Almost global existence for the nonlinear Klein-Gordon equation in the nonrelativistic limit

TL;DR

This paper proves that small solutions to the one-dimensional nonlinear Klein-Gordon equation with a convolution potential remain small over long times uniformly for large measure sets of the parameter c, in the nonrelativistic limit.

Contribution

It establishes almost global existence results for the nonlinear Klein-Gordon equation in the nonrelativistic limit with uniform bounds across large measure parameter sets.

Findings

Solutions with small initial data stay small for long times.

The results are uniform in the parameter c for c ≥ 1.

The set of c values for which the results hold has large measure.

Abstract

We study the one-dimensional nonlinear Klein-Gordon (NLKG) equation with a convolution potential, and we prove that solutions with small $H^s$ norm remain small for long times. The result is uniform with respect to $c \geq 1$, which however has to belong to a set of large measure.

Equations

$$\frac{1}{c^2}{u}_{tt}-u_{xx}+c^{2}u+V\ast u+{\partial }_{u}f(u)$$

$$V(x)$$

$$\mathcal{V}:={\mathcal{V}}_{s,M}$$

$${\omega }_j:={\omega }_j(c)$$

$$=c^2+\frac{{\lambda }_j}{2}-\frac{{\lambda }_j^2}{2c^2}\frac{1}{(1+\sqrt{1+{\lambda }_j/c^2}{)}^2},$$

$$\psi$$

$$H(\psi ,\bar{\psi })$$

$$H_0(\psi ,\bar{\psi })$$

$$N(\psi ,\bar{\psi })$$

$$\sim \sum _{l\ge 4}{\int }_I{N}_l(x){\left({\left(\frac{c}{(c^2-\Delta +\tilde{V}{)}^{1/2}}\right)}^{1/2}(\psi +\bar{\psi })\right)}^l\mathrm{d}x,$$

$${\left(\frac{c}{(c^2-\Delta +\tilde{V}{)}^{1/2}}\right)}^{1/2}:H^s\to H^s$$

$$|{\mathcal{R}}_{\gamma }\cap ([n,n+1]\times \mathcal{V})|$$

$${\mathcal{T}}_c:={\mathcal{T}}_c^{(r)}$$

$$H_r:=H\circ {\mathcal{T}}_c^{(r)}$$

$$Z^{(r)}(\psi ,\bar{\psi })$$

$$Z_n\ne 0$$

$$\underset{B_s(R/3)}{\sup }\Vert X_{R^{(r)}}(\psi ,\bar{\psi }){\Vert }_{H^s}$$

$$\underset{B_s(R/3)}{\sup }\Vert {\mathcal{T}}_c^{(r)}-id{\Vert }_{H^s}$$

$$\Vert ({\psi }_0,{\bar{\psi }}_0){\Vert }_{H^s}$$

$$\Vert (\psi (t),\bar{\psi }(t)){\Vert }_{H^s}$$

$$\Vert (I(t),\bar{I}(t)){\Vert }_{H^s}$$

$$d_{s_1}((\psi (t),\bar{\psi }(t)),I_c)$$

$$\Vert ({\psi }_0,{\bar{\psi }}_0){\Vert }_{H^s}$$

$$\Vert (\psi (t),\bar{\psi }(t)){\Vert }_{H^s}$$

$$\Vert (I(t),\bar{I}(t)){\Vert }_{H^s}$$

$$|\omega \cdot k+n|$$

$$\left|\frac{\partial p_{k,n}}{\partial v_h^{\prime }}\right|=\left|\frac{k_h{h}^{-s}}{2\sqrt{1+{\lambda }_h/c^2}}\right|$$

$$\left|\{(v_j^{\prime }{)}_{j\ge 1}:|p_{k,n}((v_j{)}_{j\ge 1})|<{\gamma }_0\}\right|$$

$$\left|\bigcup _{|n|\le rN}\bigcup _{\stackrel{0<|k|\le r}{supp(k)\subseteq \{1,\dots ,N\}}}\{(v_j^{\prime }{)}_{j\ge 1}:|p_{k,n}((v_j{)}_{j\ge 1})|<{\gamma }_0\}\right|$$

$$|{\mathcal{R}}_{\gamma }\cap ([n,n+1]\times \mathcal{V})|$$

$$\left|\sum _{j=1}^N{\omega }_j{k}_j+\sigma {\omega }_l\right|$$

$$p_{k,l}(c,(v_j{)}_{j\ge 1})$$

$$\sum _{j=1}^{N}c\sqrt{c^2+{\lambda }_j}{k}_j$$

Full text

Keywords: nonrelativistic limit, nonlinear Klein-Gordon

MSC2010: 37K55, 70H08, 70K45, 81Q05

1 Introduction

In this paper we study the nonlinear Klein-Gordon (NLKG) equation in the nonrelativistic limit, namely as the speed of light $c$ tends to infinity. Formal computations going back to the the first half of the last century suggest that, up to corrections of order ${\mathcal{O}}(c^{-2})$, the system should be described by the nonlinear Schrödinger (NLS) equation. Subsequent mathematical results have shown that the NLS describes the dynamics over time scales of order ${\mathcal{O}}(1)$.

