Da,s(s′,r),Σ_+1r(∑j∈Z2∗‖Pzj‖Da,s(s′,r),Σ_e|j|a+∑j∈Z2∗‖Pˉzj‖Da,s(s′,r),Σ_e|j|a). $
|
The following Lemma can be obtained and the proof is similar to Lemma 3.2 in [27].
Lemma 4.2. For given $ s', r > 0 $, the perturbation $ P(x, y, z, \bar{z}; \zeta) $ is real analytic for $ (x, y, z, \bar{z})\in D_{a, s} $ $ (s', r) $ and Lipschitz in the parameters $ \zeta\in \underline{\Sigma}, $ and for any $ \zeta\in \underline{\Sigma} $, its gradients with respect to $ z, \bar{z} $ satisfy
$ \partial_zP, \quad\partial_{\bar{z}}P\in {\mathcal A}(l^{a, s}, l^{a, s}), $ |
and
$ \|X_P\|_{D_{a, s+1}(s', r), \underline{\Sigma}}\leq C\varepsilon, $ |
where $ s' = \sigma_0/3 $ and $ r = \sqrt{\varepsilon}. $
In order to prove our main result (Theorem 1.1), we need to state a KAM theorem which was proved by Geng-Zhou [13]. Here we recite the theorem from [13].
Let us consider the perturbations of a family of Hamiltonian
$ H_{00} = N+\mathcal{A}+\mathcal{B}+\bar{\mathcal{B}}, $ |
where
$ N = \sum\limits_{j\in S}\widehat{\omega}_j(\xi)y_j+\sum\limits_{j\in \mathbb{Z}^2_*}\widehat{ \Omega}_j(\xi)z_j\bar{z}_j $ |
$ \mathcal{A} = \sum\limits_{d\in \mathcal{L}_1}a_d(\xi)e^{{\rm i}(x_i-x_j)} z_d\bar z_l $ |
$ \mathcal{B} = \sum\limits_{d\in \mathcal{L}_2}a_d(\xi)e^{-{\rm i}(x_i+x_j)} z_dz_l $ |
$ \bar{\mathcal{B}} = \sum\limits_{d\in \mathcal{L}_2}\bar a_d(\xi)e^{{\rm i}(x_i+x_j)} \bar z_d\bar z_l. $ |
in $ n $-dimensional angle-action coordinates $ (x, y) $ and infinite-dimensional coordinates $ (z, \bar z) $ with symplectic structure
$ \sum\limits_{j\in S}dx_j\wedge dy_j+{\rm i}\sum\limits_{j\in \mathbb{Z}^2_*} dz_j\wedge d{\bar z}_j. $ |
The tangent frequencies $ \widehat{\omega} = (\widehat{\omega}_j)_{j\in S} $ and normal ones $ \widehat{ \Omega} = (\widehat{\Omega}_{j})_{j\in \mathbb{Z}^2_*} $ depend on $ n $ parameters
$ \xi\in \Pi\subset {\Bbb R}^n, $ |
with $ \Pi $ a closed bounded set of positive Lebesgue measure.
For each $ \xi $ there is an invariant $ n $-torus $ {\mathcal T}_0^n = {\Bbb T}^n\times \{0, 0, 0\} $ with frequencies $ \widehat{\omega}(\xi). $ The aim is to prove the persistence of a large portion of this family of rotational torus under small perturbations $ H = H_{00}+P $ of $ H_{00}. $ To this end the following assumptions are made.
Assumption A1. (Non-degeneracy): The map $ \xi\mapsto \widehat{\omega}(\xi) $ is a $ C_W^4 $ diffeomorphism between $ \Pi $ and its image.
Assumption A2. (Asymptotics of normal frequencies):
$ ˆΩj=ε−ς|j|2+˜Ωj,ς>0 $
|
where $ \widetilde{\Omega}_j $ is a $ C_W^4 $ functions of $ \xi $ and $ \widetilde{\Omega}_j = O(|j|^{-\iota}), \iota > 0 $.
Assumption A3. (Melnikov conditions): Let $ B_d = \widehat{\Omega}_d $ for $ d\in \mathbb{Z}^2_*\setminus(\mathcal{L}_1\cup\mathcal{L}_2), $ and let
$ Bd=(ˆΩd+ˆωiadalˆΩl+ˆωj),d∈L1 $
|
$ Bd=(ˆΩd−ˆωiadˉalˆΩl−ˆωj),d∈L2 $
|
there exist $ \gamma', \tau > 0 $ (here $ I_2 $ is $ 2\times 2 $ identity matrix) such that
$ | \lt k, \widehat{\omega} \gt |\geq\frac{\gamma'}{|k|^\tau}, \quad k\neq 0, $ |
$ |{\rm det}( \lt k, \widehat{\omega} \gt I+B_d)|\geq\frac{\gamma'}{|k|^\tau}, $ |
$ |{\rm det}( \lt k, \widehat{\omega} \gt I\pm B_d\otimes I_2\pm I_2\otimes B_{d'})|\geq\frac{\gamma'}{|k|^\tau}, \quad k\neq 0, $ |
where $ I $ means the identity matrix.
