%Picard's Theorem and the AKNS Hierarchy %&amslatex \documentclass[12pt,reqno]{amsart} \textwidth 6in \textheight 8in \evensidemargin 0.25in \oddsidemargin 0.25in \pagestyle{plain} %%%%%%%%%THEOREMS%%%%%%%%%%%%%%%%%%%%%%%%%% \newtheorem{thm}{Theorem}[section] \newtheorem{prop}[thm]{Proposition} \newtheorem{hyp}[thm]{Hypothesis} \newtheorem{cor}[thm]{Corollary} \newtheorem{lem}[thm]{Lemma} \theoremstyle{definition} \newtheorem{defn}[thm]{Definition} \newtheorem{conj}[thm]{Conjecture} \newtheorem{exmp}[thm]{Example} \newtheorem{rem}[thm]{Remark} %%%%%%%%%%%%%%FONTS%%%%%%%%%%%%%%%%%%%%%%% \newcommand{\bbN}{{\Bbb{N}}} \newcommand{\bbR}{{\Bbb{R}}} \newcommand{\bbZ}{{\Bbb{Z}}} \newcommand{\bbC}{{\Bbb{C}}} \newcommand{\calK}{{\mathcal K}} \newcommand{\calE}{{\mathcal E}} \newcommand{\calF}{{\mathcal F}} \newcommand{\calM}{{\mathcal{M}}} \newcommand{\calD}{{\mathcal{D}}} \newcommand{\calS}{{\mathcal{S}}} \newcommand{\calP}{{\mathcal{P}}} \newcommand{\calL}{{\mathcal{L}}} \newcommand{\bb}[1]{{\mathbb{#1}}} \newcommand{\mc}[1]{{\mathcal{#1}}} \newcommand{\be}{$$} \newcommand{\ee}{$$} \newcommand{\ba}{\begin{eqnarray}} \newcommand{\ea}{\end{eqnarray}} \newcommand{\no}{\nonumber} \newcommand{\bma}{\begin{array}} \newcommand{\ema}{\end{array}} \newcommand{\lb}{\label} \newcommand{\f}{\frac} \newcommand{\frk}[1]{\mathfrak{#1}} \newcommand{\frF}{ \frk{F} } %%%%%%%%%%%%%%OPERATORNAMES AND ABBREVIATIONS%%%%%%%%%%%%%%%% \renewcommand{\Re}{\operatorname{Re}} \renewcommand{\Im}{\operatorname{Im}} \newcommand{\kdv}{\operatorname{KdV}} \newcommand{\ord}{\operatorname{ord}} \newcommand{\e}{\hbox{\rm e}} \newcommand{\romannr}[1]{\uppercase\expandafter{\romannumeral#1}} \newcommand{\tr}{\operatorname{tr}} \newcommand{\diag}{\operatorname{diag}} \newcommand{\df}{\operatorname{def}} %%%%%%%%%%%%%%%%%%%%%%%%NUMBERING%%%%%%%%%% \makeatletter \def\theequation{\thesection.\@arabic\c@equation} \makeatother %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{document} \title{A characterization of all elliptic solutions of the AKNS hierarchy} \thanks{Based upon work supported by the US National Science Foundation under Grants No. DMS-9623121 and DMS-9401816.} \thanks{{\it 1991 Mathematics Subject Classification.} Primary 35Q55, 34L40; Secondary 58F07.} \thanks{{\it Key words and phrases.} Elliptic algebro-geometric solutions, AKNS hierarchy, Floquet theory, Dirac-type operators.} \author{F.~Gesztesy${}^1$} \address{${}^1$ Department of Mathematics, University of Missouri, Columbia, MO 65211, USA.} \email{fritz@@math.missouri.edu} \author{R.~Weikard${}^2$} \address{${}^2$ Department of Mathematics, University of Alabama at Birmingham, Birmingham, AL 35294--1170, USA.} \email{rudi@@math.uab.edu} \date{May 14} %\maketitle \begin{abstract} An explicit characterization of all elliptic algebro-geometric solutions of the AKNS hierarchy is presented. More precisely, we show that a pair of elliptic functions $(p,q)$ is an algebro-geometric AKNS potential, that is, a solution of some equation of the stationary AKNS hierarchy, if and only if the associated linear differential system $J\Psi' +Q\Psi=E\Psi$, where $J=\begin{pmatrix}i&0\\0&-i\end{pmatrix}$, $Q=\begin{pmatrix}0&-iq(x)\\ip(x)&0\end{pmatrix}$, has a fundamental system of solutions which are meromorphic with respect to the independent variable for infinitely many and hence for all values of the spectral parameter $E\in\bb C$. Our approach is based on (an extension of) a classical theorem of Picard, which guarantees the existence of solutions which are elliptic of the second kind for $n^{\rm th}$-order ordinary differential equations with elliptic coefficients associated with a common period lattice. The fundamental link between Picard's theorem and elliptic algebro-geometric solutions of completely integrable hierarchies of nonlinear evolution equations has recently been established in connection with the KdV hierarchy. The current investigation appears to be the first of its kind associated with matrix-valued Lax pairs. As by-products we offer a detailed Floquet analysis of Dirac-type differential expressions with periodic coefficients specifically emphasizing algebro-geometric coefficients and a constructive reduction of singular hyperelliptic curves and their Baker-Akhiezer functions to the nonsingular case. \end{abstract} \maketitle \section{Introduction} \label{intro} Before describing our approach in some detail, we shall give a brief account of the history of the problem of characterizing elliptic algebro-geometric solutions of completely integrable systems. This theme dates back to a 1940 paper of Ince \cite{36} who studied what is presently called the Lam\'e--Ince potential \linebreak[0] $$\label{1.1} q(x)=-n(n+1)\wp(x+\omega_3), \quad n\in \bb N, \; x \in\bb R$$ in connection with the second-order ordinary differential equation $$\label{1.2} y''(E,x) + q(x) y(E,x) = Ey(E,x), \quad E\in\bb C.$$ Here $\wp(x)=\wp(x;\omega_1,\omega_3)$ denotes the elliptic Weierstrass function with fundamental periods $2\omega_1$ and $2\omega_3$ ($\Im(\omega_3/\omega_1)\ne 0$). In the special case where $\omega_1$ is real and $\omega_3$ is purely imaginary, the potential $q(x)$ in (\ref{1.1}) is real-valued and Ince's striking result \cite{36}, in modern spectral theoretic terminology, yields that the spectrum of the unique self-adjoint operator associated with the differential expression $L_2=d^2/dx^2 + q(x)$ in $L^2(\bb R)$ exhibits finitely many bands (respectively gaps), that is, \label{1.3} \sigma(L_2)=(-\infty, E_{2n}] \cup \bigcup^n_{m=1} \left[ E_{2m-1},E_{2m-2}\right], \quad E_{2n}0$. Then$S_1$and$S_3$do not intersect outside a sufficiently large disk centered at the origin. A combination of this fact and Picard's Theorem \ref{t5.1} then yields a proof of Theorem \ref{1.2} (see the proof of Theorem \ref{t5.4}). We close Section \ref{picard} with a series of remarks that put Theorem \ref{1.2} into proper perspective: Among a variety of points, we stress, in particular, its straightforward applicability based on an elementary Frobenius-type analysis, its property of complementing Picard's original result, and its connection with the Weierstrass theory of reduction of Abelian to elliptic integrals. Finally, Section \ref{Ex} rounds off our presentation with a few explicit examples. The result embodied by Theorems \ref{t1.1} and \ref{t1.2} in the context of the KdV and AKNS hierarchies, uncovers a new general principle in connection with elliptic algebro-geometric solutions of completely integrable systems: The existence of such solutions appears to be in a one-to-one correspondence with the existence of a meromorphic (with respect to the independent variable) fundamental system of solutions for the underlying linear Lax differential expression (for all values of the corresponding spectral parameter$E\in \bb C$). Even though the current AKNS case is technically much more involved than the KdV case in \cite{32a} (and despite the large number of references at the end) we have made every effort to keep this presentation self-contained. \section{The AKNS Hierarchy, Recursion Relations, and Hyperelliptic Curves} \label{akns} \setcounter{equation}{0} In this section we briefly review the construction of the AKNS hierarchy using a recursive approach. This method was originally introduced by Al'ber \cite{4} in connection with the Korteweg-de Vries hierarchy. The present case of the AKNS hierarchy was first systematically developed in \cite{27b}. Suppose that$q=iQ_{1,2}, \, p=-iQ_{2,1} \in C^{\infty} (\bb R)$(or meromorphic on$\bb C) and consider the Dirac-type matrix-valued differential expression $$\label{2.1} L=J\frac{d}{dx} + Q(x) =\begin{pmatrix}i&0\\0&-i\end{pmatrix}\frac{d}{dx}+ \begin{pmatrix}0&-iq(x)\\ip(x)&0\end{pmatrix},$$ where we abbreviate \begin{align} J&=\begin{pmatrix}i&0\\0&- i\end{pmatrix},\label{2.2} \\ Q(x)&=\begin{pmatrix}Q_{1,1}(x)&Q_{1,2}(x) \\Q_{2,1}(x)&Q_{2,2}(x)\end{pmatrix} =\begin{pmatrix}0&-iq(x)\\ip(x)&0\end{pmatrix}. \label{2.3} \end{align} In order to explicitly construct higher-order matrix-valued differential expressionsP_{n+1}$,$n\in\bb N_0$(=$\bb N \cup \{0\}$) commuting with$L$, which will be used to define the stationary AKNS hierarchy, one can proceed as follows (see \cite{27b} for more details). Define functions$f_\ell$,$g_\ell$, and$h_\ell$by the following recurrence relations, \begin{gather} f_{-1}=0,\quad g_0=1,\quad h_{-1}=0,\notag \\ f_{\ell+1}=\frac{i}{2} f_{\ell,x} - iqg_{\ell+1}, \quad g_{\ell+1,x}=pf_{\ell}+q h_{\ell}, \quad h_{\ell+1}=-\frac{i}{2}h_{\ell,x} + ip g_{\ell+1} \label{2.4} \end{gather} for$\ell=-1,0,1,...$. The functions$f_\ell$,$g_\ell$, and$h_\ell$are polynomials in the variables$p,q,p_x,q_x,...$and$c_1,c_2,...