Content-Type: multipart/mixed; boundary="-------------0001170837465"

This is a multi-part message in MIME format.

---------------0001170837465
Content-Type: text/plain; name="00-22.keywords"
Content-Transfer-Encoding: 7bit
Content-Disposition: attachment; filename="00-22.keywords"

Periodic spectrum, Zakharov-Shabat system, Gap estimates, Riesz spaces, 
Lyapunov-Schmidt decomposition
---------------0001170837465
Content-Type: application/x-tex; name="gap2.tex"
Content-Transfer-Encoding: 7bit
Content-Disposition: inline; filename="gap2.tex"

  %
  % LaTeX file for "Kappeler"
  %
   
  \documentclass[12pt]{article} %nach [12pt, leqno] = Formeln links! 

  %\setlength{\textheight}{23.5cm}

  \usepackage{amsmath}
  \usepackage{amssymb}

  \def\ni{\noindent}
  \parindent = 0.0 cm

  \setcounter{section}{0}
  \addtocounter{section}{0}
  \setcounter{subsection}{0}    %Titel 
  \setcounter{subsubsection}{0} %Theorem etc.
 
  \newtheorem{guess}{Theorem}[section]
  \newtheorem{proposition}[guess]{Proposition}
  \newtheorem{lemma}[guess]{Lemma}
  \newtheorem{corollary}[guess]{Corollary}
  \newtheorem{definition}[guess]{Definition}
  \newtheorem{notation}[guess]{Notation}
  \newtheorem{remark}[guess]{Remark}

  \numberwithin{equation}{section}
  
  % \renewcommand{\theequation}{\thesection.\arabic{equation}}
  
  %=============================== begin document ==========================

   \begin{document}
  
   \title{Estimates on periodic and Dirichlet eigenvalues
          for the Zakharov-Shabat system}


   

    
\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{2cm}

   \begin{abstract} 

Consider the $2 \times 2$ first order system due to Zakharov-Shabat,
   \[ LY:= i \left( \begin{array}{c c}
                   1 &0 \\
                   0 &-1 
             \end{array} \right)
      Y' + \left( \begin{array}{c c}
                 0       &\psi _1 \\
                 \psi _2 &0
           \end{array} \right) Y = \lambda Y
   \]
with $\psi _1, \psi _2 $ being complex valued functions of period $one$
in the weighted Sobolev space 
$H^w \equiv H^w_{\mathbb C}.$
 Denote
by $spec(\psi _1,\psi _2)$ the set of periodic eigenvalues of $L
(\psi _1,\psi _2)$ with respect to the interval $[0,2]$ and by
$spec_{Dir}(\psi _1,\psi _2)$ the set of Dirichlet eigenvalues of
$L(\psi _1,\psi _2)$ when considered on the interval $[0,1]$. It is 
well known that $spec (\psi _1,\psi _2)$ and $spec _{Dir}(\psi _1,
\psi _2)$ are discrete.

{\bf Theorem } {\it Assume that $w$ is a weight such that, 
 for some $\delta > 0$, 
$w_{-\delta }(k) = (1 + |k|)^{-\delta }w(k)$ is a weight as well.
Then for any bounded subset ${\mathbb B}$ of
$1$-periodic elements in $H^w \times H^w$ there exist $N \geq 1$
and $M \geq 1$ so that for any $|k| \geq N$, and $(\psi _1,
\psi _2) \in {\mathbb B}$, the set $spec(\psi _1,\psi _2) \cap
\{ \lambda \in {\mathbb C} \mid | \lambda - k\pi | < \pi / 2\} $
contains exactly one isolated pair of eigenvalues $\{ \lambda ^+_k,
\lambda ^-_k\}$ and $spec _{Dir}(\psi _1,\psi _2) \cap \{ \lambda
\in {\mathbb C} \mid |\lambda - k\pi | < \frac {\pi }{2} \}$
contains a single Dirichlet eigenvalue $\mu _k$. These eigenvalues
satisfy the following estimates

\begin{description}
\item[(i)] $\sum _{|k| \geq N} w(2k)^2  |\lambda ^+_k - \lambda
           ^-_k|^2 \leq M$;

\item[(ii)] $\sum _{|k| \geq N} w(2k)^2 |\frac {(\lambda ^+_k +
            \lambda ^-_k)}{2} - \mu _k |^2 \leq M.$
\end{description}

\medskip
\ni
Furthermore $spec(\psi _1,\psi _2) \backslash \{ \lambda ^\pm _k, 
|k| \geq N\}$ and $spec_{Dir}(\psi _1,\psi _2) \backslash \{
\mu _k \mid |k| \geq N \}$ are contained in $\{ \lambda \in
{\mathbb C} \mid |\lambda | < N \pi - \pi / 2\} $ and its
cardinality is $4N - 2$, respectively $2N - 1$.
}

\medskip
\ni
When $\psi _2 = \overline {\psi }_1$ (respectively $\psi _2 = -
\overline {\psi }_1), L(\psi _1, \psi _2)$ is one of the 
operators in the Lax pair for the defocusing (resp. focusing)
nonlinear Schr\"odinger equation.
   

   \end{abstract}
 
   \clearpage

   \tableofcontents

   \clearpage

   \section{Introduction}
   \label{S:introduction}

\subsection{Results}
\label{Ss:1.1 results}

Consider the Zakharov-Shabat operator (see \cite{ZS})
   \[ L(\psi _1,\psi _2):= i \left( \begin{array}{c c}
                   1 &0 \\
                   0 &-1 
             \end{array} \right)
      \frac {d}{dx} + \left( \begin{array}{c c}
                 0       &\psi _1 \\
                 \psi _2 &0
           \end{array} \right) 
   \] 
where $\psi _1, \psi _2$ are {\it $1$-periodic} elements in the 
weighted Sobolev space $H^w \equiv H^w_{\mathbb C}$ of $2$-periodic
functions
   \[ H^w:= \{ f(x) = \sum ^\infty _{-\infty } \hat {f} (k)e^{i\pi
      kx} \mid \| f\| _w < \infty \}
   \]
with
   \[ \| f\| _w:= (2 \sum _{k\in {\mathbb Z}} w(k)^2 |\hat {f}(k)|
      ^2)^{1/2}
   \]
and $w = (w(k))_{k\in {\mathbb Z}}$ a weight, i.e. a sequence of
positive numbers with $w(k) \geq 1,\, w(-k) = w(k) \, (\forall k \in
{\mathbb Z})$ and the following submultiplicative property
   \[ w(k) \leq w(k - j)w(j) \quad \forall k,j \in {\mathbb Z}.
   \]
As an example of such a weight we mention the Sobolev weights $s_N
\equiv (s_n(k))_{k\in {\mathbb Z}}, s_N(k):= \langle k\rangle ^N$,
where, for convenience,
   \[ \langle k\rangle := 1 + |k| ,
   \]
or more generally, the Abel-Sobolev weight $w_{a,b} \equiv (w_{a,b}
(k))_{k\in {\mathbb Z}}$
   \[ w_{a,b}(k):= \langle k\rangle ^a e^{b|k|} \quad (a \geq 0;
      b \geq 0) .
   \]
An element $\psi \in H^{w_{a,b}}$ is a complex valued function
$f(x) = \sum _{k\in {\mathbb Z}} \hat {f}(k) e^{i\pi kx}$,
which admits an analytic extension $f(x + iy)$ to the strip $|y|
< \frac {b}{\pi }$ such that $f(x + i\frac {b}{\pi })$ and $f(x
- i\frac {b}{\pi })$ are both in the Sobolev space $H^a_{\mathbb
C} \equiv H^a(\mathcal S^1;{\mathbb C})$. Denote by $spec(\psi _1,\psi _2)$
the periodic spectrum of $L(\psi _1,\psi _2)$ when considered
on the interval $[0,2]$ and by $spec_{Dir}(\psi _1,\psi _2)$ the
Dirichlet spectrum of $L(\psi _1,\psi _2)$ when considered on
$[0,1]$. It is well known that both, $spec(\psi _1,\psi _2)$ and
$spec _{Dir}(\psi _1,\psi _2)$ are discrete.

The main purpose of this paper is to study the asymptotics of
the large (in absolute value) eigenvalues in $spec(\psi _1,
\psi _2)$ and $spec_{Dir}(\psi _1,\psi _2)$ for $1$-periodic
functions $\psi _1,\psi _2$ in $H^w$. To formulate our first
result we need to introduce some more notation: we say that $w$
is a $\delta $-weight for $\delta > 0$ if 
   \[ w_\ast (k):= \langle k\rangle ^{-\delta } w(k)
   \]
is a weight as well. Notice that the Abel-Sobolev weight $w_{a,b}$
is a $\delta $-weight iff $0 < \delta \leq a$. Let
   \[ \delta _\ast := \delta \wedge \frac {1}{2} \left(= \inf (\delta ,
      \frac {1}{2})\right).
   \]
Further let
   \[ \rho _n:= \left( (\hat {\psi }_2(2n) + \beta ^+_0(n))
      (\hat {\psi }_1 (-2n) + \beta ^-_0(n))\right) ^{1/2}
   \]
with an arbitrary, but fixed choice of the square root and 
   \begin{align*} &\beta ^+_0(n):= \sum _{k,j\not= n} \frac {
                   \hat {\psi }_2(k+n)}{(k-n)\pi } \frac {
                   \hat {\psi }_1(-k-j)}{(j-n)\pi } \hat {\psi }
                   _2(j + n) \\
                  &\beta ^-_0(n):= \sum _{k,j\not= n} \frac {
                   \hat {\psi }_1(-k-n)}{(k-n)\pi } \frac {
                   \hat {\psi }_2(k+j)}{(j-n)\pi } \hat {\psi }
                   _1(-j - n) 
   \end{align*}
\medskip

The first result concerns the periodic eigenvalues (cf. 
section~\ref{S: 2 Periodic eigenvalues}).

\medskip

\begin{guess}
\label{Theorem1} Let $M \geq 1, \delta > 0$ and $w$ a 
$\delta $-weight. Then there exist constants $1 \leq C < \infty $
and $1 \leq N < \infty $ so that the following statements hold:

For any $|n| \geq N$ and any $1$-periodic functions $\psi _1,
\psi _2 \in H^w$ with $\| \psi _j \| _w \leq M$, the set
   $spec (\psi _1,\psi _2) \cap \{ \lambda \in {\mathbb C} \mid
      |\lambda - n\pi | < \frac {\pi }{2} \}
   $
contains exactly one isolated pair of eigenvalues $\{ \lambda ^+_k,
\lambda ^-_k\} $. These eigenvalues satisfy

\medskip
\ni 
(i) $\sum _{|n|\geq N} w(2n)^2 |\lambda ^+_n - \lambda ^-_n|^2 \leq
C$;

\medskip
\ni
(ii) $\sum _{|n| \geq N} \langle n\rangle ^{3\delta _\ast } w(2n)^2
\min _\pm |(\lambda ^+_n - \lambda ^-_n) \pm 2 \rho _n|^2 \leq C$;

\medskip
\ni
(iii) $spec(\psi _1,\psi _2) \backslash \{ \lambda ^\pm _n \mid
|n| \geq N\} $ is contained in $\{ \lambda \in {\mathbb C} \mid
|\lambda | < N \pi - \frac {\pi }{2}\} $ and its cardinality is
$4N - 2$. 
\end{guess}

\bigskip

\begin{guess}
\label{Theorem2} Let $M \geq 1, \delta > 0$ and $w$ be a 
$\delta $-weight. Then there exist constants $1 \leq C < \infty $
and $N \leq N' < \infty $ (with $N$ given by Theorem~\ref{Theorem1}) 
so that the following statements hold:

For any $|n| \geq N'$ and any $1$-periodic functions $\psi _1,\psi
_2 \in H^w$ with $\| \psi _j\| _w \leq M$, the set
$spec_{Dir}(\psi _1,\psi _2) \cap \{ \lambda \in {\mathbb C}
      \mid |\lambda - n\pi | < \frac{\pi }{2} \}$ 
contains exactly one eigenvalue denoted by $\mu _n$. These 
eigenvalues satisfy:

\medskip
\ni
(i) $\sum _{|n|\geq N'} w(2n)^2 |\lambda _n - \lambda ^+_n|^2 \leq
C$;

\medskip
\ni
(ii) $spec_{Dir}(\psi _1,\psi _2) \backslash \{ \mu _n \mid |n|
\geq N'\} $ is contained in $\{ \lambda \in {\mathbb C} \mid
|\lambda | < N' \pi - \frac {\pi }{2}\} $ and its cardinality is
$2N' - 1$.
\end{guess}

\medskip

Statement (iii) in Theorem~\ref{Theorem1} and (ii) in 
Theorem~\ref{Theorem2} are obtained in a standard way. For the
convenience of the reader we prove it in Appendix A.

In section~\ref{S: 3 Riesz' spaces and normal form of $L$}, 
we consider the Riesz spaces $E_n$, i.e. the
images of the Riesz projectors associated to $L(\psi _1,\psi _2)$
for a small circle around $n\pi $ with $|n|$ sufficiently
large. We analyze the restriction of $L - \lambda ^+_n$ to
$E_n$ and study the asymptotic properties of eigenfunctions in
$E_n$ for $|n| \rightarrow \infty $.

\vskip 1 cm

\subsection{Comments}
\label{Ss:1.2 Comments}

{\bf Operator $L(\psi _1,\psi _2)$:} The Zakharov-Shabat operator 
occurs in the Lax pair representation $\frac {dM_\pm }{dt} = [M_
\pm ,A_\pm ]$ of the focusing $(NLS_-)$ and defocusing $(NLS_+)$
nonlinear Schr\"odinger equation
   \[ i\partial _t \varphi = - \partial ^2_x \pm 2|\varphi | ^2
      \varphi ,
   \]
   \[ M_+ := L(\varphi ,\overline {\varphi }) \ ; \quad M_- :=
      L(\varphi ,- \overline {\varphi })
   \]
(whereas the operators $A_\pm $ are rather complicated third 
order operators, given in \cite{FT}). One can show that $spec \
L(\varphi ,\overline {\varphi })$ respectively $spec \ L(\varphi ,
- \overline {\varphi })$ is a complete set of conserved quantities
for $NLS_+$ respectively $NLS_-$. We mention that $L(\psi _1,
\psi _2)$ is unitarily equivalent to the $AKNS$ operator (see 
\cite{AKNS}, \cite{MA})
   \[ L_{AKNS}:= \left( \begin{array}{c c}
                 0 &-1 \\
                 1 &0 
             \end{array} \right)
      \frac {d}{dx} + \left( \begin{array}{c c}
              -q  &p \\
              p   &q
           \end{array} \right) 
   \]
where
   \[ \psi _1:= - q + ip \ ; \quad \psi _2 = -q - ip .
   \]
Hence the selfadjoint operator $M_+$ corresponds to an operator
$L_{AKNS}$ with the functions $q,p$ being {\it real valued}.

\medskip
\ni
{\bf Selfadjoint case:} We emphasize that Theorem~\ref{Theorem1}
and Theorem~\ref{Theorem2} do not require $L(\psi _1,\psi _2)$
be selfadjoint. However, in the selfadjoint case, the decay rate
of the asymptotics in Theorem~\ref{Theorem1} (ii) can be improved
from $3\delta _\ast $ to $4\delta _\ast $,
   \[ \sum _{|n|\geq N} \langle n\rangle ^{4\delta _\ast } w(2n)^2
      \min _\pm |(\lambda ^+_n - \lambda ^-_n) \pm 2 \rho _n|^2
      \leq M . 
   \]
(This is proved in section~\ref{Ss: 2.9 Improvement of Theorem 1}).

\medskip
\ni
{\bf $L^2$-case:} Theorem~\ref{Theorem1} (i) and 
Theorem~\ref{Theorem2} (i) no longer hold for $H^w = L^2$ (i.e.
$w(k) = 1 \ \forall k \in {\mathbb Z})$ as the number $N$ in
Theorem~\ref{Theorem1} cannot be chosen uniformly for $1$-periodc
functions $\psi _1,\psi _2 \in L^2$ in a $L^2$-bounded set. This
can be easily deduced from the examples considered by Li -
McLaughlin \cite{LM} : Assume that Theorem~\ref{Theorem1} (i)
holds for $L^2$. Given $M > 0$, choose $N$ as in 
Theorem~\ref{Theorem1} and $\psi _1,\psi _2 \in L^2$ with $\|
\psi _j \| \equiv \| \psi _j\| _{L^2} = M$. Define $(\psi
_{1,k}, \psi _{2,k}) = (e^{2\pi ikx} \psi _1, e^{-2\pi ikx}
\psi _2) \ (k \in {\mathbb Z}$). Then $\| \psi _{j,k}\| _{L^2} =
\| \psi _j\| _{L^2} \ (\forall k)$ and, for $n \geq N, k \geq 0$
   \[ \lambda ^\pm _{n+k} (\psi _{1,k} , \psi _{2,k}) = \lambda
      ^\pm _n (\psi _1,\psi _2) + k\pi 
   \]
which leads for appropriate choices of $\psi _1,\psi _2$ to a
contradiction. For $L$ selfadjoint, a {\it local version} of
Theorem~\ref{Theorem1} and Theorem~\ref{Theorem2} have been
established, using different methods, in \cite{GG}. Most likely,
the analysis presented in this paper can be used
to obtain a local version of Theorem~\ref{Theorem1} (i) and
Theorem~\ref{Theorem2} (i) for $L$ arbitrary.

\medskip
\ni
{\bf Submultiplicative property of weights:} Notice that the
requirement of a weight to be submultiplicative excludes weights
of super-exponential growth $\exp (a|k|^{\alpha})$ with $\alpha >1$.
Most likely, the conclusions of Theorem~\ref{Theorem1}
and Theorem~\ref{Theorem2} do not hold for such weights (cf. \cite{KM}
for the case of Schr\"odinger operators).

\medskip
\ni
{\bf Boundary conditions:} Similarly as in \cite{KM} the method
for proving Theorem~\ref{Theorem2} can be applied to a whole
class of boundary conditions (cf. section~\ref{S: 4 Dirichlet 
eigenvalues} in \cite{KM} where
this class has been described for the Schr\"odinger operator
$- \frac {d^2}{dx^2} + V$).

\medskip
\ni
{\bf Smoothness vs. decay of gap length:} For selfadjoint
Zakharov-Shabat operators $L(\psi ,\overline {\psi })$,
Theorem~\ref{Theorem1} has a partial inverse. In this case,
the eigenvalues $(\lambda ^\pm _n)_{n\in {\mathbb Z}} = spec \
L(\psi ,\overline {\psi })$ are real and can be ordered such
that 
   \[ \ldots \leq \lambda ^+_{n-1} < \lambda ^-_n \leq \lambda ^+
      _n < \lambda ^-_{n+1} \leq \ldots  \ ; \quad \lambda ^\pm
      _n = n\pi + o(1) .
   \]
Given a weight $w$ and $K \geq 0$, denote by $w_K$ the weight
$w_K(n):= \langle n\rangle ^K w(n)$.

\medskip

\begin{proposition}
\label{Proposition1} Let $w$ be a $\delta $-weight for some 
$\delta > 0, K \geq 0$ and $\varphi \in H^w$. Then $\varphi \in
H^{w_K}$ iff
   \[ \sum _{n\in {\mathbb Z}} \langle n\rangle ^{2K} |\lambda ^+
      _n - \lambda ^-_n|^2 < \infty 
   \]
where $\lambda ^\pm _n \equiv \lambda ^\pm _n(\varphi ,
\overline {\varphi })$. 
\end{proposition}

\medskip

In the non selfadjoint case, the smoothness is not characterized
by properties of the periodic spectrum alone (cf. \cite{ST} for
an analysis in the case of Schr\"odinger operators).