The nonrelativistic limit for the Klein-Gordon equation on ${\mathbb{R}}^{d}$ has been extensively studied over more then 30 years, and essentially all results only show convergence of the solutions of NLKG to the solutions of the approximate equation fortimes of order ${\mathcal{O}}(1)$. The typical results, due to Masmoudi, Machihara, Nakanishi and Ozawa, ensure convergence locally uniformly in time, either with loss of regularity (see [Tsu84], [Naj90] and [Mac01]) or without loss of regularity (see [MNO02], [MN02]).

We also mention the recent papers [LZ16] and [Pas17], which discuss the long-time convergence of solution of NLKG in the nonrelativistic limit on ${\mathbb{R}}^{d}$: however, the results proved in both papers have some limitations, either on the nonlinearity (in [LZ16] the authors studied only the quadratic NLKG) or on the particular form of the solution (in [Pas17]).

Concerning the nonrelativistic limit of the NLKG on the $d$-dimensional torus ${\mathbb{T}}^{d}$, we mention the work by Faou-Schratz [FS14], in which the authors were able to justify the approximation of the solutions of NLKG by solutions of the NLS over time scales of order ${\mathcal{O}}(1)$ through a variant of Birkhoff Normal Form theory. In [Pas17] Birkhoff Normal Form Theory was exploited in order to recover the result by Faou-Schratz, and to generalize it to all smooth compact Riemannian manifolds.

Actually, [FS14] dealt with the construction of numerical schemes which are robust in the nonrelativistic limit; we refer also to [BD12], [BZ16] and to [BFS17] for the numerical analysis of the nonrelativistic limit of the NLKG equation.

In this paper we generalize the techniques developed in [BG06] in order to prove a long-time existence result for the NLKG with a convolution potential with Dirichlet boundary conditions, uniformly in $c\geq 1$.

An immediate corollary of our result allows us to show that for any $\alpha>0$ any solution in $H^{s}$ with initial datum of size ${\mathcal{O}}(c^{-\alpha})$ remains of size ${\mathcal{O}}(c^{-\alpha})$ up to times of order $O(c^{\alpha(r+1/2)})$ for any $r\geq 1$; however, we have to assume that both the parameter $c$ and the coefficients of the potential belong to a set of large measure. The main limitation of such a result is that it holds only for solutions with initial data which are small with respect to $c$.

The new ingredient in the proof with respect to [BG06] is a diophantine type estimate for the frequencies, which holds uniformly when $c\to\infty$.

An aspect that would deserve future work is the study of the nonrelativistic limit of the NLKG without potential. This is expected to be a quite subtle problem since, for $c\neq 0$ the frequencies of NLKG are typically non resonant, while the limiting frequencies are resonant. The issue of long-time existence for small solutions of the NLKG equation without potential on compact manifolds has received a lot of interest; see for example [DS04], [Del09], [FZ10], [FHZ17] and [DI17]. However, all results in the aforementioned papers rely on a nonresonance condition which is not uniform with respect to $c$.

The paper is organized as follows. In sect. 2 we state the results of the paper, together with some examples and comments. In sect. 3 we prove our result for the NLKG with a convolution potential on $I:=[0,\pi]$.

Acknowledgements. This work is based on author’s PhD thesis. He would like to express his thanks to his supervisor Professor Dario Bambusi.

2 Statement of the Main Results

Now consider the following equation:

[TABLE]

with $c\in[1,+\infty)$, $x\in I$, $f\in C^{\infty}({\mathbb{R}})$ a real-valued function with a zero of order four at the origin, with Dirichlet boundary conditions. The potential has the form

[TABLE]

By using the same approach of [BG06], we fix a positive $s$, and for any $M>0$ we consider the probability space

[TABLE]

and we endow the product probability measure on the space of $(c,(v^{\prime}_{j})_{j})$.

We recall that in this case the frequencies are given by

[TABLE]

where $\lambda_{j}=j^{2}+v_{j}$. Now we introduce the following change of coordinates,

[TABLE]

where ${\widetilde{V}}$ is the operator that maps $u$ to $V\ast u$. The Hamiltonian of (1) now reads

[TABLE]

Therefore the Hamiltonian takes the form

[TABLE]

where

[TABLE]

where $N_{l}\in C^{\infty}$ for each $l$ (since $V\in C^{\infty}$), and

[TABLE]

is a smoothing pseudodifferential operator, which can be estimated uniformly in $c\geq 1$.

