On Reducibility of 1d Wave Equation with Quasiperiodic in Time Potentials

Abstract

In this paper we use a KAM theorem of Grébert and Thomann (Commun Math Phys 307:383–427, 2011) to prove the reducibility of the 1d wave equation with Dirichlet boundery conditions on $[0,\pi ]$ with a quasi-periodic in time potential under some symmetry assumptions. From Mathieu–Hill operator’s known results (Eastham in The spectral theory of periodic differential operators, Hafner, New York, 1974; Magnus and Winkler in Hill’s equation, Wiley-Interscience, London, 1969) and Bourgain’s techniques (Commun Math Phys 204:207–247, 1999), we prove that for any $\epsilon$ small enough, there exist a $0<m_{\epsilon }\le 1$ and one solution $u_{\epsilon }(t,x)$ with

$$\begin{aligned} \Vert u_{\epsilon }(t_n,x)\Vert _{H^1({\mathbb {T}})}\rightarrow \infty , \qquad |t_n|\rightarrow \infty , \end{aligned}$$

where $u_{\epsilon }(t,x)$ satisfies 1d wave equation

$$\begin{aligned} u_{tt}-u_{xx}+m_{\epsilon }u-\epsilon \cos 2t u=0, \end{aligned}$$

with Dirichlet boundery conditions on $[0,\pi ]$.

1 Introduction and Main Results

In this paper we consider the linear equation

$$\begin{aligned} u_{xx}-u_{xx}+(g(x)+\epsilon V(t\omega ,x; \omega ))u=0 \end{aligned}$$

(1.1)

with Dirichlet boundery conditions on $[0,\pi ]$; namely,

$$\begin{aligned} u(t,0)=u(t,\pi )=0, \end{aligned}$$

where $\epsilon >0$ is a small parameter and the frequency vector $\omega$ is regarded as a parameter in $\Pi : = [0,2\pi ]^n \subset \mathbb R^n$. g(x) is analytic on $|\mathfrak {I}x|<\rho$ and fulfills

$$\begin{aligned} g(x+2\pi )=g(x) \qquad \mathrm{and} \qquad g(-x)=g(x), \end{aligned}$$

where $\rho$ is a positive number. The function $V(\theta ,x;\omega ), (\theta ,x;\omega )\in {\mathbb {T}}^n\times {\mathbb {T}}\times \Pi$, is $C^1 -$smooth in all its variables and is analytic in $(\theta ,x)$. For $\rho >0$ it analytically in $\theta , x$ extends to the domain

$$\begin{aligned} {\mathbb {T}}^n_{\rho }\times {\mathbb {T}}_{\rho }\times \Pi , \qquad {\mathbb {T}}^n_{\rho } =\{(a+ib)\in {\mathbb {C}}^n/{2\pi {\mathbb {Z}}^n}\big ||b|<\rho \}, \end{aligned}$$

where it is bounded by $C_1$, as well as its partial derivative in $\omega$ and $|\cdot |$ denotes the sup - norm for complex vectors. In addition, $V(\theta ,-x; \omega )=V(\theta ,x;\omega )$ and $\partial _{\omega }V(\theta ,-x;\omega )=\partial _{\omega }V(\theta ,x;\omega )$ for $|\mathfrak {I}\theta |<\rho$ and $x\in {\mathbb {T}}$ and $\omega \in \Pi$. In the following we write $V(\theta ,x;\omega )$ as $V(\theta ,x)$ for simplicity.

To state our main result, we introduce another parameter. Write $g(x)=m+g_0(x)$, where m is the average over $[-\pi ,\pi ]$ of g(x) and $g_0$ has zero average. Consider now the Sturm Liouville problem

$$\begin{aligned} -\partial _{xx}\phi _j+g_0(x)\phi _j=\mu _j\phi _j, \qquad \phi _j(0)=\phi _j(\pi )=0; \end{aligned}$$

(1.2)

it is well known that all the eigenvalues are distinct, that the solutions $\{\phi _j\}_{j\ge 1}$ of (1.2) form an orthogonal basis of $L^2$ that we will assume to be normalized, and that, since $g_0(x)$ is analytic and symmetric, the $\phi _j$ are analytic and skew-symmetric. We define $\lambda _j:=\sqrt{\mu _j+m}$. It is well known that (1.1) is Hamiltonian with Hamiltonian function given by

$$\begin{aligned} H=\frac{1}{2}\langle v,v \rangle +\frac{1}{2}\langle Tu,u\rangle +\frac{\epsilon }{2}\int _{0}^{\pi }V(\omega t,x)u^{2} dx, \end{aligned}$$

(1.3)

where $v:=u_{t}$ is the momentum conjugated to u. To define formally the phase space, denote $T := -\partial _{xx}+g(x)$. Let $q\ge 0$ and denote by $\ell _q^{2}$ the space

$$\begin{aligned} \ell _q^2 := \{w=(w_j):\quad \sum _{j\ge 1}{\left| w_j \right| }^2 j^q<+\infty \}. \end{aligned}$$

The operator T has eigenfunctions $(\phi _j)_{j\ge 1}$ which satisfy

$$\begin{aligned} \left\{ \begin{array}{lll} T\phi _j=\lambda _j^2\phi _j,\quad j\ge 1,\\ \phi _j(0)=\phi _j(\pi )=0.\\ \end{array} \right. \end{aligned}$$

$\{ \phi _j \}$ form an orthogonal basis of $L^2(0,\pi )$ and $\Vert \phi _j \Vert _{L^{2}(0,\pi )}= 1$. For our aim we fix a positive number $m>0$ such that $\mu _j+m>0$ for all $j\ge 1$.

Let $u=\sum \limits _{j\ge 1}u_j \phi _j$ be a typical element of $L^2(0,\pi ).$ Then for $p\ge 0$,

$$\begin{aligned} (u_j)_{j\ge 1}\in \ell _{2p}^{2}\iff u\in \mathcal {H}^p(0,\pi )=\mathcal {D}(T^\frac{p}{2})=\{ u\in L^2(0,\pi ):\quad T^{\frac{p}{2}}u \in L^2(0,\pi ) \}. \end{aligned}$$

$\mathcal {H}^p(0,\pi )$ is a sobolev space based on T and we denote

$$\begin{aligned} \Vert u\Vert ^2_{\mathcal {H}^p}=\Vert T^{\frac{p}{2}}u\Vert ^2_{L^2(0,\pi )}=\sum _{j\ge 1}\lambda _j^{2p}|u_j|^2. \end{aligned}$$

In the above we write $\mathcal {H}^p$ instead of $\mathcal {H}^p(0,\pi )$ for simplicity. Note $\phi _j$ is skew-symmetric and periodic, we rewrite u(x) as $\tilde{u} = \sum \limits _{j\ge 1}u_j\phi _j(x), x\in [-\pi ,\pi ]$. It is clear that $\tilde{u}(x+2\pi )=\tilde{u}(x)$ and $\tilde{u}(-x)=-\tilde{u}(x)$. In the following the domain of discussed functions is changed, in $[0,\pi ]$ or in ${\mathbb {T}}$. But it is often clear from the context. For simplicity we write u(x) instead of $\tilde{u}(x)$.

For $p \ge 0$, we define the odd periodic sobolev space $\mathcal {H}_{odd}^{p}({\mathbb {T}})$, which is defined as

$$\begin{aligned} \mathcal {H}^{p}_{odd}({\mathbb {T}}) = : \{u(x)|_{x\in [0,\pi ]}\in \mathcal {H}^p(0,\pi ), u(x+2\pi )=u(x), u(-x)=-u(x)\}. \end{aligned}$$

We also denote the phase space $\mathcal {F}_p$ as the space of the functions $(u,v)\in \mathcal {F}_p:= \mathcal {H}_{odd}^{p+1}({\mathbb {T}}) \times \mathcal {H}_{odd}^{p}({\mathbb {T}}), p\ge 0$. We endow $\mathcal {F}_p$ by the norm $\Vert (u,v)\Vert _{\mathcal {F}_p}^{2}:=\Vert u\Vert _{\mathcal {H}^{p+1}({\mathbb {T}})}^{2}+\Vert v\Vert _{\mathcal {H}^{p}({\mathbb {T}})}^{2}$, where we use $\Vert v\Vert _{\mathcal {H}^p({\mathbb {T}})}$ in stead of $\Vert v\Vert _{\mathcal {H}_{odd}^p({\mathbb {T}})}$ for simplicity.

Theorem 1.1

Given g(x),$V(t\omega ,x; \omega )$ as above. Then there exists $\varepsilon _{0}>0$ so that for all $0<|\epsilon |\le \varepsilon _0$, there exists $\Pi _{\epsilon }\subset [0,2\pi )^n$ of positive measure and asymptotically full measure: $Mea(\Pi _{\epsilon })\rightarrow (2\pi )^n$ as $\epsilon \rightarrow 0$, such that for all $\omega \in \Pi _{\epsilon }$, the linear wave equation (1.1) is equivalent to an autonomous Hamiltonian system (2.2), which can be reduced in an extended phase space $(\theta ,I, z,\bar{z})\in {\mathbb {T}}^n\times {\mathbb {R}}^n \times \ell _p^2 \times \ell _p^2\ (p\ge 0)$, to a linear Hamiltonian equation with a Hamiltonian which doesn’t include the angle variable $\theta$.

In particular, we prove the following result concerning the solutions of (1.1).

Corollary 1.1

Given g(x),$V(t\omega ,x; \omega )$ as above. Let $p\ge 0$ and $(u_0,v_0)\in \mathcal {F}_p$. Then there exists $\varepsilon _{0}>0$ so that for all $0<|\epsilon |\le \varepsilon _0$ and $\omega \in \Pi _{\epsilon }$, there exists a unique solution $u(t,x)\in C({\mathbb {R}};\mathcal {H}^{p+1})\times C^1({\mathbb {R}};\mathcal {H}^{p})$, so that $(u(0,x),v(0,x))=(u_0,v_0)$. Moreover, u is almost-periodic in time and we have the bounds

$$\begin{aligned} C(g,p)^{-1}(1-\epsilon c)\Vert (u_0,v_0)\Vert _{\mathcal {F}_p} \le \Vert (u(t),v(t))\Vert _{\mathcal {F}_p} \le C(g,p)(1+c\epsilon )\Vert (u_0,v_0)\Vert _{\mathcal {F}_p}. \end{aligned}$$

Remark 1.1

In the above the constant c depends on $n,g,p,\rho$ and $\omega$. $\epsilon _0$ depends on g, n, $C_1$ and $\rho$.