\vskip 1 cm

\subsection{Method of proof}
\label{Ss: 1.3 Method of proof}

Typically, asymptotic estimates on the gap's lengths $(\lambda
^+_k - \lambda ^-_k)_{k\in{\mathbb Z}}$ of 
\par
\ni
$spec(L(\psi _1,
\psi _2))$ are obtained from asymptotic expansions of the 
eigenvalues $\lambda ^\pm _k = k\pi + \frac {c_{-1}}{k} + 
\ldots $ (cf. e.g. \cite{Ma}). This approach, however does
not allow to obtain the results of Theorem~\ref{Theorem1} and
Theorem~\ref{Theorem2} for weights with exponential decay such
as the Abel-Sobolev weight. The new feature in the proof of
our results is to use as in \cite{KM} a Lyapunov-Schmidt type
decomposition described in detail in 
section~\ref{Ss: 2.1 Lyapunov-Schmidt decomposition}.

\vskip 1 cm

\subsection{Related work}
\label{Ss: 1.4 Related work}

Similar results as the ones presented here for the
Zakharov-Shabat operator $L(\psi _1,\psi _2)$ have been obtained
previously for the Schr\"odinger operator $-\frac {d^2}{dx^2} +
V$ in \cite{KM}. In this paper we document that the same
methods, with adjustments, can be applied to $L$. At first
sight this is astonishing, as, unlike in the case of the 
Schr\"odinger operator, the distance between adjacent pairs
of eigenvalues $(\lambda ^+_n,\lambda ^-_n)$ and $(\lambda ^+
_{n+1}, \lambda ^-_{n+1})$ does {\it not} get unbounded for
$|n| \rightarrow \infty $, a fact which was used in an 
essential way in \cite{KM}. We explain in 
section~\ref{Ss: 2.1 Lyapunov-Schmidt decomposition}
how this problem for $L$ can be overcome.

A weaker version of Theorem~\ref{Theorem1} has been reported
in \cite{GKM} (cf. also \cite{GK}).

For Sobolev weights, the asymptotics of the eigenvalues 
$\lambda ^\pm _n$ and hence of the gap length $\gamma _n:= 
\lambda ^+_n - \lambda ^-_n$ have been obtained in the 
selfadjoint case by Marchenko \cite{Ma} (cf. also \cite{GG},
\cite{Gre}, \cite{Mis}, \cite{LS}). In the non selfadjoint
case only a few results have been known so far (see 
\cite{LM}, \cite{Ta1}, \cite{Ta2}).



\vskip 2 cm

\section{Periodic eigenvalues}
\label{S: 2 Periodic eigenvalues}

\subsection{Lyapunov-Schmidt decomposition}
\label{Ss: 2.1 Lyapunov-Schmidt decomposition}

Consider the Zakharov-Shabat operator 
   \[ L(\psi _1,\psi _2):= i \left( \begin{array}{c c}
                   1 &0 \\
                   0 &-1 
             \end{array} \right)
      \frac {d}{dx} + \left( \begin{array}{c c}
                 0       &\psi _1 \\
                 \psi _2 &0
           \end{array} \right) 
   \] 
where $\psi _1$ and $\psi _2$ are in $H^w$. For $\psi _1 = \psi _2
= 0$, the periodic eigenvalues are given by $\{ \lambda ^+_k,
\lambda ^-_k \mid k \in {\mathbb Z}\} $ with $\lambda ^+_k =
\lambda ^-_k = k\pi $ and an orthonormal basis of corresponding
eigenfunctions in $L^2[0,2] \times L^2[0,2]$ are given by
   \begin{equation} 
   \label{2.1}      e^+_k(x) = \frac {1}{\sqrt{2}} \left(
                    \begin{array}{c} 0 \\ 1 \end{array} 
                    \right) e^{ik\pi x} , \ 
                    e^-_k(x) = \frac {1}{\sqrt{2}} \left(
                    \begin{array}{c} 1 \\ 0 \end{array} 
                    \right) e^{-ik\pi x} .
   \end{equation} 
Considering the multiplication operator $\left( \begin{array}{c c}
0 &\psi _1 \\ \psi _2 &0 \end{array} \right) $ as a perturbation
of the Dirac operator $i \left( \begin{array}{c c}
1 &0 \\ 0 &-1 \end{array} \right) \frac {d}{dx}$ we will see that
for $k$ sufficiently large $L$ has a pair of eigenvalues near
$k\pi $, isolated from the remaining part of the spectrum of 
$L$. Our aim is to obtain an estimate for the distance between
the two eigenvalues and to compare the eigenvalues and
corresponding eigenfunctions (or root vectors) with the 
corresponding ones for $\psi _1 = \psi _2 = 0$. 

We express the eigenvalue equation
   \begin{equation} 
   \label{2.2}    LF = \lambda F
   \end{equation}
in the basis $e^+_k,e^-_k(k \in {\mathbb Z})$ defined in 
\eqref{2.1}: Given $F$ in the Sobolev space $H^1$, write
   \begin{equation}
   \label{2.3} F(x) = \sum _{k\in {\mathbb Z}} \hat {F}_2(k)
               e^+_k(x) + \hat {F}_1(-k)e^-_k(x)
   \end{equation}
and
   \begin{equation}
   \label{2.4} \psi _1(x) = \sum _{k\in {\mathbb Z}} \hat {\psi }
               _1 (k) e^{ik\pi x} \ ; \ \psi _2(x) = \sum _{k \in
               {\mathbb Z}} \hat {\psi }_2(k)e^{ik\pi x} .
   \end{equation}
Substituting \eqref{2.3} - \eqref{2.4} into \eqref{2.2} leads to
   \begin{align}
   \begin{split}
   \label{2.5} LF(x) &= \sum _{k\in {\mathbb Z}} k\pi \left(
                 \hat {F}_2(k)e^+_k(x) + \hat {F}_1(-k)e^-_k(x)
                 \right) \\
               &+ \sum _{k,j \in {\mathbb Z}} \hat {\psi }_1 (-k
                 -j) \hat {F}_2(j)e^-_k(x) + \hat {\psi }_2(k
                 + j) \hat {F}_1(-j)e^+_k(x) .
   \end{split}
   \end{align} 
Hence $\lambda $ is a periodic eigenvalue of $L(\psi _1,\psi _2)$,
when considered on the interval $[0,2]$, iff there exists $(\hat
{F}_1,\hat {F}_2) \in \ell ^2 \times \ell ^2$ with $(\hat {F}_1,
\hat {F}_2) \not= (0,0)$ such that, for all $k \in {\mathbb Z}$,
   \begin{align}
   \label{2.6} &(k\pi - \lambda )\hat {F}_2(k) + \sum _{j \in
                  {\mathbb Z}} \hat {\psi }_2(k + j)\hat {F}_1
                  (-j) = 0 \\
   \label{2.7} &(k\pi - \lambda )\hat {F}_1(-k) + \sum _{j \in
                  {\mathbb Z}} \hat {\psi }_1(- k - j)\hat {F}_2
                  (j) = 0 .
   \end{align}
Here $\ell ^2 \equiv \ell ^2({\mathbb Z};{\mathbb C})$ denotes 
the Hilbert space of complex valued $\ell ^2$-sequences $(a(k))
_{k\in {\mathbb Z}}$.
In order to solve equations \eqref{2.6} - \eqref{2.7} we
consider a Lyapunov-Schmidt type decomposition. For $n \in
{\mathbb Z}$ fixed, we look for eigenvalues near $n\pi , \lambda
= n\pi + z$, with $|z| \leq \frac {\pi }{2}$. The linear system
\eqref{2.6} - \eqref{2.7} is then decomposed into a two
dimensional system consisting of \eqref{2.6} - \eqref{2.7}
with $k = n$, referred to as the ${\mathcal Q}$-equation,
and an infinite dimensional system consisting of \eqref{2.6}
- \eqref{2.7} with $k \in {\mathbb Z} \backslash \{ n\} $, 
referred to as the ${\mathcal P}$-equation.

First we introduce some more notation. For $K \in {\mathbb Z}$
and a weight $w$ denote by $\ell ^2_w(K)$ the complex Hilbert
space $\ell ^2_w(K) \equiv \ell ^2_w(K,{\mathbb C})$,
   \[ \ell ^2_w(K):= \{ (a(k))_{k\in K} \mid \| a\| _w <
      \infty \}
   \]
where $\| a\| _w = (a,a)^{1/2}_w$ and, for $a,b \in \ell ^2_w$,
   \[ (a,b)_w:= \sum _{k \in K} w(k)^2 \overline {a(k)} b(k).
   \]
Most frequently, we will use for $K$ the set ${\mathbb Z}$ or
${\mathbb Z} \backslash n \equiv {\mathbb Z} \backslash \{ n
\} $. If necessary for clarity, we write $a_K$ for a sequence
$(a(k))_{k\in K} \in \ell ^2_w(K)$. 

For a linear operator $A : \ell ^2_{w_1}(K_1) \rightarrow \ell
^2_{w_2}(K_2)$ we denote by $A(k,j)$ its matrix elements,
   \[ (Aa)(k):= \sum _{j \in K_1} A(k,j)a(j) \quad (k \in K_2).
   \]
Further we introduce the shift operator $S$ and an
involution operator ${\mathcal J}$
   \begin{align*} &S : \ell ^2({\mathbb Z} \rightarrow
                     \ell ^2({\mathbb Z}) , \ (Sa)(k)
                     := a(k + 1) \quad \forall k \in 
                     {\mathbb Z} . \\
                 &{ J} : \ell ^2({\mathbb Z} \rightarrow
                     \ell ^2({\mathbb Z}) , \ ({\mathcal J}a)(k)
                     := a(-k) \quad \forall k \in 
                     {\mathbb Z} .
   \end{align*}
The restriction of $S$ to $\ell ^2_w(K)$ with values in $\ell ^2_
{S^nw}(K)$ is again denoted by $S$ and $S^n:= S \circ
\ldots \circ S$ denotes the $n'$th iterate of $S$. Notice that
   \[ \| S^na\| ^2_{\ell ^2_{S^nw}(K)} = \sum _{k \in K} w(k + n)
      ^2 |a(k + n)|^2 \leq \| a\| ^2_{\ell ^2_w({\mathbb Z})} .
   \]
For $(\hat {F}_2,\hat {F}_1) \in \ell ^2 \times \ell ^2$, write
   \begin{align*} &\hat {F}_2 = (x^F,\breve {F}_2) , \ x^F:=
                     \hat {F}_2(n) ; \quad \breve {F}_2:= 
                     (\hat {F}_2(k))_{k \in {\mathbb Z}\backslash
                     n} \\
                  &\hat {F}_1 = (y^F,J\breve {F}_1) , \ y^F:=
                     \hat {F}_1(-n) ; \quad \breve {F}_1:= 
                     (\hat {F}_1(k))_{k \in {\mathbb Z}\backslash
                     n} .
   \end{align*}
Using the above introduced notation, the equations \eqref{2.6} -
\eqref{2.7} read as follows:
   \begin{align}
   \label{2.8} - zx^F + \hat {\psi }_2(2n)y^F &+ \langle S^n
                  \hat {\psi }_2 , J\breve {F}_1 \rangle = 0 \\
   \label{2.9} \hat {\psi }_1(-2n)x^F - zy^F &+ \langle S^n J
                  \hat {\psi }_1 , \breve {F}_2 \rangle = 0 
   \end{align}
and 
   \begin{equation}
   \label{2.10}  \left( \begin{array}{ c}
             y^F(S^n \hat {\psi }_2)_{{\mathbb Z}\backslash n} \\
             x^F(S^n J\hat {\psi }_1)_{{\mathbb Z}\backslash n} 
             \end{array} \right)
      + (A_n - z)  \left( \begin{array}{ c}
                   \breve {F}_2 \\
                   J\breve {F}_1
           \end{array} \right) = 0 .
   \end{equation}
The equations \eqref{2.8} - \eqref{2.9} together form the 
${\mathcal Q}$-equation and \eqref{2.10} is the ${\mathcal 
P}$-equation. The operator $A_n$ is given by 
   \[ A_n = \left( \begin{array}{c c}
            \left( (k-n)\pi \delta _{kj}\right)_{k,j \in 
                 {\mathbb Z}\backslash n}  &\left( \hat {\psi }
                 _2(k + j)\right) _{k,j \in {\mathbb Z}
                 \backslash n} \\
            \left( (J\hat {\psi }_1(k + j)\right) _{k,j \in
                 {\mathbb Z} \backslash n}  &\left( (k - n)
                 \pi \delta _{kj}\right) _{k,j \in
                 {\mathbb Z} \backslash n} 
             \end{array} \right)
   \]
and $\langle \cdot ,\cdot \rangle \equiv \langle \cdot ,\cdot
\rangle _{{\mathbb Z} \backslash n}$ is defined by (no complex
conjugation)
   \[ \langle \left( \begin{array}{c}
              a_n \\
              b_n
              \end{array} \right) ,
              \left( \begin{array}{c}
              c_n \\
              d_n
              \end{array} \right) \rangle := \sum _{k\in 
      {\mathbb Z} \backslash n} \left( a_n(k) c_n(k) + b_n(k)d_n
      (k) \right) .
   \]
For $\psi _1 = \psi _2 = 0$ and $|z| \leq \frac {\pi }{2}$, the
operator $(z - A_n)$ is invertible as $(k \pi - (n\pi - z)) \not= 0$
for $k \not= n$. By a perturbation argument we will show that $(z
- A_n)$ can be inverted for $|z| \leq \frac {\pi }{2}$ and $|n|$
sufficiently large which then allows to solve the ${\mathcal
P}$-equation \eqref{2.10} for $(\breve {F}_2, J\breve {F}_1)$
for any $x^F,y^F \in {\mathbb C}$. This solution is substituted
into \eqref{2.8} - \eqref{2.9} which leads to a homogeneous
linear system of two equations for $x^F$ and $y^F$ with 
coefficients which depend on the parameter $z$. Hence $\lambda =
n\pi + z$ is a periodic eigenvalue of $L(\psi _1,\psi _2)$ iff
the corresponding determinant is equal to $0$. The nature of
the latter equation allows to obtain asymptotics for the
difference $\lambda ^+_n - \lambda ^-_n$ without having to
compute the asymptotics of $\lambda ^+_n$ and $\lambda ^-_n$
(cf. section~\ref{Ss: 2.6 $z$-equation} - \ref{Ss: 2.7 $zeta $-equation}).

\vskip 1 cm

\subsection{$P$-equation}
\label{Ss: 2.2 $P$-equation}

Let us first introduce some more notation. Denote by $\Delta _n$
the diagonal part of $A_n$ 
  \[ \Delta _n:=  \left( \begin{array}{c c}
                  D_n &0 \\
                  0 &D_n 
             \end{array} \right) ; \ 
     D_n:= \left( (k - n)\pi \delta _{kj}\right) _{k,j \in
     {\mathbb Z} \backslash n}
   \]
and set
   \[ B_n:= A_n - \Delta _n .
   \]
Notice that for $|z| \leq \frac {\pi }{2}, (z - \Delta _n)^{-1}$
is invertible. Hence we may introduce 
   \begin{equation}
   \label{2.11} T_n \equiv T_{n,z}:= B_n(z - \Delta _n)^{-1} =
                \left( \begin{array}{c c}
                0  &R^{(2)}_n \\
                R^{(1)}_n &0 
                \end{array} \right)
   \end{equation}
where $R^{(j)}_n \equiv R^{(j)}_{n,z} : \ell ^2({\mathbb Z}
\backslash n) \rightarrow \ell ^2({\mathbb Z} \backslash n)$
are defined by
   \begin{equation}
   \label{2.12} R^{(1)}_n(a):= J(\hat {\psi }_1 \ast (z - D_n)
                ^{-1})a ; \ R^{(2)}_n(a):= \hat {\psi } _2
                \ast J(z - D_n)^{-1}a .
   \end{equation}
$R^{(1)}_n$ and $R^{(2)}_n$ have the following matrix
representations
   \begin{equation}
   \label{2.13} R^{(1)}_n(k,j):= \frac {\hat {\psi }_1(-k-j)}
                {z - (j - n)\pi } ; \ R^{(2)}_n(k,j):= \frac
                {\hat {\psi }_2(k + j)}{z - (j - n)\pi } \
                (k,j \in {\mathbb Z} \backslash n) .
   \end{equation}
Formally, for any $x^F, y^F \in {\mathbb C}$, the ${\mathcal 
P}$-equation \eqref{2.10} can be solved 
   \[ \left( \begin{array}{c}
             \breve {F}_2 \\
             J\breve {F}_1
             \end{array} \right)
      = (z - A_n)^{-1} \
      \left( \begin{array}{c}
             y^F S^n \hat {\psi }_2 \\
             x^F S^n J \hat {\psi }_1
             \end{array} \right)
   \]
with
   \begin{equation}
   \label{2.14} (z - A_n)^{-1} = (z - \Delta _n)^{-1} (Id - T_n)
                ^{-1} .
   \end{equation}
To justify the formal considerations above it is to show that $(Id
- T_n)$ is invertible. Unfortunately, the norm $\| T_n\| $ of $T_n$
in ${\mathcal L}(\ell ^2_{S^nw})$ (with $\ell ^2_{S^nw} \equiv
\ell ^2_{S^nw} ({\mathbb Z} \backslash n; {\mathbb C}^2)$) does 
not become small as $|n| \rightarrow \infty $. However, it turns
out that, assuming an additional condition on the weight, the 
norm of $T^2_n$ is small for $|n| \rightarrow \infty $. The
invertibility of $(Id - T_n)$ then follows from the identity 
   \begin{equation} 
   \label{2.15} Id = (Id - T_n) \circ (Id + T_n) (Id - T^2_n)
                ^{-1} .
   \end{equation} 
Given $\varphi \in H^w$, denote by $\Phi _n$ the operator in
${\mathcal L}(\ell ^2)$ (with $\ell ^2 \equiv \ell ^2({\mathbb
Z};{\mathbb C})$) defined by $(n \in {\mathbb Z}; a \in \ell ^2
({\mathbb Z};{\mathbb C}))$
   \[ (\Phi _na)k):= \sum _{j\in {\mathbb Z}} \frac {\hat
      {\varphi }(k + j)}{\langle n - j\rangle } a(j) \quad
      (\forall k \in {\mathbb Z}), 
   \]
where $\langle k\rangle = 1 + |k|$. 

