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Nonlinear Scrodinger Equation, periodic spectrum
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\begin{document}

\title{ Symmetries of the Nonlinear
Schr\"odinger Equation}

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

\maketitle 

\begin{itemize}
\item[1.] UMR 6629 CNRS, Universite de Nantes, 
2 rue de la Houssi\`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}

\vspace{1cm}

\n {\bf Abstract:} Fundamental symmetries of the defocusing nonlinear
Schr\"odinger
equation are expressed in action-angle coordinates and
characterized in terms
of periodic and Dirichlet spectrum of the associated
Zakharov-Shabat
system. As a main application we prove a conjecture, raised by
several experts in field, that the periodic spectrum is symmetric
iff the sequence of gap lengths $(\gamma_k)_{k\in \mathbb {Z}}$ or, equivalently, the
sequence of actions $(I_k)_{k\in \mathbb {Z}}$ is symmetric with respect to
$k=0$.
\setcounter{section}{-1}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\n The defocusing nonlinear Schr\"odinger equation NLS on
the circle
\begin{equation}
\label{eq:0.1}
i \partial_t \varphi = - \partial^2_x \varphi + 2 \mid \varphi \mid^2
\varphi
\end{equation}



\n can be viewed as a completely integrable Hamiltonian system of 
infinite dimension. Indeed, on
the phase $L^2 (S^1; \mathbb{C}),$ introduce the Poisson
 bracket
\[\{F,G\} := i \int_{S^1} \Big(\frac{\partial F}{\partial \varphi
(x)}  \ \frac{\partial G}{\partial \overline{\varphi} (x)} -  \frac{\partial
F}{\partial \overline{\varphi} (x)} \ \frac{\partial G}{\partial \varphi (x)}
\Big) dx \]



\n Equation \eqref{eq:0.1} can then be written  in Hamiltonian
form as follows
\[\begin{array}{lllll}
\frac{\partial \varphi}{\partial t} & = & \{\mathcal{H}, \varphi\} & = & -i
\frac{\partial \mathcal{H}}{\partial \overline{\varphi}} \\
\frac{\partial \overline{\varphi}}{\partial t} &=& \{ \mathcal H,
\overline{\varphi}
\} &  =
& i \frac{\partial \mathcal H}{\partial \varphi}
\end{array}\]



\n where the Hamiltonian $\mathcal H$ is given by  (cf \cite{ft})
\[\mathcal H (\varphi):= \int_{S^1} \Big(\mid \frac{\partial \varphi}{\partial x}
\mid^2 + \mid \varphi \mid^4 \Big) dx. \]



\n Consider the following symmetry operators, acting on $L^2
(S^1; \mathbb{C}),$
%%%%%%%%%%%%%%%
\begin{equation}
\label{eq:0.2}
\mathcal{S}_1 (\varphi)  :=  \overline{\varphi}; \quad \mathcal{S}_2
(\varphi) = \check{\varphi}; 
\end{equation}
%%%%%%%%%%%%%%%
\begin{equation}
\label{eq:0.3}
M_{\alpha} \varphi := e^{i \alpha} \varphi;  \quad T_{\tau}
\varphi :=
\varphi (\tau + \cdot)
\end{equation}
%%%%%%%%%%%%%%%%


\n where $\check{\varphi}$ is defined by $\check{\varphi} (x) = \varphi
(-x).$
For convenience, we introduce $\mathcal{S}_3 := M_{\pi},$
i.e. $\mathcal{S}_3 (\varphi) = - \varphi.$ Notice that the
Hamiltonian $\mathcal{H}$ is invariant under $\mathcal{S}_1,
\mathcal{S}_2, M_{\alpha}$ and $T_{\tau}.$



\n Denote by ${U}(t)$ the solution operator of
\eqref{eq:0.1} for initial data in $L^2(S^1; \mathbb{C})$ (or
some Sobolev space $H^N (S^1; \mathbb{C}))$ (cf \cite{B}). It is immediate
that $U (t)$ commutes with $\mathcal{S}_2, S_3, M_{\alpha}$ and
$T_{\tau}$ and that
\begin{equation}
\label{eq:0.4}
U (t) \mathcal{S}_1 = \mathcal{S}_1 U (-t) .
\end{equation}


\n Recall that NLS admits a Lax pair representation
%%%%%%%%%%%%%%%%%%
$$\frac{dL}{dt} = [L, A] $$



\n where $L = L(\varphi)$ is the operator
\begin{equation}
\label{eq:0.5}
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}. We remark that $L(\varphi)$ is unitarily equivalent to the well
known AKNS-operator
\begin{equation}
\label{eq:0.6}
H(\varphi) := \bl {lr} 0 & -1\\ 1 & 0 \er \ \frac{d}{dx} + \bl
{rl} -q & p \\
p & q \er
\end{equation}

\n where $\varphi = -q + ip$, a fact which will be used throughout the paper.

\n Denote by $spec_{per} L (\varphi)$ the periodic spectrum of
$L (\varphi)$ when considered on the interval $[0,2]$ and by
$spec^{\pm}_{Dir} L (\varphi)$ the Dirichlet spectra of $L(\varphi)
$
when considered on the interval $[0,1]$
(cf definitions (\ref{eq:1.5}) and (\ref{eq:1.6})).
%%%%%%%%%%%%%%%%%%%%%%%%%
By elementary
considerations one shows that
%%%%%%%%%%%%%%%%%%%%%%%%%
\[\begin{array}{llllll}
spec_{per} L (\bar{\varphi}) & = & - spec_{per} L(\varphi) ;&
 spec_{per} L(\check{\varphi}) & = & - spec_{per} L(\varphi);\\
spec_{per} L (M_{\alpha} \varphi) &=& spec_{per} L (\varphi);&
 spec_{per} L (T_{\tau} \varphi)& = & spec_{per} L (\varphi);
\end{array}
\]

\n and expresses $spec^+_{Dir} L (\mathcal{S}_j \varphi)$  for $j
= 1, 2, 3$ in terms of 
$spec_{Dir}^- L (\varphi)$.
This is used to compute the symmetries
in action-angle variables. Recall from \cite{gk} that
NLS admits global Birkhoff coordinates $(z_j)_{j \in
\mathbb{Z}}$
$$
  z_j := x_j + i y_j = \sqrt{2 I_j} e^{i \theta_j}
$$

\n with $I_j = (x^2_j + y^2_j)/2 \ (j \in \mathbb{Z})$ denoting
the actions and for $\varphi$ with $I_j \neq 0$, $\theta_j$ denoting the
angle variable.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{teor}
\label{T0.0}
\n
\begin{itemize}
\item[(i)]  The actions are left
invariant by $M_{\alpha}$ and $T_{\tau}$ $\; (\forall k \in \mathbb Z)$
\[
I_k(M_{\alpha} \varphi) = I_{k} (\varphi); \quad
I_k  (T_{\tau} \varphi) = I_{k} (\varphi);
\]
 for $\mathcal{S}_j,\ j=1,2 \quad (\forall k \in \mathbb Z)$
\[I_k  (\mathcal{S}_j \varphi) = I_{-k} (\varphi).
\]

\item[(ii)]  For $k$ with $I_k \neq 0$
\[\begin{array}{lll}
\theta_k (M_{\alpha} \varphi) &\equiv & \theta_k + \alpha \quad (mod \;
2{\pi});
\\
\theta_k ( \check{\varphi}) &\equiv& \theta_{-k} (\varphi) \quad (mod \;
2{\pi}); \\
\theta_k (\overline{\varphi}) &\equiv& - \theta_{-k} (\varphi) \quad (mod \;
2{\pi}).
\end{array}\]

\end{itemize}
\end{teor}

\n As an immediate application of Theorem~\ref{T0.0} one obtains

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{coroll}
\label{C0.0}
When evaluated at $I= (I_k)_{k\in \Z}$ with $I_k =I_{-k} \;\; \forall k \in
\Z$, the NLS frequencies\; $\omega = (\omega_k)_{k\in Z}$, $\omega_k =
\frac{\partial \mathcal H}{\partial I_k}$, are symmetric,
$\omega_k(I)=\omega_{-k}(I) \;\; \forall k \in \mathbb Z$.
\end{coroll}
\n Theorem \ref{T0.0} and Corollary~\ref{C0.0} will be used for a KAM
type theorem in \cite{gk2}