Remark 1.2

In fact, the assumption on $V(\theta ,x;\omega )$ can be weakened. More clearly, we can assume that for $\rho >0$, $V(\theta ,x;\omega )$ and $\partial _{\omega }V(\theta ,x;\omega )$ analytically in $\theta$ extend to the domain

$$\begin{aligned} {\mathbb {T}}^n_{\rho }\times {\mathbb {T}}\times \Pi , \qquad {\mathbb {T}}^n_{\rho } =\{(a+ib)\in {\mathbb {C}}^n/{2\pi {\mathbb {Z}}^n}\big ||b|<\rho \} \end{aligned}$$

and $C^{\infty }$ in $x\in {\mathbb {T}}$. In addition, for $|\mathfrak {I}\theta |<\rho , x\in {\mathbb {T}}$ and $\alpha =0, \ldots , [\frac{p+1}{2}]+1$,

$$\begin{aligned} \big |\partial _{x}^{(\alpha )}(V(\theta ,x))\big |\le C_1\ \mathrm{and}\ |\partial ^{(\alpha )}_{x}(\partial _{\omega }V(\theta ,x;\omega ))|\le C_1. \end{aligned}$$

If set $g_0(x)=0$, then

Corollary 1.2

Given $m\ge 0$,$V(t\omega ,x; \omega )$ as above. Let $p\ge 0$ and $(u_0,v_0)\in \mathcal {F}_p$. Then there exists $\varepsilon _{0}>0$ so that for all $0<|\epsilon |\le \varepsilon _0$ and $\omega \in \Pi _{\epsilon }$, there exists a unique solution $u(t,x)\in C({\mathbb {R}};\mathcal {H}^{p+1})\times C^1({\mathbb {R}};\mathcal {H}^{p})$, so that $(u(0,x),v(0,x))=(u_0,v_0)$. Moreover, u is almost-periodic in time and we have the bounds

$$\begin{aligned} C(m,p)^{-1}(1-\epsilon c)\Vert (u_0,v_0)\Vert _{\mathcal {F}_p} \le \Vert (u(t),v(t))\Vert _{\mathcal {F}_p} \le C(m,p)(1+c\epsilon )\Vert (u_0,v_0)\Vert _{\mathcal {F}_p}. \end{aligned}$$

Remark 1.3

$\epsilon _0$ depends on m, n, $C_1$ and $\rho$. But if $0\le m\le M$, then $\epsilon _0$ depends on M, not on the definite choice of $m\in [0,M]$.

Following Bourgain [6] we prove the following

Corollary 1.3

For $0<|\epsilon |\le 1$, there exists a positive mass $0<m_{\epsilon }\le 1$. For the corresponding 1d wave equation

$$\begin{aligned} u_{tt}-u_{xx}+m_{\epsilon }u-\epsilon \cos 2t u=0, \end{aligned}$$

(1.4)

with Dirichlet boundery conditions on $[0,\pi ]$, there exists at least one solution $u_{\epsilon }(t,x)$ which satisfies

$$\begin{aligned} \Vert u_{\epsilon }(t_j,x)\Vert _{H^1({\mathbb {T}})}\rightarrow \infty , \qquad |t_j|\rightarrow \infty . \end{aligned}$$

Remark 1.4

Corollary 1.3 shows that one can not avoid to restrict the choice of parameters $\omega$ to a Cantor type set in Corollary 1.2.

Related Results In 1960s N. Bogolyubov [5] found that KAM - technique can be applied to prove reducibility of non-autonomous finite-dimensional linear systems to constant coefficient equations. For the large reference about the reducibility of finite-dimensional systems, refer to [8, 9, 13, 14, 23, 27, 28] and etc. Comparing with the finite-dimensional systems, the reducibility results in infinite dimensional Hamiltonian systems are relatively few.

Bambusi and Graffi [2] proved reducibility of one-dimensional Schrödinger equation

$$\begin{aligned} H(t)\psi (x,t) =\mathrm{i} \partial _t\psi (x,t), x\in {\mathbb {R}}; \qquad H(t) : = -\frac{d^2}{dx^2}+Q(x)+\epsilon V(x,\omega t), \epsilon \in {\mathbb {R}}\qquad \quad \end{aligned}$$

(1.5)

to constant in time coefficients, where $Q(x)\in C^{\infty }({\mathbb {R}}; {\mathbb {R}}), Q(x) \sim |x|^{\alpha }$ for $\alpha >2$ as $|x|\rightarrow \infty$ and $V(x,\phi )$ is a $C^{\infty }({\mathbb {R}}; {\mathbb {R}}) -$valued holomorphic function of $\phi \in {\mathbb {T}}^n$, with $|V(x,\phi )||x|^{-\beta }$ bounded as $|x|\rightarrow \infty$ for some $\beta <\frac{\alpha -2}{2}$. The main technique they used is now called Kuksin’s Lemma, which was first introduced by Kuksin [30, 31] in order to build the KAM results for Kdv equations. Their method failed in the limit case $\alpha >2$ and $\beta =\frac{\alpha -2}{2}$, which was later solved by Liu and Yuan [34].

Another interesting case is $\alpha =2$. The Eq. (1.5) with $Q(x)=x^2$ is called 1d quantum Harmonic oscillator under quasi periodic in time perturbations. It is well-known that the long-time behavior of the solution $\psi (t,x)$ of the Eq. (1.5) in this case is closely related to the spectral properties of the Floquet operator

$$\begin{aligned} K_{F}:=-\mathrm{i}\sum _{k=1}^n\omega _k\frac{\partial }{\partial \theta _k}-\frac{d^2}{dx^2}+x^2+\varepsilon V(\theta ,x). \end{aligned}$$

Wang [41] first proved the pure-point nature of the spectrum of the above Floquet operator $K_{F}$ which is defined on $L^2({\mathbb {R}}) \otimes L^2({\mathbb {T}}^n)$. Wang [41] considered the potential $V(x,\theta ) = e^{-x^2}\sum \limits _{k=1}^{n}\cos \theta _k$, which clearly has exponential decay. Later on, Grébert and Thomann [22] solved the case for $\beta <\frac{\alpha -2}{2}=0$, in which the perturbing potential $V(x,\theta )$ has polynomial decay. In [43] the above results are further improved, in which the perturbing potential $V(x,\theta )$ has only logarithmic decay, while the case for $\alpha =2$ and $\beta =0$ is still open (see Eliasson [15]).

The reducibility results in higher dimension are very few. Recently, Eliasson and Kuksin [17] (also see [18]) obtained the reducibility for the linear d-dimensional Schrödinger equation

$$\begin{aligned} \dot{u}=-\mathrm{i}(\Delta u- \epsilon V(\phi _0+t\omega , x; \omega )u), \qquad u=u(t,x), x\in {\mathbb {T}}^d. \end{aligned}$$

A direct corollary from the reducibility is that $\Vert u(t)\Vert _p\sim \Vert u(0)\Vert _p$ for small $\epsilon$ and chosen $\omega$(in large Lebesgue measure). See Bourgain [6, 7] and Wang [40, 42] for relative results for the growth of solutions for linear Schrödinger equation with time quasi-periodic and with smooth bounded potentials.

The reducibility for wave equations in any d dimension($d \ge 2$) is doubtful. See [16] for recent development for high dimensional wave equations. For 1d wave equations there are many interesting KAM or normal form results. See [1, 3, 4, 10, 19, 20, 29, 38, 44] for details.

Notations The notation “$\preceq$” used below means $\le$ modulo a multiplicative constant that, unless otherwise specified, depends on n only. $|\cdot |$ denotes the sup - norm for complex vectors.

We use the notation “Meas” to denote the Lebesgue measure in ${\mathbb {R}}^n$. The set $\Pi$ is the space of the external parameter $\xi$. Denote by $\bigtriangleup _{\xi \eta }$ the difference operator in the variable $\xi$,

$$\begin{aligned} \bigtriangleup _{\xi \eta }f=f(\cdot ,\xi )-f(\cdot ,\eta ), \end{aligned}$$

where f is a real or complex function.

For $l=(l_1,l_2,\dots ,l_k,\dots )\in \mathbb {Z}^\infty$ so that only a finite number of coordinates are nonzero, we denote by $|l|=\sum \limits _{j=1}^{\infty }|l_j|$, and its length, $\langle l \rangle =1+\vert \sum \limits _{j=1}^{\infty }jl_j\vert$. We set $\mathcal {Z}=\{(k,l)\ne 0,\vert l \vert \le 2 \} \subset \mathbb {Z}^n\times \mathbb {Z}^{\infty }$.

2 The Hamiltonians

Consider the eigenfunctions $\phi _j$ of the Sturm-Liouville problem (1.2) and their Fourier expansion $\phi _j(x)=\frac{1}{\sqrt{2\pi }}\sum \limits _{k\in \mathbb {Z}}\phi _k^je^{ik\cdot x}$, it is well-known that one has

$$\begin{aligned} |\phi _k^j|\le ce^{-\sigma \big ||k|-j\big |} \end{aligned}$$

with $0< \sigma < \rho$ and a suitable constant $c>0$ (see [11]). Remark that $\phi _0^j=0$ since $\phi _j(x)$ is anti-symmetric.

With $u=\sum \limits _{j\ge 1}\bar{q}_j\phi _j(x),v=\sum \limits _{j\ge 1}\bar{p}_j\phi _j(x)$, the Hamiltonian (1.3) is changed into

$$\begin{aligned} H=\frac{1}{2}\sum _{j\ge 1}(\bar{p}_j^2+\lambda _j^2\bar{q}_j^2)+\overline{G}(\bar{q},t), \end{aligned}$$

where

$$\begin{aligned} \overline{G}(q,t)=\frac{\epsilon }{2}\sum _{j,l\ge 1}\overline{q}_j\overline{q}_l\int _{0}^{\pi }V(\omega t,x)\phi _j(x)\phi _l(x)dx, \end{aligned}$$

and the symplectic structure is $\sum \limits _{j\ge 1} d\bar{q}_j\wedge d \bar{p}_j$. Rescaling the coordinates, set

$$\begin{aligned} q_j=\bar{q}_j{\lambda _j}^{\frac{1}{2}},\quad p_j={\lambda _j}^{-\frac{1}{2}}\bar{p}_j, \qquad j\ge 1. \end{aligned}$$

Then the new Hamiltonian is

$$\begin{aligned} H=\frac{1}{2}\sum _{j\ge 1}\lambda _j(p_j^2+q_j^2)+\widetilde{G}(g,t) \end{aligned}$$

with

$$\begin{aligned} \widetilde{G}(g,t)=\frac{\epsilon }{2}\sum _{j,l\ge 1}(\lambda _j\lambda _l)^{-\frac{1}{2}}\int _{0}^{\pi }V(\omega t,x)\phi _j(x)\phi _l(x)dx q_jq_l. \end{aligned}$$

Introducing the complex coordinates $z_j=\frac{1}{\sqrt{2}}(p_j-iq_j)$ and $\bar{z}_j=\frac{1}{\sqrt{2}}(p_j+iq_j)$ and now the symplectic structure is changed into $i\sum \limits _{j\ge 1}dz_j\wedge d\bar{z}_j$. The corresponding Hamiltonian is changed into

$$\begin{aligned} H(z,\bar{z},t)=\sum _{j\ge 1}\lambda _j z_j\bar{z}_j+G(z,\bar{z},t) \end{aligned}$$

with

$$\begin{aligned} G(z,\bar{z},t)=-\frac{\epsilon }{4}\sum _{j,l\ge 1} (\lambda _j\lambda _l)^{-1/2}\int _{0}^{\pi }V(\omega t,x)\phi _j(x)\phi _l(x)dx(z_j-\bar{z}_j)(z_l-\bar{z}_l). \end{aligned}$$