Recall that a weight $w$ is called a $\delta $-weight $(\delta
\geq 0)$ if $w_{-\delta }(k):= \langle k\rangle ^{-\delta } w
(k)$ is a weight. For convenience we denote the weight $w_{
-\delta }$ by $w_\ast $. The two key lemmas for proving that
$\lim _{n\rightarrow \infty } \| T^2_n\| = 0$ are the following
ones:

\bigskip

\begin{lemma}
\label{Lemma2.1} Let $w$ be a $\delta $-weight with $0 \leq
\delta < \frac {1}{2}$ and $n \in {\mathbb Z}$. Then there
exists $C = C(\delta )$ such that
   \[ \| \Phi _n \| _{{\mathcal L}(\ell ^2_{S^{-n}w_\ast };
      \ell ^2_{S^nw} )} \leq C \| \varphi \| _w .
   \]
\end{lemma}

\medskip
\ni
{\it Proof } For $a \in \ell ^2_{S^{-n}w_\ast }$ and $b \in \ell ^2
_{S^nw}$, 
   \begin{align*} &|(b,\Phi _na)_{S^nw}| \leq \\
                  &\leq \sum _{j,k} w(k + n) |b(k)| w_\ast(j - n)
                   |a(j)| w(k + j) |\hat {\varphi }(k + j)| \cdot \\
                  &\frac {w(k + n)}{w_\ast (j - n)w(k + j)} \frac 
                   {4}{\langle n - j\rangle }\ .
   \end{align*}
Using that $w$ is submultiplicative, one gets 
   \[ \frac {w(k + n)}{w_\ast (j - n)w(k + j)} \leq \frac {w(n - j)}
      {w_\ast (j - n)} = \langle j - n\rangle ^\delta \leq (4| n
      - j + \frac {1}{2}|)^\delta \leq 2 | n - j + \frac {1}{2}|
      ^\delta .
   \]
and hence, by the Cauchy-Schwartz inequality
   \begin{align*} &|(b,\Phi _na)_{S^nw}| \leq \\
                  &\leq \| b\| _{S^nw} \| a\| _{S^{-n}w_\ast }
                   \left( \sum _{k,j} \frac {4|\hat {\varphi }
                   (k + j)|^2 w(k + j)^2}{\langle n - j \rangle 
                   ^{2(1 - \delta )}} \right) ^{1/2} \\
                  &\leq C \| b\| _{S^nw} \| a\| _{S^{-n}w_\ast }
                   \| \varphi \| _w
   \end{align*}
with $C \equiv C(\delta ):= \left( \sum _k \frac {4}{\langle k
\rangle ^{2-2\delta }}\right) ^{1/2} < \infty $ as $\delta < \frac
{1}{2}$. 
$\blacksquare $

\bigskip

\begin{lemma}
\label{Lemma2.2} Let $\delta \geq 0, w$ be a $\delta $-weight and
$n \in {\mathbb Z}$. Then there exists $C > 0$, independent of
$\delta $, such that
   \[ \| \Phi _n\| _{{\mathcal L}(\ell ^2_{S^{n}w}; \ell ^2_{S^{-n}w
      _\ast })} \leq C \frac {\| \varphi \| _{w_\ast }}{\langle n
      \rangle ^{\delta \wedge 1}}
   \]
where as usual $\delta \wedge 1 = \min (1,\delta )$.
\end{lemma}

\medskip
\ni
{\it Proof } For $a \in \ell ^2_{S^nw}$ and $b \in \ell ^2_{S^{-n}
w_\ast }$,  
   \begin{align*} &|(b,\Phi _na)_{S^{-n}w_\ast }| \leq \\
                  &\leq \sum _{k,j} w_\ast (k - n) |b(k)| w(j + n)
                   |a(j)| w_\ast (k + j) |\hat {\varphi }(k + j)| \\
                  &\frac {w_\ast (k - n)}{w(j + n)w_\ast (k + j)} 
                   \frac {4}{\langle n - j \rangle } .
   \end{align*}
As $w_\ast $ submultiplicative and symmetric,
   \[ w_\ast (k - n) \leq w_\ast (k + j) w_\ast (j + n)
   \]
which leads to (use definition of $w_\ast $)
   \[ |(b,\Phi _n a)_{S^{-n}w_\ast }| \leq \| b\| _{S^{-n}w_\ast }
      \| a\| _{S^nw} \| \hat {\varphi }\| _{w_\ast } \left( \sum _j
      \frac {4}{\langle j + n\rangle ^{2\delta }} \frac {4}{\langle
      j - n\rangle ^2} \right) ^{1/2} .
   \]
The claimed estimate then follows from the following elementary
estimate 
   \[ \left( \sum _j \frac {1}{\langle j + n\rangle ^{2\delta }} \frac {1}
      {\langle j - n\rangle ^2} \right) ^{1/2} \leq C \frac {1}
      {\langle n\rangle ^{\delta \wedge 1}}
   \]
for some $C$, independent of $\delta $.
$\blacksquare $

\bigskip

As an application of Lemma~\ref{Lemma2.1} and \ref{Lemma2.2}
we obtain estimates for the norms of $R^{(j)}_n, T_n$ and $T^2_n$.
By definition
   \begin{equation}
   \label{2.16}  T^2_n = \left( \begin{array}{c c}
                 0     &R^{(2)}_n \\
                 R^{(1)}_n  &0
                 \end{array} \right) ^2 =
                 \left( \begin{array}{c c}
                 R^{(2)}_nR^{(1)}_n   &0  \\
                 0  &R^{(1)}_nR^{(2)}_n
                 \end{array} \right) 
   \end{equation}
and it is useful to introduce the operators
   \begin{equation}
   \label{2.17} P_n:= R^{(2)}_n R^{(1)}_n  ; \quad Q_n:=
                R^{(1)}_n R^{(2)}_n .
   \end{equation}
To make notation easier we write $\ell ^2_{S^{\pm n}w}$ for both,
$\ell ^2_{S^{\pm n}w}({\mathbb Z} \backslash n;{\mathbb C})$ and
$\ell ^2_{S^{\pm n}w}$ \par \ni
$({\mathbb Z} \backslash n ;{\mathbb C}^2)$.

\bigskip

\begin{corollary}
\label{Corollary2.3} Let $\delta \geq 0, M \geq 1$ and $w$ be a
$\delta $-weight. Then, for any $1$-periodic functions $\psi _1,
\psi _2 \in H^w$ with $\| \psi _j\| _w \leq M \ \ (j = 1,2)$, the
following statements hold:

\medskip 
\ni
\begin{description}
\item[(i)  ] If $0 \leq \delta < \frac {1}{2}$, there exists $C
\equiv C(\delta ) > 0$ so that for $1 \leq j \leq 2, n \in
{\mathbb Z}$, and $|z| \leq \frac {\pi }{2}$,
   \begin{align*} &\| R^{(j)}_n\| _{{\mathcal L}(\ell ^2_{S^{-n}
                   w_\ast }; \ell ^2_{S^nw})} \leq C M ; \\
                  &\| T_n\| _{{\mathcal L}(\ell ^2_{S^{-n}
                   w_\ast }; \ell ^2_{S^nw})} \leq C M .
   \end{align*}

\item[(ii) ] If $\delta \geq 0$, there exists $C > 0$ such that 
for $1 \leq j \leq 2, n \in {\mathbb Z}$, and $|z| \leq \frac
{\pi }{2}$,
   \begin{align*} &\| R^{(j)}_n\| _{{\mathcal L}(\ell ^2_{S^n
                   w}, \ell ^2_{S^{-n}w_\ast })} \leq \frac 
                   {C M}{\langle n\rangle ^{\delta \wedge 1}} \\
                  &\| T_n\| _{{\mathcal L}(\ell ^2_{S^n
                   w}, \ell ^2_{S^{-n}w_\ast })} \leq \frac 
                   {C M}{\langle n\rangle ^{\delta \wedge 1}} .
   \end{align*}

\item[(iii) ] If $0 \leq \delta < \frac {1}{2}$, then there exists
$C \equiv C(\delta )$ so that for $n \in {\mathbb Z}$ and $|z| \leq
\frac {\pi }{2}$,
   \begin{align*} &\| P_n\| _{{\mathcal L}(\ell ^2_{S^n
                   w})} \leq \frac {C M^2}{\langle n\rangle 
                   ^\delta }\, ; \, \quad \| Q_n\| _{{\mathcal L}(\ell ^2
                   _{S^nw})} \leq \frac {CM^2}{\langle n
                   \rangle ^\delta }\ ;\\
                  &\| P_n\| _{{\mathcal L}(\ell ^2_{S^{-n}
                   w_\ast })} \leq \frac {C M^2}{\langle n\rangle 
                   ^\delta }\, ;\, \quad \| Q_n\| _{{\mathcal L}(\ell ^2
                   _{S^{-n}w_\ast })} \leq \frac {CM^2}{\langle n
                   \rangle ^\delta }\ .
   \end{align*}
\end{description}
\end{corollary}

\medskip

{\it Proof } The claimed estimates for $R^{(j)}_n (j = 1,2)$ follow
from Lemma~\ref{Lemma2.1} and Lemma~\ref{Lemma2.2}. As $T_n =
\left( \begin{array}{c c} 0 & R^{(2)}_n \\ 
R^{(1)}_n &0
\end{array} \right)$, these estimates then imply the ones for $T_n$.
The estimates in (iii) are obtained by combining the estimates in (i)
and (ii) for $R^{(j)}_n$. 
$\blacksquare $

\bigskip

Under the assumptions of Corollary~\ref{Corollary2.3} define, for
$0 < \delta < \frac {1}{2}$ and $M \geq 1$,
   \begin{equation}
   \label{2.18} N_0 \equiv N_0(\delta ,M,w):= \max \left( 1, (2
                C M^2)^{1/\delta} \right)
   \end{equation}
with $C$ given as in Corollary~\ref{Corollary2.3} (iii).

\bigskip

\begin{proposition}
\label{Proposition2.4} Let $0 < \delta < \frac {1}{2}, M \geq 1$
and $w$ be a $\delta $-weight. Then, for any $1$-periodic functions
$\psi _1, \psi _2 \in H^w$ with $\| \psi _j\| _w \leq M, |n|
\geq N_0$ and $|z| \leq \pi / 2$,

\medskip
\begin{description}
\item[(i)  ] 
   \[ \| P_n\| _{{\mathcal L}(\ell ^2_{S^nw})} \leq \frac
      {1}{2}; \ \| Q_n\| _{{\mathcal L}(\ell ^2_{S^nw})} \leq \frac 
      {1}{2} ;
   \]

\item[(ii) ] $(Id - P_n)$ and $(Id - Q_n)$ are invertible and 
   \[ \|(Id - P_n)^{-1}\| _{{\mathcal L}(\ell ^2_{S^nw})} \leq 
      2 ; \quad \|(Id - Q_n)^{-1}\| _{{\mathcal L}(\ell ^2_{S^nw})} 
      \leq 2 .
   \]

\item[(iii)] $Id - T^2_n$ is invertible and 
   \[ \| T^2_n\| _{{\mathcal L}(\ell ^2_{S^nw})} \leq \frac {1}{2} ;
      \quad \| (Id - T^2_n)^{-1}\|_{{\mathcal L}(\ell ^2_{S^nw})}
      \leq 2 .
   \]
 
\item[(iv) ] Statements (i) - (iii) remain true if one replaces the
weight $S^nw$ by $S^{-n}w_\ast $.
\end{description}
\end{proposition}

\medskip

{\it Proof } (i) By Corollary~\ref{Corollary2.3} (iii) $P_n$
satisfies the estimate (as $0 < \delta < \frac {1}{2}$)
   \[ \| P_n\| _{{\mathcal L}(\ell ^2_{S^nw})} \leq \frac
      {CM^2}{\langle n\rangle ^\delta } .
   \]
Hence for $|n| \geq N_0$
   \[ \| P_n\| _{{\mathcal L}(\ell ^2_{S^nw})} \leq \frac 
      { C M^2}{\langle N_0 \rangle ^\delta } \leq
      \frac {1}{2} .
   \]
Similarly, one obtains $\| Q_n\| _{{\mathcal L}(\ell ^2_{S^nw})}
\leq \frac {1}{2}$.

(ii) follows immediately from (i) and (iii) follows from (i) -
(ii) and the identity $T^2_n = \left( \begin{array}{c c} 
P_n   & 0 \\ 
0   &Q_n
\end{array} \right)$. Finally, statements (i) - (iii) for the 
weight $S^{-n}w_\ast $ are proved in a similar way as for $S^n w$.
$\blacksquare $

\bigskip

Summarizing the results obtained in this section, we obtain, with
$\| \cdot \| \equiv \| \cdot \| _{{\mathcal L}(\ell ^2_{S^nw}(
{\mathbb Z} \backslash n ; {\mathbb C}^2))}$:

\medskip

\begin{corollary}
\label{Corollary2.5} Let $0 < \delta < \frac {1}{2}, M \geq 1$ and
$w$ be a $\delta $-weight. Then there exists $C > 0$ such that, for
any $1$-periodic functions $\psi _1, \psi _2 \in H^w$ with $\| \psi _j
\| _w \leq M$ $(j = 1,2)$, $|n| \geq N_0$ and $|z| \leq \pi / 2$

\begin{description}
\item[(i)  ] $\| T_n \| \leq C$;

\item[(ii) ] $(Id - T_n)$ is invertible in ${\mathcal L}(\ell ^2
_{S^nw}({\mathbb Z} \backslash n; {\mathbb C}^2))$ and $\| (Id - T_n)
^{-1}\| \leq C$;

\item[(iii)] $(z - A_n)$ is invertible in ${\mathcal L}(\ell ^2_{S^n
w}({\mathbb Z} \backslash n; {\mathbb C}^2))$ and $\| (z - A_n)^{-1}
\| \leq C$. 
\end{description}
\end{corollary}

\medskip

{\it Proof } (i) Recall that $T =
\left( \begin{array}{c c} 
0 & R^{(2)}_n \\ 
R^{(1)}_n &0
\end{array} \right) $. By standard convolution estimates, there
exists an absolute constant $C > 0$ so that for $n \in {\mathbb Z}$
and $|z| \leq \pi /2, \| \psi _j\| _w \leq M$
   \[ \| T_n\| \leq CM .
   \]

Therefore (ii) and (iii) follow immediately from \eqref{2.14} - \eqref{2.15}
and Proposition~\ref{Proposition2.4}.
$\blacksquare $

\vskip 1 cm

\subsection{$Q$-equation}
\label{Ss: 2.3 $Q$-equation}

Using the notations introduced in 
section~\ref{Ss: 2.2 $P$-equation}, we have for $|n| \geq N_0$ and 
$|z| \leq \pi /2$
   \[ (z - A_n)^{-1} = \left( \begin{array}{c c}
           (z - D_n)^{-1}(Id - P_n)^{-1}  &(z - D_n)^{-1}R^{(2)}_n
              (Id - Q_n)^{-1}  \\
           (z - D_n)^{-1} R^{(1)}_n(Id - P_n)^{-1}  &(z�- D_n)^{-1}
              (Id - Q_n)^{-1} 
           \end{array} \right) .
   \]
Hence the $P$-equation \eqref{2.10} leads to the following formulas
   \begin{align}
   \label{2.21} \breve {F}_2 &= y^F(z - D_n)^{-1}(Id - P_n)^{-1} S^n
                                \hat {\psi _2}  \\
                &+ x^F(z - D_n)^{-1}R^{(2)}_n(Id - Q_n)^{-1} S^n
                                J\hat {\psi }_1 \nonumber \\
   \label{2.22} J \breve {F}_1 &= y^F(z - D_n)^{-1} R^{(1)}_n(Id - 
                                P_n)^{-1} S^n \hat {\psi }_2 \\
                &+ x^F(z - D_n)^{-1}(Id - Q_n)^{-1} S^n J
                                \hat {\psi }_1 . \nonumber
   \end{align}
These solutions are substituted into the $Q$-equation \eqref{2.8} -
\eqref{2.9} to obtain for $|z| \leq \frac {\pi }{2}, |n| \geq n_0$
the following homogeneous system
   \begin{align} 
   \label{2.23} (- z + \alpha ^+(n,z))x^F &+ (\hat {\psi }_2(2n) +
                   \beta ^+(n,z))y^F = 0 \\
   \label{2.24} (\hat {\psi }_1(-2n) + \beta ^-(n,z))x^F &+ (-z +
                \alpha ^-(n,z))y^F = 0 , 
   \end{align}
where
   \begin{align}
   \label{2.25} &\alpha ^+(n,z):= \langle S^n \hat {\psi }_2,(z - 
                   D_n)^{-1}(Id - Q_n)^{-1} S^n J\hat {\psi }_1
                   \rangle \\
   \label{2.26} &\beta ^+(n,z):= \langle S^n \hat {\psi }_2,(z - 
                   D_n)^{-1}R^{(1)}_n (Id - P_n)^{-1} S^n \hat 
                   {\psi }_2 \rangle \\
   \label{2.27} &\alpha ^-(n,z):= \langle S^n J \hat {\psi }_1,(z - 
                   D_n)^{-1}(Id - P_n)^{-1} S^n \hat {\psi }_2
                   \rangle \\
   \label{2.28} &\beta ^-(n,z):= \langle S^n J \hat {\psi }_1,(z - 
                   D_n)^{-1}R^{(2)}_n (Id - Q_n)^{-1} S^n J \hat 
                   {\psi }_1 \rangle .
   \end{align}
Notice that $\alpha ^\pm (n,z)$ and $\beta ^\pm (n,z)$ are analytic
for $|z| < \frac {\pi }{2}$ as $R^{(j)}_n$, $P_n$ and $Q_n$ are analytic
for $|z| < \frac {\pi }{2}$. An important simplification of the 
equations \eqref{2.23} - \eqref{2.24} results from the following
observation

\bigskip

\begin{lemma}
\label{Lemma2.8} For $|z| \leq \frac {\pi }{2}$ and $|n| \geq N_0$,
   \[ \alpha ^+(n,z) = \alpha ^-(n,z) .
   \]
\end{lemma}

\medskip

{\it Proof } In view of \eqref{2.25} and \eqref{2.27} it is to show
that
   \begin{equation}
   \label{2.29} (z - D_n)^{-1}(Id - Q_n)^{-1} = \left( (Id - P_n)^{-1}
                \right) ^t (z - D_n)^{-1}
   \end{equation}
where $A^t$ denotes the transpose of $A$,
   \[ (A^t)(k,j):= A(j,k) \quad \mbox{(no complex conjugation)}.
   \]
The equation \eqref{2.29} can be reformulated, 
   \[ \left( (Id - Q_n)(z - D_n)\right) ^{-1} = \left( (z - D_n)
      (Id - P^t_n)\right) ^{-1} .
   \]
which holds iff
   \begin{equation}
   \label{2.30} Q_n(z - D_n) = (P_n(z - D_n))^t .
   \end{equation}
The identity \eqref{2.30} follows easily from 
   \begin{align*} (Q_n(z - D_n))(j,k) &= (R^{(1)}_n R^{(2)}_n)(j,k)
                    (z - (k - n)\pi ) \\                 
                  &= \sum _\ell \frac {\hat {\psi }_1(- j - \ell ) }
                    {z - (\ell - n)\pi } \cdot \hat {\psi }_2 
                    (\ell + k) 
   \end{align*}
and
   \begin{align*} (P_n(z - D_n))^t(j,k) &= (P_n(z - D_n))(k,j) \\
                  &= (R^{(2)}_nR^{(1)}_n)(k,j)(z - (j - n)\pi ) \\
                  &= \sum _\ell \frac {\hat {\psi }_2(k + \ell )}
                    {z - (\ell - n)\pi } \hat {\psi }_1(- \ell - j).
   \end{align*}
$\blacksquare $

\bigskip

In view of Lemma~\ref{Lemma2.8} we write
   \begin{equation}
   \label{2.31} \alpha (n,z):= \alpha ^+(n,z) \quad (= \alpha ^-(n,
                z)) .
   \end{equation}

\medskip

In subsequent sections we estimate the coefficients $\alpha (n,z),
\beta ^+(n,z)$ and $\beta ^-(n,z)$.