Assumption A4. (Regularity): $ \mathcal{A}+\mathcal{B}+\bar{\mathcal{B}}+P $ is real analytic in $ x, y, z, \bar z $ and Whitney smooth in $ \xi $; and we have
$ \|X_\mathcal{A}\|_{D_{a, s}(s', r), \Pi}+\|X_\mathcal{B}\|_{D_{a, s}(s', r), \Pi}+\|X_{\bar{\mathcal{B}}}\|_{D_{a, s}(s', r), \Pi} \lt 1, \quad\|X_P\|_{D_{a, s}(s', r), \Pi} \lt \varepsilon. $ |
Assumption A5. (Zero-momentum condition): The normal form part $ \mathcal{A}+\mathcal{B}+\bar{\mathcal{B}}+P $ satisfies the following condition
$ A+B+ˉB+P=∑k∈Zn,b∈Nn,α,β(A+B+ˉB+P)kbαβ(ξ)ybei<k,x>zαˉzβ $
|
and we have
$ (\mathcal{A}+\mathcal{B}+\bar{\mathcal{B}}+P)_{kb\alpha\beta}\neq 0\Rightarrow\sum\limits_{\hat{s} = 1}^n k_{\hat{s}} i_{\hat{s}}+\sum\limits_{d\in\mathbb{Z}^2_*}(\alpha_d-\beta_d)d = 0. $ |
Now we state the basic KAM theorem which is attributed to Geng-Zhou [13], and as a corollary, we get Theorem 1.1.
Theorem 5.1. ([13] Theorem 2) Assume that the Hamiltonian $ H = N+\mathcal{A}+\mathcal{B}+\bar{\mathcal{B}}+P $ satisfies $ \rm (\bf{A1})-(\bf{A5}) $. Let $ \gamma' > 0 $ be sufficiently small, then there exists $ \varepsilon > 0 $ and $ a, s > 0 $ such that if $ \|X_P\|_{D_{a, s}(s', r), \Pi} < \varepsilon $, the following holds: there exists a Cantor subset $ \Pi_{\gamma'}\subset\Pi $ with $ {\rm meas}(\Pi\setminus\Pi_{\gamma'}) = O({\gamma'}^{\varsigma}) $ ($ \varsigma $ is a positive constant) and two maps which are analytic in $ x $ and $ C_W^4 $ in $ \xi $,
$ \Psi:\mathbb{T}^n\times\Pi_{\gamma'}\rightarrow D_{a, s}(s', r), \quad \tilde\omega: \Pi_{\gamma'}\rightarrow\mathbb{R}^n, $ |
where $ \Psi $ is $ \frac{\varepsilon}{{(\gamma')}^{16}} $-close to the trivial embedding $ \Psi_0:\mathbb{T}^n\times\Pi\rightarrow\mathbb{T}^n\times\{0, 0, 0\} $ and $ \tilde\omega $ is $ \varepsilon $-close to the unperturbed frequency $ \widehat\omega $, such that for any $ \xi\in\Pi_{\gamma'} $ and $ x\in\mathbb{T}^n $, the curve $ t\rightarrow\Psi(x+\tilde\omega(\xi)t, \xi) $ is a quasi-periodic solution of the Hamiltonian equations governed by $ H = N+\mathcal{A}+\mathcal{B}+\bar{\mathcal{B}}+P. $
In order to apply the above theorem to our problem, we need to introduce a new parameter $ \bar{\omega} $ below.
Given $ \omega_-\in { R} $, for $ \omega\in \bar{\bar{ R}}: = \{\omega\in R\mid |\omega-\omega_-|\leq\varepsilon\}, $ we introduce new parameter $ \bar{\omega} $ by
$ ω=ω−+εˉω,ˉω∈[0,1]m. $
|
(5.1) |
Then the Hamiltonian (4.8) is changed into
$ H=<ˆω(ξ),ˆy>+<ˆΩ(ξ),ˆz>+A+B+ˉB+P $
|
(5.2) |
where $ \widehat{\omega}(\xi) = (\varepsilon^{-4}\omega)\oplus\breve{\omega}, \xi = \bar{\omega}\oplus \tilde{\xi}, \hat{z} = (|z_j|^2)_{j\in\mathbb{Z}^2_*}, \hat{x} = \vartheta\oplus\theta, \hat y = J\oplus I $ with
$ ˘ωi=ε−4μi+316π21λ2i[˜G2,2iiii]˜ξi+38π2∑j∈S1λiλj[˜G2,2iijj]˜ξj,i∈S, $
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(5.3) |
$ ˆΩd=ε−4μd+38π2∑j∈S1λjλd[˜G2,2jjdd]˜ξj,d∈Z2∗. $
|
(5.4) |
Denote $ \breve{\omega}(\xi) = \varepsilon^{-4}\tilde{\alpha}+A\tilde{\xi}, $ $ \widehat{\Omega}(\xi) = \varepsilon^{-4}\tilde{\beta}+B\tilde{\xi}, $ where
$ \quad\tilde{\alpha} = (\ldots, \mu_i, \ldots)_{i\in S}, \quad\tilde{\beta} = (\ldots, \mu_{d}, \ldots)_{d\in \mathbb{Z}^2_*}, $ |
$ A=(˜Gij)i,j∈S,B=(˜Gij)i∈Z2∗,j∈S, $
|
(5.5) |
with
$ ˜Gij=3⋅(2−δij)16π2λiλj[˜G2,2iijj],δij={1,i=j,0,i≠j. $
|
(5.6) |
Lemma 5.1. Let $ \Pi = [0, 1]^{m+n}, $ for any $ \varepsilon > 0 $ sufficiently small, $ r = \sqrt{\varepsilon} $, then we have
$ \|X_P\|_{D_{a, s+1}(s', r)\times \Pi}\leq C\varepsilon. $ |
The proof of the above lemma is the same as one of Lemma 4.2.
