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KAM Theory, Nonlinear Schr dinger equation, perturbation of hamiltonian system
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\begin{document}

\title{Perturbations of the defocusing NLS equation}

\author{B. Gr\'eber$\mbox{t}^{1}$, T. Kappele$\mbox{r}^2$
}

\maketitle
\begin{itemize}

\item[1.] UMR 6629 CNRS, Universit\'e de Nantes,
2 rue de la Houssini\`ere, BP 92208, 44322 Nantes cedex 3, France.
\item[2.] Institut f\"ur Mathematik, Universit\"at Z\"urich,
Winterthurerstrasse 190, CH-8057 Z\"urich, Switzerland.


\end{itemize}

\n {\bf Abstract:}  We prove that many finite dimensional tori, invariant
under the flow of the defocusing nonlinear Schr\"odinger equation,
persist under small Hamiltonian perturbations.These invariant
tori are not necessarily close to the zero solution.


%\setcounter{section}{-1}
%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%
\n Consider the defocusing nonlinear Schr\"odinger
 equation NLS with periodic boundary conditions
%%%%%%%%%%%%%%%%
\begin{equation}
\label{eq:1.1}
i \partial_t \varphi = - \partial^2_x + 2 | \varphi
 |^2 \varphi; \quad \varphi (x + 1, t) = \varphi (x, t) \quad
(x
\in
\mathbb{R}, t \in \mathbb{R}).
\end{equation}
%%%%%%%%%%%%%%%%

\n It is a completely integrable system with phase
space $H^{N} \equiv H^{N} (S^1; \mathbb{C})
\linebreak
(N
\in \mathbb{R}_{\ge 1})$ and Hamiltonian $\mathcal{H} =
\mathcal{H} (\varphi, \overline{\varphi}).$ Here, for $N\geq 0$,
%%%%%%%%%%%%%%%%%%%
\begin{equation}
\label{eq:1.2}
H^{N} (S^1; \mathbb{C}) := \{ \varphi (x) =
\sum_{k \in \mathbb{Z}} e^{2 \pi k
x} \hat{\varphi} (k) \mid || \varphi
||_{N} < \infty \}
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%
\n where
%%%%%%%%%%%%%%%%%%%%%%%
\[|| \varphi ||_{N} :=
(\sum_{k \in \mathbb{Z}} (1 + | k
| )^{2N} | \hat{\varphi}
 (k) |^2)^{1/2}\]
%%%%%%%%%%%%%%%%%%%%%%%
\n and $\hat{\varphi} (k) \quad (k \in \mathbb{Z})$
denote the Fourier coefficients of
$\varphi$, viewed  as a function of period $1$,
and, for any $\varphi \in H^{1}$,
%%%%%%%%%%%%%%%%%%%%%%%%
\begin{equation}
\label{eq:1.3}
\mathcal{H} (\varphi, \overline{\varphi}) :=
\int_{S^1} \left(\partial_{x} \varphi
\partial_x \overline{\varphi} + \varphi^2
 \overline{\varphi^2} \right) dx\ .
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\n The Poisson structure is given by the
regular Poisson bracket.
%%%%%%%%%%%%%%%%%%%%%%
\begin{equation}
\label{eq:1.4}
\{F, G\} (\varphi, \overline{\varphi}) := i
\int_{S^1} \Big(\frac{\partial F}{\partial
\varphi} \ \frac{\partial G}{\partial \bar{\varphi}} -
\frac{\partial F}{\partial \bar{\varphi}} \
\frac{\partial G}{\partial \varphi}\Big) dx
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\n where $F, G$ are functionals
on $L^2 \equiv L^2 (S^1; \mathbb{C})$ of class $C^1.$
When written in Hamiltonian form, NLS becomes on $H^1$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\[ \partial_t \varphi = -i \frac{\partial
 \mathcal{H}}{\partial \overline{\varphi}}. \]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\n It is well known that NLS admits a Lax
 pair representation
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{equation}
\label{eq:1.5}
\frac{d}{dt} L (\varphi) = [L (\varphi),
A (\varphi)]
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%
\n where $L$ is the Zakharov-Shabat operator
 (see \cite{zs})
%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{equation}
\label{eq:1.6}
L (\varphi) := i \bl {lr} 1 & 0 \\ 0 & -1 \er \
 \frac{d}{dx} + \bl {ll} 0 & \varphi \\
\overline{\varphi} & 0 \er,
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\n and $A$ is a rather complicated operator
 given in \cite{ft}. As a
consequence, the periodic spectrum,
$spec_{per} L(\varphi),$ remains invariant
under the NLS flow. Here, $spec_{per} L(\varphi)$
 denotes the spectrum of
$L(\varphi)$ when considered on the interval [0,2]
with periodic boundary
conditions. The periodic spectrum consists of
 two interlacing sequences
$(\lambda^+_j (\varphi))_{j \in \mathbb{Z}},
(\lambda^-_j (\varphi))_{j \in
\mathbb{Z}}$ of real numbers satisfying
%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{equation}
\label{eq:1.7}
\ldots < \lambda^-_j (\varphi) \le \lambda^+_j
(\varphi) < \lambda^-_{j+1} (\varphi) \le
\ldots .
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%
\n They are uniquely determined by the sequence
of the gap lengths, $\gamma (\varphi) = (\gamma_k
(\varphi))_{k \in \mathbb{Z}}$ with
$\gamma_k (\varphi) =
\lambda^+_k (\varphi) - \lambda^-_k (\varphi)$
(see \cite{gg} and also \cite{gkp}).

\vspd

\n To describe the structure of the phase
 space $H^{N} (S^1; \mathbb{C})$ we
introduce the model space
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\[l^2_{N} (\mathbb{Z}; \mathbb{R}^2) :=
 \{ (x,y) = (x_j, y_j)_{j \in
\mathbb{Z}} \mid ||(x, y)||_{N} <
\infty \}
\]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\n where
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\[|| (x, y) ||_{N} := \sum_{j \in \mathbb{Z}} (1 + | j
|)^{2N}  (x^2_j + y^2_j) < \infty.
\]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\n The space $l^2_{N} (\mathbb{Z};
\mathbb{R}^2)$ is endowed with the Poisson structure induced by the
canonical symplectic structure $\sum_{j
\in \mathbb{Z}} dx_j \wedge dy_j.$ In
\cite{gkp} (cf also  \cite{bkm2},
 \cite{kma}, \cite{kapo} and \cite{MV}) we have
proved the following
result:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{guess}
\label{T1.1} There exists a family of diffeomorphisms
 $\Phi \equiv
\Phi^{(N)}$, $N \in \mathbb{R}_{> 0}$,
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\[\Phi : l^2_{N} (\mathbb{Z}; \mathbb{R}^2)
\rightarrow H^{N} (S^1;
\mathbb{C})\]
%%%%%%%%%%%%%%%%%%%%%%%%%%%
\n such that

\begin{itemize}
\item[(1)] $\Phi$ is globally 1-1, onto, 
bianalytic and preserves the Poisson bracket.
\item[(2)] The coordinates $(x_j, y_j)_{j \in \mathbb{Z}} =
\Phi^{-1} (\varphi)$ are global Birkhoff
coordinates for NLS and its hierarchy. That is the 
transformed NLS Hamiltonian $\mathcal H \circ \Phi$ depends
only on  the actions $I_j :=
(x^2_j + y^2_j)/2$, $j \in
\mathbb{Z}$, with $(x_j, y_j)$ being canonical coordinates in 
$l^2_{N} (\mathbb{Z}; \mathbb{R}^2)$.
\item[(3)] for $N \ge
N^{\prime}$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\[\Phi^{(N)} = \Phi^{(N^{\prime})}
 |_{l^2_{N}} .\]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{itemize}
\end{guess}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\n As a consequence, the solution $\varphi (x, t)
 \equiv \varphi_t (x)$ of the
initial value problem for NLS with initial
profile $\varphi_0 = \Phi ((\sqrt{2
I_j} e^{i \theta_j})_{j \in \mathbb{Z}})$
in $H^{N} (S^1; \mathbb{C})$, with $N\geq 1$, is
given by
%%%%%%%%%%%%
\[\varphi_t  = \Phi ((\sqrt{2 I_j} \ e^{i (\theta_j + t
 \omega_j (I))})_{j \in
\mathbb{Z}})
\]
%%%%%%%%%%%%

\n where $\omega_j (I) = \frac{\partial
 \mathcal{H} (I)}{\partial I_j} \; (j \in
\mathbb{Z})$ denote the frequencies of NLS
and $\mathcal{H} = \mathcal{H}
(I)$ is the Hamiltonian of NLS when expressed
 in action variables.

\vspd

\n An asymptotic expansion of the frequencies
(cf. section 4) shows that
$\omega_k \sim \omega_{-k}$ for $| k |$
large. In order to control the
effect of these
``asymptotic'' resonnances on
perturbed equations we impose
symmetry conditions so that the action-angle
variables $(I_k, \theta_k)$ with $k < 0$ are uniquely
determined by
\linebreak
$(I_j,
\theta_j)_{j \ge 0}.$ More
precisely, we consider as phase spaces
$H^{N}_{\alpha}
(S^1; \mathbb{C}),
\alpha \in \mathbb{R},$ defined by,
%%%%%%%%%%%%
\begin{equation}
\label{eq:1.8}
H^{N}_{\alpha} (S^1; \mathbb{C}) :=
\Phi (l^2_{N; \alpha} (\mathbb{Z};
\mathbb{R}^2))
\end{equation}
%%%%%%%%%%%%%%
\n where $( \sqrt{2 I_j} e^{i \theta_j})_{j
\in \mathbb{Z}} \in l^2_{N; \alpha}$ iff $(\sqrt{2
I_j}
\ e^{i \theta_j})_{j \in \mathbb{Z}} \in l^2_{N}$
and
%%%%%%%%%%%%%%
\begin{equation}
\label{eq:1.9}
I_{-j} = I_j \ \ \forall j \ge 0
\end{equation}
%%%%%%%%%%%%%
\n and
%%%%%%%%%%%%%%%
\begin{equation}
\label{eq:1.10}
\theta_{-j} \equiv \theta_j + \alpha
\ ({\rm mod} \ 2 \pi) \quad \forall j \ge 0 \
\mbox{with} \ I_j \neq 0.
\end{equation}
%%%%%%%%%%%%%%%
\n Notice that for $\alpha \not \equiv 0
(\rm{mod} \ 2 \pi),$ \eqref{eq:1.10}
implies that $I_0 (\varphi) = 0$ for all
 $\varphi \in H^{N}_{\alpha}.$



It turns out that our analysis applies to more general
symmetric phase spaces, see remark at the end of
section 5.