\vskip 1 cm

\subsection{Estimates for $\alpha (n,z)$}
\label{Ss: 2.4 Estimates for alpha}

\begin{lemma}
\label{Lemma2.9} Let $0 < \delta \leq \frac {1}{2}$, $M \geq 1$ and
$w$ be a $\delta $-weight. Then, for any $1$-periodic functions
$\psi _1,\psi _2 \in H^w$ with $\| \psi _j\| _w \leq M, |n| \geq
N_0$  $(N_0 \equiv N_0(\delta ,M)$ given by \eqref{2.18}) and
$|z| \leq \frac {\pi }{2}$
   \[ | \alpha (n,z)| \leq \frac {4M^2}{\langle n\rangle ^{2
        \delta }} .
   \]
\end{lemma}

\medskip

{\it Proof } Write $\alpha (n,z) = \langle S^n \hat {\psi }_2,
(z - D_n)^{-1}a \rangle $ with $a:= (Id - Q_n)^{-1} S^n J \hat 
{\psi }_1 \in \ell ^2_{S^nw}$. By 
Proposition~\ref{Proposition2.4},
   \[ \| a\| _{S^nw} \leq 2\| \hat {\psi }_1\| _w \leq 2M .
   \]
Hence 
   \begin{align*} \langle n\rangle ^{2\delta } |\alpha (n,z)| 
                    &\leq \sum _{k \not= n} \frac {\langle n
                    \rangle ^ {2\delta }}{\langle n - k\rangle } 
                    |\hat {\psi }_2(k + n)| |a(k)| \\
                  &\leq \sum _{|k+n| < |n|} \frac {\langle n
                    \rangle ^{2\delta }}{\langle n\rangle }
                    |\hat {\psi }_2(k + n)| |a(k)| \\
                  &+ \sum _{|k+n| \geq |n|} \frac {\langle n\rangle
                    ^{2\delta }}{w(k + n)^2} w(k + n) |\hat {\psi }
                    _2(k + n)| w(k + n) |a(k)| \\
                  &\leq 2 \| \hat {\psi }_2\| _w \| a\| _{S^nw}
   \end{align*}
where we used that $2\delta \leq 1$ and $w(k + n) = w_\ast (k + n)
\langle k + n\rangle ^\delta \geq \langle n\rangle ^\delta $ for
$|k + n| \geq |n|$.
$\blacksquare $ 

\vskip 1 cm

\subsection{Estimates for $\beta ^\pm (n,z)$}
\label{Ss: 2.4 Estimates for beta}

In this section we provide estimates for $\beta ^\pm (n,z)$. The
$\beta ^\pm (n,z)$ - they turn out to be quite small - determine
the asymptotics of the sequence of gap lengths given in 
Theorem~\ref{Theorem1}. As $\beta ^+(n,z)$ (cf \eqref{2.26}) and
$\beta ^-(n,z)$ (cf. \eqref{2.28}) are analyzed in a similar
fashion we focus on the estimate for $\beta ^+(n,z)$. Writing
$(Id - P_n)^{-1} = \sum ^\infty _{k=0} P^k_n$ we obtain for
$\beta ^+(n,z)$ the following convergent series
   \begin{equation}
   \label{b.1} \beta ^+(n,z) = \sum ^\infty _{k=0} \beta _k(n,
               z) .
   \end{equation}
where
   \begin{equation}
   \label{b.02} \beta _k(n,z):= \langle S^n \hat {\psi }_2,(z -
                D_n)^{-1} R^{(1)}_n P^k_n S^n \hat {\psi }_2
                \rangle .
   \end{equation}
The convergence of series \eqref{b.1} follows from $\| P_n\|
_{{\mathcal L}(\ell ^2_{S^\pm n_w})} \leq \frac {1}{2}$ ($|n| \geq
N_0$, Proposition~\ref{Proposition2.4}). We begin by analyzing
$\beta _k(n,z)$. 

With $R^{(1)}_n$ defined by (cf \eqref{2.12})
   \[ (R^{(1)}_na)(j) = J(\hat {\psi }_1 \ast (z - D_n)^{-1}a)
      (j) = \sum _{\ell \not= n}  \frac {(J\hat {\psi }_1)(j
      + \ell )a(\ell )}{z - (\ell - n)\pi }
   \]
and $\inf _{|z|\leq \frac {\pi }{2}} |z - (\ell - n)\pi | \geq 
\frac {1}{2} \langle \ell - n \rangle $ (for any $\ell \not= n$) we 
get 
   \[ |(R^{(1)}_na)(j)| \leq 2 \sum _\ell \frac {|J\hat {\psi }_1
      (j + \ell )|}{\langle \ell - n\rangle } |a(\ell )| 
   \]
which leads to 
   \begin{equation}
   \label{b.2} |\beta _k(n,z)| \leq 4 \sum _j \frac {|\hat {\psi }_2
               (n + j)|}{\langle j - n\rangle } \sum _\ell \frac
               {|J \hat {\psi }_1(j + \ell )|}{\langle \ell - n
               \rangle } |(S^{-n} P^k_n S^n \hat {\psi }_2)(\ell +
               n)| .
   \end{equation}
Given three nonnegative sequences (i.e. sequences of nonnegative
numbers), $a, b, d$ in $\ell ^2({\mathbb Z})$ we define, for any
$n \in {\mathbb Z}$, the sequence $\Psi _n \equiv \Psi _n(a,b,d)$
by
   \[ \Psi _n(k + n):= \sum _j \frac {a(k + j)}{\langle j - n
      \rangle } \sum _\ell \frac {b(j + \ell )}{\langle \ell - n
      \rangle } d(\ell + n) .
   \]
Then $\Psi _n$ is a nonnegative sequence in $\ell ^2({\mathbb Z})$
and can be used to rewrite \eqref{b.2}: Introduce, for $|n| \geq
N_0$ and $|z| \leq \frac {\pi }{2}$,
   \begin{equation}
   \label{b.3} \eta_{n,0}:= 4 \Psi _n(|\hat {\psi }_2| , |J
               \hat {\psi }_1| , |\hat {\psi }_2|)
   \end{equation}
and, for $k \geq 0$,
   \begin{equation}
   \label{b.4} \eta _{n,k+1} := 4 \Psi _n(|\hat {\psi }_2|, |J
               \hat {\psi }_1|, \eta _{n,k}) .
   \end{equation}
where, for any $a = (a(j))_{j \in {\mathbb Z}} \in \ell ^2
({\mathbb Z})$, we denote by $|a|$ the sequence $(|a(j)|)_{j \in
{\mathbb Z}}$. As for any $|z| \leq \frac {\pi }{2}$,
   \begin{align*} |(S^{-n}P_nS^na)(k + n)| &= |(P_n S^na)(k)| \\
                  &= |(R^{(2)}_n R^{(1)}_n S^na)(k)| \\
                  &= |(\hat {\psi }_2 \ast (J(z - D_n)^{-1}(R
                    ^{(1)}_nS^{-n}a)))(k)| \\
                  &\leq 4 \sum _j |\hat {\psi }_2(k + j)| \frac {1}
                    {\langle j - n\rangle } \sum _\ell \frac
                    {|J\hat {\psi }_1(j + \ell)|}{\langle \ell
                    - n \rangle } |a(\ell + n)| \\
                  &= 4\Psi _n(|\hat {\psi }_2| , |J\hat {\psi }
                    _1|, |a|) 
   \end{align*}
it follows, by an induction argument, from \eqref{b.2} and
   \[ S^{-n} P^k_n S^n \hat {\psi }_2 = (S^{-n} P_n S^n)(S^{-n}
      P^{k-1}_n S^n\psi _2)
   \]
that, for any $k \geq 0$, 
   \begin{equation}
   \label{b.5} \sup _{|z| \leq \frac {\pi }{2}} |\beta _k(n,z)|
               \leq \eta _{n,k}(2n) .
   \end{equation}
To estimate $\eta _{n,k}(2n)$, we need the following auxilary 
lemma concerning the operator $\Psi _n$. For $\delta > 0$ and $w$
be a $\delta $-weight, define
   \[ \delta _\ast = \delta \wedge 1/2 .
   \]

\bigskip

\begin{lemma}
\label{Lemma2.11} Let $w$ a $\delta $-weight, and, for any $n \in
{\mathbb Z}, d_n$ a positive sequence in $\ell ^2_w$ so that
   \[ \langle n\rangle ^\alpha d_n(j) \leq d(j) \quad \forall n,
      j \in {\mathbb Z}
   \]
for some $\alpha \geq 0$ and some positive sequence $d$ in $\ell ^2
_w$. Then there exist $C \equiv C_{\delta _\ast }$, only 
depending on $\delta _\ast $, and $e \in \ell ^2_w$ so that for 
any positive sequences $a, b \in \ell ^2_w$,

\begin{description}
\item[(i)  ] 
   \begin{align*} \sum _{n \in {\mathbb Z}} \langle n\rangle ^{2(2 
                      \delta _\ast + \alpha )} &w(2n)^2 (4 \Psi 
                     _n (a,b,d_n)(2n))^2 \\
                  &\leq C \| a\| _w \| b\| _w \| d\| _ w ;
   \end{align*}

\item[(ii) ] for any $n,j \in {\mathbb Z}$, 
   \begin{align*} \langle n\rangle ^{\alpha + \delta _\ast } 4 
                     \Psi _n(a,b,d_n)(j) \leq e(j)\ ; \\
                  \| e\| _w \leq C\| a\| _w \| b\| _w \| d\| 
                     _w .
   \end{align*}
\end{description} 
\end{lemma}

\medskip

{\it Proof } Cf. Appendix B.

\bigskip

From Lemma~\ref{Lemma2.11} we obtain, in view of the definition
\eqref{b.3} - \eqref{b.4} and the estimate \eqref{b.5} the
following

\medskip

\begin{corollary}
\label{Corollary2.12} Let $M \geq 1$ and $w$ be a $\delta $-weight.
Then for any $1$-periodic functions $\psi _1, \psi _2 \in H^w$ with
$\| \psi _j\| _w \leq M$ $(j = 1,2)$

\begin{description}
\item[(i)  ] for $k \geq 0$
   \[ \sum _{|n| \geq N_0} \langle n\rangle ^{2(2+k)\delta _\ast } w
      (2n)^2 \sup _{|z| \leq \frac {\pi }{2}} |\beta _k(n,z)|^2 \leq
      C^{k+1} M^{2k+3}
   \]
where $1\leq C \equiv C_\delta < \infty $ is given by 
Lemma~\ref{Lemma2.11}

\item[(ii) ]
   \[ \sum _{|n| \geq N_0} \langle n\rangle ^{6\delta _\ast } w
      (2n)^2 \sup _{|z| \leq \frac {\pi }{2}} |\tilde {\beta }(n,z)|^2 
      \leq C'
   \]
where $\tilde {\beta }(n,z):= \sum _{k\geq 1} \beta _k(n,z)$ and $1
\leq C' < \infty $ is a constant depending only on $M$ and $\delta$.
\end{description}
\end{corollary}

\medskip

{\it Proof } We apply Lemma~\ref{Lemma2.11} to each of the $\beta 
_k$'s in an inductive fashion to obtain (i). Statement (ii) then 
follows from (i) by the Cauchy-Schwartz inequality.
$\blacksquare $

\bigskip

To simplify further the asymptotics of $\beta $ write $\beta _0(n,z)
\equiv \beta ^+_0(n,z) = \beta ^+_0(n) + z \beta ^+_\#(n,z)$
where
   \[ \beta ^\pm _0(n):= \beta ^\pm _0(n,0) ; \ \beta ^\pm _\# (n,z)
      := \int ^1_0 \partial _z \beta ^\pm _0(n,tz)dt .
   \]
As $z \mapsto \beta ^\pm _\#(n,z)$ are analytic functions in $\{ |z|
< \pi /2\} $, one deduces by Cauchy's formula
   \begin{equation}
   \label{b.20} \sup _{|z| \leq \pi /4} |\beta ^\pm _\# (n,z)| \leq
                \frac {4}{\pi } \sup _{|z| \leq \pi /2} |\beta ^\pm
                _0(n,z)| .
   \end{equation}

Summarizing our results of this section gives the following 

\medskip

\begin{proposition}
\label{Proposition2.14} Let $\delta > 0, M \geq 1, 1 \leq A < 
\infty $ and $w$ be a $\delta $-weight. Then there exists $C > 0$ 
so that for any $1$-periodic functions $\psi _1, \psi _2 \in H^w$
with $\| \psi _j\| _w \leq M$ $(j = 1,2)$,

\begin{description}
\item[(i)  ]
   \[ \sum _{|n| \geq N_0} \langle n\rangle ^{4\delta _\ast } 
      w(2n)^2 \sup _{|z| \leq \pi /4} |\beta ^\pm (n,z)|^2 \leq C
   \]

\item[(ii) ] 
   \[ \sum _{|n| \geq N_0} \langle n\rangle ^{6\delta _\ast } 
      w(2n)^2 \sup _{|z| \leq A/\langle n\rangle ^{\delta _\ast }}
      |\beta ^\pm (n,z) - \beta ^\pm _0(n)|^2 \leq C .
   \]
\end{description}
\end{proposition}   

\medskip

{\it Proof } Notice that $\beta ^\pm (n,z) = \beta ^\pm _0(n) +
z\beta ^\pm _\#(n,z) + \tilde {\beta }^\pm (n,z)$ and hence (i)
is a consequence of Corollary~\ref{Corollary2.12} and formula 
\eqref{b.20}. Statement (ii) is proved in the same fashion. As the
supremum of $|\beta ^\pm (n,z) - \beta ^\pm _0(n)|$ is only taken
over $|z| \leq \frac {A}{\langle n\rangle ^{\delta _\ast }}$, the
asymptotics of $z \beta ^\pm _\# (n,z)$ can be improved by
$\delta _\ast $ to obtain from formula \eqref{b.20}
   \[ \sum _{|n| \geq N_0} \langle n\rangle ^{6\delta _\ast } 
      w(2n)^2 \sup _{|z| \leq A/\langle n\rangle ^{\delta _\ast }}
      |z \beta ^\pm _\#(n,z)|^2 \leq C .
   \]
$\blacksquare $
   
\vskip 1 cm

\subsection{$z$-equation}
\label{Ss: 2.6 $z$-equation}

In view of \eqref{2.23} - \eqref{2.24}, and \eqref{2.31}, the
$Q$-equation leads to the following $2 \times 2$ system
   \begin{equation} 
   \label{2.50} \left( \begin{array}{c c}
                -z + \alpha (n,z)    &\hat {\psi }_2(2n) +
                  \beta ^+(n,z) \\
                \hat {\psi }_1(-2n) + \beta ^-(n,z)   &-z +
                  \alpha (n,z)
                \end{array} \right)
                \left( \begin{array}{c}
                x^F \\
                y^F
                \end{array} \right) =
                \left( \begin{array}{c}
                0 \\
                0
                \end{array} \right) .
   \end{equation}
Given $|n| \geq N$ and $|z| \leq \frac {\pi }{2}$, the number 
$\lambda = n\pi + z$ is a periodic eigenvalue of $L$ iff there
exists a nontrivial solution of \eqref{2.50} $(x^F, y^F) \in
{\mathbb C}^2 \backslash (0,0)$, or, equivalently, iff the 
determinant of the $2 \times 2$ matrix in \eqref{2.50}
vanishes,
   \begin{equation}
   \label{2.51} (z - \alpha (n,z))^2 - (\hat {\psi }_2(2n) +
                \beta ^+ (n,z))(\hat {\psi }_1(-2n) + \beta ^-
                (n,z)) = 0 .
   \end{equation}
Proceeding similarly as in \cite{KM}, equation \eqref{2.51} is
solved in two steps: For $\zeta $ with $|\zeta | \leq \frac {\pi }
{8}$ given, consider 
   \begin{equation}
   \label{2.52} z_n = \alpha (n,z_n) + \zeta .
   \end{equation}
Substituting a solution $z(\zeta ) \equiv z_n(\zeta )$ of
\eqref{2.52} into \eqref{2.51} leads to an equation for $\zeta 
\equiv \zeta _n$, 
   \begin{equation}
   \label{2.53} \zeta ^2 - \left( \hat {\psi }_2(2n) + \beta ^+(n,z
                (\zeta )\right) \left( \hat {\psi }_1(-2n) + 
                \beta ^-(n,z(\zeta ))\right) = 0\ .
   \end{equation}
Equation \eqref{2.52} is referred to as the $z$-equation and 
equation \eqref{2.53} as the $\zeta $-equation .

\medskip

In this section we deal with the $z$-equation \eqref{2.52}. To
solve it we use the contractive mapping principle. According to
Lemma~\ref{Lemma2.9} we can choose
$N_1 \geq N_0$ (with $N_0$ given by \eqref{2.18}) so that for
any $1$-periodic functions $\psi _1,\psi _2 \in H^w$ with $\| 
\psi _j\| _w \leq M$ and $|n| \geq N_1$
   \begin{equation}
   \label{2.54} \sup _{|z| \leq \frac {\pi }{2}} |\alpha (n,z)|
                < \pi / 8 .
   \end{equation}

\medskip

The following result can be proved by the same line of arguments
used in the proof of \cite[Proposition 1.6]{KM}.

\bigskip
             
\begin{proposition}
\label{Proposition2.20} Let $M \geq 1, 0 < \delta \leq 1/2$ and $w$
be a $\delta $-weight. Then, there exists $N_1 \geq N_0$ so that
for any $1$-periodic functions $\psi _1, \psi _2 \in H^w$ with
$\| \psi _j\| _w \leq M, |\zeta | \leq \frac {\pi }{8}$ and $|n|
\geq N_1$, equation \eqref{2.52} has a unique solution $z_n = z_n
(\zeta )$ satisfying $|z_n| < \pi /4$. The solution depends 
analytically on $\zeta $.
\end{proposition}
   
\vskip 1 cm

\subsection{$\zeta $-equation}
\label{Ss: 2.7 $zeta $-equation}

In this section, we improve the existence of solutions of the
$\zeta $-equation \eqref{2.53}
   \[ \zeta ^2 - \left(\hat {\psi }_2(2n) + \beta ^+(n,z(\zeta )
      \right) \left( \hat {\psi }_1(-2n) + \beta ^-(n,z(\zeta ))
      \right) = 0
   \]
using Rouch\'e's Theorem. Introduce 
   \begin{equation}
   \label{2.55} r_n:= \left( |\hat {\psi }_2(2n)| + \sup _{|z| \leq
                \frac {\pi }{2}}| \beta ^+(n,z)|\right) \vee
                \left( |\hat {\psi }_1(-2n)| + \sup _{|z| \leq 
                \frac {\pi }{2}} |\beta ^-(n,z)|\right) .
   \end{equation}

\medskip 

Using the same line of arguments used in the proof of 
\cite[Proposition 1.15]{KM} one obtains the following

\medskip

\begin{proposition}
\label{Proposition2.21} Let $M \geq 1$, $0 < \delta \leq \frac {1}
{2}$ and $w$ be a $\delta $-weight. Then there exists $N_2 \geq
N_1$ so that, for any $1$-periodic functions $\psi _1, \psi _2 \in
H^w$ with $\| \psi _j\| _w \leq M$ and $|n| \geq N_2$, equation
\eqref{2.53} has exactly two (counted with multiplicity) solutions
$\zeta ^+_n,\zeta ^-_n$ in $\overline {\mathcal D}_{r_n}$.
\end{proposition}

\vskip 1 cm

\subsection{Proof of Theorem~\ref{Theorem1}}
\label{Ss: 2.8 Proof of Theorem 1}

In this section, Theorem~\ref{Theorem1} is proved.