In this section, we prove that the Hamiltonian (5.2) satisfies the assumptions $ \rm (\bf{A1})-(\bf{A5}) $. In view of (5.5), (5.6), (2.10) and (3.45),
$ \lim\limits_{\varepsilon\rightarrow0}A = \frac{3[\phi]}{16\pi^2}\left(1λ212λ1λ2⋯2λ1λn2λ2λ11λ22⋯2λ2λn⋯⋯⋯⋯2λnλ12λnλ2⋯1λ2n \right)_{n\times n}: = \widetilde{A} = :[\phi]\widehat A, $
|
Verifying $ \rm (A1): $ From (5.3),
$ ∂ˆω∂ξ=(ε−3Im0ε−3⋅∂˜α∂ω+ε⋅∂(A˜ξ)∂ωA),forξ∈Π, $
|
where $ I_m $ denotes the unit $ m\times m $-matrix. It is obvious that $ {\rm det}{\widetilde A}\neq0. $ So $ {\rm det}A\neq0 $ can be obtained by assuming $ 0 < \varepsilon\ll1. $ Thus assumption $ \rm (A1) $ is verified.
Verifying $ \rm (A2): $ Take $ \varsigma = 4, \iota = 4, $ the proof is obvious.
Verifying $ \rm (A3): $ For (5.2), $ B_d $ is defined as follows,
$ B_d = \widehat{\Omega}_d\quad d\in \mathbb{Z}^2_*\setminus(\mathcal{L}_1\cup\mathcal{L}_2), $ |
and
$ Bd=(ˆΩd+˘ωi3[˜G2,2ijdl]√˜ξi˜ξj8π2√λiλjλdλl3[˜G2,2ijdl]√˜ξi˜ξj8π2√λiλjλdλlˆΩl+˘ωj),d∈L1 $
|
$ Bd=(ˆΩd−˘ωi3[˜G2,2dilj]√˜ξi˜ξj8π2√λiλjλdλl3[˜G2,2idjl]√˜ξi˜ξj8π2√λiλjλdλlˆΩl−˘ωj),d∈L2 $
|
where $ (i, j, l) $ is uniquely determined by $ d $. In the following, we only prove $ \rm (A3) $ for $ {\rm det}[ < k, \widehat\omega(\xi) > I\pm B_d\otimes I_2\pm I_2\otimes B_{d'}] $ which is the most complicated case. For $ k\in{\mathbb{Z}}^{m+n}, b\in{\mathbb{N}}^{m+n} $, denote
$ k=(k1,k2),b=(b1,b2),k1∈Zm,k2∈Zn,b1∈Nm,b2∈Nn. $
|
Let
$ Z(ξ)=<k,ˆω(ξ)>I±Bd⊗I2±I2⊗Bd′=(ε−4<k1,ω>+ε−4<k2,˜α>+<k2,A˜ξ>)I±Bd⊗I2±I2⊗Bd′. $
|
We need to prove that $ |{\mathcal Z}(\xi)|\geq \frac{\gamma'}{|k|^\tau}, (k\neq 0). $ For this purpose, we need to divide into the following two cases.
Case 1. When $ k_1\neq0, $ notice that
$ \frac{\partial{\left((\varepsilon^{-4} \lt k_2, \tilde{\alpha} \gt + \lt k_2, A\tilde{\xi} \gt )I\pm B_d\otimes I_2\pm I_2\otimes B_{d'}\right )}}{\partial\bar{\omega}} = \varepsilon^{-3}\cdot O(\varepsilon^{1+\rho}), $ |
and from
$ \frac{\partial{ \lt k_1, \varepsilon^{-4}\omega \gt }}{\partial \bar{\omega}}+\varepsilon^{-3}\cdot O(\varepsilon^{1+\rho}) = \varepsilon^{-3} \big(k_1+O(\varepsilon^{1+\rho})\big)\neq0, \quad 0 \lt \varepsilon\ll 1 $ |
then all the eigenvalues of $ {\mathcal Z}(\xi) $ are not identically zero.