Theorem 2.1**.**

Consider the equation (1) and fix $\gamma>0$, and $\tau>1$. Then for any $r\geq 1$ there exists $s^{\ast}>0$ and, for any $s>s^{\ast}$, there exists a set ${\mathcal{R}}_{\gamma}:={\mathcal{R}}_{\gamma,s,r}\subset\;[1,+\infty)\times{\mathcal{V}}$ satisfying

[TABLE]

and there exists $R_{s}>0$ such that for any $(c,(v_{j})_{j})\in([1,+\infty)\times{\mathcal{V}})\setminus{\mathcal{R}}_{\gamma}$ and for any $R<R_{s}$ there exist $N:=N(r,R)>0$, and a canonical transformation

[TABLE]

such that

[TABLE]

where $Z^{(r)}$ is a polynomial of degree (at most) $r+2$, such that

[TABLE]

and such that

[TABLE]

and we have that $Z^{(r)}$ depends on the actions $I=\psi\,\bar{\psi}$ only. Moreover, there exist $K^{\ast}_{s}>0$ and $K^{\prime}>0$ such that if the initial datum satisfies

[TABLE]

with $K<K^{\ast}_{s}$, then

[TABLE]

Finally, there exists a smooth torus $I_{c}$ such that for any $s_{1}<s-1/2$ there exists $K_{s_{1}}>0$ such that

[TABLE]

where $r_{1}\leq r$, and $d_{s_{1}}$ is the distance in $H^{s_{1}}$.

Remark 2.2**.**

The fact that $Z$ depends only on the actions is a direct consequence of the non-resonance property established in Theorem 3.3.

Remark 2.3**.**

It would also be interesting to study the dependence of $I_{c}$ on $c$. One could expect that it should converge to an invariant torus of the NLS with a convolution potential. We expect this fact to be true, but it needs further investigation for a proof. This is due to the fact that the NLS is the singular limit of NLKG and to the fact that $c$ is only allowed to vary in Cantor like sets, so that one can only expect a Whitney-smooth dependence.

By exploiting the same argument used to prove Theorem 2.1 one can immediately deduce the following almost global existence result for solutions with small (with respect to $c$) initial data.

Corollary 2.4**.**

Fix $\alpha>0$, $\gamma>0$, and $\tau>1$. Then for any $r\geq 1$ there exists $s^{\ast}>0$ and, for any $s>s^{\ast}$, there exists a set ${\mathcal{R}}_{\gamma,\tau,s,\alpha,r}\subset\;[1,+\infty)\times{\mathcal{V}}$ such that there exists $c^{\ast}>0$ such that for any $(c,(v_{j})_{j})\in([1,+\infty)\times{\mathcal{V}})\setminus{\mathcal{R}}_{\gamma,\tau,s,\alpha,r}$ with $c>c^{\ast}$ the following holds: there exist $K^{\ast}_{s}>0$ and $K^{\prime}>0$ such that if the initial datum satisfies

[TABLE]

with $K<K^{\ast}_{s}$, then

[TABLE]

3 Proof of Theorem 2.1

In order to prove Theorem 2.1, we need to show some nonresonance properties of the frequencies $\omega=(\omega_{j})_{j>0}$: it will be crucial that these properties hold uniformly (or at least, up to a set of small probability) in $(1,+\infty)\times{\mathcal{V}}$, since this will allow us to deduce a result which is valid in the nonrelativistic limit regime.

We use the notation “$a\lesssim b$” (resp. “$a\gtrsim b$”) to mean “there exists a constant $K>0$ independent of $c$ such that $a\leq Kb$” (resp. $a\geq Kb$).

Proposition 3.1**.**

Let $r\geq 1$, $c\geq 1$ be fixed. Then $\forall\gamma>0$ $\exists{\mathcal{V}}^{\prime}_{s,M,\gamma}\subset{\mathcal{V}}$ with $|{\mathcal{V}}\setminus{\mathcal{V}}^{\prime}_{s,M,\gamma}|={\mathcal{O}}(\gamma)$, and $\exists\tau>1$ s.t. $\forall(v_{j})_{j\geq 1}\in{\mathcal{V}}^{\prime}_{s,M,\gamma}$ and $\forall$ $N\geq 1$

[TABLE]

*for $0<|k|\leq r$, $supp(k)\subseteq\{1,\ldots,N\}$, and $\forall$ $n\in\mathbb{Z}$.

Proof.