The corresponding equations are

$$\begin{aligned} \left\{ \begin{array}{lll} \displaystyle \dot{z}_j=-i\frac{\partial H}{\partial \bar{z}_j}=(-i)\left( \lambda _j z_j+ \frac{\partial G}{\partial \bar{z}_j}\right) ,\\ \displaystyle \dot{\bar{z}}_j=i\frac{\partial H}{\partial z_j}=i\left( \lambda _j \bar{z}_j+ \frac{\partial G}{\partial z_j}\right) ,\quad j\ge 1.\\ \end{array} \right. \end{aligned}$$

(2.1)

We re-interpret (2.1) as an autonomous Hamiltonian system in an extended phase space $\mathcal {P}^{\iota }={\mathbb {T}}^n\times {\mathbb {R}}^n \times \ell _{\iota }^2 \times \ell _{\iota }^2(\iota >1)$,

$$\begin{aligned} \left\{ \begin{array}{lll} \displaystyle \dot{z}_j=(-i)\lambda _j z_j-i\frac{\partial G}{\partial \bar{z}_j},\\ \displaystyle \dot{\bar{z}}_j=i\lambda _j z_j+i\frac{\partial G}{\partial z_j},\qquad j\ge 1\\ \displaystyle \dot{\theta }_j=\omega _j,\\ \displaystyle \dot{I}_j=-\frac{\partial G}{\partial \theta _j}(\theta ,z,\bar{z}),\qquad j=1,2,\dots ,n.\\ \end{array} \right. \end{aligned}$$

(2.2)

where

$$\begin{aligned} G= & {} G(\theta ,z,\overline{z})\nonumber \\= & {} -\frac{\epsilon }{4}\sum _{j,l\ge 1} (\lambda _j\lambda _l)^{-1/2}\int _{0}^{\pi }V(\theta ,x)\phi _j(x)\phi _l(x)dx(z_j-\bar{z}_j)(z_l-\bar{z}_l). \end{aligned}$$

(2.3)

We remark that the first three equations of (2.2) are independent of I and are equivalent to (2.1). Furthermore (2.2) reads as the Hamiltonian equations associated with the Hamiltonian function

$$\begin{aligned} H=N+G, \end{aligned}$$

where

$$\begin{aligned} N=\sum _{j=1}^{n}\omega _j I_j+\sum _{j\ge 1}\lambda _j z_j\bar{z}_j \end{aligned}$$

(2.4)

and $G=G(\theta ,z,\bar{z})$. Here the external parameters are directly the frequencies $\omega =(\omega _j)_{1\le j\le n}\in \Pi : = [0,2\pi ]^n$ and the normal frequencies $\Omega _j=\lambda _j=\sqrt{\mu _j+m}$ are fixed.

3 A KAM Theorem

In this section we state a KAM Theorem which introduced by Grébert and Thomann [22] and will be used in Sect. 4 to prove Theorem 1.1. The notations we used here mostly come from [22].

3.1. Let $\Pi \in {\mathbb {R}}^n$ be a bounded closed set so that $Meas(\Pi )>0$.

Assumption 1

(Nondegeneracy) Denote by $\omega =(\omega _1,\dots ,\omega _n)$ the internal frequency. We assume that the map $\xi \mapsto \omega (\xi )$ is a homeomorphism from $\Pi$ to its image which is Lipschitz continuous and its inverse also. Moreover we assume that for all $(k,l)\in \mathcal {Z}$,

$$\begin{aligned} Meas(\{\xi :k\cdot \omega (\xi )+l\cdot \Omega (\xi )=0\})=0. \end{aligned}$$

and for all $\xi \in \Pi$,

$$\begin{aligned} l\cdot \Omega (\xi )\ne 0,\qquad \forall \ 1\le \vert l \vert \le 2. \end{aligned}$$

Assumption 2

(Spectral asymptotics) Set $\Omega _0=0$. We assume that there exists $\bar{m}>0$, so that for all $i,j\ge 0$ and uniformly on $\Pi$,

$$\begin{aligned} \vert \Omega _i-\Omega _j \vert \ge \bar{m}\vert i-j \vert . \end{aligned}$$

Moreover we assume that there exists $\beta >0$ such that the functions,

$$\begin{aligned} \xi \longmapsto j^{2\beta }\Omega _j(\xi ), \end{aligned}$$

are uniformly Lipschitz on $\Pi$ for $j\ge 1$.

In the sequel, we will use the distance

$$\begin{aligned} \vert \Omega -\Omega '\vert _{2\beta ,\Pi }=\sup _{\xi \in \Pi }\sup _{j\ge 1}j^{2\beta } \vert \Omega _j(\xi ) -\Omega _j'(\xi )\vert , \end{aligned}$$

and the semi-norm,

$$\begin{aligned} \vert \Omega \vert _{2\beta ,\Pi }^{\mathcal {L}}=\sup _{\begin{array}{c} \xi ,\eta \in \Pi \\ \xi \ne \eta \end{array}}\sup _{j\ge 1} \frac{j^{2\beta }\vert \triangle _{\xi \eta }\Omega _j\vert }{\vert \xi -\eta \vert }. \end{aligned}$$

Finally, we set

$$\begin{aligned} \vert \omega \vert _{\Pi }^{\mathcal {L}}+\vert \Omega \vert _{2\beta ,\Pi }^{\mathcal {L}}=M, \end{aligned}$$

(3.1)

where $\vert \omega \vert _{\Pi }^{\mathcal {L}}=\sup \limits _{\begin{array}{c} \xi ,\eta \in \Pi \\ \xi \ne \eta \end{array}}\max \limits _{1\le k\le n}\frac{\vert \triangle _{\xi \eta }\omega _k\vert }{\vert \xi -\eta \vert }$. For $s,r>0$, we define the (complex) neighborhood of $\mathbb {T}^n\times \{0,0,0\}$ in $\mathcal {P}$,

$$\begin{aligned} D(s,r)=\{(\theta ,I,u,v)\in \mathcal {P}\ \mathrm{s.t.}\ \vert \mathfrak {I}\theta \vert<s, |I|<r^2,\Vert u\Vert _{\iota }+\Vert v\Vert _{\iota }<r \}, \end{aligned}$$

where $|\cdot |$ denotes the sup - norm for complex vectors and $\Vert u\Vert _{\iota }^2=\sum \limits _{j\ge 1}\vert \ u_j \vert ^2 j^{\iota }$ and $\iota > 1$. Let $r>0$, then for $W=(X,Y,U,V)$ we denote

$$\begin{aligned} \vert W\vert _r=\vert X \vert +\frac{1}{r^2}\vert Y\vert +\frac{1}{r}(\Vert U\Vert _{\iota }+\Vert V\Vert _{\iota }). \end{aligned}$$

Denote by $X_P=(\partial _IP,-\partial _{\theta }P,\partial _vP,-\partial _uP)$. Then

Assumption 3

(Regularity) We assume that there exist $s,r>0$ so that

$$\begin{aligned} X_P:\qquad D(s,r)\times \Pi \longmapsto \mathcal {P}=\Pi ^n\times {\mathbb {R}}^n\times \ell _{\iota }^2 \times \ell _{\iota }^2. \end{aligned}$$

Moreover we assume that for all $\xi \in \Pi$, $X_P(\cdot ,\xi )$ is analytic in D(s, r) and that for all $\omega \in D(s,r)$, $P(\omega ,\cdot )$ and $X_P(\omega ,\cdot )$ are Lipschitz continuous on $\Pi$.

We then define the norms

$$\begin{aligned} \Vert P\Vert _{D(s,r)}:=\sup _{D(s,r)\times \Pi }\vert P\vert <+\infty , \end{aligned}$$

and

$$\begin{aligned} \Vert P\Vert _{D(s,r)}^{\mathcal {L}}:=\sup _{\begin{array}{c} \xi ,\eta \in \Pi \\ \xi \ne \eta \end{array}}\sup _{D(s,r)}\frac{\vert \triangle _{\xi \eta } P \vert }{\vert \xi -\eta \vert }, \end{aligned}$$

where $\triangle _{\xi \eta }P=P(\cdot ,\xi )-P(\cdot ,\eta )$ and we define the semi-norms

$$\begin{aligned} \Vert X_P\Vert _{r,D(s,r)}:=\sup _{D(s,r)\times \Pi }\vert X_P\vert _r<+\infty , \end{aligned}$$

and

$$\begin{aligned} \Vert X_P\Vert _{r,D(s,r)}^{\mathcal {L}}:=\sup _{\begin{array}{c} \xi ,\eta \in \Pi \\ \xi \ne \eta \end{array}}\sup _{D(s,r)}\frac{\vert \triangle _{\xi \eta } X_P\vert _r}{\vert \xi -\eta \vert }<+\infty , \end{aligned}$$

where $\triangle _{\xi \eta }X_P=X_P(\cdot ,\xi )-X_P(\cdot ,\eta )$.

We denote $\Gamma _{r,D(s,r)}^{\beta }$: Let $\beta >0$, we say that $P\in \Gamma _{r,D(s,r)}^{\beta }$ if $\langle P\rangle _{r,D(s,r)}+\langle P\rangle _{r,D(s,r)}^{\mathcal {L}}<\infty$ where the norm $\langle \cdot \rangle _{r,D(s,r)}$ is defined by the conditions

$$\begin{aligned} \Vert P\Vert _{D(s,r)}\le & {} r^2\langle P \rangle _{r,D(s,r)},\\ \max _{1\le j\le n}\Vert \frac{\partial P}{\partial I_j}\Vert _{D(s,r)}\le & {} \langle P \rangle _{r,D(s,r)},\\ \Vert \frac{\partial P}{\partial \omega _j}\Vert _{D(s,r)}\le & {} \frac{r}{j^{\beta }}\langle P \rangle _{r,D(s,r)},\qquad \forall \ j\ge 1\quad and \quad \omega _j=z_j,\overline{z}_j,\\ \Vert \frac{\partial ^{2}P}{\partial \omega _j\partial \omega _l}\Vert _{D(s,r)}\le & {} \frac{1}{(jl)^{\beta }}\langle P\rangle _{r,D(s,r)},\qquad \forall \ j,l\ge 1\quad and \quad \omega _j=z_j,\overline{z}_j. \end{aligned}$$

The semi-norm $\langle \cdot \rangle _{r,D(s,r)}^{\mathcal {L}}$ is defined by the conditions

$$\begin{aligned} \Vert P\Vert _{D(s,r)}^{\mathcal {L}}\le & {} r^2\langle P \rangle _{r,D(s,r)}^{\mathcal {L}},\\ \max _{1\le j\le n}\Vert \frac{\partial P}{\partial I_j}\Vert _{D(s,r)}^{\mathcal {L}}\le & {} \langle P \rangle _{r,D(s,r)}^{\mathcal {L}},\\ \Vert \frac{\partial P}{\partial \omega _j}\Vert _{D(s,r)}^{\mathcal {L}}\le & {} \frac{r}{j^{\beta }}\langle P \rangle _{r,D(s,r)}^{\mathcal {L}},\qquad \forall \ j\ge 1\quad and \quad \omega _j=z_j,\overline{z}_j,\\ \Vert \frac{\partial ^{2}P}{\partial \omega _j\partial \omega _l}\Vert _{D(s,r)}^{\mathcal {L}}\le & {} \frac{1}{(jl)^{\beta }}\langle P\rangle _{r,D(s,r)}^{\mathcal {L}},\qquad \forall \ j,l\ge 1\quad and \quad \omega _j=z_j,\overline{z}_j. \end{aligned}$$

Assumption 4

(Decay) $P\in \Gamma _{r,D(s,r)}^{\beta }$ for some $\beta >0$.

