# About linearization of infinite-dimensional Hamiltonian systems

Michela Procesi† Department of Mathematics and Physics, Roma 3
<EMAIL_ADDRESS>and  Laurent Stolovitch†† CNRS and Laboratoire J.-A.
Dieudonné U.M.R. 7351, Université Côte d’Azur, Parc Valrose 06108 Nice Cedex
02, France<EMAIL_ADDRESS>
###### Abstract.

This article is concerned with analytic Hamiltonian dynamical systems in
infinite dimension in a neighborhood of an elliptic fixed point. Given a
quadratic Hamiltonian, we consider the set of its analytic higher order
perturbations. We first define the subset of elements which are formally
symplectically conjugacted to a (formal) Birkhoff normal form. We prove that
if the quadratic Hamiltonian satisfies a Diophantine-like condition and if
such a perturbation is formally symplectically conjugated to the quadratic
Hamiltonian, then it is also analytically symplectically conjugated to it. Of
course what is an analytic symplectic change of variables depends strongly on
the choice of the phase space. Here we work on periodic functions with Gevrey
regularity.

††Research of L. Stolovitch was supported by the French government, through
the UCAJEDI Investments in the Future project managed by the National Research
Agency (ANR) with the reference number ANR-15-IDEX-01.

##  1\. Introduction

In finite dimension, studying the behaviour of the orbits of a vector field
(or of diffeomorphism) nearby a fixed point is a fundamental and classical
problem. The very first natural step into this understanding is to compare the
dynamical system with its linearization at the fixed point. This is done by
trying to transform the dynamical system into its linear part by a change of
coordinates. There are formal obstructions to do so, called resonances. Hence,
in general, one can merely expect the dynamical system to be transformed into
a normal form, that is supposed to capture effect the very nonlinearities,
through a formal change of coordinates. It was understood by the end of the
19th century that if the convex hull of the eigenvalues of the linear part
does not contain the origin (one says then that the linear part is in the
”Poincaré domain”), and if an higher order analytic perturbation is formally
conjugate to the linear part, then it is also analytically so. When the linear
part does not satisfy this property, then one has so-called ”small divisors”
that may forbid the transformation to be analytic. It was a major step forward
made by C.L. Siegel [Sie42], followed by H. Rüssmann [Rue77](for
diffeomorphisms) and by A.D. Brjuno [Bru72] (for vector fields) who devised a
sufficient ”small divisors condition” ensuring the analycity of a linearizing
transformation as soon as there exists a formal one. Linearizing (resp.
Normalizing) problems for diffeomorphisms were devised by J. Pöschel [Pös86]
and for commuting families by the second author [Sto15] (resp. [Sto00]). By
the end of the 70’s, it became clear to few people that some PDE’s problems
could be translated into an infinite dimensional dynamical systems to which
one would have tried to apply methods of finite dimension. In particular, we
mention the work by E. Zehnder [Zeh77] and V. Nikolenko [Nik86] who gave
results similar to finite dimensional ones. It happens that the ”small
divisors condition” they required are too strong and are rarely satisfied.
Furthermore, in general, the notion of formal normal form and formal change of
variables should be clarified (for instance if one defines formal polynomials
and formal power series it is not in general true that this space has a
Poisson algebra structure). Neverteless, in some very peculiar situation, this
problem can be handled[BS20].
Starting from the mid 80’, there has been a lot of interest in studying long
time behavior of solutions of PDEs. For those PDEs which can be considerered
as Hamiltonian (infinite dimensional) dynamical systems related to a
symplectic sturucture, one natural way to proceed is to prove the existence of
finite dimensional invariant tori in the phase space. This usually implies the
existence of quasi-periodic solutions, which are defined for all time. Lot of
progresses has been done on the problem of extending KAM theory to PDEs. This
circle of problems are very related, though distinct, to the ones solved in
this article. Indeed, here one considers a dynamical system close to an
elliptic fixed point with the purpose of conjugating it to its most simple
normal form : its linear part at the fixed point. On the other hand, in KAM
theory, one looks for the existence of a finite dimensional invariant flat
torus on which the dynamics is the linear translation by a diophantine
frequency. There is by now a wide literature dealing the subject related to
semilinear PDEs, starting from [K88, Pös90, KP96, Way90, CW93], (for instance,
see [EK10, GYX, PP16, BKM18, Y21] for more recent treatments). It has been
early understood that these results might be seen through elaborated versions
of ”Nash-Moser” theorem see for instance [Bou98, BB15, BCP, CM18]. We finally
mention [FGPr, BBHM, BM21, FG] for the case of fully-nonlinear PDEs. See also
[BMP21, CY21] and references therein for infinite-dimensional tori.

Birkhoff normal form (BNF) methods have been used in order to prove long time
existence results and control of Sobolev norms for many classes of evolution
PDEs close to an elliptic fixed point. Loosely speaking the point is to
canonically transform $H$ into a Hamiltonian Normal form which depends only on
the actions plus a remainder term whose the Taylor polynomial, at the origin
is of degree ${\mathtt{N}}+1$. If one achieves this then initial data which
are $\delta$-small (with respect to the norm on the phase space) stay small
(in the same norm) for times of order $\delta^{-\mathtt{N}}$. A more precise
formulation is given in the Strategy section below. Of course in the infinite
dimensional setting this stability time depends strongly on the choice of the
phase space as well as on the nature of the non-linear terms. A further
problem is that in general it is not obvious that one can perform even one
step of this procedure, indeed the generating function of the desired change
of variables is a formal polynomial which in infinite dimension is not
necessarily analytic. This is a particularly difficult problem in the case of
PDEs with derivatives in the nonlinearity.
Let us briefly describe some of the literature. Regarding applications to PDEs
(and particularly the NLS) the first results were given in [Bou96a] by
Bourgain, who proved that for any $\mathtt{N}$ there exists $p=p(\mathtt{N})$
such that small initial data in the $H^{p^{\prime}+p}$ norm stay small in the
$H^{p^{\prime}}$ norm, for times of order $\delta^{-\mathtt{N}}$. Afterwards,
Bambusi in [Bam99b] proved that superanalytic initial data stay small in
analytic norm for subexponentially long times. Following the strategy proposed
in [Bam03] for the Klein-Gordon equation Bambusi and Grébert in [BG03] first
considered NLS equations on $\mathbb{T}^{d}$ and then, in [BG06], proved
polynomial bounds for a class of tame-modulus PDEs. Similar results were also
proved for the Klein Gordon equation on tori and Zoll manifolds in
[DS04],[DS06],[BDGS07]. Successively Faou and Grébert in [FG13] considered the
case of analytic initial data and proved subexponential bounds on the
stability time for classes of NLS equations in $\mathbb{T}^{d}$. In [BMP18]
the first author with Biasco and Massetti studied an abstract Birkhoff normal
form on sequence spaces proving subexponential stability times for Gevrey
regular initial data. A similar result was proved in [CMW]. An interesting
feature of the last three papers is that instead on relying on tameness
properties they use the fact that the equations they study have some
symmetries, namely they are gauge and translation invariant (actually in
[BMP18] the translation invariance condition is weakened).
All the preceding results regard semilinear PDEs. Regarding equations with
derivatives in the nonlinearity, the first results were in [YZ14] for the
semilinear case. Then we mention [Del12, D15] for the Klein-Gordon equation,
[BD18] for the water waves and [FI18] for the reversible NLS equation.
Recently, Feola and Iandoli, [FI20] prove polynomial lower bounds for the
stability times of Hamiltonian NLS equations with two derivatives in the
nonlinearity. In the context of infinite chains with a finite range coupling,
similar considerations can be done and we mention [BFG88].

###  1.1. Statements

We study Hamiltonians on infinite dimensional sequence spaces, which are
higher order ($M$-regular) analytic perturbations of quadratic Hamiltonians
nearby an elliptic fixed point (i.e a zero) and satisfying the Momentum
conservation property, namely they are formally translation invariant, see
Definition 8.
We first show that the space ${\mathcal{F}}$ of formal Hamiltonians in
infinite variables $u={\left(u_{j}\right)}_{j\in\mathbb{Z}}$ satisfying this
Momentum conservation property is well defined and closed w.r.t Poisson
brackets, then we define a scaling degree (which is the homogeneity degree
minus two, see Definition 2.3, so that the degree of the Poisson bracket of
two functions is the sum of the respective degrees) so that ${\mathcal{F}}$
has a natural filtered Lie algebra structure. Thus ${\mathcal{F}}$ is
decomposed in homogeneous components ${\mathcal{F}}^{d}$ and we define
${\mathcal{F}}^{\geq d}:=\widehat{\oplus}_{h\geq d}{\mathcal{F}}^{h}$.
Given a rationally independent $\omega\in\mathbb{R}^{\mathbb{Z}}$, namely such
that all non-trivial finite rational combinations of $\omega$ are non zero, we
consider the affine space $D_{\omega}+{\mathcal{F}}^{\geq 1}$ of formal
Hamiltonians of the form

(1) $H=D_{\omega}+P\,,\quad
D_{\omega}=\sum_{j\in\mathbb{Z}}\omega_{j}|u_{j}|^{2}\,,\qquad P=O(u^{3})\,,$

and acting on this space we define the group of formal symplectic (i.e
canonical) transformations $e^{{\left\\{{\mathcal{F}}^{\geq
1},\cdot\right\\}}}$. Finally we define the space of normal forms as those
formal Hamiltonians which Poisson commute with $D_{\omega}$. We prove the
following

######  Theorem.

All Hamiltonians $H$ as above are formally symplectically conjugated to normal
form. Moreover the normal form Hamiltonian associated to $H$ is unique.

Having properly developed the formal framework, we consider the question of
formal vs. analytic linearization in the infinite dimensional setting on the
phase space of Gevrey regular functions.
In order to keep technical difficulties to a minimum, we work on Nonlinear
Schrödinger like Hamiltonians of the form with the standard symplectic
structure on $\ell_{2}=\ell_{2}(\mathbb{Z},\mathbb{C})$. As phase space we
consider the sequences of Gevrey regularity, namely we consider the weighted
space

(2)
${\mathtt{h}}_{s,p,\theta}:={\left\\{u\in\ell^{2}(\mathbb{Z},\mathbb{C}):\quad|u|^{2}_{s}:=\sum_{j\in\mathbb{Z}}\langle
j\rangle^{2p}e^{2s\langle j\rangle^{\theta}}|u_{j}|^{2}<\infty\right\\}}$

where $\langle j\rangle:=\max(|j|,1)$, $s>0$, $p\geq\frac{1}{2}$ and
$0<\theta<1$. Then, given $r>0$, we consider the space of M-regular
Hamiltonians $P\in{\mathcal{H}}_{r}({\mathtt{h}}_{s,p,\theta})$, such that the
Cauchy majorant of the map $u\to X_{P}(u)$ is analytic from the ball
$B_{r}({\mathtt{h}}_{s,p,\theta})$, centered at the origin and of radius $r$
into ${\mathtt{h}}_{s,p,\theta}$.
Now we consider a Hamiltonian as in (1), with the additional condition that
$P\in{\mathcal{H}}_{r_{0}}({\mathtt{h}}_{s_{0},p,\theta})$ and the frequency
$\omega$ is “Diophantine” in the following sense introduced by Bourgain
[Bou05]. We set

(3)
$\Omega:={\left\\{\omega={\left(\omega_{j}\right)}_{j\in\mathbb{Z}}\in\mathbb{R}^{\mathbb{Z}},\quad\sup_{j}|\omega_{j}-j^{2}|<1/2\right\\}}$

######  Definition 1.1.

Given $\gamma>0$ , we denote by ${\mathtt{D}_{\gamma}}$ the set of Diophantine
frequencies

(4)
${\mathtt{D}_{\gamma}}:={\left\\{\omega\in\Omega\,:\;|\omega\cdot\ell|>\gamma\prod_{n\in\mathbb{Z}}\frac{1}{(1+|\ell_{n}|^{2}\langle
n\rangle^{2})}\,,\quad\forall\ell\in\mathbb{Z}^{\mathbb{Z}}_{f}\setminus\\{0\\}\right\\}}.$

The map
${\left(\omega_{j}\right)}_{j\in\mathbb{Z}}\to{\left(j^{2}-\omega_{j}\right)}_{j\in\mathbb{Z}}$
identifies $\Omega$ with $[-1/2,1/2]^{\mathbb{Z}}$. Hence we endow $\Omega$
with the product topology and with the corresponding probability measure. With
respect to such measure Diophantine frequencies are typical, namely
$\Omega\setminus{\mathtt{D}}_{\gamma}$ has measure proportionally bounded by
$\gamma$ (see [BMP18][Lemma 4.1]).
Then we prove :

######  Theorem.

If $H$ is formally conjugated to $D_{\omega}$, then there exists
$r_{1}<r_{0}$, $s_{1}>s_{0}$ and a close to identity analytic symplectic
change of variables
$\Psi:B_{r_{1}}({\mathtt{h}}_{s_{1},p,\theta})\to{\mathtt{h}}_{s_{1},p,\theta}$
such that $H\circ\Psi=\sum_{j\in\mathbb{Z}}\omega_{j}|u_{j}|^{2}$.

###  1.2. Strategy

In order to describe our strategy consider a finite dimensional Hamiltonian
system with a non-degenerate elliptic fixed point, which in the standard
complex symplectic coordinates $u_{j}=\frac{1}{\sqrt{2}}(q_{j}+{\rm i}p_{j})$
is described by the Hamiltonian

(5) $H=\sum_{j=1}^{n}\omega_{j}|u_{j}|^{2}+O(u^{3})\,,\quad\mbox{where
$\omega_{j}\in\mathbb{R}$ are the {\it linear frequencies}}.$

Here if the frequencies $\omega$ are rationally independent, then one can
perform the so-called Birkhoff normal form procedure: for $\mathtt{N}\geq 1$
Hamiltonian (5) is transformed into

(6) $\sum_{j=1}^{n}\omega_{j}|u_{j}|^{2}+Z+R\,\,,$

where $Z$ depends only on the actions $(|u_{i}|^{2})_{i=1}^{n}$ while
$R=O(|u|^{\mathtt{N}+3})$ has a zero of order at least $\mathtt{N}+3$ in
${\left|u\right|}$. At each step, the generating function of the change of
variables is a polynomial, so it is analytic and generates a flow in a
sufficiently small ball $B_{\delta}$ around the origin. It is well known that
this procedure generically diverges in $\mathtt{N}$, but assuming that
$\omega$ is appropriately non resonant, say diophantine111A vector
$\omega\in\mathbb{R}^{n}$ is called diophantine when it is badly approximated
by rationals, i.e. it satisfies, for some $\gamma,\tau>0$,
${\left|k\cdot\omega\right|}\geq\gamma{\left|k\right|}^{-\tau},\quad\forall
k\in\mathbb{Z}^{n}\setminus{\left\\{0\right\\}}\,$. one can control $R$ and
hence find $\mathtt{N}=\mathtt{N}(\delta)$ which minimizes the size of the
remainder $R$. It can be shown that it is bounded by an exponentially flat
function of $\delta$, of order related to $\tau$ (for a general treatment, see
instance, [IoL05, LS10]). This phenomenon is also related to Nekhoroshev kind
of result [Pös99, BGG85, N77, Ni04, BCG].
If $H$ in (5) is ”formally linearizable”, namely there exists a formal
symplectic change of variables which conjugates $H$ to
$\sum_{j=1}^{n}\omega_{j}|u_{j}|^{2}$, and $\omega$ is Diophantine, then at
each step of the procedure described above, uwe find $Z=0$ and one can prove
convergence. In order to apply this general scheme in the infinite dimensional
setting we first discuss the BNF procedure at the level of formal power
series. Here the fundamental difference w.r.t. the finite dimensional case is
that even polynomials can be just formal power series, so it is not a priori
obvious that the space of formal power series is well defined and has a
Poisson algebra structure (which coincides with the usual one on finite
dimensional subspaces). As a simple example consider the formal power series
$H=\sum_{j}u_{j}$, then

$\\{H,\bar{H}\\}=\sum_{i}\sum_{j}\\{u_{j}\,,\bar{u}_{i}\\}=\infty\,.$

We show that for translation invariant formal Hamiltonians the Poisson
brackets are well defined (see also [FGP] ), and that formal Hamiltonians are
a filtered Lie algebra with respect to a scaling degree. Then we define a
group of formal symplectic changes of variables, and prove our BNF result. In
order to define our changes of variables and prove the group structure we
strongly rely on the properties of the scaling degree as well as on the Baker
Campbell Hausdorf formula.
Then we restrict to functions on the sequence space
${\mathtt{h}}_{s,p,\theta}$, introduce the space of regular Hamiltonians and
state the main relevant properties. All properties were proved in [BMP18] in
the more restrictive case of Gauge invariant Hamiltonians, so we follow the
same strategy; for completeness we give all the proofs in the appendix. One we
have all the basic properties needed to perform Birkhoff Normal Form, proving
that formal linearizability implies analytic linearizability becomes a
relatively straightforward induction.