Let $p_{k,n}((v_{j})_{j\geq 1})\;:=\;\sum_{j=1}^{N}\omega_{j}k_{j}+n$, and assume that $k_{h}\neq 0$ for some $h$. Then

[TABLE]

hence by Lemma 17.2 of [Rüs01]

[TABLE]

and by choosing $\gamma_{0}\;=\;\frac{\gamma}{N^{\tau}}$ with $\tau>s+r+2+max(s,2)/2$ we get that the left hand side of (20) is bounded, and therefore the thesis holds. ∎

Proposition 3.2**.**

Let $r\geq 1$ be fixed. Then $\forall\gamma>0$ there exists a set ${\mathcal{R}}_{\gamma}:={\mathcal{R}}_{\gamma,s,r}\subset\;[1,+\infty)\times{\mathcal{V}}$ satisfying

[TABLE]

and $\exists\tau>1$ such that $\forall(c,(v_{j})_{j})\in([1,+\infty)\times{\mathcal{V}})\setminus{\mathcal{R}}_{\gamma}$ and $\forall$ $N\geq 1$

[TABLE]

for $0<|k|\leq r$, $supp(k)\subseteq\{1,\ldots,N\}$, $\sigma=\pm 1$, $l\geq N$.

Proof.

Without loss of generality, we can choose $\sigma=-1$.

Now fix $k\in{\mathbb{Z}}^{N}$ with $0<|k|\leq r$, and fix $l\geq N$. Set $p_{k,l}(c,(v_{j})_{j\geq 1})\;:=\;\sum_{j=1}^{N}\omega_{j}k_{j}-\omega_{l}$. We can rewrite the function $p_{k,l}$ in the following way:

[TABLE]

where $\alpha:=(\sum_{j=1}^{N}k_{j})-1\in\{-r-1,\ldots,r-1\}$. Now we have to distinguish some cases:

Case $\alpha=0$: in this case we have that $p_{k,l}$ can be small only if $l^{2}\leq 3(N^{2}+N^{s})^{2}r^{2}$. So to obtain the result we just apply Proposition 3.1 with $N^{\prime}:=\sqrt{3}(N^{2}+N^{s})r$, $r^{\prime}=r+1$.

Case $\alpha\neq 0$, $c\leq\lambda_{N}^{1/2}r^{1/2}$: we have that

[TABLE]

so $|\sum_{j=1}^{N}\omega_{j}k_{j}-\omega_{l}|$ can be small only for $l^{2}<rN^{2}$. Therefore, in order to get the thesis we apply Proposition 3.1 with $N^{\prime}:=\sqrt{r}N$, $r^{\prime}:=r+1$.

Case $\alpha>0$, $c>\lambda_{N}^{1/2}r^{1/2}$: first notice that if we set $f(x):=\frac{x^{2}}{2\sqrt{1+x}(1+\sqrt{1+x})^{2}}$, and we put $x_{j}:=\lambda_{j}/c^{2}$, in this regime we get

[TABLE]

Now define $\tilde{p}_{k,l}(c^{2}):=\alpha c^{2}\;-\;\frac{\lambda_{l}}{1+\sqrt{1+\lambda_{l}/c^{2}}}$. One can verify that

[TABLE]

and that

[TABLE]

Besides, in an interval $\left[c^{2}_{l,\alpha}-\frac{\varrho}{\alpha(\alpha+2)},c^{2}_{l,\alpha}+\frac{\varrho}{\alpha(\alpha+2)}\right]\;=:\;[c^{2}_{l,\alpha,-},c^{2}_{l,\alpha,+}]$ we have that

[TABLE]

Then, by exploiting Lemma 17.2 of [Rüs01], we get that

[TABLE]

for any $\gamma>0$ s.t. $\gamma\frac{2(r+1)}{(r+2)\alpha}<\frac{\varrho}{\alpha(\alpha+2)}$; $\gamma<\frac{(r+2)\varrho}{2(r+1)(\alpha+2)}$.

Now, since in this regime $\left|\frac{\partial(p_{k,l}-\tilde{p}_{k,l})}{\partial c^{2}}\right|\leq\frac{1}{2}$, we can deal with $p_{k,l}$ in a similar way as before, and we can conclude that

[TABLE]

Case $\alpha<0$, $c>\lambda_{N}^{1/2}r^{1/2}$: since

[TABLE]

we have that $p_{k,l}$ can be small only if $\lambda_{N}<r\lambda_{N}$. So, in order to get the result, we apply Proposition 3.1 with $N^{\prime}:=r^{1/2}N$, $r^{\prime}:=r+1$. ∎

Theorem 3.3**.**

Let $r\geq 1$ be fixed. Then $\forall\gamma>0$ there exists a set ${\mathcal{R}}_{\gamma}:={\mathcal{R}}_{\gamma,s,r}\subset\;[1,+\infty)\times{\mathcal{V}}$ satisfying

[TABLE]

and $\exists\tau>1$ such that $\forall(c,(v_{j})_{j})\in([1,+\infty)\times{\mathcal{V}})\setminus{\mathcal{R}}_{\gamma}$ and $\forall$ $N\geq 1$

[TABLE]

for $0<|k|\leq r$, $supp(k)\subseteq\{1,\ldots,N\}$, $\sigma_{1},\sigma_{2}\in\{\pm 1\}$, $m>l\geq N$.

Proof.

If $\sigma_{i}=0$ for $i=1,2$, then we can conclude by using Proposition 3.2.