\vspd

\n Further we use Theorem \ref{T0.0} to prove the conjecture, raised by several experts
in the field, that the periodic spectrum is symmetric if and only if the sequence of the
gap lengths is symmetric. More precisely, let
 $spec L (\varphi) = (\lambda^{\pm}_k (\varphi))_{k \in
\mathbb{Z}}$ denote the periodic spectrum of $L(\varphi)$. $\lambda^{\pm}_k
(\varphi)$ are real numbers satisfying
$\lambda_{k}^- (\varphi) \le \lambda^+_k (\varphi) <
\lambda^-_{k+1} (\varphi). $
 Let $\gamma (\varphi) := (\gamma_k (\varphi))_{k \in \mathbb{Z}}$
be the sequence of gap lengths, $\gamma_k (\varphi) :=
\lambda^+_k (\varphi) - \lambda^-_k (\varphi).$
We prove
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{teor}
\label{T0.1}
For $\varphi \in L^2 (S^1;  \mathbb{C}),$ the following assertions
are equivalent:
\begin{itemize}
\item[(i)] $\lambda^{\pm}_k (\varphi) = - \lambda^{\mp}_{-k}
(\varphi)$ for all $k \ge 0.$
\item[(ii)] $\gamma_k (\varphi) = \gamma_{-k} (\varphi)$ for all $k
\ge 1.$
\end{itemize}
\end{teor}





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Symmetries and spectra}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Periodic spectrum}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\n The periodic spectrum of the Zakharov-Shabat operator $L(\varphi)$ is given
by
%%%%%%%%%%%%%%%%%%%%%%%%
\[ \begin{array}{lll}
spec_{per} L(\varphi) := \{\lambda \in \mathbb{C}& \mid & \exists
F \in H^1_{loc} (\mathbb{R}; \mathbb{C}^2), F \not \equiv 0 \; \mbox{ with }
 L(\varphi) F = \lambda F \\&& \mbox{and } \ F(x + 2) = F(x) \quad
 \forall x \in \mathbb{R} \}. \end{array}
\]
%%%%%%%%%%%%%%%%%%%%%%%%%


\n By \cite{gg} $spec_{per} L (\varphi)$ is a sequence
 of real num\-bers $(\lambda^{\pm}_k (\varphi))_{k \in
\mathbb{Z}},$ which can be ordered in such a way that $(\forall k \in \mathbb{Z})$
%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{equation}
\label{eq:1.1}
\lambda^-_k (\varphi) \le \lambda^+_k (\varphi) < \lambda^-_{k+1}
(\varphi).
\end{equation}



\n We have the following
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prop}
\label{P1.1}
Let $\varphi \in L^2 (S^1; \mathbb{C}).$ Then, for any $k \in \mathbb{Z},$

\begin{itemize}
\item[(i)]  
$\lambda^{\pm}_k (e^{i \alpha} \varphi) = \lambda^{\pm}_k (\varphi), \quad
 \lambda^{\pm}_k (\varphi) = \lambda^{\pm}_k (T_{\tau} \varphi) \qquad
(\forall \alpha \in \mathbb{R}, \tau \in \mathbb{R})$;
\item[(ii)] $\lambda^{\pm}_k (\check{\varphi}) = - \lambda^{\mp}_{-k} (\varphi), \quad
 \lambda^{\pm}_k (\bar{\varphi}) = - \lambda^{\mp}_{-k} (\varphi).$
\end{itemize}
\end{prop}



\n {\bf Proof} (i) For $\alpha \in \mathbb{R}$ arbitrary, define
$V_{\alpha} = \bl {ll}
e^{-i \alpha /2} & 0 \\
0 & e^{i \alpha / 2}
\er .$ One easily verifies that
%%%%%%%%%%%%%%%%%%%%%
\begin{equation}
\label{eq:1.2}
L (e^{i \alpha} \varphi) = V^{-1}_{\alpha} L (\varphi) V_{\alpha}
\end{equation}

\n and $L(T_{\tau} \varphi) = T_{\tau} L (\varphi) T_{-
\tau}.$ Thus both, $L (e^{i \alpha} \varphi)$ and $L(T_{\tau}
\varphi)$, are unitarily equivalent to $L(\varphi)$ and the claimed
statement follows. To prove (ii) notice that
\begin{equation}
\label{eq:1.3}
L (- \check{\varphi}) = - W^{-1} L(\varphi) W
\end{equation}

\n where $W$ is the unitary operator defined by
\[W \bl {l} Y \\ Z  \er := \bl {l} \check{Y} \\ \check{Z} \er, 
\ \mbox{ with } \ \bl {l} Y \\ Z \er \in L^2_{loc}
(\mathbb{R}; \mathbb{C}^2).\]



\n Thus
\begin{equation}
\label{eq:1.4}
spec_{per} L(-\check{\varphi}) =  - spec_{per} L(\varphi).
\end{equation}


\n Combining \eqref{eq:1.4} and (i) we obtain $\lambda^{\pm}_k
(\check{\varphi}) = - \lambda^{\mp}_{-k} (\varphi) \ (\forall k \in
\mathbb{Z}).$ Consider $\lambda \in spec_{per} L (\varphi)$ and choose $F
\in H^1_{loc} (\mathbb{R}; \mathbb{C}^2),$ satisfying $F(x +2)
= F(x) \quad (\forall x \in \mathbb R)$  and
$L(\varphi) F = \lambda F.$ As $\lambda$ is real, $L(-
\bar{\varphi}) \overline{F} = - \lambda \overline{F}$ and
thus $- \lambda \in spec_{per} L (- \bar{\varphi}).$ Combined
with (i), this leads to $\lambda^{\pm}_k (\bar{\varphi}) = -
\lambda_{-k}^{\mp} (\varphi) \ \ (\forall k \in \mathbb{Z}).$ \ \ \ \carre

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Dirichlet spectra and divisors} \hspace{8cm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\n Let $F_j (x, \lambda; \varphi) := \bl {l}
Y_j (x, \lambda; \varphi) \\
Z_j (x, \lambda; \varphi) \er, \ j =  1,2,$ be the fundamental
solutions of $H(\varphi),$ i.e. the solutions to $HF = \lambda F$
such that
%%%%%%%%%%%%%%%%%%
\[F_1 (0, \lambda; \varphi) = \bl {l} 1 \\ 0 \er, \ F_2 (0,
\lambda; \varphi) = \bl{l} 0 \\ 1 \er .\]

\n For each $x \in \mathbb R$  and $\varphi \in L^2 (S^1;  \mathbb{C}),
\ F_1 (x, \lambda; \varphi)$ and $F_2 (x, \lambda; \varphi)$ are
entire functions of $\lambda$. The two Dirichlet spectra are
defined as follows
%%%%%%%%%%%%%%%%%%
\begin{equation}
\label{eq:1.5}
spec_{Dir}^+ L (\varphi) = \{\lambda \in \mathbb{C}\ | \  Z_1 (1,
\lambda; \varphi) = 0\}
\end{equation}
%%%%%%%%%%%%%%%%%%%%
\begin{equation}
\label{eq:1.6}
spec^-_{Dir} L (\varphi) = \{\lambda \in \mathbb{C}\ | \ Y_2 (1,
\lambda; \varphi) = 0 \}.
\end{equation}

\n It is proved in \cite{gg} that $spec^+_{Dir} L (\varphi)$,
respectively $spec^-_{Dir} L (\varphi)$, consists of simple, real
eigenvalues $(\mu_k (\varphi))_{k \in \mathbb{Z}}$, respectively $(\nu_k
(\varphi))_{k \in \mathbb{Z}}.$ The numerotation is chosen in
such a way that $(\mu_k (\varphi))_{k \in \mathbb{Z}}$ and $(\nu_k
(\varphi))_{k \in \mathbb{Z}}$ are strictly increasing satisfying
$\mu_k (\varphi) \sim k \pi$ and $\nu_k (\varphi) \sim k
\pi$ for $|k|$ large.
Further introduce the function $\delta (\lambda; \varphi),$
defined by 
%%%%%%%%%%%
\begin{equation}
\label{eq:1.7}
\delta (\lambda; \varphi) = Z_2 (1, \lambda; \varphi) - Y_1 (1,
\lambda; \varphi).
\end{equation}
Notice that ($\forall k \in \mathbb Z$)
\begin{equation}
\label{eq:1.71} 
\delta (\mu_k (\varphi); \varphi)^2 = \Delta (\mu_k
(\varphi); \varphi)^2 - 4 ,
\end{equation}
 where
%%%%%%%%%%%%%%
\begin{equation}
\label{eq:1.72}
\Delta (\lambda; \varphi) = Y_1 (1,
\lambda; \varphi) + Z_2 (1, \lambda; \varphi)
\end{equation}
is the discriminant.
\vspd

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prop}
\label{P1.2}
Let $\varphi \in L^2 (S^1;  \mathbb{C}).$ Then
\begin{itemize}
\item[(i)] $\mu_k (- \varphi) = \nu_k (\varphi), \quad \nu_k (-\varphi) =
\mu_k (\varphi) \qquad (\forall \ k \in \mathbb{Z});$
\item[(ii)] $\delta (\lambda; - \varphi) = - \delta (\lambda;
\varphi).$
\end{itemize}
\end{prop}

\n {\bf Remark} For $\alpha \not \equiv 0, \pi \; (mod \; 2 \pi),\;
\mu_k (e^{i \alpha} \varphi)$ satisfies an equation in\-vol\-ving
$spec_{Dir}^{\pm} L(\varphi),$ the solution of which does not seem
to be given in form of a closed expression.

