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 $|z| \leq \pi /2$ \[ (z - A_n)^{-1} = \left( \begin{array}{c c} (z - D_n)^{-1}(Id - P_n)^{-1} &(z - D_n)^{-1}R^{(2)}_n (Id - Q_n)^{-1} \\ (z - D_n)^{-1} R^{(1)}_n(Id - P_n)^{-1} &(zī- D_n)^{-1} (Id - Q_n)^{-1} \end{array} \right) . \] 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. 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