Now, consider the case $\sigma_{1}=-1$, $\sigma_{2}=1$, and denote

[TABLE]

Now fix $\delta>3$. If $m\lesssim N^{\delta}$, then we can conclude by applying Proposition 3.1 and 3.2. So from now on we will assume that $m,l>N^{\delta}$.

We have to distinguish several cases:

Case $c<\lambda_{l}^{\alpha}$: we point out that, since

[TABLE]

we get (denote $m=l+j$)

[TABLE]

that is, the integer multiples of c are accumulation points for the differences between the frequencies as $l,m\to\infty$, provided that $\alpha<\frac{1}{6}$.

Case $c>\lambda_{m}$: in this case we have (again by denoting $m=l+j$) that $\lambda_{m}-\lambda_{l}=j(j+2l)+(v_{m}-v_{l})\;=\;2jl+j^{2}+a_{lm}$, with $|a_{lm}|\leq\frac{C}{l}$, so that

[TABLE]

If $l>2\,CN^{\tau}/\gamma$ then the term $a_{lm}$ represents a negligible correction and therefore we can conclude by applying Proposition 3.1. On the other hand, if $l\leq 2\,CN^{\tau}/\gamma$, we can apply the same Proposition with $N^{\prime}:=2CN^{\tau}/\gamma$ and $r^{\prime}:=r+2$.

Case $\lambda_{l}^{1/6}\leq c\lesssim\lambda_{l}^{1/2}$: if we rewrite the quantity to estimate

[TABLE]

where $\alpha:=\sum_{h=1}^{N}k_{h}$, we distinguish three cases:

• •

if $\alpha>0$, then we notice that

[TABLE]

[TABLE]

thus $|p_{k,l,m}|>|\lambda_{l}^{1/3}-\frac{r}{2}\lambda_{N}|>0$, since $l>N^{3}$;

• •

if $\alpha=0$, then we just notice that

[TABLE]

which is greater than $\gamma_{0}/N^{\tau}$ for $\tau>-1$, since $l>N^{3}$;

• •

if $\alpha<0$, then we just recall that $|\omega_{m}-\omega_{l}|>\gamma_{0}\lambda_{l}^{1/2}$, and by choosing $\gamma_{0}$ sufficiently small (actually $\gamma_{0}\leq|\alpha|$) we get that also in this case $p_{k,l,m}$ is bounded away from zero.

∎

The proof is based on the method of Lie transform. Let $s>s^{\ast}$ be fixed.

Given an auxiliary function $\chi$ analytic on $H^{s}$, we consider the auxiliary differential equation

[TABLE]

and denote by $\Phi^{t}_{\chi}$ its time-$t$ flow. A simple application of Cauchy inequality gives

Lemma 3.4**.**

Let $\chi$ and its symplectic gradient be analytic in $B_{s}(\rho)$. Fix $\delta<\rho$, and assume that

[TABLE]

Then, if we consider the time-$t$ flow $\Phi^{t}_{\chi}$ of $X_{\chi}$ we have that for $|t|\leq 1$

[TABLE]

The map $\Phi:=\Phi^{1}_{\chi}$ will be called the Lie transform generated by $\chi$.

Given a homogeneous polynomial $f$ of degree $m$, we denote, following [BG06], its modulus

[TABLE]

where $f_{j}$ is given by

[TABLE]

Furthermore, given a multivector

[TABLE]

we introduce the following norm

[TABLE]

Definition 3.5**.**

Let $X:H^{s}\oplus H^{s}\to H^{s}\oplus H^{s}$ be a homogeneous polynomial of degree $r$,

[TABLE]

Consider the $r$-linear symmetric form $\tilde{X}_{l}$ such that $\tilde{X}_{l}(\psi,\bar{\psi},\ldots,\psi,\bar{\psi})=X_{l}(\psi,\bar{\psi})$, and set

[TABLE]

*so that $\tilde{X}(\psi,\bar{\psi},\ldots,\psi,\bar{\psi})=X_{l}(\psi,\bar{\psi})$.

Let $s\geq 1$, then we say that X is an $s$-tame map if there exists $K_{s}>0$ such that*

[TABLE]

If a map is $s$-tame for any $s\geq 1$, then it will be said to be tame.

Definition 3.6**.**

Let us consider a vector field $X:H^{s}\oplus H^{s}\to H^{s}\oplus H^{s}$, and denote by $X_{l}$ its $l$-th component. We define its modulus by

[TABLE]

A polynomial vector field $X$ is said to hace $s$-tame modulus if its modulus $\lfloor X\rceil$ is an $s$-tame map. The set of polynomial functions $f$, whose Hamiltonian vector fields has $s$-tame modulus will be denoted by $T^{s}_{M}$. If $f\in T^{s}_{M}$ for any $s>1$, we will write $f\in T_{m}$, and say that $f$ has tame modulus.

