# Energy Transfer, Weak Resonance, and Fermi’s Golden Rule in Hamiltonian
Nonlinear Klein-Gordon Equations

Zhen Lei 111School of Mathematical Sciences; LMNS and Shanghai Key Laboratory
for Contemporary Applied Mathematics, Fudan University, Shanghai 200433, P.
R.China. 222Email<EMAIL_ADDRESS>Jie Liu 11footnotemark: 1 333Email:
<EMAIL_ADDRESS>Zhaojie Yang 11footnotemark: 1 444Email:
<EMAIL_ADDRESS>
###### Abstract

This paper focuses on a class of nonlinear Klein-Gordon equations in three
dimensions, which are Hamiltonian perturbations of the linear Klein-Gordon
equation with potential. The unperturbed dynamical system has a bound state
with frequency $\omega$, a spatially localized and time periodic solution. In
quantum mechanics, metastable states, which last longer than expected, have
been observed. These metastable states are a consequence of the instability of
the bound state under the nonlinear Fermi’s Golden Rule.

In this study, we explore the underlying mathematical instability mechanism
from the bound state to these metastable states. Besides, we derive the sharp
energy transfer rate from discrete to continuum modes, when the discrete
spectrum was not close to the continuous spectrum of the Schördinger operator
$H=-\Delta+V+m^{2}$, i.e. weak resonance regime
$\sigma_{c}(\sqrt{H})=[m,\infty)$, $0<3\omega<m$. This extends the work of
Soffer and Weinstein [32] for resonance regime $3\omega>m$ and confirms their
conjecture in [32]. Our proof relies on a more refined version of normal form
transformation of Bambusi and Cuccagna [3], the generalized Fermi’s Golden
Rule, as well as certain weighted dispersive estimates.

###### Contents

  1. 1 Introduction
    1. 1.1 Background
    2. 1.2 Related Works
    3. 1.3 Main Result
    4. 1.4 Outline of Proof
    5. 1.5 Structure of the Paper
    6. 1.6 Notations
  2. 2 Preliminary
    1. 2.1 Linear Dispersive Estimates
    2. 2.2 Singular Resolvents and Time Decay
    3. 2.3 Global Well-Posedness and Energy Conservation
  3. 3 Normal Form Transformation
    1. 3.1 Hamiltonian Structure
    2. 3.2 Lie Transform
    3. 3.3 Normal Form Transformation
  4. 4 Isolation of the Key Resonant Terms
  5. 5 Asymptotic Behavior
    1. 5.1 Dynamics of $\xi$
    2. 5.2 Dynamics of $f$
  6. 6 Error Estimates
  7. 7 Proof of Theorem 1.2
  8. A Convolution Estimates

##  1 Introduction

###  1.1 Background

The well-known Kolmogorov-Arnold-Moser (KAM) theory is concerned with the
persistence of periodic and quasi-periodic motion under the Hamiltonian
perturbation of a dynamical system, which has many applications in various
research fields, such as the N-body problem in celestial mechanics [19, 22].
For a finite dimensional integrable Hamiltonian system, the aforementioned
theory was initiated by Kolmogorov [19] and then extended by Moser [25] and
Arnold [1]. Subsequently, many efforts have been focused on generalizing the
KAM theory to infinite dimensional Hamiltonian systems (Hamiltonian PDEs),
wherein solutions are defined on compact spatial domains, such as [5, 9, 20].
In all of the above results, appropriate non-resonance conditions imply the
persistence of periodic and quasi-periodic solutions. See [22, 36] and the
references therein for a comprehensive survey.

In this study, we focus on a different phenomenon that resonance conditions
lead to the instability of periodic or quasi-periodic solutions. For example,
in quantum mechanics, under a small perturbation, an excited state could be
unstable with energy shifting to the ground state, free waves and nearby
excited states. In this process anomalously long-lived states called
metastable states in physics literature were observed. To study this
instability mechanism, Dirac, in 1927, considered an unperturbed Hamiltonian
$H_{0}$ with the initial eigenstate $i(x)$, and the perturbing Hamiltonian
$H_{1}$ with the final eigenstate $f(x)$, then calculated the transition
probability per unit time from the initial eigenstate to the final eigenstate

$\displaystyle\Gamma_{i\rightarrow
f}=\frac{2\pi}{\hbar}\left|\int_{\mathbb{R}^{3}}i(x)H_{1}(x)f(x)dx\right|^{2}\cdot\rho_{f},$
(1.1)

where $\rho_{f}$ is the density of the final states. This was utilized by
Fermi in 1934 to establish his famous theory of beta decay, where he called
the formula (1.1) “golden rule No. 2,” thus now known as “Fermi’s Golden
Rule.”

The rigorous mathematical analysis of the instability mechanism of bound
states and energy transfer rates were acquired quite late and remain largely
open. The first significant result of such problem for a nonlinear PDE was
provided by Soffer and Weinstein [32]. Consider the Cauchy problem for a
nonlinear Klein-Gordon equation with potential in the whole space,

$\begin{cases}\partial^{2}_{t}u-\Delta u+m^{2}u+V(x)u=\lambda
u^{3},&t>0,x\in\mathbb{R}^{3},\lambda\in\mathbb{R},\\\
u(x,0)=u_{0}(x),\quad\partial_{t}u(x,0)=u_{1}(x).\end{cases}$ (1.2)

This equation is a Hamiltonian system with energy

$\displaystyle\mathcal{E}[u,\partial_{t}u]\equiv\frac{1}{2}\int(\partial_{t}u)^{2}+|\nabla
u|^{2}+m^{2}u^{2}+V(x)u^{2}dx-\frac{\lambda}{4}\int u^{4}dx.$

The nonlinearity of $u^{3}$ can certainly be replaced by a general Hamiltonian
one. See [32] for more details. Assume that the potential function $V(x)$
satisfies the following properties:

######  Assumption 1.1.

Let $V(x)$ be real-valued function such that
(V1) for $\delta>5$ and $|\alpha|\leq 2,|\partial^{\alpha}V(x)|\leq
C_{\alpha}\langle x\rangle^{-\delta}$.
(V2) zero is not a resonance of the operator $-\Delta+V$.
(V3) $(-\Delta+1)^{-1}\left((x\cdot\nabla)^{l}V(x)\right)(-\Delta+1)^{-1}$ is
bounded on $L^{2}$ for $|l|\leq N_{\ast}$ with $N_{\ast}\geq 10$.
(V4) the operator $H=-\Delta+V(x)+m^{2}\triangleq B^{2}$ has a continuous
spectrum, $\sigma_{c}(H)=[m^{2},+\infty)$, and a unique strictly positive
simple eigenvalue, $\omega^{2}<m^{2}$ with the associated normalized
eigenfunction $\varphi(x)$:

$B^{2}\varphi=\omega^{2}\varphi.$

Under these assumptions, one can observe that the linear Klein-Gordon equation
(1.2), with $\lambda=0$, has a two-parameter family of spatially localized and
time-periodic solutions of the form:

$\displaystyle u(t,x)=R\cos(\omega t+\theta)\varphi(x).$

For $\lambda\neq 0$, i.e., Hamiltonian nonlinear perturbation of the linear
dispersive equation, instead of the KAM type results, Soffer and Weinstein
[32] proved that if $3\omega>m$ and the following resonance condition
(analogous Fermi’s Golden Rule Condition) was met,

$\Gamma\equiv\frac{\pi}{3\omega}\left(\mathbf{P}_{c}\varphi^{3}(x),\delta(B-3\omega)\mathbf{P}_{c}\varphi^{3}(x)\right)\equiv\frac{\pi}{3\omega}|(\mathcal{F}_{c}\varphi^{3})(3\omega)|^{2}>0.$
(1.3)

where $\mathbf{P}_{c}$ denotes the projection onto the continuous spectral
part of $B$ and $\mathcal{F}_{c}$ denotes the Fourier transform relative to
the continuous spectral part of $B$, then small global solutions to (1.2)
decayed to zero at an anomalously slow rate as time tended to infinity. More
precisely, the solution $u(t,x)$ had the following expansion as
$t\to\pm\infty$:

$\displaystyle u(t,x)=R(t)\cos(\omega t+\theta(t))\varphi(x)+\eta(t,x),$

where

$\displaystyle
R(t)=\mathcal{O}(|t|^{-\frac{1}{4}}),\theta(t)=\mathcal{O}(|t|^{\frac{1}{2}})\text{~{}and~{}}\quad\|\eta(t,\cdot)\|_{L^{8}}=\mathcal{O}(|t|^{-\frac{3}{4}}).$

This result provides the exact rate of energy transfer from discrete to
continuum modes. As a corollary, there are no small global periodic or quasi-
periodic solutions to (1.2). Besides, the instability mechanism of the bound
state is revealed through the nonlinear analogous Fermi’s Golden Rule (1.3) as
the key resonance coefficient in the dynamical equation of the discrete mode.
Thus, this non-negative coefficient reflects the nonlinear resonant
interaction between the bound state (eigenfunctions) and the radiation
(continuous spectral modes).

The Fermi’s Golden Rule condition (1.3) implies that

$3\omega\in\sigma_{c}(B)=[m,+\infty),$

which ensures a strong coupling of the discrete spectrum to the continuous
spectrum. For the weak resonance regime, i.e. the discrete spectrum $\omega$
is not close to the continuous spectrum $3\omega<m$, the Fermi’s Golden Rule
(1.3) fails, and the method proposed by Soffer and Weinstein [32] cannot be
directly applied to derive a similar result. Nevertheless, the authors in [32]
conjectured that a similar Fermi’s Golden Rule holds, and the solutions decay
to zero at certain rates for the general case. Since then, many efforts have
been focused on this conjecture, i.e. the instability mechanism and energy
transfer rate of metastable states in a weak resonance regime [2, 3]. We
emphasize that the energy transfer rate is crucial to obtain a quantitative
description on the lifespan of metastable states.

###  1.2 Related Works

For the nonlinear Klein-Gordon equations with zero potential, the global well-
posedness and scattering theory for small initial data has been established in
several works. For instance, see [18] and [29] for the three dimensional case,
[26] and [31] for the two dimensional case, and [10], [15] for the one
dimensional case with the modified scattering results. Note that these
solutions satisfy the linear dispersive estimates.

When potentials are present $V(x)\not\equiv 0$ but the operator $B$ has no
eigenvalues, the same decay results in linear Klein-Gordon equations are true
in three space dimensions, see [37]. Several interesting studies focus on
quadratic nonlinearities and variable coefficients in one dimensional case,
see [14], [23]. The lower bound of the decay rate has also been proved using
an alternative approach in a recent work by An–Soffer [2] when $B$ has one
simple eigenvalue lying close to the continuous spectrum, i.e., the case in
[32]. And in the recent interesting work [24], the authors extended the
results of [32] to quadratic nonlinearities and obtained the sharp decay
rates. Finally, in the remarkable work [3], for initial data in $H^{1}\times
L^{2}$ and a general potential which allows for multiply eigenvalues and
arbitrary gaps between the discrete and continuous spectra, Bambusi and
Cuccagana proposed a new normal form transformation preserving the Hamiltonian
structure and proved that small solutions were asymptotically free under a
non-degeneracy hypothesis. Moreover, they discovered the instability mechanism
of bound states through a nonlinear Fermi’s Golden Rule, whose coefficient was
never negative and generically strictly positive. We mention that for initial
data in $H^{1}\times L^{2}$, decay estimates do not hold due to the
conservation of energy. Nevertheless, the authors in [3] remarked that it was
possible to prove appropriate decay rates by restricting initial data to the
class in [32]. And in this paper we settle down this problem in our context.

We also mention that the Fermi’s Golden Rule was first introduced by Sigal
[30] in mathematical context, wherein the author established the instability
mechanism of quasi-periodic solutions to nonlinear Schrödinger and wave
equations. This was then further extended to study the asymptotic stability of
bound states of nonlinear Schrödinger equations by Tsai–Yau [35],
Soffer–Weinstein [33], Gang [11], Gang–Sigal [12], Gang–Weinstein [13]; see
also the recent advances by Cuccagna–Maeda [7], their survey [8] and
references therein.

###  1.3 Main Result

Our main result is the following theorem:

######  Theorem 1.2.

Let $(2N-1)\omega<m<(2N+1)\omega$ for some $N\in\mathbb{N}$ and $C_{0}$ be a
positive constant. Under Assumption 1.1, if the following resonance condition
($\gamma$ is defined in Section 4 )

$\displaystyle\gamma>0$

holds, then there exists a sufficiently small constant $\epsilon>0$ such that
for any initial data $u_{0},u_{1}$ satisfying

$\|u_{0}\|_{W^{2,2}\cap W^{2,1}}+\|u_{1}\|_{W^{1,2}\cap W^{1,1}}\leq\epsilon$

and

$\|\mathbf{P}_{c}u_{0}\|_{W^{2,2}\cap
W^{2,1}}+\|\mathbf{P}_{c}u_{1}\|_{W^{1,2}\cap W^{1,1}}\leq
C_{0}\left(\|\mathbf{P}_{d}u_{0}\|_{L^{2}}+\|\mathbf{P}_{d}u_{1}\|_{L^{2}}\right),$
(1.4)

solution $u(t,x)$ to (1.2) with $\lambda\neq 0$ has the following expansion as
$t\to\pm\infty$:

$\displaystyle u(t,x)=R(t)\cos(\omega t+\theta(t))\varphi(x)+\eta(t,x),$

where

$\displaystyle\frac{\frac{1}{C}R(0)}{(1+4N\lambda^{2N}|R(0)|^{4N}\gamma
t)^{\frac{1}{4N}}}\leq R(t)\leq\frac{CR(0)}{(1+4N\lambda^{2N}|R(0)|^{4N}\gamma
t)^{\frac{1}{4N}}},$ (1.5)
$\displaystyle\theta(t)=\mathcal{O}(|t|^{1-\frac{1}{2N}}),\quad\|\eta(t,\cdot)\|_{L^{8}}=\mathcal{O}(|t|^{-\frac{3}{4N}}).$
(1.6)

for some positive constant $C>0$. Here, $\mathbf{P}_{d}$ and $\mathbf{P}_{c}$
denote the projections onto the discrete and continuous spectral part of $B$
respectively,

######  Remark 1.3.

Evidently, the case where $N=1$ is exactly the work of Soffer-Weinstein [32].
We point out that the case of $m=(2N+1)\omega$ is beyond the scope of our
work, and it remains unsolved.

######  Remark 1.4.

We indicate that $R(0)$ can be of order one by choosing a sufficiently small
$\lambda$ and a simple scaling argument. The assumption on initial data

$\|\mathbf{P}_{c}u_{0}\|_{W^{2,2}\cap
W^{2,1}}+\|\mathbf{P}_{c}u_{1}\|_{W^{1,2}\cap
W^{1,1}}\lesssim\|\mathbf{P}_{d}u_{0}\|_{L^{2}}+\|\mathbf{P}_{d}u_{1}\|_{L^{2}},$

see (1.4), is necessary, which leads to resonance-dominated solutions with the
decay rates (1.5) and (1.6). Indeed, this assumption should also be imposed in
[32] and other related references. As pointed out by Tsai and Yau in [35],
when

$\|\mathbf{P}_{c}u_{0}\|_{W^{2,2}\cap
W^{2,1}}+\|\mathbf{P}_{c}u_{1}\|_{W^{1,2}\cap
W^{1,1}}\gg\|\mathbf{P}_{d}u_{0}\|_{L^{2}}+\|\mathbf{P}_{d}u_{1}\|_{L^{2}},$

there exist dispersion-dominated solutions with faster decay rates, even in
the case of Soffer and Weinstein [32].

######  Remark 1.5.

The decay rate (1.5) is consistent with the numerical prediction in [4].

###  1.4 Outline of Proof

Now we explain the ideas of our proof. By setting
$u_{\lambda}(t,x)=|\lambda|^{1/2}u(t,x)$, we may assume $\lambda=\pm 1$. We
adapt the framework of Hamiltonian method proposed in [3] to derive the
dynamical equations of discrete and continuous modes. By changing variables,
the nonlinear Klein-Gordon equations (1.2) can be wirtten as the following
Hamilton equations (see Section 3 for details)

$\dot{\xi}=-\mathrm{i}\frac{\partial
H}{\partial\bar{\xi}},\quad\dot{f}=-\mathrm{i}\nabla_{\bar{f}}H.$

with the corresponding Hamiltonian

$\displaystyle H$ $\displaystyle=H_{L}+H_{P},$ $\displaystyle H_{L}$
$\displaystyle=\omega\left|\xi\right|^{2}+\langle\bar{f},Bf\rangle,$
$\displaystyle H_{P}$
$\displaystyle=\int_{\mathbb{R}^{3}}\left(\sum\frac{\xi+\bar{\xi}}{\sqrt{2\omega}}\varphi(x)+U(x)\right)^{4}dx,$

where $\nabla_{\bar{f}}H$ is the gradient with respect to the $L^{2}$ metric,
and $U=B^{-\frac{1}{2}}(f+\bar{f})/\sqrt{2}\equiv P_{c}u$.

To further simplify the dynamics of discrete mode and decouple the discrete
mode from the continuous ones, we proposed a refined Birkhoff normal form
transformation compared to the one in [3]. More precisely, we prove that for
any $0\leq r\leq 2N$ there exists an analytic canonical transformation
$\mathcal{T}_{r}$ putting the system in normal form up to order $2r+4$, i.e.

$H^{(r)}:=H\circ\mathcal{T}_{r}=H_{L}+Z^{(r)}+\mathcal{R}^{(r)}$

where $Z^{(r)}$ is a polynomial of degree $2r+2$ which is in normal form (see
Definition 3.2), and $\mathcal{R}^{(r)}$ consists of error terms of higher
order, see Theorem 3.3. The normal form transformation is canonical and
preserves the Hamiltonian nature of the system. Our key observation is that
the order of normal form is actually increased by two in each step, which is
consistent with the theoretic prediction by the non-Hamiltonian method of
Soffer and Wesinsten [32]. Besides, we explore the explict forms of theses
coefficients appeared in error terms, whose structure will be crutial in error
estimates.

After applying the normal form transformation, we are able to extract the main
terms of the continuous modes $f$ and obtain the dynamics of the discret mode
as follows:

$\frac{d}{dt}|\xi|^{2}=-2\gamma|\xi|^{4N+2}+2Re(\bar{\xi}\mathcal{R}_{\xi}),$

where

$\displaystyle\gamma:=$
$\displaystyle(2N+1)\langle\Phi_{0,2N+1},\delta(B-(2N+1)\omega)\bar{\Phi}_{0,2N+1}\rangle\geq
0,\quad\Phi_{0,2N+1}\in\mathcal{S}\left(\mathbb{R}^{3},\mathbb{C}\right),$

is the coefficient of nonlinear Fermi’s Golden Rule, and $\mathcal{R}_{\xi}$
is an error term, see (4.6). Thus, under the Fermi’s Golden Rule condition
$\gamma>0$, we could expect that the decay rate of the discrete mode $\xi(t)$
is $O(|t|^{-\frac{1}{4N}})$, provided that an appropriate estimate for the
error term $\mathcal{R}_{\xi}$ is present.