$where the$c_j$denote integration constants. Assigning weight$k+1$to$p^{(k)}$and$q^{(k)}$and weight$k$to$c_k$one finds that$f_\ell$,$g_{\ell+1}$, and$h_\ell$are homogeneous of weight$\ell+1. Explicitly, one computes, \begin{align} &f_0 = -iq,\notag \\ &f_1 =\frac{1}{2} q_x+c_1(-iq),\notag \displaybreak[0]\\ &f_2=\frac{i}{4} q_{xx}-\frac{i}{2}pq^2 +c_1( \frac{1}{2} q_x)+c_2(-iq),\notag \displaybreak[2]\\ &g_0 = 1,\notag \\ &g_1 =c_1,\notag \\ &g_2 =\frac{1}{2} pq+c_2,\notag \\ &g_3 = -\frac{i}{4}(p_{_x}q - pq_x) + c_1(\frac{1}{2} pq) + c_3 , \label{2.5}\\ &h_0 =i p,\notag \\ &h_1 =\frac{1}{2} p_{_x}+ c_1(ip), \notag \\ &h_2 =-\frac{i}{4} p_{_{xx}}+\frac{i}{2}p{^2}q+ c_1( \frac{1}{2} p_{_x})+c_2(i p), \notag \\ & \text{etc}. \notag \end{align} Next one defines the matrix-valued differential expressionP_{n+1}$by $$\label{2.6} P_{n+1}=-\sum_{\ell=0}^{n+1}(g_{n-\ell+1}J+iA_{n-\ell})L^{\ell},$$ where $$\label{2.7} A_{\ell}=\begin{pmatrix}0&-f_{\ell} \\h_{\ell}&0\end{pmatrix},\quad \ell=-1,0,1,\dots .$$ One verifies that $$\label{2.8} [g_{n-\ell+1}J+iA_{n-\ell},L]=2iA_{n-\ell}L-2iA_{n-\ell+1},$$ where$[\cdot,\cdot]$denotes the commutator. This implies $$\label{2.9} [P_{n+1},L]=2i A_{n+1}.$$ The pair$(P_{n+1},L)$represents a Lax pair for the AKNS hierarchy. Introducing a deformation parameter$t$into$(p,q)$, that is,$(p(x),q(x)) \rightarrow(p(x,t),q(x,t))$, the AKNS hierarchy (cf., e.g., \cite{47b}, Chs. 3, 5 and the references therein) is defined as the collection of evolution equations (varying$n \in \bbN_0) $$\label{2.40} \frac{d}{dt}L(t)-[P_{n+1}(t),L(t)]=0$$ or equivalently, by $$\label{2.41} \mbox{AKNS}_n(p,q)=\begin{pmatrix}p_{t}(x,t)-2 h_{n+1}(x,t)\\ q_{t}(x,t)-2 f_{n+1}(x,t)\end{pmatrix}=0,$$ that is, by $$\label{2.42} \mbox{AKNS}_n(p,q)=\begin{pmatrix}p_{t}+i(H_{n,x}-2iE H_n-2p G_{n+1})\\ q_{t}-i(F_{n,x}+2iE F_n-2q G_{n+1})\end{pmatrix}=0.$$ Explicitly, one obtains for the first few equations in (\ref{2.41}) \begin{align} &\mbox{AKNS}_0(p,q)=\begin{pmatrix} p_{t}-p_{x}-2i c_1p\\ q_{t} - q_x + 2i c_1q\end{pmatrix}=0 \notag \\ &\mbox{AKNS}_1(p,q)=\begin{pmatrix} p_{t}+\frac{i}{2} p_{xx}-i p^2q-c_1 p_x-2i c_2 p \\ q_{t}-\frac{i}{2} q_{xx}+i pq^2-c_1 q_x+2i c_2 q \end{pmatrix}=0 \label{2.43}\\ &\mbox{AKNS}_2(p,q)=\begin{pmatrix} p_{t}+\frac14p_{xxx}-\frac32pp_xq+c_1(\frac{i}2p_{xx}- i p^2q) -c_2 p_x-2i c_3 p \\ q_{t}+\frac14q_{xxx}-\frac32pqq_x+c_1(-\frac{i}2q_{xx}+i pq^2) -c_2 q_x+2i c_3 q \end{pmatrix} =0 \notag \\ &\text{etc}. \notag \end{align} The stationary AKNS hierarchy is then defined by the vanishing of the commutator ofP_{n+1}$and$L$, that is, by $$\label{2.10} [P_{n+1},L]=0, \quad n\in \bb N_0,$$ or equivalently, by $$\label{2.11} f_{n+1}=h_{n+1}=0, \quad n\in \bb N_0.$$ Next, we introduce$F_n$,$G_{n+1}$, and$H_n$which are polynomials with respect to$E\in \bb C, \begin{align} F_n(E,x)&= \sum_{\ell=0}^{n}f_{n-\ell}(x)E^{\ell},\notag \\ G_{n+1}(E,x)&= \sum_{\ell=0}^{n+1} g_{n+1-\ell}(x)E^\ell,\label{2.13}\\ H_n(E,x)&= \sum_{\ell=0}^{n}h_{n-\ell}(x)E^\ell,\notag \end{align} and note that \eqref{2.11} becomes \begin{align} F_ {n,x}(E,x)&=-2 i E F_n(E,x)+2q(x) G_{n+1}(E,x),\label{2.14}\\ G_{n+1,x}(E,x)&=p(x)F_n(E,x)+q(x)H_n(E,x), \label{2.15}\\ H_{n,x}(E,x)&=2iE H_n(E,x)+2 p(x) G_{n+1}(E,x). \label{2.16} \end{align} These equations show thatG_{n+1}^2 - F_nH_n$is independent of$x$. Hence $$\label{2.17} R_{2n+2}(E)=G_{n+1}(E,x)^2- F_n(E,x)H_n(E,x)$$ is a monic polynomial in$E$of degree$2n+2$. One can use \eqref{2.14}--\eqref{2.17} to derive differential equations for$F_n$and$H_n$separately by eliminating$G_{n+1}. One obtains \begin{align} &q(2F_nF_{n,xx}-F_{n,x}^2+4(E^2-pq)F_n^2) -q_x(2F_nF_{n,x}+4iE F_n^2)=-4 q^3 R_{2n+2}(E),\label{2.18} \\ &p(2H_nH_{n,xx}-H_{n,x}^2+4(E^2-pq)H_n^2) -p_x(2H_nH_{n,x}-4iE H_n^2)=-4 p^3 R_{2n+2}(E).\label{2.19} \end{align} Next, assuming[P_{n+1},L]=0$, one infers $$\label{2.20} P_{n+1}^2=\sum_{\ell,m=0}^{n+1}(g_{n-\ell+1}J+iA_{n-\ell}) (g_{n-m+1}J+iA_{n-m})L^{\ell+m}.$$ Hence, $$\label{2.21} P_{n+1}^2=-G_{n+1}(L,x)^2+F_n(L,x)H_n(L,x)=-R_{2n+2}(L),$$ that is, whenever$P_{n+1}$and$L$commute they necessarily satisfy an algebraic relationship. In particular, they define a (possibly singular) hyperelliptic curve$\calK_n$of (arithmetic) genus$n$of the type $$\label{2.22} \calK_n: \, w^2 = R_{2n+2}(E), \quad R_{2n+2}(E)= \prod_{m=0}^ {2n+1} (E-E_m) \, \text{for some} \, \{E_m\}_{m=0}^{2n+1} \subset \bb C.$$ The functions$f_\ell$,$g_\ell$, and$h_\ell$, and hence the matrices$A_\ell$and the differential expressions$P_{\ell}$defined above, depend on the choice of the integration constants$c_1,c_2,...,c_\ell$(cf. \eqref{2.5}). In the following we make this dependence explicit and write$f_{\ell}(c_1,\dots,c_{\ell})$,$g_{\ell}(c_1,\dots,c_{\ell})$,$h_{\ell}(c_1,\dots,c_{\ell})$,$A_\ell(c_1,..., c_\ell)$,$P_\ell(c_1,...,c_\ell)$, etc. In particular, we denote homogeneous quantities, where$c_{\ell}=0, \ell \in \bb N$by$\hat f_{\ell} = f_{\ell}(0,\dots,0)$,$\hat g_{\ell} = g_{\ell}(0,\dots,0)$,$\hat h_{\ell} = h_{\ell}(0,\dots,0)$,$\hat A_{\ell} = A_{\ell}(0,\dots,0)$,$\hat P_\ell=P_\ell(0,...,0)$, etc. In addition, we note that $$\label{2.23} f_\ell(c_1,...,c_\ell) =\sum_{k=0}^\ell c_{\ell-k}\hat f_k, \quad g_\ell(c_1,...,c_\ell) =\sum_{k=0}^\ell c_{\ell-k}\hat g_k, \quad h_\ell(c_1,...,c_\ell) =\sum_{k=0}^\ell c_{\ell-k}\hat h_k,$$ and $$\label{2.24} A_\ell(c_1,...,c_\ell) =\sum_{k=0}^\ell c_{\ell-k}\hat A_k,$$ defining$c_0=1$. In particular, then $$\label{2.26} P_r(c_1,...,c_r)=\sum_{\ell=0}^r c_{r-\ell} \hat P_{\ell}.$$ Next suppose that$P_{n+1}$is any$2 \times 2$matrix-valued differential expression such that$[P_{n+1},L]$represents multiplication by a matrix whose diagonal entries are zero. This implies that the leading coefficient of$P_{n+1}$is a constant diagonal matrix. Since any constant diagonal matrix can be written as a linear combination of$J$and$I$(the identity matrix in$\bb C^2$), we infer the existence of complex numbers$\alpha_{n+1}$and$\beta_{n+1}$such that $$\label{2.27} S_1=P_{n+1} -\alpha_{n+1}\hat P_{n+1}-\beta_{n+1}L^{n+1}$$ is a differential expression of order at most$n$whenever$P_{n+1}$is of order$n+1$. Note that $$\label{2.28} [S_1,L]=[P_{n+1},L]-\alpha_{n+1}[\hat P_{n+1},L]= [P_{n+1},L]-2i\alpha_{n+1}\hat A_{n+1}$$ represents multiplication with zero diagonal elements. An induction argument then shows that there exists$S_{n+1}$such that $$\label{2.30} S_{n+1}=P_{n+1} -\sum_{\ell=1}^{n+1}(\alpha_{\ell} \hat P_{\ell}+\beta_{\ell} L^{\ell}) \text{ and } [S_{n+1},L]=[P_{n+1},L]-2i\sum_{\ell=1}^{n+1} \alpha_{\ell} \hat A_{\ell}.$$ Since the right-hand side of the last equation is multiplication with a zero diagonal,$S_{n+1}$is a constant diagonal matrix, that is, there exist complex numbers$\alpha_0$and$\beta_0$such that$S_{n+1}=\alpha_0J+\beta_0I$. Hence $$\label{2.31} P_{n+1}=\sum_{\ell=0}^{n+1}(\alpha_{\ell} \hat P_{\ell} +\beta_{\ell} L^{\ell}) \text{ and } [P_{n+1},L]=2i\sum_{\ell=0}^{n+1}\alpha_{\ell}\hat A_{\ell}.$$ Consequently, if all$\alpha_{\ell}=0$, then$P_{n+1}$is a polynomial of$L$, and$P_{n+1}$and$L$commute irrespective of$p$and$q$. If, however,$\alpha_r\neq0$and$\alpha_{\ell}=0$for$\ell >r$, then $$\label{2.32} P_{n+1}=\alpha_r P_r(\frac{\alpha_{r-1}}{\alpha_r},..., \frac{\alpha_0}{\alpha_r})+\sum_{\ell=0}^{n+1}\beta_{\ell} L^{\ell}.$$ In this case$P_{n+1}$and$L$commute if and only if $$\label{2.33} \sum_{\ell=0}^r \frac{\alpha_{\ell}}{\alpha_r}\hat A_{\ell} =A_r(\frac{\alpha_{r-1}}{\alpha_r},...,\frac{\alpha_0}{\alpha_r})=0,$$ that is, if and only if$(p,q)$is a solution of some equation of the stationary AKNS hierarchy. In this case$P_{n+1}-\sum_{\ell=0}^{n+1} \beta_{\ell} L^{\ell}$and$L$satisfy an algebraic relationship of the type \eqref{2.21}. Suppose on the other hand that$P_{n+1}$is a matrix-valued differential expression such that$(P_{n+1}-K_r(L))^2=-R_{2n+2}(L)$for some polynomials$K_r$and$R_{2n+2}$. Then$L$commutes with$(P_{n+1}-K_r(L))^2$and this enforces that$L$also commutes with$P_{n+1}-K_r(L)$and hence with$P_{n+1}$. Thus we proved the following theorem. \begin{thm} \label{t2.1} Let$L$be defined as in \eqref{2.1}. If$P_{n+1}$is a matrix-valued differential expression of order$n+1$which commutes with$L$, whose leading coefficient is different from a constant multiple of$J^{n+1}$, then there exist polynomials$K_r$and$R_{2n+2}$of degree$r\leq n+1$and$2n+2$, respectively, such that$(P_{n+1}-K_r(L))^2=-R_{2n+2}(L)$. \end{thm} Theorem \ref{t2.1} represents a matrix-valued generalization of a celebrated result due to Burchnall and Chaundy \cite{11}, \cite{12} in the special case of scalar differential expressions. By the arguments presented thus far in this section it becomes natural to make the following definition. We denote by$M_2 (\bb C)$the set of all$2\times 2$matrices over$\bb C$. \begin{defn} \label{d2.2} A function$Q:\bb R\to M_2 (\bb C)$of the type$Q=\begin{pmatrix}0&-iq\\ip&0\end{pmatrix}$is called an {\bf algebro-geometric AKNS potential} if$(p,q)$is a stationary solution of some equation of the AKNS hierarchy \eqref{2.11}. \end{defn} By a slight abuse of notation we will also call$(p,q)$an algebro-geometric AKNS potential in this case. The following theorem gives a sufficient condition for$Q$to be algebro-geometric. \begin{thm} \label{t2.3} Assume that$F_n(E,x)=\sum_{\ell=0}^n f_{n-\ell}(x)E^\ell$with$f_0(x)=-iq(x)$is a polynomial of degree$n$in$E$, whose coefficients are twice continuously differentiable complex-valued functions on$(a,b)$for some$-\infty \leq am-\lambda$) of$R$are distinct integers and\\ (ii)$b_{2\lambda-m-1}$is in the range of$R-\lambda$. \end{prop} \begin{proof} A fundamental matrix of$w'=(R/x+S+\sum_{j=0}^\infty A_{j+1}x^j)w$may be written as \eqref{05052} where$T$is in Jordan normal form. If all solutions of$L\Psi=E\Psi$and hence of$w'=(R/x +S+\sum_{j=0}^\infty A_{j+1}x^j)w$are meromorphic near$x_0$, then$T$must be a diagonal matrix with integer eigenvalues. Equation \eqref{11040} then shows that the eigenvalues of$T$are the eigenvalues of$R$and that$R$is diagonalizable. But since at least one of$p_0$and$q_0$is different from zero,$R$is not diagonalizable if$\lambda$is a double eigenvalue of$R$, a case which is therefore precluded. This proves (i). Since$T$is a diagonal matrix, equation \eqref{11041} implies $$(R+\lambda-m-j-1)\omega_{j+1}^{(2)}=b_j^{(2)}$$ for$j=0,1,...