\n {\bf Proof} (Proposition \ref{P1.2})
For $F = \bl {l} Y \\
Z \er \in H^1_{loc} (\mathbb{R}; \mathbb{R}^2)$ define
%%%%%%%%%%%%%%%%%%
\begin{equation}
\label{eq:1.8}
F^{\perp} = \bl {r} Z \\ -Y \er.
\end{equation}

\n If $H (\varphi) F = \lambda F$ we have
%%%%%%%%%%%%%%%%%%
\[H (- \varphi) F^{\perp} = \lambda F^{\perp}.\]

\n Therefore
%%%%%%%%%%%%%%%%%%
\begin{equation}
\label{eq:1.9}
F_1 (x, \lambda; - \varphi) = F^{\perp}_2 (x, \lambda; \varphi)
\end{equation}

\n and
%%%%%%%%%%%%%%%%%%
\begin{equation}
\label{eq:1.10}
F_2 (x, \lambda; - \varphi) = - F^{\perp}_1 (x, \lambda; \varphi).
\end{equation}


\n Using \eqref{eq:1.9} - \eqref{eq:1.10} one easily gets (i) and
(ii). \ \ \ \carre

\vspd
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prop}
\label{P1.3}
For $\varphi \in L^2 (S^1;  \mathbb{C}), k \in \mathbb{Z}$

\begin{itemize}
\item[(i)] $\mu_k (\bar{\varphi}) = - \nu_{-k} (\varphi), \quad
\nu_k (\bar{\varphi}) = - \mu_{-k} (\varphi);$
\item[(ii)] $\delta (\mu_k (\bar{\varphi}), \bar{\varphi}) =
- \delta (\nu_{-k} (\varphi), \varphi).$
\end{itemize}
\end{prop}

\n {\bf Proof} (i) For $F = \bl {l} Y \\ Z\er \in H^1_{loc}
(\mathbb{R}, \mathbb{R}^2)$ we define
%%%%%%%%%%%%%%%%%%
\begin{equation}
\label{eq:1.11}
\tilde{F} = \bl {l} Z \\ Y \er.
\end{equation}

\n If $H (\varphi) F = \lambda F,\; \tilde{F}$ satisfies $H
(\bar{\varphi}) \tilde{F} = - \lambda \tilde{F}.$ Therefore
%%%%%%%%%%%%%%%%%%
\begin{equation}
\label{eq:1.12}
F_1 (x, \lambda; \bar{\varphi}) = \tilde{F}_2 (x, - \lambda;
\varphi);\quad
F_2  (x, \lambda; \bar{\varphi}) = \tilde F_1 (x, - \lambda; \varphi).
\end{equation}

\n Using \eqref{eq:1.12} and the asymptotics
of $\mu_k$ and $\nu_k$, one easily obtains (i).

\n(ii) From \eqref{eq:1.12}  one
deduces $\delta (\mu_k (\bar{\varphi}); \bar{\varphi}) = -
\delta (- \mu_k (\bar{\varphi}); \varphi)$ and thus (ii) follows
using (i). \ \ \ \carre

\vspd
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prop}
\label{P1.4}
For $\varphi \in L^2 (S^1;  \mathbb{C}), k \in
\mathbb{Z},$
\begin{itemize}
\item[(i)] $\mu_k (\check{\varphi}) = - \nu_{-k} (\varphi),\quad \nu_k
(\check{\varphi}) = - \mu_{-k} (\varphi);$
\item[(ii)] $\delta (\mu_k (\check{\varphi}), \check{\varphi}) =
\delta (\nu_{-k} (\varphi), \varphi).$
\end{itemize}
\end{prop}

\n {\bf Remark} By Proposition \ref{P1.1}, \ref{P1.3} and
\ref{P1.4}, $\check{\varphi}$ and $\bar{\varphi}$ have the same
periodic and the same Dirichlet spectra. They are only distinguished
by
%%%%%%%%%%%%%%%%%%
\[\delta (\mu_k (\check{\varphi}), \check{\varphi}) = - \delta (\mu_k
(\bar{\varphi}), \bar{\varphi}) \quad (\forall k \in
\mathbb{Z}).\]

\n {\bf Proof} (Proposition \ref{P1.4})
For $F = \bl {l} Y \\ Z \er \in H^1_{loc} (\mathbb{R};
\mathbb{R}^2)$ we define
%%%%%%%%%%%%%%%%%%
\begin{equation}
\label{eq:1.14}
F^{\ast} (x) = F (1 -x).
\end{equation}


\n If $H (\varphi) F = \lambda F, \; F^{\ast}$ satisfies
%%%%%%%%%%%%%%%%%%
\begin{equation}
\label{eq:1.15}
H (- \check{\varphi}) F^{\ast} = - \lambda F^{\ast}.
\end{equation}


\noindent By the definition of $\mu_k (\varphi),$
%%%%%%%%%%%%%%%%%%
\[F_1 (1, \mu_k (\varphi); \varphi) = Y_1 (1, \mu_k (\varphi); \varphi) \bl
{l} 1 \\ 0 \er.
\]

\n Therefore $(\forall k \in \mathbb{Z})$
%%%%%%%%%%%%%%%%%%
\begin{equation}
\label{eq:1.16}
F_1 (x, - \mu_k (\varphi); \ - \check{\varphi}) = \frac{1}{Y_1(1,
\mu_k (\varphi); \varphi)} F_1 (1 - x, \mu_k (\varphi); \varphi).
\end{equation}

\n Evaluated at $x = 1,$ \eqref{eq:1.16} leads to $Z_1 (1, -
\mu_k (\varphi); - \check{\varphi}) = 0 \ (\forall k \in \mathbb{Z}).$ In
view of the asymptotics of $\mu_k$ we conclude that $\mu_k
(-\check{\varphi}) = - \mu_{-k} (\varphi).$ Statement (i) then follows
from Proposition \ref{P1.2} (i).



\n From \eqref{eq:1.16} and (i), we deduce that
%%%%%%%%%%%%%%%%%%
\begin{equation}
\label{eq:1.17}
Y_1 (1, \mu_{-k} (- \check{\varphi}); - \check{\varphi}) =
\frac{1}{Y_1 (1, \mu_k (\varphi) ; \varphi)}\; .
\end{equation}



\n Further, the Wronskian identity (see \cite{gg})
%%%%%%%%%%%%%%%%%%
\begin{equation}
\label{eq:1.18}
Y_1 (1, \lambda; \varphi) Z_2 (1, \lambda; \varphi) - Y_2 (1,
\lambda; \varphi) Z_1 (1, \lambda; \varphi) = 1
\end{equation}



\n implies that
%%%%%%%%%%%%%%%%%%
\begin{equation}
\label{eq:1.19}
Z_2 (1, \mu_k (\varphi); \varphi) = \frac{1}{Y_1 (1, \mu_k (\varphi);
\varphi)}.
\end{equation}



\n Hence \eqref{eq:1.17} and \eqref{eq:1.19} yield
%%%%%%%%%%%%%%%%%%
\begin{equation}
\label{eq:1.20}
\delta (\mu_{-k} (- \check{\varphi}); - \check{\varphi}) = - \delta
(\mu_k (\varphi); \varphi).
\end{equation}



\n Statement (ii) then follows by combining \eqref{eq:1.20} and Proposition \ref{P1.2}. \ \ \
\carre


\vspd

\n {\bf Remark} The evolution of the Dirichlet spectra under
the translation flow has been studied in \cite{bggk}.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Symmetries in action-angle coordinates}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Actions}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\n Recall from \cite{gk} (see also \cite{mv}) that, for $\varphi
\in L^2 (S^1;  \mathbb{C}),$ the actions, $I_k (\varphi), k \in
\mathbb{Z},$ are defined by
%%%%%%%%%%%%%%%%%%
\begin{equation}
\label{eq:2.1}
I_k (\varphi) := \frac{-2}{\pi} \int^{\lambda^+_k
(\varphi)}_{\lambda^-_k (\varphi)} (-1)^k \ \frac{\mu \dot{\Delta}
(\mu; \varphi)}{\mid (\Delta^2 (\mu; \varphi) - 4)^{1/2}\mid} \ d \mu
\end{equation}