To close our argument, the estimate of error term $\mathcal{R}_{\xi}$ remains
to be performed, which by (4.6) requires deriving the decay estimates of the
continuous mode $f$. Once this is achieved, a bootstrap argument would imply
Theorem 1.2. Since we are pursuing decay estimates in time, the space-time
dispersive estimates from [3] are not applicable here. Besides, we shall see
that the classical $L^{p}$ estimates also fail due to the slow decay rate of
$\xi$ and $f$, especially for the weak resonance regime $N>1$. To illustrate
this, consider

$\hat{\mathcal{R}}_{f}:=-\mathrm{i}\int_{0}^{t}e^{-\mathrm{i}B(t-s)}\partial_{\bar{f}}\mathcal{R}ds,$

which appears as an error term in the expression of the continuous mode $f$,
see (4.3). Here $\mathcal{R}$ is the error term coming from the normal form
transformation $\mathcal{R}=\sum_{d=1}^{5}\mathcal{R}_{d}.$ Denote

$\hat{\mathcal{R}}^{2}_{f}:=-\mathrm{i}\int_{0}^{t}e^{-\mathrm{i}B(t-s)}\partial_{\bar{f}}\mathcal{R}_{2}ds.$

When $N=1$ as in [32], using linear dispersive estimates, and by virtue of the
structure of $\mathcal{R}_{d}$ in Theorem 3.3, one has

$\displaystyle\|B^{-1/2}\hat{\mathcal{R}}^{2}_{f}\|_{L^{8}}$
$\displaystyle\lesssim\int_{0}^{t}\min\left\\{|t-s|^{-\frac{9}{8}},|t-s|^{-\frac{3}{8}}\right\\}|s|^{-\frac{1}{2}}[\xi]_{\frac{1}{4}}^{2}\left(\|B^{-1/2}f\|_{L^{8}}+\|B^{1/2}f\|_{L^{4}}\right)ds,$
$\displaystyle\|B^{1/2}\hat{\mathcal{R}}^{2}_{f}\|_{L^{4}}$
$\displaystyle\lesssim\int_{0}^{t}|t-s|^{-\frac{1}{2}}|s|^{-\frac{1}{2}}[\xi]_{\frac{1}{4}}^{2}\left(\|B^{-1/2}f\|_{L^{8}}+\|B^{1/2}f\|_{L^{4}}\right)ds,$

which yield

$\displaystyle\|B^{-1/2}\hat{\mathcal{R}}^{2}_{f}\|_{L^{8}}$
$\displaystyle\lesssim|t|^{-1+\delta_{0}}[\xi]_{\frac{1}{4}}^{2}\left([B^{-1/2}f]_{8,\frac{3}{4}}+[B^{1/2}f]_{4,1/2-\delta_{0}}\right),$
$\displaystyle\|B^{1/2}\hat{\mathcal{R}}^{2}_{f}\|_{L^{4}}$
$\displaystyle\lesssim|t|^{-\frac{1}{2}+\delta_{0}}[\xi]_{\frac{1}{4}}^{2}\left([B^{-1/2}f]_{8,\frac{3}{4}}+[B^{1/2}f]_{4,1/2-\delta_{0}}\right),$

where $[\xi]_{\alpha}=\sup_{0\leq t\leq T}\langle
t\rangle^{\alpha}|\xi(t)|,[g]_{p,\alpha}=\sup_{0\leq t\leq T}\langle
t\rangle^{\alpha}\|g\|_{L^{p}}$. Together with decay estimate of
$\xi(t)=O(|t|^{-\frac{1}{4}})$, these estimates imply the desired error
estimate of $\mathcal{R}_{\xi}$ for small data, which is similar to the
approach used in [32]. We emphasize that in the aforementioned estimates, the
decay rate $O(|t|^{-\frac{1}{4}})$ of $\xi(t)$ is necessary and cannot be
weakened. However, for the weak resonance regime $N>1$, the expected decay
rate $\xi(t)=O(|t|^{-\frac{1}{4N}})$, which is much slower than required. For
instance, the linear dispersive estimates imply

$\|B^{1/2}\hat{\mathcal{R}}^{2}_{f}\|_{L^{4}}\lesssim\int_{0}^{t}|t-s|^{-\frac{1}{2}}|s|^{-\frac{1}{2N}}[\xi]_{\frac{1}{2N}}^{2}\left(\|B^{-1/2}f\|_{L^{8}}+\|B^{1/2}f\|_{L^{4}}\right),$

where

$\frac{1}{2}+\frac{1}{2N}<1.$

Hence, the decay estimates of $B^{-1/2}f,B^{1/2}f$ cannot be closed this way.

In this paper, we overcome this difficulty by applying the weighted $L^{p}$
estimates, see Lemma 2.3, which acquires time decay at the cost of the spatial
decay. More precisely, instead of estimating $\|B^{1/2}f\|_{L^{4}}$, we
consider the weighted $L^{4}$ norm $\|\langle
x\rangle^{-\sigma}B^{1/2}f\|_{L^{4}}$, then

$\displaystyle\|\langle x\rangle^{-\sigma}B^{1/2}f_{32}\|_{L^{4}}\lesssim$
$\displaystyle\int_{0}^{t}\min\\{|t-s|^{-\frac{5}{4}},|t-s|^{-\frac{1}{2}}\\}|s|^{-\frac{1}{2N}}[\xi]^{2}_{\frac{1}{4N}}\left\|\langle
x\rangle^{\sigma}\partial_{\bar{f}}\mathcal{R}_{2}\right\|_{W^{1,\frac{4}{3}}}$
$\displaystyle\lesssim$
$\displaystyle\langle|\xi_{0}|^{4N}t\rangle^{-\frac{2N+2}{4N}}[\xi]^{2}_{\frac{1}{4N}}\big{(}[\langle
x\rangle^{-\sigma}B^{1/2}f]_{4,\frac{1}{2}}+[B^{-1/2}f]_{8,\frac{2N+1}{4N}}\big{)}.$

Such a technique is based on the spatially localized property of
$\partial_{\bar{f}}\mathcal{R}_{2}$ to absorb the weight $\langle
x\rangle^{\sigma}$ on the RHS.

###  1.5 Structure of the Paper

The remaining part of this paper is organized as follows. In Section 2, we
introduce some useful dispersive estimates and weighted inequalities for
linear equations with potential, and we present the global existence theory
and energy conservation of the nonlinear Klein-Gordon equation. In Section 3,
we begin our proof by performing Birkhoff normal form transformation. Then we
isolate the key resonant terms in the dynamical equation of the discrete mode
in Section 4. In Section 5, we derive the asymptotic behavior of discrete and
continuous modes. In Section 6, the error terms are carefully estimated. In
Section 7, we prove our main result using apriori estimates and bootstrap
arguments.

###  1.6 Notations

Throughout our paper, we use $C$ to denote an absolute positive constant that
may vary from line to line. We write $A\lesssim B$ to mean that $A\leq CB$ for
some absolute constant $C>0$. We will use $A\approx B$ in a similar standard
way.

##  2 Preliminary

In this section, we provide some useful lemmas on the linear analysis for the
Klein-Gordon equation with potential and the global well-posedness theory of
the nonlinear Klein-Gordon equation (1.2).

###  2.1 Linear Dispersive Estimates

Consider the Cauchy problem for three dimensional linear Klein-Gordon equation
with a potential

$\begin{cases}\partial_{t}^{2}u-\Delta u+m^{2}u+V(x)u=0,&\quad
t>0,x\in\mathbb{R}^{3},\\\
u(0,x)=u_{0},\quad\partial_{t}u(0,x)=u_{1}.\end{cases}$ (2.1)

Denote $B^{2}=-\Delta+m^{2}+V(x)$, then equation (2.1) can be solved as

$\displaystyle u(t,x)=\cos{Bt}\ u_{0}+\frac{\sin Bt}{B}\ u_{1}.$

For $V(x)=0$, i.e. free Klein-Gordon case, the standard $L^{p}$ dispersive
estimates follow from an oscillatory integration method and the conservation
of the $L^{2}$ norm. More precisely, the $L^{p}$ norm of the solution to
$u(t,x)$ satisfies the dispersive decay estimate $\|u(t,\cdot)\|_{L^{p}}\leq
C|t|^{-3(\frac{1}{2}-\frac{1}{p})}$.

For $V(x)\neq 0$, if $V(x)$ satisfies some suitable decay and regularity
conditions, then the same decay rate of $u$ can be obtained by the
$W^{k,p}$-boundedness of the wave operator after being projected on the
continuous spectrum of $B$. For instance, see [17],[28],[37].

######  Lemma 2.1 ($L^{p}$ dispersive estimates).

Assume that $V(x)$ is a real-valued function and satisfies (V1),(V2). Let
$1<p\leq 2$, $\frac{1}{p}+\frac{1}{p^{\prime}}=1$, $0\leq\theta\leq 1$,
$l=0,1$, and $s=(4+\theta)(\frac{1}{2}-\frac{1}{p^{\prime}})$. Then

$\|\mathrm{e}^{\mathrm{i}Bt}B^{-l}\mathbf{P}_{\mathrm{c}}\psi\|_{l,p^{\prime}}\lesssim|t|^{-(2+\theta)(\frac{1}{2}-\frac{1}{p^{\prime}})}\|\psi\|_{s,p},\quad|t|\geq
1,$

and

$\|\mathrm{e}^{\mathrm{i}Bt}B^{-l}\mathbf{P}_{\mathrm{c}}\psi\|_{l,p^{\prime}}\lesssim|t|^{-(2-\theta)(\frac{1}{2}-\frac{1}{p^{\prime}})}\|\psi\|_{s,p},\quad
0<|t|\leq 1.$

The following weighted decay estimate of the Klein-Gordon equation was first
established by Jensen and Kato [16] for the Schrödinger equation and then
extended to the Klein-Gordon equation by Komech and Kopylova [21]. These
estimates are used to reveal the non-resonance structure of nonlinearities in
the equation satisfied by the continuous spectrum part.

######  Lemma 2.2 (Weighted $L^{2}$ estimate).

Assume $V(x)$ satisfies the hypothesis of Lemma 2.1. Then, for
$\sigma>\frac{5}{2}$, $l=0,1$, we have

$\left\|\langle
x\rangle^{-\sigma}\mathrm{e}^{\mathrm{i}Bt}B^{-l}\mathbf{P}_{\mathrm{c}}\langle
x\rangle^{-\sigma}\psi\right\|_{l,2}\lesssim\langle
t\rangle^{-\frac{3}{2}}\|\psi\|_{2}.$

The following weighted $L^{p}$ estimates, which are consequences of the
interpolation between the standard $L^{\infty}-L^{1}$ estimates and the
weighted $L^{2}$ estimate, are also necessary:

######  Lemma 2.3 (Weighted $L^{p}$ estimate).

Assume that $V(x)$ satisfies the hypothesis of Lemma 2.1. Let $1<p\leq 2$,
$\frac{1}{p}+\frac{1}{p^{\prime}}=1$, $0\leq\theta\leq 1$, $l=0,1$, and
$s=(4+\theta)(\frac{1}{2}-\frac{1}{p^{\prime}})$. Then, for
$\sigma>\frac{5}{2}$,

$\left\|\langle
x\rangle^{-\sigma}\mathrm{e}^{\mathrm{i}Bt}B^{-l}\mathbf{P}_{\mathrm{c}}\langle
x\rangle^{-\sigma}\psi\right\|_{l,p^{\prime}}\lesssim|t|^{-(2+\theta)(\frac{1}{2}-\frac{1}{p^{\prime}})-\frac{3}{p^{\prime}}}\|\psi\|_{s,p},\quad|t|\geq
1$

and

$\left\|\langle
x\rangle^{-\sigma}\mathrm{e}^{\mathrm{i}Bt}B^{-l}\mathbf{P}_{\mathrm{c}}\langle
x\rangle^{-\sigma}\psi\right\|_{l,p^{\prime}}\lesssim|t|^{-(2-\theta)(\frac{1}{2}-\frac{1}{p^{\prime}})}\|\psi\|_{s,p},\quad
0<|t|\leq 1$

hold.

The use of the weighted $L^{p}$ estimates enables us to obtain a better time
decay of the continuous spectrum part of $u$ from its spatial decay, which
aids dealing with the asymptotic behavior of solutions to the nonlinear Klein-
Gordon equations. In particular, they help us to overcome the loss of the
derivative of $\eta_{3}$. For more details, see Section 7.

###  2.2 Singular Resolvents and Time Decay

The following local decay estimates for singular resolvents
$\mathrm{e}^{\mathrm{i}Bt}(B-\Lambda+\mathrm{i}0)^{-l}$, which was proved in
[32], are also significant. Here, $\Lambda$ is a point in the interior of the
continuous spectrum of $B(\Lambda>m)$.

######  Lemma 2.4 (Decay estimates for singular resolvents).

Assume that $V(x)$ is a real-valued function and satisfies (V1)-(V3). Let
$\sigma>16/5$. Then for any point $\Lambda>m$ in the continuous spectrum of
$B$, we have for $l=1,2:$

$\displaystyle\left\|\langle
x\rangle^{-\sigma}\mathrm{e}^{\mathrm{i}Bt}(B-\Lambda+\mathrm{i}0)^{-l}\mathbf{P}_{\mathrm{c}}\langle
x\rangle^{-\sigma}\psi\right\|_{2}\lesssim\langle
t\rangle^{-\frac{6}{5}}\|\psi\|_{1,2},\quad t>0,$ $\displaystyle\left\|\langle
x\rangle^{-\sigma}\mathrm{e}^{\mathrm{i}Bt}(B-\Lambda-\mathrm{i}0)^{-l}\mathbf{P}_{\mathrm{c}}\langle
x\rangle^{-\sigma}\psi\right\|_{2}\lesssim\langle
t\rangle^{-\frac{6}{5}}\|\psi\|_{1,2},\quad t<0.$

###  2.3 Global Well-Posedness and Energy Conservation

The global well-posedness of (1.2) with small initial data is well-known.

######  Theorem 2.5.

Assume $V\in L^{p}$ with $p>3/2$. Then, there exists $\varepsilon_{0}>0$ and
$C>0$, such that for any $\left\|\left(u_{0},u_{1}\right)\right\|_{H^{1}\times
L^{2}}\leq\epsilon<\varepsilon_{0}$, equation (1.2) admits exactly one
solution $u\in C^{0}\left(\mathbb{R};H^{1}\right)\cap
C^{1}\left(\mathbb{R};L^{2}\right)$ such that
$(u(0),\partial_{t}u(0))=\left(u_{0},u_{1}\right)$. Furthermore, the map
$\left(u_{0},u_{1}\right)\mapsto(u(t),\partial_{t}u(t))$ is continuous from
the ball $\left\|\left(u_{0},u_{1}\right)\right\|_{H^{1}\times
L^{2}}<\varepsilon_{0}$ to $C^{0}\left(I;H^{1}\right)\times
C^{0}\left(I;L^{2}\right)$ for any bounded interval $I$. Moreover, the energy

$\displaystyle\mathcal{E}[u,\partial_{t}u]\equiv\frac{1}{2}\int(\partial_{t}u)^{2}+|\nabla
u|^{2}+m^{2}u^{2}+V(x)u^{2}dx-\frac{\lambda}{4}\int u^{4}dx.$

is conserved and

$\|(u(t),v(t))\|_{H^{1}\times L^{2}}\leq
C\left\|\left(u_{0},v_{0}\right)\right\|_{H^{1}\times L^{2}}.$

We refer to [6] for details.

##  3 Normal Form Transformation

In this section, we present a new normal form transformation which is a
refined version of Theorem 4.9 in [3]. The main differences are as follows:
(i) we find that the order of normal form actually increase by two in each
step, which is consistent with the result in [32]; (ii) we give explicit forms
of these coefficients appeared in error terms, whose structure will be crucial
in the subsequent error estimates.