\bigskip

{\it Proof of Theorem~\ref{Theorem1} (i) } Let $z^\pm _n:= z
(\zeta ^\pm _n)  = \zeta ^\pm _n + \alpha (n,
z^\pm _n)$ where $\zeta ^\pm _n$ are the two solutions of the
$\zeta $-equation provided by Proposition~\ref{Proposition2.21}
($|n| \geq N_2$). Then, for $|n| \geq N_2$
   \begin{equation}
   \label{2.57} |z^+_n - z^-_n| \leq |\zeta ^+_n - \zeta ^-_n|
                + \sup _{|z| \leq \frac {\pi }{4}} |\frac {d}{dz}
                \alpha (n,z)| |z^+_n - z^-_n| .
   \end{equation}
As $N_2 \geq N_1$ and $|n| \geq N_2$ one has by the analyticity of 
$z \mapsto \alpha (n,z)$ and \eqref{2.54}
   \[ \sup _{|z| \leq \frac {\pi }{4}} |\frac {d}{dz} \alpha (n,
      z)| \leq \frac {1}{2} .
   \]
Together with $|\zeta ^+_n - \zeta ^-_n| \leq |\zeta ^+_n| + 
|\zeta ^-_n| \leq 2r_n$ equation \eqref{2.57} then leads to
   \[ |z^+_n - z^-_n| \leq 4r_n .
   \]
By the definition \eqref{2.55} of $r_n$, the estimates of 
$\beta ^\pm _n$ in Proposition~\ref{Proposition2.14} (i) and
the identity $\lambda ^+_n -\lambda ^-_n = z^+_n - z^-_n$, the
latter equation implies that there exists $C \geq 1$ such
that, for any $1$-periodic functions $\psi _1, \psi _2 \in H^w,
\| \psi _j\| _w \leq M$,
   \[ \sum _{|n| \geq N_2} w(2n)^2 |\lambda ^+_n - \lambda ^-_n|
      ^2 \leq C.
   \]
$\blacksquare $

\bigskip

Towards the proof of Theorem~\ref{Theorem1} (ii), rewrite 
equation \eqref{2.53},
   \begin{equation}
   \label{2.68} (\zeta ^\pm _n)^2 - \rho ^2_n = \eta (n,z
                (\zeta ^\pm _n))
   \end{equation}
where
   \[ \rho _n = \left( (\hat {\psi }_2 (2n) + \beta ^+_0(n))(\hat
      {\psi }_1(-2n) + \beta ^-_0(n))\right) ^{1/2}
   \]
with an arbitrary but fixed choice of the square root and
   \begin{align}
   \label{2.69} \eta (n,z) &= \hat {\psi }_2(2n)(\beta ^-(n,z) -
                   \beta ^-_0(n)) \nonumber \\
                &+ \hat {\psi }_1(-2n)(\beta ^+(n,z) - \beta ^+_0
                   (n)) \\
                &+ (\beta ^-(n,z) - \beta ^-_0(n))(\beta ^+(n,z)
                   - \beta ^+_0(n)) \nonumber .
   \end{align}
In view of the definition \eqref{2.55} and as $w$ is assumed to be
a $\delta $-weight, we have for some constant $C_1 \geq 1$ 
depending on $\delta $ and $M$ 
   \begin{equation}
   \label{2.70} r_n \leq \frac {C_1}{\langle n\rangle ^{\delta 
               _\ast }} \quad (\forall |n| \geq N_0) .
   \end{equation}
By Lemma~\ref{Lemma2.9} there exists $C_2 \geq 1$ depending on 
$\delta $ and $M$ such that for $|n| \geq N_0$ and $|z| \leq \pi 
/2$
   \begin{equation}
   \label{2.71} |\alpha (n,z)| \leq \frac {C_2}{\langle n\rangle 
                ^{\delta _\ast }} .
   \end{equation}
Let $A = C_1 + C_2$ and define
   \begin{equation}
   \label{2.72} s_n:= \sup _{|z| \leq 2A/\langle n\rangle ^{\delta
                _\ast }} |\eta (n,z)| .
   \end{equation}
Notice that by Proposition~\ref{Proposition2.14} (ii), there exists
$C > 0$ so that
   \begin{equation}
   \label{2.73} \sum _{|n| \geq N_2} \langle n\rangle ^{3\delta 
                _\ast } w(2n)^2 s_n \leq C .
   \end{equation}
Choose $N_3 \geq N_2$, depending on $\delta $ and $M$, so that
   \begin{equation}
   \label{2.74} \frac {\langle n\rangle ^{\delta _\ast }}{A} 
                \sqrt{s_n} < \frac {1}{2} , \quad \forall |n| 
                \geq N_3 .
   \end{equation}

\bigskip

\begin{lemma}
\label{Lemma2.22} Let $M \geq 1, 0 < \delta $ and $w$ be a 
$\delta $-weight. For $1$-periodic functions $\psi _1, \psi _2$ in
$H^w$ with $\| \psi _j\| _w \leq M (j = 1,2)$ and $|n| \geq N_3$,
   \[ |\zeta ^+_n - \rho _n| + |\zeta ^-_n + \rho _n| \leq 6 
      \sqrt{s_n } 
   \]
{\bf or }
   \[ |\zeta ^+_n - \rho _n| + |\zeta ^-_n - \rho _n| \leq 6 
      \sqrt{s_n } .
   \]
\end{lemma}

\medskip

{\it Proof } W.l.o.g. assume that $\delta \leq 1/2$ and hence
$\delta = \delta _\ast $. By \eqref{2.68} we have for $|n| \geq
N_3$
   \begin{equation}
   \label{2.75} (\zeta ^\pm _n - \rho _n)(\zeta ^\pm _n + \rho 
                _n) = \eta (n,z (\zeta ^\pm _n)) .
   \end{equation}
By definition, $z(\zeta ^\pm _n) = \zeta ^\pm _n + \alpha (n,z
(\zeta ^\pm _n))$ and therefore from \eqref{2.70}, \eqref{2.71}
and Proposition~\ref{Proposition2.21}, we conclude for $|n| \geq
N_3$,
   \begin{equation}
   \label{2.76} |z(\zeta ^\pm _n)| \leq \frac {A}{\langle n
                \rangle ^\delta } .
   \end{equation} 
From the definition of $s_n$ (see \eqref{2.72}) and \eqref{2.75}
we deduce
   \begin{equation}
   \label{2.77} |\zeta ^\pm _n - \rho _n| |\zeta ^\pm _n + \rho
                _n| \leq s_n .
   \end{equation}
Thus $\min _\pm |\zeta ^+_n \pm \rho _n| \leq \sqrt{s_n}$ and 
$\min _\pm  |\zeta ^-_n \pm \rho _n| \leq s^{1/2}_n$. We distinguish
two cases:

\medskip

{\bf case 1 } $|\rho _n| \leq 2 \sqrt{s_n}$. In this case $|\zeta
^\pm _n - \rho _n| \leq \sqrt{s_n}$ implies
   \[ |\zeta ^\pm _n + \rho _n| \leq |\zeta ^\pm _n - \rho _n| +
      2 |\rho _n| \leq 5 \sqrt{s_n}
   \]
and, similarly, $|\zeta ^\pm _n + \rho _n| \leq \sqrt{s_n}$ implies
$|\zeta ^\pm _n - \rho _n| \leq 5 \sqrt{s_n}$, thus 
Lemma~\ref{Lemma2.22} is proves in case 1.

\medskip

{\bf case 2 } $|\rho _n| > 2 \sqrt{s_n}$. It suffices to show that 
it is impossible to have $\max _\pm |\zeta ^\pm _n - \rho _n| \leq 
\sqrt{s_n}$, or $\max _\pm |\zeta ^\pm _n + \rho _n| \leq \sqrt{s_n}$.
To the contrary, assume that
   \begin{equation}
   \label{2.101} \max _\pm |\zeta ^\pm _n - \rho _n| \leq \sqrt
                 {s_n} .
   \end{equation}
(The other case is treated in the same way.) By \eqref{2.101},
$|\zeta ^\pm _n + \rho _n| \geq 2 |\rho _n| - \sqrt{s_n} > \frac
{3}{2} |\rho _n|$, hence 
   \begin{equation}
   \label{2.78} |\zeta ^+_n + \zeta ^-_n| \geq |\zeta ^+_n + \rho 
                _n| - |\zeta ^-_n - \rho _n| > |\rho _n| .
   \end{equation}
Divide
   \[ (\zeta ^+_n)^2 - (\zeta ^-_n)^2 = \eta (n,z(\zeta ^+_n)) -
      \eta (n,z(\zeta ^-_n)) 
   \]
by $\zeta ^+_n + \zeta ^-_n$ and use \eqref{2.78} and \eqref{2.76}
to deduce
   \begin{equation}
   \label{2.79} |\zeta ^+_n - \zeta ^-_n| \leq \frac {1}{|\rho _n|}
                \sup _{|z| \leq A/\langle n\rangle ^\delta }
                |\frac {d\eta}{dz} (n,z)|  |z(\zeta ^+_n) - z(\zeta
                ^-_n)| .
   \end{equation}
To arrive at a contradiction we first show that $\zeta ^+_n - 
\zeta ^-_n = 0$. As $|z(\zeta ^+_n) - z(\zeta ^-_n)| \leq |\zeta ^+
_n - \zeta ^-_n| + \sup _{|z| \leq \pi / 2} |\frac {d}{dz} \alpha
(n,z) | |z(\zeta ^+_n) - z(\zeta ^-_n)|$, \eqref{2.54} leads to
$(|n| \geq N_2)$
   \begin{equation}
   \label{2.80} |z(\zeta ^+_n) - z(\zeta ^-_n)| \leq 2|\zeta ^+_n
                - \zeta ^-_n| .
   \end{equation}
On the other hand, as $z \mapsto \eta (n,z)$ is analytic in $\{ z,
|z| < \pi / 2 \}$, we have by Cauchy's inequality, 
   \begin{align}
   \begin{split}
   \label{2.81} \sup _{|z| \leq \frac {A}{\langle n\rangle ^\delta }}
                   |\frac {d}{dz} \eta (n,z)| &\leq \frac {\langle n
                   \rangle ^\delta }{A} \sup _{|z| \leq \frac {2A}
                   {\langle n\rangle ^\delta }} |\eta (n,z)| \\
                &\leq \frac {\langle n\rangle ^\delta }{A} s_n .
   \end{split}
   \end{align}
Combining \eqref{2.79} - \eqref{2.80} with \eqref{2.74} we obtain,
   \begin{align*} |\zeta ^+_n - \zeta ^-_n| &\leq \frac {2}{|\rho
                     _n|} \frac {\langle n\rangle ^\delta }{A}
                     s_n |\zeta ^+_n - \zeta ^-_n| \\
                  &\leq \frac {\langle n\rangle ^\delta }{A}
                     \sqrt{s_n} |\zeta ^+_n - \zeta ^-_n| \leq
                     \frac {1}{2} | \zeta ^+ _n - \zeta ^-_n |
   \end{align*} 
and we conclude that $\zeta ^+_n = \zeta ^-_n \equiv \zeta _n$. This
contradicts the assumption $|\rho _n| > 2 \sqrt{s_n}$ as one can 
see in the following way: By the equation \eqref{2.68}, $2\zeta _n =
\frac {d}{d\zeta } \eta (n,z(\zeta _n)) = \frac {d}{dz} \eta (n,z
(\zeta _n)) \cdot \frac {d}{d\zeta } z(\zeta _n)$. By \eqref{2.81},
$|\frac {d}{dz} \eta (n,z(\zeta _n)) | \leq \frac {\langle n
\rangle ^\delta }{A} s_n$ and by \eqref{2.54}, $|\frac {d}{d\zeta }
z(\zeta )| = |\frac {d}{d\zeta }(\zeta + \alpha (n,z(\zeta )))|
\leq 1 + \frac {1}{2} \leq 2$, hence
   \begin{equation}
   \label{2.82} |\zeta _n| \leq \frac {\langle n\rangle ^\delta }
                {A} s_n 
   \end{equation}
and, by \eqref{2.78},
   \[ |\rho _n| < 2|\zeta _n| \leq 2 \frac {\langle n\rangle 
      ^\delta }{A} s_n < \sqrt{s_n} .
   \]
where for the last inequality we used \eqref{2.74}.
$\blacksquare $

\bigskip

{\it Proof of Theorem~\ref{Theorem1} (ii): } Let $N_3$ be given
by \eqref{2.74}. Recall that
  \[ \lambda ^+_n - \lambda ^-_n = z^+_n - z^-_n = \zeta ^+_n -
     \zeta ^-_n + \alpha (n,z(\zeta ^+_n)) - \alpha (n,z(\zeta ^-
     _n)) .
   \]
By Lemma~\ref{Lemma2.22}, for $|n| \geq N_3$, 
$$\min _\pm |(\zeta ^+_n
- \zeta ^-_n) \pm 2\rho _n |\leq 6 \sqrt{s_n}.$$
By the analyticity of $\alpha(n,z)$ and Lemma~\ref{Lemma2.9}, for $|n| \geq N_0$,  
   \[ \sup _{|z| \leq \pi /4} |\frac {d}{dz} \alpha (n,z)| \leq
      \frac {C}{\langle n\rangle ^{2\delta }} .
   \]
Combining these two estimates, we get for $|n| \geq N_3$,
   \begin{align*} \min _\pm |(\lambda ^+_n - \lambda ^-_n) &\pm 2 
                     \rho _n| \leq \min _\pm |(\zeta ^+_n - \zeta 
                     ^-_n) \pm 2 \rho _n| \\
                  &+ \left( \sup _{|z| \leq \pi /2} |\frac {d}{dz}
                     \alpha (n,z)|\right) |\lambda ^+_n - \lambda 
                     ^-_n| \\
                  &\leq 6\sqrt{s_n} + C \frac {|\lambda ^+_n - 
                     \lambda ^-_n|}{\langle n\rangle ^{2\delta }}\ .
   \end{align*}
Hence, by \eqref{2.73} and Theorem~\ref{Theorem1} (i),
   \[ \sum _{|n| \geq N_3} \langle n\rangle ^{3\delta } w(2n)^2
      \min _\pm |(\lambda ^+_n - \lambda ^-_n) \pm 2\rho _n|^2
      \leq C .
   \]
$\blacksquare $

\vskip 1 cm

\subsection{Improvement of Theorem~\ref{Theorem1} for $L$ 
            selfadjoint }
\label{Ss: 2.9 Improvement of Theorem 1}

For $\psi $ a $1$-periodic functions in $H^w$, the operator $L(\psi ,
\overline {\psi })$ is selfadjoint. In this section we show that in 
this case the decay rate of the asymptotics in 
Theorem~\ref{Theorem1}  (ii) can be improved as follows :

\medskip

\begin{guess}
\label{Theorem2.23} Let $M \geq 1, \delta > 0$ and $w$ be a 
$\delta $-weight. Then there exist constants $1 \leq C < \infty $
and $1 \leq N < \infty $ so that for any $|n| \geq N$ and any
$1$-periodic function $\psi \in H^w$ with $\| \psi \| _w \leq M$,
   \[ \sum _{|n| \geq N} \langle n\rangle ^{4\delta _\ast } w
      (2n)^2 \min _\pm |(\lambda ^+_n - \lambda ^-_n) \pm 2 \rho 
      _n|^2 \leq C .
   \]
\end{guess}

\medskip

{\it Proof } Using the definition \eqref{2.13} with $\psi _1 = 
\psi $ and $\psi _2 = \overline {\psi }$ we get $\overline {R^{(1)}
_n(k,j)}(\overline {z}) = R^{(2)}_n(k,j)(z)$ and thus 
$\overline {\beta ^-(n,z)} = \beta ^+(n,\overline {z})$. As
the eigenvalues $\lambda ^\pm _n = n\pi + z(\zeta ^\pm _n)$ of
$L(\psi ,\overline {\psi })$ are real, equation \eqref{2.53}
then reads (with $|n| \geq N_2$ and $N_2$ as in 
Proposition~\ref{Proposition2.21})
   \begin{align}
   \begin{split}
   \label{2.83} (\zeta ^\pm _n)^2 &= |\hat {\psi}(2n) + \beta ^+ (n,
                   z(\zeta ^\pm _n))|^2 \\
                &= |\hat {\psi }(2n) + \beta ^+_0(n) + \tilde
                   {\beta }^+(n,z(\zeta ^\pm _n))|^2
   \end{split}
   \end{align}
and $\rho _n$ is given by (with an appropriate choice of the square
root)
   \[ \rho _n = | \hat {\psi }(2n) + \beta ^+_0(n)| .
   \]
Let $t_n:= \sup _{|z| \leq A / \langle n\rangle ^{\delta _\ast }}
|\tilde {\beta }^+(n,z)|$, where $A:= C_1 + C_2$ and $C_1$ are 
$C_2$ are defined by \eqref{2.70}, \eqref{2.71}. By
Proposition~\ref{Proposition2.14} (ii),
   \begin{equation}
   \label{2.84} \sum _{|n| \geq N_0} \langle n \rangle ^{6 \delta
                _\ast } w(2n)^2 t^2_n \leq C .
   \end{equation}
From \eqref{2.83} we deduce $\min _\pm |\zeta ^+_n \pm \rho _n|
\leq t_n$ and $\min _\pm |\zeta ^-_n \pm \rho _n| \leq t_n$.
Substituting Lemma~\ref{Lemma2.24} below for Lemma~\ref{Lemma2.22},
Theorem~\ref{Theorem2.23} follows in the same way as
Theorem~\ref{Theorem1} (ii).
$\blacksquare $

\bigskip

Define $N_4 \geq N_2$ such that
   \[ 12 \langle n \rangle ^\delta t_n < A \quad \forall |n| \geq
      N_4 .
   \]

\medskip 

\begin{lemma}
\label{Lemma2.24} Let $M \geq 1, \delta > 0$ and $w$ be a 
$\delta $-weight. For any $1$-periodic function $\psi $ in $H^w$
with $\| \psi \| _w \leq M$ and $|n| \geq N_4$,
   \[ |\zeta ^+_n - \rho _n | + |\zeta ^-_n + \rho _n| \leq 6 t_n
   \]
or
   \[ |\zeta ^+_n + \rho _n | + |\zeta ^-_n - \rho _n| \leq 6 t_n .
   \]

\end{lemma}

\medskip

{\it Proof } The proof is similar to the one of 
Lemma~\ref{Lemma2.22}.
$\blacksquare $

\vskip 2 cm

\section{Riesz spaces and normal form of $L$}
\label{S: 3 Riesz' spaces and normal form of $L$}

\subsection{Riesz spaces}
\label{Ss: 3.1 Riesz spaces}

Let $M \geq 1, \delta > 0$ and $w$ be a $\delta $-weight. By
Theorem~\ref{Theorem1}, there exists $1 \leq N < \infty $ so
that for any $1$-periodic functions $\psi _1, \psi _2$ in $H^w$
with $\| \psi _j\| _w \leq M$, the operator $L = L(\psi _1,
\psi _2)$ has two (counted with multiplicity) periodic 
eigenvalues $\lambda ^+_n,\lambda ^-_n$ near $n\pi $.