The subspaces
$H^{N}_{\alpha} (S^1; \mathbb{C})$ are
invariant under the NLS-flow. A way
 to prove this (cf section 3) is to show
that the symmetries of the NLS
Hamiltonion $\mathcal{H}$ when expressed
 in
action-variables, imply that
 $\mathcal{H} (I^{\prime}) = \mathcal{H} (I)$ where
$I^{\prime} = (I^{\prime}_k)_{k \in
\mathbb{Z}}$ is given by $I^{\prime}_k := I_{-k}
\ (k
\in \mathbb{Z}).$ As a  consequence the frequencies
$\omega_j = \frac{\partial
\mathcal{H}}{\partial I_j}$ are
symmetric at points
where $I^{\prime} = I$ and we have
the following
%%%%%%%%%%%%%%
%%%%%%%%%%%%%%
\begin{prop}
\label{P1.1}
\begin{itemize}
\item[(i)] $\mathcal{H} (I^{\prime})
 = \mathcal{H} (I);$
\item[(ii)] if $I^{\prime} = I,$ then \
$\omega_j (I) = \omega_{-j} (I)
\ \forall j \ge 1.$
\end{itemize}
\end{prop}
%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%
\n Moreover, in section 3 (see also
\cite{gk3}) we prove
%%%%%%%%%%%%%%%
\[H^{N}_{\alpha} (S^1; \mathbb{C}) =
\{\varphi \in H^{N} (S^1;
\mathbb{C})\mid e^{i \alpha} \check{\varphi}
 \equiv \varphi\}
\]
%%%%%%%%%%%%%%%
\n where $\check{\varphi} (x) = \varphi (-x).$
 In particular, $H^{N}_{\pi} \cap
C^{\infty}$ (resp. $H^{N}_0 \cap C^{\infty})$ is
 the phase-space of $\varphi \in
H^{N} \cap C^{\infty}$ satisfying a
generalized Dirichlet [ Neumann ]
condition, i.e. for
$k \ge 0 $
$$ \partial^{2k}_{x}
\varphi (0) = \partial^{2k}_x \varphi (1) = 0 \quad  [\
\partial^{2k+1}_x \varphi (0) = \partial^{2k+1}_x \varphi
(1) = 0\ ]\ .$$
 By a slight abuse of notation, the
restriction of $\Phi$ to $l^2_{N; \alpha}
(\mathbb{Z};
\mathbb{R}^2)$ is again denoted by $\Phi.$



\n For $\alpha \in \mathbb{R} / 2 \pi
\mathbb{Z},$ a finite
subset
$A
\subseteq \mathbb{Z}_{\ge 0}$ (with $0 \not
\in A$ if $\alpha \not \equiv 0$ mod $2\pi$)
and $I_A \in (\mathbb{R}_{>0})^{| A |}$
we denote by $T^{\alpha}_{I_A}$ the $|
 A |$ dimensional torus of the
model space $l^2 (\mathbb{Z};
\mathbb{R}^2),$ defined by
%%%%%%%%%%%%%%%%%%
\begin{equation}
\label{eq:1.11}
\begin{array}{lll}
T^{\alpha}_{I_A} &:= & \{(\sqrt{2 J_j}
e^{i \theta_j})_{j \in \mathbb{Z}} | J_j
= J_{-j} = I_j \ (j \in A); \\ \\
&& J_j = J_{-j} = 0 \ (j \not \in A); \; \theta_j =
\theta_{-j} + \alpha \ (j \in A) \}
\end{array}
\end{equation}
%%%%%%%%%%%%%%%%%%%

\n and by $\mathcal{T}^{\alpha}_{I_A}$ the $|
 A |$ dimensional torus in
$H^{N}_{\alpha}$, invariant under NLS,
%%%%%%%%%%%%%%%%%%%%
\begin{equation}
\label{eq:1.13}
\mathcal{T}^{\alpha}_{I_A}
:= \Phi (T^{\alpha}_{I_A}).
\end{equation}
%%%%%%%%%%%%%%%%%%%
A potential $\varphi \in H^{N} (S^1;
\mathbb{C})$ is said to be  {\it symmetric} if
$I_{-j}=I_j$ for any $j\geq 1$ and is a {\it finite gap}
[ $K$-gap with $K\in \mathbb Z_{\geq 1}$ ] potential 
if there exists a finite subset $B \subset \mathbb Z$
[ of cardinality $|B|=K$ ] so that 
$$
I_j(\varphi ) = 0 \mbox{ iff } j\in \mathbb Z \setminus B \ .
$$
Hence any element in $\mathcal{T}^{\alpha}_{I_A}$ is  a 
symmetric $2 |A|$ gap potential (if $0 \not \in
A)$ or
$2 |A| -1$ gap potential (if $0 \in A$) and thus in particular
smooth by Theorem~\ref{T1.1} (cf also \cite{gk4}).

\n For $\Gamma \subseteq
(\mathbb{R}_{>0})^{| A |}$ compact and of
positive Lebesgue measure introduce the following union
of tori
%%%%%%%%%%%%%%%%%%%%
\begin{equation}
\label{eq:1.14}
\mathcal{T}^{\alpha}_{\Gamma} := \cup_{I_A \in
\Gamma}
\mathcal{T}^{\alpha}_{I_A}.
\end{equation}
%%%%%%%%%%%%%%%%%%%%


\n We will consider Hamiltonian perturbations,
$\mathcal{H}_{\varepsilon} = \mathcal{H} +
\varepsilon K$ on $H^{N}_{\alpha} (S^1;
\mathbb{C})$ with the following properties:

\begin{itemize}
\item[(P1)] $K$ is real analytic on some symmetric
neighborhood
$U_{\Gamma}$ of
\linebreak
$\{ (\varphi, \overline{\varphi})\mid \varphi
\in \mathcal{T}^{\alpha}_{\Gamma} \}$ in $(H^{N}
(S^1; \mathbb{C}))^2.$
\footnote{$U_{\Gamma}$ is said to be symmetric iff
$(e^{i
\alpha}
\check{\varphi}, e^{- i \alpha} \check{\psi}) \in
U_{\Gamma}$ for any $(\varphi, \psi) \in U_{\Gamma}.$}

\item[(P2)] $\frac{\partial K}{\partial \varphi},
\frac{\partial K}{\partial \psi}$ are bounded as
functions from $U_{\Gamma}$ into $H^{N} (S^1;
\mathbb{C})$ and verify the normalization condition
%%%%%%%%%%%%%%%%%%%%%
\[\sup \left\{ || \frac{\partial K}{\partial
\varphi} (\varphi, \psi) ||_{N} + ||
\frac{\partial K}{\partial \psi} (\varphi, \psi)
||_{N} \mid (\varphi, \psi) \in U_{\Gamma}
\right\}
\le 1.
\]
%%%%%%%%%%%%%%%%%%%%%
\item[(P3)] $K$ satisfies the symmetry condition,
$((\varphi, \psi) \in U_{\Gamma})$
%%%%%%%%%%%%%%%%%%%%
\[K(\varphi, \psi) = K(e^{i \alpha} \check{\varphi},
e^{-i\alpha} \check{\psi}). \]
%%%%%%%%%%%%%%%%%%%
\end{itemize}

\n Notice that, together with Proposition
\ref{P1.1}, condition (P3) insures that solutions of
$\frac{\partial \varphi}{\partial t} = i \frac{\partial
\mathcal{H}_{\varepsilon}}{\partial \varphi}$ for initial
data in  $H^{N}_{\alpha} (S^1;
\mathbb{C})$ evolve in the same space
$H^{N}_{\alpha} (S^1; \mathbb{C}).$

\vspd

\n Our KAM Theorem states that, for $ \varepsilon$
small enough, many of the NLS-invariant tori
$\mathcal{T}^{\alpha}_{I_A}$ persist under
perturbation of the NLS Hamiltonian by $\varepsilon
K$ with $K$ satisfying (P1), (P2), and (P3).
Moreover these tori and their linear flows are only
slightly perturbed.

\n Denote by $T^n$ the $n-$dimensional torus
$(\mathbb{R}/\mathbb{Z})^{n}.$

\vspd
%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%
\begin{guess}
\label{T1.2}
Let $N \ge 1,  A, \Gamma, \alpha,
U_{\Gamma}$ be given as above. Then, for  $K$
satisfying (P1), (P2) and (P3), there exists
$\varepsilon_0$ so that for any $\varepsilon$
with $|
\varepsilon
|
\le \varepsilon_0$

\begin{itemize}
\item[(i)] there exists a Cantor set
$\Gamma_{\varepsilon} \subset \Gamma$ with {\textrm meas}
$(\Gamma \setminus \Gamma_{\varepsilon})
\stackrel{\varepsilon \rightarrow 0}{\rightarrow}
0,$
\item[(ii)] there exists a Lipschitz family of real
analytic torus embeddings
%%%%%%%%%%%%%%%%%%%%%%
\[\Psi : T^{| A |} \times
\Gamma_{\varepsilon} \rightarrow U_{\Gamma} \cap
\{(\varphi, \overline{\varphi}) | \varphi \in
H^{N}_{\alpha}\}\]
%%%%%%%%%%%%%%%%%%%%%%
\n and

\item[(iii)] there exists a Lipschitz map $f :
\Gamma_{\varepsilon} \rightarrow \mathbb{R}^{| A
|}$
\end{itemize}

\n such that for $I_A \in \Gamma_{\varepsilon}$
and $\theta_A \in T^{| A |}, \Psi
(\theta_A + t f (I_A), I_A)$ is a quasiperiodic
solution of $\partial_t \varphi = i \frac{\partial
\mathcal{H}}{\partial \bar{\varphi}} + i \varepsilon
\frac{\partial K}{\partial \overline{\varphi}}.$
Moreover, the deformed invariant tori, $\Psi
(T^{| A |}
\times \{I_A \}),$ are linearly stable.
\end{guess}
%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%

\vspd

\n {\bf Remarks:}

\begin{itemize}
\item[1.] Theorem \ref{T1.2} generalizes results
due  to Kuksin-P\"oschel \cite{kp}
\linebreak
which concern the
special case where $\Gamma \subseteq
\mathbb{R}^{| A |}_+$ is contained in a
sufficiently small neighborhood of $0 \in
\mathbb{R}^{| A |}$ and the phase space
consists of elements satisfying generalized
Dirichlet boundary conditions. In this situation, action-angle variables
are not needed as the Fourier
coefficients $(\hat{\varphi} (k))_{k \in \mathbb{Z}}$
are a sufficiently good approximation of the
Birkhoff coordinates close to the origin.
\item[2.] Similarly, the results of \cite{cw} and
their generalization by \cite{b}, while not directly
comparable with our Theorem \ref{T1.2}, concern only
small perturbations of NLS around $\varphi = 0$.
\item[3.] Our results and methods continue the
investigation in \cite{kapo} on the Korteweg-de
Vries equation. The purpose of this paper is to
document similar features of the defocusing NLS equation.
\item[4.] Theorem \ref{T1.2} has been announced in
\cite{gk1}.
\end{itemize}

\section{Birkhoff normal form}

\n By Theorem \ref{T1.1}, we already know that
$\mathcal{H},$ the Hamiltonian of NLS, admits a
Birkhoff normal form in a neighborhood $U \subset
(H^{N} (S^1; \mathbb{C}))^2$ of $(0,0).$ In
\cite{kp}, S. Kuksin and J. P\"oschel have computed
explicitely the first few coefficients of the
Birkhoff normal form of $\mathcal{H}$ when
restricted to the reduced phase space formed by
functions satisfying generalized Dirichlet boundary
conditions.

\vspd

\n Using the fact that NLS admits Birkhoff
coordinates near $\varphi = 0,$ the coefficients of the
Birkhoff normal form of $\mathcal{H}$ up to order 2
(included) can be read off by considering
$\mathcal{H} (\varphi, \overline{\varphi})$ as a function
of the Fourier coefficients $\hat{\varphi} (k) \; (k \in
\mathbb{Z})$ of $\varphi \in H^{N} (S^1; \mathbb{C})$
the full phase space of NLS.

\vspd

\n By a slight abuse of notation, we denote also by
$\mathcal{H}$ the pull back, $\Phi^{\ast}
\mathcal{H} := \mathcal{H} \cdot \Phi,$ of
$\mathcal{H}$ by the Birkhoff map $\Phi$. Of course,
$\mathcal{H}$ is a function of the actions alone,
$\mathcal{H} = \mathcal{H} ((I_j)_{j \in
\mathbb{Z}}).$
%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%
\begin{prop}
\label{P2.1}
(cf [KP]) In a neighborhood of $I = 0,$
%%%%%%%%%%%%%%%%%%%%%
\begin{equation}
\label{eq:2.1}
\mathcal{H} = \Lambda_2 + \Lambda_4 +
0_6 (I)
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%
\n where
%%%%%%%%%%%%%%%%%%%%%
\begin{equation}
\label{eq:2.2}
\Lambda_2 (I) := \sum_{j \in
\mathbb{Z}} (2 \pi j)^2 I_j
\end{equation}
%%%%%%%%%%%%%%%%%%%%%
\begin{equation}
\label{eq:2.3}
\Lambda_4 (I) := 2 \sum_{j \neq k} I_k
I_j + \sum_{j \in \mathbb{Z}} I^2_j
\end{equation}
%%%%%%%%%%%%%%%%%%%%%
\n and where $0_6 (I)$ is a remainder term of order
6, i.e. $0_6 (I) = \sum_{| \alpha | \ge 3}
\linebreak
a_{\alpha} I^{\alpha}.$ (The action variables have
to be counted as being of order 2.)
\end{prop}
%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%
\vspd