###  3.1 Hamiltonian Structure

Now we consider 3D nonlinear Klein Gordon equation (NLKG)

$\displaystyle u_{tt}-\Delta
u+Vu+m^{2}u=u^{3},\quad(t,x)\in\mathbb{R}\times\mathbb{R}^{3},$ (3.1)

which is an Hamiltonian perturbation of linear Klein-Gordon equation with
potential. More precisely, in
$H^{1}\left(\mathbb{R}^{3},\mathbb{R}\right)\times
L^{2}\left(\mathbb{R}^{3},\mathbb{R}\right)$ endowed with the standard
symplectic form, namely

$\Omega\left(\left(u_{1},v_{1}\right);\left(u_{2},v_{2}\right)\right):=\left\langle
u_{1},v_{2}\right\rangle_{L^{2}}-\left\langle
u_{2},v_{1}\right\rangle_{L^{2}}$

we consider the Hamiltonian

$\displaystyle H$ $\displaystyle=H_{L}+H_{P},$ $\displaystyle H_{L}$
$\displaystyle:=\int_{\mathbb{R}^{3}}\frac{1}{2}\left(v^{2}+|\nabla
u|^{2}+Vu^{2}+m^{2}u^{2}\right)dx,$ $\displaystyle H_{P}$
$\displaystyle:=\int_{\mathbb{R}^{3}}-\frac{1}{4}u^{4}dx.$

The corresponding Hamilton equations are
$\dot{v}=-\nabla_{u}H,\dot{u}=\nabla_{v}H$, where $\nabla_{u}H$ is the
gradient with respect to the $L^{2}$ metric, explicitly defined by

$\left\langle\nabla_{u}H(u),h\right\rangle=d_{u}H(u)h,\quad\forall h\in H^{1}$

and $d_{u}H(u)$ is the Frechét derivative of $H$ with respect to $u$. It is
easy to see that the Hamilton equations are explicitly given by

$\displaystyle\left(\dot{v}=\Delta
u-Vu-m^{2}u+u^{3},\dot{u}=v\right)\Longleftrightarrow\ddot{u}=\Delta
u-Vu-m^{2}u+u^{3}.$

Write

$u=q\varphi+P_{c}u,\quad v=p\varphi+P_{c}v,$

with a slightly abuse of notations, from now on we denote

$B:=P_{c}\left(-\Delta+V+m^{2}\right)^{1/2}P_{c},$

and define the complex variables

$\xi:=\frac{q\sqrt{\omega}+\mathrm{i}\frac{p}{\sqrt{\omega}}}{\sqrt{2}},\quad
f:=\frac{B^{1/2}P_{c}u+\mathrm{i}B^{-1/2}P_{c}v}{\sqrt{2}}.$ (3.2)

Then, in terms of these variables the symplectic form has the form

$\begin{array}[]{r}\Omega\left(\left(\xi^{(1)},f^{(1)}\right);\left(\xi^{(2)},f^{(2)}\right)\right)=2\operatorname{Re}\left[\mathrm{i}\left(\xi^{(1)}\bar{\xi}^{(2)}+\left\langle
f^{(1)},\bar{f}^{(2)}\right\rangle\right)\right]\\\
=-\mathrm{i}\sum_{j}\left(\bar{\xi}^{(1)}\xi^{(2)}-\xi^{(1)}\bar{\xi}^{(2)}\right)-\mathrm{i}\left(\left\langle
f^{(2)},\bar{f}^{(1)}\right\rangle-\left\langle
f^{(1)},\bar{f}^{(2)}\right\rangle\right)\end{array}$

and the Hamilton equations take the form

$\dot{\xi}=-\mathrm{i}\frac{\partial
H}{\partial\bar{\xi}},\quad\dot{f}=-\mathrm{i}\nabla_{\bar{f}}H.$

where

$\begin{gathered}H_{L}=\omega\left|\xi\right|^{2}+\langle\bar{f},Bf\rangle,\\\
H_{P}(\xi,f)=\int_{\mathbb{R}^{3}}\left(\sum\frac{\xi+\bar{\xi}}{\sqrt{2\omega}}\varphi(x)+U(x)\right)^{4}dx\end{gathered}$

with $U=B^{-\frac{1}{2}}(f+\bar{f})/\sqrt{2}\equiv P_{c}u$. The Hamiltonian
vector field $X_{H}$ of a function is given by

$X_{H}(\xi,\bar{\xi},f,\bar{f})=\left(-\mathrm{i}\frac{\partial
H}{\partial\bar{\xi}},\mathrm{i}\frac{\partial
H}{\partial\xi},-\mathrm{i}\nabla_{\bar{f}}H,\mathrm{i}\nabla_{f}H\right).$

The associate Poisson bracket is given by

$\displaystyle\\{H,K\\}:=$ $\displaystyle\mathrm{i}\left(\frac{\partial
H}{\partial\xi}\frac{\partial K}{\partial\bar{\xi}}-\frac{\partial
H}{\partial\bar{\xi}}\frac{\partial K}{\partial\xi}\right)$
$\displaystyle+\mathrm{i}\left\langle\nabla_{f}H,\nabla_{\bar{f}}K\right\rangle-\mathrm{i}\left\langle\nabla_{\bar{f}}H,\nabla_{f}K\right\rangle.$

Denote $z=(\xi,f),\mathbf{f}=(f,\bar{f}),$ and
$\mathcal{P}^{k,s}=\mathbb{C}\times
P_{c}H^{k,s}\left(\mathbb{R}^{3},\mathbb{C}\right)$, where

$H^{k,s}\left(\mathbb{R}^{3},\mathbb{C}\right)=\left\\{f:\mathbb{R}^{3}\rightarrow\mathbb{C}\text{
s.t. }\|f\|_{H^{s,k}}:=\left\|\langle
x\rangle^{s}(-\Delta+1)^{k/2}f\right\|_{L^{2}}<\infty\right\\}.$

###  3.2 Lie Transform

Consider a function $\chi$ of the form

$\displaystyle\chi(z)\equiv\chi(\xi,f)=\chi_{0}(\xi,\bar{\xi})+\sum_{|\mu|+|\nu|=M_{0}+1}\xi^{\mu}\overline{\xi^{\nu}}\int_{\mathbb{R}^{3}}\mathbf{\Phi}_{\mu\nu}\cdot\mathbf{f}dx$
(3.3)

where
$\mathbf{\Phi}_{\mu\nu}\cdot\mathbf{f}:=\Phi_{\mu\nu}f+\Psi_{\mu\nu}\bar{f}$
with
$\Phi_{\mu\nu},\Psi_{\mu\nu}\in\mathcal{S}\left(\mathbb{R}^{3},\mathbb{C}\right)$
and where $\chi_{0}$ is a homogeneous polynomial of degree $M_{0}+2$. The
Hamiltonian vector field satisfies $X_{\chi}\in
C^{\infty}\left(\mathcal{P}^{-\kappa,-s},\mathcal{P}^{k,\tau}\right)$ for any
$k,\kappa,s,\tau\geq 0$. Moreover we have

$\displaystyle\left\|X_{\chi}(z)\right\|_{\mathcal{P}^{k,\tau}}\leq
C_{k,s,\kappa,\tau}\|z\|_{\mathcal{P}^{-\kappa,-s}}^{M_{0}+1}.$ (3.4)

Since $X_{\chi}$ is a smooth polynomial it is also analytic. Denote by
$\phi^{t}$ the flow generated by $X_{\chi}$. For fixed $\kappa,s,$ by (3.4)
$\phi^{t}$ is well defined up to any fixed time $\bar{t}$, in a sufficiently
small neighborhood $\mathcal{U}^{-\kappa,-s}\subset\mathcal{P}^{-\kappa,-s}$
of the origin. Set $\phi:=\left.\phi^{1}\equiv\phi^{t}\right|_{t=1}$. The
canonical transformation $\phi$ will be called the Lie transform generated by
$\chi$.

######  Lemma 3.1.

Given a functional $\chi$ of the form (3.3). Assume
$\Phi_{\mu\nu},\Psi_{\mu\nu}\in$
$\mathcal{S}\left(\mathbb{R}^{3},\mathbb{C}\right)$ for all $\mu$ and $\nu$.
Let $\phi$ be its Lie transform. Denote $z^{\prime}=\phi(z),z\equiv(\xi,f)$
and $z^{\prime}\equiv\left(\xi^{\prime},f^{\prime}\right)$. Then, there exists
a sufficiently small neighborhood
$\mathcal{U}^{-\kappa,-s}\subset\mathcal{P}^{-\kappa,-s}$ of the origin, such
that the following expansions hold:

$\displaystyle\xi^{\prime}=\xi+\sum_{k=1}^{\infty}\sum_{i=0}^{k}\sum_{\mu+\nu=kM_{0}+1-i}a_{i\mu\nu}\xi^{\mu}\bar{\xi}^{\nu}\prod_{j=1}^{i}\int\mathbf{\Phi}_{\mu\nu}^{ij}\cdot\mathbf{f}dx,$
(3.5) $\displaystyle
f^{\prime}=f+\sum_{k=1}^{\infty}\sum_{i=0}^{k-1}\sum_{\mu+\nu=kM_{0}+1-i}b_{i\mu\nu}\xi^{\mu}\bar{\xi}^{\nu}\prod_{j=1}^{i}\int\mathbf{\Lambda}_{\mu\nu}^{ij}\cdot\mathbf{f}dx\Psi_{\mu\nu}^{i}.$
(3.6)

where $a_{i\mu\nu}$ and $b_{i\mu\nu}$ are constants and
$\mathbf{\Phi}_{\mu\nu}^{ij},\mathbf{\Lambda}_{\mu\nu}^{ij},\Psi_{\mu\nu}^{i}\in\mathcal{S}\left(\mathbb{R}^{3},\mathbb{C}\right).$

###### Proof.

Let $z(t)=(\xi(t),f(t))=\phi^{t}(z)$, then $z(0)=z,z(1)=z^{\prime}$. By virtue
of the Lie transform and the analyticity of the Hamiltonian vector field
$X_{\chi}$, we have

$\xi^{\prime}=\xi+\sum_{k=1}^{\infty}\frac{1}{k!}\underbrace{\\{\chi_{r},\dots,\\{\chi_{r}}_{k\text{
times}},\xi\\}\\},$ (3.7)

and it is easy to show by induction that

$\displaystyle\underbrace{\\{\chi_{r},\dots\\{\chi_{r}}_{k\text{
times}},\xi\\}\\}=$
$\displaystyle\sum_{i=0}^{k}\sum_{\mu+\nu=kM_{0}+1-i}a_{i\mu\nu}\xi^{\mu}\bar{\xi}^{\nu}\prod_{j=1}^{i}\int\mathbf{\Phi}_{\mu\nu}^{ij}\cdot\mathbf{f}dx.$

Similarly, since

$\frac{df}{dt}=-i\sum_{\mu+\nu=M_{0}+1}\xi^{\mu}\bar{\xi}^{\nu}\Psi_{\mu\nu},$
(3.8)

we have

$f^{\prime}=f-i\sum_{\mu+\nu=M_{0}+1}\int_{0}^{1}\xi^{\mu}(t)\bar{\xi}^{\nu}(t)dt\Psi_{\mu\nu}.$
(3.9)

By virtue of the Lie transform, we have

$\xi^{\mu}(t)\bar{\xi}^{\nu}(t)=\sum_{k=0}^{\infty}\frac{t^{k}}{k!}\underbrace{\\{\chi_{r},\dots\\{\chi_{r}}_{k\text{
times}},\xi^{\mu}\bar{\xi}^{\nu}\\}\\},$ (3.10)

hence

$f^{\prime}=f-i\sum_{\mu+\nu=M_{0}+1}\sum_{k=0}^{\infty}\frac{1}{(k+1)!}\underbrace{\\{\chi_{r},\dots\\{\chi_{r}}_{k\text{
times}},\xi^{\mu}\bar{\xi}^{\nu}\\}\\}\Psi_{\mu\nu}.$ (3.11)

The rest follows similarly. ∎

###  3.3 Normal Form Transformation

######  Definition 3.2.

A polynomial $Z$ is in normal form if we have

$Z=Z_{0}+Z_{1}$

where $Z_{0}$ is a linear combination of monomials $|\xi|^{2\mu}$, and $Z_{1}$
is a linear combination of monomials of the form

$\xi^{\mu}\overline{\xi^{\nu}}\int\Phi(x)f(x)dx,\quad\xi^{\mu^{\prime}}\overline{\xi^{\nu^{\prime}}}\int\Phi(x)\bar{f}(x)dx$

with indexes satisfying

$\omega(\mu-\nu)<-m,\quad\omega\left(\mu^{\prime}-\nu^{\prime}\right)>m$

and $\Phi\in\mathcal{S}\left(\mathbb{R}^{3},\mathbb{C}\right).$

######  Theorem 3.3.

For any $n>0,s>0$ and any integer $r$ with $0\leq r\leq 2N$, there exist open
neighborhoods of the origin $\mathcal{U}_{r,n,s}\subset\mathcal{P}^{1/2,0}$,
$\mathcal{U}_{r}^{-n,-s}\subset\mathcal{P}^{-n,-s}$, and an analytic canonical
transformation
$\mathcal{T}_{r}:\mathcal{U}_{r,n,s}\rightarrow\mathcal{P}^{1/2,0}$, such that
$\mathcal{T}_{r}$ puts the system in normal form up to order $2r+4$. More
precisely, we have

$H^{(r)}:=H\circ\mathcal{T}_{r}=H_{L}+Z^{(r)}+\mathcal{R}^{(r)}$

where: (i) $Z^{(r)}$ is a polynomial of degree $2r+2$ which is in normal form,
(ii) $I-\mathcal{T}_{r}$ extends into an analytic map from
$\mathcal{U}_{r}^{-n,-s}$ to $\mathcal{P}^{n,s}$ and

$\left\|z-\mathcal{T}_{r}(z)\right\|_{\mathcal{P}^{n,s}}\lesssim\|z\|_{\mathcal{P}^{-n,-s}}^{3}.$

(iii) we have $\mathcal{R}^{(r)}=\sum_{d=0}^{5}\mathcal{R}_{d}^{(r)}$ with the
following properties:
(iii.0) we have

$\mathcal{R}_{0}^{(r)}=\sum_{\mu+\nu=2r+4}a_{\mu\nu}^{(r)}\left(\xi\right)\xi^{\mu}\bar{\xi}^{\nu}$

where $a_{\mu\nu}^{(r)}\in
C^{\infty}(\mathbb{C}),\overline{a_{\mu\nu}^{(r)}}=a_{\nu\mu}^{(r)}$
satisfying the following expansion with a sufficiently large integer
$M^{\star}>0$:

$a_{\mu\nu}^{(r)}(\xi)=\sum_{k=0}^{M^{\star}}\sum_{|\alpha+\beta|=2k}a^{(r)}_{\mu\nu\alpha\beta}\xi^{\alpha}\bar{\xi}^{\beta},$
(3.12)

(iii.1) we have

$\mathcal{R}_{1}^{(r)}=\sum_{\mu+\nu=2r+3}\xi^{\mu}\bar{\xi}^{\nu}\int_{\mathbb{R}^{3}}\mathbf{\Phi}_{\mu\nu}^{(r)}\left(x,\xi\right)\cdot\mathbf{f}(x)dx$

where the map

$\mathbb{C}\ni\xi\longmapsto\mathbf{\Phi}_{\mu\nu}^{(r)}(\cdot,\xi)\in\left(H^{n,s}\right)^{2}\text{
is
}C^{\infty},\quad\mathbf{\Phi}_{\mu\nu}^{(r)}=(\Phi_{\mu\nu}^{(r)},\overline{\Phi_{\nu\mu}^{(r)}})$

satisfying the following expansion:

$\mathbf{\Phi}_{\mu\nu}^{(r)}(\cdot,\xi)=\sum_{k=0}^{M^{\star}}\sum_{|\alpha+\beta|=2k}\mathbf{\Phi}^{(r)}_{\mu\nu\alpha\beta}(x)\xi^{\alpha}\bar{\xi}^{\beta}$
(3.13)

with
$\mathbf{\Phi}^{(r)}_{\mu\nu\alpha\beta}(x)\in\mathcal{S}\left(\mathbb{R}^{3},\mathbb{C}\right)$.
(iii.2-4) for $d=2,3,4$, we have

$\mathcal{R}_{d}^{(r)}=\int_{\mathbb{R}^{3}}F_{d}^{(r)}\left(x,z\right)[U(x)]^{d}dx+\sum_{k}\prod_{l=1}^{d}\int_{\mathbb{R}^{3}}\mathbf{\Lambda}_{dlk}^{(r)}(x,z)\cdot\mathbf{f}dx,$
(3.14)

where $F_{4}^{(r)}\equiv 1;$ for $d=2,3,$ $F_{d}^{(r)}(x,z)\in\mathbb{R}$ is a
linear combination of terms of the form

$\displaystyle\sum_{k=0}^{M^{\star}}\sum_{i=0}^{k}\sum_{\mu+\nu=4-d+2k-i}\xi^{\mu}\overline{\xi^{\nu}}\prod_{j=1}^{i}\int\mathbf{\Phi}^{ij}_{\mu\nu}(x)\cdot\mathbf{f}dx\Psi^{i}_{\mu\nu}(x),$
(3.15)

and
$\mathbf{\Lambda}_{dlk}^{(r)}(x,z)=(\Lambda_{dlk}^{(r)},\overline{\Lambda_{dlk}^{(r)}})\
(d=2,3,4)$ is a linear combination of terms of the form

$\displaystyle\sum_{k=0}^{M^{\star}}\sum_{i=0}^{k}\sum_{\mu+\nu=1+2k-i}\xi^{\mu}\overline{\xi^{\nu}}\prod_{j=1}^{i}\int\tilde{\mathbf{\Phi}}^{ij}_{\mu\nu}(x)\cdot\mathbf{f}dx\tilde{\Psi}^{i}_{\mu\nu}(x),$
(3.16)

with
$\mathbf{\Phi}^{ij}_{\mu\nu}(x),\tilde{\mathbf{\Phi}}^{ij}_{\mu\nu}(x),\Psi^{i}_{\mu\nu}(x),\tilde{\Psi}^{i}_{\mu\nu}(x)\in\mathcal{S}\left(\mathbb{R}^{3},\mathbb{C}\right),$
and the sum over index $k$ in (3.14) is a finite sum.
(iii.5) for $d=5$, we have

$\left\|\nabla_{z,\bar{z}}\mathcal{R}_{5}^{(r)}\right\|_{\left(\mathcal{P}^{n,s}\right)^{2}}\lesssim|\xi|^{M^{\star}}.$

######  Remark 3.4.

For $f\in X=W^{2,2}\left(\mathbb{R}^{3},\mathbb{C}\right)\cap
W^{2,1}\left(\mathbb{R}^{3},\mathbb{C}\right),$ since
$H^{k,s}\left(\mathbb{R}^{3},\mathbb{C}\right)\subset X$ for $k,s$ large, we
also have

$\left\|(\xi,f)-\mathcal{T}_{r}(\xi,f)\right\|_{\mathbb{C}\times
X}\lesssim\|(\xi,f)\|_{\mathcal{P}^{-n,-s}}^{3},$

###### Proof.

We prove Theorem 3.3 by induction. We note that with some slightly abuses of
notations, we denote $a$ with indexs as some constant, and denote $\Phi$ or
$\Psi$ with indexs as some Schwartz function, they may change line from line,
depending on the context. We also note that the sum with no upper index always
denotes finite sum.
(Step 0) First, when $r=0$, Theorem 3.3 holds with
$\mathcal{T}_{0}=I,Z^{(0)}=0,\mathcal{R}^{(0)}=H_{P}.$ And we have

$\displaystyle R_{0}^{(0)}$
$\displaystyle=\sum_{\mu+\nu=4}\xi^{\mu}\bar{\xi}^{\nu}\int_{\mathbb{R}^{3}}\frac{\varphi^{4}}{2\sqrt{\omega^{4}}}dx,$
$\displaystyle R_{1}^{(0)}$
$\displaystyle=\sum_{\mu+\nu=3}\xi^{\mu}\bar{\xi}^{\nu}\int_{\mathbb{R}^{3}}\frac{3\varphi^{3}}{2\sqrt{\omega^{3}}}\left(B^{-\frac{1}{2}}f+B^{-\frac{1}{2}}\bar{f}\right)dx,$
$\displaystyle R_{d}^{(0)}$
$\displaystyle=\int_{\mathbb{R}^{3}}F_{d}^{(0)}U^{d}dx,\quad
F_{d}^{(0)}=\sum_{\mu+\nu=4-d}\frac{C_{4}^{d}}{\sqrt{2}^{4-d}\sqrt{\omega^{4-d}}}\xi^{\mu}\bar{\xi}^{\nu}\varphi^{4-d}(d=2,3),F_{4}^{(0)}=1.$

Thus,
$a_{\mu\nu}^{(0)}\triangleq\int_{\mathbb{R}^{3}}\frac{\varphi^{4}}{2\sqrt{\omega^{4}}}dx$
and
$\mathbf{\Phi}_{\mu\nu}^{(0)}\triangleq\left(\frac{3}{2\sqrt{\omega^{3}}}B^{-\frac{1}{2}}\left(\varphi^{3}\right),\frac{3}{2\sqrt{\omega^{3}}}B^{-\frac{1}{2}}\left(\varphi^{3}\right)\right).$

(Step $r\to r+1$) Now we assume that the theorem holds for some $0\leq r\leq
2N$, we shall prove this for $r+1$. More precisely, define

$\displaystyle\mathcal{R}_{02}^{(r)}=\mathcal{R}_{0}^{(r)}-\sum_{\mu+\nu=2r+4}a_{\mu\nu}^{(r)}(0)\xi^{\mu}\bar{\xi}^{\nu},$
$\displaystyle\mathcal{R}_{12}^{(r)}=\mathcal{R}_{1}^{(r)}-\sum_{\mu+\nu=2r+3}\xi^{\mu}\bar{\xi}^{\nu}\int_{\mathbb{R}^{3}}\mathbf{\Phi}_{\mu\nu}^{(r)}(x,0)\cdot\mathbf{f}(x)dx.$

By (3.12) and (3.13), we have

$\displaystyle\mathcal{R}_{02}^{(r)}+\mathcal{R}_{12}^{(r)}=$
$\displaystyle\sum_{\mu+\nu=2r+6}a_{\mu\nu}^{(r+1)}(\xi)\xi^{\mu}\bar{\xi}^{\nu}$
$\displaystyle+\sum_{\mu+\nu=2r+5}\xi^{\mu}\bar{\xi}^{\nu}\int_{\mathbb{R}^{3}}\mathbf{\Phi}_{\mu\nu}^{(r+1)}(x,\xi)\cdot\mathbf{f}(x)dx$

where the coefficients
$a_{\mu\nu}^{(r+1)}(\xi),\mathbf{\Phi}_{\mu\nu}^{(r+1)}(x,\xi)$ satisfy
(3.12)-(3.13) respectively, with $r$ replaced by $r+1$.