Denote $M=\vert \omega \vert _{\Pi }^{\mathcal {L}}+\vert \Omega \vert _{2\beta ,\Pi }^{\mathcal {L}}$.

Theorem 3.1

(Grébert and Thomann [22]) Suppose that N is a family of Hamiltonian of the form (2.4) on the phase space $\mathcal {P}$ depending on parameters $\xi \in \Pi$ so that Assumptions 1 and 2 are satisfied. Then there exists $\epsilon _0>0$ and $s>0$ so that every perturbation $H=N+P$ of N which satisfies Assumptions 3 and  4 and the smallness condition

$$\begin{aligned} \epsilon =(\Vert X_P\Vert _{r,D(s,r)}+\langle P\rangle _{r,D(s,r)})+ \frac{\alpha }{M}(\Vert X_P\Vert _{r,D(s,r)}^{\mathcal {L}}+\langle P\rangle _{r,D(s,r)}^{\mathcal {L}})\le \epsilon _0\alpha , \end{aligned}$$

for some $r>0$ and $0<\alpha \le 1$, the following holds.

There exist

• (i) a Cantor set $\Pi _{\alpha }\subset \Pi$ with $Meas(\Pi \backslash \Pi _{\alpha })\rightarrow 0$ as $\alpha \rightarrow 0$;

• (ii) a Lipschitz family of real analytic,symplectic coordinate transformations

  $$\begin{aligned} \Phi :D(s/2,r/2)\times \Pi _{\alpha }\longmapsto D(s,r); \end{aligned}$$

• (iii) a Lipschitz family of new normal form

  $$\begin{aligned} N^{*}=\sum _{j=1}^{n}\omega _j^{*}(\xi )I_j+\sum _{j\ge 1}\Omega _j^{*}(\xi )z_j\bar{z}_j \end{aligned}$$

  defined on $D(s/2,r/2)\times \Pi _{\alpha }$ such that

  $$\begin{aligned} H\circ \Phi =N^{*}+R^{*}, \end{aligned}$$

  where $R^{*}$ is analytic on D(s / 2, r / 2) and globally of order 3 at ${\mathbb {T}}^n\times \{0,0,0\}$. That is the Taylor expansion of $R^{*}$ only contains monomials $I^mz^q\bar{z}^{\bar{q}}$ with $2\vert m \vert +\vert q+\bar{q}\vert \ge 3$. Moreover each symplectic coordinate transformation is close to the identity

  $$\begin{aligned} \Vert \Phi -Id\Vert _{r,D(\frac{s}{2},\frac{r}{2})}\le c\epsilon , \end{aligned}$$

  and the new frequencies are close to the original ones

  $$\begin{aligned} |\omega ^{*}-\omega |_{\Pi _{\alpha }}+|\Omega ^{*}-\Omega |_{2\beta ,\Pi _{\alpha }} \le c\epsilon , \end{aligned}$$

  and the new frequencies satisfy a non resonance condition

  $$\begin{aligned} |k\cdot \omega ^{*}(\xi )+l\cdot \Omega ^{*}(\xi )|\ge \frac{\alpha }{2}\cdot \frac{\langle l \rangle }{1+|k|^{\tau }},\qquad \langle k,l \rangle \in \mathcal {Z},\xi \in \Pi _{\alpha }. \end{aligned}$$

As a consequence, for each $\xi \in \Pi _{\alpha }$ the forms $\Phi ({\mathbb {T}}^n\times \{0,0,0\})$ is still invariant under the flow of the perturbed Hamiltonian $H=N+P$, the flow is linear on these tori and furthermore all these tori are linearly stable.

4 Proof of Theorem 1.1 and Corollary 1.1

To prove Theorem 1.1, we need to check Assumptions 1 - 4 in Theorem 3.1 hold. For Assumption 1, we establish the following lemmas. The proofs for Lemma 4.1, Lemma 4.2 and Lemma 4.3 are similar. We only present the proof for Lemma 4.2 and delay it in the “Appendix”, which is the most complex case.

Lemma 4.1

For $\omega \in \mathcal {J}_1$ and any $k\in {\mathbb {Z}}^n$, $j\ge 1$,

$$\begin{aligned} k\cdot \omega +\Omega _j\ne 0, \end{aligned}$$

where $Meas(\Pi {\setminus } \mathcal {J}_1)=0$.

Lemma 4.2

For $\omega \in \mathcal {J}_2$ and any $k\in {\mathbb {Z}}^n$, $i\ne j,\ i,j \ge 1$,

$$\begin{aligned} k\cdot \omega +\Omega _i-\Omega _j\ne 0, \end{aligned}$$

where $Meas(\Pi {\setminus } \mathcal {J}_2)=0$.

Lemma 4.3

For $\omega \in \mathcal {J}_3$ and any $k\in {\mathbb {Z}}^n$, $i,j\ge 1$,

$$\begin{aligned} k\cdot \omega +\Omega _i+\Omega _j\ne 0, \end{aligned}$$

where $Meas(\Pi {\setminus } \mathcal {J}_3)=0$.

For Assumption 2, we have

Lemma 4.4

For any $i>j\ge 1$, there exists $c(m)>0$ so that

$$\begin{aligned} |\Omega _i-\Omega _j|\ge c(m)|i-j|. \end{aligned}$$

Assumption 3 is clear. For Assumption 4, we have the following lemma.

Lemma 4.5

For $\iota >1$,

$$\begin{aligned} \Vert G\Vert _{D(\rho ,r)}\le & {} c(\iota ,g, \rho )\epsilon r^2, \nonumber \\ \big |\frac{\partial G}{\partial w_k}\big |\le & {} \frac{c(\iota , g, \rho )\epsilon r}{k^{1/2}}, \qquad w_k=z_k\ \mathrm{{or}}\ \bar{z}_k, \\ \big |\frac{\partial ^2 G}{\partial w_m\partial w_n}\big |\le & {} \frac{c(\iota , g, \rho )\epsilon }{m^{1/2}n^{1/2}},\qquad m,n\ge 1, w_m=z_{m}\ \mathrm{{or}}\ \bar{z}_m. \nonumber \end{aligned}$$

(4.1)

Proof

We only give the estimate for (4.1) and the proofs for the others are similar. From $|\phi _{k}^j|\le ce^{-\sigma ||k|-j|}$ with $0<\sigma <\rho$, we have for $x\in [0,\pi ]$,

$$\begin{aligned} |\phi _j(x)|\le C_{\rho }. \end{aligned}$$

(4.2)

From the assumption on $V(\theta ,x)$ and (4.2) and a straightforward computation, we obtain

$$\begin{aligned} \int _{0}^{\pi }V(\theta , x)\phi _{j}(x)\phi _{l}(x)dx|\le C_{\rho }. \end{aligned}$$

Thus, for $|\mathfrak {I}\theta |<\rho$, $|I|<r^2$, $|z|_{\iota }<r$, $|\bar{z}|_{\iota }<r$,

$$\begin{aligned} \big |\frac{\partial G}{\partial z_{k}}\big |\le & {} \frac{\epsilon }{2}\frac{C_g}{k^{1/2}}\sum \limits _{l}\frac{1}{\lambda _{l}^{1/2}}C_{\rho }|z_l-\bar{z}_{l}|\\\le & {} \frac{c(\iota ,g,\rho )\epsilon r}{k^{1/2}}, \end{aligned}$$

where the second is from $\iota >1$ and the definition of $|z|_{\iota }$. $\square$

Similarly,

Lemma 4.6

For $\iota >1$,

$$\begin{aligned} \Vert G\Vert ^{\mathcal {L}}_{D(\rho ,r)}\le & {} c(\iota ,g, \rho )\epsilon r^2, \\ \big |\frac{\partial G}{\partial w_k}\big |^{\mathcal {L}}\le & {} \frac{c(\iota , g, \rho )\epsilon r}{k^{1/2}}, \qquad w_k=z_k\ \mathrm{{or}}\ \bar{z}_k, \\ \big |\frac{\partial ^2 G}{\partial w_m\partial w_n}\big |^{\mathcal {L}}\le & {} \frac{c(\iota , g, \rho )\epsilon }{m^{1/2}n^{1/2}},\qquad \;\; m,n\ge 1, w_m=z_{m}\ \mathrm{{or}}\ \bar{z}_m. \end{aligned}$$

Lemma 4.7

For $\iota >1$, $G\in \Gamma _{r,D(\rho ,r)}^{1/2}$ and

$$\begin{aligned} \langle G\rangle _{r,D(\rho ,r)}, \langle G\rangle _{r,D(\rho ,r)}^{\mathcal {L}}\le c(\iota ,g,\rho )\epsilon . \end{aligned}$$

Clearly, $M=|\omega |_{\Pi }^{\mathcal {L}}+|\Omega |_{1,\Pi }^{\mathcal {L}}=1$. To check the smallness conditions for Theorem 3.1, one needs to compute $\Vert X_{G}\Vert _{r, D(\rho ,r)}$ and $\Vert X_{G}\Vert ^{\mathcal {L}}_{r, D(\rho ,r)}$ . The following Lemma 4.8 and Lemma serve for Lemma 4.10.

Lemma 4.8

If $\iota >1$, $u(x)\in \mathcal {H}_{odd}^{\frac{\iota }{2}}({\mathbb {T}})$, then there exists a positive constant $C_{*}(\iota ,g,\rho )>1$, such that

$$\begin{aligned} \frac{1}{C_{*}(\iota ,g,\rho )}\Vert u(x)\Vert _{H^{\frac{\iota }{2}}({\mathbb {T}})}\le \Vert u(x)\Vert _{\mathcal {H}^{\frac{\iota }{2}}(0,\pi )}\le C_{*}(\iota ,g, \rho )\Vert u(x)\Vert _{H^{\frac{\iota }{2}}({\mathbb {T}})}, \end{aligned}$$

where $u(x)=\sum \limits _{k\ge 1}u_k\phi _k(x)$, $x\in {\mathbb {T}}$.