\n where $\Delta (\mu; \varphi)$ is the discriminant given by
%%%%%%%%%%%%%%%%%%
\begin{equation}
\label{eq:2.2}
\Delta (\mu; \varphi) = Y_1 (1, \mu; \varphi) + Z_2 (1, \mu; \varphi).
\end{equation}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prop}
\label{P2.1}
For $\varphi \in L^2 (S^1;  \mathbb{C})$ and $k \in \mathbb{Z}$

\begin{itemize}
\item[(i)]  $I_k (e^{i \alpha} \varphi) = I_k (\varphi) \quad \forall \alpha \in \mathbb{R};$
\item[(ii)] $I_k (\bar{\varphi}) = I_{-k} (\varphi);$
\item[(iii)] $I_k (\check{\varphi}) = I_{-k} (\varphi);$
\item[(iv)] $I_k (T_{\tau} \varphi) = I_k (\varphi) \quad \forall \tau \in \mathbb{R}.$
\end{itemize}
\end{prop}



\n {\bf Proof} Recall from \cite{gg} that $\Delta^2 (\mu;
\varphi) - 4$ has a representation as an infinite product 
%%%%%%%%%%%%%%%%%%
\begin{equation}
\label{eq:2.3}
\Delta^2 (\mu; \varphi) -4 = - 4 (\lambda^+_0 (\varphi) - \mu)
(\lambda^-_0 (\varphi) - \mu) \times 
\prod_{k \in \mathbb{Z}^{\ast}} \ \frac{(\lambda^+_k (\varphi) -
\mu)(\lambda^-_k (\varphi) - \mu)}{k^2 \pi^2}
\end{equation}

\n where, in order
that the above infinite product $\prod_{k \in \mathbb{Z}^{\ast}} a_k$
 be absolutely convergent, it has to be computed as
 $\prod_{k \ge 1} a_k \cdot a_{-k}$ .
Furthermore, for $\mu \in (\lambda^-_k (\varphi), \lambda^+_k
(\varphi)),$ sign $\Delta (\mu; \varphi) = (-1)^k.$ Therefore
$\Delta (\cdot; \varphi)$, and thus $(I_k (\varphi))_{k \in
\mathbb{Z}}$, is uniquely determined by $spec_{per} L (\varphi).$
In particular, statements (i) and (iv) follow from Proposition
\ref{P1.1}. 
To prove (ii) and (iii) notice that, 
by Proposition \ref{P1.1}, for $\psi \in \{
\overline{\varphi}, \check{\varphi}\},\; spec_{per} L (\psi) = -
spec_{per} L (\varphi)$. Therefore $\Delta (\lambda; \psi) = \Delta
(-\lambda; \varphi)$ and thus, by \eqref{eq:2.1}, one easily
obtains for any $k \in \mathbb{Z}, \; I_k (\psi) = I_{-k} (\varphi).$
\carre


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Angles}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\n Denote by $\mathcal{M} (\varphi)$ the hyperelliptic Riemann
surface, $y = \sqrt{\Delta (\lambda; \varphi)^2 - 4}$, and by
$\mathcal{P} = (\lambda, y)$ an arbitrary point on
$\mathcal{M} (\varphi).$ Let $\beta_j = \beta_j (\varphi) \quad (\forall j \in
\mathbb{Z})$ be Abelian differential 1-forms of the third kind
on $\mathcal{M} (\varphi),$ uniquely determined by the
normalization conditions (\cite{gk}, \cite{mv}),
\[\int_{a_k} \beta_j = 2 \pi \ \delta_{kj}, \]

\n where the cycle $a_k \equiv a_k (\varphi) \quad (\forall k \in \mathbb{Z})$
is a contour on the upper sheet of $\mathcal{M} (\varphi)$ around
$[\lambda_k^-, \lambda^+_k]$ with clockwise orientation (see
\cite{gk}).
For $k \in \mathbb{Z}$ with $I_k (\varphi) \neq 0,$ the angle
$\theta_k (\varphi)$ is given by
%%%%%%%%%%%%%%%%%%
\begin{equation}
\label{eq:2.4}
\theta_k (\varphi) = \int^{\mathcal{P}_k (\varphi)}_{\lambda^-_k
(\varphi)} \sum_{j \in \mathbb{Z}} \beta_j (\varphi),
\end{equation}

\n where $\mathcal{P}_k (\varphi)$ denotes the point $\left( \mu_k
(\varphi), \delta (\mu_k (\varphi); \varphi) \right)$ on $\mathcal{M}
(\varphi)$ (cf. \eqref{eq:1.71}). As path of integration in \eqref{eq:2.4} one chooses the
straight line on the sheet of $\mathcal{M} (\varphi)$ containing
$\mathcal{P}_k (\varphi),$ connecting $\lambda^-_k$ and
$\mathcal{P}_k.$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prop}
\label{P2.2}
For $\varphi \in L^2 (S^1;  \mathbb{C}),\; \alpha \in
\mathbb{R},$ and $k \in \mathbb{Z}$ with $I_k (\varphi) \neq 0$
%%%%%%%%%%%%%%%%%%
\[ \theta_k (e^{i \alpha} \varphi) \equiv \theta_k (\varphi) + \alpha
\ \ {(\rm{mod }\ 2 \pi)}.\]
\end{prop}



\n {\bf Proof} Denote by $R_{\alpha}$ the action of
$M_{\alpha}$ expressed in Birkhoff coordinates, i.e.
$R_{\alpha} = \Phi^{-1} M_{\alpha} \Phi$ where $\Phi$ is the
Birkhoff map,
$\varphi = \Phi \left( (\sqrt{2I_j} e^{i \theta_j})_{j \in \mathbb Z} \right)$
 (cf \cite{gk}).
Notice that $R_{\alpha + \beta } = \Phi^{-1} M_{\alpha} M_{\beta} \Phi =
 R_{\alpha} R_{ \beta }$, $R_0 = Id,$ and 
$$ \frac{d}{d \alpha } R_{\alpha} = 
 \lim_{ \beta \to 0} \frac {R_{\alpha + \beta } - R_{\alpha}}{\beta} =
\lim_{\beta \to 0} \frac { R_{ \beta } - Id}{\beta} {R_{\alpha} = 
i \cdot R_{\alpha}}.
$$
Hence $R_{\alpha} = e^{ i \alpha} Id.$ \ \ \ \carre
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prop}
\label{P2.3}
For $\varphi \in L^2 (S^1;  \mathbb{C}),$
\begin{itemize}
\item[(i)] $\theta_k (\check{\varphi}) \equiv \theta_{-k} (\varphi)\ ({\rm mod }\
2\pi)  \quad \forall \ k \in \mathbb{Z} $ with $I_{-k} (\varphi) \neq 0;$
\item[(ii)] $\theta_k (\bar{\varphi}) \equiv - \theta_{-k} (\varphi) \
({\rm mod }\ 2 \pi) \quad \forall \ k \in \mathbb{Z}$ with $I_{-k} (\varphi)  \neq 0.$
\end{itemize}
\end{prop}

\noindent {\bf Proof} For $k \in \mathbb{Z}$ with $I_k
(-\check{\varphi}) \neq 0$ we have, by \eqref{eq:2.4} and
Proposition \ref{P1.1},
%%%%%%%%%%%%%%%%%%
\begin{equation}
\label{eq:2.6}
\theta_k (-\check{\varphi}) = \int^{\mathcal{P}_k (-
\check{\varphi})}_{\lambda^-_k (-\check{\varphi})} \sum_{j \in
\mathbb{Z}} \beta_j (- \check{\varphi}) = \int^{\mathcal{P}_k
(-\check{\varphi})}_{- \lambda^+_{-k} (\varphi)} \sum_{j \in
\mathbb{Z}} \beta_j (- \check{\varphi}).
\end{equation}

\n To compute the latter integral, introduce the map $\sigma :
\mathcal{M} (- \check{\varphi}) \rightarrow \mathcal{M} (\varphi)$
 defined by
%%%%%%%%%%%%%%%%%%
\begin{equation}
\label{eq:2.7}
\sigma (\lambda , y) = (-\lambda, - y).
\end{equation}

\n Notice that for $(\lambda , y) \in \mathcal{M} (- \check{\varphi})$,
$\sigma (\lambda , y) \in  \mathcal{M} (\varphi)$ as $\Delta^2 (\lambda;
- \check{\varphi}) = \Delta^2 (- \lambda; \varphi)$ (cf formula (\ref{eq:2.3}) and
Proposition~\ref{P1.1}).