In Appendix~\ref{AppendixA} we introduce the periodic and
antiperiodic boundary conditions $bc$ $Per^+$ and $bc$ $Per^-$. We
point out that
   \[ spec L = spec L_{Per^+} \cup spec L_{Per^-} 
   \]
and introduce the Riesz projectors $\Pi _{2n} : L^2([0,1];
{\mathbb C}^2) \rightarrow L^2([0,1]; {\mathbb C}^2)$,
corresponding to $bc Per ^+$ and $\Pi _{2n-1} : L^2([0,1];
{\mathbb C}^2) \rightarrow L^2([0,1];{\mathbb C}^2)$ ,
corresponding to $bc Per ^-$ ($n \in {\mathbb Z}$). Further
denote by $E_n$ the ${\mathbb C}$-vector spaces
   \[ E_n:= \Pi _n(L^2([0,1];{\mathbb C}^2)) \quad (|n| \geq
      N) .
   \]
Notice that $\dim _{\mathbb C} E_n = tr \Pi _n = 2$ $\forall |n| 
\geq N$. If $\lambda ^+_n \not= \lambda ^-_n$ or $\lambda ^+_n
= \lambda ^-_n$ is of geometric multiplicity two, there exists
a basis of $E_n$ consisting of eigenfunctions $F^+$ and $F^-$
corresponding to the eigenvalues $\lambda ^\pm _n$. If $\lambda
^+_n = \lambda ^-_n$ is of geometric multiplicity $1$, $E_n$ is
the root space of $\lambda ^+_n$. Denote by $F$ a
$L^2$-normalized eigenfunction of $L$ corresponding to the
eigenvalue $\lambda = n\pi + z$,
   \[ (L - \lambda )F = 0 , \quad \| F\| = 1 
   \]
where $\| \cdot \| $ denotes the $L^2$-norm in $L^2([0,1];
{\mathbb C}^2)$. Then 
   \begin{align*} F(x) = &x^F e ^+_n(x) + y^F e ^-_n
                     (x) + 
                   \sum _{k\not= n} (\breve {F}_2(k)e^+_k
                     (x) + \breve {F}_1(-k)e^-_k(x))
   \end{align*}
where 
   \[ \left( \begin{array}{c} \breve {F}_2 \\
                              J\breve {F}_1
             \end{array} \right) =
      \left( \begin{array}{c c} V_{11} & V_{12} \\
                                V_{21} & V_{22}
             \end{array} \right)
      \left( \begin{array}{c} y^F \\
                              x^F
             \end{array} \right)
   \]
with
   \begin{align*} V_{11} &= (z - D_n)^{-1}(Id - P_n)^{-1}
                     S^n \hat {\psi }_2 \\
                 V_{12} &= (z - D_n)^{-1}R^{(2)}_n (Id - Q_n)
                     ^{-1} S^n J \hat {\psi }_1 \\
                 V_{21} &= (z - D_n)^{-1}R^{(1)}_n (Id - P_n)
                     ^{-1} S^n \hat {\psi }_2 \\
                 V_{22} &= (z - D_n)^{-1}(Id - Q_n)^{-1}
                     S^n J \hat {\psi }_1  .
   \end{align*}

\bigskip

\begin{proposition}
\label{Proposition3.1} Let $0 < \delta \leq 1, M \geq 1$ and $w$
be a $\delta $-weight. Then there exist $C \equiv C(\delta ,M) 
\geq 1$ and $N \equiv N(M,\delta ) \geq 1$ such that for 
$1$-periodic functions $\psi _1, \psi _2 \in H^w$ with $\| \psi 
_j \| _w \leq M$ and $|n| \geq N$

\begin{description}
\item[(i)  ] $\frac {1}{2} \leq |x^F|^2 + |y^F|^2 \leq 1$ 

\item[(ii) ] $\| \breve {F}_2\| \leq 2 \frac {C}{\langle n \rangle
^\delta }; \ \| J \breve {F}_1 \| \leq 2 \frac {C}{\langle n
\rangle ^\delta } $ \\ where $\| \cdot \| $ stands for the 
$\ell ^2$-norm. 
\end{description}
\end{proposition}

\medskip

{\it Proof } As $\| F\| = 1$, we have 
   \[ \| F\| ^2 = |x^F|^2 + |y^F|^2 + \| \breve {F}_2\| ^2 +
      \| J \breve {F}_1\| ^2 = 1 .
   \]
Hence 
   \[ |x^F|^2 + |y^F|^2 \leq 1 .
   \]
Further, by Proposition~\ref{Proposition2.4}, for $|n| \geq N_0$
   \[ \| (Id - P_n)^{-1}\| _{{\mathcal L}(\ell ^2_{S^nw})} \leq 
      2 ; \ \| (Id - Q_n)^{-1}\| _{{\mathcal L}(\ell ^2_{S^nw})}
      \leq 2 .
   \]
By Corollary~\ref{Corollary2.3}, there exists $C > 1$ such that
   \[ \| R^{(j)}_n\| _{{\mathcal L}(\ell ^2_{S^nw},\ell ^2)} 
      \leq \frac {C}{\langle n\rangle ^{\delta \wedge 1}}
   \]
and by the definition of $D_n$, for some $1 \leq C < \infty $,
   \begin{align*} &\| (z - D_n)^{-1}\| _{{\mathcal L}(\ell ^2
                     _{S^nw},\ell ^2)} \leq \frac {C}{\langle
                     n\rangle ^{1 \wedge \delta }} \\
                  &\| (z - D_n)^{-1}\| _{{\mathcal L}(\ell ^2,
                     \ell ^2)} \leq 1 .
   \end{align*}
Hence for $|n| \geq N_0$
   \begin{align*} &\| V_{11}\| + \| V_{22}\| \leq \frac {C}
                     {\langle n\rangle ^{\delta \wedge 1}} \\
                  &\| V_{12}\| + \| V_{21}\| \leq \frac {C}
                     {\langle n\rangle ^{\delta \wedge 1}} 
   \end{align*}
for some $1 < C < \infty $ and one concludes that
   \begin{align*} &\| \breve {F}_2\| \leq \frac {C}{\langle n
                     \rangle ^{\delta \wedge 1}} (|x^F| + |y
                     ^F|) \leq 2 \frac {C}{\langle n \rangle
                     ^{\delta \wedge 1}} \\
                  &\| J \breve {F}_1\| \leq \frac {C}{\langle n
                     \rangle ^{\delta \wedge 1}} (|x^F| + |y
                     ^F|) \leq 2 \frac {C}{\langle n \rangle
                     ^{\delta \wedge 1}} .
   \end{align*}
By choosing $N \geq N_0$ sufficiently large we have, for $|n| 
\geq N$,
   \[ \| \breve {F}_2\| ^2 + \| J \breve {F}_1\| ^2 \leq \frac
      {1}{2}
   \]
and hence $\frac {1}{2} \leq |x^F|^2 + |y^F|^2$.
$\blacksquare $

\vskip 1 cm

\subsection{Normal form of $L$}
\label{Ss: 3.2 Normal form of $L$}

In this section we want to derive a normal form of the
restriction of $L$ to the Riesz spaces $E_n$. For this purpose
introduce an orthonormal basis of $E_n$ as follows: Choose
$F \equiv F^+$ to be an $L_2$-normalized eigenfunction of $L$
corresponding to the eigenvalue $\lambda ^+ \equiv \lambda ^+_n$
and $\Phi \in E_n$ with
   \[ (\Phi ,F) = 0 ; \ \| \Phi \| = 1
   \]
where, as usual, $(\Phi ,F) = \int ^1_0 \overline {\Phi (x)}F
(x)dx$. In case $\lambda ^+$ is a double eigenvalue,
   \begin{equation}
   \label{3.1} \left( \begin{array}{c} L\Phi \\
                                       LF
                      \end{array} \right) =
               \left( \begin{array}{c c} \lambda ^+ 
                      &\xi \\
                      0 &\lambda ^+
                      \end{array} \right)
               \left( \begin{array}{c} 
                      \Phi \\
                      F
                      \end{array} \right)
   \end{equation}
where $\xi \equiv \xi _n$ vanishes iff $\lambda ^+$ is of geometric
multiplicity two.

In case $\lambda ^-_n \not= \lambda ^+_n$, choose an 
$L_2$-normalized eigenfunction $F^-$ of $\lambda ^- \equiv \lambda 
^-_n$. Then
   \[ F^- = a F + b \Phi ; \ |a|^2 + |b|^2 = 1 ; \ b \not= 0 .
   \]
With $\Phi = \frac {1}{b} F^- - \frac {a}{b}F$,
   \begin{align*} L \Phi &= \lambda ^- \frac {1}{b} F^- + \lambda
                     ^+ \frac {a}{b} F \\
                  &= \lambda ^-(\frac {1}{b} F^- - \frac {a}{b}
                     F) - \gamma \frac {a}{b} F
   \end{align*}
where $\gamma \equiv \gamma _n := \lambda ^+ - \lambda ^-$. Hence
   \begin{equation}
   \label{3.2} \left( \begin{array}{c} L\Phi \\
                                       LF
                      \end{array} \right) =
               \left( \begin{array}{c c} \lambda ^- 
                      &\xi \\
                      0 &\lambda ^+
                      \end{array} \right)
               \left( \begin{array}{c} 
                      \Phi \\
                      F
                      \end{array} \right)
   \end{equation}
with $\xi \equiv \xi _n:= - \gamma \frac {a}{b}$. Notice that
\eqref{3.1} and \eqref{3.2} have the same form. We refer to this
form as the normal form of the restriction of $L$ to the Riesz
space $E_n$.

In the remaining part of this section we want to estimate the
size of $(\xi _n)_{|n| \geq N}$. To this end, we write the equation
$(L - \lambda ^-)\Phi = \xi F$ in the basis $e^+_k, e^-_k (k \in
{\mathbb Z})$. With $\Phi = x^\Phi e^+_n +  y^\Phi e^-_n +
\sum _{k\not= n} \breve {\Phi }_2 (k) e^+_k + \breve {\Phi }_1(-k)
e^-_k$ and $F = x^F e^+_n + y^F e^-_n + \sum _{k\not= n} 
\breve {F}_2(k) e^+_k + \breve {F}_1(-k)e^-_k$, we then obtain
the following inhomogeneous system (cf. \eqref{2.8} - 
\eqref{2.10})
   \begin{align}
   \label{3.3}  - z^- x^\Phi + \hat {\psi }_2(2n)y^\Phi &+ \langle
                   S^n \hat {\psi }_2, J\breve {\Phi }_1 \rangle
                   = \xi x^F \\
   \label{3.4}  \hat {\psi }_1(-2n)x^\Phi - z^- y^\Phi &+ \langle
                   S^n J\hat {\psi }_1, \breve {\Phi }_2 \rangle 
                   = \xi y^F \\
   \label{3.5} \left( \begin{array}{c} y^\Phi (S^n \hat {\psi }_2)
                      _{{\mathbb Z} \backslash n} \\
                      x^\Phi (S^n J \hat {\psi }_1)_{{\mathbb Z}
                      \backslash n}  
                      \end{array} \right) &+ (A_n - z^-)
               \left( \begin{array}{c}  
                      \breve {\Phi }_2 \\
                      J \breve {\Phi }_1
                      \end{array} \right) = \xi 
               \left( \begin{array}{c} 
                      \breve {F}_2 \\
                      J \breve {F}_1
                      \end{array} \right)
   \end{align}
where, as usual, $\lambda ^-_n \equiv \lambda ^- = n\pi + z^-$.
We use the above system to obtain an estimate for $\xi \equiv
\xi _n$. 

Write $\breve {\Phi } = (\breve {\Phi }_2,J\breve {\Phi }_1)$ and
$\breve {F} = (\breve {F}_2, J\breve {F}_1)$. Recall that $w$ is
assumed to be a $\delta $-weight and hence by
Corollary~\ref{Corollary2.5}, equation \eqref{3.5} (with $|n| \geq N_0$)
can be solved for $\breve {\Phi }$,
   \[ \breve {\Phi } = (z^- - A_n)^{-1}
      \left( \begin{array}{c}
             y^\Phi (S^n \hat {\psi }_2)_{{\mathbb Z} \backslash
                n} \\
             x^\Phi (S^n J \hat {\psi }_1)_{{\mathbb Z}
                \backslash n}
             \end{array} \right) 
      - \xi (z^- - A_n)^{-1} \breve {F} .
   \]
In this form, $\breve {\Phi }$ is substituted into \eqref{3.3} -
\eqref{3.4} to obtain (cf. Corollary~\ref{Corollary2.5})
   \begin{align}
   \begin{split}
   \label{3.6} & \left( \begin{array}{c c}
                 -z^- + \alpha (n,z^-)  & \hat {\psi }_2(2n) +
                    \beta ^+(n,z^-) \\
                 \hat {\psi }_1(-2n) + \beta ^-(n,z^-)  &-z^-
                    + \alpha (n,z^-)
                 \end{array} \right) 
                 \left( \begin{array}{c}
                 x^\Phi \\
                 y^\Phi 
                 \end{array} \right) \\
               &= \xi
                 \left( \begin{array}{c}
                 x^F \\
                 y^F
                 \end{array} \right)
                 + \xi \langle
                 \left( \begin{array}{c}
                 S^n \hat {\psi }_2 \\
                 S^n J \hat {\psi }_1
                 \end{array} \right) ,
                 (z^- - A_n) ^{-1} \breve {F} \rangle .  
   \end{split}
   \end{align}
Denote the right side of \eqref{3.6} by $RS$. By
Corollary~\ref{Corollary2.5} $(z^- - A_n)^{-1}$ is uniformly
bounded for $|n|$ sufficiently large and by
Proposition~\ref{Proposition3.1}, for $|n| \geq N$,
   \[ \frac {1}{2} \leq |x^F|^2 + |y^F|^2 ; \ \| \breve {F}\|
      \leq \frac {C}{\langle n \rangle ^\delta } .
   \]
Hence $RS$ can be estimated from
below: There exists $1 \leq C \equiv C_{\delta ,M} < \infty $
so that for $|n| \geq N$ ($N$ as in 
Proposition~\ref{Proposition3.1}) 
   \[ RS \geq |\xi | (\frac {1}{\sqrt{2}} - \frac {C}{\langle
      n\rangle ^\delta }) .
   \]
By choosing $N$ larger if necessary, we can assume that
   \begin{equation}
   \label{3.7} \frac {1}{\sqrt{2}} - \frac {C}{\langle n \rangle
               ^\delta } \geq \frac {1}{2} \quad \forall |n|
               \geq N
   \end{equation}
and \eqref{3.6} leads to 
   \begin{equation}
   \label{3.8} |\xi _n| \leq 4(|\zeta ^-_n| + |\hat {\psi }_1(-2n)|
               + |\hat {\psi }_2(2n)| + |\beta ^+(n,z^-)| + 
               |\beta ^-(n,z^-)|)
   \end{equation}
where we used that $|x^\Phi |^2 + |y^\Phi |^2 \leq 1$ and $\zeta ^-
_n = z^- - \alpha (n,z^-)$ with $z^- \equiv z(\zeta ^-_n)$.

\bigskip

In view of Proposition~\ref{Proposition2.14} and 
Lemma~\ref{Lemma2.22}, one then concludes from \eqref{3.8} the
following

\medskip

\begin{proposition}
\label{Proposition3.2} Let $M \geq 1, 0 < \delta $, and $w$ be a
$\delta $-weight. Then there exist $1 \leq N < \infty , 1 \leq C
= C_\delta < \infty $ such that for any $1$-periodic functions
$\psi _1, \psi _2 \in H^w$ with $\| \psi _j\| _w \leq M$
   \[ \sum _{|n| \geq N} w(2n)^2 | \xi _n|^2 \leq C .
   \]
\end{proposition}

\vskip 2 cm

\section{Dirichlet eigenvalues}
\label{S: 4 Dirichlet eigenvalues}

\subsection{Dirichlet boundary value problem}
\label{Ss: 4.1 Dirichlet boundary value problem}

Consider the Zakharov-Shabat operator $L \equiv L(\psi _1,\psi _2)$
on the interval $[0,1]$.

\medskip

\begin{definition} 
$F = (F_1,F_2) \in H^1([0,1];{\mathbb C}^2)$
satisfies {\it Dirichlet boundary conditions } if 
   \begin{equation}
   \label{4.1} F_1(0) - F_2(0) = 0 ; \ F_1(1) - F_2(1) = 0 .
   \end{equation}
\end{definition}


We mention that the Dirichlet boundary conditions take a more
familiar form when the operator $L$ is written as an $AKNS$
operator $L_{AKNS}$
   \begin{equation}
   \label{4.2} L_{AKNS} = \left( \begin{array}{c c}
                          0 &-1 \\
                          1 & 0 
                          \end{array} \right)
               \frac {d}{dx} + \left( \begin{array}{c c}
                               -q &p \\
                               p  &q
                               \end{array} \right)
   \end{equation}
where $(\psi _1,\psi _2)$ and $(p,q)$ are related by
   \[ \psi _1 = -q + ip ; \ \psi _2 = - q - ip .
   \]
If $F = (F_1,F_2) \in H^1 ([0,1];{\mathbb C}^2)$ satisfies $LF
= \lambda F$, then $L \tilde {F} = \lambda \tilde {F}$ where
$\tilde {F} = (\tilde {F}_1,\tilde {F}_2)$ is given by
   \[ \tilde {F}_1 = \frac {1}{\sqrt {2}i} (F_1 + F_2) ; \
      \tilde {F}_2 = \frac {1}{\sqrt {2}}(F_2 - F_1) . 
   \]
The Dirichlet boundary conditions \eqref{4.1} then take the
familiar form 
   \[ \tilde {F}_2(0) = 0 ; \ \tilde {F} _2(1) = 0 .
   \]
For the remaining part of section 4, let $M \geq 1, \delta > 0$,
and a $\delta $-weight $w$ be given as well as arbitrary
$1$-periodic functions $\psi _1,\psi _2 \in H^w$ with $\| \psi _j
\| _w < M$. In Appendix~\ref{AppendixA} we have introduced, for
$|n| \geq N$ with $N$ given by Theorem~\ref{Theorem1}, the
Riesz projectors $\Pi _{2n}, \Pi _{2n-1}$ corresponding to
periodic resp. antiperiodic boundary value problem on $[0,1]$
for $L$ and the two dimensional subspaces $E_n = Range (\Pi _n)$.

The following proposition assures that there exists a 
$1$-dimensional subspace of $E_n$ which satisfies Dirichlet
boundary conditions. Let $(F,\Phi )$ denote the orthonormal
basis of $E_n \subseteq L^2([0,1];{\mathbb C}^2)$, introduced
in section~\ref{Ss: 3.2 Normal form of $L$}.