Case 2. When $ k_1 = 0, $ then
$ Z(ξ)=(ε−4<k1,ω>+ε−4<k2,˜α>+<k2,A˜ξ>)I±Bd⊗I2±I2⊗Bd′=(ε−4<k2,˜α>+<k2,A˜ξ>)I±Bd⊗I2±I2⊗Bd′, $
|
We assert that all the eigenvalues of $ {\mathcal Z}(\xi) $ are not identically zero. Here we're just proving it for $ d, d'\in\mathcal{L}_1 $, and everything else is similar. Let
$ B_d = \varepsilon^{-4} B_d^1+B_d^2, \quad \forall d\in\mathcal{L}_1 $ |
where
$ B1d=(μd+μi00μl+μj), $
|
$ B2d=(−3[˜G2,2iiii]˜ξi16π2λ2i+3∑κ∈S([˜G2,2κκii]λiλκ+[˜G2,2κκdd]λκλd)˜ξκ8π23[˜G2,2ijdl]√˜ξi˜ξj8π2√λiλjλdλl3[˜G2,2ijdl]√˜ξi˜ξj8π2√λiλjλdλl−3[˜G2,2jjjj]˜ξj16π2λ2j+3∑κ∈S([˜G2,2κκjj]λκλj+[˜G2,2κκll]λκλl)˜ξκ8π2). $
|
Then
$ Z(ξ)=ε−4(<k2,˜α>I±B1d⊗I2±I2⊗B1d′)+(<k2,A˜ξ>I±B2d⊗I2±I2⊗B2d′). $
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In view of $ |i|^2+|d|^2 = |j|^2+|l|^2 $ and (2.10), (3.45),
$ \lim\limits_{\varepsilon\rightarrow0}B_d^1 = \left(|i|2+|d|200|i|2+|d|2 \right): = \widehat {B_d^1}, $
|
$ limε→0B2d=(−3[ϕ]˜ξi16π2λ2i+3[ϕ]∑κ∈S(1λκλi+1λκλd)˜ξκ8π23[ϕ]√˜ξi˜ξj8π2√λiλjλdλl3[ϕ]√˜ξi˜ξj8π2√λiλjλdλl−3[ϕ]˜ξj16π2λ2j+3[ϕ]∑κ∈S(1λκλj+1λκλl)˜ξκ8π2):=~B2d:=[ϕ]^B2d, $
|
Thus,
$ limε→0Z(ξ)=ε−4(<k2,ˆα>I±^B1d⊗I2±I2⊗^B1d′)+[ϕ](<k2,ˆA˜ξ>I±^B2d⊗I2±I2⊗^B2d′)=ε−4(<k2,ˆα>±(|i|2+|d|2)±(|i′|2+|d′|2))I+[ϕ]<ˆAk2±(1λi+1λd)ˆβ±(1λj+1λl)ˆβ,˜ξ>I±(−3[ϕ]˜ξi16π2λ2i3[ϕ]√˜ξi˜ξj8π2λiλj3[ϕ]√˜ξi˜ξj8π2λiλj−3[ϕ]˜ξj16π2λ2j)⊗I2±I2⊗(−3[ϕ]˜ξi′16π2λ2i′3[ϕ]√˜ξi′˜ξj′8π2λi′λj′3[ϕ]√˜ξi′˜ξj′8π2λi′λj′−3[ϕ]˜ξj′16π2λ2j′):=ˆZ(ξ) $
|
with $ \hat{\alpha} = (\lambda_{i_1}, \lambda_{i_2}, \ldots, \lambda_{i_n}) $, $ \hat{\beta} = (\frac{3}{8\pi^2\lambda_{i_1}}, \frac{3}{8\pi^2\lambda_{i_2}}, \ldots, \frac{3}{8\pi^2\lambda_{i_n}}) $ and $ \tilde\xi = (\tilde\xi_{i_1}, \tilde\xi_{i_2}, \ldots, \tilde\xi_{i_n}) $. The eigenvalues of $ \widehat{\mathcal Z}(\xi) $ are
$ ε−4(<k2,ˆα>±(|i|2+|d|2)±(|i′|2+|d′|2))+[ϕ]<ˆAk2±(1λi+1λd)ˆβ±(1λj+1λl)ˆβ,˜ξ>±3[ϕ]32π2[(−˜ξiλ2i−˜ξjλ2j±√˜ξi2λ4i+14˜ξi˜ξjλ2iλ2j+˜ξj2λ4j)±(−˜ξi′λ2i′−˜ξj′λ2j′±√˜ξi′2λ4i′+14˜ξi′˜ξj′λ2i′λ2j′+˜ξj′2λ4j′)]. $
|
Similar to [10], we know that all the eigenvalues are not identically zero. Thus all the eigenvalues of $ {\mathcal Z}(\xi) $ are not identically zero as $ 0 < \varepsilon\ll1. $ Moreover, they are similar to $ d\in\mathcal{L}_1, d'\in\mathcal{L}_2 $ or $ d\in\mathcal{L}_2, d'\in\mathcal{L}_2 $, and omit them here.
Hence all eigenvalues of $ {\mathcal Z}(\xi) $ are not identically zero for $ k\neq 0. $ According to Lemma 3.1 in [10], $ {\rm det}({\mathcal Z}(\xi)) $ is polynomial function in $ \xi $ of order at most four. Thus
$ \left|\partial_{\xi}^4({\rm det}({\mathcal Z}(\xi)))\right|\geq\frac{1}{2}|k|\neq 0. $ |
By excluding some parameter set with measure $ O(\sqrt[4]{\gamma'}) $, we get
$ \left|{\rm det}({\mathcal Z}(\xi))\right|\geq\frac{\gamma'}{|k|^\tau}, \quad k\neq 0. $ |
$ \rm (A3) $ is verified.
Verifying $ \rm (A4): $ Assumption $ \rm (A4) $ can be verified easily fulfilled by Lemma 5.1.
Verifying $ \rm (A5): $ The proof is similar to [27].
By applying Theorem 5.1([13] Theorem 2), we get Theorem 1.1.