\n {\bf Proof:} Substitute $\varphi (x) = \sum_{k \in
\mathbb{Z}} \hat{\varphi} (k) e^{2i\pi k x}$ into
\eqref{eq:1.3},
%%%%%%%%%%%%%%%%%%%%%%
\begin{equation}
\label{eq:2.4}
\mathcal{H} (\varphi, \overline{\varphi}) = \sum_{k \in
\mathbb{Z}} (2 \pi k)^2 \hat{\varphi} (k)
\overline{\hat{\varphi}(k)} + G (\varphi,
\overline{\varphi})
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%
\n where
%%%%%%%%%%%%%%%%%%%%%%%
\begin{equation}
\label{eq:2.5}
G  (\varphi, \overline{\varphi}) = \sum_{k + l = j + m}
\hat{\varphi} (k) \hat{\varphi} (l)
\overline{\hat{\varphi} (j)} \ \overline{\hat{\varphi}
(m)}.
\end{equation}
%%%%%%%%%%%%%%%%%%%%
\n Notice that \eqref{eq:2.4} is already in Birkhoff
normal form up to order 2. Recall that the Poisson
bracket with respect to $\hat{\varphi} (k),
\overline{\hat{\varphi} (k)} \ (k \in \mathbb{Z})$ is
given by
%%%%%%%%%%%%%%%%%%%%%%%
\[\{F, G\} = i \sum_{k \in \mathbb{Z}}
\Big(\frac{\partial F}{\partial
\hat{\varphi}(k)}
\frac{\partial G}{\partial \overline{\hat{\varphi} (k)}}
 -
\frac{\partial F}{\partial \overline{\hat{\varphi} (k)}}
\frac{\partial G}{\partial \hat{\varphi}(k)}
\Big)
\]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\n and that near $\varphi = 0,$ the action $I_j$ is
close to
$| \hat{\varphi}(j)|^2 (j \in \mathbb{Z})$. Using
the fact that the coefficients in the Birkhoff
normal form are unique, we then conclude
that  $\Lambda_1 (I)$ can be read off from
\eqref{eq:2.4}
%%%%%%%%%%%%%%%%%%%%%%%%
\[\Lambda_1 (I) = \sum_{k \in \mathbb{Z}} 4 \pi^2 k^2
I_k. \]
%%%%%%%%%%%%%%%%%%%%%%
\n Similarly, we can read off the coefficients of
the 4'th order from \eqref{eq:2.4}. In fact we only
have to consider
%%%%%%%%%%%%%%%%%%%%%
\[S := \sum_{\{k, l\} = \{j, m\}} \hat{\varphi} (k)
\hat{\varphi} (l) \overline{\hat{\varphi} (j)}
\overline{\hat{\varphi} (m)}. \]
%%%%%%%%%%%%%%%%%%%%
\n This sum is split into two parts, $S = S_1 +
S_2,$ where $S_1 := \sum_{k \in \mathbb{Z}} |
\hat{\varphi} (k)|^4$ includes the terms in $S$
with $k = l$ and $S_2 := 2 \sum_{k \neq l} |
\hat{\varphi}(k)|^2 | \hat{\varphi} (l)|^2.$

\vspd

\n Therefore arguing as above, we obtain
\eqref{eq:2.3}. \ \ \ \carre

\vspd

\n From Proposition \ref{P2.1}, we obtain an
expansion of the NLS-frequencies,
$\omega_j (I) :=
\frac{\partial \mathcal{H}}{\partial I_j} \ (j \in
\mathbb{Z})$ at $I = 0:$

\vspd
%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%
\begin{coroll}
\label{C2.1}
In a neighborhood of $I = 0,$
%%%%%%%%%%%%%%%%%%%%%
\begin{equation}
\label{eq:2.6}
\omega_j (I) = (2 \pi j)^2 + 2 (2 \sum_{k \in
\mathbb{Z}} I_k - I_j) + 0_4 (I);
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%
\begin{equation}
\label{eq:2.7}
\frac{\partial \omega_j}{\partial I_k}  = 2
(2 - \delta_{jk}) + 0_2 (I);
\end{equation}
%%%%%%%%%%%%%%%%%%%%%
\n As a consequence, for any finite set $A \subseteq
\mathbb{Z}$ with $| A | \ge 2,$
%%%%%%%%%%%%%%%%%%%%%%%
\begin{equation}
\label{eq:2.8}
\det \left( \left(\frac{\partial \omega_j}{\partial
I_k} \right)_{j, k
\in A} \right) |_{I = 0} = - (-1)^{| A |} (2
(| A | -2) + 3) \neq 0.
\end{equation}
%%%%%%%%%%%%%%%%%%%%
\end{coroll}
%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%
\vspd

\n {\bf Proof:} The asymptotics \eqref{eq:2.6} and
\eqref{eq:2.7} follow immediately from Proposition
\ref{P2.1}.
Towards \eqref{eq:2.8}, notice that at $I=0$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\[\frac{1}{2} (\frac{\partial \omega_j}{\partial
I_k})_{j, k \in A} = 2 E_{| A |} - Id_{| A
|},\]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\n where $E_{| A |}$ is the $|A|
\times |A|$ matrix whose entries are all
equal to 1 and where $Id_{| A |}$ is the $|
A | \times | A |$ identity matrix. By
substracting the first column from the other ones
and then expanding with respect to the last column,
we obtain, with $d_n := \det (2 E_n - I_n),$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\[d_n = - d_{n-1} + (1)^{n+1} 2. \]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\n Hence, for $n \ge 2$, we get $d_n = (-1)^{n+1} (2
(n-2) + 3).$ \ \ \ \carre
%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Symmetries of NLS}
%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\n In this section, we recall results of \cite{gk3}
where  symmetries of NLS are expressed
in  action-angle coordinates and prove Proposition
\ref{P1.1} stated in the introduction. We briefly
describe the steps needed to prove that the reduced
phase space
$H^{N}_{\alpha} (S^1;
\mathbb{C})$ (defined in \eqref{eq:1.8}) is equal to
$\{ \varphi \in H^{N} (S^1; \mathbb{C}) | e^{i
\alpha} \check{\varphi} = \varphi \}.$

\vspd

\n Introduce the
symmetry operator acting on $L^2,
\; S_{\alpha} (\varphi) := e^{i \alpha} \check{\varphi}$
where, as above, $\check{\varphi}$ is defined by
$\check{\varphi} (x) = \varphi (-x)$ and $\alpha \in
\mathbb{R}.$

\vspd

\n In view of \eqref{eq:1.1}, the NLS flow
commutes with $S_{\alpha}$ and from \eqref{eq:1.3}
it follows that the Hamiltonian $\mathcal{H} (\varphi)
\equiv \mathcal{H} (\varphi, \overline{\varphi})$ is
invariant under
$S_{\alpha}$,
%%%%%%%%%%%%%%%%%%%%%
\begin{equation}
\label{eq:3.1}
\mathcal{H} (S_{\alpha} \varphi) = \mathcal{H} (\varphi).
\end{equation}
%%%%%%%%%%%%%%%%%%%
\n Concerning the periodic spectrum we have (cf
\cite{gk3}) the following
%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%
\begin{lemma}
\label{L3.1}
For $\varphi \in L^2$ and $\alpha \in
\mathbb{R},$
%%%%%%%%%%%%%%%%%%%%%%%
\[\lambda^{\pm}_k (S_{\alpha} (\varphi)) = -
\lambda^{\mp}_{-k} (\varphi) \ (k \in \mathbb{Z}).\]
%%%%%%%%%%%%%%%%%%%%%%%%%
\n In particular, $spec_{per} (L (S_{\alpha}
(\varphi))) = - spec_{per} (\varphi).$
\end{lemma}
%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%
\n Using this lemma, the definition of the action
and angle variables (see \cite{gkp}) we obtained
in \cite{gk3}
%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%
\begin{lemma}
\label{L3.2}
For $\varphi \in L^2$ and $\alpha \in
\mathbb{R},$
\begin{itemize}
\item[(i)] $I_k (S_{\alpha} (\varphi)) = I_{-k} (\varphi)
\ \ (k \in \mathbb{Z});$
\item[(ii)] $\theta_k (S_{\alpha} (\varphi)) \equiv
\theta_{-k} (\varphi) + \alpha \ (\rm{mod} \;  2
\pi)
\quad \forall k \in \mathbb{Z}$ with $I_k (S_{\alpha}
(\varphi)) \neq 0.$
\end{itemize}
\end{lemma}
%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%
\n As an application of Lemma \ref{L3.2} we have
%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prop}
\label{P3.3}
For all $\alpha \in \mathbb{R},$
%%%%%%%%%%%%%%%%%%%%%%%%%
\[H^{N}_{\alpha} (S^1; \mathbb{C}) = \{ \varphi \in
H^{N} (S^1; \mathbb{C}) | e^{i \alpha}
\check{\varphi} = \varphi \}. \]
%%%%%%%%%%%%%%%%%%%%%%%%%
\end{prop}
%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%&
\vspd

\n {\bf Proof:} Introduce $G^{N}_{\alpha} := \{
\varphi \in H^{N} (S^1; \mathbb{C}) | e^{i \alpha}
\check{\varphi} = \varphi \}.$ Using Lemma \ref{L3.2} and
Theorem \ref{T1.1}, we  get $G^{N}_{\alpha} \subset
H^{N}_{\alpha}.$ Conversely, if $\varphi \in
H^{N}_{\alpha},$ it follows from Lemma \ref{L3.2}
that $I_k (S_{\alpha} (\varphi)) = I_k (\varphi) \ (k \in
\mathbb{Z})$ and $\theta_k (S_{\alpha} (\varphi))
\equiv \theta_k (\varphi) (\rm{mod} \ 2 \pi)
\linebreak (\forall k
\in \mathbb{Z}$ with $I_k (\varphi) \neq 0).$ Since
 the Birkhoff map $\Phi$
is one to one (cf Theorem \ref{T1.1}), $S_{\alpha}
(\varphi) =
\varphi,$ i.e. $\varphi \in G^{N}_{\alpha}.$ \ \ \ \carre

\vspd

\n Lemma 3.2 can also be used to prove Proposition
\ref{P1.1}, which is stated in the introduction.

\vspf

\n {\bf Proof} (of Propostion \ref{P1.1}). As in
the introduction, denote by $I^{\prime} (\varphi) =
(I^{\prime}_k (\varphi))_{k \in \mathbb{Z}}$ the
sequence $I^{\prime}_k (\varphi) := I^{\prime}_{-k}
(\varphi)\; (k \in \mathbb{Z}).$ The formula in
Lemma \ref{L3.2} (i) then reads
%%%%%%%%%%%%%%%%%%%%%%%
\begin{equation}
\label{eq:3.2}
I (S_{\alpha} (\varphi)) = I^{\prime} (\varphi).
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%
\n On the other hand, when expressing the
Hamiltonian with respect to action coordinates,
formula \eqref{eq:3.1} reads
%%%%%%%%%%%%%%%%%%%%%%%%
\begin{equation}
\label{eq:3.3}
\mathcal{H} (I (S_{\alpha} (\varphi)) = \mathcal{H} (I)
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%
\n Combining \eqref{eq:3.2} and \eqref{eq:3.3} leads
to the claimed identity $\mathcal{H} (I^{\prime}) =
\mathcal{H} (I).$ This proves (i). Statement (ii)
follows immediately from (i). \ \ \ \carre

\vspf

 
   
   \section{Frequencies of NLS}
   \label{S:4. Frequencies of NLS}

In this section we obtain formulas for the NLS-frequencies
   \[ \omega _n:= \frac {\partial \mathcal H}{\partial I_n} \quad
      (n \in {\mathbb Z})
   \]
in terms of 
differentials defined on the Riemann surface associated to the spectral data
of a potential $\varphi$.
These formulas then lead to the asymptotics of $\omega _n$ for
$n \rightarrow \pm \infty $ which will be needed for the KAM
result.