We delay the above proof in the “Appendix”.

Lemma 4.9

For $\iota >1$, then for $|\mathfrak {I}\theta |<\rho$, $V(\theta ,x)\in H^{\frac{\iota }{2}}({\mathbb {T}})$ and

$$\begin{aligned} \Vert V(\theta ,x)\Vert _{H^{\frac{\iota }{2}}({\mathbb {T}})}\le c(\iota ,\rho ). \end{aligned}$$

Lemma 4.10

For $\iota >1$, $V(\theta , x)\cdot u(x) \in \mathcal {H}_{odd}^{\frac{\iota }{2}}$ and

$$\begin{aligned} \Vert V(\theta , x)\cdot u(x)\Vert _{\mathcal {H}^{\frac{\iota }{2}}}\preceq c(\iota ,g,\rho )\Vert u(x)\Vert _{\mathcal {H}^{\frac{\iota }{2}}}, \end{aligned}$$

where $|\mathfrak {I}\theta |<\rho$ and $u(x)\in \mathcal {H}_{odd}^{\frac{\iota }{2}}$.

Proof

From the assumption on $V(\theta ,x)$ and u, we obtain that $V(\theta ,x)u(x)$ is also an odd periodic function on $x\in {\mathbb {T}}$. Thus, for $|\mathfrak {I}\theta |<\rho$ and $x\in {\mathbb {T}}$,

$$\begin{aligned} \Vert V(\theta ,x)u\Vert _{\mathcal {H}^{\frac{\iota }{2}}}\le & {} C_{*}(\iota ,g,\rho ) \Vert V(\theta ,x) u(x)\Vert _{H^{\frac{\iota }{2}}({\mathbb {T}})}\\\le & {} C_*(\iota ,g,\rho ) \Vert V(\theta ,x)\Vert _{H^{\frac{\iota }{2}}({\mathbb {T}}^)}\Vert u\Vert _{H^{\frac{\iota }{2}}({\mathbb {T}})}\\\le & {} C_*(\iota ,g,\rho ) c(\iota ,\rho )\Vert u\Vert _{H^{\frac{\iota }{2}}({\mathbb {T}})}\\\preceq & {} c(\iota ,g,\rho ) \Vert u\Vert _{\mathcal {H}^{\frac{\iota }{2}}}. \end{aligned}$$

The first inequality is from Lemma 4.8 and $V(\theta ,x)u(x)$ is an odd periodic function on $x\in {\mathbb {T}}$. The second one is from $\iota >1$. Note $\iota /2>1/2$, $H^{\frac{\iota }{2}}({\mathbb {T}})$ is a Banach algebra. The third is from the assumption on $V(\theta ,x)$. We recall $V(\theta ,x)$ analytically in $\theta , x$ extends to the domain

$$\begin{aligned} {\mathbb {T}}^n_{\rho }\times {\mathbb {T}}_{\rho }\times \Pi , \qquad {\mathbb {T}}^n_{\rho } =\{(a+ib)\in {\mathbb {C}}^n/{2\pi {\mathbb {Z}}^n}\big ||b|<\rho \}, \end{aligned}$$

and it is bounded by $C_1$. From this, we have $\Vert V(\theta ,x)\Vert _{H^{\frac{\iota }{2}}({\mathbb {T}})}\le c(\iota ,\rho )$ for $|\mathfrak {I}\theta |<\rho$. The last inequality comes from Lemma 4.8 again. $\square$

Similarly,

Lemma 4.11

For $\iota >1$, $\partial _{\omega }V(\theta , x;\omega )\cdot u(x) \in \mathcal {H}_{odd}^{\frac{\iota }{2}}$ and

$$\begin{aligned} \Vert \partial _{\omega }V(\theta , x;\omega )\cdot u(x)\Vert _{\mathcal {H}^{\frac{\iota }{2}}}\preceq c(\iota ,g,\rho )\Vert u(x)\Vert _{\mathcal {H}^{\frac{\iota }{2}}}, \end{aligned}$$

where $|\mathfrak {I}\theta |<\rho$ and $u(x)\in \mathcal {H}_{odd}^{\frac{\iota }{2}}$.

Lemma 4.12

For $\iota >1$, then

$$\begin{aligned} \frac{1}{r}\Vert G_{w}\Vert _{\iota }\le c(g)^{\frac{\iota }{2}}\epsilon , \end{aligned}$$

where $(\theta , I, z, \bar{z})\in D(\rho ,r)$ and $w=z$ or $\bar{z}$.

Proof

For cutting the notation, denote $w(x)=V(\theta ,x)u$, where we forget other variables except x. Recall $\frac{\partial G}{\partial z_j}=\frac{i\epsilon }{\sqrt{2\lambda _j}}\int _0^{\pi }w(x)\phi _j(x)dx$,

$$\begin{aligned} \Vert G_z\Vert _{\iota }^2= & {} \frac{\epsilon ^2}{2}\sum \limits _{j\ge 1}\frac{1}{\lambda _j}(\int _0^{\pi }w(x)\phi _{j}(x)dx)^2 j^{\iota } \nonumber \\\preceq & {} c(g)^{\iota +1}\epsilon ^2\Vert w(x)\Vert _{\mathcal {H}^{\frac{\iota }{2}}}^2 \nonumber \\ \underline{Lemma~4.10}\preceq & {} c(g)^{\iota +1}\epsilon ^2 c(\iota ,g,\rho )\Vert u\Vert _{\mathcal {H}^{\frac{\iota }{2}}}^2. \end{aligned}$$

(4.3)

Note $u=\sum \limits _{j\ge 1}\frac{i}{\sqrt{2\lambda _j}}(z_j-\bar{z}_j)\phi _j(x)$, then

$$\begin{aligned} (4.3)\preceq & {} c(\iota ,g,\rho ) c(g)^{\iota +1}\epsilon ^2 (\sum \limits _{k\ge 1}{\lambda _k}^{\iota -1}|z_k|^2+\sum \limits _{k\ge 1}{\lambda _k}^{\iota -1}|\bar{z}_k|^2)\\\preceq & {} c(g)^{\iota +1}\epsilon ^2c(\iota ,g,\rho )c(g)^{\iota -1}r^2. \end{aligned}$$

Thus,

$$\begin{aligned} \frac{1}{r}\Vert G_{z}\Vert _{\iota }\preceq c(\iota ,g,\rho )\epsilon . \end{aligned}$$

$\square$

Similarly,

Lemma 4.13

For $\iota >1$, then

$$\begin{aligned} \frac{1}{r^2}|G_{\theta }|\le c(\iota ,g,\rho )\epsilon , \end{aligned}$$

where $(\theta , I, z, \bar{z})\in D(\rho /2, r)$.

Combining all the lemmas,

Lemma 4.14

For $\iota >1$, then

$$\begin{aligned} \Vert X_{G}\Vert _{r,D(\rho /2,r)}\le c(\iota ,g,\rho )\epsilon . \end{aligned}$$

Similarly,

Lemma 4.15

For $\iota >1$, then

$$\begin{aligned} \Vert X_{G}\Vert ^{\mathcal {L}}_{r,D(\rho /2,r)}\le c(\iota ,g,\rho )\epsilon . \end{aligned}$$

From

$$\begin{aligned} \Vert X_{G}\Vert _{r,D(\rho /2,r)}+\langle G\rangle _{r, D(\rho /2,r)}+\alpha \left( \Vert X_{G}\Vert ^{\mathcal {L}}_{r,D(\rho /2,r)}+\langle G\rangle _{r, D(\rho /2,r)}^{\mathcal {L}}\right) \le 2c(g, \iota ,\rho )\epsilon , \end{aligned}$$

if choose $\alpha =\epsilon ^{1/2}$ and $\epsilon \le \varepsilon _0(g,\iota ,\rho ,n,\tau ) : = \left( \frac{\epsilon _0}{2c(g,\iota ,\rho )}\right) ^2$, we have

$$\begin{aligned} \Vert X_{G}\Vert _{r,D(\rho /2,r)}+\langle G\rangle _{r, D(\rho /2,r)}+\alpha \left( \Vert X_{G}\Vert ^{\mathcal {L}}_{r,D(\rho /2,r)}+\langle G\rangle _{r, D(\rho /2,r)}^{\mathcal {L}}\right) \le \epsilon _0\alpha , \end{aligned}$$

where $\epsilon _0$ depends on $n, \tau$ and $\rho$. Now we can use Theorem 3.1 if we set $s=\rho /2$.

Proposition 4.1

There exists $\varepsilon _0\sim g,\iota ,n, \tau , \rho$ and if $0<\epsilon \le \varepsilon _0$, then

• (i) a Cantor set $\Pi _{\epsilon }\subset \Pi$ with $Meas(\Pi {\setminus } \Pi _{\epsilon })\rightarrow 0$ as $\epsilon \rightarrow 0;$

• (ii) a Lipschitz family of real analytic, symplectic and linear coordinate transform $\Phi : \Pi _{\epsilon }\times \mathcal {P}^0\rightarrow \mathcal {P}^0$ of the form $\Phi (I,\theta ,\zeta )=(I+\frac{1}{2} \zeta \cdot M_{\omega }(\theta )\zeta , \theta , L_{\omega }(\theta )\zeta )$, where $\zeta =(z,\bar{z})$, $L_{\omega }(\theta )$ and $M_{\omega }(\theta )$ are linear bounded operators from $\ell _{p}^2\times \ell _{q}^2$ into itself for all $q\ge 0$ and $L_{\omega }(\theta )$ is invertible.

• (iii) a Lipschitz family of new normal forms

  $$\begin{aligned} N^{*}(\omega )=\sum \limits _{j=1}^n \omega _jI_j+\sum \limits _{j\ge 1}\Omega _j^{*}(\omega )z_j\bar{z}_j \end{aligned}$$

  defined on $D(\rho /4,r/2)\times \Pi _{\epsilon };$ such that on $\Pi _{\epsilon }\times \mathcal {P}^0$, $H\circ \Phi =N^{*}$. Moreover,

  $$\begin{aligned} \Vert \Phi ^{\pm }-\mathrm{Id}\Vert _{r, D(\rho /4, r/2)}\le c(n,g,\iota ,\rho )\epsilon , \end{aligned}$$

  the new external frequencies are close to the original ones $|\Omega ^{*}-\Omega |_{1,\Pi _{\epsilon }}\le c(n,g,\iota ,\rho )\epsilon$, and the new frequencies satisfy a non-resonance condition

  $$\begin{aligned} |k\cdot \omega +l\cdot \Omega ^{*}(\omega )|\ge \frac{\epsilon ^{1/2}}{2}\frac{\langle l\rangle }{1+|k|^{\tau }}, \end{aligned}$$

  where $(k,l)\in \mathcal {Z}, \xi \in \Pi _{\epsilon }$.