$. Statement (ii) is just the special case where$j=2\lambda-m-1$. Conversely, assume that (i) and (ii) are satisfied. If the recurrence relations \eqref{11040}, \eqref{11041} are satisfied, that is, if$w$is a formal solution of$w'=(R/x+S+\sum_{j=0}^\infty A_{j+1}x^j)w$then it is also an actual solution near$x_0$(see, e.g., Coddington and Levinson \cite{CL}, Sect. 4.3). Since$R$has distinct eigenvalues it has linearly independent eigenvectors. Using these as the columns of$\Omega_0$and defining$T=\Omega_0^{-1}R\Omega_0$yields \eqref{11040}. Since$Tis a diagonal matrix, \eqref{11041} is equivalent to the system \begin{align} b_j^{(1)}&=(R-\lambda-j-1)\omega_{j+1}^{(1)}, \label{05053}\\ b_j^{(2)}&=(R+\lambda-m-j-1)\omega_{j+1}^{(2)}. \label{05054} \end{align} Next, we note thatR-\lambda-j-1$is invertible for all$j\in\bb N_0$. However,$R+\lambda-m-j-1$is only invertible if$j\neq 2\lambda-m-1$. Hence a solution of the proposed form exists if and only if$b_{2\lambda-m-1}^{(2)}$is in the range of$R-\lambda$, which is guaranteed by hypothesis (ii). \end{proof} Note that $$B_0=-E J\Omega_0-A_1\Omega_0$$ is a first-order polynomial in$E$. As long as$R$has distinct eigenvalues and$j\leq 2\lambda -m-1$, we may compute$\Omega_j$recursively from \eqref{05053} and \eqref{05054} and$B_j$from the equality on the right in \eqref{11041}. By induction one can show that$\Omega_j$is a polynomial of degree$j$and that$B_j$is a polynomial of degree$j+1$in$E$. This leads to the following result. \begin{thm} \label{t05051} Suppose$Q$is a meromorphic potential of$L\Psi=E\Psi$. The equation$L\Psi=E\Psi$has a fundamental system of solutions which are meromorphic with respect to the independent variable for all values of the spectral parameter$E\in\bb C$whenever this is true for a sufficiently large finite number of distinct values of$E$. \end{thm} \begin{proof} By hypothesis,$Q$has countably many poles. Let$x_0$be any one of them. Near$x_0$the functions$p$and$q$have the Laurent expansions \eqref{11042}. The associated matrix$R$has eigenvalues$\lambda$and$m-\lambda$, which are independent of$E$. The vector$v=(q_0,-\lambda)^t$spans$R-\lambda$, and the determinant of the matrix whose columns are$v$and$b_{2\lambda-m-1}$is a polynomial in$E$of degree$2\lambda-m$. Our hypotheses and Proposition \ref{p552} imply that this determinant has more than$2\lambda-m$zeros and hence is identically equal to zero. This shows that$b_{2\lambda-m-1}$is a multiple of$v$for every value of$E$. Applying Proposition \ref{p552} once more then shows that all solutions of$L\Psi=E\Psi$are meromorphic near$x_0$for all$E\in \bb C$. Since$x_0$was arbitrary, this concludes the proof. \end{proof} Next, let$\{E_0,...,E_{2n+1}\}$be a set of not necessarily distinct complex numbers. We recall (cf. (\ref{2.22})), $$\label{6.1} {\calK}_n=\{P=(E,V)\, | \, V^2 = R_{2n+2}(E)=\prod_{m=0}^{2n+1} (E-E_m)\}.$$ We introduce the meromorphic function$\phi(\cdot,x)$on$\calK_n$by $$\label{6.2} \phi(P,x) = \frac{V+G_{n+1}(E,x)}{F_n(E,x)},\quad P=(E,V) \in \calK_n.$$ We remark that$\phi$can be extended to a meromorphic function on the compactification (projective closure) of the affine curve$\calK_n$. This compactification is obtained by joining two points to$\calK_n. Next we define \begin{align} \psi_1(P,x,x_0) &= \exp \left\{\int_{x_0}^x\, dx' [-iE+q(x') \phi(P,x')] \right\}, \label{6.3} \\ \psi_2 (P,x,x_0) &= \phi(P,x) \psi_1 (P,x,x_0), \label{6.4} \end{align} where the simple Jordan arc fromx_0$to$x$in (\ref{6.3}) avoids poles of$q$and$\phi$. One verifies with the help of \eqref{2.14}--\eqref{2.17}, that $$\label{6.7} \phi_x(P,x)=p(x)-q(x)\phi(P,x)^2+2iE\phi(P,x).$$ From this and \eqref{2.6} we find $$\label{6.5} L \Psi(P,x,x_0) = E \Psi(P,x,x_0), \quad P_{n+1}\Psi(P,x,x_0)=iV\Psi(P,x,x_0),$$ where $$\label{6.6} \Psi(P,x,x_0)=\begin{pmatrix} {\psi}_1(P,x,x_0) \\{\psi}_2(P,x,x_0) \end{pmatrix}.$$ One observes that the two branches$\Psi_{\pm} (E,x,x_0) =(\psi_{\pm,1}(E,x,x_0),\psi_{\pm,2}(E,x,x_0))^t$of$\Psi(P,x,x_0)$represent a fundamental system of solutions of$Ly=Ey$for all$E\in\bb C\setminus \{ \{E_m\}_{m=0}^{2n+1} \cup\{\mu_j(x_0)\}_{j=1}^n \}$, since $$\label{6.8} W(\Psi_-(E,x,x_0), \Psi_+(E,x,x_0))=\frac{2V_+(E)}{F_n(E,x_0)}.$$ Here$W(f,g)$denotes the determinant of the two columns$f$and$g$and$V_+(\cdot)$(resp.$V_-(\cdot))$denotes the branch of$V(\cdot)$on the upper (resp. lower) sheet of$\calK_n$(we follow the notation established in \cite{27b}). In the special case where$\calK_n$is nonsingular, that is,$E_m\neq E_{m'}$for$m\neq m'$, the explicit representation of$\Psi(P,x,x_0)$in terms of the Riemann theta function associated with$\calK_n$immediately proves that$\Psi_{\pm}(E,x,x_0)$are meromorphic with respect to$x\in \bbC$for all$E\in \bbC \setminus \{\{E_m\}_{m=0}^{2n+1} \cup\{\mu_j(x_0)\}_{j=1}^n\}$. A detailed account of this theta function representation can be found, for instance, in Theorem~3.5 of \cite{27b}. In the following we demonstrate how to use gauge transformations to reduce the case of singular curves$\calK_n$to nonsingular ones. Let$(p,q)$be meromorphic on$\bb C$, the precise conditions on$(p,q)$being immaterial (at least, temporarily) for introducing gauge transformations below. Define$L$and$Q$as in (\ref{2.1}), (\ref{2.3}) and consider the formal first-order differential system$L\Psi=E\Psi$. Introducing, $$\label{6.11} A(E,x)=\begin{pmatrix}iE&-q(x)\\-p(x)&-iE\end{pmatrix},$$$L\Psi=E\Psi$is equivalent to$\Psi_x (E,x)+A(E,x)\Psi(E,x)=0$. Next we consider the gauge transformation, \begin{gather} \tilde \Psi(E,x)=\Gamma(E,x)\Psi(E,x), \label{6.13} \\ \tilde A(E,x)=\begin{pmatrix}iE& -\tilde q(x)\\ -\tilde p(x)&-iE\end{pmatrix} =\Gamma(E,x)A(E,x)\Gamma(E,x)^{-1} - \Gamma_x(E,x)\Gamma(E,x)^{-1}, \label{6.14} \end{gather} implying $$\label{6.15} \tilde \Psi_x(E,x)+\tilde A(E,x)\tilde \Psi(E,x)=0, \, \, \text{that is,} \, \, \tilde L \tilde \Psi(E,x)=E\tilde \Psi(E,x),$$ with$\tilde L$defined as in (\ref{3.0}), (\ref{3.1}) replacing$(p,q)$by$(\tilde p, \tilde q)$. In the following we make the explicit choice (cf., e.g., \cite{39b}), $$\label{6.16} \Gamma(E,x)=\begin{pmatrix}E-\tilde E- \tfrac{i}{2}q(x)\phi^{(0)}(\tilde E,x)& \tfrac{i}{2}q(x)\\ \tfrac{i}{2}\phi^{(0)}(\tilde E,x)&-\tfrac{i}{2}\end{pmatrix}, \quad E\in \bbC \setminus \{\tilde E\}$$ for some fixed$\tilde E \in \bbC$and $$\label{6.19} \phi^{(0)}(\tilde E,x)=\psi^{(0)}_2(\tilde E,x)/\psi^{(0)}_1(\tilde E,x),$$ where$\Psi^{(0)}(\tilde E,x)=(\psi^{(0)}_1(\tilde E,x), \psi^{(0)}_2(\tilde E,x))^t$is any solution of$L\Psi=\tilde E\Psi. Using (\ref{6.7}), equation \eqref{6.14} becomes \begin{align} \tilde p(x) &= \phi^{(0)}(\tilde E,x), \label{6.17} \\ \tilde q(x) &= -q_x(x)-2i\tilde E q(x)+q(x)^2 \phi^{(0)}(\tilde E,x). \label{6.18} \end{align} Moreover, one computes for\tilde \Psi = (\tilde \psi_1,\tilde \psi_2)^t$in terms of$\Psi=(\psi_1,\psi_2)^t, \begin{align} \tilde \psi_1(E,x)&=(E-\tilde E)\psi_1(E,x) + \frac{i}{2}q(x) (\psi_2(E,x)-\phi^{(0)}(\tilde E,x)\psi_1(E,x)), \label{6.20} \\ \tilde \psi_2(E,x)&= -\frac{i}{2} (\psi_2(E,x)-\phi^{(0)}(\tilde E,x)\psi_1(E,x)). \end{align} In addition, we note that $$\label{6.21} \det(\Gamma(E,x))=-\frac{i}{2}(E-\tilde E)$$ and therefore, $$\label{6.22} W(\tilde \Psi_1(E),\tilde \Psi_2(E))=-\frac{i}{2}(E-\tilde E) W(\Psi_1(E),\Psi_2(E)),$$ where\Psi_j(E,x)$,$j=1,2$are two linearly independent solutions of$L\Psi=E\Psi$. Our first result proves that gauge transformations as defined in this section leave the class of meromorphic algebro-geometric potentials of the AKNS hierarchy invariant. \begin{thm} \label{t6.1} Suppose$(p,q)$is a meromorphic algebro-geometric AKNS potential. Fix$\tilde E\in \bbC$and define$(\tilde p,\tilde q)$as in (\ref{6.17}), (\ref{6.18}), with$\phi^{(0)}(\tilde E,x)$defined as in (\ref{6.19}). Suppose$\phi^{(0)}(\tilde E,x)$is meromorphic in$x$. Then$(\tilde p,\tilde q)$is a meromorphic algebro-geometric AKNS potential. \end{thm} \begin{proof} The upper right entry$G_{1,2}(E,x,x')$of the Green's matrix of$L$is given by $$G_{1,2}(E,x,x')=\frac{i\psi_{+,1}(E,x,x_0)\psi_{-,1}(E,x',x_0)} {W(\Psi_-(E,\cdot,x_0),\Psi_+(E,\cdot,x_0))}, \quad x \geq x'.