\medskip

   \subsection{Differential forms}
   \label{Ss:4.1 Abelian differentials}
   
Let $\varphi \in L^2(S^1; {\mathbb C})$ and $(\lambda ^\pm _k)_{k \in
{\mathbb Z}} \equiv spec \left( L(\varphi ) \right)$. The periodic
eigenvalues can be used to define the discriminant $\Delta (\lambda )$
by the following product representation (cf e.g. \cite{gg} or \cite{gkp})
   \[ \Delta (\lambda ) - 2 = - 2 (\lambda ^+_0 - \lambda )(\lambda
      ^-_0 - \lambda ) \underset{\underset {k \not= 0}{k \in 2
	  {\mathbb Z}}}{\prod } \ \frac {(\lambda ^+_k - \lambda )(
	  \lambda ^-_k - \lambda )}{k ^2 \pi ^2}
   \]
and
   \[ \Delta (\lambda ) + 2 = 2 \underset {k \in  2 {\mathbb Z}+1}
      {\prod } \ \frac {(\lambda ^+_k - \lambda )(\lambda ^-_k -
	  \lambda )}{k ^2 \pi ^2} .
   \]
These infinite products converge
\footnote{ Given a sequence $(a_k)_{k\in \mathbb Z}$,
we say that the infinite product $\prod_{k\in \mathbb Z} a_k$ is
convergent if $\lim_{N\to \infty}\prod_{|k|\leq N} a_k$
exists. } as the eigenvalues $\lambda ^\pm _k$
satisfy the asymptotic estimate $\lambda ^\pm _k = k \pi + \ell ^2
(k)$, meaning that
   \begin{equation}
   \label{4.1} (\lambda ^\pm _k - k \pi )_{k \in {\mathbb Z}} \in
               \ell ^2({\mathbb Z}) .
   \end{equation}
Denote by $\Sigma _\varphi $ the hyperelliptic Riemann surface 
   \[ \Sigma _\varphi := \{ (\lambda , y) \in {\mathbb C}^2
      \big\arrowvert y^2 = \Delta (\lambda )^2 - 4 \} .
   \]
Let $\Sigma ^c_\varphi $ be the canonical sheet of $\Sigma _\varphi $
determined by the normalization 
   \begin{equation}
   \label{4.2} i \sqrt{\Delta (\lambda )^2 - 4} > 0 \quad \mbox{ for any } 
               \lambda ^+_0 < \lambda < \lambda ^-_1 .
   \end{equation}
On $\Sigma ^c_\varphi , \sqrt[c]{\Delta (\lambda )^2 - 4}$ is
then given by
   \[ \sqrt[c]{\Delta (\lambda )^2 - 4} = 2 i \sqrt[s]{(\lambda ^-
      _0 - \lambda )(\lambda ^+_0 - \lambda )} \ \underset {k \not=
	  0}{\prod } \ \frac {\sqrt[s]{(\lambda ^-_k - \lambda )
	  (\lambda ^+_k - \lambda )}}{k \pi }
   \]
where for $a, b \in {\mathbb R}$ with $a \leq b, \sqrt[s]{(a - 
\lambda )(b - \lambda )}$ denotes the {\it standard} square root,
defined on ${\mathbb C} \backslash [a, b]$ and
determined by
   \[ \sqrt[s]{(a - \lambda )(b - \lambda )} < 0 \quad \mbox{ for  } 
      b < \lambda < \infty .
   \]
For any $k \in {\mathbb Z}$, let $a_k$ be a counterclockwise 
oriented cycle on $\Sigma ^c_\varphi $ around the gap 
$[\lambda ^-_k, \lambda ^+_k]$
 and denote by $\beta _n$ $(n \in {\mathbb Z})$
the  differential  on $\Sigma _\varphi $ (cf
\cite{gkp}, \cite{MV})
   \[ \beta _n:= \frac {\psi _n(\lambda )}{\sqrt{\Delta (\lambda )
      ^2 - 4}} \ d \lambda  
   \]
where $\psi _n(\lambda )$ is an entire function given by,
for $n \not= 0$,
   \[ \psi _n(\lambda ):= - 2 \ \frac {\nu ^n_0 - \lambda }{\tau
      _n - \lambda } \ \underset {k \not= 0}{\prod } \ \frac {
	  \nu ^n_k - \lambda }{k \pi }
   \]
whereas for $n = 0$,
   \[ \psi _0(\lambda ):= - 2 \underset {k \not= 0}{\prod } \
      \frac {\nu ^0_k - \lambda }{k \pi } .
   \]
The zeroes $\nu ^n_k$ $(k \not= n)$ of $\beta _n$ are uniquely
determined by the normalization conditions
   \[ \int _{a_k} \beta _n = 2 \pi \delta _{nk} 
   \]
and $\nu ^n_n:= \tau _n$ with $\tau _n = (\lambda ^+_n + 
\lambda ^-_n) / 2$. For any $k \in {\mathbb Z}$,
   \[ \lambda ^-_k \leq \nu ^n_k \leq \lambda ^+_k .
   \]
From the asymptotics \eqref{4.1} of the eigenvalues it follows
that
   \[ \nu ^n_k = k \pi + \ell ^2(k)
   \]
insuring the convergence of the infinite products defining the
$\beta _n$.

 \smallskip
 
The one-forms $\beta _n$ are holomorphic except at the 
points at infinity $\infty ^\pm $ on each of the two sheets
$\Sigma ^\pm _\varphi $ of $\Sigma _\varphi $. In the case
where $\varphi$ is a finite gap potential, the $\beta _n$
have a simple pole at infinity and thus are Abelian differentials
of the third kind.




\smallskip

To compute the asymptotic expansion of $\beta _n$ at $\infty $
on $\Sigma ^c_\varphi $ we need first to establish an auxilary
result for {\it finite gap} potentials: Let $(x_k)_{k \in {\mathbb Z}}$ be a sequence of real 
numbers with $\lambda ^-_k \leq x_k \leq \lambda ^+_k$ and 
define
   \[ g(\lambda ):= 2(x_0 - \lambda ) \underset {k \not= 0}
      {\prod } \ \frac {x_k - \lambda }{k \pi } .
   \]
As $x_k = k \pi + \ell ^2(k)$, the infinite product above
converges and $g$ defines an entire function (cf \cite{gkp}
or \cite{gg}).

\bigskip

\begin{lemma}
\label{4.1 Lemma} For any finite gap potential $\varphi $ and any
entire function $g$ as above, $i\frac { g(\lambda )}{\sqrt{\Delta
(\lambda )^2 - 4}}$ admits 
on $\Sigma ^c_\varphi $ an asymptotic expansion for 
$|\lambda | \rightarrow \infty $ of the form
   \begin{align}
   \begin{split}
   \label{4.4} i \frac { g(\lambda )}{\sqrt[c]{\Delta (\lambda )
                  ^2 - 4}} = 1 &+ a \lambda ^{-1} 
			   + \frac {1}{2} \left( a^2 + \sum _k (\tau ^2_k - x^2
			    _k) + \frac {1}{4} \sum_k \gamma ^2_k \right) \lambda
				  ^{-2} \\
			   &+ 0 \left( \lambda ^{-3} \right)
   \end{split}
   \end{align}
where
   \begin{equation}
   \label{4.5} a:= \sum _{k \in {\mathbb Z}} (\tau _k - x_k)
   \end{equation}  
\end{lemma}

\smallskip

{\bf Remark } Note that the sums in \eqref{4.4} and \eqref{4.5}
are finite as $\varphi$ is assumed to be a finite gap potential.

\smallskip

{\it Proof } Using the definition of $\sqrt[c]{\Delta (\lambda )^2 - 4}$
one gets
   \[ \frac { g(\lambda )}{\sqrt[c]{\Delta (\lambda )^2 - 4}} \ = - i
      \underset {k \in {\mathbb Z}}{\prod } \ \frac {x_k - \lambda }
	  {\sqrt[s]{(\lambda ^+_k - \lambda )(\lambda ^-_k - \lambda )}} .
   \]
With $z = - \lambda ^{-1}$ as a local parameter near $\infty $ on $\Sigma
^c_\varphi $,
   \[ \frac {g(\lambda )}{\sqrt[c]{\Delta (\lambda )^2 - 4}} = - i 
      \underset {k \in {\mathbb Z}}{\prod } \ \frac {1 + z x_k }{\sqrt
	  [+]{(1 + z \lambda ^+_k)(1 + z \lambda ^-_k)}}
   \]
where, in view of the definition of the standard root $\sqrt[s]{(\lambda
^+_k - \lambda )(\lambda ^-_k - \lambda )}$,
   \[ \sqrt[s]{(\lambda ^+_k - \lambda )(\lambda ^-_k - \lambda )} \ =
      - \lambda \sqrt[+]{(1 - \lambda ^+_k / \lambda )(1 - \lambda ^-_k
	  / \lambda ) }
   \]
for $|\lambda |$ large enough. In particular, for any $k$ with $\lambda ^+
_k = \lambda ^-_k$ one gets
   \[ \sqrt[s]{(\lambda ^+_k - \lambda )(\lambda ^-_k - \lambda )} \ =
      \lambda ^+_k - \lambda . 
   \]
Hence $f(z):= i \frac {g(\lambda )}{\sqrt[c]{\Delta (\lambda )^2 - 4}}
\big\arrowvert _{\lambda = - z^{-1}}$ is given by
   \[ f(z) = \underset {k \in {\mathbb Z}}{\prod } \ f_k(z)
   \]
with
   \[ f_k(z) = \ \frac {1 + x_k z}{\sqrt[+]{(1 + \lambda ^+_k z)
      (1 + \lambda ^-_k z)}}
   \]
and $f_k(z) \equiv 1$ for $|k|$ so sufficiently large. Therefore $f(z)$
admits a Taylor expansion at $z = 0$,
   \[ f(z) = f(0) + f'(0) z + \frac {f''(0)}{2} \ z^2 + 0(z^3)
   \]
where $f'(0)$ and $f''(0)$ are given by
   \[ f'(0) = \sum _{k \in {\mathbb Z}} x_k - \tau _k \quad (=: - a)
   \]
and
   \[ f''(0) = a^2 + \sum _{k \in {\mathbb Z}} \left( \tau ^2_k +
      \gamma ^2_k / 4 - x^2_k \right) .
   \]
The sums in the expressions for $f'(0)$ and $f''(0)$ are
finite as $\varphi $ is assumed to be a finite gap potential.
$\blacksquare $

\bigskip

Lemma~\ref{4.1 Lemma} is now applied to complete the asymptotic
expansion of the one-form $\beta _n$.

\medskip

\begin{corollary}
\label{4.2 Corollary} For any finite gap potential $\varphi ,
\ \frac {\psi _n(\lambda )}{\sqrt[c]{\Delta (\lambda )^2 - 4}}$
admits for $|\lambda | \rightarrow \infty $ an asymptotic
expansion of the form
   \begin{align*} i\frac {\psi _n}{\sqrt[c]{\Delta (\lambda )^2 - 4}}
                     =  & \frac {1}{\lambda } + (\tau _n
					 + \alpha _n) \frac {1}{\lambda ^2} 
				  + \Big( \tau ^2_n + \alpha _n \tau _n + 
				  \frac {1}
				     {2} \alpha ^2_n + \frac {1}{2} \sum _k \big(
					 \tau _k - (\nu ^n_k) ^2 \big) \\ +& 
					 \frac {1}{8}
					 \sum _k \gamma ^2_k \Big) \frac {1}
					 {\lambda 
					 ^3} 
				  + 0 \big( \frac {1}{\lambda ^4} \big)  
   \end{align*}
where $\alpha _n:= \sum _k (\tau _k - \nu ^n_k)$.
\end{corollary}

\smallskip

{\it Proof } Let $\varphi $ be a finite gap potential.
 By definition,
   \[ {\psi _n}(\lambda) \ = \ \frac {
      1}{\lambda - \tau _n } \ 2(\nu ^n _0 - \lambda )\underset {k\neq 0}{\prod } \
	  \frac {\nu ^n _k - \lambda }{k\pi} .
   \]
Thus combining Lemma~\ref{4.1 Lemma} with
   \[ \frac {1}{\lambda - \tau _n } = \frac {1}{\lambda } \ \sum _{k
      \geq 0} (\tau _n / \lambda )^k
   \]
the claimed formula follows. 
$\blacksquare $

\bigskip

   \subsection{NLS-frequencies}
   \label{Ss:4.1 Formulas for NLS-frequencies}
   
To obtain formulas for the NLS-frequencies we use that according to 
\cite{gkp} (see also \cite{MV}), the NLS-Hamiltonian appears in
the expansion at infinity of the discriminant $\Delta (\lambda )$,
or more conveniently of $ch^{-1} \left( \frac {\Delta (\lambda )}
{2} \right) $ where $ch^{-1}$ denotes the branch of $arccosh$
defined on ${\mathbb C} \backslash (- \infty , 1)$ which
for $|z| > 1$ is given by
   \[ ch^{-1}(z) = \log \left( z + z \sqrt[+]{1 - \frac {1}{z^2}}
      \right) 
   \]
with $\log $ denoting the principal branch of the logarithm. 