Proof

As [22], we need to prove

• (i) the symplectic coordinate transform $\Phi$ is quadratic and have the specific form in Proposition 4.1;

• (ii) the new normal form still has the same frequencies vector $\omega ;$

• (iii) the new Hamiltonian reduces to the new normal form, i.e. $R^{*}=0$.

• (iv) the symplectic coordinate transform $\Phi$, which is defined by Theorem 3.1 on each $\mathcal {P}^{\iota }(\iota >1)$, extends to $\mathcal {P}^0 := {\mathbb {T}}^n\times {\mathbb {R}}^n\times \ell _{0}^2\times \ell _{0}^2$.

The proofs for $(i) - (iv)$ are similar as [22]. $\square$

Theorem 1.1 is a direct result of Proposition 4.1.

Before we prove Corollary 1.1, we give a preparation lemma. For simplicity we denote $\bar{p}=2p-1\ge 0$. Denote $|\zeta -\zeta '|_{\bar{p}} : = \Vert z-z'\Vert _{\bar{p}}+\Vert \bar{z}-\bar{z}'\Vert _{\bar{p}}$ and $|\zeta |_{\bar{p}} : = \Vert z\Vert _{\bar{p}}+\Vert \bar{z}\Vert _{\bar{p}}$.

Lemma 4.16

When $0<\epsilon \le \varepsilon _0\ll 1$ and $\bar{p}\ge 0$, for the initial data $(\theta (0), I(0), \zeta (0))$ and any $t\in {\mathbb {R}}$,

$$\begin{aligned} |I(t)|\le cr^2\epsilon +(r/4)^2, \theta (t)=\omega t, \end{aligned}$$

and

$$\begin{aligned} |\zeta (0)|_{\bar{p}}(1-c\epsilon )\le |\zeta (t)|_{\bar{p}}\le |\zeta (0)|_{\bar{p}}(1+c\epsilon ), \end{aligned}$$

where $\theta (0)=0$, $|I(0)|<(r/4)^2$, $\zeta (0)\in \ell _{\bar{p}}^2$ and $|\zeta (0)|_{\bar{p}}=ar(a<1/4)$ and c depends on n, g, p and $\rho$, which is independent of a.

Proof

When $0<\epsilon \le \varepsilon _0$, Proposition 4.1 holds true. From $|\zeta (0)|_{\bar{p}}=ar$, then $\Vert \zeta (0)\Vert _{\bar{p}}\le |\zeta (0)|_{\bar{p}}=ar<r/4$. Now from Proposition 4.1, one obtains

$$\begin{aligned} \Vert \Phi _{\omega }^{-1}-\mathrm{Id}\Vert _{r, D(\rho /4, r/2)}\le c\epsilon . \end{aligned}$$

Note $(\theta (0), I(0), \zeta (0))\in D(\rho /4, r/2)$, we induce

$$\begin{aligned} \frac{1}{r^2}|I'(0)-I(0)|+\frac{1}{r}|\zeta (0)-\zeta '(0)|_{\bar{p}}\le c\epsilon . \end{aligned}$$

(4.4)

It follows

$$\begin{aligned} |\zeta (0)-\zeta '(0)|_{\bar{p}}\le cr\epsilon . \end{aligned}$$

(4.5)

From the motion equation of $N^{*}$ and (4.5), we deduce that for any $t\in {\mathbb {R}}$,

$$\begin{aligned} |\zeta (0)|_{\bar{p}}-cr\epsilon \le |\zeta '(t)|_{\bar{p}}=|\zeta '(0)|_{\bar{p}}\le |\zeta (0)|_{\bar{p}}+cr\epsilon . \end{aligned}$$

(4.6)

From (4.4) and the motion equation of $N^{*}$,

$$\begin{aligned} |I'(t)|=|I'(0)|\le |I(0)|+|I'(0)-I(0)|\le (r/4)^2+cr^2\epsilon , \end{aligned}$$

and $\theta '(t)=\omega t$. On the other hand, from (4.6) $\Vert \zeta '(t)\Vert _{\bar{p}}\le |\zeta '(t)|_{\bar{p}}<\frac{r}{2}$ if $\epsilon \ll 1$. This means $(\theta '(t), I'(t), \zeta '(t))\in D(\rho /4, r/2)$. From Proposition 4.1 again,

$$\begin{aligned} \Vert (\theta (t), I(t), \zeta (t))-(\theta '(t), I'(t), \zeta '(t))\Vert _{\bar{p}, r}\le c\epsilon . \end{aligned}$$

Therefore,

$$\begin{aligned} |\zeta (0)|_{\bar{p}}-cr\epsilon \le |\zeta (t)|_{\bar{p}}\le |\zeta (0)|_{\bar{p}}+cr\epsilon . \end{aligned}$$

Substitute $\epsilon$ by $a\epsilon$ and note $0<a<\frac{1}{4}$, then

$$\begin{aligned} |\zeta (0)|_{\bar{p}}(1-c\epsilon )\le |\zeta (t)|_{\bar{p}}\le |\zeta (0)|_{\bar{p}}(1+c\epsilon ). \end{aligned}$$

The remained estimates for I(t) and $\theta (t)$ are clear.

Proof of Corollary 1.1. Choose $(u,v)\in \mathcal {H}_{odd}^{p+1}\times \mathcal {H}_{odd}^{p}(p\ge 0)$. From $\Vert u\Vert _{\mathcal {H}^{p+1}}^2=\sum \limits _{k\ge 1}\lambda _k^{2p+2}|\frac{q_{k}}{\sqrt{\lambda _k}}|^2$ and $\Vert v\Vert _{\mathcal {H}^{p}}^2=\sum \limits _{k\ge 1}\lambda _k^{2p}|\sqrt{\lambda _{k}}p_{k}|^2$ and $\lambda _k\ge c_{g}^{-1}k(k\ge 1)$, we obtain

$$\begin{aligned} \Vert z\Vert _{2p+1}\le c_{g}^{p}\Vert (u,v)\Vert _{\mathcal {F}_{p}}, \end{aligned}$$

where $z_{k}=\frac{1}{\sqrt{2}}(p_k-iq_k)$. Similarly, $\Vert \bar{z}\Vert _{2p+1}\le c_{g}^{p}\Vert (u,v)\Vert _{\mathcal {F}_{p}}$. Therefore,

$$\begin{aligned} |\zeta |_{2p+1}\le 2c_g^{p}\Vert (u,v)\Vert _{\mathcal {F}_{p}}. \end{aligned}$$

(4.7)

On the other hand, from $\Omega _j=\lambda _j=\sqrt{\mu _j+m}\le c_gj$, we have

$$\begin{aligned} \Vert u\Vert _{\mathcal {H}^{p+1}}\le c_g^p\Vert q\Vert _{2p+1}. \end{aligned}$$

Similarly,

$$\begin{aligned} \Vert v\Vert _{\mathcal {H}^{p}}\le c_{g}^{p}\Vert p\Vert _{2p+1}. \end{aligned}$$

Thus,

$$\begin{aligned} \Vert z\Vert _{2p+1}\ge \frac{1}{\sqrt{2}c_g^{p}}\Vert (u,v)\Vert _{\mathcal {F}_{p}}. \end{aligned}$$

Similarly,

$$\begin{aligned} \Vert \bar{z}\Vert _{2p+1}\ge \frac{1}{\sqrt{2}c_g^p}\Vert (u,v)\Vert _{\mathcal {F}_{p}}. \end{aligned}$$

Therefore,

$$\begin{aligned} |\zeta |_{2p+1}\ge \frac{\sqrt{2}}{c_g^{p}}\Vert (u,v)\Vert _{\mathcal {F}_{p}}. \end{aligned}$$

(4.8)

Now choose $(u_0,v_0)\in \mathcal {H}^{p+1}\times \mathcal {H}^{p}(p\ge 0)$ and $0<\epsilon \le \varepsilon _0$ and $\omega \in \Pi _{\epsilon }$, where $\Vert (u_0,v_0)\Vert _{\mathcal {F}_{p}}<\frac{r}{10c_g^p}$. It means that $|\zeta (0)|_{2p+1}<\frac{r}{5}$. For the motion equation (2.2), we choose $\theta (0)=0$ and $|I(0)|<(r/4)^2$. Since $0<\epsilon \le \varepsilon _0$ and $\omega \in \Pi _{\epsilon }$, from Lemma 4.16 we deduce that for any $t\in {\mathbb {R}}$,

$$\begin{aligned} |\zeta (0)|_{2p+1}(1-c\epsilon )\le |\zeta (t)|_{2p+1}\le |\zeta (0)|_{2p+1}(1+c\epsilon ) \end{aligned}$$

(4.9)

and $\theta (0)=0$. From (4.7),(4.8) and (4.9), we obtain

$$\begin{aligned} C(g,p)^{-1}(1-c\epsilon )\Vert (u_0,v_0)\Vert _{\mathcal {F}_p}\le \Vert (u(t),v(t))\Vert _{\mathcal {F}_{p}}\le C(g,p)(1+c\epsilon )\Vert (u_0,v_0)\Vert _{\mathcal {F}_p}, \end{aligned}$$

where $C(g,p)=\sqrt{2}c_g^{2p}$ and c depends on n, g, p and $\rho$.$\square$

Remark 4.1

It seems that we need the condition $\Vert (u_0,v_0)\Vert _{\mathcal {F}_{p}}<\frac{r}{10c_g^p}$ in the proof. Note that the special form of G(see (2.3)), r can be chosen arbitrary large. This means this condition always holds true once we choose r large enough.

5 Proof of Corollary 1.3

In this section we will prove Corollary 1.3 and explain Remark 1.4.

We first introduce the Schrödinger operator

$$\begin{aligned} Ly = -y''+q(t)y, \qquad -\infty<t<\infty , \end{aligned}$$

(5.1)

generated in $L^2(-\infty , \infty )$ with the Mathieu potential $q(t)=\epsilon \cos 2t$, $\epsilon \ne 0$. Let the pair $\{\lambda _k^{-}, \lambda _{k}^{+}\}$ denote, respectively, the kth periodic eigenvalues of the Mathieu–Hill operator on the interval $[0,\pi ]$ with the periodic boundary conditions

$$\begin{aligned} y(0)=y(\pi ), \qquad y'(0) =y'(\pi ),\qquad \mathrm{for\ k\ even }, \end{aligned}$$

and the anti-periodic boundary conditions

$$\begin{aligned} y(0)=-y(\pi ), \qquad y'(0) =-y'(\pi ), \qquad \mathrm{for\ k\ odd }. \end{aligned}$$

It is well-known that these sequences of eigenvalues are ordered as follows:

$$\begin{aligned} \lambda _0<\lambda ^{-}_1\le \lambda ^{+}_1<\lambda _2^{-}\le \lambda _{2}^{+}<\lambda _3^{-}\le \lambda _3^{+}<\lambda _4^{-}\le \lambda _4^{+}<\cdots . \end{aligned}$$

For $\lambda$ in any of the open intervals $(\lambda _0, \lambda _1^{-})$, $(\lambda _{k-1}^{+}, \lambda _k^{-})(k\ge 2)$, all solutions of (5.1) are bounded in $(-\infty , \infty )$, and these intervals are called the stability intervals of (5.1). For $\lambda$ outside these intervals, all nontrivial solutions of (5.1) are unbounded in $(-\infty , \infty )$. Therefore, the intervals $(\lambda _k^{-}, \lambda _k^+)(k\in {\mathbb {N}})$ are called the instability intervals of the operator L, while $(-\infty , \lambda _0)$ is called the zeroth instability interval. See [35] for further background.