Remark 3.7**.**

The property of having tame modulus depends on the coordinate system.

Definition 3.8**.**

Let $X$ be an $s$-tame vector field homogeneous polynomial of degree $r$. The infimum of the constants $K_{s}$ such that the inequality

[TABLE]

holds will be called tame $s$ norm of $X$, and will be denoted by $|X|^{T}_{s}$.

The tame $s$ norm of a polynomial Hamiltonian f of degree $r+1$ is given by

[TABLE]

where the $\sup$ is taken over all multivectors $\phi=(\phi^{(1)},\ldots,\phi^{(r)})$ such that $\phi^{(j)}\neq 0$ for any $j$.

Definition 3.9**.**

Let $f\in T^{s}_{M}$ be a non-homogeneous polynomial, and consider its Taylor expansion

[TABLE]

where $f_{m}$ is homogeneous of degree $m$. Let $R>0$, then we denote

[TABLE]

Such a definition extends naturally to analytic functions such that (29) is finite. The set of functions of class $T^{s}_{M}$ for which (29) is finite will be denoted by $T_{s,R}$.

With the above definitions,

[TABLE]

It is easy to check that the set $T_{s,R}$ endowed with the norm (29) is a Banach space.

Now we introduce the Fourier projection

[TABLE]

and we split the variables $(\psi,\bar{\psi})$ into

[TABLE]

The use of Fourier projection is important in view of the following result, whose proof can be found in Appendix A of [BG06].

Lemma 3.10**.**

Fix $N$, and consider the decomposition $\psi=\psi_{l}+\psi_{h}$ as above. Let $f\in T^{s}_{M}$ be a polynomial of degree less or equal than $r+2$. Assume that $f$ has a zero of order three in the variables $(\psi_{h},\bar{\psi}_{h})$, then one has

[TABLE]

Lemma 3.11**.**

Let $f,g\in T^{s}_{M}$ be homogeneous polynomial of degrees $n+1$ and $m+1$ respectively. Then one has $\{f,g\}\in T^{s}_{M}$, and

[TABLE]

The proof of this lemma can be found again in Appendix A of [BG06].

Remark 3.12**.**

Given $g$ analytic on $H^{s}\oplus H^{s}$, consider the differential equation

[TABLE]

where by $X_{g}$ we denote the vector field of $g$. Now define

[TABLE]

In the new variables $(\phi,\bar{\phi})$ defined by $(\psi,\bar{\psi})=\Phi(\phi,\bar{\phi})$ equation (32) is equivalent to

[TABLE]

Using the relation

[TABLE]

we formally get

[TABLE]

In order to estimate the terms appearing in (34) we exploit the following results

Lemma 3.13**.**

Let $h,g\in T_{s,R}$, then for any $d\in(0,R)$ one has that $\{h,g\}\in T_{s,R-d}$, and

[TABLE]

Proof.

Write $h=\sum_{j}h_{j}$ and $g=\sum_{k}g_{k}$, with $h_{j}$ homogeneous of degree $j$ and similarly for $g$. Then we have

[TABLE]

where $\{h_{j},g_{k}\}$ has degree $j+k-2$. Therefore by (31) in Lemma 3.11

[TABLE]

where we exploited the inequality $k(R-d)^{k-1}<R^{k}/d$, which holds for any positive $R$ and $d\in(0,R)$. ∎

Lemma 3.14**.**

Let $g,\chi\in T_{s,R}$ be analityc functions; denote by $g_{l}$ the functions defined recursively by (34); then for any $d\in(0,R)$ one has that $g_{l}\in T_{s,R-d}$, and

[TABLE]

Proof.

Fix $l$, and denote $\delta:=d/l$. We look for a sequence $C^{(l)}_{m}$ such that

[TABLE]

By (37) in Lemma 3.13 we can define the sequence

[TABLE]

One has

[TABLE]

and using the inequality $l^{l}<l!e^{l}$ one can conclude. ∎

Lemma 3.15**.**

*Let $f\in T^{s}_{M}$ be a polynomial which is at most quadratic in the variables $(\psi_{h},\bar{\psi}_{h})$.

Then $\forall\gamma>0$ there exists a set ${\mathcal{R}}_{\gamma}\subset\;[1,+\infty)\times{\mathcal{V}}$ satisfying*

[TABLE]

and $\exists\tau>1$ such that $\forall(c,(v_{j})_{j})\in([1,+\infty)\times{\mathcal{V}})\setminus{\mathcal{R}}_{\gamma}$ and $\forall$ $N\geq 1$ the following holds: there exist $\chi,Z\in T_{s,R}$ such that

[TABLE]

and such that $Z$ depends only on the actions and satisfies (12). Moreover, $\chi$ and $Z$ satisfy the following estimates

[TABLE]

Proof.