\smallskip

Denote by $H_j$ $(j \geq 1)$ the NLS-hierarchy. Recall that $H_1, H_2$
are given by 
   \begin{align*} H_1 &\equiv H_1(\varphi ):= \int ^1_0 \varphi 
                     \overline {\varphi }\ dx \quad \mbox{ for } \varphi \in H^0\\
		H_2 &\equiv H_2(\varphi ):= i \int '_0 \varphi '
				     \overline {\varphi }\ dx \quad
		\mbox{ for } \varphi \in H^{1}		      
  \end{align*}
with $\varphi '(x) = \frac {d}{dx} \varphi (x)$, whereas $H_3$ is the
NLS-Hamiltonian (denoted by $\mathcal H$ in the introduction)
   \[ H_3 \equiv H_3 (\varphi ):= \int ^1_0 \left( | \varphi '|^2 + 
      | \varphi |^4 \right) dx 
    \quad \mbox{ for } \varphi \in H^{1}\ .
   \]

According to \cite{MV} (cf also \cite{gkp}) one has

\medskip

\begin{lemma}
\label{4.3 Lemma} At any finite gap potential,
   \[ ch^{-1} \left( \frac {\Delta (\lambda )}{2} \right) = - i 
      \lambda + \frac { i H_1}{2 \lambda } +  \frac {iH_2}{4 \lambda 
	  ^2} + \frac {i H_3}{8 \lambda ^3} + 0 \left( \frac {1}{\lambda
	  ^4} \right)
   \]
for $\lambda $ near $\infty $.
\end{lemma}

\medskip

Recall that for any $N>0$, $(\lambda , \varphi ) \mapsto \Delta (\lambda , \varphi )$
is analytic on ${\mathbb C} \times H^N(S^1; {\mathbb C})$. As the 
Birkhoff map $\varphi \mapsto \left( x_k(\varphi ), y_k(\varphi )
\right) _{k \in {\mathbb Z}}$ is a conformal diffeomorphism from 
$H^N(S^1; {\mathbb C})$ to $\ell ^2_N({\mathbb Z}; {\mathbb R}^2)$ and
$\Delta (\lambda , \varphi )$ is a spectral invariant, $\Delta
(\lambda )$ is a function of the actions alone and $\left( \lambda ,
(I_k)_{k \in {\mathbb Z}} \right) \mapsto \Delta (\lambda )$ is
analytic on ${\mathbb C} \times \ell ^1({\mathbb Z}; {\mathbb R})$.
Hence, for any $n \in {\mathbb Z}$, 
   \[ \lambda \mapsto \frac {\partial \Delta (\lambda )}{\partial
      I_n }
   \]
is an entire function in $\lambda $.

For $K\geq 1$, denote by $G_K$ the following set of finite gap
potentials
$$
G_K :=\{ \varphi \in L^2(S^1; {\mathbb C})\mid \lambda_k^+ = \lambda_k^-
\mbox{ iff } |k|>K \}\ .
$$
For $\varphi \in G_K$ and $|n|\leq K$, $\lambda \mapsto \frac {\partial 
\Delta (\lambda )}{\partial I_n }$ vanishes at each $\lambda_k^+$ with
$|k|>K$ and thus 
      
   \[ \eta _n:= - 2 \frac { \frac {\partial \Delta (\lambda )}{\partial
      I_n}} {\sqrt{\Delta (\lambda )^2 - 4}} \ d \lambda 
   \]
is an Abelian differential on $\Sigma _\varphi $ which is holomorphic except
possibly at $\infty ^\pm $.

\smallskip

In \cite{gkp} (cf also \cite{MV}) it is proved that $H_1$ can be
expressed in terms of the action variables $(I_n)_{n \in {\mathbb Z}}$
as follows
   \[ H_1 = \sum _{k \in {\mathbb Z}} I_k .
   \]
Hence for any $n \in {\mathbb Z}$,
   \[ \frac {\partial H_1}{\partial I_n} = 1 .
   \]
Lemma~\ref{4.3 Lemma} then leads to an expansion of $\eta _n$ at 
infinity on $\Sigma ^c_\varphi $. Introduce, for any $n \in {\mathbb Z}$,
   \[ w_n:= \frac {\partial H_2}{\partial I_n} \ \mbox { and } \
      \omega _n:= \frac {\partial H_3}{\partial I_n} .
   \]

\medskip

\begin{corollary}
\label{4.4 Corollary} Let $\varphi $ be a finite gap potential 
in $G_K$ for some $K\geq 1$. Then for
any $|n|\leq K$, $\eta _n$ is a holomorphic one-form on $\Sigma _\varphi $
except at $\infty ^\pm $ where it has a simple pole. At 
infinity $\infty^c$ of the canonical sheet $\Sigma ^c_\varphi $, $\eta _n$ admits
an asymptotic expansion of the form 
   \begin{equation}
   \label{4.6} \eta _n = \frac{1}{i} \left( \frac {1}{\lambda } + \frac {w_n}
               {2} \frac {1}{\lambda ^2} + \frac {\omega _n}{4}
			   \frac {1}{\lambda ^3} + 0 \left( \frac {1}{\lambda ^4}
			   \right) \right) \ d \lambda
   \end{equation}
\end{corollary}

\smallskip

{\it Proof } We already know that $\eta _n$ is holomorphic on $\Sigma
_\varphi \backslash \{ \infty ^\pm \} $. Next, let us prove the expansion \eqref{4.6}.
 By a straightforward computation one sees that
on $\Sigma ^c$, for $|\lambda |$ sufficiently large 
   \begin{equation}
   \label{4.7} \eta _n = \frac {\partial }{\partial I_n} \left( - 2
               h(\lambda ) \right) \ d \lambda 
   \end{equation}
where $h$ is given by  
   \[ h(\lambda ):= \log \left( \Delta (\lambda ) - \sqrt[c]{\Delta
      (\lambda)^2 - 4} \ \right) 
   \]
   with $\log$ denoting the principal branch of the logarithm.
Recall
that (cf \cite{gg}, \cite{gkp}) as $|\lambda | \rightarrow \infty $,
   \[ \Delta (\lambda ) = 2 \cos \lambda + o(1) .
   \]
Hence $\Delta (iy) = 2 ch y + o(1)$ and 
   \[ \Re \left( \Delta (iy)\right) > 0 \ \mbox { and } \ |\Delta (iy)| > 4
   \]
for $y$ large enough. By the definition of the canonical root it then
follows that
   \[ \sqrt[c]{\Delta (iy)^2 - 4 } \ = - \Delta (iy) \sqrt[+]{1 - 4 /
      \Delta (iy)^2}
   \]
and thus
   \[ h(y) = ch^{-1} \left( \frac {\Delta (iy)}{2} \right) 
   \]
for $y$ sufficiently large. 

\smallskip

In view of Lemma~\ref{4.3 Lemma}, it then follows that
   \[ - \eta _n = i \left( \frac {1}{\lambda } + \frac {w_n}{2 \lambda
      ^2} + \frac {\omega _n}{4 \lambda ^3} + 0 \left( \frac {1}
	  {\lambda ^4} \right) \right) \ d \lambda
   \]
as claimed. In the same fashion one shows that $\eta _n$ admits an
expansion at infinity on the second sheet of $\Sigma _\varphi $ of the
same type as \eqref{4.6}. This implies that $\eta _n$ has simple
poles at $\infty ^\pm$.
$\blacksquare $

\bigskip

It turns out that the one forms $\eta _n$ can be identified with the
one-forms $\beta _n$ introduced in subsection~\ref{Ss:4.1 Abelian 
differentials}.

\medskip

\begin{proposition}
\label{4.5 Proposition} Let $\varphi $ be a finite gap potential 
in $G_K$ for some $K\geq 1$. Then for
any $|n|\leq K$
   \[ \beta _n = \eta _n
   \]
\end{proposition}

\smallskip

{\it Proof } Let $\varphi \in G_K$. By 
Corollary~\ref{4.4 Corollary},  $\eta _n$ ($|n|\leq K$)  is a
meromorphic one-form on $\Sigma _\varphi $ which is holomorphic except
at $\infty ^\pm $ where it has simple poles. To compute $\int _{a_k}
\eta _n$, first remark that
$$
\int _{a_k} \eta _n = -2\int _{\Gamma _k}
\frac {\frac{\partial \Delta }{\partial I_n}}
		{\sqrt[c]{\Delta (\lambda )^2 - 4}}  \
					 d \lambda
$$
where ${\Gamma _k}$ is a counterclockwise oriented circuit around 
$[\lambda_k^-, \lambda_k^+]$ in $\mathbb C$. As on ${\Gamma _k}$,
$$\frac {\partial }{\partial \lambda } ch^{-1} \left( \frac{\Delta (\lambda)}{2}\right)
=\frac { \dot \Delta (\lambda )}{\sqrt[c]
{\Delta (\lambda )^2 - 4}} \ \mbox{ and }\
 \frac {\partial }{\partial I_n } ch^{-1} \left( \frac{\Delta (\lambda)}{2}\right)
 =\frac {\frac{\partial \Delta }{\partial I_n}(\lambda )}
{\sqrt[c]{\Delta (\lambda )^2 - 4}}$$
one gets,
integrating by parts,
   \begin{align*} \int _{a_k} \eta _n &= 2 \int _{\Gamma _k} \lambda
                     \frac {\partial }{\partial \lambda } \left(
					 \frac {\frac{\partial \Delta }{\partial I_n}}
					 {\sqrt[c]{\Delta (\lambda )^2 - 4}} \right) \
					 d \lambda \\
				  &= 2 \frac {\partial }{\partial I_n} \int _{\Gamma _k}
				     \frac {\lambda \dot \Delta (\lambda )}{\sqrt[c]
					 {\Delta (\lambda )^2 - 4}} \ d \lambda \\
				  &= \frac {\partial }{\partial I_n} (2 \pi \ I_k) =
				     2 \pi \delta _{nk}
   \end{align*}
where we used that by definition
   \[ I_k = \frac {1}{\pi } \int _{\Gamma _k} \ \frac {\lambda \
      \dot \Delta (\lambda )}{\sqrt[c]{\Delta (\lambda )^2 - 4 }} \
	  d \lambda .
   \]

\smallskip

By the construction of the functions $\psi _n$,
   \[ \int _{a_k} \beta _n = 2 \pi \delta _{nk}
   \]
hence for any $k \in {\mathbb Z}$
   \[ \int _{a_k} \beta _n = \int _{a_k} \eta _n .
   \]
Further $\eta _n$ and $\beta _n$ are holomorphic differentials except
at $\infty ^\pm $ where they admit simple poles. By Cauchy's theorem,
   \begin{align*} \lim _{j \rightarrow \infty } \int _{C(0,j)} \beta
                     _n &= \sum _{k \in {\mathbb Z}} \int _{a_k}
					 \beta _n = \sum _{k \in {\mathbb Z}} 2\pi
					 \delta _{nk} \\
				  &= \sum _{k \in {\mathbb Z}} \int _{a_k} \eta _n
				     = \lim _{j \rightarrow \infty } \int _{C(0,j)}
					 \eta _n
   \end{align*}
with $C(0,j)$ denoting the counterclockwise oriented circle of radius
$j$ centered at $0$ on $\Sigma ^c_\varphi $. Hence $\beta _n$ and $\eta
_n$ have the same residue at $\infty ^+$ and, arguing in the same 
way, at $\infty ^-$. Thus $\beta _n - \eta _n$ is a holomorphic 
differential on $\Sigma $ satisfying 
   \[ \int _{a_k} (\beta _n - \eta _n) = 0 \quad \forall k \in
      {\mathbb Z} .
   \]
This implies that $\beta _n = \eta _n$.
$\blacksquare $

\bigskip

Determining the expansion of $\beta _n$ on $\Sigma ^c_\varphi $ for
$|\lambda | \rightarrow \infty $ and comparing it with $\eta _n$
leads to a formula for the frequency $\omega _n$.