Set

$\displaystyle K_{r+1}:=$
$\displaystyle\sum_{\mu+\nu=2r+4}a_{\mu\nu}^{(r)}(0)\xi^{\mu}\bar{\xi}^{\nu}+\sum_{\mu+\nu=2r+3}\xi^{\mu}\bar{\xi}^{\nu}\int_{\mathbb{R}^{3}}\mathbf{\Phi}_{\mu\nu}^{(r)}(x,0)\cdot\mathbf{f}(x)d,$

which is real-valued. Then, we solve the following homologic equation

$\displaystyle\left\\{H_{L},\chi_{r+1}\right\\}+Z_{r+1}=K_{r+1},$

with $Z_{r+1}$ in normal form. Thus,

$\displaystyle Z_{r+1}=$
$\displaystyle\sum_{\mu+\nu=2r+4,\mu=\nu}a_{\mu\mu}^{(r)}(0)|\xi|^{2\mu}+\sum_{\begin{subarray}{c}\mu+\nu=2r+3\\\
\omega\cdot(\mu-\nu)<-m\end{subarray}}\xi^{\mu}\bar{\xi}^{\nu}\int\Phi_{\mu\nu}^{(r)}(x,0)f(x)dx$
$\displaystyle+\sum_{\begin{subarray}{c}\mu+\nu=2r+3\\\
\omega\cdot(\mu-\nu)>m\end{subarray}}\xi^{\mu}\bar{\xi}^{\nu}\int\overline{\Phi_{\nu\mu}^{(r)}}(x,0)\bar{f}(x)dx,$

and

$\displaystyle\chi_{r+1}=$ $\displaystyle
i\sum_{\begin{subarray}{c}\mu+\nu=2r+4\\\ \omega\cdot(\mu-\nu)\neq
0\end{subarray}}\frac{a_{\mu\nu}^{(r)}(0)}{\omega\cdot(\mu-\nu)}\xi^{\mu}\bar{\xi}^{\nu}+i\sum_{\begin{subarray}{c}\mu+\nu=2r+3\\\
\omega\cdot(\mu-\nu)>-m\end{subarray}}\xi^{\mu}\bar{\xi}^{\nu}\int
R_{\nu\mu}\Phi_{\mu\nu}^{(r)}(x,0)fdx$
$\displaystyle-i\sum_{\begin{subarray}{c}\mu+\nu=2r+3\\\
\omega\cdot(\mu-\nu)<m\end{subarray}}\xi^{\mu}\bar{\xi}^{\nu}\int
R_{\mu\nu}\overline{\Phi_{\nu\mu}^{(r)}}(x,0)\bar{f}dx,$

where the operator

$R_{\mu\nu}:=(B-\omega\cdot(\mu-\nu))^{-1}.$

Let $\phi_{r+1}$ be the Lie transform generated by $\chi_{r+1}$, i.e.
$\phi_{r+1}=\phi_{r+1}^{t}|_{t=1}$, where

$\frac{d\phi_{r+1}^{t}}{dt}=X_{\chi_{r+1}}=(-i\partial_{\bar{\xi}}\chi_{r+1},-i\nabla_{\bar{f}}\chi_{r+1}).$

Then, for $z^{\prime}=(\xi^{\prime},f^{\prime})=\phi_{r+1}(\xi,f),$ Lemma 3.1
holds, i.e.

$\displaystyle\xi^{\prime}=\xi+\sum_{k=1}^{\infty}\sum_{i=0}^{k}\sum_{\mu+\nu=(2r+2)k+1-i}a_{i\mu\nu}\xi^{\mu}\bar{\xi}^{\nu}\prod_{j=1}^{i}\int\mathbf{\Phi}_{\mu\nu}^{ij}\cdot\mathbf{f}dx,$
(3.17) $\displaystyle
f^{\prime}=f+\sum_{k=1}^{\infty}\sum_{i=0}^{k-1}\sum_{\mu+\nu=(2r+2)k+1-i}\xi^{\mu}\bar{\xi}^{\nu}\prod_{j=1}^{i}\int\mathbf{\Lambda}_{\mu\nu}^{ij}\cdot\mathbf{f}dx\Psi_{\mu,\nu}^{i}.$
(3.18)

Recall that

$R^{(r)}=K_{r+1}+R^{(r)}_{02}+R^{(r)}_{12}+\sum_{d=2}^{5}R^{(r)}_{d},$

and $K_{r+1}=Z_{r+1}+\\{H_{L},\chi_{r+1}\\}$, we have

$\displaystyle H^{(r+1)}\triangleq$ $\displaystyle
H^{(r)}\circ\phi_{r+1}=H\circ(\mathcal{T}_{r}\circ\phi_{r+1})\equiv
H\circ\mathcal{T}_{r+1}$ $\displaystyle=$ $\displaystyle
H_{L}\circ\phi_{r+1}+Z^{(r)}\circ\phi_{r+1}+R^{(r)}\circ\phi_{r+1}$
$\displaystyle=$ $\displaystyle H_{L}+Z^{(r)}+Z_{r+1}$
$\displaystyle+[H_{L}\circ\phi_{r+1}-(H_{L}+\\{\chi_{r+1},H_{L}\\})]$ (3.19)
$\displaystyle+Z^{(r)}\circ\phi_{r+1}-Z^{(r)}$ (3.20)
$\displaystyle+(K_{r+1}\circ\phi_{r+1}-K_{r+1})$ (3.21)
$\displaystyle+(R^{(r)}_{02}+R^{(r)}_{12})\circ\phi_{r+1}$ (3.22)
$\displaystyle+\sum_{d=2}^{5}R^{(r)}_{d}\circ\phi_{r+1}.$ (3.23)

We define $Z^{(r+1)}=Z^{(r)}+Z_{r+1}$ in the normal form of order $2r+4$. For
the term (3.19), similar to the proof of Lemma 3.1, we have

$\displaystyle H_{L}\circ\phi_{r+1}-(H_{L}+\\{\chi_{r+1},H_{L}\\})$
$\displaystyle=$
$\displaystyle\sum_{k=2}^{\infty}\frac{1}{k!}\underbrace{\\{\chi_{r+1},\dots\\{\chi_{r+1}}_{k\text{
times}},H_{L}\\}\\}$ $\displaystyle=$
$\displaystyle\sum_{k=2}^{\infty}\sum_{i=0}^{k}\sum_{\mu+\nu=2(r+1)k+2-i}a_{i\mu\nu}\xi^{\mu}\bar{\xi}^{\nu}\prod_{j=1}^{i}\int\mathbf{\Phi}_{\mu\nu}^{ij}\cdot\mathbf{f}dx$
$\displaystyle=$
$\displaystyle\sum_{k=2}^{M^{*}}\left(\sum_{\mu+\nu=2(r+1)k+2}a_{0\mu\nu}\xi^{\mu}\bar{\xi}^{\nu}+\sum_{\mu+\nu=2(r+1)k+1}a_{1\mu\nu}\xi^{\mu}\bar{\xi}^{\nu}\int\mathbf{\Phi}^{11}_{\mu\nu}\cdot\mathbf{f}dx\right)$
$\displaystyle+\sum_{k=2}^{M^{*}}\sum_{i=2}^{k}\sum_{\mu+\nu=2(r+1)k+2-i}a_{i\mu\nu}\xi^{\mu}\bar{\xi}^{\nu}\prod_{j=1}^{i}\int\mathbf{\Phi}_{\mu\nu}^{ij}\cdot\mathbf{f}dx+\mathcal{O}(|\xi|^{M^{*}}).$

Thus (3.19) can be absorbed into $R^{(r+1)}_{0},R^{(r+1)}_{1}$,
$R^{(r+1)}_{2}$ and $R^{(r+1)}_{5}$.

The terms (3.20), (3.21) and (3.22) can be handled similarly.

For the term (3.23), denote $f^{\prime}=f+G_{f},U^{\prime}=U+G_{U}$, then for
$d=2,3,4$, we have

$\displaystyle R^{(r)}_{d}\circ\phi_{r+1}$
$\displaystyle=\int_{\mathbb{R}^{3}}F_{d}^{(r)}\left(x,z^{\prime}\right)(U+G_{U})^{d}dx+\sum_{k}\prod_{l=1}^{d}\int_{\mathbb{R}^{3}}\mathbf{\Lambda}_{dlk}^{(r)}(x,z^{\prime})\cdot\left(\mathbf{f}+\mathbf{G}_{f}\right)dx$
$\displaystyle=\sum_{j=0}^{d}\left[\int
F_{d}^{(r)}\left(x,z^{\prime}\right)U^{j}G_{U}^{d-j}dx+\sum_{k,l_{i}}\prod_{i=1}^{j}\int_{\mathbb{R}^{3}}\mathbf{\Lambda}_{d,l_{i},k}^{(r)}(x,z^{\prime})\cdot\mathbf{f}dx\prod_{l\neq
l_{i}}\int_{\mathbb{R}^{3}}\mathbf{\Lambda}_{d,l}^{(r)}(x,z^{\prime})\cdot\mathbf{G}_{f}dx\right]$
$\displaystyle:=\sum_{j=0}^{d}H_{dj}.$

By (3.18), we have

$\displaystyle
G_{f}=\sum_{k=1}^{\infty}\sum_{i=0}^{k-1}\sum_{\mu+\nu=(2r+2)k+1-i}\xi^{\mu}\bar{\xi}^{\nu}\prod_{j=1}^{i}\int\mathbf{\Lambda}_{\mu\nu}^{ij}\cdot\mathbf{f}dx\Psi_{\mu,\nu}^{i},\quad
G_{U}=(G_{f}+\overline{G_{f}})/\sqrt{2B}.$

Therefore, by (3.15), (3.16) and (3.17), we derive

$\displaystyle H_{d0}$ $\displaystyle=\int
F_{d}^{(r)}\left(x,z^{\prime}\right)G_{U}^{d}dx+\sum_{k}\prod_{l=1}^{d}\int_{\mathbb{R}^{3}}\mathbf{\Lambda}_{dlk}^{(r)}(x,z^{\prime})\cdot\mathbf{G}_{f}dx$
$\displaystyle=\sum_{k=0}^{M^{*}}\sum_{i=0}^{k}\sum_{\mu+\nu=4+(2r+2)d+2k-i}a_{i\mu\nu}\xi^{\mu}\bar{\xi}^{\nu}\sum\prod_{j=1}^{i}\int\mathbf{\Phi}_{\mu\nu}^{ij}\cdot\mathbf{f}dx+\mathcal{O}(|\xi|^{M^{*}}),$
$\displaystyle H_{d1}$ $\displaystyle=\int
F_{d}^{(r)}\left(x,z^{\prime}\right)UG_{U}^{d-1}dx+\sum_{k}\sum_{i=1}^{d}\int_{\mathbb{R}^{3}}\mathbf{\Lambda}_{dik}^{(r)}(x,z^{\prime})\cdot\mathbf{f}dx\prod_{l\neq
i}\int_{\mathbb{R}^{3}}\mathbf{\Lambda}_{dlk}^{(r)}(x,z^{\prime})\cdot\mathbf{G}_{f}dx$
$\displaystyle=\sum_{k=0}^{M^{*}}\sum_{i=0}^{k}\sum_{\mu+\nu=3+(2r+2)(d-1)+2k-i}a_{i\mu\nu}\xi^{\mu}\bar{\xi}^{\nu}\sum\prod_{j=1}^{i+1}\int\mathbf{\Phi}_{\mu\nu}^{ij}\cdot\mathbf{f}dx+\mathcal{O}(|\xi|^{M^{*}}),$

and for $2\leq j\leq d$,

$\displaystyle H_{dj}$ $\displaystyle=\int
F_{d}^{(r)}\left(x,z^{\prime}\right)U^{j}G_{U}^{d-j}dx+\sum_{k,l_{i}}\prod_{i=1}^{j}\int_{\mathbb{R}^{3}}\mathbf{\Lambda}_{d,l_{i},k}^{(r)}(x,z^{\prime})\cdot\mathbf{f}dx\prod_{l\neq
l_{i}}\int_{\mathbb{R}^{3}}\mathbf{\Lambda}_{dlk}^{(r)}(x,z^{\prime})\cdot\mathbf{G}_{f}dx$
$\displaystyle=\int_{\mathbb{R}^{3}}F_{j}^{(r+1)}\left(x,z\right)U^{j}dx+\sum_{k}\prod_{l=1}^{j}\int_{\mathbb{R}^{3}}\mathbf{\Lambda}_{jlk}^{(r+1)}(x,z)\cdot\mathbf{f}dx+\mathcal{O}(|\xi|^{M^{*}})$

where

$\displaystyle
F_{j}^{(r+1)}=F_{d}^{(r)}\left(x,z^{\prime}\right)G_{U}^{d-j}-\mathcal{O}(|\xi|^{M^{*}})=\sum_{k=0}^{M^{*}}\sum_{i=0}^{k}\sum_{\mu+\nu=4-j+(2r+2)(d-j)+2k-i}a_{i\mu\nu}\xi^{\mu}\bar{\xi}^{\nu}\sum\prod_{l=1}^{i}\int\mathbf{\Phi}_{\mu\nu}^{il}\cdot\mathbf{f}dx\psi^{l}_{\mu\nu}(x),$

note that $F_{4}^{(r+1)}\equiv 1.$ Thus $H_{dj}$ can be absorbed into
$R^{(r+1)}$. Finally, it is direct to see $R^{(r)}_{5}\circ\phi_{r+1}$ can be
absorbed into $R^{(r+1)}_{5}.$ ∎

##  4 Isolation of the Key Resonant Terms

Apply Theorem 3.3 for $r=2N$, we obtain the new Hamiltonian

$\displaystyle
H=H_{L}(\xi,\mathbf{f})+Z_{0}(\xi)+Z_{1}(\xi,\mathbf{f})+\mathcal{R},$

where

$Z_{1}(\xi,\mathbf{f}):=\langle G,f\rangle+\langle\bar{G},\bar{f}\rangle,$
$G:=\sum_{(\mu,\nu)\in
M}\xi^{\mu}\bar{\xi}^{\nu}\Phi_{\mu\nu}(x),\Phi_{\mu\nu}\in\mathcal{S}\left(\mathbb{R}^{3},\mathbb{C}\right),$

with

$M=\\{(\mu,\nu)\mid\mu+\nu=2r+1,0\leq r\leq 2N,\omega(\mu-\nu)<-m\\}.$

Then, the corresponding Hamiltonian equations are

$\displaystyle\dot{f}$
$\displaystyle=-\mathrm{i}(Bf+\bar{G})-\mathrm{i}\partial_{\bar{f}}\mathcal{R},$
(4.1) $\displaystyle\dot{\xi}$
$\displaystyle=-\mathrm{i}\omega\xi-\mathrm{i}\partial_{\bar{\xi}}Z_{0}-\mathrm{i}\left\langle\partial_{\bar{\xi}}G,f\right\rangle-\mathrm{i}\left\langle\partial_{\bar{\xi}}\bar{G},\bar{f}\right\rangle-\mathrm{i}\partial_{\bar{\xi}}\mathcal{R}.$
(4.2)

Define

$M_{1}=\\{(\mu,\nu)\mid\mu+\nu=2N+1,\quad\omega(\mu-\nu)<-m\\},$

and

$G=\sum_{(\mu,\nu)\in
M}\xi^{\mu}\bar{\xi}^{\nu}\Phi_{\mu\nu}=\sum_{(\mu,\nu)\in
M_{1}}\xi^{\mu}\bar{\xi}^{\nu}\Phi_{\mu\nu}+\sum_{(\mu,\nu)\in M\backslash
M_{1}}\xi^{\mu}\bar{\xi}^{\nu}\Phi_{\mu\nu}:=\mathcal{M}_{G}+\mathcal{R}_{G}.$