$$ Combining (\ref{6.2})--(\ref{6.4}) and (\ref{6.7}), its diagonal (where$x=x'$) equals $$\label{6.26} G_{1,2}(E,x,x)=\frac{iF_n(E,x)}{2V_+(E)}.$$ The corresponding diagonal of the upper right entry$\tilde G_{1,2}(E,x,x)$of the Green's matrix of$\tilde Lis computed to be $$\label{6.29} \tilde G_{1,2}(E,x,x)=\frac{i\tilde\psi_{+,1}(E,x)\tilde\psi_{-,1}(E,x)} {W(\tilde \Psi_-(E,\cdot),\tilde \Psi_+(E,\cdot))} =\frac{i\tilde F_{n+1}(E,x)}{2(E-\tilde E)V_+(E)},$$ where \begin{align}\label{01052} \tilde F_{n+1}(E,x)=&2iF_n(E,x) \{(E-\tilde E) +\tfrac{i}{2}q(x)[\phi_+(E,x)-\phi^{(0)}(\tilde E,x)]\} \times \notag \\ &\times \{(E-\tilde E)+\tfrac{i}{2}q(x) [\phi_-(E,x) -\phi^{(0)}(\tilde E,x)]\}, \end{align} using \eqref{6.8}, \eqref{6.20}, \eqref{6.22}, and $$\label{6.30} \psi_{+,1}(E,x,x_0)\psi_{-,1}(E,x,x_0)=\frac{F_n(E,x)}{F_n(E,x_0)}.$$ By \eqref{6.2},\phi_+(E,x)+\phi_-(E,x) =2G_{n+1}(E,x)/F_n(E,x)$and$\phi_+(E,x)\phi_-(E,x) =H_n(E,x)/F_n(E,x)$. From this, \eqref{2.5}, and \eqref{2.13}, it follows that$\tilde F_{n+1}(\cdot,x)$is a polynomial of degree$n+1$with leading coefficient $$\label{6.28a} i q_x(x)-2\tilde E q(x) -i q(x)^2 \phi^{(0)}(\tilde E,x) =-i\tilde q(x).$$ Finally, using$\tilde L\tilde\Psi_{\pm}=E \tilde \Psi_{\pm}$, that is, (\ref{6.15}), one verifies that$\tilde G_{1,2}(E,x,x)$satisfies the differential equation, $$\label{6.33} \tilde q\bigl(2\tilde G_{1,2}\tilde G_{1,2,xx}-\tilde G_{1,2,x}^2 +4(E^2-\tilde p\tilde q)\tilde G_{1,2}^2\bigr) -\tilde q_x \bigl(2\tilde G_{1,2}\tilde G_{1,2,x} +4iE\tilde G_{1,2}^2\bigr)=\tilde q^3.$$ Hence$\tilde F_{n+1}(E,x)$satisfies the hypotheses of Corollary \ref{c2.3a} (with$n$replaced by$n+1$and$(p,q)$replaced by$(\tilde p,\tilde q)$) and therefore,$(\tilde p, \tilde q)$is a meromorphic algebro-geometric AKNS potential. \end{proof} We note here that \eqref{6.33} implies also that$\tilde F_{n+1}satisfies \begin{align} &\tilde q(2\tilde F_{n+1}\tilde F_{n+1,xx}-\tilde F_{n+1,x}^2 +4(E^2-\tilde p\tilde q)\tilde F_{n+1}^2) -\tilde q_x(2\tilde F_{n+1}\tilde F_{n+1,x}+4iE \tilde F_{n+1}^2) \notag\\ =&-4 \tilde q^3 (E-\tilde E)^2 R_{2n+2}(E), \label{02051} \end{align} that is,(\tilde p,\tilde q)$is associated with the curve $$\label{02052} \tilde \calK_{n+1}=\{(E,V) \, | \, V^2=(E-\tilde E)^2 R_{2n+2}(E)\}.$$ \begin{cor} \label{c6.2} Suppose$(p,q)$is a meromorphic algebro-geometric AKNS potential associated with the hyperelliptic curve $$\calK_n=\{(E,V) \, | \,V^2=R_{2n+2}(E)=\prod^{2n+1}_{m=0}(E-E_m)\},$$ which has a singular point at$(\tilde E,0)$, that is,$R_{2n+2}$has a zero of order$r\geq2$at the point$\tilde E$. Choose $$\label{6.35} \phi^{(0)}(\tilde E,x)=\frac{G_{n+1}(\tilde E,x)} {F_n(\tilde E,x)}$$ (cf. (\ref{6.2})) and define$(\tilde p,\tilde q)$as in (\ref{6.17}), (\ref{6.18}). Then$\phi^{(0)}(\tilde E,\cdot)$is meromorphic and the meromorphic algebro-geometric AKNS potential$(\tilde p,\tilde q)$is associated with the hyperelliptic curve $$\label{6.36} \tilde \calK_{\tilde n}=\{(E,V) \, | \,V^2=\tilde R_{2n-2s+4}(E) =(E-\tilde E)^{2-2s} R_{2n+2}(E)\}$$ for some$2\leq s\leq (r/2)+1$. In particular,$\tilde \calK_{\tilde n}$and$\calK_n$have the same structure near any point$E\neq \tilde E$. \end{cor} \begin{proof} Since$V^2=R_{2n+2}(E)$we infer that$V_+(E)$has at least a simple zero at$\tilde E$. Hence $$\label{01051} \phi_\pm(E,x)-\phi^{(0)}(\tilde E,x)=\frac{\pm V_+(E)}{F_n(E,x)} +\frac{G_{n+1}(E,x)F_n(\tilde E,x)-G_{n+1}(\tilde E,x)F_n(E,x)} {F_n(E,x)F_n(\tilde E,x)}$$ also have at least a simple zero at$\tilde E$. From \eqref{01052} one infers that$\tilde F_{n+1}(E,x)$has a zero of order at least$2$at$\tilde E$, that is, $$\label{02053} \tilde F_{n+1}(E,x)=(E-\tilde E)^s \tilde F_{n+1-s}(E,x), \,\, s\geq 2.$$ Define$\tilde n=n+1-s$. Then$\tilde F_{\tilde n}$still satisfies the hypothesis of Corollary \ref{c2.3a}. Moreover, inserting \eqref{02053} into \eqref{02051} shows that$(E-\tilde E)^{2s}$must be a factor of$(E-\tilde E)^2 R_{2n+2}(E)$. Thus,$2s\leq r+2and hence %\begin{align} $$\tilde q(2\tilde F_{\tilde n}\tilde F_{\tilde n,xx} -\tilde F_{\tilde n,x}^2+4(E^2-\tilde p\tilde q)\tilde F_{\tilde n}^2) -\tilde q_x(2\tilde F_{\tilde n}\tilde F_{\tilde n,x} +4iE \tilde F_{\tilde n}^2) =-4 \tilde q^3 \tilde R_{2\tilde n+2}(E), \label{02054}$$ %\end{align} where $$\label{02055} \tilde R_{2\tilde n+2}(E)=(E-\tilde E)^{2-2s} R_{2n+2}(E)$$ is a polynomial inE$of degree$0<2n-2s+4<2n+2$. This proves \eqref{6.36}. \end{proof} In view of our principal result, Theorem \ref{t5.4}, our choice of$\phi^{(0)}(\tilde E,x)$led to a curve$\tilde \calK_{\tilde n}$which is less singular at$\tilde E$than$\calK_n$, without changing the structure of the curve elsewhere. By iterating the procedure from$\calK_n$to$\tilde\calK_{\tilde n}$one ends up with a curve which is nonsingular at$(\tilde E,0)$. Repeating this procedure for each singular point of$\calK_n$then results in a nonsingular curve$\hat \calK_{\hat n}$and a corresponding Baker-Akhiezer function$\hat \Psi(P,x,x_0)$which is meromorphic with respect to$x\in \bbC$(this can be seen by using their standard theta function representation, cf., e.g., \cite{27b}). Suppose that$\hat \calK_{\hat n}$was obtained from$\calK_n$by applying the gauge transformation $$\Gamma(E,x)=\Gamma_N(E,x)...\Gamma_1(E,x),$$ where each of the$\Gamma_j$is of the type \eqref{6.16}. Then the branches of $$\label{02056} \Psi(P,x)=\Gamma(E,x)^{-1} \hat \Psi(P,x,x_0) =\Gamma_1(E,x)^{-1}...\Gamma_N(E,x)^{-1} \hat \Psi(P,x,x_0)$$ are linearly independent solutions of$L\Psi=E\Psi$for all$E\in \bbC \setminus \{E_0,...,E_{2n+1},\mu_1(x_0),...,\linebreak[0] \mu_n(x_0)\}$. These branches are meromorphic with respect to$x$since $$\label{02057} \Gamma_j(E,x)^{-1}=\frac{2i}{E-\tilde E} \begin{pmatrix}-\tfrac{i}{2}&-\tfrac{i}{2}q(x)\\ -\tfrac{i}{2}\phi^{(0)}(\tilde E,x) &E-\tilde E-\tfrac{i}{2}q(x)\phi^{(0)}(\tilde E,x) \end{pmatrix}$$ maps meromorphic functions to meromorphic functions in view of the fact that$q$and$\phi^{(0)}(\tilde E,\cdot)=G_{n+1}(\tilde E,\cdot)/ F_n(\tilde E,\cdot)$are meromorphic. Combining these findings and Theorem \ref{t05051} we thus proved the principal result of this section. \begin{thm} \label{t6.3} Suppose$(p,q)$is a meromorphic algebro-geometric AKNS potential. Then the solutions of$L\Psi=E\Psi$are meromorphic with respect to the independent variable for all values of the spectral parameter$E\in \bbC$. \end{thm} \begin{rem} In the case of the KdV hierarchy, Ehlers and Kn\"orrer \cite{17b} used the Miura transformation and algebro-geometric methods to prove results of the type stated in Corollary \ref{c6.2}. An alternative approach in the KdV context has recently been found by Ohmiya \cite{49c}. The present technique to combine gauge transformations, the polynomial recursion approach to integrable hierarchies based on hyperelliptic curves (such as the KdV, AKNS, and Toda hierarchies), and the fundamental meromorphic function$\phi(\cdot,x)$on$\calK_n$(cf. (\ref{6.2})), yields a relatively straightforward and unified treatment, further details of which will appear elsewhere. To the best of our knowledge this is the first such approach for the AKNS hierarchy. A systematic study of the construction used in Theorem \ref{t6.3} yields explicit connections between the$\tau$-function associated with the possibly singular curve$\calK_n$and the Riemann theta function of the nonsingular curve$\hat \calK_{\hat n}$. This seems to be of independent interest and will be pursued elsewhere. \end{rem} \section{Floquet Theory} \label{floquet} \setcounter{equation}{0} Throughout this section we will assume the validity of the following basic hypothesis. \begin{hyp} \label{h3.1} Suppose that$p, q \in L^1_{\text{loc}} (\bb R)$are complex-valued periodic functions of period$\Omega>0$and that$L$is a$2 \times 2$matrix-valued differential expression of the form $$\label{3.0} L=J\frac{d}{dx}+Q,$$ where $$\label{3.1} J=\begin{pmatrix}i&0\\0&-i\end{pmatrix}, \quad Q=\begin{pmatrix}0&-iq\\ip&0\end{pmatrix}.$$ \end{hyp} We note that $$\label{3.2} -J^2=I \quad \text{ and } \quad JQ+QJ=0,$$ where$I$is the$2\times2$identity matrix in$\bb C^2$. Given Hypothesis \ref{h3.1}, we uniquely associate the following densely and maximally defined closed linear operator$H$in$L^2(\bb R)^2$with the matrix-valued differential expresssion$L$, $$\label{3.2a} Hy=Ly, \, {\mathcal D}(H)=\{y \in L^2 (\bb R)^2 \, | \, y \in AC_{\text{loc}}(\bb R)^2, \, Ly \in L^2(\bb R)^2\}.