##  2\.  Formal Birkhoff Normal on sequence spaces

As usual given a vector $k\in\mathbb{Z}^{\mathbb{Z}}$,
$|k|:=\sum_{j\in\mathbb{Z}}|k_{j}|$. We denote ${\mathbb{N}}^{\mathbb{Z}}_{f}$
to be the set of finitely supported sequences of non negative integers,
similarly for $\mathbb{Z}^{\mathbb{Z}}_{f}$. If $j\in\mathbb{Z}$ then ${\bf
e}_{j}\in\mathbb{Z}^{\mathbb{Z}}_{f}$ denotes the vector the $j$-coordinate of
which is $1$, while the others are zero.

######  Definition 2.1 (Formal power series).

We consider the space ${\mathcal{F}}$ of formal power series expansions in
$u\in\mathbb{C}^{\mathbb{Z}}$:

$H(u)=\sum_{\begin{subarray}{c}{{\alpha}},{{\beta}}\in{\mathbb{N}}^{\mathbb{Z}}_{f}\end{subarray}}H_{{{\alpha}},{{\beta}}}u^{{\alpha}}\bar{u}^{{\beta}}\,,\qquad
u\in\mathbb{C}^{\mathbb{Z}},\quad
u^{{\alpha}}:=\prod_{j\in\mathbb{Z}}u_{j}^{{{\alpha}}_{j}}\quad|v|:=\sum_{i}|v_{i}|$

with the following properties:

  1. (1)

$H_{0,0}=0$, $H_{{\bf e}_{0},0}=H_{0,{\bf e}_{0}}=0$

  2. (2)

Reality condition:

(7) $H_{{{\alpha}},{{\beta}}}=\overline{H}_{{{\beta}},{{\alpha}}}\,;$

  3. (3)

Momentum conservation:

(8)
$H_{{{\alpha}},{{\beta}}}=0\quad\mbox{if}\;\,\pi({{\alpha}},{{\beta}}):=\sum_{j\in\mathbb{Z}}j({{\alpha}}_{j}-{{\beta}}_{j})\neq
0$

######  Remark 2.2.

The condition (3) means that the formal Hamiltonian is invariant w.r.t. the
symmetry $u_{j}\to e^{{\rm i}j\tau}u_{j}$, $\tau\in\mathbb{R}$.

We shall denote

${\mathcal{M}}:=\\{({{\alpha}},{{\beta}})\in{\mathbb{N}}^{\mathbb{Z}}_{f}:\qquad\pi({{\alpha}},{{\beta}})=0\\}$

so that $H\in{\mathcal{F}}$ can be written as

$\sum_{({{\alpha}},{{\beta}})\in{\mathcal{M}}}H_{{{\alpha}},{{\beta}}}u^{{\alpha}}\bar{u}^{{\beta}}$

Finally we define

(9)
${\mathcal{K}}:=\left\\{Z\in{\mathcal{F}}\,:\,Z(u)=\\!\\!\sum_{\begin{subarray}{c}{{\alpha}}\in{\mathbb{N}}^{\mathbb{Z}}_{f}\end{subarray}}Z_{{{\alpha}},{{\alpha}}}|u|^{2{{\alpha}}}\right\\}\,,\qquad{\mathcal{R}}:=\left\\{R\in{\mathcal{F}}\,:\,R(u)=\\!\\!\\!\\!\\!\\!\\!\\!\sum_{{{\alpha}},{{\beta}}\in{\mathcal{M}}:\,{{\alpha}}\neq{{\beta}}}\\!\\!\\!\\!\\!\\!\\!\\!R_{{{\alpha}},{{\beta}}}u^{{\alpha}}\bar{u}^{{\beta}}\right\\}$

and we can decompose ${\mathcal{F}}={\mathcal{K}}\oplus{\mathcal{R}}$ as each
element of ${\mathcal{F}}$ can uniquely be expressed in term of monomials the
coefficients of which is either zero or not zero.

######  Definition 2.3 ( scaling degree).

For $d\in{\mathbb{N}}$, we denote by ${\mathcal{F}}^{d}\subset{\mathcal{F}}$
the vector space of homogeneous formal polynomials of degree ${\mathtt{d}}+2$,
and define

${\mathcal{F}}^{\leq d}=\oplus_{h\leq
d}{\mathcal{F}}^{h}\,,\quad{\mathcal{F}}^{>d}:=\widehat{\oplus}_{h>d}{\mathcal{F}}^{h}\,,\quad{\mathcal{F}}^{\geq
d}:={\mathcal{F}}^{>d}\oplus{\mathcal{F}}^{d}\,,{\mathcal{F}}={\mathcal{F}}^{\leq
d}\oplus{\mathcal{F}}^{>d},\dots$

We define the projections associated to these direct sum decompositions

$\Pi^{({\mathtt{d}})}H=\sum_{|{\alpha}|+|{\beta}|={\mathtt{d}}+2}H_{{\alpha},{\beta}}u^{\alpha}\bar{u}^{\beta}\,,\quad\Pi^{(>{\mathtt{d}})}H=\sum_{|{\alpha}|+|{\beta}|>{\mathtt{d}}+2}H_{{\alpha},{\beta}}u^{\alpha}\bar{u}^{\beta}\,,\dots$

Elements of ${\mathcal{F}}^{\geq d}$ (resp. ${\mathcal{F}}^{>d}$) are said to
be of scaling order $\geq d+2$ (resp. $>d+2$). In the sequel, for simplicity,
we shall just say that an element of $f\in{\mathcal{F}}^{\geq d}$ is of ”order
d” and we shall say that $f$ is ”exactly of order d” if it has a non vanishing
component $\Pi^{({\mathtt{d}}))}f$ in ${\mathcal{F}}^{d}$. Finally we define

$\Pi^{\mathcal{K}}H=\sum_{{\alpha}}H_{{\alpha},{\alpha}}|u|^{2{\alpha}}\,,\quad\Pi^{\mathcal{R}}H=\sum_{{\alpha}\neq{\beta}}H_{{\alpha},{\beta}}u^{\alpha}\bar{u}^{\beta}.$

We denote by ${\mathcal{K}}^{d}:={\mathcal{F}}^{d}\cap{\mathcal{K}}$ and
similarly for ${\mathcal{R}}$ and $\geq{\mathtt{d}},\leq{\mathtt{d}}$. Note
that ${\mathcal{F}}=\widehat{\oplus}_{d}{\mathcal{F}}^{d}$.

######  Remark 2.4.

Of course, since we are in infinite dimension, even if the ${\mathcal{F}}^{d}$
are homogeneous they are only formal polynomials. However if we restrict to
monomials $u^{{\alpha}}\bar{u}^{{\beta}}$ with
$|{{\alpha}}_{j}|+|{{\beta}}_{j}|=0$ for all $j>N$ we are working on the usual
space of polynomials on which we have the standard symplectic structure ${\rm
i}\sum_{j\leq N}du_{j}\wedge d\bar{u}_{j}$. We now show that such structure
extends to ${\mathcal{F}}$.

######  Proposition 2.5.

The following Formula (10) is well defined and endows ${\mathcal{F}}$ with a
Poisson algebra structure which is a filtered Lie algebra w.r.t. the
${\mathcal{F}}^{\geq d}$’s.

(10) ${\left\\{F,G\right\\}}:={\rm
i}\sum_{({{\alpha}}^{(i)},{{\beta}}^{(i))}\in{\mathcal{M}}}F_{{{\alpha}}^{1},{{\beta}}^{1}}G_{{{\alpha}}^{2},{{\beta}}^{2}}\sum_{j}{\left({{\alpha}}^{(1)}_{j}{{\beta}}^{(2)}_{j}-{{\beta}}^{(1)}_{j}{{\alpha}}^{(2)}_{j}\right)}u^{{{\alpha}}^{(1)}+{{\alpha}}^{(2)}-{\bf
e}_{j}}\bar{u}^{{{\beta}}^{(1)}+{{\beta}}^{(2)}-{\bf e}_{j}}$

Before proving our assertion we need a technical lemma. Let ${\bf
e}_{j}\in{\mathbb{N}}^{\mathbb{Z}}_{f}$ be the $j$th vector of the standard
basis.

######  Lemma 2.6.

1) Given $\alpha\in{\mathbb{N}}^{\mathbb{Z}}_{f}$ there is only a finite
number of pairs $\alpha^{(1)},\alpha^{(2)}\in{\mathbb{N}}^{\mathbb{Z}}_{f}$
with $\alpha=\alpha^{(1)}+\alpha^{(2)}$. 2) Given
$({{\alpha}},{{\beta}})\in{\mathcal{M}}$ there is only a finite number of
pairs $({{\alpha}}^{(1)},{{\beta}}^{(1)}),\
({{\alpha}}^{(2)},{{\beta}}^{(2)})\in{\mathcal{M}}$ and indices
$j\in\mathbb{Z}$ such that:

  1. i)

$({{\alpha}},{{\beta}})=({{\alpha}}^{(1)},{{\beta}}^{(1)})+({{\alpha}}^{(2)},{{\beta}}^{(2)})-({\bf
e}_{j},{\bf e}_{j})$

  2. ii)

one has
${{\alpha}}_{j}^{(1)}{{\beta}}_{j}^{(2)}+{{\alpha}}_{j}^{(2)}{{\beta}}_{j}^{(1)}\neq
0$.

###### Proof.

1) is clear since for all $j$ one has $0\leq(\alpha_{1})_{j}\leq\alpha_{j}$.
2) By item 1) we may divide
$({{\alpha}},{{\beta}})=(a^{(1)},b^{(1)})+(a^{(2)},b^{(2)})$ in a finite
number of ways. Then the pairs
$({{\alpha}}^{(1)},{{\beta}}^{(1)}),({{\alpha}}^{(2)},{{\beta}}^{(2)})$ can
only have one of the following forms (up to exchanging the indices)

A) $\quad({{\alpha}}^{(1)},{{\beta}}^{(1)})=(a^{(1)},b^{(1)})+({\bf
e}_{j},{\bf
e}_{j})\,,\quad({{\alpha}}^{(2)},{{\beta}}^{(2)})=(a^{(2)},b^{(2)})$

B) $\quad({{\alpha}}^{(1)},{{\beta}}^{(1)})=(a^{(1)},b^{(1)})+({\bf
e}_{j},0)\,,\quad({{\alpha}}^{(2)},{{\beta}}^{(2)})=(a^{(2)},b^{(2)})+(0,{\bf
e}_{j})$,
for some index $j\in\mathbb{Z}$.
If we are in case $A)$ then by condition $ii)$ we have
$j\in$Supp$(a^{(2)}+b^{(2)})$, which restricts to a finite number of possible
$j^{\prime}s$. Otherwise in case $B)$ by momentum conservation e have
$j=-\pi(a^{(1)},b^{(1)})={\pi}(a^{(2)},b^{(2)})$ and again $j$ is restricted
to a finite number of possible choices. ∎

###### Proof of Proposition 2.5.

The fact that the Poisson bracket is well defined follows immediately from the
previous Lemma. Indeed by construction

$\\{F,G\\}=\sum_{{{\alpha}},{{\beta}}}P_{{{\alpha}},{{\beta}}}{u^{{\alpha}}}{\bar{u}^{{\beta}}}\in{\mathcal{F}}$

where $P_{{{\alpha}},{{\beta}}}=0$ if $\pi({{\alpha}},{{\beta}})\neq 0$ and
otherwise

(11) $P_{{{\alpha}},{{\beta}}}={{\rm
i}}\sum_{j}\sum_{\begin{subarray}{c}{{\alpha}}^{(i)},{{\beta}}^{(i)}\in{\mathbb{N}}^{\mathbb{Z}}_{f}:\;\pi({{\alpha}}^{(i)},{{\beta}}^{(i)})=0\\\
{{\alpha}}={{\alpha}}^{(1)}+{{\alpha}}^{(2)}-{\bf
e}_{j}\,,{{\beta}}={{\beta}}^{(1)}+{{\beta}}^{(2)}-{\bf
e}_{j}\end{subarray}}F_{{{\alpha}}^{1},{{\beta}}^{1}}G_{{{\alpha}}^{2},{{\beta}}^{2}}{\left({{\alpha}}^{(1)}_{j}{{\beta}}^{(2)}_{j}-{{\beta}}^{(1)}_{j}{{\alpha}}^{(2)}_{j}\right)}\,.$

Then item 2 of the previous Lemma implies that $P_{{\alpha},{\beta}}$ above is
given by a finite sum.
The fact that it endows ${\mathcal{F}}$ with a Poisson algebra structure
follows from the fact that the infinitely many identities defining such a
structure involve only a finite number of elements $u_{i},\bar{u}_{i}$ and
then we are in the canonical Poisson algebra.
The filtered Lie algebra property comes from the fact that in (11) we get
$|\alpha|+|\beta|=|{{\alpha}}^{(1)}|+|{{\alpha}}^{(2)}|+|{{\beta}}^{(1)}|+|{{\beta}}^{(2)}|-2$,
this shows that if $F\in{\mathcal{F}}^{\geq{\mathtt{d}}_{1}}$, and
$G\in{\mathcal{F}}^{\geq{\mathtt{d}}_{2}}$ then

(12)
$|\alpha|+|\beta|\geq{\mathtt{d}}_{1}+2+{\mathtt{d}}_{2}+2-2={\mathtt{d}}_{1}+{\mathtt{d}}_{2}+2.$

so $\\{F,G\\}\in{\mathcal{F}}^{\geq{\mathtt{d}}_{1}+{\mathtt{d}}_{2}}$. ∎

######  Remark 2.7.

Let $H_{i}\in{\mathcal{F}}^{\geq{\mathtt{d}}_{i}}$ be a sequence of formal
Hamiltonians with ${\mathtt{d}}_{i+1}\geq{\mathtt{d}}_{i}$ for all $i\geq 1$.
Then the series

$H=\sum_{i=1}^{\infty}H_{i}\in{\mathcal{F}}^{\geq{\mathtt{d}}_{1}}$

is well defined since for any ${\mathtt{d}}\geq{\mathtt{d}}_{0}$ the
projection

$\Pi^{({\mathtt{d}})}H=\Pi^{({\mathtt{d}})}\sum_{i:{\mathtt{d}}_{i}\leq{\mathtt{d}}}H_{i}$

is a finite sum.

We say that a linear operator $L:{\mathcal{F}}\to{\mathcal{F}}$ is of order
(or increase the order by) ${\mathtt{d}}$ if for all $h$

$L:{\mathcal{F}}^{\geq h}\to{\mathcal{F}}^{\geq h+{\mathtt{d}}}\,.$

######  Lemma 2.8.

let $L_{n}$ be a sequence of linear operators on ${\mathcal{F}}$ and let
${\mathtt{d}}_{n}$ be the order of $L_{n}$. If the sequence ${\mathtt{d}}_{n}$
increases to infinity then

$L:=\sum_{n=1}^{\infty}L_{n}\,,\qquad
T=\prod_{n=1}^{\infty}(\operatorname{id}+L_{n})-\operatorname{id}$

are linear operators on ${\mathcal{F}}$ of order ${\mathtt{d}}_{1}$.

###### Proof.

For the first statement, for all ${\mathtt{d}}\in{\mathbb{N}}$ let
$N({\mathtt{d}})$ be the largest $N$ such that
${\mathtt{d}}_{N}\leq{\mathtt{d}}$. By construction
$\Pi^{(\leq{\mathtt{d}})}L_{n}K=0$ for all $n>N({\mathtt{d}})$ and for any
$K\in{\mathcal{F}}$. Then for all $K\in{\mathcal{F}}$ and $N>N({\mathtt{d}})$
one has

$\Pi^{(\leq{\mathtt{d}})}\sum_{n=1}^{N}L_{n}K=\Pi^{(\leq{\mathtt{d}})}\sum_{n=1}^{N({\mathtt{d}})}L_{n}K\,,$

and the claim follows.
Regarding the second statement we proceed similarly

$\prod_{n=1}^{N}(\operatorname{id}+L_{n})=\prod_{n=1}^{N-1}(\operatorname{id}+L_{n})+L_{N}\prod_{n=1}^{N-1}(\operatorname{id}+L_{n})\,,$

hence, for all ${\mathtt{d}}\geq 0$ and all $N>N({\mathtt{d}})$

$\Pi^{(\leq{\mathtt{d}})}\prod_{n=1}^{N}(\operatorname{id}+L_{n})=\Pi^{(\leq{\mathtt{d}})}\prod_{n=1}^{N({\mathtt{d}})}(\operatorname{id}+L_{n})\,.$

∎

As a direct consequence we have the following.