Expanding $f$ in Taylor series, namely $f(\psi,\bar{\psi})=\sum_{j,l}f_{j,l}\psi^{j}\bar{\psi}^{l}$, and similarly for $\chi$ and $Z$, equation (39) becomes an equation for the coefficients of $f$, $\chi$ and $Z$,

[TABLE]

Then we define

[TABLE]

By construction and by Theorem 3.3 we get estimates (40) and (41). Furthermore, since $f$ is at most quadratic in $(\psi_{h},\bar{\psi}_{h})$, we obtain that $\sum_{k>N}j_{k}\leq 2$, and thus $Z$ satisfies (12). ∎

Remark 3.16**.**

Let $s>s^{\ast}$, and assume that $\chi$, $F$ are analytic on $B_{s}(R)$. Fix $d\in(0,R)$, and assume also that

[TABLE]

Then by Lemma 3.4 for $|t|\leq 1$

[TABLE]

Lemma 3.17**.**

Let $\chi\in T_{s,R}$ be the solution of the equation (39), with $f\in T^{s}_{M}$. Denote by $H_{0,l}$ the functions defined recursively via (34) from $H_{0}$. Then for any $d\in(0,R)$ one has that $H_{0,l}\in T_{s,R-d}$, and

[TABLE]

Proof.

Using (39) one gets $H_{0,1}=Z-f\in T^{s}_{M}$. Then, arguing as for (38), one can conclude. ∎

The main step of the proof of Theorem 2.1 is the following result, that allows to increase by one the order of the perturbation. As a preliminary step, we take the Taylor series of $N(\psi,\bar{\psi})$ up to order $r+2$,

[TABLE]

where $N_{l}$ is a homogeneous polynomial in $\psi$ and $\bar{\psi}$ of degree $l+2$ with variable $C^{\infty}$-coefficients (since $V\in C^{\infty}$).

Now we consider the analytic Hamiltonian

[TABLE]

Then for $R$ sufficiently small one has that

[TABLE]

Lemma 3.18**.**

*Consider the Hamiltonian (51), and fix $s>s^{\ast}$.

Then $\forall\gamma>0$ there exists a set ${\mathcal{R}}_{\gamma}\subset\;[1,+\infty)\times{\mathcal{V}}$ satisfying*

[TABLE]

and $\exists\tau>1$ such that $\forall(c,(v_{j})_{j})\in([1,+\infty)\times{\mathcal{V}})\setminus{\mathcal{R}}_{\gamma}$ and $\forall$ $N\geq 1$ the following holds: for any $m\leq r$ there exists $R^{\ast}_{m}\ll 1$ and, for any $N>1$ there exists an analytic canonical transformation

[TABLE]

such that

[TABLE]

where for any $R<R^{\ast}_{m}/N^{\tau}$ the following properties are fulfilled

the transformation ${\mathcal{T}}^{(m)}$ satisfies

[TABLE] 2. 2.

$Z^{(m)}$* is a polynomial of degree (at most) $m+2$ that depends only on the actions $I=\psi\,\bar{\psi}$ and satisfies (12); $f^{(m)}$ is a polynomial of degree (at most) $r+2$. Moreover*

[TABLE] 3. 3.

the remainder terms ${\mathcal{R}}^{(m)}_{N}$ and ${\mathcal{R}}^{(m)}_{T}$ satisfy

[TABLE]

Proof.

We argue by induction. The theorem is trivial for the case $m=0$, by setting ${\mathcal{T}}^{(0)}=id$, $Z^{(0)}=0$, $f^{(0)}=N^{(1)}$, ${\mathcal{R}}^{(m)}_{N}={\mathcal{R}}^{(m)}_{T}=0$.

Then we split $f^{(m)}$ into two parts, an effective one and a remainder. Indeed, we perform a Taylor expansion of $f^{(m)}$ only in the variables $(\psi_{h},\bar{\psi}_{h})$, namely we write

[TABLE]

where $f^{(m)}_{0}$ is the truncation of such a series at second order, and $f^{(m)}_{0}$ is the remainder. Since both $f^{(m)}_{0}$ and $f^{(m)}_{N}$ are truncations of $f^{(m)}$, one has that

[TABLE]

Now consider the truncated Hamiltonian $H_{0}+Z^{(m)}+f^{(m)}_{0}$: we look for a Lie transform ${\mathcal{T}}_{m}$ that eliminates the non-normalized part of order $m+4$ of the truncated Hamiltonian. Let $\chi_{m}$ be the analytic Hamiltonian generating ${\mathcal{T}}_{m}$. Using (34) we have

[TABLE]

with $Z^{(m)}_{l}$ the $l$-th term in the expansion of the Lie transform of $Z^{(m)}$, and similarly for the other quantities. It is easy to see that the terms in the first line are already normalized, that the term in the (60) is the non-normalized part of order $m+3$ that will vanish through the choice of a suitable $\chi_{m}$, and that the last lines contains all the terms having a zero of order $m+4$ at the origin.