$$ One easily verifies that$L$is unitarily equivalent to $$\label{3.3} \begin{pmatrix}0&-1\\1&0\end{pmatrix}\frac{d}{dx}+ \frac12 \begin{pmatrix}(p+q)&i(p-q)\\i(p-q)&-(p+q)\end{pmatrix},$$ a form widely used in the literature. We consider the differential equation$Ly=Ey$where$L$satisfies Hypothesis \ref{h3.1} and where$E$is a complex spectral parameter. Define$\phi_0(E,x,x_0,Y_0)=\e^{E(x-x_0)J}Y_0$for$Y_0\in M_2(\bb C)$. The matrix function$\phi(E,\cdot,x_0,Y_0)$is the unique solution of the integral equation $$\label{3.4} Y(x)=\phi_0(E,x,x_0,Y_0)+\int_{x_0}^x \e^{E(x-x')J} JQ(x') Y(x') dx'$$ if and only if it satisfies the initial value problem $$\label{3.5} JY'+QY=EY, \quad Y(x_0)=Y_0.$$ Since $$\label{3.6} \frac{\partial \phi_0}{\partial x_0}(E,x,x_0,Y_0) =E J \e^{E(x-x_0)J}Y_0=E \e^{E(x-x_0)J} JY_0,$$ differentiating \eqref{3.4} with respect to$x_0yields \begin{align} \frac{\partial \phi}{\partial x_0}(E,x,x_0,Y_0) =& \e^{E(x-x_0)J}(EJ-JQ(x_0))Y_0 \notag \\ &+\int_{x_0}^x \e^{E(x-x')J} J Q(x') \frac{\partial \phi}{\partial x_0}(E,x',x_0,Y_0,) dx', \label{3.7} \end{align} that is, $$\label{3.8} \frac{\partial \phi}{\partial x_0}(E,x,x_0,Y_0) =\phi(E,x,x_0,(E+Q(x_0))JY_0),$$ taking advantage of the fact that\eqref{3.4}$has unique solutions. In contrast to the Sturm-Liouville case, the Volterra integral equation \eqref{3.4} is not suitable to determine the asymptotic behavior of solutions as$E$tends to infinity. The following treatment circumvents this difficulty and closely follows the outline in \cite{44e}, Section~1.4. Suppose$L$satifies Hypothesis \ref{h3.1},$p,q \in C^n(\bb R), and then define recursively, \begin{align} a_1(x)&=iQ(x), \notag \\ b_k(x)&=-i\int_0^x Q(t) a_k(t) dt, \label{3.9} \\ a_{k+1}(x)&=-a_{k,x}(x)+iQ(x) b_k(x), \quad k=1, \dots,n. \notag \end{align} Next letA:\bb R^2\to \bb M_2(\bb C)be the unique solution of the integral equation $$\label{3.10} A(x,y)=a_{n+1}(x-y)+\int_0^y Q(x-y') \int_{y-y'}^{x-y'} Q(x') A(x',y-y') dx' dy'.$$ Introducing $$\label{3.11} \hat a_n(E,x)=\int_0^x A(x,y) \e^{-2iEy}dy, \quad \hat b_n(E,x)=-i\int_0^x Q(y) \hat a_n(E,y) dy$$ and \begin{align} u_1(E,x)&=I+\sum_{k=1}^n b_k(x) (2iE)^{-k} +\hat b_n(E,x)(2iE)^{-n}, \label{3.12} \\ u_2(E,x)&=\sum_{k=1}^n a_k(x) (2iE)^{-k} +\hat a_n(E,x)(2iE)^{-n}, \label{3.13} \end{align} we infer that \begin{align} Y_1(E,x)&=\e^{iEx} \{(I+iJ)u_1(E,x)+(I-iJ)u_2(E,x)\}, \label{3.14}\\ Y_2(E,x)&=\e^{-iEx} \{(I-iJ)u_1(-E,x)-(I+iJ)u_2(-E,x)\} \label{3.15} \end{align} satisfy the differential equation $$\label{3.16} JY'+QY=EY.$$ Since|A(x,y)|$is bounded on compact subsets of$\bb R^2$one obtains the estimates $$\label{3.17} |\e^{iEx}\hat a_n(E,x)|, \quad |\e^{iEx} \hat b_n(E,x)|\leq CR^2 \e^{|x\Im(E)|}$$ for a suitable constant$C>0$as long as$|x|$is bounded by some$R>0$. The matrix$\hat Y(E,x,x_0)=(Y_1(E,x-x_0)+Y_2(E,x-x_0))/2$is also a solution of$JY'+QY=EY$and satisfies$\hat Y(E,x_0,x_0)=I+Q(x_0)/(2E)$. Therefore, at least for sufficiently large$|E|$, the matrix function $$\label{3.18} \phi(E,\cdot,x_0,I)= \hat Y(E,\cdot,x_0) \hat Y(E,x_0,x_0)^{-1}$$ is the unique solution of the initial value problem$JY'+QY=EY$,$Y(x_0)=I$. Hence, if$p, q \in C^2(\bb R), one obtains the asymptotic expansion \begin{align} \phi(E,x_0 + \Omega,x_0,I)=& \begin{pmatrix} \e^{-iE\Omega}&0\\0&\e^{iE\Omega}\end{pmatrix} +\frac{1}{2iE}\begin{pmatrix}\beta\e^{-iE\Omega}&2q(x_0)\sin(E\Omega)\\ 2p(x_0)\sin(E\Omega) &-\beta\e^{iE\Omega}\end{pmatrix} \notag \\ &+O(\e^{|\Im(E)|\Omega}E^{-2}), \label{3.19} \end{align} where $$\beta=\int_{x_0}^{x_0+\Omega} p(t)q(t) dt. \label{3.19a}$$ From this result we infer in particular that the entries of\phi(\cdot,x_0 +\Omega,x_0,I)$, which are entire functions, have order one whenever$q(x_0)$and$p(x_0)$are nonzero. Denote by$T$the operator defined by$Ty=y(\cdot+\Omega)$on the set of$\bb C^2$-valued functions on$\bb R$and suppose$L$satisfies Hypothesis \ref{h3.1}. Then$T$and$L$commute and this implies that$T(E)$, the restriction of$T$to the (two-dimensional) space$V(E)$of solutions of$Ly=Ey$, maps$V(E)$into itself. Choosing as a basis of$V(E)$the columns of$\phi(E,\cdot,x_0,I)$, the operator$T(E)$is represented by the matrix$\phi(E,x_0 + \Omega,x_0,I)$. In particular,$\det (T(E))=\det(\phi(E,x_0 +\Omega,x_0,I))=1$. Therefore, the eigenvalues$\rho(E)$of$T(E)$, the so called Floquet multipliers, are determined as solutions of $$\label{3.20} \rho^2 -\tr (T(E)) \rho +1=0.$$ These eigenvalues are degenerate if and only if$\rho^2 (E)=1$which happens if and only if the equation$Ly=Ey$has a solution of period$2\Omega$. Hence we now study asymptotic properties of the spectrum of the densely defined closed realization$H_{2\Omega,x_0}$of$L$in$L^2([x_0,x_0 + 2\Omega])^2given by \begin{align} H_{2\Omega,x_0}y=Ly, \, {\mathcal D} (H_{2\Omega,x_0}) = \{& y \in L^2([x_0,x_0+2\Omega])^2 \, | \, y\in AC([x_0,x_0 + 2\Omega])^2, \notag \\ & y(x_0+2\Omega)=y(x_0), \, Ly \in L^2([x_0,x_0+2\Omega])^2 \}. \label{3.21} \end{align} Its eigenvalues, which are called the (semi-)periodic eigenvalues ofL$, and their multiplicities are given, respectively, as the zeros and their multiplicities of the function$\tr (T(E))^2 -4$. The asymptotic behavior of these eigenvalues is described in the following result. \begin{thm} \label{t3.2} Suppose that$p,q \in C^2(\bb R)$. Then the eigenvalues$E_j$,$j\in\bb Z$of$H_{2\Omega,x_0}$are$x_0$-independent and satisfy the asymptotic behavior $$\label{3.22} E_{2j},E_{2j-1}=\frac{j\pi}{\Omega}+O(\frac1{|j|})$$ as$|j|$tends to infinity, where all eigenvalues are repeated according to their algebraic multiplicities. In particular, all eigenvalues of$H_{2\Omega,x_0}$are contained in a strip $$\label{3.23} \Sigma=\{E\in\bb C \, | \, |\Im(E)|\leq C\}$$ for some constant$C>0$. \end{thm} \begin{proof} Denoting$A(E,x)=(E+Q(x))J(cf.(\ref{6.11})), equation \eqref{3.8} implies \begin{align} &\partial \phi(E,x, x_0,I)/\partial x =-A(E,x)\phi(E,x,x_0,I), \label{3.31} \\ &\partial \phi(E,x,x_0,I)/\partial x_0 =\phi(E,x,x_0,I)A(E,x) \label{3.32} \end{align} and hence $$\label{3.24a} \partial \tr(T(E))/\partial x_0 =0.$$ Thus the eigenvalues ofH_{2\Omega,x_0}$are independent of$x_0$. According to \eqref{3.19},$\tr (T(E))$is asymptotically given by $$\label{3.24} \tr (T(E))=2\cos (E\Omega)+\beta \sin(E\Omega)E^{-1} + O(\e^{|\Im(E)|\Omega}E^{-2}).$$ Rouch\'e's theorem then implies that two eigenvalues$E$lie in a circle centered at$j\pi/a$with radius of order$1/|j|$. To prove that the eigenvalues may be labeled in the manner indicated, one again uses Rouch\'e's theorem with a circle of sufficiently large radius centered at the origin of the$E$-plane in order to compare the number of zeros of$\tr (T(E))^2 -4$and$4\cos(E\Omega)^2 -4$in the interior of this circle. \end{proof} The conditional stability set$\mc S(L)$of$L$in \eqref{3.0} is defined to be the set of all spectral parameters$E$such that$Ly=Ey$has at least one bounded nonzero solution. This happens if and only if the Floquet multipliers$\rho (E)$of$Ly=Ey$have absolute value one. Hence $$\label{3.25} \mc S(L)=\{E\in\bb C \, | \, -2 \leq \tr (T(E))\leq 2\}.$$ It is possible to prove that the spectrum of$H$coincides with the conditional stability set$\mc S(L)$of$L$, but since we do not need this fact we omit a proof. In the following we record a few properties of$\mc S(L)$to be used in Sections \ref{fingap} and \ref{picard}. \begin{thm} \label{t3.3} Assume that$p,q \in C^2(\bb R)$. Then the conditional stability set$\mc S(L)$consists of a countable number of regular analytic arcs, the so called spectral bands. At most two spectral bands extend to infinity and at most finitely many spectral bands are closed arcs. The point$E$is a band edge, that is, an endpoint of a spectral band, if and only if$\tr (T(E))^2 -4$has a zero of odd order. \end{thm} \begin{proof} The fact that$\mc S(L)$is a set of regular analytic arcs whose endpoints are odd order zeros of$(\tr (T(E)))^2 -4$and hence countable in number, follows in standard manner from the fact that$\tr (T(E))$is entire with respect to$E$. (For additional details on this problem, see, for instance, the first part of the proof of Theorem 4.2 in \cite{w3}.) From the asymptotic expansion \eqref{3.19} one infers that$\tr (T(E))$is approximately equal to$2\cos(E\Omega)$for$|E|$sufficiently large. This implies that the Floquet multipliers are in a neighborhood of$\e^{\pm iE\Omega}$. If$E_0\in\mc S(L)$and$|E_0|$is sufficiently large, then it is close to a real number. Now let$E=|E_0|\e^{it}$, where$t\in(-\pi/2,3\pi/2]$. Whenever this circle intersects$\mc S(L)$then$t$is close to$0$or$\pi$. When$t$is close to$0$, the Floquet multiplier which is near$\e^{iE\Omega}$moves radially inside the unit circle while the one close to$\e^{-iE\Omega}$leaves the unit disk at the same time. Since this can happen at most once, there is at most one intersection of the circle of radius$|E_0|$with$\mc S(L)$in the right half-plane for$|E|$sufficiently large. Another such intersection may take place