######  Corollary 2.9.

Given $G\in{\mathcal{F}}^{\geq{\mathtt{d}}}$, with ${\mathtt{d}}\geq 1$ we
define

(13)
$\operatorname{ad}_{G}:=\\{G,\cdot\\}\,,\quad\Phi_{G}:=\exp(\\{G,\cdot\\})=\sum_{k\geq
0}\frac{\operatorname{ad}_{G}^{k}}{k!}\,,$

then $\operatorname{ad}_{G}$ and $\Phi_{G}-\operatorname{id}$ are operators of
order ${\mathtt{d}}$, namely

$\operatorname{ad}_{G},\Phi_{G}-\operatorname{id}:{\mathcal{F}}^{\geq
h}\to{\mathcal{F}}^{\geq h+{\mathtt{d}}}\,.$

Similarly for any sequence $b_{k}$ one has that

$\sum_{k\geq{\mathtt{n}}}b_{k}{\operatorname{ad}_{G}^{k}}:{\mathcal{F}}^{\geq
h}\to{\mathcal{F}}^{\geq h+{\mathtt{d}}{\mathtt{n}}}\,.$

######  Definition 2.10.

Given $G\in{\mathcal{F}}^{\geq 1}$ we call the operator $\Phi_{G}$ defined in
(13) a formal symplectic change of variables on ${\mathcal{F}}$.

The following Lemma ensures the group structure of the formal symplectic
changes of variables

######  Lemma 2.11 (Baker-Campbell-Haussdorf).

Given $F\in{\mathcal{F}}^{\geq{\mathtt{d}}_{1}}$ and
$G\in{\mathcal{F}}^{\geq{\mathtt{d}}_{2}}$, with ${\mathtt{d}}_{i}\geq 1$,
then there exists $K\in{\mathcal{F}}^{\geq 1}$, such that

$e^{\\{G,\\}}e^{\\{F,\\}}=e^{\\{K,\\}}\,,\qquad
K-F-G\in{\mathcal{F}}^{\geq{\mathtt{d}}_{1}+{\mathtt{d}}_{2}}$

###### Proof.

By the Baker-Campbell-Hausdorff formula ([Se92][p.29]) one has

$K:=\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n}\sum_{r_{i}+s_{i}>0}\frac{[G^{r_{1}}F^{s_{1}}\dots
G^{r_{n}}F^{s_{n}}]}{\left(\sum_{i=1}^{n}(r_{i}+s_{i})\right)\prod_{i=1}^{n}r_{i}!s_{i}!}$

where

(14) $[G^{r_{1}}F^{s_{1}}\dots
G^{r_{n}}F^{s_{n}}]:=\begin{cases}\operatorname{ad}_{G}^{r_{1}}\operatorname{ad}_{F}^{s_{1}}\dots\operatorname{ad}_{G}^{r_{n}}F\quad\mbox{
if}\;s_{n}=1\\\
\operatorname{ad}_{G}^{r_{1}}\operatorname{ad}_{F}^{s_{1}}\dots\operatorname{ad}_{F}^{s_{n-1}}G\quad\mbox{
if}\;s_{n}=0\,,\;\mbox{and}\;r_{n}=1\\\
0\quad\quad\quad\quad\mbox{otherwise}\end{cases}$

Recalling that $F\in{\mathcal{F}}^{\geq{\mathtt{d}}_{1}}$ and
$G\in{\mathcal{F}}^{\geq{\mathtt{d}}_{2}}$, each term
$\operatorname{ad}_{G}^{r_{1}}\operatorname{ad}_{F}^{s_{1}}\dots\operatorname{ad}_{G}^{r_{n}}F$
(resp.
$\operatorname{ad}_{G}^{r_{1}}\operatorname{ad}_{F}^{s_{1}}\dots\operatorname{ad}_{F}^{s_{n-1}}G$)
is of order
$\left(\sum_{i=1}^{n}r_{i}\right){\mathtt{d}}_{2}+(\sum_{i=1}^{n}s_{i}){\mathtt{d}}_{1}\geq
n\min({\mathtt{d}}_{1},{\mathtt{d}}_{2})$. Hence setting $N({\mathtt{d}})$ to
be the largest $N$ such that
$N\min({\mathtt{d}}_{1},{\mathtt{d}}_{2})\leq{\mathtt{d}}$n

$\Pi^{\leq{\mathtt{d}}}K=\Pi^{\leq{\mathtt{d}}}\sum_{n=1}^{N({\mathtt{d}})}\frac{(-1)^{n-1}}{n}\sum_{r_{i}+s_{i}>0}\frac{[G^{r_{1}}F^{s_{1}}\dots
G^{r_{n}}F^{s_{n}}]}{\left(\sum_{i=1}^{n}(r_{i}+s_{i})\right)\prod_{i=1}^{n}r_{i}!s_{i}!}$

Moreover if $n\geq 2$ then the Hamiltonian in (14) is of order
$\geq{\mathtt{d}}_{1}+{\mathtt{d}}_{2}$, so
$K-F-G\in{\mathcal{F}}^{\geq{\mathtt{d}}_{1}+{\mathtt{d}}_{2}}$. ∎

######  Lemma 2.12.

Given a sequence of generating functions
$G_{i}\in{\mathcal{F}}^{{\geq}{\mathtt{d}}_{i}}$ with
${\mathtt{d}}_{i+1}>{\mathtt{d}}_{i}\geq 1$ then there exists
${\mathcal{G}}\in{\mathcal{F}}^{\geq d_{1}}$ such that the composition

$\prod_{i}e^{\\{G_{i},\\}}=e^{\\{{\mathcal{G}},\\}}$

###### Proof.

By Lemma 2.8 with $L_{n}=e^{\\{G_{n},\\}}-\operatorname{id}$ we know that
$\prod_{i}e^{\\{G_{i},\\}}$ is a well defined operator of ${\mathcal{F}}$.
Using Lemma 2.11 we can define $F_{k}\in{\mathcal{F}}^{\geq 1}$ iteratively so
that

$e^{\\{F_{k},\cdot\\}}=e^{\\{G_{k},\cdot\\}}e^{\\{F_{k-1},\cdot\\}}$

since $e^{\\{G_{k},\cdot\\}}-\operatorname{id}$ is of order ${\mathtt{d}}_{k}$
there exists $N({\mathtt{d}})$ such that if $k>N({\mathtt{d}})$ then

$\Pi^{(\leq{\mathtt{d}})}F_{k}=\Pi^{(\leq{\mathtt{d}})}F_{N({\mathtt{d}})}$

Then ${\mathcal{G}}=\lim_{k\to\infty}F_{k}$ is well defined. ∎

For any vector $\omega\in\mathbb{R}^{\mathbb{Z}}$ such that

$\omega\cdot\ell\neq
0\,,\quad\forall\ell\in\mathbb{Z}^{\mathbb{Z}}_{f}\setminus\\{0\\}\,,$

we define the non-resonant quadratic Hamiltonian

$D_{\omega}:=\sum_{j}\omega_{j}|u_{j}|^{2}\,.$

######  Lemma 2.13.

The operator $\operatorname{ad}_{D_{\omega}}$ is invertible on
${\mathcal{R}}^{({\mathtt{d}})}$ for all ${\mathtt{d}}$.

###### Proof.

Given $F\in{\mathcal{R}}^{({\mathtt{d}})}$, we have

${\left\\{D_{\omega},G\right\\}}={\rm
i}\sum_{\begin{subarray}{c}{{\alpha}}^{(2)},{{\beta}}^{(2)}\in{\mathbb{N}}^{\mathbb{Z}}\,,\\\
|{{\alpha}}^{(2)}|+|{{\beta}}^{(2)}|<\infty\,,\,\pi({{\alpha}}^{(2)},{{\beta}}^{(2)})=0\end{subarray}}G_{{{\alpha}}^{2},{{\beta}}^{2}}\left(\sum_{j}\omega_{j}{\left({{\beta}}^{(2)}_{j}-{{\alpha}}^{(2)}_{j}\right)}\right)u^{{{\alpha}}^{(2)}}\bar{u}^{{{\beta}}^{(2)}}=F$

Hence, we have $G:=\operatorname{ad}_{D_{\omega}}^{-1}(F)$ with for all
${{\alpha}}^{(2)},{{\beta}}^{(2)}\in{\mathbb{N}}^{\mathbb{Z}}$,
${{\alpha}}^{(2)}\neq{{\beta}}^{(2)}$ with
$|{{\alpha}}^{(2)}|+|{{\beta}}^{(2)}|<\infty$ and
$\pi({{\alpha}}^{(2)},{{\beta}}^{(2)})=0$,

$G_{{{\alpha}}^{2},{{\beta}}^{2}}:=F_{{{\alpha}}^{2},{{\beta}}^{2}}\left(\sum_{j}{{\rm
i}}\omega_{j}{\left({{\beta}}^{(2)}_{j}-{{\alpha}}^{(2)}_{j}\right)}\right)^{-1},\quad
G_{\alpha^{2},\alpha^{2}}=0.$

∎

######  Proposition 2.14 (Birkhoff Normal Form).

Given any formal Hamiltonian of the form

(15) $H=D_{\omega}+Z+R$

where $Z\in{\mathcal{K}}^{\geq 2}$ and $R\in{\mathcal{F}}^{\geq{\mathtt{d}}}$
with ${\mathtt{d}}\geq 1$, then

  1. (1)

Formal Normal Form: there exists $S\in{\mathcal{F}}^{\geq{\mathtt{d}}}$ such
that

$e^{\\{S,\cdot\\}}H=D_{\omega}+\widetilde{Z}\,,\quad\widetilde{Z}-Z\in{\mathcal{K}}^{\geq{\mathtt{d}}}\,.$

  2. (2)

Uniqueness: if $G\in{\mathcal{F}}^{\geq 1}$ is such that
$e^{\\{G,\cdot\\}}H\in{\mathcal{K}}$ then
$e^{\\{G,\cdot\\}}H=e^{\\{S,\cdot\\}}H$. Hence to each $H$ as above we can
associate a unique $Z_{H}\in{\mathcal{K}}^{\geq 2}$ such that
$e^{\\{S,\cdot\\}}H=D_{\omega}+Z_{H}$.

###### Proof.

For item $(1)$ Let us first consider the case ${\mathtt{d}}\geq 2$. we start
with a Hamiltonian $H_{0}\in{\mathcal{F}}$ of the form
$D_{\omega}+Z_{0}+P_{0}$ with $P_{0}\in{\mathcal{F}}^{\geq d}$ and we
iteratively construct a sequence of generating functions
$S_{i}\in{\mathcal{R}}^{\geq 2i+d}$ and Hamiltonians $H_{i}$ by setting

$\\{D_{\omega},S_{i}\\}=\Pi^{{\mathcal{R}}}H_{i}\,,\quad
H_{i+1}=e^{\\{S_{i},\cdot\\}}H_{i}\,.$

We now show inductively that for each $i$

$\Pi^{(<2i+d)}\Pi^{{\mathcal{R}}}H_{i}=0\,,\quad S_{i}\in{\mathcal{R}}^{\geq
2i+d}$

so in other words

$H_{i}=D_{\omega}+Z_{i}+P_{i}\,,\quad
Z_{i}\in{\mathcal{K}}\cap{\mathcal{F}}^{\leq 2i+d-1}\,,\quad
P_{i}\in{\mathcal{F}}^{\geq 2i+d}\,.$

For $i=0$ we just set $Z_{0}=\Pi^{<d}Z$ and $P_{0}=R+\Pi^{\geq d}Z$. By
induction we assume that $P_{i}\in{\mathcal{F}}^{\geq 2i+d}$. Then by Lemma
2.13, $S_{i}\in{\mathcal{F}}^{\geq 2i+d}$.

$\displaystyle e^{\\{S_{i},\cdot\\}}H_{i}$
$\displaystyle=D_{\omega}+Z_{i}+P_{i}+\\{S_{i},D_{\omega}\\}+\sum_{h=2}^{\infty}\frac{\operatorname{ad}_{S_{i}}^{h-1}}{h!}\\{S_{i},D_{\omega}\\}+\sum_{k=1}^{\infty}\frac{\operatorname{ad}_{S_{i}}^{k}}{k!}(Z_{i}+P_{i})$
$\displaystyle=D_{\omega}+Z_{i}+\Pi^{\mathcal{K}}P_{i}-\sum_{k=1}^{\infty}\frac{\operatorname{ad}_{S_{i}}^{k}}{(k+1)!}\Pi^{{\mathcal{R}}}P_{i}+\sum_{k=1}^{\infty}\frac{\operatorname{ad}_{S_{i}}^{k}}{k!}(Z_{i}+P_{i})\,.$

So we may set

$Z_{i+1}:=Z_{i}+\Pi^{(<2i+d+2)}\Pi^{\mathcal{K}}P_{i}\,,\quad
P_{i+1}=e^{\\{S_{i},\cdot\\}}H_{i}-D_{\omega}-Z_{i+1}$

and verify that $P_{i+1}\in{\mathcal{F}}^{\geq 2i+d+2}$ by applying
Proposition 2.5 and noticing that, since $4i+2{\mathtt{d}}\geq
2i+{\mathtt{d}}+2$, the term of lowest degree is $\\{S_{i},Z\\}$.
Then we set

$\widetilde{Z}=\lim_{i\to\infty}Z_{i}=Z_{0}+\sum_{i=0}^{\infty}\Pi^{\leq
2i+d+1}\Pi^{\mathcal{K}}P_{i}=Z_{0}+\Pi^{\mathcal{K}}\sum_{i=0}^{\infty}\Pi^{(<2i+d+2)}\Pi^{(\geq
2i+d)}P_{i}\,,$

which is well defined by Remark 2.7. Finally by Lemma 2.12 we can define
$S\in{\mathcal{F}}^{\geq{\mathtt{d}}}$ so that

$e^{\\{S,\cdot\\}}=\prod_{i=0}^{\infty}e^{\\{S_{i},\cdot\\}}\,.$

If ${\mathtt{d}}=1$ we perform a preliminary step in order to increase the
degree by one and then we start the procedure explained above. We start with
$H=D_{\omega}+P$, with $P:=R+Z$ . As before we fix $S\in{\mathcal{R}}^{\geq
1}$ so that $\\{D_{\omega},S\\}=\Pi^{\mathcal{R}}H$ we set

$\displaystyle
H_{0}:=e^{\\{S,\cdot\\}}H=D_{\omega}+\Pi^{\mathcal{K}}P-\sum_{k=1}^{\infty}\frac{\operatorname{ad}_{S}^{k}}{(k+1)!}\Pi^{{\mathcal{R}}}P+\sum_{k=1}^{\infty}\frac{\operatorname{ad}_{S}^{k}}{k!}P\,.$

then fixing $Z_{0}:=\Pi^{\leq 2}\Pi^{\mathcal{K}}P$ and
$P_{0}:=H_{0}-D_{\omega}-Z_{0}$ we are in the setting of the previous case.

Regarding item $(2)$ we remark that If $e^{\\{S_{1},\cdot\\}}$ transforms a
normal form $D_{{\omega}}+K_{1}$ into a normal form $D_{{\omega}}+K_{2}$, then

$e^{\\{S_{1},\cdot\\}}(D_{\omega}+K_{1})=D_{\omega}+K_{1}+\sum_{h=1}^{\infty}\frac{\operatorname{ad}_{S_{1}}^{h-1}}{h!}\\{S_{1},D_{\omega}+K_{1}\\}=D_{\omega}+K_{2}.$

Since ${\mathcal{K}}={\mathcal{K}}^{\geq 2}$ and $S\in{\mathcal{H}}^{\geq 1}$,
comparing homogeneous terms of degree $1$ we get $\\{S_{1},D_{\omega}\\}=0$ so
we should have $S_{1}^{(1)}\in{\mathcal{K}}$ which can only be possible if
$S_{1}^{(1)}=0$. Comparing homogeneous terms of degree $2$, we obtain
$K_{1}^{(2)}-K_{2}^{(2)}+\\{S_{1}^{(2)},D_{{\omega}}\\}=0$. Recalling that
$\\{S_{1}^{(2)},D_{{\omega}}\\}\in{\mathcal{R}}$ we have
$K_{1}^{(2)}-K_{2}^{(2)}\in{\mathcal{K}}\cap{\mathcal{R}}$ is zero and
$S_{1}^{(2)}\in{\mathcal{K}}$. Assuming that
$K_{1}^{(j)}=K_{2}^{(j)}\in{\mathcal{K}}$ and $S_{1}^{(j)}\in{\mathcal{K}}$
for $2\leq j\leq m$. Then we have

$\displaystyle K_{1}^{(m+1)}-K_{2}^{(m+1)}$ $\displaystyle+$
$\displaystyle\\{S_{1}^{(m+1)},D_{{\omega}}\\}+\sum_{h=2}^{\infty}\frac{1}{h!}\left(\sum_{j_{1}+\cdots+j_{l}=m+1}\\{S_{1}^{(j_{1})},\\{S_{1}^{(j_{2})},\cdots\\{S_{1}^{(j_{h})},D_{\omega}\\}\\}\\}\right.$
$\displaystyle+$
$\displaystyle\left.\sum_{j_{1}+\cdots+j_{h}+j_{h+1}=m+1}\\{S_{1}^{(j_{1})},\\{S_{1}^{(j_{2})},\cdots\\{S_{1}^{(j_{h})},K_{1}^{(j_{h+1})}\\}\\}\\}\right)=0$

By induction and since $D_{{\omega}}$ is non resonant, then both sums above
are zero. Hence, we the same reasoning as above, we obtain
$K_{1}^{(m+1)}=K_{2}^{(m+1)}\in{\mathcal{K}}$ and
$S_{1}^{(m+1)}\in{\mathcal{K}}$. The result follows from Proposition 2.12.