\n In view of Proposition \ref{P1.1}
%%%%%%%%%%%%%%%%%%
\[\int_{\sigma (a_k (- \check{\varphi}))} \beta_{-j} (\varphi) =
\int_{a_{-k} (\varphi)} \beta_{-j} (\varphi) = 2 \pi \delta_{jk}.
\]

\n On the other hand, changing coordinates according to $\sigma$,
%%%%%%%%%%%%%%%%%%
\[\int_{\sigma (a_k (- \check{\varphi}))}  \sigma^{\ast} \beta_j
(-\check{\varphi}) = \int_{a_k (-\check{\varphi})} \beta_j (-
\check{\varphi}) = 2 \pi \delta_{kj}.
\]

\n As $\sigma^{\ast} \beta_j (- \check{\varphi})$ is uniquely
determined by its nomalization conditions we conclude that
%%%%%%%%%%%%%%%%%%
\[\sigma^{\ast} \beta_j (- \check{\varphi}) = \beta_{-j} (\varphi). \]

\n Further, using Proposition \ref{P1.2} and Proposition
\ref{P1.4} we get
%%%%%%%%%%%%%%%%%%
\[\begin{array}{lll}
\sigma (\mathcal{P}_k (-\check{\varphi})) & = & \sigma (\mu_k
(- \check{\varphi}), \delta (\mu_k (- \check{\varphi}); -
\check{\varphi}))\\
&= &\sigma (- \mu_{-k} (\varphi), - \delta (\mu_{-k} (\varphi); \varphi)) \\
&=&  (\mu_{-k} (\varphi), \delta (\mu_{-k} (\varphi); \varphi)) =
\mathcal{P}_{-k} (\varphi).
\end{array}\]


\n Therefore, by a change of variable of integration in
\eqref{eq:2.6} we obtain
%%%%%%%%%%%%%%%%%%
\begin{equation}
\label{eq:2.8}
\theta_k (- \check{\varphi}) =
\int^{\mathcal{P}_{-k}(\varphi)}_{\lambda^+_{-k} (\varphi)} \sum_{j
\in \mathbb{Z}} \ \beta_j (\varphi).
\end{equation}

\n By contour integration,
%%%%%%%%%%%%%%%%%%
\begin{equation}
\label{eq:2.9}
\int^{\mathcal{P}_k}_{\lambda^-_k} \sum_{j \in \mathbb{Z}}
\beta_j - \int^{\mathcal{P}_k}_{\lambda^+_k} \sum_{j \in
\mathbb{Z}} \beta_j \equiv \int^{\lambda^+_k}_{\lambda^-_k}
\sum_{j \in \mathbb{Z}} \beta_j \ ({\rm mod }\; 2 \pi),
\end{equation}

\n where in the last integral the path of integration is chosen
to be the straight line on the upper sheet connecting
$\lambda^-_k$ with $\lambda^+_k$. Notice that by the
normalization conditions, $\int_{a_k} \beta_j = 2 \pi
\delta_{jk},$ we get $\forall k \in \mathbb{Z}$
%%%%%%%%%%%%%%%%%%
\begin{equation}
\label{eq:2.10}
\int^{\lambda^+_k}_{\lambda^-_k} \sum_{j \in \mathbb{Z}}
\beta_j \equiv \pi \ \ ({\rm mod }\ 2 \pi).
\end{equation}

\n Therefore combining \eqref{eq:2.4}, \eqref{eq:2.8},
\eqref{eq:2.9} and \eqref{eq:2.10} we obtain $\theta_k (-
\check{\varphi}) \equiv \theta_{-k} (\varphi) + \pi$, which leads to
(i) using Proposition \ref{P2.2} with $\alpha = \pi.$

\vspd
\n In order to prove (ii), recall from Proposition \ref{P1.1} that $\lambda^{\pm}_k
(\check{\varphi}) = \lambda^{\pm}_k (\bar{\varphi})$ and thus, in particular,
$\beta_j (\check{\varphi}) = \beta_j (\bar{\varphi}) \; (\forall j \in \mathbb Z).$
Further, by Proposition \ref{P1.3} and \ref{P1.4},
$$
\mu_k (\check{\varphi}) = \mu_k (\bar{\varphi});
\quad \delta (\mu_k (\check{\varphi}); \check{\varphi}) = - \delta (\mu_k
(\bar{\varphi}); \bar{\varphi}).
$$

\n From \eqref{eq:2.4} it thus follows that
%%%%%%%%%%%%%%%%%%
\begin{equation}
\label{eq:2.11}
\theta_k (\bar{\varphi}) = \int^{f ( \mathcal{P}_k
(\check{\varphi}))}_{\lambda^-_k (\check{\varphi})} 
\sum_{j \in \mathbb{Z}} \beta_j (\check{\varphi})
\end{equation}

\n where $f (\lambda, \delta) = (\lambda, - \delta)$ is the
involution on $\mathcal{M} (\check{\varphi})$ exchanging the two
sheets. As $-\beta_j (\varphi)$ satisfies the normalization
conditions, 
%%%%%%%%%%
\[ 
- \int_{f (a_k (\varphi))} \beta_j(\varphi) = 2 \pi \delta_{kj},
\]

we conclude that $f^{\ast} \beta_j (\varphi) = -
\beta_j (\varphi).$ Therefore, by a change of variable of
integration in \eqref{eq:2.11}, we get
%%%%%%%%%%%%%%%%%%
\begin{equation}
\label{eq:2.12}
\theta_k (\bar{\varphi}) = - \int^{\mathcal{P}_k
(\check{\varphi})}_{\lambda^-_k (\check{\varphi})} \sum_{j \in
\mathbb{Z} } \beta_j (\check{\varphi}) \equiv - \theta_k
(\check{\varphi}).
\end{equation}

\n Combining \eqref{eq:2.12} and (i) we obtain (ii). \ \ \
\carre
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prop}
Let $\varphi \in L^2 (S^1;  \mathbb{C}), \tau \in \mathbb{R}$ and
$k \in \mathbb{Z}$ with $I_k (\varphi) \neq 0.$ Then

\[ \theta_k (T_{\tau} \varphi) = \theta_k(\varphi) + 2 \pi k \tau. \]
\end{prop}



\n {\bf Proof} By continuity, it suffices to consider $\varphi
\in H^1 (S^1;  \mathbb{C}).$ The translation flow $T_{\tau}
\varphi (\cdot) = \varphi (\tau + \cdot)$ is the Hamiltonian flow
associated with the Hamiltonian (see \cite{ft} or \cite{gg}).
%%%%%%%%%%%%%%%%%%
\begin{equation}
\label{eq:2.13}
\mathcal{H}_1 (\varphi) = i/2 \int_{S^1}
(\bar{\varphi} \varphi^{\prime} - \varphi \overline{\varphi^{\prime}})
\ dx
\end{equation}

\n which commutes with the NLS-Hamiltonian. Thus for
$\varphi = \Phi ((\sqrt{2 I_j} e^{i \theta_j})_{j \in \mathbb{Z}})$
$(\Phi =$ Birkhoff map)
%%%%%%%%%%%%%%%%%%
\begin{equation}
\label{eq:2.14}
\theta_k (T_{\tau} \varphi) = \theta_k (\varphi) + w_k (I) \tau
\end{equation}

\n where
%%%%%%%%%%%%%%%%%%
\begin{equation}
\label{eq:2.15}
w_k (I) := \frac{\partial \mathcal{H}_1}{\partial I_k} (I) \qquad (\forall k \in
\mathbb{Z})
 \end{equation}

\n are the frequencies of the translation flow. Since $\theta_k
(T_1 \varphi) = \theta_k (\varphi),$ there exists for any $k \in \mathbb{Z}$ and
 $I \in \ell ^1 (\mathbb{Z}; \mathbb{C}),$ an
integer $n_k (I) \in \mathbb{Z}$ such that $w_k (I) = n_k (I).$
Furthermore, since $w_k(I)$
is continuous and $n_k (I)$ takes discrete values, $n_k (I)$
does not depend on $I$, i.e. $n_k (I) \equiv n_k.$ From Lemma
\ref{L2.5} below we deduce that for a 1-gap potential $\varphi$
with $I_j (\varphi) = 0 \quad (\forall j \neq k)$ and $I_k (\varphi) > 0$
the frequency $w_k(I)$ is given by $w_k (I) = 2 \pi k.$ \ \ \
\carre
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{lemma}
\label{L2.5}
For $k \in \mathbb{Z}$
$$
\{ \varphi \in L^2 (S^1;  \mathbb{C}) \mid \gamma_j (\varphi) = \gamma
\delta_{kj}, \ j \in \mathbb{Z} \} 
= \{ \varphi (x) = c e^{ 2i \pi k x} \mid c \in \mathbb{C}
 ,\ \ |c| = \gamma/2\}.$$
\end{lemma}