Now we want to determine $\chi_{m}$ in order to solve the so-called “homological equation”

[TABLE]

with $Z_{m+1}$ depending only on the actions and satisfying (12). The existence of $\chi_{m}$ and $Z_{m+1}$ is ensured by Lemma 3.15, and by applying (41) and (56) we get

[TABLE]

In particular, in view of (46), we can deduce (55) at level $m+1$. Now define $Z^{(m+1)}:=Z^{(m)}+Z_{m+1}$, and $f_{C}^{(m+1)}:=\eqref{nonnorm2s}$. By 62, recalling that $R<R_{m}^{\ast}/N^{\tau}$, we can deduce (56) at level $m+1$. Moreover, provided that $R_{m}^{\ast}<2^{-(m+1)/2}$, one has

[TABLE]

By (38) and (56) one thus gets

[TABLE]

Write now $f_{C}^{(m+1)}=f^{(m+1)}+{\mathcal{R}}_{m,T}$, where $f^{(m+1)}$ is the Taylor polynomial of order $r+2$ of $f_{C}^{(m+1)}$, and where ${\mathcal{R}}_{m,T}$ has a zero of order $r+3$ at the origin. Clearly $f^{(m+1)}$ satisfies (57) at level $m+1$, since it is a truncation of $f_{C}^{(m+1)}$. The remainder may be bounded by using Lagrange and Cauchy estimates,

[TABLE]

Now define ${\mathcal{R}}_{T}^{(m+1)}:={\mathcal{R}}_{T}^{(m)}\circ{\mathcal{T}}_{m}+{\mathcal{R}}_{m,T}$. By (46) we can deduce (58) at level $m+1$. Then set ${\mathcal{R}}_{N}^{(m+1)}:=({\mathcal{R}}_{N}^{(m)}+f_{N}^{(m)})\circ{\mathcal{T}}_{m}$. By (57) and (59), together with (46) and (30) in Lemma 3.10, we obtain (59) at level $m+1$. ∎

Now we conclude the proof of Theorem 2.1.

By taking the canonical transformation ${\mathcal{T}}^{(r)}$ defined in the iterative Lemma 3.18 we have that

[TABLE]

with $Z^{(r)}$ depending only on the actions and satisfying (12), and for any $R<R^{\ast}_{m}/N^{\tau}$ the following holds

[TABLE]

To conclude we have just to choose $N$ and $s$ sich that ${\mathcal{R}}_{N}^{(r)}$ and ${\mathcal{R}}_{T}^{(r)}$ are of the same order of magnitude. First take $N=R^{-a}$, with $a$ still to be determined; then, in order to obtain that ${\mathcal{R}}_{T}^{(r)}$ is of order ${\mathcal{O}}(R^{r+3/2})$ we choose $a:=\frac{1}{2\tau(r+2)}$. By taking $s>2\tau r(r+2)+1$ we get that also $N^{(1,r)}$ is of the same order of magnitude.

Now take $K^{\ast}=1/24$, and construct the canonical transformation $(\psi,\bar{\psi})={\mathcal{T}}^{(r)}(\psi^{\prime},\bar{\psi}^{\prime})$. Denote by $I^{\prime}$ the actions expressed in the variable $(\psi^{\prime},\bar{\psi}^{\prime})$, and define the function ${\mathcal{N}}(\psi^{\prime},\bar{\psi}^{\prime}):=\|I^{\prime}\|^{2}_{s}$. By (14) one has that ${\mathcal{N}}(\psi^{\prime}_{0},\bar{\psi}^{\prime}_{0})\leq\frac{32}{31}R^{2}$, provided that $R$ is sufficiently small. Since

[TABLE]

and therefore, as far as ${\mathcal{N}}(\psi^{\prime},\bar{\psi}^{\prime})<\frac{64}{9}R^{2}$,

[TABLE]

Denote by $T_{f}$ the escape time of $(\psi^{\prime},\bar{\psi}^{\prime})$ from $B_{s}(R/3)$; observe that for all times smaller than $T_{f}$, (65) holds. So one has

[TABLE]

which shows that $T_{f}$ should be of order (at least) $R^{r+1/2}$. Going back to the original variables one gets (16). To show (17), one has to recall that

[TABLE]

and that by (14) and (16) one can estimate the first and the third term; the second term can be bounded by computing the time derivative of $\|I^{\prime}\|^{2}_{s}$ with the Hamiltonian, and observing that it is of order ${\mathcal{O}}(R^{r+5/2})$.

Now, consider the initial actions $(I_{0},\bar{I}_{0}):=(I(0),\bar{I}(0))$. By passing to the Fourier transform,

[TABLE]

we have that for any $r_{1}\leq r$

[TABLE]

If we define the torus

[TABLE]

we get

[TABLE]

and by using (66) we obtain

[TABLE]

which is convergent for $s_{1}<s-1/2$.