Since $(2N-1)\omega<m<(2N+1)\omega$, we have

$\mathcal{M}_{G}=\bar{\xi}^{2N+1}\Phi_{0,2N+1}.$

Let $\eta=e^{\mathrm{i}\omega t}\xi,$ by (4.1) and Duhamel’s formula, we have

$\displaystyle f(t)$
$\displaystyle=e^{-\mathrm{i}Bt}f(0)+\int_{0}^{t}e^{-\mathrm{i}B(t-s)}(-\mathrm{i}\bar{G}-\mathrm{i}\partial_{\bar{f}}\mathcal{R})ds$
(4.3)
$\displaystyle=e^{-\mathrm{i}Bt}f(0)-\mathrm{i}e^{-\mathrm{i}Bt}\sum_{(\mu,\nu)\in
M}\int_{0}^{t}e^{\mathrm{i}(B-\omega(\nu-\mu))s}\eta^{\nu}\bar{\eta}^{\mu}\bar{\Phi}_{\mu\nu}ds-\mathrm{i}\int_{0}^{t}e^{-\mathrm{i}B(t-s)}\partial_{\bar{f}}\mathcal{R}ds$
$\displaystyle:=\mathcal{M}_{f}+\mathcal{R}_{f},$ (4.4)

where

$\displaystyle\mathcal{M}_{f}$ $\displaystyle=-\sum_{(\mu,\nu)\in
M_{1}}e^{-\mathrm{i}\omega(\nu-\mu)t}\eta^{\nu}\bar{\eta}^{\mu}(B-\omega(\nu-\mu)-\mathrm{i}0)^{-1}\bar{\Phi}_{\mu\nu},$
$\displaystyle=-\xi^{2N+1}(B-(2N+1)\omega-i0)^{-1}\bar{\Phi}_{0,2N+1}$

is the main term and

$\displaystyle\mathcal{R}_{f}=$ $\displaystyle
e^{-\mathrm{i}Bt}f(0)-\mathrm{i}\int_{0}^{t}e^{-\mathrm{i}B(t-s)}\partial_{\bar{f}}\mathcal{R}ds$
$\displaystyle+\sum_{(\mu,\nu)\in
M_{1}}\eta^{\nu}(0)\bar{\eta}^{\mu}(0)e^{-\mathrm{i}Bt}(B-\omega(\nu-\mu)-\mathrm{i}0)^{-1}\bar{\Phi}_{\mu\nu}$
$\displaystyle+e^{-\mathrm{i}Bt}\sum_{(\mu,\nu)\in
M_{1}}\int_{0}^{t}e^{\mathrm{i}(B-\omega(\nu-\mu))s}\frac{d}{ds}(\eta^{\nu}\bar{\eta}^{\mu})(B-\omega(\nu-\mu)-\mathrm{i}0)^{-1}\bar{\Phi}_{\mu\nu}ds$
$\displaystyle-\mathrm{i}e^{-\mathrm{i}Bt}\sum_{(\mu,\nu)\in M\backslash
M_{1}}\int_{0}^{t}e^{\mathrm{i}(B-\omega(\nu-\mu))s}\eta^{\nu}\bar{\eta}^{\mu}\bar{\Phi}_{\mu\nu}ds$

are error terms, which will be estimated later, see Section 6. Substitute
(4.4) into (4.2), we obtain

$\dot{\xi}=-\mathrm{i}\omega\xi-\mathrm{i}\partial_{\bar{\xi}}Z_{0}-\mathrm{i}\left\langle\partial_{\bar{\xi}}\mathcal{M}_{G},\mathcal{M}_{f}\right\rangle-\mathrm{i}\left\langle\partial_{\bar{\xi}}\bar{\mathcal{M}}_{G},\bar{\mathcal{M}}_{f}\right\rangle+\mathcal{R}_{\xi},$
(4.5)

where

$\mathcal{R}_{\xi}=-\mathrm{i}\left\langle\partial_{\bar{\xi}}\mathcal{R}_{G},\mathcal{M}_{f}\right\rangle-\mathrm{i}\left\langle\partial_{\bar{\xi}}\bar{\mathcal{R}}_{G},\bar{\mathcal{M}}_{f}\right\rangle-\mathrm{i}\left\langle\partial_{\bar{\xi}}G,\mathcal{R}_{f}\right\rangle-\mathrm{i}\left\langle\partial_{\bar{\xi}}\bar{G},\bar{\mathcal{R}}_{f}\right\rangle-\mathrm{i}\partial_{\bar{\xi}}\mathcal{R}$
(4.6)

can be treated as error terms. Hence, we compute

$\displaystyle\frac{d}{dt}|\xi|^{2}$
$\displaystyle=2\operatorname{Re}(\bar{\xi}\dot{\xi})$
$\displaystyle=-2(2N+1)|\xi|^{4N+2}\operatorname{Im}\langle\Phi_{0,2N+1},(B-(2N+1)\omega-i0)^{-1}\bar{\Phi}_{0,2N+1}\rangle+2Re(\bar{\xi}\mathcal{R}_{\xi}).$

Note that by using Plemelji formula

$\frac{1}{x\mp i0}=\operatorname{P.V}\frac{1}{x}\pm\mathrm{i}\pi\delta(x),$

we have

$\displaystyle\gamma:=$
$\displaystyle(2N+1)\operatorname{Im}\langle\Phi_{0,2N+1},(B-(2N+1)\omega-i0)^{-1}\bar{\Phi}_{0,2N+1}\rangle$
$\displaystyle=$
$\displaystyle(2N+1)\langle\Phi_{0,2N+1},\delta(B-(2N+1)\omega)\bar{\Phi}_{0,2N+1}\rangle$
$\displaystyle\geq$ $\displaystyle 0,$

and

$\frac{d}{dt}|\xi|^{2}=-2\gamma|\xi|^{4N+2}+2Re(\bar{\xi}\mathcal{R}_{\xi}).$
(4.7)

Throughout this paper, we assume the following non-degenerate assumption.

######  Assumption 4.1 (Fermi’s Golden Rule).

$\gamma>0.$

######  Remark 4.2.

The Fermi’s Golden Rule condition implies that
$|\xi|\approx\frac{|\xi_{0}|}{(1+|\xi_{0}|^{4N}t)^{\frac{1}{4N}}}$ if we could
neglect the error term $2Re(\bar{\xi}\mathcal{R}_{\xi})$, which will be
justified in Section 6 .

##  5 Asymptotic Behavior

In this section, we derive the asymptotic behavior of $\xi$ and $f$. Before
proceeding, we introduce some useful notations. Let $T>0$ be fixed, denote
$\langle t\rangle=(1+t^{2})^{1/2}$ and

$\begin{gathered}{[\xi]_{\frac{1}{4N}}(T)=\sup_{0\leq t\leq
T}\langle|\xi_{0}|^{4N}t\rangle^{\frac{1}{4N}}|\xi(t)|},\end{gathered}$

We define the norm $\|\cdot\|_{X}$ as

$\|f\|_{X}=\|f\|_{W^{2,2}}+\|f\|_{W^{2,1}}.$

###  5.1 Dynamics of $\xi$

The following theorem allows to treat $\mathcal{R}_{\xi}$ perturbatively in
the dynamics of $\xi$:

######  Theorem 5.1.

Suppose that $\gamma>0$ and the error term $\mathcal{R}_{\xi}$ satisfies

$|\mathcal{R}_{\xi}(t)|\leq
Q_{0}(1+4N\gamma|\xi_{0}|^{4N}t)^{-\frac{4N+1}{4N}-\delta}$

for some small constant $Q_{0}$ and $\delta>0$, then

$|\xi(t)|^{4N}\leq(1+4N\gamma|\xi_{0}|^{4N}t)^{-1}\left(|\xi_{0}|^{4N}+\frac{C(\gamma^{-1/2}Q_{0}^{1/2}|\xi_{0}|^{\frac{4N-1}{2}}+\gamma^{-\frac{4N}{4N+1}}Q_{0}^{\frac{4N}{4N+1}})}{(1+4N\gamma|\xi_{0}|^{4N}t)^{\delta/2}}\right)$
(5.1)

for some absolute constant $C$. In addition, if
$Q_{0}=O(|\xi_{0}|^{4N+1+\epsilon})$ and $|\xi_{0}|$ is sufficiently small,
then

$|\xi(t)|^{4N}\geq(1+4N\gamma|\xi_{0}|^{4N}t)^{-1}\left(|\xi_{0}|^{4N}-\frac{|\xi_{0}|^{4N+\epsilon}}{(1+4N\gamma|\xi_{0}|^{4N}t)^{\delta}}\right).$
(5.2)

###### Proof.

We use standard comparison theorem to prove Theorem 5.1. Define
$r=|\xi|^{4N}$, by (4.7) we have

$-4N\gamma r^{2}-4N|\mathcal{R}_{\xi}|r^{\frac{4N-1}{4N}}\leq
r^{\prime}\leq-4N\gamma r^{2}+4N|\mathcal{R}_{\xi}|r^{\frac{4N-1}{4N}},$ (5.3)

Define

$h(t)=(1+4N\gamma r_{0}t)^{-1}(r_{0}+C_{0}(1+4N\gamma r_{0}t)^{-\delta/2}),$

then

$h^{\prime}(t)=-\frac{4N\gamma r_{0}^{2}}{(1+4N\gamma
r_{0}t)^{2}}-\frac{4N(1+\delta/2)\gamma r_{0}C_{0}}{(1+4N\gamma
r_{0}t)^{2+\delta/2}}.$

Hence,

$\displaystyle h^{\prime}+4N\gamma h^{2}$
$\displaystyle=\frac{4N(1-\delta/2)\gamma r_{0}C_{0}}{(1+4N\gamma
r_{0}t)^{2+\delta/2}}+\frac{4N\gamma C_{0}^{2}}{(1+4N\gamma
r_{0}t)^{2+\delta}}$ $\displaystyle\geq\frac{4N\gamma C_{0}^{2}}{(1+4N\gamma
r_{0}t)^{2+\delta}}.$

Note that

$h^{\frac{4N-1}{4N}}\leq(1+4N\gamma
r_{0}t)^{-\frac{4N-1}{4N}}(r_{0}^{\frac{4N-1}{4N}}+C_{0}^{\frac{4N-1}{4N}}(1+4N\gamma
r_{0}t)^{-\frac{(4N-1)\delta}{8N}}),$

then

$4N|\mathcal{R}_{\xi}|h^{\frac{4N-1}{4N}}\leq\frac{4NQ_{0}r_{0}^{\frac{4N-1}{4N}}+4NQ_{0}C_{0}^{\frac{4N-1}{4N}}}{(1+4N\gamma
r_{0}t)^{2+\delta}}.$

Hence, if we choose $C_{0}$ such that

$4N\gamma C_{0}^{2}\geq
8NQ_{0}r_{0}^{\frac{4N-1}{4N}}+8NQ_{0}C_{0}^{\frac{4N-1}{4N}},$ (5.4)

then

$h^{\prime}+4N\gamma h^{2}>4N|\mathcal{R}_{\xi}|h^{\frac{4N-1}{4N}}.$ (5.5)

By comparing (5.3) and (5.5) and note that $r_{0}\leq h_{0}$, we get $r(t)\leq
h(t)$ and the desired estimate is proved. To achieve the condition (5.4), it
suffices to choose
$C_{0}=C(\gamma^{-1/2}Q_{0}^{1/2}r_{0}^{\frac{4N-1}{8N}}+\gamma^{-\frac{4N}{4N+1}}Q_{0}^{\frac{4N}{4N+1}})$
for some large constant $C$.

For the lower bound of $|\xi|$, note that

$r^{\prime}\geq-4N\gamma r^{2}-4N|\mathcal{R}_{\xi}|r^{\frac{4N-1}{4N}}.$
(5.6)

Choose

$\tilde{h}(t)=(1+4Nr_{0}\gamma
t)^{-1}\left(r_{0}-r_{0}^{(1+\epsilon/8N)}(1+4Nr_{0}\gamma
t)^{-\delta}\right),$

similarly we compute

$\displaystyle\tilde{h}^{\prime}+4N\gamma\tilde{h}^{2}$
$\displaystyle=-\frac{4N\gamma(1-\delta)r_{0}^{2+\frac{\epsilon}{8N}}}{(1+4N\gamma
r_{0}t)^{2+\delta}}+\frac{4N\gamma r_{0}^{2+\frac{\epsilon}{4N}}}{(1+4N\gamma
r_{0}t)^{2+2\delta}}$
$\displaystyle\lesssim-\frac{r_{0}^{2+\frac{\epsilon}{8N}}}{(1+4N\gamma
r_{0}t)^{2+\delta}},$

and

$\displaystyle\tilde{h}^{\frac{4N-1}{4N}}<(1+4N\gamma
r_{0}t)^{-\frac{4N-1}{4N}}r_{0}^{\frac{4N-1}{4N}},$

hence

$\displaystyle
4N|\mathcal{R}_{\xi}|\tilde{h}^{\frac{4N-1}{4N}}\lesssim\frac{r_{0}^{2+\frac{\epsilon}{4N}}}{(1+4N\gamma
r_{0}t)^{2+\delta}}.$

Therefore if we choose $|\xi_{0}|$ to be sufficiently small, then

$\tilde{h}^{\prime}<-4N\gamma\tilde{h}^{2}-4N|\mathcal{R}_{\xi}|\tilde{h}^{\frac{4N-1}{4N}}.$
(5.7)

Using comparison theorem we get $r\geq\tilde{h}.$ ∎

###  5.2 Dynamics of $f$

By (4.3), $f$ can be decomposed as

$\displaystyle f(t)$
$\displaystyle=e^{-\mathrm{i}Bt}f(0)+\int_{0}^{t}e^{-\mathrm{i}B(t-s)}(-\mathrm{i}\bar{G}-\mathrm{i}\partial_{\bar{f}}\mathcal{R})ds.$

Denote

$\hat{\mathcal{R}}_{f}:=-\mathrm{i}\int_{0}^{t}e^{-\mathrm{i}B(t-s)}\partial_{\bar{f}}\mathcal{R}ds.$

######  Proposition 5.2.

Let $\sigma>\frac{5}{2}$, suppose that the error term $\hat{\mathcal{R}}_{f}$
satisfies

$\displaystyle\|B^{-1/2}\hat{\mathcal{R}}_{f}\|_{L^{8}}\lesssim\langle
t\rangle^{-\frac{2N+1}{4N}}\mathcal{P}_{1}+\langle|\xi_{0}|^{4N}t\rangle^{-\frac{2N+1}{4N}}\mathcal{P}_{2},$
(5.8) $\displaystyle\|\langle
x\rangle^{-\sigma}B^{1/2}\hat{\mathcal{R}}_{f}\|_{L^{4}}\lesssim\langle
t\rangle^{-\frac{1}{2}}\mathcal{P}_{3}+\langle|\xi_{0}|^{4N}t\rangle^{-\frac{1}{2}}\mathcal{P}_{4},$
(5.9)

for some constants
$\mathcal{P}_{1},\mathcal{P}_{2},\mathcal{P}_{3},\mathcal{P}_{4}$, then the
following estimates hold

$\displaystyle\|B^{-1/2}f\|_{L^{8}}\lesssim\langle
t\rangle^{-\frac{2N+1}{4N}}\Big{(}\|B^{-1/2}f(0)\|_{X}+\mathcal{P}_{1}\Big{)}+\langle|\xi_{0}|^{4N}t\rangle^{-\frac{2N+1}{4N}}\Big{(}[\xi]^{2N+1}_{\frac{1}{4N}}+\mathcal{P}_{2}\Big{)},$
$\displaystyle\|\langle x\rangle^{-\sigma}B^{1/2}f\|_{L^{4}}\lesssim\langle
t\rangle^{-\frac{1}{2}}\Big{(}\|B^{-1/2}f(0)\|_{X}+\mathcal{P}_{3}\Big{)}+\langle|\xi_{0}|^{4N}t\rangle^{-\frac{1}{2}}\Big{(}[\xi]^{2N+1}_{\frac{1}{4N}}+\mathcal{P}_{4}\Big{)}.$

###### Proof.

By Lemma 2.1 (choosing $l=0,p^{\prime}=8,\theta=1$), we have

$\displaystyle\|B^{-1/2}e^{-\mathrm{i}Bt}f(0)\|_{L^{8}}\lesssim\langle
t\rangle^{-9/8}\|B^{-1/2}f(0)\|_{W^{2,\frac{8}{7}}}\lesssim\langle
t\rangle^{-9/8}\|B^{-1/2}f(0)\|_{X}.$

Similarly, (choosing $l=0,p^{\prime}=4,\theta=0$)

$\displaystyle\|B^{1/2}e^{-\mathrm{i}Bt}f(0)\|_{L^{4}}\lesssim\langle
t\rangle^{-1/2}\|B^{1/2}f(0)\|_{W^{1,\frac{4}{3}}}\lesssim\langle
t\rangle^{-1/2}\|B^{-1/2}f(0)\|_{X}.$

Thus, it suffices to prove that

$\displaystyle\left\|B^{-1/2}\int_{0}^{t}e^{-\mathrm{i}B(t-s)}\bar{G}ds\right\|_{L^{8}}\lesssim\langle|\xi_{0}|^{4N}t\rangle^{-\frac{2N+1}{4N}}[\xi]^{2N+1}_{\frac{1}{4N}},$
(5.10)

and

$\displaystyle\left\|\langle
x\rangle^{-\sigma}B^{1/2}\int_{0}^{t}e^{-\mathrm{i}B(t-s)}\bar{G}ds\right\|_{L^{4}}\lesssim\langle|\xi_{0}|^{4N}t\rangle^{-\frac{2N+1}{4N}}[\xi]^{2N+1}_{\frac{1}{4N}}.$
(5.11)

Note that by definition,

$G=\sum_{(\mu,\nu)\in
M}\xi^{\mu}\bar{\xi}^{\nu}\Phi_{\mu\nu}(x),\Phi_{\mu\nu}\in\mathcal{S}\left(\mathbb{R}^{3},\mathbb{C}\right),$

with $M=\\{(\mu,\nu)\mid\mu+\nu=2r+1,0\leq r\leq 2N,\omega(\mu-\nu)<-m\\}.$
Since $(2N-1)\omega<m<(2N+1)\omega$, we have

$\mu+\nu\geq 2N+1,$

which implies (5.10) and (5.11) by using Lemma A.1 in Appendix. ∎

##  6 Error Estimates

In this section, we estimate error terms $\hat{\mathcal{R}}_{f}$ and
$\mathcal{R}_{\xi}$ by using the asymptotic behavior of $\xi$ and $f$. Once
this was achieved, we can use bootstrap arguments to finish our proof, see
Section 7. To proceed, we assume that there exist positive constants
$A_{f}(T),B_{f}(T),C_{f}(T),D_{f}(T)$ such that for $0\leq t\leq T,$ it holds
that

$\displaystyle\|B^{-1/2}f\|_{L^{8}}\lesssim\langle
t\rangle^{-\frac{2N+1}{4N}}A_{f}(T)+\langle|\xi_{0}|^{4N}t\rangle^{-\frac{2N+1}{4N}}B_{f}(T),$
(6.1) $\displaystyle\|\langle
x\rangle^{-\sigma}B^{1/2}f\|_{L^{4}}\lesssim\langle
t\rangle^{-\frac{1}{2}}C_{f}(T)+\langle|\xi_{0}|^{4N}t\rangle^{-\frac{1}{2}}D_{f}(T).$
(6.2)

Recall that

$\hat{\mathcal{R}}_{f}=-\mathrm{i}\int_{0}^{t}e^{-\mathrm{i}B(t-s)}\partial_{\bar{f}}\mathcal{R}ds,$

we first prove estimates of
$\partial_{\bar{f}}\mathcal{R}=\sum_{d=1}^{5}\partial_{\bar{f}}\mathcal{R}_{d}.$