\medskip

\begin{proposition}
\label{Proposition4.1} For any $|n| \geq N$, there exists $G =
(G_1,G_2) \in E_n$
   \[ G = \alpha F + \beta \Phi ; \ |\alpha |^2 + |\beta |^2 = 
      1
   \]
which satisfies Dirichlet boundary conditions
   \[ G_1(0) - G_2(0) = 0 ; \ G_1(1) - G_2(1) = 0 .
   \]
\end{proposition}

\medskip

{\it Proof } First consider the case where $F$ satisfies $F_1(0) -
F_2(0) = 0$. As $F$ is either periodic or antiperiodic we conclude
that $F_1(1) - F_2(1) = 0$ as well and thus $G:= F$ has the
required properties. If $F_1(0) - F_2(0) \not= 0$, notice that
   \[ \tilde {G}(x):= \left( F_1(0) - F_2(0)\right) \Phi (x) -
      \left( \Phi _1(0) - \Phi _2(0) \right) F(x)
   \]
satisfies Dirichlet boundary conditions. As $\tilde {G} 
\not\equiv 0$, we may define
   \[ G:= \frac {\tilde {G}}{\| \tilde {G}\| } .
   \]
$\blacksquare $

\bigskip

By \eqref{3.1} - \eqref{3.2}, $L\Phi = \lambda ^- \Phi + \xi F$
and $LF = \lambda ^+F$, hence, with $\gamma \equiv \gamma _n =
\lambda ^+ - \lambda ^-$ and $\lambda \equiv \lambda ^+$
   \begin{align}
   \begin{split}
   \label{4.5} LG &= \alpha \lambda F + \beta L\Phi \\
               &= \lambda G - \beta \gamma \Phi + \beta \xi F .
   \end{split}
   \end{align}
For $|n| \geq N$ sufficiently large, $\xi \equiv \xi _n$ and 
$\gamma \equiv \gamma _n$ are small and $G$ is almost a
Dirichlet eigenfunction. In the next sections we prove that
$\lambda \equiv \lambda ^+_n$ and $G$ are good approximations
of the Dirichlet eigenvalue $\mu \equiv \mu _n$ respectively
Dirichlet eigenfunction $H$.

\vskip 1 cm

\subsection{Decomposition}
\label{Ss: 4.2 Decomposition}

Let $L_{Dir}$ denote the closed operator $L_{Dir} = L(\psi _1,
\psi _2)$ with domain 
   \[ dom L_{Dir}:= \{ F \in H^1[0,1] \mid F_1(0) - F_2(0) = 0;
      F_1(1) - F_2(1) = 0 \} .
   \]
Let us fix $n$ with $|n| \geq N$ ($N$ as in 
Theorem~\ref{Theorem1}). $\Pi _{Dir} \equiv \Pi _{n,Dir}$ denotes
the Riesz projector
   \[ \Pi _{Dir}:= \frac {1}{2\pi i} \int _{|z-n\pi | = \frac 
      {\pi }{2}} (z - L_{Dir})^{-1} dz
   \]
acting on $L^2([0,1];{\mathbb C}^2)$ (cf. 
Appendix~\ref{AppendixA}). Let $\Omega _{Dir}:= Id - \Pi _{Dir}$.

Notice that
   \[ Range \Pi _{Dir} = \{ a H \mid a \in {\mathbb C} \}
   \]
where $H \in dom L_{Dir}$ is an $L^2[0,1]$-normalized eigenfunction for the
Dirichlet eigenvalue $\mu \equiv \mu _n$,
   \[ L_{Dir} H = \mu H ; \ \| H\| = 1 .
   \]
Let $\chi\in {\mathbb C}$ with $|\chi| \leq 1$ defined by
$\Pi _{Dir} G = \chi H$ where $G$ is given by Proposition~\ref{Proposition4.1}.
We have
   \[ G =\chi H  + \Omega _{Dir} G. 
   \]
As $G$ and $H$ are in $dom L_{Dir}$, 
$\Omega _{Dir} G \in dom L_{Dir}$ and
   \begin{equation}
   \label{4.8} L_{Dir} G = \chi \mu H +  L_{Dir}
               \Omega _{Dir} G = \chi \mu H + \Omega _{Dir} L_{Dir}
               \Omega _{Dir} G\, .
   \end{equation}
Where for the last equality we have used,  
$\Pi _{Dir} L_{Dir} \Omega _{Dir} G =0$, as $L_{Dir}$ and $\Pi _{Dir}$ 
commute on $dom L_{Dir}$ and 
$\Pi _{Dir} \Omega _{Dir} =0.$

On the other hand
by \eqref{4.5},
   \[ LG = \lambda G + R\, ; \ R = - \beta \gamma \Phi + \beta
      \xi F
   \]
and thus, with $G = \chi H + \Omega _{Dir} G$,
   \begin{equation}
   \label{4.9} L_{Dir} G = \lambda \chi H + \lambda \Omega _{Dir}
               G + (\Pi _{Dir} + \Omega _{Dir}) R .
   \end{equation}
Comparing the decompositions of the right sides of \eqref{4.8}
and \eqref{4.9} leads to the following

\medskip

\begin{lemma}
\label{Lemma4.2} 
   \begin{align}
   \label{4.10} \chi (\mu - \lambda )H &= \Pi _{Dir} R ; \\
   \label{4.11} (L_{Dir} - \lambda )(\Omega _{Dir} G) &= 
                   \Omega _{Dir} R
   \end{align}
where $R$ is given by
   \begin{equation}
   \label{4.12} R = - \beta \gamma \Phi + \beta \xi F .
   \end{equation}
\end{lemma}

\vskip 1 cm


\subsection{Proof of Theorem 1.2}
\label{Ss: 4.3 Proof of Theorem 2}

The equations \eqref{4.10} - \eqref{4.12} are now used to obtain
estimates for $|\mu _n - \lambda ^+_n |\ (|n| \geq N)$. For this
we need to establish that $|\chi| \leq 1$ is bounded away from $0$
and that $\| \Pi _{Dir} R\| $ is small. The latter is easily
seen as $\| R\| \leq |\gamma | + |\xi |$. To verify that $|\chi|$
is bounded away from $0$ we show that $\Omega _{Dir} G = G -
\chi G$ is small. This is proved by using equation \eqref{4.11}.

\bigskip

\begin{lemma}
\label{Lemma4.3} There exists $N \geq 1$ so that
   \[ | \chi _n| \geq \frac {1}{2} \quad \forall |n| \geq N .
   \]
\end{lemma}

\medskip

{\it Proof } As $G = \chi H + \Omega _{Dir} G$,
   \[ |\chi | \| H\| = \| G\| - \| \Omega _{Dir} G\| = 1 - \| 
      \Omega _{Dir} G\| .
   \]
By Lemma~\ref{Lemma4.2} and Lemma~\ref{LemmaA.2} (for 
\eqref{4.16}), Proposition~\ref{Proposition3.2} (for 
\eqref{4.17}), and Theorem~\ref{Theorem1} (i) (for 
\eqref{4.18}) there exist $1 \leq N < \infty $ and $1 \leq C <
\infty $ so that for $|n| \geq N$
   \begin{equation}
   \label{4.16} \| \Omega _{Dir} G\| = \| (L_{Dir} - \lambda )
                ^{-1} (\Omega _{Dir} R)\| \leq C\| R\| \leq C
                (| \xi _n| + |\gamma _n|)
   \end{equation}
   \begin{equation}
   \label{4.17} |\xi _n| \leq \frac {C}{\langle n\rangle 
                ^\delta }  
   \end{equation}
   \begin{equation}
   \label{4.18} |\gamma _n| \leq \frac {C}{\langle n\rangle
                ^\delta } ,
   \end{equation}
(where for the last two inequalities we used that $w$ is a 
$\delta $-weight). Combining the above inequalities shows that
for $|n|$ large enough
   \[ | \chi_n| \geq \frac {1}{2}  .
   \]
$\blacksquare $

\bigskip

{\it Proof of Theorem~\ref{Theorem2} } By \eqref{4.10},
   \[ |\chi | |\mu - \lambda | \| H\| = \| \Pi _{Dir} R\| .
   \]
By Lemma~\ref{Lemma4.3} there exists $N \geq 1$ so that for
$|n| \geq N$
   \[ | \mu _n - \lambda ^+_n| \leq 2C  \left( |\xi _n|
        + |\gamma _n| \right) 
   \]
where we have used that $\| \Pi _{Dir}\| \leq C$ (cf. 
Lemma~\ref{LemmaA.2}). The claimed estimate then follows from
the estimates of $\xi _n$ (Proposition~\ref{Proposition3.2})
and of $\gamma _n$ (Theorem~\ref{Theorem1} (i)).
$\blacksquare $

\vskip 2 cm

\appendix
\setcounter{equation}{0}

\section{Appendix A: Spectral properties of $L(\psi _1,\psi _2)$}
\label{AppendixA} 


In this appendix we consider the operator $L(\psi _1,\psi _2)$
($\psi _1, \psi _2$ $1$-periodic functions in $L^2([0,2],
{\mathbb C}^2)$) with various boundary conditions. For $bc \in
\{ Dir, Per ^\pm , Per\}$ denote by $L_{bc}$ the 
Zakharov-Shabat operator $L_{bc} = L(\psi _1,\psi _2)$ with the
following domains:
   \begin{align*} dom L_{Dir}:= &\{ F \in H^1[0,1] \mid F_1(0) - F_2
                     (0) = 0 ;\ F_1(1) - F_2(1) = 0\} ;\\
                  dom L_{Per^+}:= &\{ F \in H^1[0,1] \mid F(0) = F(1)
                     \} ; \\
                  dom L_{Per^-}:= &\{ F \in H^1[0,1] \mid F(0) = - F
                     (1) \} .
   \end{align*}
The operator $L \equiv L_{Per}$ is defined on the interval $[0,2]$
and has the following domain,
   \[ dom L_{Per}:= \{ F \in H^1[0,2] \mid F(0) = F(2)\} .
   \]
Let $spec _{bc} \equiv spec(L_{bc})$ be the spectrum of $L_{bc}$.
For potentials $\psi _1 = \psi _2 \equiv 0$, i.e. $L_0:= L(0,0)
= i \left( \begin{array}{c c} 1  & 0 \\      
                              0  &-1
           \end{array} \right) \frac {d}{dx}$, $spec _{bc}(L_0)$ can
be given explicitely:
   \begin{align}
   \label{A.1} spec_{Dir}(L_0) &= \{ k \pi \mid k \in {\mathbb Z}
                  \} ; \\
   \label{A.2} spec_{Per^+}(L_0) &= \{ 2k \pi \mid k \in {\mathbb Z}
                  \} ; \\
   \label{A.3} spec_{Per^-}(L_0) &= \{ 2(k + 1)\pi \mid k \in
                  {\mathbb Z} \} ; \\
   \label{A.4} spec_{Per}(L_0) &= \{ k \pi \mid k \in {\mathbb Z} \} .
   \end{align}

\medskip

\begin{proposition}
\label{PropositionA.1} Let $\delta > 0, M \geq 1$ and $w$ be a 
$\delta $-weight. There exists an even integer $N$ such that for
any $1$-periodic functions $\psi _1$ and $\psi _2$ in $H^w, \|
\psi _j\| _w \leq M$, the following statements hold:

\begin{description}
\item[(i)  ] for $bc \in \{ Dir, Per^\pm ,Per\} $,
   \[ spec _{bc} \subset \{ \lambda \in {\mathbb C}\mid |\lambda |
      < N \pi - \pi / 2\} \cup \left( \bigcup _{|k| \geq N} \{ 
      \lambda \in {\mathbb C}\mid |\lambda - k\pi | < \pi / 2\} 
      \right) ;
   \]

\item[(ii) ] for $|k| \geq N ,\ spec _{Per} \cap \{ \lambda \in
{\mathbb C} \mid |\lambda - k\pi | < \pi / 2\} $ contains exactly
one isolated pair of eigenvalues;

\item[(iii)] for $|k| \geq N$ and $ bc := Per^+ $  ($k$ 
even) and $bc := Per^- $ ($k$ odd), $spec _{bc}
\cap \{ \lambda \in {\mathbb C}\mid |\lambda - k\pi | < \pi / 2\} $
contains exactly one isolated pair of eigenvalues;

\item[(iv) ] for $|k| \geq N, spec _{Dir} \cap \{ \lambda \in
{\mathbb C} \mid |\lambda - k\pi | < \pi / 2\} $ contains exactly
one  eigenvalue;


\item[(v) ] the cardinality $N_{bc}$ of $spec_{bc } \cap \{
\lambda \in {\mathbb C} \mid |\lambda | < N\pi - \pi / 2\} $ is
equal to $4N - 2$ for $bc = Per, 2N - 1$ for $bc = Dir, 2N - 2$
for $bc = Per^+$ and $2N$ for $bc = Per^-$.
\end{description}
\end{proposition}

\medskip 

As $spec _{Per^+} \cup spec _{Per^-} \subseteq spec_{Per}$,
Proposition~\ref{PropositionA.1} implies 
   \[ spec _{Per} = spec _{Per^+} \cup spec_{Per^-} .
   \]

\medskip

{\it Proof } Define for $n \geq 1$, the union of contours,
   \[ \mathcal{R}_n = \{ \lambda \in {\mathbb C} \mid | \lambda | =
      n \pi - \pi / 2\} \cup \left( \bigcup _{|k| > n} \{ 
      \lambda \in {\mathbb C} \mid |\lambda - k \pi | = \pi
      / 2\} \right) .
   \]
By \eqref{A.1} - \eqref{A.4}, $(L_0 - \lambda ) : dom (L_{bc})
\rightarrow L^2$ is invertible for any $\lambda \in \mathcal{R}_n$, hence
   \begin{equation}
   \label{A.5} (L - \lambda ) = (L_0 - \lambda )(Id + Q
               _\lambda )
   \end{equation}
where 
   \[ Q_\lambda = (L_0 - \lambda )^{-1} 
      \left( \begin{array}{c c} 
             0  &\psi_1 \\
             \psi _2 &0 
             \end{array} \right) .
   \]
Using the orthogonal decomposition of $L^2$ by the eigenfunctions
of $(L_0)_{bc}$ and the assumption that $w$ is a $\delta $-weight,
one gets (with ${\mathcal L} \equiv {\mathcal L}(L^2)$)
   \begin{equation}
   \label{A.6} \| Q_\lambda \| _{\mathcal L} \leq M \left(
               \sum _{k \in {\mathbb Z}} \frac {1}{|k|^{2\delta }
               |k\pi - \lambda |^2} \right) ^{1/2} .
   \end{equation}
As $\max _{k \in {\mathbb Z}} \left( \frac {1}{|k|^{2\delta }
\langle k - n\rangle ^{1/2}} \right) ^{1/2} \leq \frac {1}{\langle
n\rangle ^{\delta \wedge 1/4}}$, one deduces from \eqref{A.6}
that, for $\lambda \in \mathcal{R}_n$,
   \begin{equation}
   \label{A.7} \| Q_\lambda \| _{\mathcal L} \leq \frac {M}
               {\langle n \rangle ^{\delta \wedge 1/4}} \left(
               \sum _{k \in {\mathbb Z}} \frac {1}{\langle k
               \rangle ^{3/2}} \right) ^{1/2} .
   \end{equation}
Let $N$ be an even integer such that 
   \[ \frac {M}{\langle n \rangle ^{\delta \wedge 1/4}} 
      \left( \sum _{k \in {\mathbb Z}} \frac {1}{\langle k
               \rangle ^{3/2}} \right) ^{1/2} \leq 1/2 .
   \]
Then, for $\lambda \in \mathcal{R}_n$ with $n \geq N$
   \begin{equation}
   \label{A.8} \| Q_\lambda \| _{\mathcal L} \leq 1/2 .
   \end{equation}
Combining \eqref{A.5} and \eqref{A.8}, one deduces that $(L -
\lambda ) : dom (L_{bc}) \rightarrow L^2$ is invertible for
any $\lambda \in \mathcal{R}_n (n \geq N)$ and any $1$-periodic functions
$\psi _1, \psi _2$ in $H^w$ with $\| \psi _j\| _w \leq M$. In
particular, $\mathcal{R}_n$ $(n \geq N)$ is contained in the resolvent set 
of $L_{bc}(t \psi _1, t\psi _2)$ for any $0 \leq t \leq 1$.
Hence the number of eigenvalues of $L_{bc}(t\psi _1, t\psi _2)$
in each connected component of the interior of $\mathcal{R}_n$ stays the
same for any $0 \leq t \leq 1$. To see that all eigenvalues are inside
$\mathcal{R}_n$ one chooses $n$ bigger and bigger. 
$\blacksquare $

\bigskip 

It follows
from Proposition~\ref{PropositionA.1} that the Riesz projectors
$\Pi _n$ and $\Pi _{n,Dir}$ are well defined (for any $|n| \geq
N$ and $1$-periodic functions $\psi _1, \psi _2$ with $\| \psi 
_j\| _w \leq M$)
   \begin{align*} \Pi _n:= &\frac {1}{2\pi i} \int _{|\lambda -
                     n\pi | = \pi / 2} (z - L_{Per^+})^{-1} dz \quad
                     (n \mbox { even }, |n| \geq N) , \\
                  \Pi _n:= &\frac {1}{2\pi i} \int _{|\lambda -
                     n\pi | = \pi / 2} (z - L_{Per^-})^{-1} dz \quad
                     (n \mbox { odd }, |n| \geq N) 
   \end{align*}
and
   \[ \Pi _{n,Dir} := \frac {1}{2\pi i} \int _{|\lambda -
      n\pi | = \pi / 2} (z - L_{Dir})^{-1} dz \ (|n| \geq N) ,
   \]
where the contours $\{ \lambda \mid |\lambda - n\pi | = \pi / 2\} $
in the integrals above are counterclockwise oriented. Furthermore,
using \eqref{A.5} and \eqref{A.8}, one deduces 

\medskip

\begin{lemma}
\label{LemmaA.2} Assume that the assumptions of
Proposition~\ref{PropositionA.1} hold. Then there exists a 
constant $1 \leq C \leq \infty $ such that for any $|n| \geq N$
(with $N$ as in Proposition~\ref{PropositionA.1})
   \[ \| \Pi _n\| _{{\mathcal L}(L^2[0,1])} \leq C
   \]
and 
   \[ \| \Pi _{n,Dir} \| _{{\mathcal L}(L^2[0,1])} \leq C .
   \]
\end{lemma}

\vskip 1 cm

\section{Appendix B: Proof of Lemma 2.8}
\label{AppendixB}

W.l.o.g. we may assume that $\delta _\ast = \delta $.