\medskip

\begin{guess}
\label{4.6 Theorem} The frequencies of the $2^{nd}$ and $3^{rd}$
Hamiltonian $H_2$ resp. $H_3$ in the NLS-hierarchy defined for
potentials in the appropriate Sobolev spaces are
   \begin{align}
   \label{4.10} &w_n:= \frac {\partial H_2}{\partial I_n} = 2 (\tau
                   _n + \alpha _n) \\
   \label{4.12} &\omega _n:= \frac {\partial H_3}{\partial I_n} =
                   \left( (2 \tau _n)^2 + 4 \alpha _n \tau _n +
				   2 \alpha ^2_n + 2 \sum _k (\tau ^2_k - (\nu ^n_k
				     )^2) + \frac {1}{2} \sum _k \gamma ^2
				   _k \right) 
   \end{align}
where
   \[ \alpha _n:= \sum _{k \in {\mathbb Z}} (\tau _k - \nu ^n_k)
   \]
\end{guess} 

\smallskip

{\it Proof } Let us fix $n\in \mathbb Z$.
It has been shown in \cite{gkp} that the $\nu ^n_k$ as
well as $\tau _k:= (\lambda ^+_k + \lambda ^-_k) / 2$ and $\gamma ^2
_k:= (\lambda ^+_k - \lambda ^-_k)^2$ are real analytic functions
of $\varphi $. The $\nu ^n_k$'s satisfy the asymptotics 
   \[ \nu ^n_k - \tau _k = \gamma _k \ell ^2 (k)
   \]
and for any $\varphi \in L^2(S^1; {\mathbb C})$ one has (cf \cite{gk4})
   \[ (\gamma _k)_{k \in {\mathbb Z}} \in \ell ^2_N ({\mathbb Z};
      {\mathbb R}) \Longleftrightarrow \varphi \in H^N .
   \]
Hence left and right hand side of the identities \eqref{4.10} and 
\eqref{4.12} are well defined on appropriate Sobolev spaces and 
continuous. As the set of finite gap potentials is dense in any
Sobolev space $H^N$ (cf \cite{gk5}),
it suffices to prove these identities
for the dense set $\cup_{K\geq 1}G_K$ of finite gap potentials (cf Theorem~\ref{T1.1}). 
Assume that $\varphi \in G_K$
with $|n|\leq K$. By Corollary~\ref{4.2 Corollary}, for $|
\lambda |$ sufficiently large
   \begin{align*} i\frac {\psi _n }{\sqrt[c]{\Delta (\lambda )^2 - 4}}
                     = & \frac {1}{\lambda } + (\tau _n
					 + \alpha _n) \frac {1}{\lambda ^2} + 
		 \Big( \tau ^2_n + \alpha _n \tau _n  + \frac {1}{2}
				     \alpha ^2_n \\+& \frac {1}{2} \sum _k \Big( \tau ^2
					 _k - (\nu ^n_k) ^2 \Big) + \frac {1}{8} \sum _k
					 \gamma ^2_k \Big) \frac {1}{\lambda ^3} 
				  + 0 \Big( \frac {1}{\lambda ^4} \Big) \ .
   \end{align*}
Comparing this with the expansion of $\eta _n$ given in 
Corollary~\ref{4.4 Corollary},
   \[ i\eta _n =  \left( \frac {1}{\lambda } + \frac {w_n}{2}
      \frac {1}{\lambda ^2} + \frac {\omega _n}{4} \frac {1}
	  {\lambda ^3} + 0 \left( \frac {1}{\lambda ^4} \right) 
	  \right) \ d \lambda
   \]
leads in view of Proposition~\ref{4.5 Proposition} to the claimed
identities.
$\blacksquare $

\bigskip

Formula \eqref{4.12} can be  simplified by the following 
observation.


\begin{lemma}
\label{4.7 Lemma} For any $\varphi \in L^2(S^1; {\mathbb C})$ and $n
\in {\mathbb Z}$,
   \[ \alpha _n = n \pi - \tau _n .
   \]
\end{lemma}

\smallskip

{\it Proof } The frequencies $w_n$ of $H_2$ are the frequencies of
the translation flow $T_t \varphi := \varphi (t + \cdot )$ which 
can be computed to be (cf \cite{gk3})
   \[ w_n = 2 \pi n .
   \]
Combined with formula \eqref{4.10}, $w_n = 2(\tau _n + \alpha _n)$,
we obtain the claimed identity.
$\blacksquare $

\bigskip

{\bf Remark } Lemma~\ref{4.1 Lemma} can be applied to obtain an
asymptotic expansion of $\frac {\dot \Delta (\lambda )}{\sqrt[c]
{\Delta (\lambda )^2 - 4}} \ d \lambda $ in view of the product
representation of $\dot \Delta (\lambda ) = \frac {d}{d \lambda }
\Delta (\lambda )$, 
   \[ \dot \Delta (\lambda ) = 2 (\dot \lambda _0 - \lambda )
      \underset {k \not= 0}{\prod } \ \frac {\dot \lambda _k - 
	  \lambda }{k \pi } .
   \]
Noticing that for any finite gap potential $\varphi$ and $|\lambda |$ large,
   \[ \frac {\dot \Delta (\lambda )}{\sqrt[c]{\Delta (\lambda )^2 -
      4}} \ d \lambda = d \left( ch^{-1} \left( \frac {\Delta 
	  (\lambda )}{2} \right) \right)
   \]
one can use the asymptotic expansion of $ch^{-1} \left( \frac {
\Delta (\lambda )}{2} \right) $ established in Lemma~\ref{4.3 Lemma}
to see that for a finite gap potential,
   \begin{align*} i \frac {\dot \Delta (\lambda )}{\sqrt[c]{\Delta 
                     (\lambda )^2 - 4} } \ d \lambda =  \Big(
					 1 &+ \frac {H_1}{2} \frac {1}{\lambda ^2} +
					 \frac {H_2}{2} \frac {1}{\lambda ^3} + \frac
					 {3}{8} H_3 \frac {1}{\lambda ^4} \\ 
				  &+ 0 \big( \frac {1}{\lambda ^5} \big) \Big) \
				     d \lambda .
   \end{align*}
Comparing the two asymptotic expansions then leads as in the proof of
Theorem~\ref{4.6 Theorem} to formulas expressing the Hamiltonians
$H_j$ in terms of the spectral data $(\lambda ^\pm _k)_{k \in
{\mathbb Z}}$ and $(\dot \lambda _k) _{k \in {\mathbb Z}}$. For 
$H_0 = \frac {1}{2} \| \varphi \| ^2$ one obtains in this way
   \[ \| \varphi \| ^2 = \sum _k \left( (\lambda ^+_k)^2 + 
      (\lambda ^-_k)^2 - 2 \dot \lambda ^2_k \right) .
   \]
   
\bigskip

   \subsection{Asymptotics of the NLS-frequencies}
   \label{Ss:4.3 Asymptotics of the NLS-frequencies}
   
Theorem~\ref{4.6 Theorem} leads to  asymptotics for the
NLS-frequencies needed for the KAM result.

\begin{guess}
\label{4.8 Theorem} For $\varphi $ in $H^1(S^1; {\mathbb C})$,
   \[ \omega _n = (2 \pi n)^2 + 0(1) 
   \]
locally uniformly on $H^1(S^1; {\mathbb C})$.
\end{guess}

\smallskip

{\it Proof } By Theorem~\ref{4.6 Theorem} and Lemma~\ref{4.7 Lemma}
   \begin{align*} \omega _n = &4 \tau _n \pi n + 2(\pi n - \tau _n)^2
                  + 2 \sum _k (\tau _k - \nu ^n_k)(\tau _k + \nu ^n
				     _k) + \frac {1}{2} \sum _k \gamma ^2_k .
   \end{align*}
By \cite{gkp} 
   \[ \nu ^n_k - \tau _k = \gamma _k \ell ^2(k)
   \]
locally uniformly on $L^2(S^1; {\mathbb C})$ and by \cite{gg} (cf also
\cite{Ma}, \cite{gk4})
   \[ \tau _k = k \pi + 0 \left( \frac {1}{k} \right) \ \mbox { and } \
      (k \gamma _k)_{k \in {\mathbb Z}} \in \ell ^2
   \]
locally uniformly on $H^1(S^1; {\mathbb C})$. Hence   
   $$4 \tau _n \pi n = (2 \pi n)^2 + 0(1)\ , $$
   $$\sum _{\tau _k} ( \tau _k - \nu ^n_k)(\tau _k + \nu
				     ^n_k) = \sum _k k \gamma _k \ell ^2(k) = 0(1)$$
				     and
				$$  \sum _k \gamma ^2_k = 0(1) 
   $$
locally uniformly on $H^1(S^1; {\mathbb C})$. Combining these estimates
leads to the claimed asymptotics. 
$\blacksquare $







%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%
\section{KAM Theorem for NLS}
%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%
\n The KAM Theorem for NLS presented in this
paper (Theorem
\ref{T1.2}), is derived from an abstract KAM-Theorem
with parameters in infinite dimension due to Kuksin
\cite{ku1} (cf also \cite{ku2} and \cite{ku3}) and, in a refined form, to P\"oschel
\cite{po}. For the convenience of the reader we
present  P\"oschel's version in the following
paragraph (cf also \cite{kapo} and notation
established there).

\vspd

\n Unlike as in the case of the Korteweg-de Vries
equation, this abstract KAM theorem is well suited
for NLS once the equations under consideration are
expressed in Birkhoff coordinates (cf Theorem
\ref{T1.1}). In paragraph 5.2 we verify that the
assumptions of the abstract KAM theorem are satisfied
in the case of NLS by using the results of sections 2
and 4.
%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%
\subsection{Abstract KAM Theorem}
%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%
\n  For $n \ge 1$ and $N \in \mathbb{R}_{>0}$
fixed, introduce the real phase space
%%%%%%%%%%%%%%%
\[\mathcal{S}^{N} := \mathbb{T}^n \times
\mathbb{R}^n \times l^2_{N} (\mathbb{N};
\mathbb{R}) \times l^2_{N} (\mathbb{N};
\mathbb{R}) \]
%%%%%%%%%%%%%%%%%%%%%%%%
\n and denote by $\mathcal{S}^{N}_{\mathbb{C}}$
its complexification.

\vspd

\n Using standard coordinates $(x, y, u, v)$ for
$\mathcal{S}^{N}$ the canonical symplectic
structure takes the form
%%%%%%%%%%%%%%%%%%%%
\[\sum^n_{j=1} dx_j \wedge dy_j +
\sum^{\infty}_{j=1} du_j \wedge dv_j.\]
%%%%%%%%%%%%%%%%%%%%%%
\n We perturb a family of infinite dimensional
integrable Hamiltonians, para\-metrized by $\zeta \in
\Pi,$
%%%%%%%%%%%%%%%%%%%
\begin{equation}
\label{eq:5.1}
\mathcal{N} \equiv \mathcal{N} (x, y, u, v; \zeta)
:= \sum^n_{j=1} \omega_j (\zeta) y_j + \frac{1}{2}
\sum^{\infty}_{j =1} \Omega_j (\zeta)(u^2_j +
v^2_j).
\end{equation}
%%%%%%%%%%%%%%%%%%%
\n Here $\Pi \subseteq \mathbb{R}^n$ is a compact
parameter set of positive Lebesgue measure and
$\omega_j (\cdot) \; (1 \le j \le n), \;
\Omega_j (\cdot) \; (j \in \mathbb{N})$ are real
functions defined on $\Pi.$

\vspd

\n Notice that for every value $\zeta \in \Pi,$
%%%%%%%%%%%%%%%%%%
\[\mathcal{T}_0 := \mathbb{T}^n \times \{0\} \times
\{0\} \times \{0\}\]
%%%%%%%%%%%%%%%%%%%
\n is an invariant torus for
$\mathcal{N}$ and the Hamiltonian flow on
$\mathcal{T}_0$, induced by $\mathcal{N},$ is
rotational with frequency vector
$\omega (\zeta) := (\omega_j (\zeta))_{1 \le j \le
n}.$ Our aim is to prove the persistence of
$\mathcal{T}_0$ under small Hamiltonian
perturbations, $H = \mathcal{N} + P$ of
$\mathcal{N}$ for many parameter values $\zeta$.