######  Lemma 6.1.

The following estimates hold:

$\displaystyle\|B^{-1/2}\partial_{\bar{f}}\mathcal{R}_{1}\|_{W^{2,8/7}}$
$\displaystyle\lesssim\langle|\xi_{0}|^{4N}t\rangle^{-\frac{4N+3}{4N}}[\xi]^{4N+3}_{\frac{1}{4N}},$
$\displaystyle\|B^{-1/2}\partial_{\bar{f}}\mathcal{R}_{2}\|_{W^{2,8/7}}$
$\displaystyle\lesssim\langle|\xi_{0}|^{4N}t\rangle^{-\frac{2}{4N}}[\xi]^{2}_{\frac{1}{4N}}\Big{(}\|B^{-1/2}f\|_{L^{8}}+\|\langle
x\rangle^{-\sigma}B^{1/2}f\|_{L^{4}}\Big{)},$
$\displaystyle\|B^{-1/2}\partial_{\bar{f}}\mathcal{R}_{3}\|_{W^{2,8/7}}$
$\displaystyle\lesssim\langle|\xi_{0}|^{4N}t\rangle^{-\frac{1}{4N}}[\xi]_{\frac{1}{4N}}\|B^{-1/2}f\|_{L^{8}}\Big{(}\|B^{-1/2}f\|_{L^{8}}+\|\langle
x\rangle^{-\sigma}B^{1/2}f\|_{L^{4}}\Big{)},$
$\displaystyle\|B^{-1/2}\partial_{\bar{f}}\mathcal{R}_{4}\|_{W^{2,8/7}}$
$\displaystyle\lesssim\|B^{1/2}f\|_{L^{2}}^{4/3}\|B^{-1/2}f\|_{L^{8}}^{5/3}$
$\displaystyle\|B^{-1/2}\partial_{\bar{f}}\mathcal{R}_{5}\|_{W^{2,8/7}}$
$\displaystyle\lesssim\langle|\xi_{0}|^{4N}t\rangle^{-\frac{M^{\star}}{4N}}[\xi]^{M^{\star}}_{\frac{1}{4N}}.$

###### Proof.

The estimates of $\mathcal{R}_{1}$ and $\mathcal{R}_{5}$ are direct. For
$\mathcal{R}_{d},d=2,3,$ by Theorem 3.3, $F^{(2N)}_{d}(x,z),$ are linear
combinations of terms of the form
$\xi^{\mu}\bar{\xi}^{\nu}\prod_{j}^{i}\int\mathbf{\Phi}_{j}\cdot\mathbf{f}dx\Psi,$
where $\mu+\nu\geq 4-d+2k-i,0\leq i\leq
k,\mathbf{\Phi}_{j},\Psi\in\mathcal{S}\left(\mathbb{R}^{3},\mathbb{C}\right).$
And $\Lambda^{(2N)}_{dlk}(x,z)$ are linear combinations of terms of the form
$\xi^{\mu}\bar{\xi}^{\nu}\prod_{j}^{i}\int\mathbf{\Phi}_{j}\cdot\mathbf{f}dx\Psi,$
where $\mu+\nu\geq 1+2k-i,0\leq i\leq
k,\mathbf{\Phi}_{j},\Psi\in\mathcal{S}\left(\mathbb{R}^{3},\mathbb{C}\right).$
Hence, $\partial_{\bar{f}}\mathcal{R}_{d}$ are linear combinations of terms of
forms

$\xi^{\mu}\bar{\xi}^{\nu}\prod_{j}^{i}\int\mathbf{\Phi}_{j}\cdot\mathbf{f}dx\int\Psi
U^{d}dx\Psi^{\prime},\quad\mu+\nu\geq 4-d+2(k+1)-i,0\leq i\leq k,$

or

$\xi^{\mu}\bar{\xi}^{\nu}\prod_{j}^{i}\int\mathbf{\Phi}_{j}\cdot\mathbf{f}dxB^{-1/2}\left(\Psi
U^{d-1}\right),\quad\mu+\nu\geq 4-d+2k-i,0\leq i\leq k,$

or

$\xi^{\mu}\bar{\xi}^{\nu}\prod_{j}^{d-1+i}\int\mathbf{\Phi}_{j}\cdot\mathbf{f}dx\Psi,\quad\mu+\nu\geq
4-d+2k-i,0\leq i\leq k.$

In each form, $\mu+\nu\geq 4-d$ and $f$ has at least order $d-1.$ Then the
rest follows directly by Hölder’s inequality and Leibnitz rule. For $d=4,$ we
have

$\displaystyle\|B^{-1}\left(\left(B^{-1/2}f+B^{-1/2}\bar{f}\right)^{3}\right)\|_{W^{2,8/7}}$
$\displaystyle\lesssim$
$\displaystyle\|\left(B^{-1/2}f+B^{-1/2}\bar{f}\right)^{3}\|_{W^{1,8/7}}$
$\displaystyle\lesssim$
$\displaystyle\|B^{-1/2}f\|_{W^{1,2}}^{4/3}\|B^{-1/2}f\|_{L^{8}}^{5/3}.$

∎

######  Corollary 6.2.

Assuming (6.1) and (6.2), it holds that

$\displaystyle\|B^{-1/2}\partial_{\bar{f}}\mathcal{R}_{1}\|_{W^{2,8/7}}$
$\displaystyle\lesssim\langle|\xi_{0}|^{4N}t\rangle^{-\frac{4N+3}{4N}}[\xi]^{4N+3}_{\frac{1}{4N}},$
$\displaystyle\|B^{-1/2}\partial_{\bar{f}}\mathcal{R}_{2}\|_{W^{2,8/7}}$
$\displaystyle\lesssim\langle|\xi_{0}|^{4N}t\rangle^{-\frac{1}{4N}}\langle
t\rangle^{-\frac{2N+1}{4N}}[\xi]^{2}_{\frac{1}{4N}}\Big{(}A_{f}+|\xi_{0}|^{-1}C_{f}\Big{)}+\langle|\xi_{0}|^{4N}t\rangle^{-\frac{2N+2}{4N}}[\xi]^{2}_{\frac{1}{4N}}\big{(}B_{f}+D_{f}\big{)},$
$\displaystyle\|B^{-1/2}\partial_{\bar{f}}\mathcal{R}_{3}\|_{W^{2,8/7}}$
$\displaystyle\lesssim\langle|\xi_{0}|^{4N}t\rangle^{-\frac{1}{4N}}\langle
t\rangle^{-\frac{2N+1}{4N}}[\xi]_{\frac{1}{4N}}A_{f}\big{(}A_{f}+C_{f}+D_{f}\big{)}+\langle|\xi_{0}|^{4N}t\rangle^{-\frac{2N+2}{4N}}[\xi]_{\frac{1}{4N}}B_{f}\big{(}B_{f}+C_{f}+D_{f}\big{)},$
$\displaystyle\|B^{-1/2}\partial_{\bar{f}}\mathcal{R}_{4}\|_{W^{2,8/7}}$
$\displaystyle\lesssim\langle
t\rangle^{-\frac{2N+2}{4N}}\|B^{1/2}f\|_{L^{2}}^{4/3}A_{f}^{5/3}+\langle|\xi_{0}|^{4N}t\rangle^{-\frac{2N+2}{4N}}\|B^{1/2}f\|_{L^{2}}^{4/3}B_{f}^{5/3},$
$\displaystyle\|B^{-1/2}\partial_{\bar{f}}\mathcal{R}_{5}\|_{W^{2,8/7}}$
$\displaystyle\lesssim\langle|\xi_{0}|^{4N}t\rangle^{-\frac{M^{\star}}{4N}}[\xi]^{M^{\star}}_{\frac{1}{4N}}.$

###### Proof.

The proof is direct by substituting (6.1) and (6.2) into Lemma 6.1, we have
used the fact that
$\langle|\xi_{0}|^{4N}t\rangle^{-\frac{1}{4N}}|\xi_{0}|\leq\langle
t\rangle^{-\frac{1}{4N}}$. ∎

Similarly, we have the following lemma. The proof also relies on the explicit
forms of $\partial_{\bar{f}}\mathcal{R}_{d}$ and standard Hölder’s inequality.
Here we omit the details.

######  Lemma 6.3.

Let $\sigma>\frac{5}{2}$, the following estimates hold:

$\displaystyle\|\langle
x\rangle^{\sigma}B^{1/2}\partial_{\bar{f}}\mathcal{R}_{1}\|_{W^{1,4/3}}\lesssim\langle|\xi_{0}|^{4N}t\rangle^{-\frac{4N+3}{4N}}[\xi]^{4N+3}_{\frac{1}{4N}},$
$\displaystyle\|\langle
x\rangle^{\sigma}B^{1/2}\partial_{\bar{f}}\mathcal{R}_{2}\|_{W^{1,4/3}}\lesssim\langle|\xi_{0}|^{4N}t\rangle^{-\frac{2}{4N}}[\xi]^{2}_{\frac{1}{4N}}\Big{(}\|B^{-1/2}f\|_{L^{8}}+\|\langle
x\rangle^{-\sigma}B^{1/2}f\|_{L^{4}}\Big{)},$ $\displaystyle\|\langle
x\rangle^{\sigma}B^{1/2}\partial_{\bar{f}}\mathcal{R}_{3}\|_{W^{1,4/3}}\lesssim\langle|\xi_{0}|^{4N}t\rangle^{-\frac{1}{4N}}[\xi]_{\frac{1}{4N}}\|B^{-1/2}f\|_{L^{8}}\Big{(}\|B^{-1/2}f\|_{L^{8}}+\|\langle
x\rangle^{-\sigma}B^{1/2}f\|_{L^{4}}\Big{)},$
$\displaystyle\|B^{1/2}\partial_{\bar{f}}\mathcal{R}_{4}\|_{W^{1,4/3}}\lesssim\|B^{1/2}f\|_{L^{2}}\|B^{-1/2}f\|_{L^{8}}^{2},$
$\displaystyle\|\langle
x\rangle^{\sigma}B^{1/2}\partial_{\bar{f}}\mathcal{R}_{5}\|_{W^{1,4/3}}\lesssim\langle|\xi_{0}|^{4N}t\rangle^{-\frac{M^{\star}}{4N}}[\xi]^{M^{\star}}_{\frac{1}{4N}},$

######  Corollary 6.4.

Let $\sigma>\frac{5}{2}$ and assume (6.1) and (6.2) hold , then

$\displaystyle\|\langle
x\rangle^{\sigma}B^{1/2}\partial_{\bar{f}}\mathcal{R}_{1}\|_{W^{1,4/3}}\lesssim\langle|\xi_{0}|^{4N}t\rangle^{-\frac{4N+3}{4N}}[\xi]^{4N+3}_{\frac{1}{4N}},$
$\displaystyle\|\langle
x\rangle^{\sigma}B^{1/2}\partial_{\bar{f}}\mathcal{R}_{2}\|_{W^{1,4/3}}\lesssim\langle
t\rangle^{-\frac{1}{2}}[\xi]^{2}_{\frac{1}{4N}}\big{(}A_{f}+C_{f}\big{)}+\langle|\xi_{0}|^{4N}t\rangle^{-\frac{2N+2}{4N}}[\xi]^{2}_{\frac{1}{4N}}\big{(}B_{f}+D_{f}\big{)},$
$\displaystyle\|\langle
x\rangle^{\sigma}B^{1/2}\partial_{\bar{f}}\mathcal{R}_{3}\|_{W^{1,4/3}}\lesssim\langle
t\rangle^{-\frac{2N+1}{4N}}[\xi]_{\frac{1}{4N}}A_{f}\big{(}A_{f}+C_{f}+D_{f}\big{)}+\langle|\xi_{0}|^{4N}t\rangle^{-\frac{2N+2}{4N}}[\xi]_{\frac{1}{4N}}B_{f}\big{(}B_{f}+C_{f}+D_{f}\big{)},$
$\displaystyle\|B^{1/2}\partial_{\bar{f}}\mathcal{R}_{4}\|_{W^{1,4/3}}\lesssim\langle
t\rangle^{-\frac{2N+2}{4N}}\|B^{1/2}f\|_{L^{2}}A_{f}^{2}+\langle|\xi_{0}|^{4N}t\rangle^{-\frac{2N+2}{4N}}\|B^{1/2}f\|_{L^{2}}B_{f}^{2},$
$\displaystyle\|\langle
x\rangle^{\sigma}B^{1/2}\partial_{\bar{f}}\mathcal{R}_{5}\|_{W^{1,4/3}}\lesssim\langle|\xi_{0}|^{4N}t\rangle^{-\frac{M^{\star}}{4N}}[\xi]^{M^{\star}}_{\frac{1}{4N}},$

Now we can derive estimates of $\hat{\mathcal{R}}_{f}$ and
$\mathcal{R}_{\xi}$:

######  Proposition 6.5.

For $\sigma>\frac{5}{2}$, the following estimates hold:

$\displaystyle\left\|B^{-1/2}\hat{\mathcal{R}}_{f}\right\|_{L^{8}}\lesssim\langle
t\rangle^{-\frac{2N+1}{4N}}\mathcal{P}_{1}+\langle|\xi_{0}|^{4N}t\rangle^{-\frac{2N+2}{4N}}\mathcal{P}_{2},$
$\displaystyle\left\|\langle
x\rangle^{-\sigma}B^{1/2}\hat{\mathcal{R}}_{f}\right\|_{L^{4}}\lesssim\langle
t\rangle^{-\frac{1}{2}}\mathcal{P}_{3}+\langle|\xi_{0}|^{4N}t\rangle^{-\frac{1}{2}}\mathcal{P}_{4},$

where $\mathcal{P}_{1},\mathcal{P}_{2},\mathcal{P}_{3},\mathcal{P}_{4}$ are
defined by (6.3), (6.4), (6.5) and (6.6) respectively.

###### Proof.

By Lemma 2.1, Lemma 6.1 and the corollary thereafter, we have

$\displaystyle\left\|B^{-1/2}\hat{\mathcal{R}}_{f}\right\|_{L^{8}}\lesssim$
$\displaystyle\int_{0}^{t}\left\|e^{-\mathrm{i}B(t-s)}B^{-1/2}\partial_{\bar{f}}\mathcal{R}\right\|_{L^{8}}ds$
$\displaystyle\lesssim$
$\displaystyle\int_{0}^{t}\min\\{|t-s|^{-9/8},|t-s|^{-3/8}\\}\|B^{-1/2}\partial_{\bar{f}}\mathcal{R}\|_{W^{2,8/7}}ds$
$\displaystyle\lesssim$
$\displaystyle\int_{0}^{t}\min\\{|t-s|^{-9/8},|t-s|^{-3/8}\\}\Big{(}\langle
s\rangle^{-\frac{2N+1}{4N}}\langle|\xi_{0}|^{4N}s\rangle^{-\frac{1}{4N}}\mathcal{P}_{1}+\langle|\xi_{0}|^{4N}s\rangle^{-\frac{2N+2}{4N}}\mathcal{P}_{2}\Big{)}ds$
$\displaystyle\lesssim$ $\displaystyle\langle
t\rangle^{-\frac{2N+1}{4N}}\langle|\xi_{0}|^{4N}t\rangle^{-\frac{1}{4N}}\mathcal{P}_{1}+\langle|\xi_{0}|^{4N}t\rangle^{-\frac{2N+2}{4N}}\mathcal{P}_{2},$

where we can choose

$\displaystyle\mathcal{P}_{1}=$
$\displaystyle[\xi]^{2}_{\frac{1}{4N}}A_{f}+[\xi]^{2}_{\frac{1}{4N}}|\xi_{0}|^{-1}C_{f}+[\xi]_{\frac{1}{4N}}A_{f}^{2}+[\xi]_{\frac{1}{4N}}A_{f}C_{f}+[\xi]_{\frac{1}{4N}}A_{f}D_{f}+|\xi_{0}|^{4/3}A_{f}^{5/3},$
(6.3) $\displaystyle\mathcal{P}_{2}=$
$\displaystyle[\xi]^{2N+3}_{\frac{1}{4N}}+[\xi]^{2}_{\frac{1}{4N}}B_{f}+[\xi]^{2}_{\frac{1}{4N}}D_{f}+[\xi]_{\frac{1}{4N}}B_{f}^{2}+[\xi]_{\frac{1}{4N}}B_{f}C_{f}+[\xi]_{\frac{1}{4N}}B_{f}D_{f}+|\xi_{0}|^{4/3}B_{f}^{5/3}.$
(6.4)

Similarly, by Lemma 2.3, Lemma 6.3 and the corollary thereafter, we have

$\displaystyle\left\|\langle
x\rangle^{-\sigma}B^{1/2}\hat{\mathcal{R}}_{f}\right\|_{L^{4}}\lesssim$
$\displaystyle\int_{0}^{t}\left\|\langle
x\rangle^{-\sigma}e^{-\mathrm{i}B(t-s)}B^{1/2}\partial_{\bar{f}}\mathcal{R}\right\|_{L^{4}}ds$
$\displaystyle\lesssim$
$\displaystyle\sum_{d\in\\{1,2,3,5\\}}\int_{0}^{t}\min\\{|t-s|^{-5/4},|t-s|^{-1/2}\\}\|\langle
x\rangle^{\sigma}B^{1/2}\partial_{\bar{f}}\mathcal{R}_{d}\|_{W^{1,4/3}}ds$
$\displaystyle+\int_{0}^{t}|t-s|^{-1/2}\|B^{1/2}\partial_{\bar{f}}\mathcal{R}_{4}\|_{W^{1,4/3}}ds$
$\displaystyle\lesssim$ $\displaystyle\langle
t\rangle^{-\frac{1}{2}}\mathcal{P}_{3}+\langle|\xi_{0}|^{4N}t\rangle^{-\frac{1}{2}}\mathcal{P}_{4},$

where we choose

$\displaystyle\mathcal{P}_{3}=$
$\displaystyle[\xi]^{2}_{\frac{1}{4N}}A_{f}+[\xi]^{2}_{\frac{1}{4N}}C_{f}+[\xi]_{\frac{1}{4N}}A_{f}^{2}+[\xi]_{\frac{1}{4N}}A_{f}C_{f}+[\xi]_{\frac{1}{4N}}A_{f}D_{f}+|\xi_{0}|A_{f}^{2},$
(6.5) $\displaystyle\mathcal{P}_{4}=$
$\displaystyle[\xi]^{2N+3}_{\frac{1}{4N}}+[\xi]^{2}_{\frac{1}{4N}}B_{f}+[\xi]^{2}_{\frac{1}{4N}}D_{f}+[\xi]_{\frac{1}{4N}}B_{f}^{2}+[\xi]_{\frac{1}{4N}}B_{f}C_{f}+[\xi]_{\frac{1}{4N}}B_{f}D_{f}+|\xi_{0}|^{-(2N-1)}B_{f}^{2}.$
(6.6)

∎

We finally come to the following error estimates:

######  Proposition 6.6.

For $\sigma>\frac{5}{2}$, the following estimates hold:

$\displaystyle\|\langle
x\rangle^{-\sigma}B^{-1/2}\mathcal{R}_{f}\|_{L^{2}}\lesssim\langle
t\rangle^{-\frac{2N+1}{4N}}\langle|\xi_{0}|^{4N}t\rangle^{-\frac{1}{4N}}\left(\|B^{-1/2}f(0)\|_{X}+|\xi_{0}|^{2N+1}+\mathcal{P}_{1}\right)+\langle|\xi_{0}|^{4N}t\rangle^{-\frac{2N+2}{4N}}\mathcal{P}_{2},$
(6.7) $\displaystyle|\mathcal{R}_{\xi}|\lesssim\langle
t\rangle^{-\frac{4N+2}{4N}}\mathcal{P}_{5}+\langle
t\rangle^{-\frac{2N+1}{4N}}\langle|\xi_{0}|^{4N}t\rangle^{-\frac{2N+1}{4N}}\mathcal{P}_{6}+\langle|\xi_{0}|^{4N}t\rangle^{-\frac{4N+2}{4N}}\mathcal{P}_{7},$
(6.8)

where $\mathcal{P}_{5},\mathcal{P}_{6},\mathcal{P}_{7}$ are defined in (6.9),
(6.10) and (6.11) respectively.