∎

######  Corollary 2.15.

For any $H$ as in (15), if for $G\in{\mathcal{F}}^{\geq 1}$ one has
$e^{\\{G,\cdot\\}}H=D_{\omega}+Z+R$ with $R\in{\mathcal{F}}^{\geq d_{1}}$ then
$Z-Z_{H}\in{\mathcal{K}}^{\geq d_{1}}$.

###### Proof.

By Proposition 2.14 $(1)$ there exists $S\in{\mathcal{F}}^{\geq d_{1}}$ which
normalizes $D_{\omega}+Z+R$ to $D_{\omega}+\widetilde{Z}$ with
$\widetilde{Z}-Z\in{\mathcal{F}}^{\geq d_{1}}$. By Lemma 2.12 there exists
$G_{1}\in{\mathcal{F}}$ such that
$e^{\\{G_{1},\cdot\\}}=e^{\\{S,\cdot\\}}e^{\\{G,\cdot\\}}$. Since $G_{1}$ puts
$H$ in normal form, by Proposition 2.14 $(2)$, $\widetilde{Z}=Z_{H}$ and the
result follows. ∎

######  Definition 2.16.

We say that $H$ is formally linearizable if $Z_{H}=0$.

######  Corollary 2.17.

If $H$ is formally linearizable and there exists a formal symplectic change of
variables with $e^{\\{S,\cdot\\}}H=D_{\omega}+Z+R$ with
$R\in{\mathcal{F}}^{\geq d}$ and $Z\in{\mathcal{K}}^{<d}$ (this last condition
does not imply any loss of generality) then $Z=0$.

###### Proof.

This follows directly from Corollary 2.15. ∎

If we know a priori that $H$ is formally linearizable then we get a faster
growth of the degree of $P_{i}$.

######  Lemma 2.18.

. If $H_{0}\in{\mathcal{F}}$ of the form $D_{\omega}+P_{0}$ with
$P_{0}\in{\mathcal{F}}^{\geq 1}$ is formally linearizable then the sequence of
generating functions

$\\{D_{\omega},S_{i}\\}=\Pi^{{\mathcal{R}}}H_{i}\,,\quad
H_{i+1}=e^{\\{S_{i},\cdot\\}}H_{i}\,.$

satisfies

$H_{i}=D_{\omega}+P_{i}\,,\quad P_{i}\in{\mathcal{F}}^{\geq 2^{i}}\,.$

###### Proof.

By induction we assume that $P_{i}\in{\mathcal{F}}^{\geq 2^{i}}$. Then by
construction $S_{i}\in{\mathcal{F}}^{\geq 2^{i}}$.

$\displaystyle e^{\\{S_{i},\cdot\\}}H_{i}$
$\displaystyle=D_{\omega}+P_{i}+\\{S_{i},D_{\omega}\\}+\sum_{h=2}^{\infty}\frac{\operatorname{ad}_{S_{i}}^{h-1}}{h!}\\{S_{i},D_{\omega}\\}+\sum_{k=1}^{\infty}\frac{\operatorname{ad}_{S}^{k}}{k!}P_{i}$
$\displaystyle=D_{\omega}+\Pi^{\mathcal{K}}P_{i}-\sum_{k=1}^{\infty}\frac{\operatorname{ad}_{S_{i}}^{k}}{(k+1)!}\Pi^{{\mathcal{R}}}P_{i}+\sum_{k=1}^{\infty}\frac{\operatorname{ad}_{S_{i}}^{k}}{k!}P_{i}\,$
$\displaystyle=:D_{\omega}+\Pi^{<2^{i+1}}\Pi^{\mathcal{K}}P_{i}+P_{i+1}\,.$

By Proposition 2.5 the two series in the formula above are in
${\mathcal{F}}^{2^{i+1}}$ so to prove our claim we only need to show
$\Pi^{\mathcal{K}}\Pi^{<2^{i+1}}P_{i}=0$. This is a consequence of Corollary
2.17. ∎

##  3\. Regular Hamiltonians

We now revisit the formal Birkhoff normal form in the case of analytic
Hamiltonians. We start by introducing an appropriate functional setting.

###  3.1. Spaces of Hamiltonians

Let us consider the weighted space

${\mathtt{h}}_{s}={\mathtt{h}}_{s,p,\theta}:={\left\\{u\in\ell^{2}(\mathbb{Z},\mathbb{C}):\quad|u|^{2}_{s}:=\sum_{j\in\mathbb{Z}}\langle
j\rangle^{2p}e^{2s\langle j\rangle^{\theta}}|u_{j}|^{2}<\infty\right\\}}$

where $\langle j\rangle:=\max(|j|,1)$, $p\geq\frac{1}{2}$ and $0<\theta\leq
1$. The spaces ${\mathtt{h}}_{s,p,\theta}$ are contained in
$\ell^{2}(\mathbb{C})$, so we endow them with the standard symplectic
structure coming from the Hermitian product on $\ell^{2}(\mathbb{C})$.
We identify $\ell^{2}(\mathbb{C})$ with
$\ell^{2}(\mathbb{R})\times\ell^{2}(\mathbb{R})$ through
$u_{j}={\left(x_{j}+iy_{j}\right)}/\sqrt{2}$ and induce on
$\ell^{2}(\mathbb{C})$ the structure of a real symplectic Hilbert space222We
recall that given a complex Hilbert space $H$ with a Hermitian product
$(\cdot,\cdot)$, its realification is a real symplectic Hilbert space with
scalar product and symplectic form given by  $\langle u,v\rangle=2{\rm
Re}(u,v)\,,\quad\omega(u,v)=2{\rm Im}(u,v)\,.$ by setting, for any
$(u^{(1)},u^{(2)})\in\ell^{2}(\mathbb{C})\times\ell^{2}(\mathbb{C})$,

$\langle
u^{(1)},u^{(2)}\rangle=\sum_{j}{\left(x_{j}^{(1)}x_{j}^{(2)}+y_{j}^{(1)}y_{j}^{(2)}\right)}\,,\quad\omega(u^{(1)},u^{(2)})=\sum_{j}{\left(y_{j}^{(1)}x_{j}^{(2)}-x_{j}^{(1)}y_{j}^{(2)}\right)},$

which are the standard scalar product and symplectic form
$\Omega=\sum_{j}dy_{j}\wedge dx_{j}$.

Given $H\in{\mathcal{F}}$, we define its majorant as

(16)
$\underline{H}(u)=\sum_{\begin{subarray}{c}{{\alpha}},{{\beta}}\in{\mathbb{N}}^{\mathbb{Z}}\,,\\\
|{{\alpha}}|+|{{\beta}}|<\infty\end{subarray}}|H_{{{\alpha}},{{\beta}}}|u^{{\alpha}}\bar{u}^{{\beta}}\,.$

######  Definition 3.1 (M-regular Hamiltonians).

For $r>0$, let ${\mathcal{H}}_{r,s}$ be the subspace of ${\mathcal{F}}$ of
formal power series $H$ such that $\underline{H}$ is pointwise absolutely
convergent on $B_{r}({\mathtt{h}}_{s})$, the ball of radius $r$ centered at
the origin of ${\mathtt{h}}_{s}$, and

$|H|_{B_{r}({\mathtt{h}}_{s})}\equiv\|H\|_{r,s}:=r^{-1}{\left(\sup_{{\left|u\right|}_{{\mathtt{h}}_{s}}\leq
r}{\left|{X}_{{\underline{H}}}\right|}_{{\mathtt{h}}_{s}}\right)}<\infty\,.$

Note that in ${\mathcal{F}}$ one has $H(0)=0$ so this is actually a norm.

We shall show in the next subsection that $H\in{\mathcal{H}}_{r,s}$ guarantees
that the Hamiltonian flow of $H$ exists at least locally and generates a
symplectic transformation on ${\mathtt{h}}_{s}$, i.e. ${\mathtt{h}}_{s}$ is an
invariant subspace for the dynamics.

######  Theorem 3.2 (Main).

Consider a Hamiltonian of the form

$\sum_{j\in\mathbb{Z}}\omega_{j}|u_{j}|^{2}+P_{0}\,,\quad
P_{0}\in{\mathcal{H}}_{{\mathtt{r}},s_{0}}\cap{\mathcal{F}}^{\geq 1}$

where $\omega\in{\mathtt{D}_{\gamma}}$. Assume that there exists
$G\in{\mathcal{F}}^{\geq 1}$ such that

$e^{\\{G,.\\}}H=\sum_{j\in\mathbb{Z}}\omega_{j}|u_{j}|^{2}\,,$

then there exists $r_{1}<{\mathtt{r}}$, $s_{1}>s_{0}$ and a close to identity
change of variables $\Psi$

$\Psi:B_{r_{1}}({\mathtt{h}}_{s_{1}})\to{\mathtt{h}}_{s_{1}}$

such that $H\circ\Psi=\sum_{j\in\mathbb{Z}}\omega_{j}|u_{j}|^{2}$.

###  3.2. Poisson structure and homological equation

The following Lemmata are proved in [BMP18] under the extra assumption of mass
conservation, we discuss the proof in our slightly more general setting in the
apendix.

######  Lemma 3.3.

If $H\in{\mathcal{H}}_{r,s}\cap{\mathcal{F}}^{\geq{\mathtt{d}}}$, then for all
${r^{*}}\leq r$ one has

${\left\|H\right\|}_{{r^{*}},s}\leq{\left(\frac{{r^{*}}}{r}\right)}^{{\mathtt{d}}}{\left\|H\right\|}_{r,s}\,.$

######  Lemma 3.4.

If $H\in{\mathcal{H}}_{r,s}$, then for all $s_{1}\geq s$ one has

${\left\|H\right\|}_{r,s_{1}}\leq{\left\|H\right\|}_{r,s}\,.$

######  Lemma 3.5 (Poisson brakets and Hamiltonian flow).

Let $0<\rho<r$, and $F,G\in{\mathcal{H}}_{r+\rho,\eta}({\mathtt{h}}_{s})$,
then

(17) ${\left\|\\{F,G\\}\right\|}_{r,s}\leq
4{\left(1+\frac{r}{\rho}\right)}{\left\|F\right\|}_{r+\rho,s}{\left\|G\right\|}_{r+\rho,s}\,.$

For $S\in{\mathcal{H}}_{r+\rho,\eta}({\mathtt{h}}_{s})$ with

(18)
${\left\|S\right\|}_{r+\rho,s}\leq\delta:=\frac{\rho}{8e{\left(r+\rho\right)}}.$

Then the time $1$-Hamiltonian flow $\Psi^{1}_{S}:B_{r}({\mathtt{h}}_{s})\to
B_{r+\rho}({\mathtt{h}}_{s})$ is well defined, analytic, symplectic with

(19) $\sup_{u\in
B_{r}({\mathtt{h}}_{s})}{\left\|\Psi^{1}_{S}(u)-u\right\|}_{{\mathtt{h}}_{s}}\leq(r+\rho){\left\|S\right\|}_{r+\rho,s}\leq\frac{\rho}{8e}.$

For any $H\in{\mathcal{H}}_{r+\rho,s}$ we have that
$H\circ\Psi^{1}_{S}=e^{{\left\\{S,\cdot\right\\}}}H\in{\mathcal{H}}_{r,s}$ and

(20) $\displaystyle{\left\|e^{{\left\\{S,\cdot\right\\}}}H\right\|}_{r,s}$
$\displaystyle\leq 2{\left\|H\right\|}_{r+\rho,s}\,,$ (21)
$\displaystyle{\left\|{\left(e^{{\left\\{S,\cdot\right\\}}}-\operatorname{id}\right)}H\right\|}_{r,s}$
$\displaystyle\leq\delta^{-1}{\left\|S\right\|}_{r+\rho,s}{\left\|H\right\|}_{r+\rho,s}\,,$
(22)
$\displaystyle{\left\|{\left(e^{{\left\\{S,\cdot\right\\}}}-\operatorname{id}-{\left\\{S,\cdot\right\\}}\right)}H\right\|}_{r,s}$
$\displaystyle\leq\frac{1}{2}\delta^{-2}{\left\|S\right\|}_{r+\rho,s}^{2}{\left\|H\right\|}_{r+\rho,s}$

More generally for any $h\in{\mathbb{N}}$ and any sequence
$(c_{k})_{k\in{\mathbb{N}}}$ with $|c_{k}|\leq 1/k!$, we have

(23) ${\left\|\sum_{k\geq
h}c_{k}\operatorname{ad}^{k}_{S}{\left(H\right)}\right\|}_{r,s}\leq
2\|H\|_{r+\rho,s}\big{(}\|S\|_{r+\rho,s}/2\delta\big{)}^{h}\,,$

where $\operatorname{ad}_{S}{\left(\cdot\right)}:={\left\\{S,\cdot\right\\}}$.

######  Lemma 3.6.

Fix $s\geq 0$ and ${\sigma}>0$ and $\omega\in{\mathtt{D}_{\gamma}}$. For any
$R\in{\mathcal{H}}_{r,s}^{d}$ with $d\geq 1$ and such that
$\Pi_{{\mathcal{K}}}R=0$, the Homological equation $L_{\omega}S=R$ has a
unique solution $S=L_{\omega}^{-1}R\in{\mathcal{H}}_{r,s+{\sigma}}^{d}$ such
that $\Pi_{{\mathcal{K}}}S=0$ and moreover

(24)
${\left\|L_{\omega}^{-1}R\right\|}_{r,s+{\sigma}}\leq\gamma^{-1}e^{{\mathcal{C}_{1}}{\sigma}^{-\frac{3}{\theta}}}{\left\|R\right\|}_{r,s}$

###  3.3. Poof of the main Theorem

The theorem follows by the following holomorphic version of Lemma 2.18. If
$H_{0}\in{\mathcal{F}}\cap{\mathcal{H}}_{{\mathtt{r}},s_{0}}$ of the form
$D_{\omega}+P_{0}$ with $P_{0}\in{\mathcal{F}}^{\geq 1}$ is formally
linearizable.

Fix $0<r_{0}<{\mathtt{r}}$ and $s_{0}>0$ so that

${\varepsilon}_{0}:=\gamma^{-1}\|P_{0}\|_{r_{0},s_{0}}\leq\gamma^{-1}\frac{r_{0}}{{\mathtt{r}}}\|P_{0}\|_{{\mathtt{r}},s_{0}}$

is appropriately small. More precisely, fix ${\mathtt{C}}=1+\pi^{2}/6$ and
assume

(25)
${\varepsilon}_{0}^{-1}\geq{\mathtt{K}}\sup_{n}e^{{\mathcal{C}}_{2}(s_{0})n^{\frac{6}{\theta}}}n^{2}\max(e^{n-\chi^{n}},e^{-(2-\chi)\chi^{n}}).$

where ${\mathtt{K}}$ is an appropriately large absolute constant while
${\mathcal{C}}_{2}(s_{0})={\mathcal{C}}_{1}{{\mathtt{C}}}^{\frac{3}{\theta}}{s_{0}}^{-\frac{3}{\theta}}$.
Let

$r_{i}=r_{i-1}-\rho_{i-1}\,,\quad
s_{i}=s_{i-1}+{\sigma}_{i-1}\,,\quad{\mathtt{d}}_{i}=2^{i}\,,\rho_{i}=\frac{r_{0}}{2{\mathtt{C}}\langle
i\rangle^{2}}\,,\quad{\sigma}_{i}=\frac{s_{0}}{{\mathtt{C}}\langle
i\rangle^{2}}$

so that $r_{i}\to r_{0}/2$ and $s_{i}\to 2s_{0}$.

Fix $1<\chi<2$ such that333for example if $\chi=15/14$ the sup on the left
hand side is smaller than $-0,2$.

(26) $\sup_{n\geq
0}2^{n+1}\ln(1-\frac{1}{2{\mathtt{C}}n^{2}})+\chi^{n}(\chi-1)\leq-0.1\,$

######  Lemma 3.7.

The sequence of generating functions and Hamiltonians of Lemma 2.18.