Proof of Lemma 4.1. Case 1. Similar to Lemma 3.1 in [27], there exists a set $ R^{3, 1} $ so that $ \forall \omega\in [\varrho, 2\varrho]^m\setminus R^{3, 1} $, Lemma 4.1$\rm(i)$ is true, and $ {\rm meas} R^{3, 1}\leq\frac{\gamma}{9}\varrho^{m} $. We omit the proof.
Case 2. Assume $ i+j+d+l = 0, |i|^2+|j|^2+|d|^2+|l|^2\neq0 $ and $ \#(S\cap\{i, j, d, l\})\geq 2 $. First of all, we have $ \left||i|^2+|j|^2+|d|^2+|l|^2\right|\geq 1. $ Denote $ f(\varepsilon) = \mu_i+\mu_j+ \mu_d+\mu_l, $ then by $ \mu_j = \lambda_j+\frac{\varepsilon}{2\lambda_j}[\phi]+\frac{1}{\lambda_j}\varepsilon^{(1+\rho)}\mu_j^* $ we have
$ f(\varepsilon) = |i|^2+|j|^2+|d|^2+|l|^2+\varepsilon[\phi] (\frac{1}{2\lambda_i}+\frac{1}{2\lambda_j}+\frac{1}{2\lambda_d}+\frac{1}{2\lambda_l})+\varepsilon^{(1+\rho)} (\frac{\mu_i^*}{\lambda_i}+\frac{\mu_j^*}{\lambda_j}+\frac{\mu_d^*}{\lambda_d}+\frac{\mu_l^*}{\lambda_l}). $ |
Case 1.1. For $ k = 0, $ then
$ |f(\varepsilon)+ \lt k, \omega \gt | = |f(\varepsilon)|\geq 1-C\varepsilon\geq \frac{\varrho}{C_*} $ |
when $ \varepsilon $ small enough and $ C_* $ large enough.
Case 1.2. For $ k\neq0, $ denote
$ {\mathcal I}_{ijdl, k}^{3, 2} = \left\{\omega\in [\varrho, 2\varrho]^{m}:|f(\varepsilon)+ \lt k, \omega \gt | \lt \frac{\varrho }{C_*|k|^{m+1}}\right\}, $ |
and
$ R^{3, 2} = \bigcup\limits_{0\neq k\in\Bbb{Z}^m}\bigcup\limits_{i, j, d, l}{\mathcal I}_{ijdl, k}^{3, 2}. $ |
Case 1.2.1. When $ \#(S\cap\{i, j, d, l\}) = 4 $. Denote
$ {\mathcal I}_{ijdl, k}^{3, 2, 1} = \left\{\omega\in [\varrho, 2\varrho]^{m}:|f(\varepsilon)+ \lt k, \omega \gt | \lt \frac{\varrho }{C_*|k|^{m}}\right\}, $ |
$ R^{3, 2, 1} = \bigcup\limits_{0\neq k\in\Bbb{Z}^m}\bigcup\limits_{i\in S, j\in S, d\in S, l\in S}{\mathcal I}_{ijdl, k}^{3, 2, 1}, $ |
we have
$ {\rm meas}{\mathcal I}_{ijdl, k}^{3, 2, 1}\leq \frac{2\varrho^m }{C_*|k|^{m+1}}. $ |
Let
$ |k|_{\infty} = \max\{|k_1|, |k_{2}|, \ldots, |k_{m}|\}, $ |
in view of
$ \sum\limits_{|k|_\infty = p}1\leq 2m(2p+1)^{m-1}, $ |
$ |k|_\infty\leq |k|\leq m|k|_\infty, $ |
we have
$ measR3,2,1=meas⋃0≠k∈Zm⋃i∈S,j∈S,d∈S,l∈SI3,2,1ijdl,k≤∑0≠k∈Zmn42ϱmC∗|k|m+1≤C″1C∗ϱm∑0≠k∈Zm1|k|m+1≤C′1C∗ϱm∞∑p=1(2p+1)m−1p−(m+1)≤C1C∗ϱm $
|
where the constant $ C_1 $ depends on $ n, m. $ Thus
$ {\rm meas} R^{3, 2, 1}\leq\frac{\gamma}{27}\varrho^m $ |
provided $ C_* $ large enough.
Case 1.2.2. When $ \#(S\cap\{i, j, d, l\}) = 3 $. Assume $ i, j, d\in S, l\in\mathbb{Z}_*^2 $ without loss of generality. Then $ l = -i-j-d $ is at most $ n^3 $ different values. Denote
$ {\mathcal I}_{ijdl, k}^{3, 2, 2} = \left\{\omega\in [\varrho, 2\varrho]^{m}:|f(\varepsilon)+ \lt k, \omega \gt | \lt \frac{\varrho }{C_*|k|^{m}}\right\}, $ |
$ R^{3, 2, 2} = \bigcup\limits_{0\neq k\in\Bbb{Z}^m}\bigcup\limits_{i\in S, j\in S, d\in S, l = -i-j-d}{\mathcal I}_{ijdl, k}^{3, 2, 2}, $ |
then
$ {\rm meas}{\mathcal I}_{ijdl, k}^{3, 2, 2}\leq \frac{2\varrho^m }{C_*|k|^{m+1}}. $ |
We obtain
$ {\rm meas} R^{3, 2, 2} = {\rm meas}\bigcup\limits_{ 0\neq k\in {\Bbb Z}^{m}}\bigcup\limits_{i\in S, j\in S, d\in S, l = -i-j-d}{\mathcal I}_{ijdl, k}^{3, 2, 2}\leq\sum\limits_{0\neq k\in {\Bbb Z}^{m}}n^6\frac{2\varrho^m }{C_*|k|^{m+1}}\leq \frac{C_2}{C_*}\varrho^m $ |
where the constant $ C_2 $ depends on $ n, m. $ Thus
$ {\rm meas} R^{3, 2, 2}\leq\frac{\gamma}{27}\varrho^m $ |
provided $ C_* $ large enough.