For the sake of convenience we first prove

######  Lemma 6.7.

$\displaystyle|\partial_{\bar{\xi}}\mathcal{R}|\lesssim\langle
t\rangle^{-\frac{4N+2}{4N}}\left([\xi]_{\frac{1}{4N}}A_{f}^{2}+A_{f}^{3}\right)+\langle|\xi_{0}|^{4N}t\rangle^{-\frac{4N+3}{4N}}\left([\xi]^{4N+3}_{\frac{1}{4N}}+[\xi]^{4N+2}_{\frac{1}{4N}}(A_{f}+B_{f})+[\xi]_{\frac{1}{4N}}B_{f}^{2}+B_{f}^{3}\right).$

###### Proof.

By using Leibnitz rule and expicit formula of $\mathcal{R}(iii.0-iii.5)$, we
have

$\displaystyle|\partial_{\bar{\xi}}\mathcal{R}_{0}|\lesssim|\xi|^{4N+3}\lesssim\langle|\xi_{0}|^{4N}t\rangle^{-\frac{4N+3}{4N}}[\xi]^{4N+3}_{\frac{1}{4N}},$
$\displaystyle|\partial_{\bar{\xi}}\mathcal{R}_{1}|\lesssim|\xi|^{4N+2}\|B^{-1/2}f\|_{L^{8}}\lesssim\langle|\xi_{0}|^{4N}t\rangle^{-\frac{6N+3}{4N}}[\xi]^{4N+2}_{\frac{1}{4N}}(A_{f}+B_{f}),$
$\displaystyle|\partial_{\bar{\xi}}\mathcal{R}_{2}|\lesssim|\xi|\|B^{-1/2}f\|_{L^{8}}^{2}\lesssim\langle
t\rangle^{-\frac{4N+2}{4N}}[\xi]_{\frac{1}{4N}}A_{f}^{2}+\langle|\xi_{0}|^{4N}t\rangle^{-\frac{4N+3}{4N}}[\xi]_{\frac{1}{4N}}B_{f}^{2},$
$\displaystyle|\partial_{\bar{\xi}}\mathcal{R}_{3}|\lesssim\|B^{-1/2}f\|_{L^{8}}^{3}\lesssim\langle
t\rangle^{-\frac{6N+3}{4N}}A_{f}^{3}+\langle|\xi_{0}|^{4N}t\rangle^{-\frac{6N+3}{4N}}B_{f}^{3},$
$\displaystyle|\partial_{\bar{\xi}}\mathcal{R}_{4}|\lesssim\|B^{-1/2}f\|_{L^{8}}^{4}\lesssim\langle
t\rangle^{-\frac{8N+4}{4N}}A_{f}^{4}+\langle|\xi_{0}|^{4N}t\rangle^{-\frac{8N+4}{4N}}B_{f}^{4},$
$\displaystyle|\partial_{\bar{\xi}}\mathcal{R}_{5}|\lesssim|\xi|^{M^{*}}\lesssim\langle|\xi_{0}|^{4N}t\rangle^{-\frac{M^{\star}}{4N}}[\xi]^{M^{\star}}_{\frac{1}{4N}}.$

Hence, we get desired estimate of $\partial_{\bar{\xi}}\mathcal{R}$. ∎

###### Proof of Proposition 6.6.

By Hölder’s inequality,

$\displaystyle\left\|\langle
x\rangle^{-\sigma}B^{-1/2}\mathcal{R}_{f}\right\|_{L^{2}}$
$\displaystyle\lesssim$
$\displaystyle\left\|e^{-\mathrm{i}Bt}B^{-1/2}f(0)\right\|_{L^{8}}+\left\|\int_{0}^{t}e^{-\mathrm{i}B(t-s)}B^{-1/2}\partial_{\bar{f}}\mathcal{R}ds\right\|_{L^{8}}$
$\displaystyle+\sum_{(\mu,\nu)\in
M_{1}}\left|\eta^{\nu}(0)\bar{\eta}^{\mu}(0)\right|\left\|\langle
x\rangle^{-\sigma}e^{-\mathrm{i}Bt}(B-\omega(\nu-\mu)-\mathrm{i}0)^{-1}B^{-1/2}\bar{\Phi}_{\mu\nu}\right\|_{L^{2}}$
$\displaystyle+\sum_{(\mu,\nu)\in M_{1}}\int_{0}^{t}\left\|\langle
x\rangle^{-\sigma}e^{-\mathrm{i}B(t-s)}(B-\omega(\nu-\mu)-\mathrm{i}0)^{-1}B^{-1/2}\bar{\Phi}_{\mu\nu}\right\|_{L^{2}}\left|\frac{d}{ds}(\eta^{\nu}\bar{\eta}^{\mu})\right|ds$
$\displaystyle+\sum_{(\mu,\nu)\in M\backslash M_{1}}\int_{0}^{t}\left\|\langle
x\rangle^{-\sigma}e^{-\mathrm{i}B(t-s)}B^{-1/2}\bar{\Phi}_{\mu\nu}\right\|_{L^{2}}\left|\eta^{\nu}\bar{\eta}^{\mu}\right|ds$
$\displaystyle:=$ $\displaystyle I+II+III+IV+V.$

In the proof of Proposition 5.2, we have showed that

$I=\|e^{-\mathrm{i}Bt}B^{-1/2}f(0)\|_{L^{8}}\lesssim\langle
t\rangle^{-9/8}\|B^{-1/2}f(0)\|_{X}.$

By Proposition 6.5,

$\displaystyle
II=\left\|B^{-1/2}\hat{\mathcal{R}}_{f}\right\|_{L^{8}}\lesssim\langle
t\rangle^{-\frac{2N+1}{4N}}\mathcal{P}_{1}+\langle|\xi_{0}|^{4N}t\rangle^{-\frac{2N+2}{4N}}\mathcal{P}_{2}.$

By singular resolvents estimates Lemma 2.4, we have

$III\lesssim\langle t\rangle^{-6/5}|\xi_{0}|^{2N+1}.$

For the term $IV$, since
$\left|\frac{d}{ds}(\eta^{\nu}\bar{\eta}^{\mu})\right|\lesssim|\eta|^{2N}|\dot{\eta}|$,
and

$\displaystyle|\dot{\eta}|=|\dot{\xi}+\mathrm{i}\omega\xi|$
$\displaystyle\leq|\partial_{\bar{\xi}}Z_{0}|+|\left\langle\partial_{\xi}G,f\right\rangle|+|\left\langle\partial_{\bar{\xi}}\bar{G},\bar{f}\right\rangle|+|\partial_{\bar{\xi}}\mathcal{R}|$
$\displaystyle\lesssim\langle
t\rangle^{-\frac{4N+2}{4N}}\left([\xi]_{\frac{1}{4N}}A_{f}^{2}+A_{f}^{3}\right)+\langle|\xi_{0}|^{4N}t\rangle^{-\frac{3}{4N}}\left([\xi]^{3}_{\frac{1}{4N}}+[\xi]_{\frac{1}{4N}}B_{f}^{2}+B_{f}^{3}\right),$

it holds that

$\displaystyle IV\lesssim$ $\displaystyle\int_{0}^{t}\langle
t-s\rangle^{-6/5}\langle
s\rangle^{-\frac{4N+2}{4N}}[\xi]^{2N}_{\frac{1}{4N}}\left([\xi]_{\frac{1}{4N}}A_{f}^{2}+A_{f}^{3}\right)ds$
$\displaystyle+\int_{0}^{t}\langle
t-s\rangle^{-6/5}\langle|\xi_{0}|^{4N}s\rangle^{-\frac{2N+3}{4N}}[\xi]^{2N}_{\frac{1}{4N}}\left([\xi]^{3}_{\frac{1}{4N}}+[\xi]_{\frac{1}{4N}}B_{f}^{2}+B_{f}^{3}\right)ds$
$\displaystyle\lesssim$ $\displaystyle\langle
t\rangle^{-\frac{4N+2}{4N}}[\xi]^{2N}_{\frac{1}{4N}}\left([\xi]_{\frac{1}{4N}}A_{f}^{2}+A_{f}^{3}\right)+\langle|\xi_{0}|^{4N}t\rangle^{-\frac{2N+3}{4N}}[\xi]^{2N}_{\frac{1}{4N}}\left([\xi]^{3}_{\frac{1}{4N}}+[\xi]_{\frac{1}{4N}}B_{f}^{2}+B_{f}^{3}\right).$

Similarly,

$\displaystyle V\lesssim$ $\displaystyle\int_{0}^{t}\langle
t-s\rangle^{-6/5}\langle|\xi_{0}|^{4N}s\rangle^{-\frac{2N+3}{4N}}ds[\xi]^{2N+3}_{\frac{1}{4N}}$
$\displaystyle\lesssim$
$\displaystyle\langle|\xi_{0}|^{4N}t\rangle^{-\frac{2N+3}{4N}}[\xi]^{2N+3}_{\frac{1}{4N}}.$

Combining the above estimates and recall the formula of
$\mathcal{P}_{1},\mathcal{P}_{2}$, we get desired estimates for
$\mathcal{R}_{f}.$

Then using formula (4.6) of $\mathcal{R}_{\xi}$, also combining the estimates
of $\mathcal{R}_{f}$ and Lemma 6.7, we get estimates of $\mathcal{R}_{\xi}$:

$\displaystyle|\mathcal{R}_{\xi}|\lesssim$
$\displaystyle\langle|\xi_{0}|^{4N}t\rangle^{-\frac{4N+3}{4N}}[\xi]^{4N+3}_{\frac{1}{4N}}+\langle|\xi_{0}|^{4N}t\rangle^{-\frac{1}{2}}[\xi]^{2N}_{\frac{1}{4N}}\|\langle
x\rangle^{-\sigma}B^{-1/2}\mathcal{R}_{f}\|_{L^{2}}+|\partial_{\bar{\xi}}\mathcal{R}|$
$\displaystyle\lesssim$ $\displaystyle\langle
t\rangle^{-\frac{4N+2}{4N}}\mathcal{P}_{5}+\langle
t\rangle^{-\frac{2N+1}{4N}}\langle|\xi_{0}|^{4N}t\rangle^{-\frac{2N+1}{4N}}\mathcal{P}_{6}+\langle|\xi_{0}|^{4N}t\rangle^{-\frac{4N+2}{4N}}\mathcal{P}_{7},$

where

$\displaystyle\mathcal{P}_{5}$
$\displaystyle=[\xi]_{\frac{1}{4N}}A_{f}^{2}+A_{f}^{3},$ (6.9)
$\displaystyle\mathcal{P}_{6}$
$\displaystyle=[\xi]^{2N}_{\frac{1}{4N}}\left(\|B^{-1/2}f(0)\|_{X}+|\xi_{0}|^{2N+1}+\mathcal{P}_{1}\right),$
(6.10) $\displaystyle\mathcal{P}_{7}$
$\displaystyle=[\xi]^{2N}_{\frac{1}{4N}}\mathcal{P}_{2}+[\xi]^{4N+2}_{\frac{1}{4N}}(A_{f}+B_{f})+[\xi]_{\frac{1}{4N}}B_{f}^{2}+B_{f}^{3}.$
(6.11)

∎

##  7 Proof of Theorem 1.2

Now we shall prove Theorem 1.2 by bootstrap arguments. Denote the original
variables to be $(\xi^{\prime},f^{\prime})=\mathcal{T}_{2N}(\xi,f)$, then by
(1.4) the initial data $(\xi^{\prime}_{0},f^{\prime}(0))$ satisfies

$\displaystyle\|B^{-1/2}f^{\prime}(0)\|_{X}\lesssim|\xi^{\prime}_{0}|\lesssim\epsilon.$

By Theorem 3.3 (ii), it is equivalent to

$\displaystyle\|B^{-1/2}f(0)\|_{X}\lesssim|\xi_{0}|\lesssim\epsilon.$

Then by energy conservation, we have

$|\xi(t)|+\|B^{1/2}f\|_{L^{2}}\lesssim|\xi(0)|+\|B^{1/2}f(0)\|_{L^{2}}\leq(1+C_{0})|\xi_{0}|.$
(7.1)

######  Proposition 7.1.

Assume $[\xi]_{\frac{1}{4N}}(T)\lesssim|\xi_{0}|$, then (6.1) and (6.2) hold
with

$\displaystyle A_{f}(T)$ $\displaystyle\lesssim|\xi_{0}|,$ $\displaystyle
B_{f}(T)$ $\displaystyle\lesssim|\xi_{0}|^{2N+1},$ $\displaystyle C_{f}(T)$
$\displaystyle\lesssim|\xi_{0}|,$ $\displaystyle D_{f}(T)$
$\displaystyle\lesssim|\xi_{0}|^{2N+1},$

consequently,

$\displaystyle|\mathcal{R}_{\xi}|\lesssim\langle
t\rangle^{-\frac{4N+2}{4N}}|\xi_{0}|^{3}+\langle
t\rangle^{-\frac{2N+1}{4N}}\langle|\xi_{0}|^{4N}t\rangle^{-\frac{2N+1}{4N}}|\xi_{0}|^{2N+1}+\langle|\xi_{0}|^{4N}t\rangle^{-\frac{4N+2}{4N}}|\xi_{0}|^{4N+3}.$

###### Proof.

The proof consists of a bootstrap argument using Proposition 5.2 and 6.5.
First, by (7.1), the result (6.1) and (6.2) hold for small $t<1$. Then, assume
this is true for $0<t<t^{*}<T$ with

$\displaystyle A_{f}(T)$ $\displaystyle\leq K|\xi_{0}|,$ $\displaystyle
B_{f}(T)$ $\displaystyle\leq K|\xi_{0}|^{2N+1},$ $\displaystyle C_{f}(T)$
$\displaystyle\leq K|\xi_{0}|,$ $\displaystyle D_{f}(T)$ $\displaystyle\leq
K|\xi_{0}|^{2N+1},$

where $K$ is a large constant to be chosen. By
$[\xi]_{\frac{1}{4N}}(T)\lesssim|\xi_{0}|$ and Proposition 6.5, (5.8) and
(5.9) hold for $0<t<t^{*}$ with

$\displaystyle\mathcal{P}_{1}$
$\displaystyle\lesssim|\xi_{0}|^{2}A_{f}+|\xi_{0}|C_{f}+|\xi_{0}|A_{f}^{2}+|\xi_{0}|A_{f}C_{f}+|\xi_{0}|A_{f}D_{f}+|\xi_{0}|^{4/3}A_{f}^{5/3}\lesssim
K^{2}|\xi_{0}|^{2},$ $\displaystyle\mathcal{P}_{2}$
$\displaystyle\lesssim|\xi_{0}|^{2N+3}+|\xi_{0}|^{2}B_{f}+|\xi_{0}|^{2}D_{f}+|\xi_{0}|B_{f}^{2}+|\xi_{0}|B_{f}C_{f}+|\xi_{0}|B_{f}D_{f}+|\xi_{0}|^{4/3}B_{f}^{5/3}\lesssim
K^{2}|\xi_{0}|^{2N+3},$ $\displaystyle\mathcal{P}_{3}$
$\displaystyle\lesssim|\xi_{0}|^{2}A_{f}+|\xi_{0}|^{2}C_{f}+|\xi_{0}|A_{f}^{2}+|\xi_{0}|A_{f}C_{f}+|\xi_{0}|A_{f}D_{f}+|\xi_{0}|A_{f}^{2}\lesssim
K^{2}|\xi_{0}|^{3},$ $\displaystyle\mathcal{P}_{4}$
$\displaystyle\lesssim|\xi_{0}|^{2N+3}+|\xi_{0}|^{2}B_{f}+|\xi_{0}|^{2}D_{f}+|\xi_{0}|B_{f}^{2}+|\xi_{0}|B_{f}C_{f}+|\xi_{0}|B_{f}D_{f}+|\xi_{0}|^{-(2N-1)}B_{f}^{2}\lesssim
K^{2}|\xi_{0}|^{2N+3}.$

By Proposition 5.2, we have

$\displaystyle\|B^{-1/2}f\|_{L^{8}}\lesssim\langle
t\rangle^{-\frac{2N+1}{4N}}\Big{(}|\xi_{0}|+K^{2}|\xi_{0}|^{2}\Big{)}+\langle|\xi_{0}|^{4N}t\rangle^{-\frac{2N+1}{4N}}\Big{(}|\xi_{0}|^{2N+1}+K^{2}|\xi_{0}|^{2N+3}\Big{)},$
$\displaystyle\|\langle x\rangle^{-\sigma}B^{1/2}f\|_{L^{4}}\lesssim\langle
t\rangle^{-\frac{1}{2}}\Big{(}|\xi_{0}|+K^{2}|\xi_{0}|^{3}\Big{)}+\langle|\xi_{0}|^{4N}t\rangle^{-\frac{1}{2}}\Big{(}|\xi_{0}|^{2N+1}+K^{2}|\xi_{0}|^{2N+3}\Big{)}.$

By choosing $K$ large and $|\xi_{0}|$ small we complete the bootstrap
argument. In addition, we have