$\\{D_{\omega},S_{i}\\}=\Pi^{{\mathcal{R}}}H_{i}\,,\quad
H_{i}=e^{\\{S_{i-1},\cdot\\}}H_{i-1}\,.$

satisfies

$H_{i}=D_{\omega}+P_{i}\,,\quad
P_{i}\in{\mathcal{F}}^{\geq{\mathtt{d}}_{i}}\cap{\mathcal{H}}_{r_{i},s_{i}}\,.$

with the bounds

$\|S_{i-1}\|_{r_{i-1},s_{i}}\leq\gamma^{-1}e^{{\mathcal{C}}_{1}{\sigma}_{i-1}^{-\frac{3}{\theta}}}\|P_{i-1}\|_{r_{i-1},s_{i-1}}\,,\quad\|P_{i}\|_{r_{i},s_{i}}\leq\|P_{0}\|_{r_{0},s_{0}}e^{-\chi^{i}}\,.$

Moreover each $S_{i-1}$ defines a symplextic analytic change of variables
$\Psi_{i-1}:B_{r_{i}}({\mathtt{h}}_{s})\to B_{r_{i-1}}({\mathtt{h}}_{s})$ for
all $s\geq s_{i}$ satisfying

(27) $\sup_{|u|_{s}\leq r_{i}}|\Psi_{i}(u)-u|_{s}\leq 2^{-i}r_{0}$

Finally setting

$\Phi_{i}=\Psi_{1}\circ\Psi_{2}\circ\dots\Psi_{i}$

we have that $\Phi_{i}\to\Phi_{\infty}$ where $\Phi_{\infty}$ is an invertible
symplectic map $B_{r_{0}/2}({\mathtt{h}}_{2s_{0}})\to
B_{r_{0}}({\mathtt{h}}_{2s_{0}})$ such that

$H_{0}\circ\Phi_{\infty}=D_{\omega}$

###### Proof.

By induction. Let us denote
$\gamma^{-1}\|P_{0}\|_{r_{0},s_{0}}:={\varepsilon}_{0}$. Fix $k\geq 0$ and
assume that for all $i\leq k$ the Lemma holds. By definition

$S_{k}=\operatorname{ad}_{D_{\omega}}^{-1}\Pi_{\mathcal{R}}P_{k}\,.$

For all $s\geq s_{k}+{\sigma}_{k}\equiv s_{k+1}$, by Lemma 3.6 and (25)

$\|S_{k}\|_{r_{k},s}\leq\|S_{k}\|_{r_{k},s_{k+1}}\leq\gamma^{-1}e^{{\mathcal{C}}_{1}{\sigma}_{k}^{-\frac{3}{\theta}}}\|P_{k}\|_{r_{k},s_{k}}\leq{\varepsilon}_{0}e^{{\mathcal{C}}_{2}(s_{0})k^{\frac{6}{\theta}}}e^{-\chi^{k}}\leq\frac{1}{16e2{\mathtt{C}}k^{2}}\leq\frac{\rho_{k}}{8er_{k}}$

so, by Lemma 3.5 the time one flow
$\Psi^{1}_{S_{k}}:B_{r_{k+1}}({\mathtt{h}}_{s})\to
B_{r_{k}}({\mathtt{h}}_{s})$ is well defined analytic, symplectic and, by (19)
satisfies

(28) $\sup_{u\in
B_{r_{k+1}}({\mathtt{h}}_{s})}{\left|\Phi^{1}_{S_{k}}(u)-u\right|}_{{\mathtt{h}}_{s}}\leq
r_{k}{\left\|S_{k}\right\|}_{r_{k},s}\leq
C{\varepsilon}_{0}r_{0}k^{-2}e^{{\mathcal{C}}_{2}(s_{0})k^{\frac{6}{\theta}}}e^{-\chi^{k}}\stackrel{{\scriptstyle\eqref{piccolo1}}}{{\leq}}2^{-k}r_{0}\,.$

Recalling that

$\displaystyle H_{k+1}:=e^{\\{S_{k},\cdot\\}}H_{k}$
$\displaystyle=D_{\omega}+P_{k}+\\{S_{k},D_{\omega}\\}+\sum_{h=2}^{\infty}\frac{\operatorname{ad}_{S_{k}}^{h-1}}{h!}\\{S_{k},D_{\omega}\\}+\sum_{h=1}^{\infty}\frac{\operatorname{ad}_{S}^{h}}{h!}P_{k}$
$\displaystyle=D_{\omega}+\Pi^{\mathcal{K}}P_{k}-\sum_{h=1}^{\infty}\frac{\operatorname{ad}_{S_{k}}^{h}}{(h+1)!}\Pi^{{\mathcal{R}}}P_{k}+\sum_{h=1}^{\infty}\frac{\operatorname{ad}_{S_{k}}^{h}}{h!}P_{k}\,$
$\displaystyle=:D_{\omega}+\Pi^{<2^{k+1}}\Pi^{\mathcal{K}}P_{k}+P_{k+1}\,.$

and that in Lemma 2.18 we have proved that
$\Pi^{<2^{k+1}}\Pi^{\mathcal{K}}P_{k}=0$, we get

$P_{k+1}=\Pi^{\geq
2^{k+1}}\Pi^{\mathcal{K}}P_{k}-\sum_{h=1}^{\infty}\frac{\operatorname{ad}_{S_{k}}^{h}}{(h+1)!}\Pi^{{\mathcal{R}}}P_{k}+\sum_{h=1}^{\infty}\frac{\operatorname{ad}_{S_{k}}^{h}}{h!}P_{k}$

Now

$\displaystyle\|\Pi^{\geq
2^{k+1}}\Pi^{\mathcal{K}}P_{k}\|_{r_{k+1},s_{k+1}}\leq{\left(\frac{r_{k+1}}{r_{k}}\right)}^{{\mathtt{d}}_{k+1}}\|P_{k}\|_{r_{k},s_{k}}$
$\displaystyle\leq{\varepsilon}_{0}(1-\frac{1}{2{\mathtt{C}}k^{2}})^{2^{k+1}}e^{-\chi^{k}}$
$\displaystyle\|\sum_{h=1}^{\infty}\frac{\operatorname{ad}_{S_{k}}^{h}}{(h+1)!}\Pi^{{\mathcal{R}}}P_{k}+\sum_{h=1}^{\infty}\frac{\operatorname{ad}_{S_{k}}^{h}}{h!}P_{k}\|_{r_{k+1},s_{k+1}}$
$\displaystyle\leq\frac{16er_{k}}{\rho_{k}}\|P_{k}\|_{r_{k},s_{k}}\|S_{k}\|_{r_{k+1},s_{k+1}}$
$\displaystyle\leq
C{\varepsilon}^{2}_{0}e^{{\mathcal{C}}_{2}(s_{0})k^{\frac{6}{\theta}}}e^{-2\chi^{k}}k^{2}$

The bound on $P_{k+1}$ follows from (25) and (26) which imply

${\varepsilon}_{0}(1-\frac{1}{2{\mathtt{C}}k^{2}})^{2^{k+1}}e^{-\chi^{k}}+C{\varepsilon}^{2}_{0}e^{{\mathcal{C}}_{2}(s_{0})k^{\frac{6}{\theta}}}e^{-2\chi^{k}}k^{2}\leq{\varepsilon}_{0}e^{-\chi^{k+1}}$

In order to prove the convergence we remark that all the $\Psi_{i}$ map
$B_{r_{i}}({\mathtt{h}}_{2s_{0}})\to B_{r_{i-1}}({\mathtt{h}}_{2s_{0}})$,
consequently $\Phi_{i}$ maps $B_{r_{i}}({\mathtt{h}}_{2s_{0}})\to
B_{r_{0}}({\mathtt{h}}_{2s_{0}})$ and, by (27), it is a Cauchy sequence. ∎

##  Appendix A Technical Lemmata

In the following, we adapt material from [BMP18] to non mass conservation
situation.

###  A.1. Proof of Lemmata 3.3 and 3.4

We follow here [BMP18][Appendix B. Proof of lemma 3.1]. For any
$H\in{\mathcal{H}}_{r,s}$ (we recall that this space depends on two extra
parameters $p\geq\frac{1}{2}$ and $0<\theta\leq 1$) we define a map

$B_{1}(\ell^{2})\to\ell^{2}\,,\quad
y={\left(y_{j}\right)}_{j\in\mathbb{Z}}\mapsto{\left(Y^{(j)}_{H}(y;r,s)\right)}_{j\in\mathbb{Z}}$

by setting

(29)
$Y^{(j)}_{H}(y;r,s):=\sum_{\ast}|H_{{{\alpha}},{{\beta}}}|\frac{({{\alpha}}_{j}+{{\beta}}_{j})}{2}c^{(j)}_{r,s}({{\alpha}},{{\beta}})y^{{{\alpha}}+{{\beta}}-e_{j}}$

where $e_{j}$ is the $j$-th basis vector in ${\mathbb{N}}^{\mathbb{Z}}$, while
the coefficient

(30)
$c^{(j)}_{r,s}({\alpha},{\beta})=r^{|{\alpha}|+|{\beta}|-2}{\left(\frac{\langle
j\rangle^{2}}{\prod_{i}\langle
i\rangle^{{{\alpha}}_{i}+{{\beta}}_{i}}}\right)}^{p}e^{-s(\sum_{i}\langle
i\rangle^{\theta}({{\alpha}}_{i}+{{\beta}}_{i})-2\langle j\rangle^{\theta})}$

For brevity, we set

$\sum_{\ast}:=\sum_{{{\alpha}},{{\beta}}:\;\pi({\alpha},{\beta})=0}\,.$

The vector field $Y_{H}$ is a majorant analytic function on $\ell^{2}$ which
has the same norm as $H$. Since the majorant analytic functions on a given
space have a natural ordering this gives us a natural criterion for
immersions, as formalized in the following Lemma.

######  Lemma A.1.

Let $r,{r^{*}}>0,\,s,s^{\prime}\geq 0.$ The following properties hold.

  1. (1)

The norm of $H$ can be expressed as

(31) ${\left\|H\right\|}_{r,s}=\sup_{|y|_{\ell^{2}}\leq
1}{\left|Y_{H}(y;r,s)\right|}_{\ell^{2}}$

  2. (2)

Given $H^{(1)}\in{\mathcal{H}}_{{r^{*}},s^{\prime}}$ and
$H^{(2)}\in{\mathcal{H}}_{r,s}\,,$
such that for all ${{\alpha}},{{\beta}}\in{\mathbb{N}}^{\mathbb{Z}}_{f}$ and
$j\in\mathbb{Z}$ with ${{\alpha}}_{j}+{{\beta}}_{j}\neq 0$ one has

$|H^{(1)}_{{{\alpha}},{{\beta}}}|c^{(j)}_{{r^{*}},s^{\prime}}({{\alpha}},{{\beta}})\leq
c|H^{(2)}_{{{\alpha}},{{\beta}}}|c^{(j)}_{r,s}({{\alpha}},{{\beta}}),$

for some $c>0,$ then

${\left\|H^{(1)}\right\|}_{{r^{*}},s^{\prime}}\leq
c{\left\|H^{(2)}\right\|}_{r,s}\,.$

###### Proof of Lemma 3.3.

Recalling (30), we have

$\frac{c^{(j)}_{{r^{*}},s}({{\alpha}},{{\beta}})}{c^{(j)}_{r,s}({{\alpha}},{{\beta}})}={\left(\frac{{r^{*}}}{r}\right)}^{|{{\alpha}}|+|{{\beta}}|-2}\,.$

Since $|{{\alpha}}|+|{{\beta}}|-2\geq{\mathtt{d}}$, the inequality follows by
Lemma A.1 with $H^{(1)}=H^{(2)}$ and $s=s^{\prime}$. ∎

In order to prove Lemma 3.4 we need some notations and results proven in
[Bou05] and [CLSY].

######  Definition A.2.

Given a vector
$v={\left(v_{i}\right)}_{i\in\mathbb{Z}}\in{\mathbb{N}}^{\mathbb{Z}}_{f}$ with
$|v|\geq 2$ we denote by $\widehat{n}=\widehat{n}(v)$ the vector
${\left(\widehat{n}_{l}\right)}_{l\in I}$ (where $I\subset{\mathbb{N}}$ is
finite) which is the decreasing rearrangement of

$\\{{\mathbb{N}}\ni h>1\;\;\mbox{
repeated}\;v_{h}+v_{-h}\;\mbox{times}\\}\cup{\left\\{1\;\;\mbox{
repeated}\;v_{1}+v_{-1}+v_{0}\;\mbox{times}\right\\}}$

######  Remark A.3.

A good way of envisioning this list is as follows. Given an infinite set of
variables ${\left(x_{i}\right)}_{i\in\mathbb{Z}}$ and a vector
$v={\left(v_{i}\right)}_{i\in\mathbb{Z}}\in{\mathbb{N}}^{\mathbb{Z}}_{f}$
consider the monomial $x^{v}:=\prod_{i}x_{i}^{v_{i}}$. We can write

$x^{v}=\prod_{i}x_{i}^{v_{i}}=x_{j_{1}}x_{j_{2}}\cdots
x_{j_{|v|}}\,,\quad\mbox{ with}\quad j_{k}\in\mathbb{Z}$

then $\widehat{n}(v)$ is the decreasing rearrangement of the list
${\left(\langle j_{1}\rangle,\dots,\langle j_{|v|}\rangle\right)}$.

######  Example A.4.

Let us set

$v_{-1}=2,v_{0}=3,v_{1}=1,v_{3}=1,v_{4}=2.$

Hence, $1$ is repeated 6 times, $3$ is repeated 1 time, and $4$ is repeated 2
times :

$\hat{n}_{1}=4,\hat{n}_{2}=4,\hat{n}_{3}=3,\hat{n}_{4}=\dots=\hat{n}_{9}=1$

Given ${{\alpha}},{{\beta}}\in{\mathbb{N}}^{\mathbb{Z}}_{f}$ with
$|{{\alpha}}|+|{{\beta}}|\geq 2$ from now on we define

$\widehat{n}=\widehat{n}({{\alpha}}+{{\beta}})\,\qquad\mbox{and set}\quad
N:=|{{\alpha}}|+|{{\beta}}|$

which is the cardinality of $\widehat{n}.$ We observe that, $N\geq 2$ and
since

(32)
$0=\sum_{i\in\mathbb{Z}}i{\left({{\alpha}}_{i}-{{\beta}}_{i}\right)}=\sum_{h>0}h{\left({{\alpha}}_{h}-{{\beta}}_{h}-{{\alpha}}_{-h}+{{\beta}}_{-h}\right)}\,,$

there exists a choice of ${\sigma}_{i}=\pm 1,0$ such that444A given $h>1$
appears ${{{\alpha}}_{h}+{{\beta}}_{h}+{{\alpha}}_{-h}+{{\beta}}_{-h}}$ times
in the list $\widehat{n}$. Thus in order to get the summand
$h{\left({{\alpha}}_{h}-{{\beta}}_{h}-{{\alpha}}_{-h}+{{\beta}}_{-h}\right)}$
we assign to the $\widehat{n}_{l}$ with $\widehat{n}_{l}=h$ the sign
${\sigma}_{l}=+$, ${\alpha}_{h}+{\beta}_{-h}$ times and the sign
${\sigma}_{l}=-$, ${\alpha}_{-h}+{\beta}_{h}$ times. Let us now consider the
case $h=1$. By construction, $1$ appears
${{{\alpha}}^{(1)}+{{\beta}}^{(1)}+{{\alpha}}_{-1}+{{\beta}}_{-1}+{\alpha}_{0}+{\beta}_{0}}$
times in $\widehat{n}$. Thus in order to obtain the summand
${\left({{\alpha}}^{(1)}-{{\beta}}^{(1)}-{{\alpha}}_{-1}+{{\beta}}_{-1}\right)}$
we assign to the $\widehat{n}_{l}$ with $\widehat{n}_{l}=1$ the sign
${\sigma}_{l}=+$, ${\alpha}_{1}+{\beta}_{-1}$ times, the sign
${\sigma}_{l}=-$, ${\alpha}_{-1}+{\beta}_{1}$ times and ${\sigma}_{l}=0$ the
remaining ${\alpha}_{0}+{\beta}_{0}$ times.

(33) $\sum_{l}\sigma_{l}\widehat{n}_{l}=0.$

with $\sigma_{l}\neq 0$ if $\widehat{n}_{l}\neq 1$. Hence,

(34) $\widehat{n}_{1}\leq\sum_{l\geq 2}\widehat{n}_{l}.$

Indeed, if $\sigma_{1}=\pm 1$, the inequality follows directly from (33); if
$\sigma_{1}=0$, then $\widehat{n}_{1}=1$ and consequently
$\widehat{n}_{l}=1\,\forall l$. Since $|{\alpha}|+|{\beta}|\geq 2$, the list
$\widehat{n}$ has at least two elements, so the inequality is achieved.