Case 1.2.3. When $ \#(S\cap\{i, j, d, l\}) = 2 $. Assume $ i, j\in S, d, l\in\mathbb{Z}_*^2 $ without loss of generality. Then we have $ l = -i-j-d $ and
$ f(ε)=|i|2+|j|2+|d|2+|i+j+d|2+ε[ϕ](12λi+12λj+12λd+12λl)+ε(1+ρ)(μ∗iλi+μ∗jλj+μ∗dλd+μ∗lλl)=g+ε[ϕ](12λi+12λj+12λd+12λl)+ε(1+ρ)(μ∗iλi+μ∗jλj+μ∗dλd+μ∗lλl) $
|
where $ g = |i|^2+|j|^2+|d|^2+|i+j+d|^2\in\mathbb{Z}^{+}. $ Denote
$ {\mathcal I}_{ijdl, k}^{3, 2, 3} = \left\{\omega\in [\varrho, 2\varrho]^{m}:|f(\varepsilon)+ \lt k, \omega \gt | \lt \frac{\varrho }{C_*|k|^{m+1}}\right\}, $ |
$ R^{3, 2, 3} = \bigcup\limits_{0\neq k\in\Bbb{Z}^m}\bigcup\limits_{i\in S, j\in S, d\in\mathbb{Z}^2_*, l = -i-j-d}{\mathcal I}_{ijdl, k}^{3, 2, 3}. $ |
For given $ i, j, g, $ denote
$ d^*_{ijg} = \left\{d\in\mathbb{Z}^2_*: g = |i|^2+|j|^2+|d|^2+|i+j+d|^2\right\} $ |
$ \mu^*_{ijg, 1} = \sup\limits_{d\in{d^*_{ijg}}}\left\{\frac{\mu^*_d}{\lambda_d}+\frac{\mu^*_{-i-j-d}}{\lambda_{-i-j-d}}\right\}, \quad\quad\mu^*_{ijg, 2} = \inf\limits_{d\in{d^*_{ijg}}}\left\{\frac{\mu^*_d}{\lambda_d}+\frac{\mu^*_{-i-j-d}}{\lambda_{-i-j-d}}\right\} $ |
$ g^{*} = g+\varepsilon[\phi] (\frac{1}{2\lambda_i}+\frac{1}{2\lambda_j}+\frac{1}{2\lambda_d}+\frac{1}{2\lambda_l}) $ |
$ {\mathcal I}_{ijg, k}^{3, 2, 3, 1} = \left\{\omega\in [\varrho, 2\varrho]^{m}:| \lt k, \omega \gt +g^{*}+\varepsilon^{(1+\rho)}(\frac{\mu_i^*}{\lambda_i}+\frac{\mu_j^*}{\lambda_j}+\mu_{ijg, 1}^*)| \lt \frac{\varrho } {C_*|k|^{m+1}}\right\}, $ |
$ {\mathcal I}_{ijg, k}^{3, 2, 3, 2} = \left\{\omega\in [\varrho, 2\varrho]^{m}:| \lt k, \omega \gt +g^{*}+\varepsilon^{(1+\rho)}(\frac{\mu_i^*}{\lambda_i}+\frac{\mu_j^*}{\lambda_j}+\mu_{ijg, 2}^*)| \lt \frac{\varrho } {C_*|k|^{m+1}}\right\}, $ |
then for $ l = -i-j-d, d\in d^*_{ijg}, $ from $ \varepsilon^{(1+\rho)}(\frac{\mu^*_d}{\lambda_d}+\frac{\mu^*_{-i-j-d}}{\lambda_{-i-j-d}}) $ is sufficiently small,
$ {\mathcal I}_{ijdl, k}^{3, 2, 3}\subset{\mathcal I}_{ijg, k}^{3, 2, 3, 1}\bigcup{\mathcal I}_{ijg, k}^{3, 2, 3, 2}. $ |
Thus
$ \bigcup\limits_{l = -i-j-d}\bigcup\limits_{d\in {d^*_{ijg}}}{\mathcal I}_{ijdl, k}^{3, 2, 3}\subset({\mathcal I}_{ijg, k}^{3, 2, 3, 1}\bigcup{\mathcal I}_{ijg, k}^{3, 2, 3, 2}). $ |
We get
$ {\rm meas}{\mathcal I}_{ijg, k}^{3, 2, 3, 1}\leq\frac{2\varrho^{m}}{C_*|k|^{m+2}}, \quad {\rm meas}{\mathcal I}_{ijg, k}^{3, 2, 3, 2}\leq\frac{2\varrho^{m}}{C_*|k|^{m+2}}. $ |
When $ |g| > |k|\varrho+4 $, the sets $ {\mathcal I}_{ijg, k}^{3, 2, 3, 1}, {\mathcal I}_{ijg, k}^{3, 2, 3, 2} $ are empty. So let
$ R^{3, 2, 3} = \bigcup\limits_{ 0\neq k\in {\Bbb Z}^{m}}\bigcup\limits_{i\in S, j\in S}\bigcup\limits_{d\in \mathbb{Z}^2_*}\bigcup\limits_{l = -i-j-d}{\mathcal I}_{ijdl, k}^{3, 2, 3} \subset\bigcup\limits_{ 0\neq k\in {\Bbb Z}^{m}}\bigcup\limits_{i\in S, j\in S}\bigcup\limits_{g\in\mathbb{Z}}({\mathcal I}_{ijg, k}^{3, 2, 3, 1}\bigcup{\mathcal I}_{ijg, k}^{3, 2, 3, 2}), $ |
then
$ measR3,2,3≤meas⋃0≠k∈Zm⋃i∈S,j∈S⋃g∈Z(I3,2,3,1ijg,k⋃I3,2,3,2ijg,k)=meas⋃0≠k∈Zm⋃i∈S,j∈S⋃1≤|g|≤|k|ϱ+4(I2,3,1ijg,k⋃I2,3,2ijg,k)≤∑0≠k∈Zm4n2(|k|ϱ+4)2ϱmC∗|k|m+2≤C3C∗ϱm, $
|
where the constant $ C_3 $ depends on $ n, m. $ Thus
$ {\rm meas} R^{3, 2, 3}\leq\frac{\gamma}{27}\varrho^m $ |
provided $ C_* $ large enough. Denote
$ R^{3, 2} = R^{3, 2, 1}\cup R^{3, 2, 2}\cup R^{3, 2, 3}, $ |
then we have $ {\rm meas} R^{3, 2}\leq\frac{\gamma}{9}\varrho^{m} $.