$\displaystyle\mathcal{P}_{5}$
$\displaystyle=[\xi]_{\frac{1}{4N}}A_{f}^{2}+A_{f}^{3}\lesssim|\xi_{0}|^{3},$
(7.2) $\displaystyle\mathcal{P}_{6}$
$\displaystyle=[\xi]^{2N}_{\frac{1}{4N}}\left(\|B^{-1/2}f(0)\|_{X}+|\xi_{0}|^{2N+1}+\mathcal{P}_{1}\right)\lesssim|\xi_{0}|^{2N+1},$
(7.3) $\displaystyle\mathcal{P}_{7}$
$\displaystyle=[\xi]^{2N}_{\frac{1}{4N}}\mathcal{P}_{2}+[\xi]^{4N+2}_{\frac{1}{4N}}(A_{f}+B_{f})+[\xi]_{\frac{1}{4N}}B_{f}^{2}+B_{f}^{3}\lesssim|\xi_{0}|^{4N+3},$
(7.4)

by Proposition 6.6 we get estimates of $|\mathcal{R}_{\xi}|$. ∎

###### Proof of Theorem 1.2.

We use bootstrap argument again to show that

$\displaystyle[\xi]_{\frac{1}{4N}}(T)\leq 8C^{*}|\xi_{0}|$ (7.5)

holds for all $T>0$, where $C^{*}=[1+(4N\gamma)^{-1}]^{\frac{1}{4N}}(1+C_{0})$
is an absolute constant. Note that by energy conservation, (7.5) holds for
$T\leq|\xi_{0}|^{-4N(1-\delta_{0})}$. For
$T\geq|\xi_{0}|^{-4N(1-\delta_{0})},$ we assume that

$\displaystyle[\xi]_{\frac{1}{4N}}(T^{*})\leq 8C^{*}|\xi_{0}|$ (7.6)

for some $|\xi_{0}|^{-4N(1-\delta_{0})}\leq T^{*}<T$. Then, by Proposition
7.1, we have

$|\mathcal{R}_{\xi}|\lesssim\langle|\xi_{0}|^{4N}t\rangle^{-\frac{4N+2}{4N}}\left(|\xi_{0}|^{(4N+2)(1-\delta_{0})}+|\xi_{0}|^{(4N+2)(1-\delta_{0}/2)}+|\xi_{0}|^{4N+3}\right)$

for $0\leq t\leq T^{*}$. Choosing $\delta_{0}<\frac{1}{4N+2}$, we have
$|\mathcal{R}_{\xi}|\lesssim\langle|\xi_{0}|^{4N}t\rangle^{-\frac{4N+2}{4N}}|\xi_{0}|^{4N+1+\delta_{1}}$
for some $\delta_{1}>0.$ Hence, we can apply Theorem 5.1 at
$t_{0}=|\xi_{0}|^{-4N(1-\delta_{0})}$ with
$Q_{0}\approx|\xi_{0}|^{4N+1+\delta_{1}}.$ Note that

$\displaystyle|r(t_{0})-r(0)|$
$\displaystyle\leq\int_{0}^{t_{0}}\Big{(}4N\gamma
r^{2}(t)+4N|\mathcal{R}_{\xi}|r^{\frac{4N-1}{4N}}(t)\Big{)}dt$
$\displaystyle\leq
C\int_{0}^{t_{0}}\Big{(}|\xi_{0}|^{8N}+|\xi_{0}|^{4N+1+\delta_{1}}|\xi_{0}|^{4N-1}\Big{)}dt$
$\displaystyle\leq C|\xi_{0}|^{4N(1+\delta_{0})}$
$\displaystyle\leq\frac{1}{2}r(0),$

we have
$\frac{1}{2}|\xi_{0}|^{4N}\leq|\xi(t_{0})|^{4N}\leq\frac{3}{2}|\xi_{0}|^{4N}.$
Therefore, we get

$\displaystyle|\xi(t)|^{4N}\leq$ $\displaystyle
4(1+4N\gamma|\xi_{0}|^{4N}t)^{-1}\left(|\xi_{0}|^{4N}+\frac{C(C^{*})|\xi_{0}|^{4N+\delta_{1}/2}}{(1+4N\gamma|\xi_{0}|^{4N}t)^{\delta/2}}\right)$
$\displaystyle\leq$ $\displaystyle
5\left(C^{*}\right)^{4N}(1+|\xi_{0}|^{4N}t)^{-1}|\xi_{0}|^{4N}.$

The last inequality holds by using

$(1+4N\gamma|\xi_{0}|^{4N}t)^{-1}\leq\left(C^{*}\right)^{4N}(1+|\xi_{0}|^{4N}t)^{-1}$

and choosing $|\xi_{0}|$ to be sufficiently small such that
$C(C^{*})|\xi_{0}|^{\delta_{1}/2}\leq 1$.

Next, we shall prove the lower bound of $|\xi|$. We have proved that
$|\xi(t)|^{4N}\approx|\xi_{0}|^{4N}\approx(1+|\xi_{0}|^{4N}t)^{-1}|\xi_{0}|^{4N}$
for $0\leq T\leq|\xi_{0}|^{-4N(1-\delta_{0})}$. For
$T\geq|\xi_{0}|^{-4N(1-\delta_{0})},$ by Proposition 7.1 we have

$|\mathcal{R}_{\xi}|\lesssim\langle|\xi_{0}|^{4N}t\rangle^{-\frac{4N+2}{4N}}|\xi_{0}|^{4N+1+\delta_{1}},$

thus using Lemma 5.1 we also obtain
$|\xi(t)|^{4N}\gtrsim(1+|\xi_{0}|^{4N}t)^{-1}|\xi_{0}|^{4N}$. Combining above
estimates we obtain

$|\xi(t)|\approx\frac{|\xi_{0}|}{(1+|\xi_{0}|^{4N}t)^{\frac{1}{4N}}}.$

By Proposition 7.1, we also have

$\|B^{-1/2}f\|_{L^{8}}\lesssim\langle
t\rangle^{-\frac{2N+1}{4N}}|\xi_{0}|+\langle|\xi_{0}|^{4N}t\rangle^{-\frac{2N+1}{4N}}|\xi_{0}|^{2N+1}.$

By the property of normal form transformation, for the original variables
$(\xi^{\prime},f^{\prime})=\mathcal{T}_{2N}(\xi,f)$, we have

$|\xi^{\prime}(t)|\approx\frac{|\xi^{\prime}_{0}|}{(1+|\xi^{\prime}_{0}|^{4N}t)^{\frac{1}{4N}}}$

and

$\|B^{-1/2}f^{\prime}\|_{L^{8}}\lesssim\langle
t\rangle^{-\frac{2N+1}{4N}}|\xi^{\prime}_{0}|+\langle|\xi_{0}^{\prime}|^{4N}t\rangle^{-\frac{2N+1}{4N}}|\xi^{\prime}_{0}|^{3}.$

Write $\xi^{\prime}(t)=\rho(t)e^{-\mathrm{i}(\omega t+\theta(t))}$, then
$q(t)=\sqrt{\frac{2}{\omega}}\operatorname{Re}\xi(t)=\sqrt{\frac{2}{\omega}}\rho(t)\cos(\omega
t+\theta(t))$, and

$u(t,x)=R(t)\cos(\omega t+\theta(t))\varphi(x)+\eta(t,x)$

with $R(t)=\sqrt{\frac{2}{\omega}}\rho(t).$ Hence, the above estimates are
equivalent to

$|R(t)|\approx\frac{|R(0)|}{(1+|R(0)|^{4N}t)^{\frac{1}{4N}}}$

and

$\|\eta(t)\|_{L^{8}}\lesssim\frac{|R(0)|}{(1+t)^{\frac{2N+1}{4N}}}+\frac{|R(0)|^{3}}{(1+|R(0)|^{4N}t)^{\frac{3}{4N}}}.$

In addition, since

$\dot{\xi}^{\prime}+\mathrm{i}\omega\xi^{\prime}=\dot{\rho}e^{-\mathrm{i}(\omega
t+\theta)}-\mathrm{i}\dot{\theta}\xi^{\prime},$

where
$|\dot{\xi}^{\prime}+\mathrm{i}\omega\xi^{\prime}|=\mathcal{O}(|t|^{-\frac{3}{4N}}),|\dot{\rho}|=\mathcal{O}(|t|^{-\frac{3}{4N}})$,
we get $|\dot{\theta}|=\mathcal{O}(|t|^{-\frac{1}{2N}})$, hence
$\theta(t)=\mathcal{O}(|t|^{1-\frac{1}{2N}}),$ which completes the proof of
Theorem 1.2. ∎

## Acknowledgment

We would like to thank H. Jia for introducing the problem to us and for the
useful discussions. The authors were in part supported by NSFC (Grant No.
11725102), National Support Program for Young Top-Notch Talents, and Shanghai
Science and Technology Program (Project No. 21JC1400600 and No. 19JC1420101).

##  Appendix A Convolution Estimates

The following lemma deals with the decay rate of the convolution of two
functions, which is used frequently in the asymptotic analysis in Section 5
and Section 6.

######  Lemma A.1.

Let $0<\delta<1$ and
$0<f(t)\lesssim\min\\{|t|^{-(1+\delta)},|t|^{-(1-\delta)}\\}$, then

$\int_{0}^{t}f(t-s)\langle|\xi_{0}|^{4N}s\rangle^{-\alpha}ds\lesssim\langle|\xi_{0}|^{4N}t\rangle^{-\alpha},\quad\forall\
0\leq\alpha\leq 1+\delta.$ (A.1)

###### Proof.

Note that $f$ is integrable on the whole line, thus it suffices to consider
$t\geq 2|\xi_{0}|^{-4N}$. Then we have

$\displaystyle\int_{0}^{t}f(t-s)\langle|\xi_{0}|^{4N}s\rangle^{-\alpha}ds\leq$
$\displaystyle\int_{0}^{|\xi_{0}|^{-4N}}f(t-s)\langle|\xi_{0}|^{4N}s\rangle^{-\alpha}ds+\int_{|\xi_{0}|^{-4N}}^{t/2}f(t-s)\langle|\xi_{0}|^{4N}s\rangle^{-\alpha}ds$
$\displaystyle+\int_{t/2}^{t}f(t-s)\langle|\xi_{0}|^{4N}s\rangle^{-\alpha}ds$
$\displaystyle\triangleq$ $\displaystyle I+II+III,$

where

$\displaystyle I$ $\displaystyle\lesssim
t^{-(1+\delta)}|\xi_{0}|^{-4N}\lesssim\langle|\xi_{0}|^{4N}t\rangle^{-(1+\delta)},$
$\displaystyle II$ $\displaystyle\lesssim
t^{-(1+\delta)}\int_{|\xi_{0}|^{-4N}}^{t/2}\langle|\xi_{0}|^{4N}s\rangle^{-\alpha}ds\lesssim\begin{cases}t^{-(1+\delta)}|\xi_{0}|^{-4N\alpha}t^{1-\alpha}\lesssim(|\xi_{0}|^{4N}t)^{-\alpha}&if\quad\alpha<1,\\\
t^{-(1+\delta)}|\xi_{0}|^{-4N\alpha}|\xi_{0}|^{4N(\alpha-1)}\lesssim(|\xi_{0}|^{4N}t)^{-(1+\delta)}&if\quad\alpha>1,\end{cases}$
$\displaystyle III$
$\displaystyle\lesssim\langle|\xi_{0}|^{4N}t\rangle^{-\alpha}\int_{t/2}^{t}f(t-s)ds\lesssim\langle|\xi_{0}|^{4N}t\rangle^{-\alpha}.$

Combining the above estimates, we obtain the desired decay estimate. ∎

Similarly, we can also prove the following lemma:

######  Lemma A.2.

For $\alpha>1$, we have

$\int_{0}^{t}(t-s)^{-1/2}\langle|\xi_{0}|^{4N}s\rangle^{-\alpha}ds\lesssim|\xi_{0}|^{-2N}\langle|\xi_{0}|^{4N}t\rangle^{-1/2}.$
(A.2)

## References

  * [1] V. I. Arnold, Proof of a theorem of A. N. Kolmogorov on the preservation of conditionally periodic motions under a small perturbation of the Hamiltonian, Uspehi Mat. Nauk 18 1963, no. 5 (113), 13-40.
  * [2] X. An, A. Soffer, Fermi’s golden rule and $H^{1}$ scattering for nonlinear Klein-Gordon equations with metastable states, Discrete Contin. Dyn. Syst. 40 (2020), no. 1, 331-373.
  * [3] D. Bambusi and S. Cuccagna, On dispersion of small energy solutions to the nonlinear Klein-Gordon equation with a potential, Amer. J. Math., 133 (2011), 1421-1468.
  * [4] Piotr Bizoń, Maciej Dunajski, Michał Kahl, Michał Kowalczyk, Sine-Gordon on a wormhole. (English summary) Nonlinearity 34 (2021), no. 8, 5520–5537.
  * [5] J. Bourgain, Quasi-periodic solutions of Hamiltonian evolution equations, Lecture Notes in Pure and Appl. Math. 186, Dekker, New York 1997.
  * [6] T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations, Oxford Lecture Series in Mathematics and Its Applications, vol. 13, The Clarendon Press, New York, 1998.
  * [7] S. Cuccagna and M. Maeda, Coordinates at small energy and refined profiles for the Nonlinear Schrödinger Equation. Preprint arXiv:2004.01366.
  * [8] S. Cuccagna and M. Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Preprint arXiv:2009.00573.
  * [9] W. Craig and C. E. Wayne. Newton’s method and periodic solutions of nonlinear wave equations, Comm. Pure Appl. Math., 46(11):1409-1498, 1993.
  * [10] J.M. Delort, Existence globale et comportement asymptotique pour l’ équation de Klein–Gordon quasi-linéaire à données petites en dimen-sion1, Ann. Sci. Éc. Norm. Supér. 34 (2001) 1-61.
  * [11] Z. Gang, Perturbation expansion and Nth order Fermi golden rule of the nonlinear Schrödinger equations, J. Math. Phys., 48:5 (2007)
  * [12] Z. Gang and I. M. Sigal, Relaxation of solitons in nonlinear Schrödinger equations with potential, Adv. Math. 216:2 (2007), 443-490.
  * [13] Z. Gang and M. I. Weinstein, Dynamics of Nonlinear Schrödinger / Gross-Pitaevskii Equations; Mass Transfer in Systems with Solitons and Degenerate Neutral Modes, Anal. PDE 1 (2008), no. 3, 267-322.
  * [14] P. Germain, F. Pusateri, Quadratic Klein-Gordon equations with a potential in one dimension, Preprint arXiv:2006.15688.
  * [15] N. Hayashi, P. Naumkin, Quadratic nonlinear Klein-Gordon equation in one dimension, J. Math. Phys. 53(10), 103711, 36, 2012.
  * [16] A. Jensen, T. Kato, Spectral properties of Schrödinger operators and time-decay of the wave functions, Duke Math. J. 46, 583-611 (1979)
  * [17] J.-L. Journé, A. Soffer, C.D. Sogge, Decay estimates for Schrödinger operators, Commun. Pure Appl. Math. 44, 573-604 (1991)
  * [18] S. Klainerman, The null condition and global existence to nonlinear wave equations, Nonlinear systems of partial differential equations in applied mathematics, Part 1 (Santa Fe, N.M., 1984), 293-326, Lectures in Appl. Math., 23, Amer. Math. Soc., Providence, RI, 1986.
  * [19] A. N. Kolmogorov, On conservation of conditionally periodic motions for a small change in Hamilton’s function, Dokl. Akad. Nauk SSSR (N.S.) 98, (1954). 527-530.
  * [20] S.B. Kuksin, Nearly integrable infinite-dimensional Hamiltonian systems, Lecture Notes in Math. 1556, Springer Verlag, Berlin-Heidelberg-New York, 1993.
  * [21] A. I. Komech and E. A. Kopylova, Weighted energy decay for 3d klein–gordon equation, Journal of Differential Equations 248 (2010), no. 3, 501-520.
  * [22] R. De La Llave, A tutorial on KAM theory. Smooth ergodic theory and its applications, 175–292, Proc. Sympos. Pure Math., 69, Amer. Math. Soc., Providence, RI, 2001.
  * [23] H. Lindblad, J. Lührmann, A. Soffer, Asymptotics for 1D Klein-Gordon equations with variable coefficient quadratic nonlinearities, Arch. Ration. Mech. Anal. 241 (2021), no. 3, 1459-1527.
  * [24] T. Léger and F. Pusateri, Internal modes and radiation damping for quadratic Klein-Gordon in 3D. Preprint arxiv:2112.13163.
  * [25] J. Moser, On invariant curves of area-preserving mappings of an annulus, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II 1962 (1962), 1-20.
  * [26] T. Ozawa, K. Tustaya, and Y. Tsutsumi, Global existence and asymptotic behavior of solutions for the Klein-Gordon equations with quadratic nonlinearity in two space dimensions, Math. Z. 222, 341-362 (1996)
  * [27] M. Reed, B. Simon, Modern Methods of Mathematical Physics, Volume 1, Functional Analysis, Academic Press 1972
  * [28] I. Rodnianski and W. Schlag, Time decay for solutions of Schrödinger equations with rough and time-dependent potentials, Invent. Math. 155 (2004), no. 3, 451-513.
  * [29] J. Shatah. Normal forms and quadratic nonlinear Klein-Gordon equations, Comm. Pure Appl. Math. 38 (1985), no. 5, 685-696.
  * [30] I. M. Sigal, Nonlinear wave and Schrödinger equations. I. Instability of periodic and quasiperiodic solutions, Comm. Math. Phys. 153 (1993), no. 2, 297-320.
  * [31] J. C. H. Simon and E. Taflin, The Cauchy problem for nonlinear Klein-Gordon equations, Comm. Math. Phys. 152 (1993) 433-478.
  * [32] A. Soffer and M. I. Weinstein, Resonance, radiation damping and instability in Hamitonian nonlinear wave equations, Invent. Math. 136, 9-74(1999)
  * [33] A. Soffer and M. I. Weinstein, Selection of the ground state for nonlinear Schrödinger equations, Rev. Math. Phys. 16:8 (2004), 977-1071.
  * [34] E.M. Stein, Harmonic analysis real-variable methods orthogonality and oscillatory integrals, monographs in harmonic analysis, 1993.
  * [35] T.P. Tsai, H.T. Yau, Asymptotic dynamics of nonlinear schrödinger equations: resonance dominated and radiation dominated solutions, Comm. Pure Appl. Math. 55, 153-216 (2002)
  * [36] C. E. Wayne. An introduction to KAM theory, In Dynamical Systems and Probabilistic Methods in Partial Differential Equations, pages 3-29. Amer. Math. Soc, Providence, RI, 1996.
  * [37] K. Yajima, $W^{k,p}$-continuity of wave operators for Schrödinger operators, J. Math. Soc. Japan 47, 551-581 (1995)