\vspd

\n We need to make the following three assumptions:

\vspd

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%
\n {\bf Assumption A:} Asymptotics of the
exterior frequencies $(\Omega_j (\xi))_{j \ge 1}$
%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%

{\em
The frequencies $\Omega_j (\zeta)$ are real valued
functions of $\zeta$ of the form, $\Omega_j (\zeta) =
\overline{\Omega}_j + \tilde{\Omega}_j (\zeta), \;
\overline{\Omega}_j$ being independent of $\zeta$,
with the following properties:\\ 
There exist two real
numbers $d > 1$ and $\delta < d-1$ so that
$\overline{\Omega}_j$ admits a finite expansion in
$j$ of order $d$

$$\overline{\Omega}_j = c j^d + \ldots
$$

\n where $c \neq 0$ and the dots stand for an expansion in lower
order terms in $j$
and, $(j^{- \delta}
\tilde{\Omega}_j (\zeta))_{j \ge 1}$ is uniformly
Lipschitz when considered as a map

$$\Pi \rightarrow l^{\infty}
(\mathbb{N}),
\
\zeta
\mapsto (j^{- \delta} \tilde{\Omega}_j (\zeta))_{j
\ge 1}. $$
}
\vspd

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%
\n {\bf Assumption B:} Kolmogorov condition and Melnikov
condition
%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%

{\em
The map, $\zeta \rightarrow \omega (\zeta),$
between
$\Pi$ and  its image is a homeomorphism which is
Lipschitz continuous in both directions. Moreover
for every $k \in \mathbb{Z}^n$ and $l \in
\mathbb{Z}^{\mathbb{N}}$ with $1 \le | l | :=
\sum^{\infty}_{j = 1} | l_j | \le 2,$ the
resonance set
%%%%%%%%%%%%%%%%%%%%%%%%%
\[\mathcal{R}_{k, l} := \{ \zeta \in \Pi \mid \langle
k,
\omega (\zeta) \rangle + \langle l, \Omega (\zeta)
\rangle = 0 \}\]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\n has Lebesgue measure zero.
}
%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%

\medskip

\n The third assumption concerns the perturbation
given by the Hamiltonian $P$. We assume that $P$ is
defined on  a neighborhood $U$ of $\mathcal{T}_0$ in
$\mathcal{S}^{N}_{\mathbb{C}}$. The corresponding
Hamiltonian vectorfield is denoted as follows
%%%%%%%%%%%%%%%%%%%%%
\[X_P = (\partial_y P, - \partial_x P, \partial_v P,
- \partial_u P). \]
%%%%%%%%%%%%%%%%%%%%%%%
\vspd
%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%
\n {\bf Assumption }${\mathbf C_{U, N }:}$ Regularity of the perturbation
%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%

{\em
In the neighborhood $U$ of $\mathcal{T}_0$ in
$\mathcal{S}^{N}_{\mathbb{C}},$ the Hamiltonian
vectorfield $X_{P}$ has the following properties:

\vspd

\n (C.1)$ \ \
X_P \ \mbox{takes values in} \
\mathcal{S}_{\mathbb{C}}^{N}, \; X_p : U \times \Pi
\rightarrow
\mathcal{S}^{N}_{\mathbb{C}},$
and $X_p (\cdot; \zeta)$ is real analytic for any
given $\zeta \in \Pi;$

\vspd

\n (C.2)
$
X_P (x, y, u, v; \cdot)$ is uniformly
Lipschitz on  $\Pi$ for any given
$(x, y, u, v)
\linebreak
\in U.$}

\vspd

\n To state the abstract KAM Theorem we need to
introduce various domains and norms. For $s > 0$ and
$r > 0$ denote by $D(s, r)$ the complex
neighborhoods of $\mathcal{T}_0,$
%%%%%%%%%%%%%%%%%%%%%
\[D(s, r) := \{ | Imx | < s \} \times \{ |
y | < r^2 \} \times \{ || u ||_{N} + || v ||_{N} < r \}\]
\[\subseteq \mathcal{S}^{N}_{\mathbb{C}} :=
\mathbb{C}^n \times \mathbb{C}^n \times l^2_{N,
\mathbb{C}} \times l^2_{N, \mathbb{C}} \]
%%%%%%%%%%%%%%%%%%%%%%%
\n where, in $\mathbb{C}^n,$ we use the norm $| z
| = max_{1 \le j \le n} | z_j |$ and
$l^2_{N, \mathbb{C}} := l^2_{N} (\mathbb{N};
\mathbb{C}).$ On $\mathcal{S}^{N}_{\mathbb{C}}$ we
introduce the weighted norm $|| W
||_{r,N}$ of $W = (X, Y, U, V),$
%%%%%%%%%%%%%%%%%%%%%%%%%%%
\[|| W ||_{r, N} = | X | +
\frac{1}{r^2} | Y | + \frac{1}{r}
|| U ||_{N} + \frac{1}{r} ||
V ||_{N}\]
%%%%%%%%%%%%%%%%%%%%%%%%
\n For a map $W : U \times \Pi \rightarrow
\mathcal{S}^{N}_{\mathbb{C}}$ with $U$ a
neighborhood of the form $D(s,r)$ we introduce the
norms
%%%%%%%%%%%%%%%%%%%%%%%
\[\begin{array}{lll}
|| W ||^{sup}_{r, N; U \times \Pi}
&:=& sup_{(w, \xi) \in U \times \Pi} || W(w,
\xi) ||_{r, N} \\
|| W ||^{lip}_{r, N; U \times \Pi} &:=
& sup_{\stackrel{\xi, \zeta \in \Pi}{\xi \neq
\zeta}} \ \frac{|| \Delta_{\xi \zeta} W
||^{sup}_{r, N; U}}{| \xi - \zeta |}.
\end{array} \]
%%%%%%%%%%%%%%%%%%%%%%%%%%%
\n where $(\Delta_{\xi \zeta} W) (w) = W(w, \xi) -
W (w, \zeta).$ Similarly we define
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\[|  \tilde{\Omega} |^{lip}_{- \delta ; \Pi} :=
sup \left\{ \frac{j^{-\delta} \Delta_{\xi
\zeta} \tilde{\Omega}_j}{|\xi - \zeta|} \mid
\xi, \zeta \in \Pi; \; \xi \neq \zeta; \; j \ge
1\right\}.\]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\n By Assumptions A and B, there exist constants $M <
\infty$, $L < \infty$ such that
%%%%%%%%%%%%%%%%%%%%%%
\[| \omega |^{lip}_{\Pi} + | \Omega
|^{lip}_{- \delta; \Pi} \le M; \quad | \omega^{-1}
|^{lip}_{\omega (\Pi)} \le L < \infty.\]
%%%%%%%%%%%%%%%%%%%%%%%%%%%

\n The following result is due to Kuksin \cite{ku2}
and is stated here in a refined version, due to
\cite{po}.

\vspd
%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%
\begin{guess}
\label{T5.1}
Suppose $\mathcal{N}$ is a family of Hamiltonians of
the form (\ref{T5.1}) defined on $\mathcal{S}^{N}
\times \Pi$ so that Assumptions A and B are
satisfied. Given a neighborhood $D(s,r)$ of
$\mathcal{T}_0$ with $s > 0, r > 0$ there exists a
positive  constant $\gamma$, depending only on $n,
d,$ the frequencies $\omega$ and $\Omega$, and $ s,$
and such that for every perturbation
$H =
\mathcal{N} + P$ of $\mathcal{N}$, defined on $D(s,r)
\times
\Pi$ with $P$ satisfying Assumption $C_{D(s,r), N}$
and the smallness condition
%%%%%%%%%%%%%%%%%%%%
\begin{equation}
\label{eq:5.2}
\varepsilon := || X_P ||^{sup}_{r,N;
D(s,r) \times \Pi} + \frac{\beta}{M} || X_P
||^{lip}_{r,N; D(s,r) \times \Pi} \le \beta
\gamma
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%
\n for some $0 < \beta < 1,$ the following holds:\\
There exist

\begin{itemize}
\item[(i)] a Cantor set $\Pi_{\beta} \subset \Pi,$
depending on the perturbation $P$, with $lim_{\beta
\rightarrow 0}$ meas $(\Pi \setminus \Pi_{\beta}) =
0$ uniformly for $P$ satisfying Assumption
$C_{D(s,r), N}$ and \eqref{eq:5.2}
\item[(ii)] a Lipschitz family of real analytic
torus embeddings $\Psi : \mathbb{T}^n \times
\Pi_{\beta} \rightarrow \mathcal{S}^{N}$
\item[(iii)] a Lipschitz map $f: \Pi_{\beta}
\rightarrow \mathbb{R}^n$
\end{itemize}

\n such that
for each $\xi \in \Pi_{\beta},$ the restriction
$\Psi |_{\mathbb{T}^n \times \{ \xi \}}$ is a real
analytic embedding of a rotational torus with
frequencies $f(\xi)$ for the Hamiltonian $H =
\mathcal{N} + P$ at $\xi.$ In other words, $t
\rightarrow \Psi (\theta + t f (\xi), \xi)$ is a
real analytic, quasi-periodic solution for the
Hamiltonian $H(\cdot, \xi)$ for every $\theta \in
\mathbb{T}^n$ and $\xi \in \Pi_{\beta}$\ .

\vspd

\n Moreover, each embedding is analytic on $D (s/2)
:= \{ | Im \ x \ | < s/2 \}$ and
%%%%%%%%%%%%%%%%%%%%%%%%%
\[|| \Psi - \Psi_0 ||^{sup}_{r, N;
D(s/2) \times \Pi_{\beta}} + \frac{\beta}{M}
|| \Psi - \Psi_0 ||^{lip}_{r, p; D
(s/2) \times
\Pi_{\beta}} \le \frac{c \varepsilon}{\beta}\]
\[| f - \omega |^{sup}_{\Pi_{\beta}} +
\frac{\beta}{M}
| f - w |^{lip}_{\Pi_P} \le c
\varepsilon\]
%%%%%%%%%%%%%%%%%%%%%
\n where $\Psi_0 : \mathbb{T}^{n} \times \Pi
\rightarrow \mathcal{T}_0, (x, \xi) \mapsto (x,
0, 0, 0)$ is the trivial embedding and $c > 0$ is a
constant which depends on the same parameters as
$\gamma$.
\end{guess}
%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%
\subsection{Proof of KAM Theorem for NLS}
%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%
\n In this paragraph we prove Theorem 1.2 as given
in the introduction, by applying the  abstract KAM
Theorem stated in paragraph 5.1 (Theorem \ref{T5.1}).

\vspd

\n We follow closely the line of arguments presented
in \cite{kapo} for the KdV equation.

\vspd

\n To simplify notations we consider in this
paragraph the phase space $l^2_{N; \alpha}
\linebreak
(\mathbb{Z}; \mathbb{R}^2)$ with $\alpha =
0.$ In this case $  l^2_{N; 0} (\mathbb{Z}; \mathbb{R}^2)$ may
be identifyed with $l^2_{N} (\mathbb{Z}_{\ge 0};
\mathbb{R}^2)$. (In the case $\alpha \ne 0$,
$l^2_{N; \alpha} (\mathbb{Z}; \mathbb{R}^2) \simeq
l^2_{N} (\mathbb{Z}_{\ge 1}; \mathbb{R}^2)).$

\vspd

\n As a first step, apply the Birkhoff map $\Phi$
of Theorem \ref{T1.1}, restricted to $l^2_{N}
(\mathbb{Z}_{\ge 0}; \mathbb{R}^2),$
%%%%%%%%%%%%%%%%%%
\[\tilde{\Phi} : l^2_{N} (\mathbb{Z}_{\ge 0};
\mathbb{R}^2) \rightarrow H^{N}_0 (S^1;
\mathbb{C}).\]
%%%%%%%%%%%%%%%%%%%
\n Again, to simplify notation, let us assume that
$A = \{0, 1, \ldots , n-1\}$. Let $\Gamma \subseteq
(\mathbb{R}_+)^A$ be a set of positive Lebesgue
measure and set
%%%%%%%%%%%%%%%%%
\[T_{\Gamma} := \bigcup_{I_A \in \Gamma} \{ (
\sqrt{2 \mathcal{J}_j} e^{i \theta_j})_{j \ge 0}
\mid \mathcal{J}_j  = I_j, j \in A; \; \mathcal{J}_j
= 0, j \not \in A \}.\]
%%%%%%%%%%%%%%%%%%%%%%%
\n Since $\tilde{\Phi}$ is analytic, there exists a
neighborhood $V_{\Gamma}$  of $T_{\Gamma}$ in
$l^2_{N} (\mathbb{Z}_{\ge 0}; \mathbb{C}^2)$ which
is mapped bianalytically onto a neighborhood $U :=
\tilde{\Phi} (V_{\Gamma})$ of
$\{(\varphi, \overline{\varphi}), \varphi \in
\mathcal{T}^{0}_{\Gamma} \} \ \mbox{in} \
H^{N} (S^1; \mathbb{C}) \times H^{N} (S^1;
\mathbb{C}).
$