Case 3. Similar to Case 2, there exists a set $ R^{3, 3} $ so that $ \forall\omega\in [\varrho, 2\varrho]^m\backslash R^{3, 3} $, Lemma 4.1$(\rm {iii})$ is true, and $ {\rm meas} R^{3, 3}\leq\frac{\gamma}{9}\varrho^{m} $. We omit the proof.
Denote
$ \overline R = [\varrho, 2\varrho]^m\setminus\Big( R^{3, 1}\cup R^{3, 2}\cup R^{3, 3}\Big), $ |
then it satisfies as required and
$ {\rm meas}\overline R\geq(1-\frac{\gamma}{3})\varrho^m. $ |
$ \mathbb{N} $ is the set of natural Numbers, $ \mathbb{Z} $ is the set of integers, $ \mathbb{Z}^n $ is an n-dimensional integer space, $ \mathbb{R} $ is the set of real Numbers, $ \mathbb{R}^n $ is an n-dimensional Euclidean space, $ \mathbb{T}^n $ is an n-dimensional torus.
We would like to thank the referees for their valuable comments and suggestions to improve our paper. This paper is partially supported by the National Natural Science Foundation of China (Grant Nos.11701567, 11601270) and the Fundamental Research Funds for the Central Universities(Grant Nos.19CX02048A, 17CX02048).
The authors declare that they have no competing interests in this paper.
[1] |
Wilson B, Dorman M (2008) Cochlear implants: A remarkable past and a brilliant future. Hearing Res 242: 3-21. doi: 10.1016/j.heares.2008.06.005
![]() |
[2] | Zrenner E (2013) Fighting Blindness with Microelectronics. Sci Transl Med 5(210): 210ps16. |
[3] |
Perlmutter J, Mink J (2006) Deep brain stimulation. Annu Rev Neurosci 29: 229-257. doi: 10.1146/annurev.neuro.29.051605.112824
![]() |
[4] | Priori A, Foffani G, Rossi L, et al. (2012) Adaptive deep brain stimulation (aDBS) controlled by local field potential oscillations. Exp Neurol 245: 77-86. |
[5] |
Gallentine W, Mikati M (2009) Intraoperative electrocorticography and cortical stimulation in children. J Clin Neurophysiol 26: 95-108. doi: 10.1097/WNP.0b013e3181a0339d
![]() |
[6] |
Keefer E, Botterman B, Romero M, et al. (2008) Carbon nanotube coating improves neuronal recordings. Nat Nanotechnol 3: 434-439. doi: 10.1038/nnano.2008.174
![]() |
[7] |
Ben-Jacob E, Hanein Y (2008) Carbon nanotube micro-electrodes for neuronal interfacing. J Mater Chem 18: 5181-5186. doi: 10.1039/b805878b
![]() |
[8] | Nick C, Joshi R, Schneider J, et al. (2012) Three-dimensional carbon nanotube electrodes for extracellular recording of cardiac myocytes. Biointerphases 7: 58-64. |
[9] |
Brüggemann D, Wolfrum B, Maybeck V, et al. (2011) Nanostructured gold microelectrodes for extracellular recording from electrogenic cells. Nanotechnology 22: 265104. doi: 10.1088/0957-4484/22/26/265104
![]() |
[10] |
Nick C, Quednau S, Sarwar R, et al. (2014) High Aspect Ratio Gold Nanopillars on Microelectrodes for Neural Interfaces. Microsyst Technol 20: 1849-1857. doi: 10.1007/s00542-013-1958-x
![]() |
[11] | Kim D, Abidian M, Martin D (2004) Conducting Polymers Grown in Hydrogel Scaffolds Coated on Neural Prosthetic Devices. J Biomed Mater Res A 71: 577-585. |
[12] | Poppendieck W, Hoffmann K-P (2009) In 4th European Conference of the International Federation for Medical and Biological Engineering Springer: 2409-2412. |
[13] |
Potter S, DeMarse T (2001) A new approach to neural cell culture for long-term studies. J Neurosci Methods 110: 17-24. doi: 10.1016/S0165-0270(01)00412-5
![]() |
[14] |
Potter S (2001) Distributed processing in cultured neuronal networks In M.A.L. Nicoleleis, (ed.), Advances in Neural Population Coding (Progress in Brain Research), 130: 49-62. doi: 10.1016/S0079-6123(01)30005-5