From Eastham [12] the Mathieu–Hill operator L has only simple eigenvalues both for periodic and anti - periodic boundary conditions, that is, all instability intervals of the operator are open. From [35](see Theorem 2.12, page 40), we have

Lemma 5.1

Let $y''+[\lambda +Q(t)]y=0,$ where Q(t) is periodic with period $\pi$ and assume that the second derivative $Q''$ exists and is continuous. Let $\int _0^{\pi }Q(t)dt=0$, and let $C=\frac{1}{\pi }\int _0^{\pi }Q^2(t)dt$. Then for large k,

$$\begin{aligned} \lambda _{2k-1}^{-}= & {} (2k-1)^2+\frac{C}{(4k)^2}+o(\frac{1}{k^2}),\\ \lambda _{2k-1}^{+}= & {} (2k-1)^2+\frac{C}{(4k)^2}+o(\frac{1}{k^2}),\\ \lambda _{2k}^{-}= & {} (2k)^2+\frac{C}{(4k)^2}+o(\frac{1}{k^2}),\\ \lambda _{2k}^{+}= & {} (2k)^2+\frac{C}{(4k)^2}+o(\frac{1}{k^2}). \end{aligned}$$

Proof of Corollary 1.3. We construct $u_1(t)$ firstly. As we mentioned above, for $\lambda _1$ inside $(\lambda _{k}^-,\lambda _{k}^+)$, all nontrivial solutions of

$$\begin{aligned} \left( -\frac{d^2}{dt^2}+\epsilon \cos 2t\right) v(t)=\lambda _1 v(t) \end{aligned}$$

(5.2)

are unbounded in $(-\infty , \infty )$, where k will be chosen in the following. From [35] one can choose $u_1(t)=e^{\sigma t}p(t)$, where $\sigma =a+bi, a\ne 0$ and p(t) is periodic with period $\pi$ and $u_1$ is the solution of (5.2). From Bourgain [6], we choose $u_2(x)=\sin kx$, which is the solution of

$$\begin{aligned} -u''(x)= \lambda _2 u(x), \qquad \lambda _2=k^2, \end{aligned}$$

under the boundary conditions $u(0)=u(\pi )=0$. Then from a straightforward computation, $u(x,t)=u_1(t)u_2(x)=e^{\sigma t}p(t)\sin kx$ is the solution of 1d wave equation (1.4) with Dirichlet boundery conditions on $[0,\pi ]$, where $m_{\epsilon }=\lambda _1-\lambda _2$. From Lemma 5.1 and a straightforward computation, we obtain $C=\epsilon ^2/2$. By Lemma 5.1 if choose k large enough, we have $0<\lambda _1-\lambda _2\le C\le 1$. Furthermore,

$$\begin{aligned} \Vert u(t,x)\Vert _{H^1({\mathbb {T}})}=|k| e^{a t}|p(t)|/\sqrt{2}. \end{aligned}$$

Since p(t) is nontrivial periodic function, there exists $t_0\in [0,\pi ]$ so that $p(t_0)\ne 0$. If $a>0$, one can choose $t_j=t_0+j\pi$. It deduces that

$$\begin{aligned} \Vert u(t_j,x)\Vert _{H^1({\mathbb {T}})}\rightarrow \infty , \qquad j\rightarrow \infty . \end{aligned}$$

If $a<0$, the proof is similar.

Now choose $m\in (0,1]$, $p=1$ and $V(t\omega )=-\cos \omega t$, where $\omega \in [0,2\pi ]$. From Corollary 1.2, there exists $0<\epsilon _0\ll 1$, which is independent of m and $\omega$, so that for all $0<|\epsilon |\le \epsilon _0$ and $\omega \in \Pi _{\epsilon }\subset [0,2\pi ]$, the solution u(t) of 1d wave equation

$$\begin{aligned} u_{tt}-u_{xx}+mu-\epsilon \cos \omega t u=0, \end{aligned}$$

with Dirichlet boundery conditions on $[0,\pi ]$ satisfy

$$\begin{aligned} \Vert u(t)\Vert _{H^1({\mathbb {T}})}\le c(m,\omega ) \Vert (u_0,v_0)\Vert _{\mathcal {F}_1}, \qquad t\in {\mathbb {R}}, \end{aligned}$$

(5.3)

where $(u(0), v(0))\in \mathcal {F}_1$. On the other hand, for $\epsilon _0$ mentioned above, by Corollary 1.3 there exists $m_{\epsilon _0}$. Consider the Eq. (1.4) with $m_{\epsilon }=m_{\epsilon _0}$ under Dirichlet boundary conditions, there exists at least one solution $u_{\epsilon _0}(t,x)$ satisfying

$$\begin{aligned} \Vert u_{\epsilon _0}(t_j,x)\Vert _{H^1({\mathbb {T}})}\rightarrow \infty , \qquad |t_j|\rightarrow \infty . \end{aligned}$$

(5.4)

If $\omega =2\in \Pi _{\epsilon _0}$, then $u_{\epsilon _0}(t,x)$ satisfies (5.3) with $m=m_{\epsilon _0}$. This contradicts with (5.4) and thus shows us $\omega =2\notin \Pi _{\epsilon _0}$.

Appendix

1.1 Proof of Lemma 4.2

To prove Lemma 4.2, we need a series of lemmas. In the following we suppose $i>j\ge 1$ and let $b=i-j\ge 1$. Denote

$$\begin{aligned} \mathcal {I}_1=\bigcup \limits _{k\ne 0}\bigcup \limits _{1\le b\le c_0(g)(1+2\pi |k|)}\{\omega \in \Pi :|k\cdot \omega +b|<\frac{\gamma }{|k|^{\tau }}\}, \end{aligned}$$

where $c_0(g)$ depends on g only. Clearly,

Lemma 6.1

If $\tau >n+1$, then

$$\begin{aligned} Meas(\mathcal {I}_1)\le c_1(n,\tau ,g)\gamma . \end{aligned}$$

From a straightforward computation, we have

Lemma 6.2

For any $i>j\ge 1$, $b\ge c_0(g)(1+2\pi |k|)$, then

$$\begin{aligned} |k\cdot \omega +\Omega _i-\Omega _j|\ge 1. \end{aligned}$$

In the next, we only need to consider the case $k\ne 0$, $i>j\ge 1$ and $1\le b\le c_0(g)(1+2\pi |k|)$. Clearly,

Lemma 6.3

When $j\ge j_0>0$,

$$\begin{aligned} |\Omega _i-\Omega _j-b|\le \frac{bc_2(g)}{j}, \end{aligned}$$

where $j_0$ depends on g only.

Lemma 6.4

For $k\ne 0$, if $j\ge \frac{c_3(g)|k|^{\tau +1}}{\gamma }+j_0$ and $1\le b\le c_0(g)(1+2 \pi |k|)$, then for $\omega \in \Pi {\setminus } \mathcal {I}_1$,

$$\begin{aligned} |k\cdot \omega +\Omega _i-\Omega _j|\ge \frac{\gamma }{2|k|^{\tau }}>0. \end{aligned}$$

Proof

From Lemma 6.3, if $j\ge \frac{c_3(g)|k|^{\tau +1}}{\gamma }+j_0$, then

$$\begin{aligned} |k\cdot \omega +\Omega _i-\Omega _j|\ge & {} |k\cdot \omega +b|-\frac{bc_2(g)}{j},\\\ge & {} \frac{\gamma }{|k|^{\tau }}-\frac{bc_2(g)}{j},\\\ge & {} \frac{\gamma }{2|k|^{\tau }}. \end{aligned}$$

$\square$

Denote

$$\begin{aligned} \mathcal {I}_{ij,k}^2 := \{\omega \in \Pi : |k\cdot \omega +\Omega _i-\Omega _j |<\frac{\gamma ^2}{|k|^{\tau +n+3}}\} \end{aligned}$$

and

$$\begin{aligned} \mathcal {I}_2=\bigcup \limits _{k\ne 0}\bigcup \limits _{\begin{array}{c} 1\le b\le c_0(g)(1+2\pi |k|)\\ 1\le j\le \frac{c_3(g)|k|^{\tau +1}}{\gamma }+j_0 \end{array}}\mathcal {I}_{ij,k}^2. \end{aligned}$$

The following two lemmas are easy.

Lemma 6.5

$$\begin{aligned} Meas(\mathcal {I}_2)\le c_4(n,\tau ,g)\gamma . \end{aligned}$$

Note $0<\gamma <1$,

Lemma 6.6

If $\omega \in \Pi {\setminus } (\mathcal {I}_1\cup \mathcal {I}_2)$, for $k\ne 0$, $1\le b\le c_0(g)(1+2\pi |k|)$,

$$\begin{aligned} |k\cdot \omega +\Omega _i-\Omega _j|\ge \frac{\gamma ^2}{2|k|^{\tau +n+3}}>0, \end{aligned}$$

where $Meas(\mathcal {I}_1\cup \mathcal {I}_2)\le c_5(n,\tau ,g)\gamma$.