in the left half-plane. Hence at most two arcs extend to infinity and there are no closed arcs outside a sufficiently large disk centered at the origin. Since there are only countably many endpoints of spectral arcs, and since outside a large disk there can be no closed spectral arcs, and at most two arcs extend to infinity, the conditional stability set consists of at most countably many arcs. \end{proof} Subsequently we need to refer to components of vectors in$\bb C^2$. If$y\in\bb C^2$, we will denote the first and second components of$y$by$y_1$and$y_2$, respectively, that is,$y=(y_1,y_2)^t$, where the superscript $t$'' denotes the transpose of a vector in$\bb C^2$. The boundary value problem$Ly=zy$,$y_1(x_0)=y_1(x_0+\Omega)=0$in close analogy to the scalar Sturm-Liouville case, will be called the Dirichlet problem for the interval$[x_0,x_0+\Omega]$and its eigenvalues will therefore be called Dirichlet eigenvalues (associated with the interval$[x_0,x_0+\Omega]$). In the corresponding operator theoretic formulation one introduces the following closed realization$H_{D,x_0}$of$L$in$L^2([x_0,x_0+\Omega])^2, \begin{align} H_{D,x_0} y=Ly, \, {\mathcal D} (H_{D,x_0}) = \{& y \in L^2([x_0,x_0+\Omega])^2 \, | \, y\in AC([x_0,x_0 + \Omega])^2, \notag \\ & y_1(x_0)=y_1(x_0+\Omega)=0, \, Ly \in L^2([x_0,x_0+\Omega])^2 \}. \label{3.26} \end{align} The eigenvalues ofH_{D,x_0}$and their algebraic multiplicities are given as the zeros and their multiplicities of the function $$\label{3.31a} g(E,x_0)=(1,0)\phi(E,x_0+\Omega,x_0,I)(0,1)^t,$$ that is, the entry in the upper right corner of$\phi(E,x_0+\Omega,x_0,I)$. \begin{thm} \label{t3.4} Suppose$p,q \in C^2(\bb R)$. If$q(x_0)\neq0$then there are countably many Dirichlet eigenvalues$\mu_j(x_0)$,$j\in\bb Z$, associated with the interval$[x_0,x_0+\Omega]$. These eigenvalues have the asymptotic behavior $$\label{3.27} \mu_j(x_0)=\frac{j\pi}{\Omega}+O(\frac1{|j|})$$ as$|j|$tends to infinity, where all eigenvalues are repeated according to their algebraic multiplicities. \end{thm} \begin{proof} From the asymptotic expansion \eqref{3.19} we obtain that $$\label{3.28} g(E,x_0)=\frac{-iq(x_0)}{E}\sin(E\Omega)+O(\e^{|\Im(E)|\Omega}E^{-2}).$$ Rouch\'e's theorem implies that one eigenvalue$E$lies in a circle centered at$j\pi/ \Omega$with radius of order$1/|j|$and that the eigenvalues may be labeled in the manner indicated (cf. the proof of Theorem \ref{t3.2}). \end{proof} We now turn to the$x$-dependence of the function$g(E,x)$. \begin{thm} \label{t3.5} Assume that$p,q \in C^1(\bb R)$. Then the function$g(E,\cdot)satisfies the differential equation \begin{align} &q(x)(2g(E,x) g_{xx}(E,x) -g_x(E,x)^2+4(E^2- p(x)q(x))g(E,x)^2) \notag \\ &-q_x(x)(2g(E,x) g_x(E,x)+4iE g(E,x)^2) =-q(x)^3(\tr (T(E))^2 -4). \label{3.30} \end{align} \end{thm} \begin{proof} Sinceg(E,x)=(1,0)\phi(E,x,x+\Omega,I) (0,1)^twe obtain from \eqref{3.31} and \eqref{3.32}, \begin{align} g_x(E,x)=&(1,0)(\phi(E,x,x+\Omega,I) A(E,x)-A(E,x) \phi(E,x,x+\Omega,I))(0,1)^t, \label{3.33} \\ g_{xx}(E,x)=&(1,0)(\phi(E,x,x+\Omega,I) A(E,x)^2- 2A(E,x)\phi(E,x,x+\Omega,I) A(E,x) \notag \\ &+A(E,x)^2\phi(E,x,x+\Omega,I) +\phi(E,x,x+\Omega,I) A_x(E,x) \notag \\ &-A_x(E,x)\phi(E,x,x+\Omega,I))(0,1)^t, \label{3.34} \end{align} where we used periodicity ofA$, that is,$A(E,x+\Omega)=A(E,x)$. This yields the desired result upon observing that$\tr(\phi(z,x+\Omega,x,I))=\tr (T(E))$is independent of$x$. \end{proof} \begin{defn} \label{d3.6} The algebraic multiplictiy of$E$as a Dirichlet eigenvalue$\mu(x)$of$H_{D,x}$is denoted by$\delta(E,x)$. The quantities $$\label{3.35} \delta_i(E)=\min\{\delta(E,x)\, | \,x\in\bb R\},$$ and $$\label{3.36} \delta_m(E,x)=\delta(E,x)-\delta_i(E)$$ will be called the immovable part and the movable part of the algebraic multiplicity$\delta(E,x)$, respectively. The sum$\sum_{E\in\bb C} \delta_m(E,x)$is called the number of movable Dirichlet eigenvalues. \end{defn} If$q(x)\neq0$the function$g(\cdot,x)is an entire function with order of growth equal to one. The Hadamard factorization theorem then implies $$\label{3.37} g(E,x)=F_D(E,x) D(E),$$ where \begin{align} F_D(E,x)&=g_D(x) \e^{h_D(x)E} E^{\delta_m(0,x)} \prod_{\lambda\in {\bb C} \setminus \{0\}} (1-(E/ \lambda))^{\delta_m(z,x)} \e^{\delta_m(z,x)E}, \label{3.38} \\ D(E)&=\e^{d_0 E} E^{\delta_i(E)}\prod_{\lambda\in {\bb C} \setminus \{0\}} (1-(E/ \lambda))^{\delta_i(E)} \e^{\delta_i(E)E},\label{3.39} \end{align} for suitable numbersg_D(x)$and$h_D(x)$and$d_0. Define $$\label{3.39a} U(E)=(\tr (T(E))^2 -4)/D(E)^2.$$ Then Theorem \ref{t3.2} shows that \begin{align} &-q(x)^3 U(E)\notag\\ =&q(x)(2F_D(E,x)F_{D,xx}(E,x)- F_{D,x}(E,x)^2+ 4(E^2-q(x)p(x))F_D(E,x)^2)\notag \\ &-q_x(x)(2F_D(E,x)F_{D,x}(E,x_0)+4iE F_D(E,x)^2). \label{3.40} \end{align} As a function ofE$the left-hand side of this equation is entire (see Proposition 5.2 in \cite{w3} for an argument in a similar case). Introducing$s(E)=\ord_E(\tr (T(E))^2 -4)$we obtain the following important result. \begin{thm} \label{t3.7} Under the hypotheses of Theorem \ref{t3.5},$s(E)-2\delta_i(E)\geq0$for every$E\in\bb C$. \end{thm} We now define the sets${\calE}_1=\{E\in\bb C \, | \, s(E)>0, \, \delta_i(E)=0\}$and${\calE}_2=\{E\in\bb C \, | \, s(E)-2\delta_i(E)>0\}$. Of course,$\calE_1$is a subset of$\calE_2$which, in turn, is a subset of the set of zeros of$\tr (T(E))^2-4$and hence isolated and countable. \begin{thm} \label{t3.8} Assume Hypothesis \ref{h3.1} and that$Ly=Ey$has degenerate Floquet multipliers$\rho$(equal to$\pm1$) but two linearly independent Floquet solutions. Then$E$is an immovable Dirichlet eigenvalue, that is,$\delta_i(E)>0$. Moreover,$\calE_1$is contained in the set of all those values of$E$such that$Ly=Ey$does not have two linearly independent Floquet solutions. \end{thm} \begin{proof} If$Ly=Ey$has degenerate Floquet multipliers$\rho (E)$but two linearly independent Floquet solutions then every solution of$Ly=Ey$is Floquet with multiplier$\rho (E)$. This is true, in particular, for the unique solution$y$of the initial value problem$Ly=Ey$,$y(x_0)=(0,1)^t$. Hence$y(x_0+\Omega)=(0,\rho)^t$and$y$is a Dirichlet eigenfunction regardless of$x_0$, that is,$\delta_i(E)>0$. If$E\in \calE_1$then$s(E)>0$and$Ly=Ey$has degenerate Floquet multipliers. Since$\delta_i(E)=0$, there cannot be two linearly independent Floquet solutions. \end{proof} Spectral theory for nonself-adjoint periodic Dirac operators has very recently drawn considerable attention in the literature and we refer the reader to \cite{32e} and \cite{59c}. \section{Floquet Theory and Algebro-Geometric Potentials} \label{fingap} \setcounter{equation}{0} In this section we will obtain necessary and sufficient conditions in terms of Floquet theory for a function$Q:\bb R\to M_2(\bb C)$which is periodic with period$\Omega>0$and which has zero diagonal entries to be algebro-geometric (cf. Definition \ref{d2.2}). Throughout this section we assume the validity of Hypothesis \ref{h3.1}. We begin with sufficient conditions on$Q$and recall the definition of$U(E)$in \eqref{3.39a}. \begin{thm} \label{t4.1} Suppose that$p,q \in C^2(\bb R)$are periodic with period$\Omega>0$. If$U(E)$is a polynomial of degree$2n+2$then the following statements hold. \\ (i)$\deg(U)$is even, that is,$n$is an integer. \\ (ii)The number of movable Dirichlet eigenvalues (counting algebraic multiplicities) equals$n$. \\ (iii)$\mc S(L)$consists of finitely many regular analytic arcs. \\ (iv)$p,q\in C^\infty(\bb R)$. \\ (v) There exists a$2\times2$matrix-valued differential expression$P_{n+1}$of order$n+1$with leading coefficient$J^{n+2}$which commutes with$L$and satisfies $$\label{4.1} P_{n+1}^2=\prod_{E\in F_2} (L-E)^{s(E)-2\delta_i(E)}.