######  Lemma A.5.

Given ${{\alpha}},{{\beta}}$ such that
$\sum_{i}i({{\alpha}}_{i}-{{\beta}}_{i})=0$, and $|{\alpha}|+|{\beta}|\geq 2$,
we have that setting $\widehat{n}=\widehat{n}({{\alpha}}+{{\beta}})$

(35) $\sum_{i}\langle
i\rangle^{\theta}({{\alpha}}_{i}+{{\beta}}_{i})=\sum_{l\geq
1}\widehat{n}_{l}^{\theta}\geq
2\widehat{n}^{\theta}_{1}+(2-2^{\theta}){\sum_{l\geq
3}\widehat{n}_{l}^{\theta}}.$

###### Proof.

The lemma above was proved in [Bou05] for $\theta=\frac{1}{2}$ and for general
$0<\theta<1$ in [CLSY][Lemma 2.1], in the case of zero mass and momentum. For
completeness we give below a proof , using only momentum conservation.
We start by noticing that if $|{\alpha}|+|{\beta}|=2$ then $\widehat{n}$ has
cardinality equal to two and (35) becomes $\widehat{n}_{1}+\widehat{n}_{2}\geq
2\widehat{n}_{1}$. Now, by (34), momentum conservation implies that
$\widehat{n}_{1}=\widehat{n}_{2}$ and hence (35).
If $|{\alpha}|+|{\beta}|\geq 3$ we write

$\sum_{i}\langle
i\rangle^{\theta}({{\alpha}}_{i}+{{\beta}}_{i})-2\widehat{n}_{1}^{\theta}=\sum_{l\geq
2}\widehat{n}_{l}^{\theta}-\widehat{n}_{1}^{\theta}\geq\sum_{l\geq
2}\widehat{n}_{l}^{\theta}-(\sum_{l\geq 2}\widehat{n}_{l})^{\theta}$

since the cardinality of $\widehat{n}$ is at least three we may write

$\sum_{l\geq 2}\widehat{n}_{l}^{\theta}-(\sum_{l\geq
2}\widehat{n}_{l})^{\theta}=\widehat{n}_{2}^{\theta}+\sum_{l\geq
3}\widehat{n}_{l}^{\theta}-(\widehat{n}_{2}+\sum_{l\geq
3}\widehat{n}_{l})^{\theta}$

Now setting, for $x_{i}\geq 1$, $i=2,\ldots,N$,

$f(x_{2},\dots,x_{N}):=x_{2}^{\theta}+(2^{\theta}-1)\sum_{l\geq
3}x_{l}^{\theta}-(x_{2}+\sum_{l\geq 3}x_{l})^{\theta}.$

Hence, we have $\partial_{x_{2}}f\geq 0$ for $x_{2}\geq x_{3}\geq 1$. Then

$f(x_{2},\dots,x_{N})\geq
f(x_{3},x_{3},x_{4},\dots,x_{N})=:f_{3}(x_{3},\dots,x_{N})\,.$

Now we set

$f_{n}(x_{n},\dots,x_{N}):=f(\underbrace{x_{n},\dots,x_{n}}_{n-1},x_{n+1},\dots,x_{N})=(1+(2^{\theta}-1)(n-2))x_{n}^{\theta}+\sum_{\ell\geq
n+1}x_{\ell}-((n-1)x_{n}+\sum_{\ell\geq n+1}x_{\ell})^{\theta}$

so that $f(x_{2},\dots,x_{N})\geq f_{3}(x_{3},\dots,x_{N})$. Assume
inductively that for some $3\leq n<N$, one has $f(x_{2},\dots,x_{N})\geq
f_{3}(x_{3},\dots,x_{N})\geq\dots\geq f_{n}(x_{n},\dots,x_{N})$. By direct
computation 555recalling that the $x_{\ell}>0$ and that
$1+(2^{\theta}-1)k-(k+1)^{\theta}\geq 0$, with $k=n+2>1$

$\displaystyle\partial_{x_{n}}f_{n}$
$\displaystyle=\theta\Big{[}\frac{(1+(2^{\theta}-1)(n-2))}{x_{n}^{1-\theta}}-\frac{n-1}{((n-1)x_{n}+\sum_{\ell\geq
n+1}x_{\ell})^{1-\theta}}\Big{]}$ $\displaystyle\geq\theta
x_{n}^{\theta-1}\Big{[}{(1+(2^{\theta}-1)(n+2))}-{(n-1)^{\theta}}\Big{]}\geq
0\,,$

so that the minimum is attained in $x_{n}=x_{n+1}$ and
$f(x_{2},\dots,x_{N})\geq f_{n+1}(x_{n+1},\dots,x_{N})$. In conclusion

$f(x_{2},\dots,x_{N})\geq f(x_{N},\dots,x_{N})\geq 0$

where the last inequality follows by recalling that
$1+(2^{\theta}-1)k-(k+1)^{\theta}\geq 0$ for $k\geq 1$. ∎

The Lemma proved above, is fundamental in discussing the properties of
${\mathcal{H}}_{r}({\mathtt{h}}_{p,s,a})$ with $s>0$, indeed it implies

(36) $\sum_{i}\langle i\rangle^{\theta}({{\alpha}}_{i}+{{\beta}}_{i})-2\langle
j\rangle^{\theta}\geq(2-2^{\theta}){\left(\sum_{l\geq
3}\widehat{n}_{l}^{\theta}\right)}\geq 0$

for all ${{\alpha}},{{\beta}}$ such that ${{\alpha}}_{j}+{{\beta}}_{j}\neq 0$.
Indeed, this follows from the fact that $\langle j\rangle\leq\widehat{n}_{1}$.

###### Proof of Lemma 3.4.

In all that follows we shall use systematically the fact that our Hamiltonians
are momentum preserving, are zero at the origin and have no linear term so
that ${\left|{{\alpha}}\right|}+{\left|{{\beta}}\right|}\geq 2$.
We need to show that

(37)
$\frac{c^{(j)}_{r,s+{\sigma}}({{\alpha}},{{\beta}})}{c^{(j)}_{r,s}({{\alpha}},{{\beta}})}=\exp(-{\sigma}(\sum_{i}\langle
i\rangle^{\theta}({{\alpha}}_{i}+{{\beta}}_{i})-2\langle
j\rangle^{\theta})\leq 1\,.$

The first identity comes form (30), while the last inequality follows by (36)
of Lemma A.5 ∎

###  A.2. Proof of Lemma 3.5

We recall the following classical result.

######  Lemma A.6.

Let $0<r_{1}<r.$ Let $E$ be a Banach space endowed with the norm
$|\cdot|_{E}$. Let $X:B_{r}\to E$ a vector field satisfying

$\sup_{B_{r}}|X|_{E}\leq\delta_{0}\,.$

Then the flow $\Phi(u,t)$ of the vector field666Namely the solution of the
equation $\partial_{t}\Phi(u,t)=X(\Phi(u,t))$ with initial datum
$\Phi(u,0)=u.$ is well defined for every

$|t|\leq T:=\frac{r-r_{1}}{\delta_{0}}$

and $u\in B_{r_{1}}$ with estimate

$|\Phi(u,t)-u|_{E}\leq\delta_{0}|t|\,,\qquad\forall\,|t|\leq T\,.$

###### Proof of Lemma 3.5.

The estimate for the Poisson bracket is proven in [BBP13]. In order to prove
the other estimates we use Lemma A.6, with $E\to{\mathtt{h}}_{s}$, $X\to
X_{S}$, $\delta_{0}\to(r+\rho)|S|_{r+\rho},$ $r\to r+\rho,$ $r_{1}\to r,$
$T\to 8e.$ finally we do not write the dependence on $s$ which is fixed.

Then the fact that the time $1$-Hamiltonian flow
$\Phi^{1}_{S}:B_{r}({\mathtt{h}}_{s})\to B_{r+\rho}({\mathtt{h}}_{s})$ is well
defined, analytic, symplectic follows, since

$\sup_{u\in
B_{r+\rho}({\mathtt{h}}_{s})}|X_{S}|_{{\mathtt{h}}_{s}}\leq(r+\rho)|S|_{r+\rho}<\frac{\rho}{8e}\,.$

Regarding the estimate (19), again by Lemma A.6 (choosing $t=1$), we get

$\sup_{u\in
B_{r}({\mathtt{h}}_{s})}{\left|\Phi^{1}_{S}(u)-u\right|}_{{\mathtt{h}}_{s}}\leq(r+\rho)|S|_{r+\rho}<\frac{\rho}{8e}\,.$

Estimates (20),(21),(22) directly follow by (23) with $h=0,1,2,$ respectively
and $c_{k}=1/k!$, recalling that by Lie series

$H\circ\Phi^{1}_{S}=e^{\rm ad_{S}}H=\sum_{k=0}^{\infty}\frac{{\rm
ad}_{S}^{k}H}{k!}=\sum_{k=0}^{\infty}\frac{H^{(k)}}{k!}\,,$

where $H^{(i)}:={\rm ad}_{S}^{i}(H)={\rm ad}_{S}(H^{(i-1)})$, $H^{(0)}:=H$.
Let us prove (23). Fix $k\in{\mathbb{N}},$ $k>0$ and set

$r_{i}:=r+\rho(1-\frac{i}{k})\,\,,\qquad i=0,\ldots,k\,.$

Note that, by the immersion properties of the norm in Lemma 3.3,

(38) $\|S\|_{r_{i}}\leq\|S\|_{r+\rho}\,,\qquad\forall\,i=0,\ldots,k\,.$

Noting that

(39) $1+\frac{kr_{i}}{\rho}\,\leq
k{\left(1+\frac{r}{\rho}\right)}\,,\qquad\forall\,i=0,\ldots,k\,,$

by using $k$ times (17) we have

$\displaystyle\|{H^{(k)}}\|_{r}$ $\displaystyle=$
$\displaystyle\|\\{S,{H^{(k-1)}}\\}\|_{r}\leq
4(1+\frac{kr}{\rho})\|{H^{(k-1)}}\|_{r_{k-1}}\|S\|_{r_{k-1}}$
$\displaystyle\stackrel{{\scriptstyle\eqref{morello}}}{{\leq}}$
$\displaystyle\|H\|_{r+\rho}\|S\|_{r+\rho}^{k}4^{k}\prod_{i=1}^{k}(1+\frac{kr_{i}}{\rho})\stackrel{{\scriptstyle\eqref{dave}}}{{\leq}}\|H\|_{r+\rho}\left(4k{\left(1+\frac{r}{\rho}\right)}\|S\|_{r+\rho}\right)^{k}\,.$

Then, using $k^{k}\leq e^{k}k!,$ we get

$\displaystyle\left\|\sum_{k\geq h}c_{k}{H^{(k)}}\right\|_{r}$
$\displaystyle\leq$ $\displaystyle\sum_{k\geq
h}|c_{k}|\|{H^{(k)}}\|_{r}\leq\|H\|_{r+\rho}\sum_{k\geq
h}\left(4e{\left(1+\frac{r}{\rho}\right)}\|S\|_{r+\rho}\right)^{k}$
$\displaystyle=$ $\displaystyle\|H\|_{r+\rho}\sum_{k\geq
h}(\|S\|_{r+\rho}/2\delta)^{k}\stackrel{{\scriptstyle\eqref{stima
generatrice}}}{{\leq}}2\|H\|_{r+\rho}(\|S\|_{r+\rho}/2\delta)^{h}\,.$

Finally, if $S$ and $H$ satisfy momentum conservation so does each ${\rm
ad}_{S}^{k}H$, $k\geq 1$, hence $H\circ\Phi^{1}_{S}$ too. ∎

###  A.3. Proof of lemma 3.6

Here we strongly use the fact that we are working with a dispersive PDE on the
circle with superlinear dispersion law.

By Lemma A.1 (2), we have

${\left\|L_{\omega}^{-1}R\right\|}_{r,s+{\sigma}}\leq\gamma^{-1}K{\left\|R\right\|}_{r,s}$

where

$K=\gamma\sup_{\begin{subarray}{c}j:{{\alpha}}_{j}+{{\beta}}_{j}\neq 0\\\
\pi({\alpha},{\beta})=0\end{subarray}}\frac{e^{-{\sigma}{\left(\sum_{i}\langle
i\rangle^{\theta}({{\alpha}}_{i}+{{\beta}}_{i})-2\langle
j\rangle^{\theta}\right)}}}{{\left|\omega\cdot{{\left({{\alpha}}-{{\beta}}\right)}}\right|}}\,.$

Therefore proving (24) amounts to showing that

(40) $K\leq e^{{\mathcal{C}}_{1}{\sigma}^{-\frac{3}{\theta}}}\,.$

We divide in two cases regarding whether the inequality

(41)
${\left|\sum_{i}{{\left({{\alpha}}_{i}-{{\beta}}_{i}\right)}i^{2}}\right|}\leq
2\sum_{i}{\left|{{\alpha}}_{i}-{{\beta}}_{i}\right|}\,,$

holds or not. We remark that

(42)
${\left|\sum_{i}{{\left({{\alpha}}_{i}-{{\beta}}_{i}\right)}i^{2}}\right|}\geq
2\sum_{i}{\left|{{\alpha}}_{i}-{{\beta}}_{i}\right|}\qquad\Longrightarrow\qquad{\left|\omega\cdot{\left({{\alpha}}-{{\beta}}\right)}\right|}\geq
1\,,$

indeed denoting $\omega_{j}=j^{2}+\xi_{j}$ with
${\left|\xi_{j}\right|}\leq\frac{1}{2}$,

${\left|\omega\cdot{\left({{\alpha}}-{{\beta}}\right)}\right|}\geq
2\sum_{j}{\left|{{\alpha}}_{j}-{{\beta}}_{j}\right|}-\frac{1}{2}\sum_{j}{\left|{{\alpha}}_{j}-{{\beta}}_{j}\right|}\geq
1.$

Of course if
${\left|\omega\cdot{\left({{\alpha}}-{{\beta}}\right)}\right|}\geq 1$, by (36)
and (37) we get

$\gamma\frac{e^{-{\sigma}{\left(\sum_{i}\langle
i\rangle^{\theta}({{\alpha}}_{i}+{{\beta}}_{i})-2\langle
j\rangle^{\theta}\right)}}}{{\left|\omega\cdot{{\left({{\alpha}}-{{\beta}}\right)}}\right|}}\leq
1\,$

and the bound (40) is trivially achieved.
Otherwise, to deal with the case in which (41) holds, we need some notation.
Given $u\in\mathbb{Z}^{\mathbb{Z}}_{f}$, consider the set

$M(u):={\left\\{j\neq
0\,,\quad\mbox{repeated}\quad{\left|u_{j}\right|}\;\mbox{times}\right\\}}\,,$

where $D(u)<\infty$ is its cardinality. Define the vector $m=m(u)$ as the
reordering of the elements of the set above such that
$|m_{1}|\geq|m_{2}|\geq\dots\geq|m_{D}|\geq 1.$
Given ${{\alpha}}\neq{{\beta}}\in{\mathbb{N}}^{\mathbb{Z}}_{f}$ with
$|{{\alpha}}|+|{{\beta}}|\geq 3$ we consider $m=m({{\alpha}}-{{\beta}})$ and
$\widehat{n}=\widehat{n}({{\alpha}}+{{\beta}}).$ If we denote by $D$ the
cardinality of $m$ and $N$ the one of $\widehat{n}$ we have

(43) $D+{{\alpha}}_{0}+{{\beta}}_{0}\leq N$

and

(44)
$(|m_{1}|,\dots,|m_{D}|,\underbrace{1,\;\dots\;,1}_{N-D\;\rm{times}})\,\preceq\,{\left(\widehat{n}_{1},\dots\widehat{n}_{N}\right)}\,.$

######  Example A.7.

Let set $v={\alpha}+\beta$ and $u={\alpha}-\beta$ with

$\displaystyle{\alpha}_{-5}=1,{\alpha}_{-2}=2,\alpha_{0}=2,\alpha_{4}=1$
$\displaystyle\beta_{-5}=1,\beta_{-3}=2,\beta_{0}=3,\beta_{6}=1$
$\displaystyle\pi({\alpha},\beta)=(-5)(1-1)+(-3)(-2)+(-2)(2)+4(1)+6(-1)=0$
$\displaystyle v_{-5}=2,v_{-3}=2,v_{-2}=2,v_{0}=5,v_{4}=1,v_{6}=1$
$\displaystyle u_{-5}=0,u_{-3}=-2,u_{-2}=2,u_{0}=-1,u_{4}=1,u_{6}=-1$
$\displaystyle\widehat{n}(v)=(6,5,5,4,3,3,2,2,1,1,1,1,1),N=13(=\text{Card}(\hat{n}))$
$\displaystyle M(u)=\\{-3,-3,-2,-2,4,6\\},m(u)=\\{6,4,-3,-3,-2,-2\\},D(u)=6.$

Therefore, we have $D(u)+\alpha_{0}+\beta_{0}=8\leq 13=N(\hat{n}(v))$. Hence,
(43) holds.
Futhermore, $(6,4,3,3,2,2,1,1,1,1,1,1,1)\leq\hat{n}(v)$, that is (44).