\n {\bf Proof} A straightforward computation proves that for
$\varphi \in L^2 (S^1;  \mathbb{C})$ and $j, k \in \mathbb{Z}$
arbitrary,
%%%%%%%%%%%%%%%%%%
\begin{equation}
\label{eq:2.16}
\gamma_j (e^{-2i \pi k x} \varphi) = \gamma_{j-k} (\varphi).
\end{equation}

\n Further one knows (cf \cite{gg})
%%%%%%%%%%%%%%%%%%
\begin{equation}
\label{eq:2.17}
\{ \varphi \in L^2  (S^1;  \mathbb{C}) \mid \gamma_j (\varphi) =
\gamma \delta_{0j} ,\ \forall j\in \mathbb{Z} \} = 
\{\varphi (x) = c \mid c \in \mathbb{C} ,\ |c|
= \gamma/2 \}.
\end{equation}

\n Combining \eqref{eq:2.16} and \eqref{eq:2.17}, Lemma
\ref{L2.5} follows. \ \ \ \carre

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Applications}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Symmetries of the Hamiltonian and its frequencies}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\n As already mentioned in the introduction, the NLS
Hamiltionian $\mathcal{H}$ is invariant under $\mathcal{S}_2.$
When $\mathcal{H}$ is expressed with respect to action
variables, $\mathcal{H} = \mathcal{H} (I)$, this invariance of
$\mathcal{H}$ leads to our first application.
 For $I = (I_k)_{k \in \mathcal{Z}},$ denote by $\mathcal{J}
(I)$ the sequence given by
%%%%%%%%%%%%%%%%%%
\[\mathcal{J} (I)_k := I_{-k} \qquad (\forall k \in \mathbb{Z}). \]

\n Denote by $\omega_j := \frac{\partial \mathcal{H}}{\partial
I_j}$ the j'th frequency.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prop}
\label{P3.1}
\n
\begin{itemize}
\item[(i)] $\mathcal{H} (\mathcal{J} (I)) = \mathcal{H} (I)$;
\item[(ii)] $\omega_j (I) = \omega_{-j} (I) \quad (\forall j \ge 1)$ for
$I$ with $\mathcal{J} (I) = I.$
\end{itemize}
\end{prop}

\n {\bf Proof} (i) follows from combining the two identities
$\mathcal{H} (\check{\varphi}) = \mathcal{H}(\varphi)$ and $I
(\check{\varphi}) = \mathcal{J} (I(\varphi))$
(Proposition \ref{P2.1}).\\
(ii) Write $\mathcal{H}$ as a function of 
$r_k := \frac{I_k + I_{-k}}{2}\quad (k \ge 0)$ and $\rho_k := \frac{I_k - I_{-k}}{2}
\quad (k \ge 1)$. By (i), $\mathcal{H}$ is an even function of the $\rho_k$'s and thus 
$\frac{\partial \mathcal{H}}{\partial {\rho_{k}}} = 0 $ at points where 
$\mathcal{J} (I) = I$.  \ \ \ \carre


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Symmetric phase spaces}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\n For $\alpha \in \mathbb{R},$ introduce the subspace
%%%%%%%%%%%%%%%%%%
\begin{equation}
\label{eq:3.1}
P_{\alpha} := \{\varphi \in L^2 (S^1;  \mathbb{C}) \mid e^{i \alpha}
\check{\varphi} = \varphi \}.
\end{equation}

\n Notice that for $\alpha = \pi$, respectively $\alpha = 0$,
$P_{\alpha} \cap C^{\infty}$ is the phase space 
consisting of elements $\varphi \in
C^{\infty}$ satisfying a generalized Dirichlet respectively Neumann
condition, i.e. for $\forall k \ge 0,$
%%%%%%%%%%%%%%%%%%
\[\partial^{2k}_x \varphi (0) = \partial^{2k}_x \varphi(1) = 0 \quad
(\mbox{Dirichlet}); \quad \partial^{2k+1}_x \varphi (0) = \partial^{2k+1}_x
\varphi (1) = 0 \quad (\mbox{Neumann}).
\]


\n Next, introduce the subspace
$$
Q_{\alpha} := \{\varphi \in L^2 (S^1;  \mathbb{C}) \mid e^{i \alpha}
\bar{\varphi} = \varphi \}.
$$
\n Notice that $ e^{i \alpha} \bar{\varphi} = \varphi $ means that $\varphi$ is of the form
$\varphi =  e^{i \alpha /2} f(x)$ with $f(x)$ a real valued function.



\n Proposition \ref{T3.1} provides a charaterization of
$P_{\alpha}$ and $Q_{\alpha}$ in terms of action-angle variables. Recall that $\Phi$
denotes the Birkhoff map,\; $\varphi =\Phi \left( (\sqrt{2 I_k} e^{i
\theta_k})_{k \in \mathbb{Z}} \right).$ 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prop}
\label{T3.1}
The following statements hold:

\vspd
$\begin{array}{ll}
(i)\qquad\ \ \ 
P_{ 0} =& \{\Phi \left( (\sqrt{2 I_k} e^{i
\theta_k})_{k \in \mathbb{Z}} \right) \mid I_k = I_{-k} \quad \forall k \ge1;\\ & 
\theta_k \equiv \theta_{-k}\ ({\rm mod }\ 2 \pi), \quad \forall k \ge 1
\mbox{ with }  I_k \neq 0   \};\\ & \\


(ii)\;
 For\; \alpha \not \equiv 0 &
\ ({\rm mod}\ 2 \pi), \\
\qquad \qquad P_{\alpha} =& \{ \Phi ((\sqrt{2 I_k} e^{i \theta_k})_{k \in
\mathbb{Z}}) \mid I_0 = 0;\ I_k = I_{-k}, 
\quad \forall k \ge 1;\\ &  \theta_k \equiv \theta_{-k} + \alpha \
({\rm mod }\ 2 \pi), \quad \forall k \ge 1 \mbox{ with } I_k \neq 0  \}
;\\ & \\



(iii)\quad \ \ \ 
Q_{\alpha} =
&\{\Phi ((\sqrt{2 I_k} e^{i \theta_k})_{k \in \mathbb{Z}})
\mid I_k = I_{-k} \quad \forall k \ge 1\\ & \theta_k \equiv - \theta_{-k} + \alpha
 \ ({\rm mod }
\ 2 \pi) \quad \forall k \ge 0 \mbox{ with } I_k \neq 0   \}.\end{array}$

\end{prop}

\n {\bf Proof} (i) Let us denote by $\tilde{P}_{0}$
the set on the right side of equality (i). If $\varphi \in
P_{ 0},$ Proposition \ref{P2.1} and \ref{P2.3} imply
$\varphi \in \tilde{P}_{0}.$ Conversely, if $\varphi \in
\tilde{P}_{0}$ then by Proposition \ref{P2.1},
\ref{P2.3}
$I_k (\check{\varphi}) = I_k (\varphi) \quad (\forall k \in
\mathbb{Z}) \ \mbox{and} \ \theta_k (\check{\varphi}) = \theta_k
(\varphi) \quad \forall k \in \mathbb{Z}$ with $I_k \neq 0$. Thus
$\check{\varphi} = \varphi$ since $\Phi$ is one to one.
\vspd

\n (ii) Let us denote $\tilde{P_{\alpha}}$ the set on the
right side of the equality (ii). If $\varphi \in
\tilde{P_{\alpha}}$ then, by Proposition \ref{P2.1},
\ref{P2.2}, \ref{P2.3}, $I_k (e^{i \alpha}\check{\varphi}) = I_k
(\varphi)\ (\forall k \in \mathbb{Z})$ and $\theta_k (e^{i \alpha}
\check{\varphi}) = \theta_k (\varphi)\ (\forall k \in \mathbb{Z}$ with
$I_k \neq 0)$ and thus $\varphi \in P_{\alpha}.$
Conversely, if $\varphi \in
P_{\alpha}$ we have $I_k (\varphi) = I_{-k} (\varphi) \quad (\forall k \in \mathbb{Z})$
and $\theta_k (\varphi)
\equiv \theta_{-k} (\varphi) + \alpha \ ({\rm mod }\ 2 \pi) \ \
(\forall k \in \mathbb{Z}$ with $I_k (\varphi) \neq 0).$ In
particular, if $I_{0} (\varphi) \neq 0,\ \theta_{0} (\varphi) \equiv
\theta_0 (\varphi) + \alpha \ ({\rm mod }\ 2 \pi)$. As $\alpha \not
\equiv 0 \ ({\rm mod }\ 2 \pi),$ it follows that $I_0 (\varphi) = 0$ and
thus $\varphi \in \tilde{P_{\alpha}}.$