![]() |
[15] |
Hoogerwerf A, Wise K (1994) A Three-Dimensional Microelectrode Array for Chronic Neural Recording. IEEE Trans Biomed Eng 41: 1136-1146. doi: 10.1109/10.335862
![]() |
[16] |
Nordhausen C, Maynard E, Normann R (1996) Single unit recording capabilities of a 100 microelectrode array. Brain Res 726: 129-140. doi: 10.1016/0006-8993(96)00321-6
![]() |
[17] |
Gabay T, Ben-David M, Kalifa I, et al. (2007) Electro-chemical and biological properties of carbon nanotube based multi-electrode arrays. Nanotechnology 18: 035201-035206. doi: 10.1088/0957-4484/18/3/035201
![]() |
[18] | Fuchsberger K, Le Goff A, Gerwig R, et al. (2010) Integration of Carbon Nanotubes in Microelectrode Arrays by Microcontact Printing and Electropolymerization for Neurostimulation and Biosensing Applications. In 7th Meeting on Substrate-Integrated Microelectrodes, Reutlingen, Germany; 267-268. |
[19] |
Gabriel G, Gomez-Martinez R, Villa R (2008) Single walled carbon nanotubes deposited on surface electrodes to improve interface impedance. Physiol Meas 29: 203-212. doi: 10.1088/0967-3334/29/6/S18
![]() |
[20] | Nick C, Thielemann C (2014) Are Carbon Nanotube Microelectrodes Manufactured from Dispersion Stable Enough for Neural Interfaces? Bio Nano Sci 4: 216-225. |
[21] |
Bauer L, Birenbaum N, Meyer G (2004) Biological applications of high aspect ratio nanoparticles. J Mater Chem 14: 517-526. doi: 10.1039/b312655b
![]() |
[22] |
Wang H-W, Shieh C-F, Chen H-Y, et al. (2006) Standing [111] gold nanotube to nanorod arrays via template growth. Nanotechnology 17: 2689-2694. doi: 10.1088/0957-4484/17/10/041
![]() |
[23] |
Pancrazio J, Whelan J, Borkholder D, et al. (1999) Development and application of cell-based biosensors. Ann Biomed Eng 27: 697-711. doi: 10.1114/1.225
![]() |
[24] | Joye N, Schmid A, Leblebici Y (2008) An electrical model of the cell-electrode interface for high-density microelectrode arrays. In Proceedings of the 30th Annual International Conference of the IEEE Engineering in Medicine and Biology Society: 559-562. |
[25] |
Sessler F, Hsu F, Felder T, et al. (1998) Effects of ethanol on rat so mato sensory cortical neurons. Brain Res 804: 266-274. doi: 10.1016/S0006-8993(98)00680-5
![]() |
[26] |
Massobrio P, Massobrio G, Martinoia S (2007) Multi-program approach for simulating recorded extracellular signals generated by neurons coupled to microelectrode arrays. Neurocomputing 70: 2467-2476. doi: 10.1016/j.neucom.2006.09.008
![]() |
[27] | Gabay T (2009) Carbon Nanotube Microelectrode Arrays for Neuronal Patterning and Recording (PhD thesis) Tel-Aviv University. |
[28] |
Bauerdick S, Burkhardt C, Kern D, et al. (2003) Substrate-Integrated Microelectodes with Improved Charge Transfer Capacity by 3-Dimensional Micro-Fabrication. Biomed Microdevices 5: 93-99. doi: 10.1023/A:1024526626016
![]() |
[29] |
Sorkin R, Gabay T, Blinder P, et al. (2006) Compact self-wiring in cultured neural networks. J Neural Eng 3: 95-101. doi: 10.1088/1741-2560/3/2/003
![]() |
[30] |
Sorkin R, Greenbaum A, David-Pur M, et al. (2009) Process entanglement as a neuronal anchorage mechanism to rough surfaces. Nanotechnology 20: 015101. doi: 10.1088/0957-4484/20/1/015101
![]() |
[31] | CNT Expertise Centre (2009) Available from: http://www.nanocyl.com/en/CNT-Expertise-Centre/Carbon-Nanotubes. |
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