\medskip

(i) As $w_\ast $ is submultiplicative, one has 
   \begin{align}
   \begin{split}
   \label{B.1} w(2n) &= \langle 2n\rangle ^\delta w_\ast (2n) 
               \leq 2^\delta \langle n\rangle ^\delta w_\ast(n+k)
               w_\ast (k + j) w_\ast (j + n)
   \end{split}
   \end{align}
and, by assumption, $\langle n\rangle ^\alpha d_n(k) \leq d(k) \
(\forall k)$. This leads to 
   \begin{align} \begin{split} \label{B.10}
   \langle n \rangle ^{2\delta + \alpha } &w(2n)
                    \Psi _n(a,b,d_n)(2n) \\  
     &\leq \langle n\rangle ^{2\delta } w(2n) \sum
                    _k \frac {a(k + n)}{\langle k - n\rangle }
                    \sum _j \frac {b(k + j)}{\langle j - n
                    \rangle } d(j + n) \\
                 &\leq \sum _{k,j} K_n(k,j) \tilde {a}(k + n)
                    \tilde {b}(k,j) \tilde {d}(j + n) 
 \end{split}  \end{align}
where for any $u \in \ell ^2_w$ we denote by $\tilde {u}$ the
$\ell ^2$-sequence $\tilde {u}(j):= w(j)u(j)$ and $K_n(k,j)$ is
given by 
   \[ K_n(k,j):= \frac {2^\delta \langle n\rangle ^{3\delta }}
      {\langle k - n\rangle \langle j - n\rangle \langle k + n
      \rangle ^\delta \langle k + j \rangle ^\delta \langle
      j + n\rangle ^\delta } .
   \]
Notice that $K_n(k,j)$ is symmetric in $k$ and $j$. To estimate
$K_n(k,j)$ we need to consider four different regions:

\medskip

{\it Estimate of $K_n(k,j)$ in $|k - n| < \frac {|n|}{2}, |j - n|
< \frac {|n|}{2}$: } In this case
   \[ |k + n| \geq |n| ; \ |j + n| \geq |n| ; \ |k + j| \geq 2|n| 
      - |k - n| - |j - n| \geq |n| ,
   \]
hence
   \[ K_n(k,j) \leq \frac {2^\delta }{\langle k - n\rangle \langle
      j - n\rangle } \leq \frac {1}{\langle k - n\rangle ^2} +
      \frac {1}{\langle j - n\rangle ^2 } .
   \]

\medskip

{\it Estimate of $K_n(k,j)$ in $|k - n| \geq \frac {|n|}{2}, |j -
n| < \frac {|n|}{2}$: } In this case
   \[ |k - n| \geq \frac {|n|}{2} ; \ |j + n| > |n| ,
   \]
hence
   \[ K_n(k,j) \leq \frac {2^\delta }{\langle j - n\rangle 
      \langle k + j\rangle ^\delta } .
   \]

\medskip

{\it Estimate of $K_n(k,j)$ in $|k - n| < \frac {|n|}{2}, |j - n|
\geq \frac {|n|}{2}$: } Using the symmetry of $K_n(k,j)$ in $k$ and
$j$, the latter estimate leads to
   \[ K_n(k,j) \leq \frac {2^\delta }{\langle k - n\rangle \langle
      k+ j\rangle ^\delta } .
   \]

\medskip

{\it Estimate of $K_n(k,j)$ in $|k - n| \geq \frac {|n|}{2}, |j - n|
\geq \frac {|n|}{2}$ : } We get
   \[ K_n(k,j) \leq \frac {16^\delta }{\langle k - n\rangle ^{1-
      \delta } \langle k + j\rangle ^\delta \langle j + n\rangle
      ^\delta } .
   \]
Combining the above estimates one obtains for $k,j,n \in {\mathbb
Z}$,
   $$ K_n(k,j) \leq \frac {1}{\langle k - n\rangle ^2} +
               \frac {1}{\langle j - n\rangle ^2} + \frac {2}
               {\langle k + j\rangle ^\delta } \frac {1}{\langle 
               k - n\rangle } + \frac {4}{\langle k - n\rangle
               ^{1 - \delta } \langle k + j\rangle ^\delta 
               \langle j + n\rangle ^\delta } .
   $$
Therefore 
   \begin{align} &\sum _{k,j} K_n(k,j) \tilde {a}(k + n) \tilde
                     {b}(k + j)\tilde {d}(j + n) \nonumber \\
   \label{B.3}   &\leq \left( \tilde {a} \ast \frac {1}{\langle
                     k\rangle ^2} (J \tilde {b} \ast \tilde {d})
                     \right) (2n) + \left( \tilde {d} \ast \frac
                     {1}{\langle k\rangle ^2} (J \tilde {b} \ast
                     \tilde {a}) \right) (2n) + \\
                  &+ 2\left( \tilde {a} \ast \frac {1}{\langle k\rangle }
                     (\frac {J\tilde {b}}{\langle k\rangle ^\delta }
                     \ast \tilde {d})\right) (2n) + 4\left( \tilde
                     {a} \ast \frac {1}{\langle k\rangle ^{1-\delta }}
                     (\frac {J\tilde {b}}{\langle k\rangle ^\delta }
                     \ast \frac {\tilde {d}}{\langle k\rangle 
                     ^\delta })\right) (2n) \nonumber
   \end{align}
where for $u \in \ell ^2({\mathbb Z})$ and $\eta \geq 0, \frac {u}
{\langle k\rangle ^\eta }$ denotes the sequence given by $\left( \frac {u}
{\langle k\rangle ^\eta }\right) (j):= \frac {u(j)}{\langle j\rangle ^\eta }$ ($\forall j$).
Using the standard convolution estimates $\| u \ast v\| _{\ell ^2} 
\leq \| u\| _{\ell ^1} \| v\| _{\ell ^2}$ and  $\| u \ast v\| _{\ell ^\infty} 
\leq \| u\| _{\ell ^2} \| v\| _{\ell ^2}$ for the first two terms
on the right side of \eqref{B.3}, Corollary~\ref{CorollaryB.2} (i)
for the third term and Corollary~\ref{CorollaryB.2} (ii) for the
last term on the right side of \eqref{B.3}, one obtains from
\eqref{B.10}
   \[ \sum _n \left( \langle n\rangle ^{2\delta +\alpha } w(2n)
      \Psi  _n(a,b,d_n)(2n)\right) ^2 \leq C\| a\| _w \| b\| _w
      \| d\| _w 
   \]
for a constant $1 \leq C \leq C_\delta < \infty $ only depending on 
$\delta $.

\medskip

(ii) Using \eqref{B.1} and the assumption $\langle n\rangle ^\alpha
d_n(k) \leq d(k) \ (\forall k)$ we get
   \begin{align*} \langle n\rangle ^{\delta + \alpha } &w(n + \ell)
                     \Psi _n(a,b,d_n)(\ell + n) \leq \\
                  &\leq \langle n\rangle ^\delta w(n + \ell ) \sum
                     _{k,j} \frac {a(k + \ell)}{\langle k - n
                     \rangle } \frac {b(k + j)}{\langle k - j
                     \rangle } d(j + n) \\
                  &\leq \sum _{k,j} H_n(\ell ,k,j) \tilde {a}(k +
                     \ell ) \tilde {b}(k + j)\tilde {d}(j + n)
   \end{align*}
where $H_n(\ell ,k,j)$ is given by
   \[ H_n(\ell ,k,j):= \frac {\langle n\rangle ^\delta \langle 
      \ell + n\rangle ^\delta }{\langle k - n\rangle \langle j - n
      \rangle \langle k + \ell \rangle ^\delta \langle k + j
      \rangle ^\delta \langle j + n\rangle ^\delta } .
   \]
To estimate $H_n(\ell ,k,j)$ we need to consider two different
regions:

\medskip

{\it Estimate of $H_n(\ell ,k,j)$ in $|j - n| < \frac {|n|}{2}$: }
In this case
   \[ |j + n| > |n| ; \ \langle \ell + n\rangle ^\delta \leq
      \langle \ell + k\rangle ^\delta \langle - k + n\rangle
      ^\delta ,
   \]
hence
   \begin{align*} H_n(\ell ,k,j) &\leq \frac {1}{\langle k - n
                     \rangle ^{1 - \delta } \langle j - n\rangle
                     \langle k + j\rangle ^\delta \langle j + n
                     \rangle ^\delta } \\
                  &\leq \frac {1}{\langle k - n\rangle ^{1-\delta }
                     \langle k + j\rangle ^\delta \langle j + n
                     \rangle ^\delta } .
   \end{align*}

\medskip

{\it Estimate of $H_n(\ell ,k,j)$ in $|j - n| > \frac {|n|}{2}$: }
In this case
   \[ 2|j - n| > |n| ; \ \langle \ell + n\rangle ^\delta \leq
      \langle \ell + k \rangle ^\delta \langle - k + n \rangle
      ^\delta ,
   \]
hence  
   \begin{align*} H_n(\ell ,k,j) &\leq \frac {2^\delta }{\langle 
                     k - n \rangle ^{1 - \delta } 
                     \langle j - n\rangle ^{1 - \delta }
                     \langle k + j\rangle ^\delta \langle j + n
                     \rangle ^\delta } \\
                  &\leq \frac {2^\delta }{\langle k - n\rangle 
                     ^{1-\delta }
                     \langle k + j\rangle ^\delta \langle j + n
                     \rangle ^\delta } .
   \end{align*}
Hence in both cases we obtain the same estimate. Define $\tilde
{e}(\ell + n) \equiv w(\ell + n)e(\ell + n)$ by
   \[ \tilde {e}(\ell + n):= \sum _{k,j} \left( \frac {1}{\langle
      k - n\rangle \langle j + n\rangle ^\delta \langle k + j
      \rangle ^\delta } \right) \tilde {a}(k + \ell) \tilde {b}
      (\ell + j) \tilde {d}(j + n) .
   \]
Then we have 
   \[ \langle n\rangle ^{\delta + \alpha } w(n + \ell ) \Psi _n
      (a,b,d_n)(\ell + n) \leq w(\ell + n)e(\ell + n)
   \]
and
   \[ \tilde {e}(\ell ) = \left( \tilde {a} \ast \frac {1}{\langle
      k \rangle ^{1-\delta }}(\frac {J\tilde {b}}{\langle k\rangle
      ^\delta } \ast \frac {\tilde {d}}{\langle k\rangle ^\delta })
      \right) (\ell ) .
   \]
By Corollary~\ref{CorollaryB.2} (ii),
   \[ \| \tilde {e}\| _{\ell ^2} \leq C\| a\| _w \| b\| _w \| d\| 
      _w 
   \]
for some constant $1 \leq C \equiv C_\delta < \infty $.
$\blacksquare $

\bigskip

It remains to establish the auxilary results used in the proof of
Lemma~\ref{Lemma2.11}. First we need the following

\medskip  

\begin{lemma}
\label{LemmaB.1} Let $0 < \eta \leq 1$. Then 

\begin{description}
\item[(i)  ] $\| \frac {a}{\langle k\rangle ^\eta }\| _{\ell ^p} \leq
C_{p,\eta } \| a\| _{\ell ^2} \quad \forall a \in \ell ^2$ and $p >
\frac {2}{2\eta + 1}$ 

\item[(ii) ] $\| \frac {a}{\langle k\rangle ^\eta }\| _{\ell ^1} \leq
C_{q,\eta } \| a\| _{\ell ^q} \quad \forall a \in \ell ^q$ with $1 
\leq q < \frac {1}{1 - \eta }$.
\end{description}
\end{lemma}

\medskip

{\it Proof } (i) follows from H\"older's inequality with $\alpha =
\frac {2}{p}$ and $\beta = \frac {2}{2 - p}$,
   \begin{align*} \left( \sum _k (\frac {a(k)}{\langle k\rangle
                     ^\eta })^p \right)^{1/p} &= \left( \sum _k a(k)
                     ^p \frac {1}{\langle k\rangle ^{\eta p}}
                     \right) ^{1/p} \\ 
                  &\leq \left( \sum  |a(k)|^2 \right) ^{1/2} \left(
                     \sum _k \frac {1}{\langle k\rangle ^{\eta p
                     \beta }}
                     \right) ^{1/\beta p} 
   \end{align*}
where $\eta p\beta = \eta p \frac {2}{2-p} > 1$ or $2\eta p > 2 - p$
as $(2\eta + 1)p > 2$ by assumption.

\medskip

(ii) follows from H\"older's inequality with $\alpha = q$ and $\frac
{1}{\beta } = 1 - \frac {1}{q} = \frac {q - 1}{q}$
   \[ \sum _k \frac {|a(k)|}{\langle k\rangle ^\eta } \leq \left(
      \sum _k a(k)^q \right) ^{1/q} \ \left( \sum _k (\frac {1}{
      \langle k \rangle ^\eta })^{\frac {q}{q-1}} \right) ^{\frac
      {q-1}{q}}
   \]
where $\eta \frac {q}{q - 1} > 1$ or $\eta q > q - 1$ as $1 > (1 -
\eta )q$ by assumption.$\blacksquare $

\medskip

Recall Young's inequality
   \[ \| u \ast v\| _q \leq C_{r,p,q} \| u\| _p \| v\| _r 
   \]
where $r,p, q \geq 1$ with $\frac {1}{p} + \frac {1}{r} = 1 +
\frac {1}{q}$.

\medskip

\begin{corollary}
\label{CorollaryB.2} Let $0 < \delta \leq \frac {1}{2}$

\begin{description}
\item[(i)  ] $\| \frac {1}{\langle k\rangle } \left( \frac {a}{\langle k
\rangle ^\delta } \ast b \right)\| _{\ell ^1} \leq C_\delta \| a\| _{\ell ^2}
\| b\| _{\ell ^2} \quad \forall a,b \in \ell ^2 $

\item[(ii) ] $\| \frac {1}{\langle k\rangle ^{1-\delta }} \left(
\frac {a}{\langle k \rangle ^\delta } \ast \frac {b}{\langle k\rangle  
^\delta } \right) \| _{\ell ^1} \leq C_\delta \| a\| _{\ell ^2}
\| b\| _{\ell ^2} \quad \forall a,b \in \ell ^2 $
\end{description}
\end{corollary}

\medskip

{\it Proof } (i) Let $\frac {1}{p}:= \frac {1}{2} + \frac {\delta }
{2}$ and $\frac {1}{q}:= \frac {\delta }{2}$. Then $\frac {1}{p} +
\frac {1}{2} = 1 + \frac {\delta }{2} = 1 + \frac {1}{q}$ and
hence by Young's inequality 
   \[ \| \frac {a}{\langle k\rangle ^\delta } \ast b\| _{\ell ^q}
     \leq C \ \| \frac {a}{\langle k\rangle ^\delta }\| _{\ell ^p}
     \| b\| _{\ell ^2} .
   \]
As $p = \frac {2}{1 + \delta } > \frac {2}{1 + 2\delta }$, 
Lemma~\ref{LemmaB.1} (i) can be applied,
   \[ \| \frac {a}{\langle k\rangle ^\delta }\| _{\ell ^p} \leq C
      \| a\| _{\ell ^2} ,
   \]
and as $q = \frac {2}{\delta } < \infty $, Lemma~\ref{LemmaB.1} (ii)
gives
   \[ \| \frac {1}{\langle k\rangle } \left( \frac {a}{\langle k \rangle 
      ^\delta } \ast b \right) \| _{\ell ^1} \leq C \| a\| _{\ell ^2 } \| b
      \| _{\ell ^2}
   \]
as claimed.

\medskip

(ii) By Lemma~\ref{LemmaB.1}, for $ \frac{1}{p}:= \frac {1}{2} +
\frac {2\delta }{3}\ (< \frac {1}{2} + \delta )$
   \[ \| \frac {a}{\langle k \rangle ^\delta } \| _{\ell ^p} \leq
      C \| a\| _{\ell ^2} .
   \]
By Young's inequality with $2 \cdot \frac {1}{p} = 1 + \frac {1}{q}$
or $\frac {1}{q} = \frac {4\delta }{3} < 1$ (as $0 < \delta \leq
\frac {1}{2}$)
   \[ \| \frac {a}{\langle k \rangle ^\delta } \ast \frac {b}{\langle
      k \rangle ^\delta } \| _{\ell ^q} \leq C \| a\| _{\ell ^2} \| b
      \| _{\ell ^2} .
   \]
By Lemma~\ref{LemmaB.1} (ii) with $\eta = 1 - \delta $ (hence $1 \leq
q = \frac {3}{4\delta } < \frac {1}{\delta } = \frac {1}{1 - \eta }$)
   \[ \| \frac {1}{\langle k\rangle ^{1-\delta }} \left( \frac {a}{\langle
      k \rangle ^\delta } \ast \frac {b}{\langle k\rangle ^\delta} \right) \|
      _{\ell ^1} \leq C \| a\| _{\ell ^2} \| b\| _{\ell ^2}
   \]
where $1 \leq C \equiv C_\delta < \infty $ depends only on $\delta $.
$\blacksquare $
    
\clearpage

\bibliographystyle{plain}
\begin{thebibliography}{999999}

\bibitem[AKNS]{AKNS} M.I. Ablowitz, D.I. Kaup, A.C. Newell, H.Segur:
              {\em The inverse scattering transform - Fourier 
              analysis for nonlinear problems}. Stud. Appl. Math. 
              54 (1974), p. 249 - 315.

\bibitem[FT]{FT} L.D. Faddeev, L.A. Takhtajan: {\em Hamiltonian 
            Methods in the Theory of solitons}. Springer Verlag
            (1987).

\bibitem[GG]{GG} B. Gr\'ebert, J.C. Guillot: {\em Gaps of one 
            dimensional periodic AKNS systems}. Forum Math. 5
            (1993), p. 459 - 504.

\bibitem[Gre]{Gre} B. Gr\'ebert: {\em Probl\`emes spectraux
             inverses pour les syst\`emes AKNS sur la droite
             r\'eelle}. Th\`ese de l'Universit\'e de Paris-Nord,
             1990.

\bibitem[GK]{GK} B. Gr\'ebert, T. Kappeler: {\em Gap estimates of
            the spectrum of the Zakhavov-Shabat system}. Preprint (1997).

\bibitem[GKM]{GKM} B. Gr\'ebert, T. Kappeler, B. Mityagin: {\em
             Gap estimates of the spectrum of the Zakhavov-Shabat
             system}. Appl. Math. Lett. 11 (1998), p. 95 - 97.

\bibitem[KM]{KM} T. Kappeler, B. Mityagin: {\em Estimates for 
            periodic and Dirichlet eigenvalues of the 
            Schr\"odinger operator}. Preprint.

\bibitem[LS]{LS} B.M. Levitan, I.S. Sargsjan: {\em Operators of
            Sturm-Liouville and Dirac}. Nauka, Moscow, 1988.

\bibitem[LM]{LM} Y. Li, D.W. McLaughlin: {\em Morse and Melnikov
            functions for NLS PDE's}. Com. Math. Phys. 162 (1994),
            p. 175 - 214.

\bibitem[MA]{MA} Y.C. Ma, M.I. Ablowitz: {\em The periodic cubic
            Schr\"odinger equation}. Stud. Appl. Math. 65 (1981),
            p. 113 - 158.

\bibitem[Ma]{Ma} V.A. Marchenko: {\em Sturm-Liouville operators and 
            applications}. Operator theory: Advances and 
            Applications 22, Birkh\"auser, 1986.

\bibitem[Mis]{Mis} T.V. Misyura: {\em Properties of the spectra of
             periodic and antiperiodic boundary value problems
             generated by Dirac operators I, II}. (In Russian),
             Teor. Funktsii Funktsii Anal. i. Prilozhen 30 (1978),
             p. 90 - 101; 31 (1979), p. 102 - 109; and {\em
             Finite-zone Dirac operators}. (In Russian), Teor.
             Funktsii Funktsional Anal. i. Prilozhen 33 (1980),
             p. 107 - 111.

\bibitem[ST]{ST} J.J. Sansuc, V. Tkachenko: {\em Spectral
            properties of non-selfadjoint Hill's operators with
            smooth potentials}. In A. Boutet de Monvel and 
            V. Mar\u{c}enko (eds.), Algebraic and geometric
            methods in mathematical physics, p. 371 - 385, Kluwer,
            1996.

\bibitem[Ta1]{Ta1} V. Tkachenko: {\em Non-selfadjoint periodic
             Dirac operators}. Preprint.

\bibitem[Ta2]{Ta2} V. Tkachenko: {\em Non-selfadjoint periodic
             Dirac operatos with finite band spectrum}. Preprint.

\bibitem[ZS]{ZS} V. Zakharov, A. Shabat: {\em A scheme for integrating
             nonlinear equations of mathematical physics by the
             method of the inverse scattering problem}. Functional 
             Anal. Appl. 8 (1974), p. 226 - 235. 

\end{thebibliography}
   \end{document}
---------------0001170837465--