\n The statement (iii) is proved in a similar
way. \ \ \ \carre

\vspd

\n Further, introduce the subspace
$E_{\alpha} := \{\varphi \in L^2 (S^1;  \mathbb{C}) \mid  e^{i \alpha}\check{\varphi} =
\bar{\varphi}\}$ and let $E :=E_{0}$. Notice that $E$    
is also given by
%%%%%%%%%%%%%%%%%%
\begin{equation}
\label{eq:3.2}
E = \{ \varphi \in L^2 (S^1;  \mathbb{C}) \mid \mbox{Re}  \varphi \
\mbox{is even;} \quad \mbox{Im} \varphi \ \mbox{is odd}\}
\end{equation}

\n and recall from \cite{gg} that
%%%%%%%%%%%%%%%%%%
\begin{equation}
\label{eq:3.3}
\begin{array}{lll}
E&=& \{\varphi \in L^2 (S^1;  \mathbb{C}) \, \mid \, spec_{per} L(\varphi)
= spec^+_{Dir} L (\varphi) \cup spec^-_{Dir} L(\varphi) \} \\
&=& \{ \varphi \in L^2 (S^1;  \mathbb{C}) \, \mid \, \forall k \in
\mathbb{Z}, \ \{\lambda^+_k (\varphi), \lambda^-_k (\varphi)\} = \{\mu_k
(\varphi), \nu_k (\varphi)\}\}.
\end{array}
\end{equation}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prop}
\label{P3.e} 

$$E_{\alpha} = \{ \Phi ((\sqrt{2I_k} e^{i \theta_k})_{k \in
\mathbb{Z}}) \mid \theta_k \equiv \alpha /2 \ ({\rm mod }\ \pi) \quad
\forall k \in \mathbb{Z} \mbox{ with } I_k \neq 0 \}.
$$
\end{prop}

\n {\bf Proof} The statement is proved in a similar way as Proposition \ref{T3.1}.
\ \ \ \ \carre

\vspd


\n {\bf Remark} In particular for $\alpha = 0,$  Proposition \ref{P3.e}, 
combined with (\ref{eq:3.3}), shows
that
$$
\begin{array}{l} \{\varphi \in L^2 (S^1;  \mathbb{C}) \, \mid \, spec_{per} L(\varphi)
= spec^+_{Dir} L (\varphi) \cup spec^-_{Dir} L(\varphi) \}
 =\\ \{ \Phi ((\sqrt{2I_k} e^{i \theta_k})_{k \in
\mathbb{Z}}) \mid \theta_k \equiv 0\ ({\rm mod }\ \pi) \quad
\forall k \in \mathbb{Z} \mbox{ with } I_k \neq 0 \}.
\end{array}
$$



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Symmetric spectrum} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\n In this subsection we present two spectral results.
Recall from \cite{gg} that the sequence of the gap lengths,
$(\gamma_k (\varphi))_{k \in \mathbb Z}  =  (\lambda^+_k (\varphi) - \lambda^-_k (\varphi))_{k
\in \mathbb{Z}},$ uniquely determines the periodic spectrum of
$L(\varphi).$ Similarly, the sequence of actions $(I_k (\varphi))_{k
\in \mathbb{Z}}$ determines uniquely the periodic spectrum of
$L(\varphi)$ (cf
 \cite{gk}, \cite{mv}, \cite{bbgk}). In Theorem \ref{T3.2}, and
\ref{T3.3} below, we prove that the periodic spectrum is
symmetric if and only if the sequence of the actions is
symmetric (Theorem \ref{T3.2}) or, if and only if the sequence
of the gap lengths is symmetric (Theorem \ref{T3.3}).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{teor}
\label{T3.2}
For any $\varphi \in L^2 (S^1;  \mathbb{C}),$ the following two
statements are equivalent:

\begin{itemize}
\item[(i)] $\lambda^{\pm}_k (\varphi) = - \lambda^{\mp}_{-k}
(\varphi) \ \ \forall k \in \mathbb{Z};$
\item[(ii)] $I_k (\varphi) = I_{-k} (\varphi) \ \  \forall k \in
\mathbb{Z}.$
\end{itemize}
\end{teor}



\n {\bf Proof} In view of the formula for the actions, (i)
implies (ii). Conversely, assume (ii) holds. 
As the Birkhoff map is bijective, there exists 
$\varphi_0 \in L^2 (S^1;  \mathbb{C})$
such that $I_k (\varphi_0) = I_k (\varphi) \ \forall k \in \mathbb{Z}
$ and $\theta_k (\varphi_0) \equiv \theta_{-k} (\varphi_0)\ ({\rm mod }\
2 \pi) \ \ (\forall k \in \mathbb{Z} \mbox{ with } I_k \neq 0 ).$ By Proposition \ref{T3.1}
(i), it follows that $\check{\varphi}_0 = \varphi_0.$ By Proposition
\ref{P1.1} (ii), this then implies that $spec_{per} L (\varphi_0)$
is symmetric. Since $\varphi_0$ and $\varphi$ are isospectral,
$spec_{per} L (\varphi)$ is symmetric. \ \ \ \carre

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{teor}
\label{T3.3}
For any $\varphi \in L^2 (S^1;  \mathbb{C})$, the following two
statements are equivalent:

\begin{itemize}
\item[(i)] $\lambda^{\pm}_k (\varphi) = - \lambda^{\mp}_{-k}
(\varphi), \ \ \forall k \in \mathbb{Z};$
\item[(ii)] $\gamma_k (\varphi) = \gamma_{-k} (\varphi), \ \ \forall
k \in \mathbb{Z}.$
\end{itemize}
\end{teor}

\n {\bf Proof} As we did not succeed to deduce the symmetry of
the sequence of gap lengths,
$\gamma_k = \gamma_{-k}\ (\forall k \in \mathbb{Z})$,
 from the symmetry of the sequence of actions, $ I_k = I_{-k} \quad (\forall k \in
\mathbb{Z})$,
 we use, instead of Birkhoff coordinates,
the coordinates defined in \cite{bggk}. In that paper, 
it is proved that there exist angles, $(\tilde{\theta}_k
(\varphi))_{k \in \mathbb{Z}} \in (\mathbb{R}/ \pi
\mathbb{Z})^{\infty},$ depending
continuously on $\varphi \in L^2 (S^1;  \mathbb{C})$ for $\varphi$
with $\gamma_k (\varphi) \neq 0$, such that the map 
$\tilde{\Phi} (\varphi) :=
(\tilde{\Phi} (\varphi))_{k \in \mathbb{Z}}
$
from $L^2 (S^1;  \mathbb{C})$ into $\ell^2 (\mathbb{Z};
\mathbb{R}^2),$ defined by $\tilde{\Phi}_k (\varphi)
= \frac{\gamma_k (\varphi)}{2} (\cos 2 \tilde{\theta}_k (\varphi),
\sin 2 \tilde{\theta}_k(\varphi)),$ is a homeomorphism. Arguing in the same way as in
the proof of Theorem~\ref{T3.2}, Theorem~\ref{T3.3} then follows from
Proposition~\ref{P3.B} below.
\quad \carre

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prop}
\label{P3.B}
For $\varphi \in L^2 (S^1; 
\mathbb{C})$ and $k \in \mathbb{Z},\ \tilde{\theta}_k
(\check{\varphi}) \equiv \tilde{\theta}_{-k} (\varphi)\ ({\rm mod }\ \pi)$.
\end{prop}

\n{\bf Proof} The statement is proved in Appendix A. \quad \carre 



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Additional symmetries of the spectrum} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Theorem~\ref{T3.3} suggests that the 0's gap plays a special role for symmetry
properties of the spectrum. Indeed, if we consider the action $\rho$
of $\mathbb Z$ on $L^2 (S^1;  \mathbb{C})$ given by 
$\rho(j)\varphi(x) = e^{-2\pi ijx}\varphi (x)$, one verifies easily (cf (\ref{eq:2.16}))
that the following statements are equivalent


\begin{itemize}
\item[(i)] $\lambda^{\pm}_{j+k} (\varphi) = - \lambda^{\mp}_{j-k}
(\varphi) +2j\pi, \ \ \forall k \in \mathbb{Z};$
\item[(ii)] $\gamma_{j+k} (\varphi) = \gamma_{j-k} (\varphi), \ \ \forall
k \in \mathbb{Z}.$
\end{itemize}





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\appendix
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\renewcommand{\thesection}{\Alph{section}}