In the following, denote $\mathcal {I}_1^{(l)}=\mathcal {I}_1(\gamma _{l})=\bigcup \limits _{k\ne 0}\bigcup \limits _{1\le b\le c_0(g)(1+2\pi |k|)}\{\omega \in \Pi : |k\cdot \omega +b|<\frac{\gamma _{l}}{|k|^{\tau }}\}$ and $\mathcal {I}_2^{(l)}=\mathcal {I}_2(\gamma _{l})$. From the above, $Meas(\mathcal {I}_1(\gamma _l))\le c_1(n,\tau ,g)\gamma _{l}$ and $Meas(\mathcal {I}_2(\gamma _l))\le c_4(n,\tau ,g)\gamma _{l}$. Choose $\gamma _{l}=\frac{1}{c_1(n,\tau ,g)+c_4(n,\tau ,g)}\cdot \frac{1}{2^{l}}$. It is clear that $Meas(\mathcal {I}_1(\gamma _{l}))\rightarrow 0$ and $Meas(\mathcal {I}_2(\gamma _{l}))\rightarrow 0$. Define $\mathcal {J}_{2,1}^{(l)}=\Pi {\setminus } \mathcal {I}_{1}^{(l)}$ and $\mathcal {J}_{2,1}=\bigcup \limits _{l\ge 1}\mathcal {J}_{2,1}^{(l)}.$ Then

$$\begin{aligned} Meas(\Pi {\setminus } \mathcal {J}_{2,1})= & {} Meas(\Pi \cap (\mathcal {J}_{2,1})^c)\\\le & {} Meas(\Pi \cap (\mathcal {J}_{2,1}^{(l)})^c)=Meas(\mathcal {I}_1^{(l)})\rightarrow 0. \end{aligned}$$

This follows $Meas(\Pi {\setminus } \mathcal {J}_{2,1})=0$. Similarly, denote $\mathcal {J}_{2,2}^{(l)}=\Pi {\setminus } \mathcal {I}_{2}^{l}$ and $\mathcal {J}_{2,2}=\bigcup \limits _{l\ge 1}\mathcal {J}_{2,2}^{(l)}.$ As above $Meas(\Pi {\setminus } \mathcal {J}_{2,2})=0$. Denote $\mathcal {J}_2=\mathcal {J}_{2,1}\cap \mathcal {J}_{2,2}.$

Lemma 6.7

For $\omega \in \mathcal {J}_{2}$ and $k\ne 0$, $1\le b\le c_2(g)(1+2\pi |k|)$, $i>j\ge 1$,

$$\begin{aligned} k\cdot \omega +\Omega _i-\Omega _j\ne 0, \end{aligned}$$

where $Meas(\Pi {\setminus } \mathcal {J}_2)=0$.

Proof

For $\omega \in \mathcal {J}_2$ and $k\ne 0$, there exist $l_0, l_1\ge 1$ such that $\omega \in \mathcal {J}_{2,1}^{(l_0)}$ and $\omega \in \mathcal {J}_{2,2}^{(l_1)}$. Without losing generality, suppose $l_0\ge l_1$. Clearly, $\mathcal {I}_1^{(l_0)}\subset \mathcal {I}_{1}^{(l_1)}$ and $\mathcal {I}_2^{(l_0)}\subset \mathcal {I}_2^{(l_1)}$. It then follows $\omega \in \Pi {\setminus } (\mathcal {I}_1^{(l_0)}\cup \mathcal {I}_2^{(l_0)})$. From Lemma 6.6, for $k\ne 0$, $1\le b\le c_0(g)(1+2\pi |k|)$,

$$\begin{aligned} |k\cdot \omega +\Omega _i-\Omega _j|\ge \frac{\gamma _{l_0}^2}{2|k|^{\tau +n+3}}>0. \end{aligned}$$

$\square$

combining with Lemma 6.2 and Lemma 6.7, we complete the proof of Lemma 4.2.

1.2 Proof of Lemma 4.8

We need some preparations lemmas for Lemma 4.8. For the following lemma, we denote $A_{mn} : = (\frac{m}{n})^{\frac{\iota }{2}}e^{-\frac{\rho }{2}|m-n|}$ for $m\ne n$, while $A_{mn} : = 0$ if $m=n$. The proof for the following lemma is an easy exercise in the analysis.

Lemma 6.8

$$\begin{aligned} \sum \limits _{m\ge 1}A_{mn}\le C(\iota ,\rho ), \qquad \sum \limits _{n\ge 1}A_{mn}\le C(\iota ,\rho ). \end{aligned}$$

The techniques in the proof of the following lemma have been used in [4, 30] and [39].

Lemma 6.9

$$\begin{aligned} \Delta : = \sum \limits _{m\ge 1}m^{\iota }\big ( \sum \limits _{n\ge 1,n\ne m}|u_n|e^{-\frac{\rho }{2}|n-m|}\big )^2\le C_1(\iota ,g,\rho )\Vert u\Vert ^2_{\mathcal {H}^{\frac{\iota }{2}}}. \end{aligned}$$

Proof

From Lemma 6.8, we have

$$\begin{aligned} \Delta\le & {} \sum \limits _{m\ge 1}\big ( \sum \limits _{n\ge 1,n\ne m}|u_n|n^{\frac{\iota }{2}}A_{mn}\big )^2\\\le & {} \sum \limits _{m\ge 1}\big (\sum \limits _{n\ge 1,n\ne m}|u_n|^2n^{\iota }A_{mn}\big )(\sum \limits _{n\ge 1,n\ne m}A_{mn})\\\le & {} C(\iota ,\rho )\sum \limits _{m\ge 1}\big (\sum \limits _{n\ge 1,n\ne m}|u_n|^2n^{\iota }A_{mn}\big )\\\le & {} C(\iota ,\rho )\sum \limits _{m\ge 1}A_{mn}\sum \limits _{n\ge 1}|u_n|^2n^{\iota }\\\le & {} C^2(\iota ,\rho )\sum \limits _{n\ge 1}|u_n|^2n^{\iota }\\\le & {} C_1(\iota ,g,\rho )\Vert u\Vert _{\mathcal {H}^{\frac{\iota }{2}}}^2. \end{aligned}$$

For the last inequality, we use

$$\begin{aligned} \sum \limits _{n\ge 1}n^{\iota }|u_n|^2\le & {} (c(g))^{\iota }\Vert u\Vert ^2_{\mathcal {H}^{\frac{\iota }{2}}}, \qquad \iota >0, \end{aligned}$$

(6.1)

(6.1) follows from the definition of $\Vert u\Vert _{\mathcal {H}^{\frac{\iota }{2}}}$ and $\frac{1}{c(g)}j\le \lambda _j\le c(g)j$ for some $c(g)\ge 1$. $\square$

Proof of Lemma 4.8 From $\iota >1$, $u(x)=\sum \limits _{n\ge 1}u_n\phi _n(x)$ and $\phi _{n}(x)=\sum \limits _{m\ge 1}\phi _n^m \sin mx$, we have

$$\begin{aligned} u(x)=\sum \limits _{m\ge 1}(\sum \limits _{n\ge 1}u_n\phi _n^m)\sin mx. \end{aligned}$$

(6.2)

If write u(x) in another form, namely,

$$\begin{aligned} u(x)=\sum \limits _{m\in {\mathbb {Z}}}\tilde{u}_m e^{imx}, \end{aligned}$$

(6.3)

note u(x) is skew-symmetric, comparing (6.2) with (6.3), we have $\tilde{u}_m=\frac{1}{2i}\sum \limits _{n\ge 1}u_n\phi _n^m$ for $m>0$ and $\tilde{u}_{-m}=-\tilde{u}_{m}$. Therefore,

$$\begin{aligned} \Vert u\Vert _{H^{\frac{\iota }{2}}({\mathbb {T}})}^2= & {} 2\sum \limits _{m\ge 1}m^{\iota }|\tilde{u}_m|^2 \nonumber \\= & {} \frac{1}{2} \sum \limits _{m\ge 1}m^{\iota } |\sum \limits _{n\ge 1}u_n \phi _n^m|^2. \end{aligned}$$

(6.4)

From $|\phi _n^m|\le C_1(g,\rho )e^{-\frac{\rho }{2}|n-m|}$(see [11]), we obtain

$$\begin{aligned} (6.4)\le & {} \frac{1}{2}C_1^2(g,\rho )\sum \limits _{m\ge 1}m^{\iota } \big (\sum \limits _{n\ge 1}e^{-\frac{\rho }{2}|n-m|}|u_n|\big )^2\\\le & {} \frac{1}{2} C_1^2(g,\rho )\sum \limits _{m\ge 1}m^{\iota }\big (\sum \limits _{n\ge 1, n\ne m}e^{-\frac{\rho }{2}|n-m|}|u_n|+|u_m|\big )^2\\\le & {} C_1^2(g,\rho )\sum \limits _{m\ge 1}m^{\iota } |u_m|^2+C_1^2(g,\rho ) \Delta \\\le & {} C_1^2(g,\rho )c^{\iota }(g)\Vert u\Vert ^2_{\mathcal {H}^{\frac{\iota }{2}}} +C^2_1(g,\rho )c_1(\iota ,g,\rho )\Vert u\Vert ^2_{\mathcal {H}^{\frac{\iota }{2}}}\\\le & {} C_2(\iota ,g,\rho )\Vert u\Vert ^2_{\mathcal {H}^{\frac{\iota }{2}}}. \end{aligned}$$

It follows

$$\begin{aligned} \Vert u\Vert ^2_{H^{\frac{\iota }{2}}({\mathbb {T}})}\le C_2(\iota ,g,\rho )\Vert u\Vert ^2_{\mathcal {H}^{\frac{\iota }{2}}}. \end{aligned}$$

The other half is similar. First,

$$\begin{aligned} \Vert u\Vert ^2_{\mathcal {H}^{\frac{\iota }{2}}}\le c(g)^{\iota }\sum \limits _{n\ge 1}n^{\iota }|u_n|^2. \end{aligned}$$

As above, write $u(x)=\sum \limits _{m\in {\mathbb {Z}}}\tilde{u}_me^{imx}$. Note $u(-x)=-u(x)$, we have

$$\begin{aligned} u(x)=2i\sum \limits _{m>0}\tilde{u}_m\sin mx. \end{aligned}$$

Denote $\sin mx=\sum \limits _{n\ge 1}\hat{s}_n(m)\phi _n(x)$, where

$$\begin{aligned} |\hat{s}_n(m)|\le C_2(g,\rho )e^{-\frac{\rho }{2}|n-m|}, \end{aligned}$$

we obtain

$$\begin{aligned} u(x)=2i \sum \limits _{n\ge 1}\big (\sum \limits _{m\ge 1}\tilde{u}_m\hat{s}_n(m)\big )\phi _n(x). \end{aligned}$$

On the other hand, $u(x)=\sum \limits _{n\ge 1}u_n\phi _n(x)$. It follows

$$\begin{aligned} u_n=2i\sum \limits _{m\ge 1}\tilde{u}_m\hat{s}_n(m). \end{aligned}$$

Thus,

$$\begin{aligned} \sum \limits _{n\ge 1}n^{\iota }|u_n|^2\preceq & {} \sum \limits _{n\ge 1}n^{\iota }\left( \sum \limits _{m\ge 1}|\tilde{u}_m||\hat{s}_n(m)|\right) ^2\\\preceq & {} C^2_2(g,\rho )\sum \limits _{n\ge 1} n^{\iota }\left( \sum \limits _{m\ge 1}|\tilde{u}_m|e^{-\frac{\rho }{2}|n-m|} \right) ^2\\\preceq & {} C^2_2(g,\rho )C(\iota ,\rho )\Vert u(x)\Vert ^2_{H^{\frac{\iota }{2}}({\mathbb {T}})}. \end{aligned}$$

It follows

$$\begin{aligned} \Vert u\Vert ^2_{\mathcal {H}^{\frac{\iota }{2}}}\le C_3(g,\iota ,\rho )\Vert u(x)\Vert _{H^{\frac{\iota }{2}}({\mathbb {T}})}^2. \end{aligned}$$

Choosing $C_{*}(g,\iota ,\rho )>1$ suitably, we complete the proof.$\square$