######  Lemma A.8.

Assume that $g$ defined on $\mathbb{Z}$ is non negative, even and not
decreasing on ${\mathbb{N}}.$ Then, if ${{\alpha}}\neq{{\beta}}$,

(45) $\sum_{i\in\mathbb{Z}}g(i)|{{\alpha}}_{i}-{{\beta}}_{i}|\leq
2g(m_{1})+\sum_{l\geq 3}g(\widehat{n}_{l})\,.$

###### Proof.

By definition of $m({\alpha}-{\beta})$ and setting ${\sigma}_{l}={\rm
sign}({{\alpha}}_{m_{l}}-{{\beta}}_{m_{l}})\,,$ we have

(46) $\displaystyle\sum_{i\in\mathbb{Z}}g(i)({{\alpha}}_{i}-{{\beta}}_{i})$
$\displaystyle=$ $\displaystyle g(0)({{\alpha}}_{0}-{{\beta}}_{0})+\sum_{l\geq
1}{\sigma}_{l}g(m_{l})\,.$

Hence

$\displaystyle\sum_{i\in\mathbb{Z}}g(i)|{{\alpha}}_{i}-{{\beta}}_{i}|$
$\displaystyle=$ $\displaystyle g(0)|{{\alpha}}_{0}-{{\beta}}_{0}|+\sum_{l\geq
1}g(m_{l})$ $\displaystyle\leq$ $\displaystyle
g(1)({{\alpha}}_{0}+{{\beta}}_{0})+2g(m_{1})+\sum_{l\geq 3}g(m_{l})$

and (45) follows by (43) and (44). ∎

By (46)

(47)
${\color[rgb]{.5,0,.5}0=}\sum_{i\in\mathbb{Z}}{\left({{\alpha}}_{i}-{{\beta}}_{i}\right)}i=\sum_{l}{\sigma}_{l}m_{l}$

and

(48)
${\sum_{i}{{\left({{\alpha}}_{i}-{{\beta}}_{i}\right)}i^{2}}}=\sum_{l}{\sigma}_{l}m^{2}_{l}\,.$

Analogously

(49)
${\sum_{i}{{\left|{{\alpha}}_{i}-{{\beta}}_{i}\right|}}}=D+|{{\alpha}}_{0}-{{\beta}}_{0}|\stackrel{{\scriptstyle\eqref{cappella}}}{{\leq}}N\,.$

Finally note that

(50) $\sigma_{l}\sigma_{l^{\prime}}=-1\qquad\Longrightarrow\qquad m_{l}\neq
m_{l^{\prime}}\,.$

######  Lemma A.9.

Given ${{\alpha}}\neq{{\beta}}\in{\mathbb{N}}^{\mathbb{Z}}_{f},$ such that
$\pi({{\alpha}}-{{\beta}})=0$, $N\geq 3,D\geq 1$ and satisfying (41), we have

(51) ${\left|m_{1}\right|}\leq 7\sum_{l\geq 3}\widehat{n}_{l}^{2}\,.$

###### Proof.

The case $D=1$ is not compatible with momentum conservation. Let us now
consider the case $D=2$, i.e.

${{\alpha}}-{{\beta}}={\sigma}_{1}{\bf e}_{m_{1}}+{\sigma}_{2}{\bf
e}_{m_{2}}+({{\alpha}}_{0}-{{\beta}}_{0}){\bf e}_{0}\,.$

If ${\sigma}_{1}{\sigma}_{2}=-1$, momentum conservation imposes $m_{1}=m_{2}$
but this contradicts (50). In the case ${\sigma}_{1}{\sigma}_{2}=1,$ by
momentum conservation we have $m_{1}=-m_{2}$. Then conditions (41) and (49)
imply that

$m_{1}^{2}+m_{2}^{2}\leq
2(D+|{{\alpha}}_{0}-{{\beta}}_{0}|)\stackrel{{\scriptstyle\eqref{enne}}}{{\leq}}2N\leq
6(N-2)\leq 6\sum_{l=3}^{N}\widehat{n}_{l}^{2}$

since $\widehat{n}_{l}\geq 1$.

Let us now consider the case $D\geq 3$. By (41),(48) and (49)

$\displaystyle m_{1}^{2}+{\sigma}_{1}{\sigma}_{2}m_{2}^{2}$
$\displaystyle\leq$ $\displaystyle
2(D+|{{\alpha}}_{0}-{{\beta}}_{0}|)+\sum_{l=3}^{D}m_{l}^{2}\leq
2N+\sum_{l=3}^{D}m_{l}^{2}\leq
2N+\sum_{l=3}^{N}\widehat{n}_{l}^{2}{\leq}7\sum_{l=3}^{N}\widehat{n}_{l}^{2}\,.$

since (recall $N\geq 3$) $2N\leq 6(N-2)\leq
6\sum_{l=3}^{N}\widehat{n}_{l}^{2}$.

If $\sigma_{1}\sigma_{2}=1$ then

${\left|m_{1}\right|},{\left|m_{2}\right|}\leq\sqrt{7\sum_{l\geq
3}\widehat{n}_{l}^{2}}.$

If ${\sigma}_{1}{\sigma}_{2}=-1$

$(|m_{1}|+|m_{2}|)(|m_{1}|-|m_{2}|)=m_{1}^{2}-m_{2}^{2}\leq 7\sum_{l\geq
3}\widehat{n}_{l}^{2}.$

Now, if ${\left|m_{1}\right|}\neq{\left|m_{2}\right|}$ then

${\left|m_{1}\right|}+{\left|m_{2}\right|}\leq 7\sum_{l\geq
3}\widehat{n}_{l}^{2}.$

Conversely, if ${\left|m_{1}\right|}={\left|m_{2}\right|}$, by (50),
$m_{1}\neq m_{2}$, hence $m_{1}=-m_{2}$. By substituting this relation into
(47), we have

$2{\left|m_{1}\right|}\leq\sum_{l\geq 3}{\left|m_{l}\right|}\leq\sum_{l\geq
3}\widehat{n}_{l}^{2}\,,$

concluding the proof. ∎

Now the key to proving Lemma A.10 is the following.

######  Lemma A.10.

Consider ${{\alpha}},{{\beta}}\in{\mathcal{M}}$ with ${\alpha}\neq{\beta}$ and
$|{\alpha}|+|{\beta}|\geq 3$. If (41) holds then for all $j$ such that
${{\alpha}}_{j}+{{\beta}}_{j}\neq 0$ one has

(52) $\sum_{i}{\left|{{\alpha}}_{i}-{{\beta}}_{i}\right|}\langle
i\rangle^{\theta/2}\leq
C_{*}{\left(\sum_{i}{\left({{\alpha}}_{i}+{{\beta}}_{i}\right)}\langle
i\rangle^{\theta}-2\langle j\rangle^{\theta}\right)}\,,\qquad
C_{*}=\frac{7}{2-2^{\theta}}$

###### Proof.

Let us first consider the case $D=0$, this means that
${\alpha}-{\beta}=({\alpha}_{0}-{\beta}_{0}){\bf e}_{0}$ and the left hand
side of (52) reads $|{\alpha}_{0}-{\beta}_{0}|$. By (36) and $N\geq 3$ the
right hand side of (52) is at least $2-2^{\theta}$, so if
$|{\alpha}_{0}-{\beta}_{0}|\leq 7$ the result is trivial. Otherwise we have
two cases, if $j=0$

$|{\alpha}_{0}-{\beta}_{0}|\leq 2(|{\alpha}_{0}-{\beta}_{0}|-2\langle
j\rangle^{\theta})\leq
2{\left(\sum_{i}{\left({{\alpha}}_{i}+{{\beta}}_{i}\right)}\langle
i\rangle^{\theta}-2\langle j\rangle^{\theta}\right)}\,,$

Otherwise we remark that if $j\neq 0$, ${\alpha}_{j}+{\beta}_{j}\neq 0$ and
${\alpha}_{j}-{\beta}_{j}=0$, then ${\alpha}_{j}+{\beta}_{j}\geq 2$, then

$|{\alpha}_{0}-{\beta}_{0}|\leq({\alpha}_{0}+{\beta}_{0})+({\alpha}_{j}+{\beta}_{j}-2)\langle
j\rangle^{\theta}\leq\sum_{i}{\left({{\alpha}}_{i}+{{\beta}}_{i}\right)}\langle
i\rangle^{\theta}-2\langle j\rangle^{\theta}\,.$

Now we consider indices ${\alpha},{\beta}$ such that $N\geq 3,D\geq 1$. Here
we apply Lemma A.9 Given
${{\alpha}},{{\beta}}\in{\mathbb{N}}^{\mathbb{Z}}_{f},$ as above we consider
$m=m({{\alpha}}-{{\beta}})$ and
$\widehat{n}=\widehat{n}({{\alpha}}+{{\beta}}).$

We have777Using that for $x,y\geq 0$ and $0\leq c\leq 1$ we get $(x+y)^{c}\leq
x^{c}+y^{c}.$

(53) $\displaystyle\sum_{i}{\left|{{\alpha}}_{i}-{{\beta}}_{i}\right|}\langle
i\rangle^{\theta/2}$
$\displaystyle\stackrel{{\scriptstyle\eqref{pula}}}{{\leq}}$ $\displaystyle
2{\left|m_{1}\right|}^{\frac{\theta}{2}}+\sum_{l\geq
3}\widehat{n}_{l}^{\frac{\theta}{2}}$
$\displaystyle\stackrel{{\scriptstyle\eqref{chiappechiare}}}{{\leq}}$
$\displaystyle 2{\left(7\sum_{l\geq
3}\widehat{n}_{l}^{2}\right)}^{\frac{\theta}{2}}+\sum_{l\geq
3}\widehat{n}_{l}^{\frac{\theta}{2}}$ $\displaystyle\leq$
$\displaystyle+2(7)^{\frac{\theta}{2}}\sum_{l\geq
3}\widehat{n}_{l}^{\theta}+\sum_{l\geq 3}\widehat{n}_{l}^{\frac{\theta}{2}}$
$\displaystyle\leq$
$\displaystyle\frac{2\sqrt{7}+1}{2-2^{\theta}}{\left((2-2^{\theta}){\left(\sum_{l\geq
3}\widehat{n}_{l}^{\theta}\right)}\right)}\,,$

Then by Lemma A.5 and (53) we get

$\displaystyle\sum_{i}{\left|{{\alpha}}_{i}-{{\beta}}_{i}\right|}\langle
i\rangle^{\theta/2}$ $\displaystyle\leq$
$\displaystyle\frac{7}{2-2^{\theta}}{\left(\sum_{i}\langle
i\rangle^{\theta}{\left({{\alpha}}_{i}+{{\beta}}_{i}\right)}-2\widehat{n}_{1}^{\theta}\right)}$
$\displaystyle\leq$ $\displaystyle\frac{7}{2-2^{\theta}}{\left[\sum_{i}\langle
i\rangle^{\theta}{\left({{\alpha}}_{i}+{{\beta}}_{i}\right)}-2\langle
j\rangle^{\theta}\right]}\,,$

proving (52). ∎

###### Conclusion of the proof of Lemma 3.6.

By applying Lemma A.10, since $\omega\in{\mathtt{D}_{\gamma}}$ we get:

$\displaystyle\gamma\frac{e^{-{\sigma}{\left(\sum_{i}\langle
i\rangle^{\theta}({{\alpha}}_{i}+{{\beta}}_{i})-2\langle
j\rangle^{\theta}\right)}}}{{\left|\omega\cdot{{\left({{\alpha}}-{{\beta}}\right)}}\right|}}\stackrel{{\scriptstyle\eqref{diofantinoBIS}}}{{\leq}}e^{-{\sigma}{\left(\sum_{i}\langle
i\rangle^{\theta}({{\alpha}}_{i}+{{\beta}}_{i})-2\langle
j\rangle^{\theta}\right)}}\prod_{i}{\left(1+({{\alpha}}_{i}-{{\beta}}_{i})^{2}\langle
i\rangle^{{2}}\right)}$
$\displaystyle\stackrel{{\scriptstyle\eqref{adele}}}{{\leq}}e^{-\frac{{\sigma}}{C_{*}}\sum_{i}{\left|{{\alpha}}_{i}-{{\beta}}_{i}\right|}\langle
i\rangle^{\frac{\theta}{2}}}\prod_{i}{\left(1+({{\alpha}}_{i}-{{\beta}}_{i})^{2}\langle
i\rangle^{{2}}\right)}$
$\displaystyle\leq\exp{\sum_{i}{\left[-\frac{{\sigma}}{C_{*}}{\left|{{\alpha}}_{i}-{{\beta}}_{i}\right|}\langle
i\rangle^{\frac{\theta}{2}}+\ln{{\left(1+{\left({{\alpha}}_{i}-{{\beta}}_{i}\right)}^{2}\langle
i\rangle^{{2}}\right)}}\right]}}$ (54)
$\displaystyle=\exp{\sum_{i}f_{i}({\left|{{\alpha}}_{i}-{{\beta}}_{i}\right|})}$

where, for $0<{\sigma}\leq 1$, $i\in\mathbb{Z}$ and $x\geq 0$, we defined

$f_{i}(x):=-\frac{{\sigma}}{C_{*}}x\langle
i\rangle^{\frac{\theta}{2}}+\ln{{\left(1+x^{2}\langle
i\rangle^{{2}}\right)}}\,.$

Finally, we have

######  Lemma A.11 ([BMP18]Lemma 7.2).

Setting

$i_{\sharp}:=\left(\frac{24C_{*}}{{\sigma}\theta}\ln\frac{12C_{*}}{{\sigma}\theta}\right)^{\frac{2}{\theta}}\,,$

we get

(55) $\sum_{i}f_{i}(|\ell_{i}|)\leq 18i_{\sharp}\ln i_{\sharp}$

for every $\ell\in\mathbb{Z}^{\mathbb{Z}}_{f}$.