$$ \end{thm} \begin{proof} The asymptotic behavior of Dirichlet and periodic eigenvalues (Theorems \ref{t3.2} and \ref{t3.4}) shows that$s(E)\leq 2$and$\delta(E,x) \leq1$when$|E|$is suitably large. Since$(U(E)$is a polynomial,$s(E)>0$implies that$s(E)=2\delta_i(E)=2$. If$\mu(x)$is a Dirichlet eigenvalue outside a sufficiently large disk, then it must be close to$m\pi/ \Omega$for some integer$m$and hence close to a point$E$where$s(E)=2\delta_i(E)=2$. Since there is only one Dirichlet eigenvalue in this vicinity we conclude that$\mu(x)=E$is independent of$x$. Hence, outside a sufficiently large disk, there is no movable Dirichlet eigenvalue, that is,$F_D(\cdot,x)$is a polynomial. Denote its degree, the number of movable Dirichlet eigenvalues, by$\tilde n$. By \eqref{3.40}$U(E)$is a polynomial of degree$2\tilde n+2$. Hence$\tilde n=n$and this proves parts (i) and (ii) of the theorem. Since asymptotically$s(E)=2$, we infer that$s(E)=1$or$s(E)\geq 3$occurs at only finitely many points$E$. Hence, by Theorem \ref{t3.3}, there are only finitely many band edges, that is,$\mc S(L)$consists of finitely many arcs, which is part (iii) of the theorem. Let$\gamma(x)$be the leading coefficient of$F_D(\cdot,x)$. From equation \eqref{3.40} we infer that$-\gamma(x)^2/q(x)^2$is the leading coefficient of$U(E)$and hence$\gamma(x)=ci q(x)$for a suitable constant$c$. Therefore,$F(\cdot,x)=F_D(\cdot,x)/c$is a polynomial of degree$n$with leading coefficient$iq(x)$satisfying the hypotheses of Theorem \ref{t2.3}. This proves that$p, q\in C^\infty(\bb R)$and that there exists a$2 \times 2$matrix-valued differential expression$P_{n+1}$of order$n+1$and leading coefficient$J^{n+2}$which commutes with$L$. The differential expressions$P_{n+1}$and$L$satisfy$P_{n+1}^2=R_{2n+2}(L)$, where $$\label{4.2} R_{2n+2}(E)=U(E)/(4c^2)=\prod_{\lambda\in F_2} (E-\lambda)^{s(\lambda)-2\delta_i(\lambda)},$$ concluding parts (iv) and (v) of the theorem. \end{proof} \begin{thm} \label{t4.2} Suppose that$p, q \in C^2(\bb R)$are periodic of period$\Omega > 0$and that the differential equation$Ly=Ey$has two linearly independent Floquet solutions for all but finitely many values of$E$. Then$U(E)$is a polynomial. \end{thm} \begin{proof} Assume that$U(E)$in \eqref{3.39a} is not a polynomial. At any point outside a large disk where$s(E)>0$we have two linearly independent Floquet solutions and hence, by Theorem \ref{t3.8},$\delta_i(E)\geq1$. On the other hand, we infer from Theorem \ref{t3.2} that$s(E)\leq 2$and hence$s(E)-2\delta_i(E)=0$. Therefore,$s(E)-2\delta_i(E)>0$happens only at finitely many points and this contradiction proves that$U(E)$is a polynomial. \end{proof} \begin{thm} \label{t4.3} Suppose that$p, q \in C^2(\bb R)$are periodic of period$\Omega > 0$and that the associated Dirichlet problem has$n$movable eigenvalues for some$n \in \bb N$. Then$U(E)$is a polynomial of degree$2n+2$. \end{thm} \begin{proof} If there are$n$movable Dirichlet eigenvalues, that is, if$\deg(F_D(\cdot,x))=n$then \eqref{3.40} shows that$U(E)=(\tr (T(E))^2 -4)/D(E)^2$is a polynomial of degree$2n+2$. \end{proof} Next we prove that$U(E)$being a polynomial, or the number of movable Dirichlet eigenvalues being finite, is also a necessary condition for$Q$to be algebro-geometric. \begin{thm} \label{t4.4} Suppose$L$satisfies Hypothesis \ref{h3.1}. Assume there exists a$2 \times 2$matrix-valued differential expression$P_{n+1}$of order$n+1$with leading coefficient$J^{n+2}$which commutes with$L$but that there is no such differential expression of smaller order commuting with$L$. Then$U(E)$is a polynomial of degree$2n+2$. \end{thm} \begin{proof} Without loss of generality we may assume that$P_{n+1} =\hat P_{c_1,...,c_{n+1}}$for suitable constants$c_j$. According to the results in Section \ref{akns}, the polynomial $$\label{4.3} F_n(E,x) =\sum_{\ell=0}^n f_{n-\ell}(c_1,...,c_{n-\ell})(x)E^{\ell}$$ satisfies the hypotheses of Theorem \ref{t2.3}. Hence the coefficients$f_{\ell}$and the functions$p$and$q$are in$C^\infty(\bb R)$. Also the$f_{\ell}$, and hence$P_{n+1}$, are periodic with period$\Omega$. Next, let$\mu(x_0)$be a movable Dirichlet eigenvalue. Since$\mu(x)$is a continuous function of$x\in \bb R$and since it is not constant, there exists an$x_0 \in \bb R$such that$s(\mu(x_0))=0$, that is,$\mu(x_0)$is neither a periodic nor a semi-periodic eigenvalue. Suppose that for this choice of$x_0$the eigenvalue$\mu:=\mu(x_0)$has algebraic multiplicity$k$. Let$V=\ker ((H_{D,x_0}-\mu)^k)$be the algebraic eigenspace of$\mu$. Then$V$has a basis$\{y_1,...,y_k\}$such that$(H_{D,x_0}-\mu)y_j=y_{j-1}$for$j=1,...,k$, agreeing that$y_0=0$. Moreover, we introduce$V_m := \text{span} \, \{y_1,...,y_m\}$and$V_0=\{0\}$. First we show by induction that there exists a number$\nu$such that$(T-\rho)y, (P_{n+1}-\nu)y \in V_{m-1}$, whenever$y\in V_m$. Let$m=1$. Then$(H_{D,x_0}-\mu)y=0$implies$y=\alpha y_1$for some constant$\alpha$and hence$y$is a Floquet solution with multiplier$\rho=y_{1,2}(x_0+\Omega)$, that is,$(T-\rho)y=0$. (We define, in obvious notation,$y_{j,k}$,$k=1,2$to be the$k$-th component of$y_j$,$1\leq j \leq m$.) Since$P_{n+1}$commutes with both$L$and$T$, we find that$P_{n+1}y$is also a Floquet solution with multiplier$\rho$. Since$s(\mu)=0$, the geometric eigenspace of$\rho$is one-dimensional and hence$P_{n+1}y=\nu y$for a suitable constant$\nu$. Now assume that the statement is true for$1\leq m0$,$\Re(\omega_3) \ge 0$,$\Im(\omega_3/\omega_1)>0$. The fundamental period parallelogram then consists of the points$E=2\omega_1 s +2\omega_3 t$, where$0\leq s,t<1$. We introduce$\theta\in(0,\pi)$by $$\label{5.3} e^{i\theta} = \frac{\omega_3}{\omega_1} \left|\frac{\omega_1}{\omega_3} \right|$$ and for$j=1,3$, $$\label{5.4} Q_j(\zeta)=t_j Q(t_j\zeta+x_0),$$ where$t_j=\omega_j/|\omega_j|$. Subsequently, the point$x_0$will be chosen in such a way that no pole of$Q_j$,$j=1,3$lies on the real axis. (This is equivalent to the requirement that no pole of$Q$lies on the line through the points$x_0$and$x_0+ 2\omega_1$nor on the line through$x_0$and$x_0+2\omega_3$. Since$Q$has only finitely many poles in the fundamental period parallelogram this can always be achieved.) For such a choice of$x_0$we infer that the entries of$Q_j(\zeta)$are real-analytic and periodic of period$\Omega_j =2|\omega_j|$whenever$\zeta$is restricted to the real axis. Using the variable transformation$x=t_j\zeta+x_0$,$\psi(x)=\chi(\zeta)$one concludes that$\psi$is a solution of $$\label{5.5} J\psi'(x)+Q(x)\psi(x)=E\psi(x)$$ if and only if$\chi$is a solution of $$\label{5.6} J\chi'(\zeta) + Q_j(\zeta)\chi(\zeta) = \lambda \chi(\zeta),$$ where$\lambda=t_j E$. Theorem \ref{t3.2} is now applicable and yields the following result. \begin{thm}\label{t5.3} Let$j=1$or$3$. Then all$4\omega_j$-periodic (i.e., all$2\omega_j$-periodic and all$2\omega_j$-semi-periodic) eigenvalues associated with$Q$lie in the strip$S_j$given by $$\label{5.7} S_j=\{E\in\bb C \, | \, |\Im(t_jE)|\le C_j \}$$ for suitable constants$C_j>0$. The angle between the axes of the strips$S_1$and$S_3$equals$\theta \in (0,\pi)$. \end{thm} Theorem \ref{t5.3} applies to any elliptic potential$Q$whether or not it is algebro-geometric. Next we present our principal result, a characterization of all elliptic algebro-geometric potentials of the AKNS hierarchy. Given the preparations in Sections \ref{gauge}--\ref{fingap}, the proof of our principal result, Theorem \ref{t5.4} below, will be fairly short. \begin{thm}\label{t5.4}$Q$is an elliptic algebro-geometric AKNS potential if and only if it is a Picard-AKNS potential. \end{thm} \begin{proof} The fact that any elliptic algebro-geometric AKNS potential is a Picard potential is a special case of Theorem~\ref{6.3}. Conversely, assume that$Q$is a Picard-AKNS potential. Choose$R>0$large enough such that the exterior of the closed disk$\overline{D(0,R)}$of radius$R$centered at the origin contains no intersection of$S_1$and$S_3$(defined in \eqref{5.7}), that is, $$\label{5.21} (\bb C\backslash \overline{D(0,R)}) \cap(S_1\cap S_3)=\emptyset.$$ Let$\rho_{j,\pm}(\lambda)$be the Floquet multipliers of$Q_j$, that is, the solutions of $$\label{5.22} \rho^2_j-\tr (T_j) \rho_j+1=0.$$ Then \eqref{5.21} implies that for$E\in\bb C \backslash\overline{D(0,R)}$at most one of the eigenvalues$\rho_1(t_1E)$and$\rho_3(t_3E)$can be degenerate. In particular, at least one of the operators$T_1$and$T_3$has distinct eigenvalues. Since by hypothesis$Q$is Picard, Picard's Theorem \ref{t5.1} applies with$A=-J(Q-E)$and guarantees the existence of two linearly independent solutions$\psi_1(E,x)$and$\psi_2(E,x)$of$J\psi' + Q\psi=E\psi$which are elliptic of the second kind. Then$\chi_{j,k}(\zeta)=\psi_k(t_j\zeta+x_0)$,$k=1,2$are linearly independent Floquet solutions associated with$Q_j$. Therefore the points$\lambda$for which$J \chi'+Q_j \chi=\lambda \chi$has only one Floquet solution are necessarily contained in$\overline{D(0,R)}$and hence finite in number. This is true for both$j=1$and$j=3$. Applying Theorem \ref{t4.2} then proves that both$Q_1$and$Q_3$are algebro-geometric. This implies that$Q$itself is algebro-geometric. \end{proof} The following corollary slightly extends the class of AKNS potentials$Q(x)$considered thus far in order to include some cases which are not elliptic but very closely related to elliptic$Q(x)$. Such cases have recently been considered by Smirnov \cite{58g}. \begin{cor}\label{c5.5} Suppose $$Q(x)=\begin{pmatrix} 0&-i q(x) \e^{-2(ax+b)}\\ip(x)\e^{2(ax+b)}&0 \end{pmatrix}$$ where$a,b\in\bb C$and$p,q$are elliptic functions with a common period lattice. Then$Q$is an algebro-geometric AKNS potential if and only if$J\Psi'+Q\Psi=E\Psi$has a meromorphic fundamental system of solutions (with respect to the independent variable) for all values of the spectral parameter$E\in\bb C$. \end{cor} \begin{proof} Suppose that for all values of$E$the equation$L\Psi=J\Psi'+Q\Psi =E\Psi$has a meromorphic fundamental system of solutions. Let $$\label{05051} \mathcal T= \begin{pmatrix}\e^{ax+b} & 0 \\ 0 & \e^{-ax+b} \end{pmatrix}.$$ Then$\mathcal T L \mathcal T^{-1}=\tilde L+ia I=Jd/dx+\tilde Qia I$, where $$\tilde Q=\begin{pmatrix}0 &-iq\\ ip & 0\end{pmatrix}.