\vspd

\n Choosing $V_{\Gamma}$ and/or $U_{\Gamma}$
(defined in the Introduction) smaller, if necessary,
we may assume that $U = U_{\Gamma}.$ Consider the
pull back $\tilde{\mathcal{H}} = \mathcal{H} \circ
\tilde{\Phi}$ of $\mathcal{H}$ on $U$. Notice that
$\tilde{\mathcal{H}}$ depends only on the actions
$(I_j)_{j \ge 0}.$ Using Taylor's formula and the
definition of the frequencies, $\omega_j (I) :=
\frac{\partial \mathcal{H}}{\partial I_j} (I),$ we
obtain
%%%%%%%%%%%%%%%%%%
\begin{equation}
\label{eq:5.3}
\tilde{\mathcal{H}} (I + I_0) = \mathcal{H} (I_0) +
\sum_{j
\ge 0} \omega_j (I_0) I_j + Q
\end{equation}
%%%%%%%%%%%%%%%%%%%%%
\n where $Q := \sum_{i, j \ge 0} Q_{ij} (I_0, I) I_i
I_j$ with
%%%%%%%%%%%%%%%%%%%
\begin{equation}
\label{eq:5.4}
Q_{ij} (I_0, I) := \int_0^1 (1 - t) \frac{\partial^2
\mathcal{H}}{\partial I_i \partial I_j} (I_0 + tI)
dt.
\end{equation}
%%%%%%%%%%%%%%%%%%%%%
\n As a second step we introduce symplectic polar
coordinates near the tori in the family $T_{\Gamma}:$
Near $T_{\Gamma}$, introduce new coordinates $(x,
y, u, v) = \Psi^{-1} (c,b)
\linebreak
((c,b) \in l^2_{N}
(\mathbb{Z}_{\ge 0}; \mathbb{R}^2))$ depending on
the parameter $\xi = (\xi_0, \ldots , \xi_{n-1})
\in
\Gamma,$ by setting $(0 \le j \le n-1)$
%%%%%%%%%%%%%%%%%%%%%%%%
\[\sqrt{\xi_j + y_j} \ e^{-ix_j} := c_j + ib_j, \;
\sqrt{\xi_j + y_j} \ e^{ix_j} = c_j - ib_j \]
%%%%%%%%%%%%%%%%%%%%%%%%%
\n and $(j \ge 1)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%
\[u_j := c_{n-1+j}; \quad  v_j := b_{n-1+j} \ (j \ge
1).\]
%%%%%%%%%%%%%%%%%%%%%%%%%%%
\n For each $\xi \in \Gamma,$ this transformation
is real analytic and symplectic on $D(s, r)$ for $s >
0$ and $r >0$ arbitrary but small enough so that $D
(s, r) \subset \Psi (V_{\Gamma}).$ Actually, $\xi$
parametrizes the invariant $n$-tori and $y =
(y_j)_{0 \le j \le n-1}$ are shifted action
variables contained in a (small) neighborhood of
$0$. Using the expansion of $\tilde{\mathcal{H}}$ in
\eqref{eq:5.3} and setting $I_0 := (\xi, 0),  \quad
\tilde H = \tilde{\mathcal{H}} \circ
\Psi$ is, up to a  constant depending only on
$\xi,$ given on $D(s,r)$ by
%%%%%%%%%%%%%%%%%%%%%%%%%%%
\[\tilde H = \mathcal{N} (y,
u, v;
\xi) + Q (y, u, v; \xi)\]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\n where 
$$\mathcal{N} := \sum^{n - 1}_{j = 0}
\omega_j (\zeta) y_j + \frac{1}{2}
\sum^{\infty}_{j=1}
\Omega_j (\zeta) (u^2_j + v^2_j)$$
 with $\Omega_j
(\xi) := \omega_{n-1 +j} (\xi)$
$(j \ge 1)$
and $Q$ denotes the higher order rest. 

As the
notation indicates, $\mathcal{N}$ will play the role
of the integrable normal form, which is perturbed by
$P := Q + \varepsilon R$ with $R := K \circ
\tilde{\Phi} \circ \Psi.$ We now verify the
assumptions A, B and C of Theorem \ref{T5.1}

\vspd

\n Write $\omega_j (\xi) = \overline{\Omega}_j +
\tilde{\Omega}_j (\xi)$ with 
$$\overline{\Omega}_j :=
4 \pi^2 (n - 1 + j)^2$$
 and
%%%%%%%%%%%%%%%%%%%
\[\tilde{\Omega}_j (\xi) := \Omega_j (\xi) -
\overline{\Omega}_j = \frac{\partial
\mathcal{H}}{\partial I_{n - 1 + j}} (\xi, 0) - 4
\pi^2 (n-1+j)^2.\]
%%%%%%%%%%%%%%%%%%%
\n By Theorem~\ref{4.8 Theorem}, $\tilde{\Omega} :
\xi
\mapsto (\tilde{\Omega}_j (\xi))_{j \ge 1}$
maps $\Gamma$ into $l^{\infty} (\mathbb{Z}_{\ge 1};
\mathbb{R})$ and is analytic on a neighborhood of
$\Gamma.$ Thus $\tilde{\Omega}$ is real analytic on
some complex neighborhood of $\Gamma$ (cf
\cite{kapo}, Appendix D, Analyticity Lemma).



\n Hence the map is also Lipschitz by Cauchy's
estimate. In all, we conclude that assumption A is
satisfiied with $d = 2$ and $\delta = 0.$

\vspd

\n To verify Assumption B, recall from Corollary
2.2 that on $\Gamma$
%%%%%%%%%%%%%%%%%%%%%
\begin{equation}
\label{eq:5.5}
\det ((\frac{\partial \omega_j}{\partial \xi_k})_{0
\le j, k \le n-1} ) \not \equiv 0.
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%
\n In particular, for any given $\eta > 0$ we may
excise from $\Gamma$ a relative open subset
$\Gamma_{\eta}$ of Lebesgue measure $< \eta$ so that
on  $\Gamma \setminus \Gamma_{\eta},$ the
determinant \eqref{eq:5.5} in bounded and uniformly
bounded away from $0$. Moreover, we may cover
$\Gamma
\setminus \Gamma_{\eta}$ by finitely many closed
subsets $\Gamma_i$ so that on each such subset, the
map $\xi \mapsto \omega (\xi)$ is a bianalytic
homeomorphism on its image in $\mathbb{R}^n.$
We consider each of these parameter sets
$\Gamma_i$ separately.



\n On each such a set we have, by Corollary
2.2,
%%%%%%%%%%%%%%%%%%%%%
\[k \cdot \omega (\xi) + l \Omega (\xi) \not \equiv
0 \]
%%%%%%%%%%%%%%%%%%%%
\n for every $k \in \mathbb{Z}^n$ and $l \in
\mathbb{Z}^{\infty}$ with $1 \le | l | \le 2.$
Since each such expression is analytic in $\xi,$
its zero set is a set of measure zero.
Thus, Assumption B is satisfied for each subset
$\Gamma_i.$

\vspd

\n Moreover, we have for some $M < \infty, L <
\infty$
%%%%%%%%%%%%%%%%
\[| \omega |^{lip}_{\Pi} + | \Omega
|^{lip}_{\Pi} \le M < \infty\]
%%%%%%%%%%%%%%%%%%%%%%
\n and
%%%%%%%%%%%%%%%%%%%%%%%
\[| \omega^{-1} |^{lip}_{\omega (\Pi_i \setminus
\Pi_{\eta})} \le L < \infty.\]
%%%%%%%%%%%%%%%%%%%%%%%
\n It remains to check Assumption C for the
perturbation $P := Q + \varepsilon R.$

\n Following the same line of arguments as in
\cite{kapo} we prove for any $\beta \le M$, $s > 0$, $r
> 0$ small enough and $\eta > 0,$
%%%%%%%%%%%%%%%%%%%%%
\begin{equation}
\label{eq:5.6}
|| X_P ||^{sup}_{r, N; D (s,r) \times
\Gamma \setminus \Gamma_{\eta}}  + \frac{\beta}{M}
|| X_P ||^{lip}_{r, N; D (s,r) \times
\Gamma \setminus \Gamma_{\eta}} \le  C (r^2 +
\frac{\varepsilon}{r^2}).
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%
\n In particular, \eqref{eq:5.6} says that
%%%%%%%%%%%%%%%%%%%
\[X_P : U \times \Gamma \setminus \Gamma_{\eta}
\rightarrow \mathcal{S}^{N}_{\mathbb{C}}\]
%%%%%%%%%%%%%%%%%%%%
\n with $U \subseteq D (s, r).$ Thus $X_p$ has the
required regularity properties on $\Gamma \setminus
\Gamma_{\eta}$ for each $\eta > 0.$ To satisfy the
smallness condition \eqref{eq:5.2} of Theorem
\ref{T5.1} for the perturbation $P$, choose $r^2 :=
\sqrt{\varepsilon}, \beta = \frac{2c}{\gamma}
\sqrt{\varepsilon}$ with $c$ being the constant from
\eqref{eq:5.5} and $\gamma$ chosen as in  Theorem
\ref{T5.1}. We then obtain
%%%%%%%%%%%%%%%%%%%%%%%
\[|| X_P ||^{sup}_{r, N; D (s, r)
\times \Gamma \setminus \Gamma_{\eta}} +
\frac{\beta}{M} || X_P ||^{lip}_{r, p;
D (s, r) \times \Gamma \setminus \Gamma_{\eta}} \le
\gamma \beta\]
%%%%%%%%%%%%%%%%%%%%%%%%
\n for all $\varepsilon$ as required.

\vspd

\n In view of the considerations above, Theorem
\ref{T5.1} can be applied and Theorem \ref{T1.2} is
proved. \ \ \ \carre

\vspd

\n {\bf Remark 1:}  Theorem \ref{T1.2} remains true for
a larger class of phase spaces (cf \cite{gk2}),
%%%%%%%%%%%%%%%%%%%%%%%%%
\[G^{N}_{\eta, \pm} (S^1; \mathbb{C}) := \Phi (\{
(x, y) \in l^2_{N} \mid S^{\pm}_{\eta} (x, y) =
(x, y) \} \]
%%%%%%%%%%%%%%%%%%%%%%%%
\n where $\eta : l^1_{2N} (\mathbb{Z};
\mathbb{R}) \rightarrow l^{\infty} (\mathbb{Z}_{\ge
1}, \mathbb{R})$, $I := (I_k)_{k \in \mathbb{Z}}
\mapsto \eta (I)$ is an arbitrary real analytic map
and $S^{\pm}_{\eta}$ is the real analytic
isomorphism,
$S^{\pm}_{\eta} : l^2_{N} (\mathbb{Z};
\mathbb{R}^2)  \rightarrow l^2_{N} (\mathbb{Z};
\mathbb{R}^2),$ given in  action-angle coordinates by
%%%%%%%%%%%%%%%%%%%%%%%
\[\begin{array}{lll}
I^{\prime}_k & := & I_{-k} \;  \ (k \in \mathbb{Z}
\setminus \{ 0 \}); \quad I^{\prime}_0 := \pm I_0 \\
\theta^{\prime}_k &:=& \theta_{-k} + \eta_k (I)
\; \
\mbox{(for} \ k \ge 1 \ \mbox{with} \ I_k \neq 0); \\
\theta^{\prime}_k &:=& \theta_{-k} - \eta_{-k} (I)
\; \
\mbox{(for} \ k \le  -1 \ \mbox{with} \ I_k \neq 0);
\\
\theta^{\prime}_0 &:= & \theta_0 \;  \ \mbox{(if} \
I_0
\neq 0).
\end{array}
\]
%%%%%%%%%%%%%%%%%%%%%%%



\n As for $\varphi \in G^{N}_{\eta, \pm} (S^1;
\mathbb{C}),$ one has $I_k = I_{-k} \quad (k \ge 1)$,
the frequencies satisfy $\omega_k (I) = \omega_{-k}
(I)$ $ (k \ge 1)$ by Proposition \ref{P1.1}. Therefore,
mutatis mutandis, Theorem \ref{T1.2} holds for
$G^{N}_{\eta} (S^1; \mathbb{C})$ as stated in
\cite{gk2}.



\n However, we have succeeded in characterizing
$G^{N}_{\eta}$ explicitly in terms of the
potential $\varphi$ only in the case where $\eta$ is a
constant,  $\eta_j (I) \equiv \alpha$ with $\alpha
\in
\mathbb{R}$ independent of $j \ge 1.$

\vspd
\n {\bf Remark 2:}
Theorem~\ref{T1.1}, and in turn Theorem~\ref{T1.2}, hold for  more
general phase spaces. However the version of Theorem~\ref{T1.1}
for Abel-Sobolev spaces $H^{N,a}(S^1; \mathbb C)$ in the form announced in
\cite{gk2} is not true in view of the fact that nontrivial
finite gap potentials are not entire functions.



%\newpage

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\end{document}



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