###### Proof.

First of all we note that

$\sum_{i}f_{i}(|\ell_{i}|)=\sum_{i\ \text{s.t.}\ \ell_{i}\neq
0}f_{i}(|\ell_{i}|)$

since $f_{i}(0)=0.$ We have that888Using that $\ln(1+y)\leq 1+\ln y$ for every
$y\geq 1.$

$f_{i}(x)\leq-\frac{{\sigma}}{C_{*}}\langle
i\rangle^{\frac{\theta}{2}}x+{2}\ln(x)+2\ln\langle
i\rangle+1\,,\qquad\forall\,x\geq 1\,.$

Now,

$\max_{x\geq 1}\left(-\frac{{\sigma}}{C_{*}}\langle
i\rangle^{\frac{\theta}{2}}x+2\ln(x)\right)=\left\\{\begin{array}[]{l}\displaystyle-\frac{{\sigma}}{C_{*}}\langle
i\rangle^{\frac{\theta}{2}}\qquad\qquad\qquad\ \,\qquad\text{if}\quad\langle
i\rangle\geq i_{0}\,,\\\ \\\
\displaystyle-{2}+{2}\ln\frac{2C_{*}}{{\sigma}}-\theta\ln\langle
i\rangle\qquad\text{if}\quad\langle i\rangle<i_{0}\,,\end{array}\right.$

where

$i_{0}:=\left(\frac{2C_{*}}{{\sigma}}\right)^{\frac{2}{\theta}}\,,$

since the maximum is achieved for $x=1$ if $\langle i\rangle\geq i_{0}$ and
$x=\frac{2C_{*}}{{\sigma}\langle i\rangle^{\theta/2}}$ if $\langle
i\rangle<i_{0}$. Note that $i_{0}\geq e.$ Then we get

$\displaystyle\sum_{i}f_{i}(|\ell_{i}|)=\sum_{i\ \text{s.t.}\ \ell_{i}\neq
0}f_{i}(|\ell_{i}|)\leq$ $\displaystyle\sum_{\langle i\rangle\geq i_{0}\
\text{s.t.}\ \ell_{i}\neq 0}\left(2\ln\langle
i\rangle+1-\frac{{\sigma}}{C_{*}}\langle
i\rangle^{\frac{\theta}{2}}\right)+\sum_{\langle
i\rangle<i_{0}}\left(2\ln\frac{2C_{*}}{{\sigma}}+\Big{(}2-\theta\Big{)}\ln\langle
i\rangle\right)\,.$

We immediately have that

$\displaystyle\sum_{\langle
i\rangle<i_{0}}\left(2\ln\frac{2C_{*}}{{\sigma}}+\Big{(}{2}-\theta\Big{)}\ln\langle
i\rangle\right)\leq 6i_{0}\left(\ln\frac{2C_{*}}{{\sigma}}+\ln i_{0}\right)$
$\displaystyle=6\left(1+\frac{2}{\theta}\right)\left(\frac{2C_{*}}{{\sigma}}\right)^{\frac{2}{\theta}}\ln\frac{2C_{*}}{{\sigma}}\,.$

Moreover, in the case $\langle i\rangle\geq i_{0}\geq e,$

$2\ln\langle i\rangle+1-\frac{{\sigma}}{C_{*}}\langle
i\rangle^{\frac{\theta}{2}}\leq 3\ln\langle
i\rangle-\frac{{\sigma}}{C_{*}}\langle
i\rangle^{\frac{\theta}{2}}=\frac{6}{\theta}\Big{(}\ln\langle
i\rangle^{\frac{\theta}{2}}-{2\mathfrak{C}}\langle
i\rangle^{\frac{\theta}{2}}\Big{)}$

where

${\mathfrak{C}}:=\frac{\theta{\sigma}(2-2^{\theta})}{84}<1\,.$

We have that999Using that, for every fixed $0<{\mathfrak{C}}\leq 1,$ we have
${\mathfrak{C}}x\geq\ln x$ for every
$x\geq\frac{2}{{\mathfrak{C}}}\ln\frac{1}{{\mathfrak{C}}}.$

$\ln\langle i\rangle^{\frac{\theta}{2}}-2{\mathfrak{C}}\langle
i\rangle^{\frac{\theta}{2}}\leq-{\mathfrak{C}}\langle
i\rangle^{\frac{\theta}{2}}\,,\qquad\text{when}\qquad\langle i\rangle\geq
i_{*}:=\left(\frac{2}{{\mathfrak{C}}}\ln\frac{1}{{\mathfrak{C}}}\right)^{\frac{2}{\theta}}\,.$

Note that

$i_{\sharp}\geq\max\\{i_{0},i_{*}\\}\,.$

Therefore

$\displaystyle\sum_{\langle i\rangle\geq i_{0}\ \text{s.t.}\ \ell_{i}\neq
0}{\left(2\ln\langle i\rangle+1-\frac{{\sigma}}{C_{*}}\langle
i\rangle^{\frac{\theta}{2}}\right)}\leq\sum_{\langle i\rangle\geq i_{0}\
\text{s.t.}\ \ell_{i}\neq 0}\frac{6}{\theta}{\left(\ln\langle
i\rangle^{\frac{\theta}{2}}-{2\mathfrak{C}}\langle
i\rangle^{\frac{\theta}{2}}\right)}$ $\displaystyle\leq$
$\displaystyle\frac{6}{\theta}\left(\sum_{\langle
i\rangle<i_{\sharp}}\ln\langle i\rangle^{\frac{\theta}{2}}-\sum_{\langle
i\rangle\geq i_{\sharp}\ \text{s.t.}\ \ell_{i}\neq
0}\Big{(}{\mathfrak{C}}\langle i\rangle^{\frac{\theta}{2}}\Big{)}\right)\leq
9i_{\sharp}\ln i_{\sharp}\,.$

In conclusion we get

$\displaystyle\sum_{i}f_{i}(|\ell_{i}|)$ $\displaystyle\leq$ $\displaystyle
6\frac{2+\theta}{\theta}\left(\frac{2C_{*}}{{\sigma}}\right)^{\frac{2}{\theta}}\ln\frac{2C_{*}}{{\sigma}}+9i_{\sharp}\ln
i_{\sharp}$ $\displaystyle\leq$ $\displaystyle
9\left(\frac{2C_{*}}{{\sigma}\theta}\right)^{\frac{2}{\theta}}\ln{\left(\frac{2C_{*}}{{\sigma}}\right)}^{\frac{\theta}{2}}+9i_{\sharp}\ln
i_{\sharp}\leq 18i_{\sharp}\ln i_{\sharp}$

∎

The inequality (24) follows from plugging (55) into (54) and evaluating the
constant. ∎

## References

  * [BBHM] Baldi P., Berti M., Haus E., Montalto R., Time quasi-periodic gravity water waves in finite depth, Invent. math., 214(2), 739-911, 2018
  * [Bam99a] D. Bambusi.  Nekhoroshev theorem for small amplitude solutions in nonlinear Schrödinger equations.  Math. Z., 230(2):345–387, 1999.
  * [Bam99b] D. Bambusi.  On long time stability in Hamiltonian perturbations of nonresonant linear PDE’s.  Nonlinearity, 12:823–850, 1999.
  * [Bam03] D. Bambusi.  Birkhoff normal form for some nonlinear PDEs.  Comm. Math. Phys., 234(2):253–285, 2003.
  * [BDGS07] D. Bambusi, J.-M. Delort, B. Grébert, and J. Szeftel.  Almost global existence for Hamiltonian semilinear Klein-Gordon equations with small Cauchy data on Zoll manifolds.  Comm. Pure Appl. Math., 60(11):1665–1690, 2007.
  * [BG03] D. Bambusi and B. Grébert.  Forme normale pour NLS en dimension quelconque.  C. R. Math. Acad. Sci. Paris, 337(6):409–414, 2003.
  * [BG06] D. Bambusi and B. Grébert.  Birkhoff normal form for partial differential equations with tame modulus.  Duke Math. J., 135(3):507–567, 2006.
  * [BS20] D. Bambusi and L. Stolovitch.  Convergence to normal forms of integrable PDEs.  Comm. Math. Phys., 376(2):1441–1470, 2020.
  * [BCG] G.Benettin, L. Chierchia, and M. Guzzo,  The Steep Nekhoroshev’s Theorem.  Commun. Math. Phys., 342, 569–601 (2016)
  * [BGG85] G. Benettin, L. Galgani and A. Giorgilli.  A proof of Nekhoroshev’s theorem for the stability times in nearly integrable Hamiltonian systems.  Celestial Mech., 37(1): 1–25, 1985.
  * [BFG88] G. Benettin, J. Fröhlich, and A. Giorgilli.  A Nekhoroshev-type theorem for Hamiltonian systems with infinitely many degrees of freedom.  Comm. Math. Phys., 119(1):95–108, 1988.
  * [BKM18] M. Berti, Th. Kappeler, and R. Montalto.  Large KAM tori for perturbations of the defocusing NLS equation.  Astérisque, (403):viii+148, 2018.
  * [BBP13] M. Berti, L. Biasco, and M. Procesi.  KAM theory for the Hamiltonian derivative wave equation.  Annales Scientifiques de l’ENS, 46(2):299–371, 2013.
  * [BB15] M. Berti and Ph. Bolle.  Quasi-Periodic Solutions of Nonlinear Wave Equations on the d-Dimensional Torus  EMS Series Lect. in Math, 2020
  * [BBP10] M. Berti, P. Bolle, and M. Procesi.  An abstract Nash-Moser theorem with parameters and applications to PDEs.  Ann. Inst. H. Poincaré Anal. Non Linéaire, 27(1):377–399, 2010.
  * [BCP] M.Berti, L.Corsi, M.Procesi.  An abstract Nash-Moser theorem and quasi-periodic solutions for NLW and NLS on compact Lie groups and homogeneous manifolds.  Com. Math. Phys. 2014\. DOI 10.1007/s00220-014-2128-4
  * [BFP] Berti M., Feola R., Pusateri F.,  Birkhoff normal form and long time existence for periodic gravity Water Waves, preprint arXiv:1810.11549.
  * [BD18] M. Berti and J.M. Delort.  Almost global existence of solutions for capillarity-gravity water waves equations with periodic spatial boundary conditions.  Springer, 2018.
  * [BM21] M. Berti, L. Franzoi, A. Maspero.  Traveling quasi-periodic water waves with constant vorticity.  ARMA 240(1): 99-202, 2021.
  * [BMP18] L. Biasco, J.E. Massetti, and M. Procesi.  An abstract Birkhoff Normal Form Theorem and exponential type stability of the 1d NLS.  Comm. Math. Phys., 375(3), 2089–2153, 2020.
  * [BMP21] L. Biasco, J.E. Massetti, and M. Procesi.  Almost-periodic invariant tori for the NLS on the circle.  Ann. Inst. H. Poincaré Anal. Non Linéaire, 38(3):711–758, 2021.
  * [BFN15] A. Bounemoura, B. Fayad, and L. Niederman.  Double exponential stability for generic real-analytic elliptic equilibrium points.  2015\.  Preprint ArXiv : arxiv.org/abs/1509.00285.
  * [Bou98] J. Bourgain.  Quasi-periodic solutions of Hamiltonian perturbations of 2D linear Schrödinger equations.  Ann. of Math. (2), 148(2):363–439, 1998.
  * [Bou96a] J. Bourgain.  Construction of approximative and almost periodic solutions of perturbed linear Schrödinger and wave equations.  Geom. Funct. Anal., 6(2):201–230, 1996.
  * [Bou05] J. Bourgain.  On invariant tori of full dimension for 1D periodic NLS.  J. Funct. Anal., 229(1):62–94, 2005.
  * [Bru72] A.D. Bruno.  Analytical form of differential equations.  Trans. Mosc. Math. Soc, 25,131-288(1971); 26,199-239(1972), 1971-1972.
  * [CM18] L. Corsi, R. Montalto  Quasi-periodic solutions for the forced Kirchhoff equation on $\mathbb{T}^{d}$ NonLinearity, 31(11), 5075-5109.
  * [CLSY] H. Cong, J. Liu, Y. Shi, and X. Yuan.  The stability of full dimensional KAM tori for nonlinear Schrödinger equation.  J. Differential Equations, 264(7):4504–4563, 2018.
  * [CMW] H. Cong, L. Mi, and P. Wang.  A Nekhoroshev type theorem for the derivative nonlinear Schrödinger equation.  J. Differential Equations, 268(9):5207–5256, 2020.
  * [CY21] Cong, H., Yuan, X.  The existence of full dimensional invariant tori for 1-dimensional nonlinear wave equation  Annales de l’Institut Henri Poincare (C) Analyse Non Lineaire, 2021, 38(3), pp. 759–786
  * [CW93] 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.
  * [DS04] J.-M. Delort and J. Szeftel.  Long-time existence for small data nonlinear Klein-Gordon equations on tori and spheres.  Int. Math. Res. Not., (37):1897–1966, 2004.
  * [DS06] J.-M. Delort and J. Szeftel.  Bounded almost global solutions for non Hamiltonian semi-linear Klein-Gordon equations with radial data on compact revolution hypersurfaces.  Ann. Inst. Fourier (Grenoble), 56(5):1419–1456, 2006.
  * [Del12] J. M. Delort.  A quasi-linear birkhoff normal forms method. application to the quasi-linear klein-gordon equation on $\mathtt{S}^{1}$.  Astérisque, 341, 2012.
  * [D15] J.-M. Delort  Quasi-Linear Perturbations of Hamiltonian Klein-Gordon Equations on Spheres,  American Mathematical Society, 2015.
  * [EK10] L. H. Eliasson and S. B. Kuksin.  KAM for the nonlinear Schrödinger equation.  Ann. of Math. (2), 172(1):371–435, 2010.
  * [FG13] E. Faou and B. Grébert.  A Nekhoroshev-type theorem for the nonlinear Schrödinger equation on the torus.  Anal. PDE, 6(6):1243–1262, 2013.
  * [FG] R. Feola - F. Giuliani,  Quasi-periodic Traveling Waves on an Infinitely Deep Perfect Fluid Under Gravity, preprint, arXiv:2005.08280 (2020).
  * [FGP] Feola R., Giuliani F., Pasquali S., On the integrability of Degasperis-Procesi equation: control of the Sobolev norms and Birkhoff resonances,  J. Differential Equations  266 (2019) 3390–3437
  * [FGPr] R. Feola - F. Giuliani - M. Procesi,  Reducibile KAM tori for the Degasperis Procesi equation,  Comm. Math. Phys., 377, (2020), 1681–1759.
  * [FI18] R. Feola and F. Iandoli.  Long time existence for fully nonlinear NLS with small Cauchy data on the circle.  Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) Vol. XXII (2021), 109-182
  * [FI20] R. Feola and F. Iandoli.  A non-linear Egorov theorem and Poincaré-Birkhoff normal forms for quasi-linear pdes on the circle.  preprint 2020, arxiv:2002.1244
  * [GYX] Geng, J., Xu, X., You, J.  An infinite dimensional KAM theorem and its application to the two dimensional cubic Schrödinger equation  Advances in Mathematics, 2011, 226(6), pp. 5361–5402
  * [KP03] Th. Kappeler and J. Pöschel.  KdV & KAM,  Ergebnisse der Mathematik und ihrer Grenzgebiete. 3., 45, Springer-Verlag, Berlin, 2003.
  * [K88] S. B. Kuksin.  Perturbation of conditionally periodic solutions of infinite-dimensional Hamiltonian systems.  Izv. Akad. Nauk SSSR Ser. Mat., 52(1):41–63, 240, 1988.
  * [KP96] Kuksin, S. and Pöschel, J.  Invariant cantor manifolds of quasi-periodic oscillations for a nonlinear Schrödinger equation  Annals of Mathematics, 143(1) 149-179 1996.
  * [Pös90] Pöschel, J.,  Small divisors with spatial structure in infinite dimensional Hamiltonian systems  Communications in Mathematical Physics, 127(2), 351-393, 1990.
  * [IoL05] G. Iooss and E. Lombardi.  Polynomial normal forms with exponentially small remainder for analytic vector fields.  J. Differential Equations, 212(1):1–61, 2005.
  * [K00] S. B. Kuksin.  Analysis of Hamiltonian PDEs.  Oxford Lecture Series in Math. and Appl., 19, Oxford University Press, Oxford, 2000.
  * [LS10] E. Lombardi and L. Stolovitch.  Normal forms of analytic perturbations of quasihomogeneous vector fields: Rigidity, invariant analytic sets and exponentially small approximation.  Ann. Scient. Ec. Norm. Sup., pages 659–718, 2010.
  * [N77] N.N. Nehorošev.  An exponential estimate of the time of stability of nearly integrable Hamiltonian systems.  Uspehi Mat. Nauk, 32, 6(198), 5–66, 1977.
  * [Ni04] Niederman L.  Exponential stability for small perturbations of steep integrable Hamiltonian systems.  Ergod. Theory Dyn. Syst., 24(2), 593–608 (2004) .
  * [Nik86] N.V. Nikolenko.  The method of Poincaré normal forms in problems of integrability of equations of evolution type.  Russian Math. Surveys, 41:5:63–114, 1986.
  * [Pös86] J. Pöschel.  On invariant manifolds of complex analytic mappings near fixed points.  Expo. Math., 4,97-109, 1986.
  * [Pös99] J. Pöschel.  On Nekhoroshev’s estimate at an elliptic equilibrium.  Internat. Math. Res. Notices, (4), 203–215, 1999. Pöschel, Jürgen
  * [PP16] C. Procesi and M. Procesi,  Reducible quasi-periodic solutions for the non linear Schrödinger equation,  BUMI 9(2),189, 2016
  * [Rue77] H. Rüssmann.  On the convergence of power series transformations of analytic mappings near a fixed point into a normal form.  Preprint I.H.E.S., M/77/178, 1–44, 1977.
  * [Se92] J.-P Serre.  Lie Algebras and Lie groups.  Lecture Notes in Mathematics, 1500, Springer-Verlag, 1992
  * [Sie42] C.L. Siegel.  Iteration of analytic functions.  Ann. of Math. (2), 43:607–612, 1942.
  * [Sto00] L. Stolovitch.  Singular complete integrabilty.  Publ. Math. I.H.E.S., 91:133–210, 2000.
  * [Sto15] L. Stolovitch.  Family of intersecting totally real manifolds of $(\mathbb{C}^{n},0)$ and germs of holomorphic diffeomorphisms.  Bull. Soc. math. France, 143(1):247–263, 2015.
  * [Way90] C. E. Wayne.  Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory.  Comm. Math. Phys., 127(3):479–528, 1990.
  * [Y21] Yuan, X.,  KAM Theorem with Normal Frequencies of Finite Limit-Points for Some Shallow Water Equations  CPAM, 2021, 74(6), pp. 1193–1281
  * [YZ14] Xiaoping Yuan and Jing Zhang.  Long time stability of Hamiltonian partial differential equations.  SIAM J. Math. Anal., 46(5):3176–3222, 2014.
  * [Zeh77] E. Zehnder.  C. L. Siegel’s linearization theorem in infinite dimensions.  _M_ anuscripta Math., 23(4) 363–371, 1977/78