\section{Appendix}

\n In this appendix, we prove Proposition~\ref{P3.B}. First, recall some
notations and results from \cite{gk} (cf also \cite{bggk}). Let $\varphi \in L^2 (S^1; 
\mathbb{C})$ and $k \in \mathbb{Z}$ with $\gamma_k (\varphi) \neq
0.$ Denote by $F^{\pm}_k (\cdot) = F^{\pm}_k (\cdot; \varphi) =
(F^{\pm}_{k,1}, F^{\pm}_{k,2})$ the eigenfunctions 
of norm 1 of the AKNS-operator $H(\varphi)$,
corresponding to the eigenvalues $\lambda^{\pm}_{k} (\varphi)$
normalized as follows,
%%%%%%%%%%%%%%%%%%
\[\begin{array}{ll}
F^{\pm}_{k,1} (0) > 0 & (\mbox{if} \ F^{\pm}_{k,1} (0) \neq
0)\\
F^{\pm}_{k,2} (0) > 0 & (\mbox{if} \ F^{\pm}_{k,1} (0)
= 0).
\end{array}  \]

\n Let $E_k (\varphi)$ be the two-dimensional subspace of $L^2
(S^1;  \mathbb{R}^2)$ generated by $F^+_k (\cdot; \varphi)$ and
$F^-_k (\cdot; \varphi)$ and denote by $G^+_k (\cdot, \varphi),
G^-_k (\cdot, \varphi)$ the basis of $E_k (\varphi)$ uniquely
determined by
%%%%%%%%%%%%%%%%%%
\begin{equation}
\label{eq:B.1}
\begin{array}{ll}
\mbox{(i)} & \parallel G^{\pm}_k (\cdot; \varphi)
\parallel_{L^2} = 1 ;\\
\mbox{(ii)} & G^-_{k,1} (0; \varphi) = 0 \ \mbox{and} \ G^-_{k,2}
(0; \varphi) > 0 ;\\
\mbox{(iii)} & \langle G^+_k (\cdot ; \varphi), G^-_k (\cdot ;
\varphi)\rangle_{L^2} = 0; \\
\mbox{(iv)} & G^+_{k,1} (0 ; \varphi) > 0.
\end{array}
\end{equation}

\n Let $\tilde{\theta}_k (\varphi) \in [0, 2 \pi)$ be the
counterclockwise oriented angle between $G^+_k (\varphi)$ and
$F^+_k (\varphi)$ and introduce $\tilde{\Phi}_k (\varphi) :=
(\tilde{\Phi}_{k,1} (\varphi), \tilde{\Phi}_{k,2} (\varphi))$ with
%%%%%%%%%%%%%%%%%%
\begin{equation}
\label{eq:B.2}
\tilde{\Phi}_{k,1} (\varphi) := \langle G^+_k (\cdot ; \varphi), (H(\varphi) -
\tau_k (\varphi)) G^+_k (\cdot, \varphi) \rangle
\end{equation}

\n and
%%%%%%%%%%%%%%%%%%
\begin{equation}
\label{eq:B.3}
\tilde{\Phi}_{k,2} (\varphi) := \langle G^+_k (\cdot; \varphi), (H (\varphi) -
\tau_k (\varphi)) G^-_k (\cdot, \varphi) \rangle,
\end{equation}

\n where $\tau_k (\varphi) := (\lambda^+_k (\varphi) + \lambda^-_k
(\varphi))/2.$ One verifies in a straightforward way that
%%%%%%%%%%%%%%%%%%
\begin{equation}
\label{eq:B.4}
\tilde \Phi_k (\varphi) = \frac{\gamma_k (\varphi)}{2} (\cos 2
\tilde{\theta}_k (\varphi), \sin 2 \tilde{\theta}_k (\varphi)).
\end{equation}

\n From \cite{bggk} we learn that $\tilde{\Phi} = (\tilde{\Phi}_k)_{k
\in \mathbb{Z}}$ is a homeomorphism from $L^2 (S^1; 
\mathbb{C})$ onto $\ell ^2 (\mathbb{Z}; \mathbb{R}^2).$


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{lemma}
\label{LB.1}
For $\varphi \in L^2 (S^1;  \mathbb{C})$ and $k \in \mathbb{Z}$
with $\gamma_k (- \check{\varphi}) \neq 0,$

\[\tilde{\theta}_k (- \check{\varphi}) \equiv \tilde{\theta}_{-k}
(\varphi) + \pi/2 \ ({\rm mod }\ \pi)
\]
\end{lemma}


\n {\bf Proof} By \eqref{eq:1.15} and Proposition \ref{P1.1}
%%%%%%%%%%%%%%%%%%
\begin{equation}
\label{eq:B.5}
\begin{array}{lll}
E_k (-\check{\varphi}) &= &span (F^+_{-k} (1 -x; \varphi), F^-_{-k} (1
-x; \varphi)) \\
&=& span (G^+_{-k} (1 - x; \varphi), G^-_{-k} (1 - x; \varphi)).
\end{array}
\end{equation}

\n Hence, as $G^{\pm}_{-k} (1; \varphi) = (-1)^k G^{\pm}_{-k} (0;
\varphi)$ (see \cite{bggk}), we obtain, using \eqref{eq:B.1},
%%%%%%%%%%%%%%%%%%
\begin{equation}
\label{eq:B.6}
G^{\pm}_k (x, - \check{\varphi}) = (-1)^k G^{\pm}_{-k} (1 - x;
\varphi).
\end{equation}

\n By \eqref{eq:B.2}, \eqref{eq:B.3} \eqref{eq:B.6} and
Proposition \ref{P1.1} one concludes that
%%%%%%%%%%%%%%%%%%
\begin{equation}
\label{eq:B.7}
\tilde \Phi_k (- \check{\varphi}) = - \tilde \Phi_{-k} (\varphi).
\end{equation}

\n In view of \eqref{eq:B.4} and \eqref{eq:B.7}, Lemma
\ref{LB.1} follows. \ \ \ \carre
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{lemma}
\label{LB.2}
For $\varphi \in L^2 (S^1;  \mathbb{C})$, $k \in \mathbb{Z}$ with
$\gamma_k (\varphi) \neq 0,$ and $\alpha \in \mathbb{R},$
%%%%%%%%%%%%%%%%%%
\[\tilde{\theta}_k (e^{i \alpha} \varphi) \equiv \theta_k (\varphi) +
\frac{\alpha}{2}\ ({\rm mod }\ \pi).
\]
\end{lemma}


\n {\bf Proof} Let $\tilde{V}_{\alpha}$ be the following
rotation matrix
%%%%%%%%%%%%%%%%%%
\begin{equation}
\label{eq:B.8}
\tilde{V}_{\alpha} := \bl{lr}
\cos \alpha / 2 & -\sin \alpha /2 \\
\sin \alpha /2 & \cos \alpha /2 \er.
\end{equation}

\n One has (cf \eqref{eq:1.2})
%%%%%%%%%%%%%%%%%%
\begin{equation}
\label{eq:B.9}
\tilde{V}_{\alpha}^{-1} H(e^{i \alpha} \varphi) \tilde{V}_{\alpha}
= H (\varphi)
\end{equation}

\n and therefore, with $\varepsilon_k = \pm 1,$
%%%%%%%%%%%%%%%%%%
\begin{equation}
\label{eq:B.10}
F^+_k (\cdot; e^{i \alpha} \varphi) = \varepsilon_k
\tilde{V}_{\alpha} F^+_k (\cdot; \varphi)
\end{equation}

\n and
%%%%%%%%%%%%%%%%%%
\begin{equation}
E_k (e^{i \alpha} \varphi) = \tilde{V}_{\alpha} E_k (\varphi) = E_k
(\varphi).
\end{equation}

\n As a consequence,
%%%%%%%%%%%%%%%%%%
\begin{equation}
\label{eq:B.12}
G^{\pm}_k (e^{i \alpha} \varphi) = G^{\pm}_k (\varphi).
\end{equation}

\n Using \eqref{eq:B.10}, \eqref{eq:B.12} and the definition of
the angle $\tilde{\theta}_k,$ one obtains
%%%%%%%%%%%%%%%%%%
\begin{equation}
\label{eq:B.13}
\tilde{\theta}_k (e^{i \alpha} \varphi) = \tilde{\theta}_k (\varphi)
+ \frac{\alpha}{2}\ ({\rm mod }\ \pi). \ \ \ \mbox{\carre}
\end{equation}



\n {\bf Proof} (Proposition~\ref{P3.B}). Combining Lemma \ref{LB.1} and
\ref{LB.2} (with $\alpha = \pi)$ Proposition~\ref{P3.B} follows. \ \ \
\carre


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\end{thebibliography}
%\setcounter{page}{20}

\end{document}



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