$$ Moreover,$L\Psi=E\Psi$is equivalent to$\tilde L(\mathcal T\Psi) =(E-ia)(\mathcal T\Psi)$. Hence the equation$\tilde L\Psi=(E-ia)\Psi$has a meromorphic fundamental system of solutions for all$E$. Consequently, Theorem \ref{t5.4} applies and yields that$\tilde Q$is an algebro-geometric AKNS potential. Thus, for some$n$there exists a differential expression$\tilde P$of order$n+1$with leading coefficient$-J^{n}$such that$[\tilde P,\tilde L]=0$. Define$P=\mathcal T^{-1} \tilde P\mathcal T$. The expression$P$is a differential expression of order$n+1$with leading coefficient$-J^{n}$and satisfies$[P,L]=\mathcal T^{-1}[\tilde P,\tilde L+ia I] \mathcal T=0$, that is,$Q$is an algebro-geometric AKNS potential. The converse follows by reversing the above proof. \end{proof} We add a series of remarks further illustrating the significance of Theorem \ref{t5.4}. \begin{rem} \label{r5.5} While an explicit proof of the algebro-geometric property of$(p,q)$is in general highly nontrivial (see, e.g., the references cited in connection with special cases such as the Lam\'e-Ince and Treibich-Verdier potentials in the introduction), the fact of whether or not$J\Psi'(x)+Q(x)\Psi(x)= E\Psi(x)$has a fundamental system of solutions meromorphic in$x$for all but finitely many values of the spectral parameter$E\in\bb C$can be decided by means of an elementary Frobenius-type analysis (see, e.g., \cite{29} and \cite{30}). To date, Theorem \ref{t5.4} appears to be the only effective tool to identify general elliptic algebro-geometric solutions of the AKNS hierarchy. \end{rem} \begin{rem} \label{r5.6} Theorem \ref{t5.4} complements Picard's Theorem \ref{t5.1} in the special case where$A(x)=-J(Q(x)-E)$in the sense that it determines the elliptic matrix functions$Q$which satisfy the hypothesis of the theorem precisely as (elliptic) algebro-geometric solutions of the stationary AKNS hierarchy. \end{rem} \begin{rem} \label{r5.7} Theorem \ref{t5.4} is also relevant in the context of the Weierstrass theory of reduction of Abelian to elliptic integrals, a subject that attracted considerable interest, see, for instance, \cite{5}, \cite{6}, \cite{6a}, Ch. 7, \cite{7}, \cite{8}, \cite{13d}, \cite{18}--\cite{20}, \cite{38}, \cite{40}, \cite{44}, \cite{44c}, \cite{58d}, \cite{58}, \cite{59}. In particular, the theta functions corresponding to the hyperelliptic curves derived from the Burchnall-Chaundy polynomials \eqref{2.22}, associated with Picard potentials, reduce to one-dimensional theta functions. \end{rem} \section{Examples} \label{Ex} \setcounter{equation}{0} With the exception of the studies by Christiansen, Eilbeck, Enol'skii, and Kostov in \cite{13d} and Smirnov in \cite{58g}, not too many examples of elliptic solutions$(p,q)$of the AKNS hierarchy associated with higher (arithmetic) genus curves of the type \eqref{2.22} have been worked out in detail. The genus$n=1$case is considered, for example, in \cite{37a}, \cite{51}. Moreover, examples for low genus$n$for special cases such as the nonlinear Schr\"odinger and mKdV equation (see \eqref{2.47} and \eqref{2.49}) are considered, for instance, in \cite{4b}, \cite{6}, \cite{44d}, \cite{45a}, \cite{50}, \cite{58f}. Subsequently we will illustrate how the Frobenius method, whose essence is captured by Proposition \ref{p552}, can be used to establish existence of meromorphic solutions and hence, by Theorem \ref{t5.4}, proves their algebro-geometric property. The notation established in the beginning of Section \ref{gauge} will be used freely in the following. \begin{exmp} \label{e7.1} Let $$p(x)=q(x)=n(\zeta(x)-\zeta(x-\omega_2)-\eta_2),$$ where$n\in\bb N$. The potential$(p,q)$has two poles in the fundamental period parallelogram. Consider first the pole$x=0$. In this case we have $$R=\begin{pmatrix} 0&n\\n&0\end{pmatrix},$$ whose eigenvalues are$\pm n$, that is,$\lambda=n$. Moreover, since$p=q$is odd, we have$p_{2j-1}=q_{2j-1}=0$. One proves by induction that$b_{2j}^{(2)}$is a multiple of$(1,1)^t$and that$b_{2j-1}^{(2)}$is a multiple of$(1,-1)^t$. Hence$b_{2n-1}^{(2)}$is a multiple of$(1,-1)^t$, that is, it is in the range of$R-n$. Hence every solution of$L\Psi=E\Psi$is meromorphic at zero regardless of$E$. Next consider the pole$x=\omega_2$and shift coordinates by introducing$\xi=x-\omega_2$. Then we have$p(x)=q(x) =n(\zeta(\xi+\omega_2)-\zeta(\xi)-\eta_2) =-p(\xi)$and hence $$R=\begin{pmatrix} 0&-n\\-n&0\end{pmatrix}.$$ One can use again a proof by induction to show that$b_{2n-1}^{(2)}$is in the range of$R-n$, which is spanned by$(1,1)^t$. Hence we have shown that the matrix $$Q(x)=\begin{pmatrix}0&-in(\zeta(x)-\zeta(x-\omega_2)-\eta_2)\\ in(\zeta(x)-\zeta(x-\omega_2)-\eta_2)&0\end{pmatrix}$$ is a Picard-AKNS and therefore an algebro-geometric AKNS potential. \end{exmp} \begin{exmp} \label{e7.2} Here we let$p=1$and$q=n(n+1)\wp(x)$, where$n\in\bb N$. Then we have just one pole in the fundamental period parallelogram. In this case we obtain $$R=\begin{pmatrix} 1&n(n+1)\\1&0\end{pmatrix}$$ and$\lambda=n+1$. Since$q$is even we infer that$q_{2j-1}=0$. A proof by induction then shows that$b_{2j}^{(2)}$is a multiple of$(n-2j,1)^t$and that$b_{2j-1}^{(2)}$is a multiple of$(1,0)^t$. In particular,$b_{2n}^{(2)}$is a multiple of$(-n,1)^t$, which spans the range of$R-\lambda$. This shows that $$Q(x)=\begin{pmatrix}0&-in(n+1)\wp(x)\\ i&0\end{pmatrix}$$ is a Picard-AKNS and hence an algebro-geometric AKNS potential. \end{exmp} Incidentally, if$p=1$, then$J\Psi'+Q\Psi=E\Psi$is equivalent to the scalar equation$\psi_2''-q\psi_2=-E^2\psi_2$where$\Psi=(\psi_1,\psi_2)^t$and$\psi_1=\psi_2'-iE \psi_2$. Therefore, if$-q$is an elliptic algebro-geometric potential of the KdV hierarchy then by Theorem 5.7 of \cite{32a}$\psi_2$is meromorphic for all values of$E$. Hence$\Psi$is meromorphic for all values of$E$and therefore$Q$is a Picard-AKNS and hence an algebro-geometric AKNS potential. Conversely if$Q$is an algebro-geometric AKNS potential with$p=1$then$-q$is an algebro-geometric potential of the KdV hierarchy (cf. \eqref{2.51}). In particular,$q(x)=n(n+1)\wp(x)$is the celebrated class of Lam\'e potentials associated with the KdV hierarchy (cf., e.g., \cite{29} and the references therein). \begin{exmp} \label{e7.3} Suppose$e_2=0$and hence$g_2=4e_1^2$and$g_3=0$. Let$u(x)=-\wp'(x)/(2e_1)$. Then, near$x=0$, $$u(x)=\frac{1}{e_1 x^3}-\frac{e_1}{5}x +O(x^3),$$ and near$x=\pm\omega_2$, $$u(x)=e_1(x\mp\omega_2)-\frac{3e_1^3}{5}(x\mp\omega_2)^5 +O((x\mp\omega_2)^7).$$ Now let$p(x)=3u(x)$and$q(x)=u(x-\omega_2).$Then$p$has a third-order pole at$0$and a simple zero at$\omega_2$while$q$has a simple zero at zero and a third-order pole at$\omega_2$. Let us first consider the point$x=0$. We have $$R=\begin{pmatrix} -2&e_1\\3/e_1&0\end{pmatrix},$$ and hence$\lambda=1$. Moreover,$p_2=q_2=0$,$p_4=-3e_1/5$, and$q_4=-3e_1^3/5$. Since$\lambda=1$we have to show that$b_3^{(2)}$is a multiple of$(q_0,-1)^t$. We get, using$p_2=q_2=0$, $$b_3^{(2)}=(q_0 E^4/6+q_4,-E^4/6-q_0p_4)^t,$$ which is a multiple of$(q_0,-1)^t$if and only if$q_4=p_4q_0^2$, a relationship which is indeed satisfied in our example. Next consider the point$x=\omega_2$. Changing variables to$\xi=x-\omega_2$and using the periodic properties of$u$we find that$p(x)=3q(\xi)$and$q(x)=p(\xi)/3$. Thus$q$has a pole at$\xi=0$and one obtains$m=2$,$p_0=3e_1$,$q_0=1/e_1$,$p_2=q_2=0$,$p_4=-9e_1^3/5$, and$q_4=-e_1/5$. Since$\lambda=3$, we have to compute again$b_3^{(2)}$and find, using$p_2=q_2=0$, $$b_3^{(2)}=(-q_0 E^4/6+3q_4,-E^4/2-q_0p_4)^t,$$ which is a multiple of$(q_0,-3)^t$if and only if$9q_4=p_4q_0^2$, precisely what we need. Hence, if$e_2=0$and$u(x)=-\wp'(x)/(2e_1)$, then $$Q(x)=\begin{pmatrix}0&-iu(x-\omega_2)\\3iu(x)&0\end{pmatrix}$$ is a Picard-AKNS and therefore an algebro-geometric AKNS potential. \end{exmp} \begin{exmp} \label{e7.4} Again let$e_2=0$. Define$p(x)=\frac23(\wp''(x)-e_1^2)$and$q(x)=-\wp(x-\omega_2)/e_1^2$. First consider$x=0$. We have$m=-3$,$p_0=4$,$q_0=1$,$p_2=q_2=0$,$p_4=-2e_1^2/5$, and$q_4=-e_1^2/5$. This yields$\lambda=1$and we need to show that$b_4^{(2)}$is a multiple of$(1,-1)^t$. We find, using$p_2=q_2=0$and$q_0=\lambda=1$, $$b_4^{(2)}=i(-E^5/24-p_4/4-q_4,E^5/24+5p_4/4-q_4)^t.$$ This is a multiple of$(1,-1)^t$if$2q_4=p_4$, which is indeed satisfied. Next consider$x=\omega_2$. Now$q$has a second-order pole, that is, we have$m=1$. Moreover, $$q(x)=\frac{-1}{e_1^2}(\frac{1}{(x-\omega_2)^2}+\frac{e_1^2}{5} (x-\omega_2)^2+ O((x-\omega_2)^2)$$ and $$p(x)=-2e_1^2+ 96 e_1^4 (x-\omega_2)^4+ O((x-\omega_2)^6).$$ We now need$b_2^{(2)}$to be a multiple of$(q_0,-2)^t$, which is satisfied for$q_2=p_2=0$. Hence, if$e_2=0$, then $$Q(x)=\begin{pmatrix}0&i\wp(x-\omega_2)/e_1^2\\ 2i(\wp''(x)-e_1^2)/3&0\end{pmatrix}$$ is a Picard-AKNS and thus an algebro-geometric AKNS potential. \end{exmp} %{\bf Acknowledgments.} \begin{thebibliography}{99} \bibitem{2} H. Airault, H. P. McKean, and J. 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