%&amslatex \documentstyle[12pt,righttag]{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{\calM}{{\cal{M}}} \newcommand{\calD}{{\cal{D}}} \newcommand{\calS}{{\cal{S}}} \newcommand{\calP}{{\cal{P}}} \newcommand{\calL}{{\cal{L}}} %%%%%%%%%%%%%%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}} %%%%%%%%%%%%%%%%%%%%%%%%NUMBERING%%%%%%%%%% \makeatletter \def\theequation{\thesection.\@arabic\c@equation} \makeatother %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{document} \title{Picard Potentials and Hill's Equation on a Torus} \thanks{Based upon work supported by the National Science Foundation under Grant No. DMS-9401816.} \author{F.~Gesztesy${}^1$} \address{${}^1$ Department of Mathematics, University of Missouri, Columbia, MO 65211, USA.} \email{mathfg@@mizzou1.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} \maketitle \begin{abstract} An explicit characterization of all elliptic (algebro-geometric) finite-gap solutions of the KdV hierarchy is presented. More precisely, we show that an elliptic function $q$ is an algebro-geometric finite-gap potential, i.e., a solution of some equation of the stationary KdV hierarchy, if and only if every solution of the associated differential equation $\psi''+q\psi=E\psi$ is a meromorphic function of the independent variable for every complex value of the spectral parameter $E$. Our result also provides an explicit condition for a classical theorem of Picard to hold. This theorem guarantees the existence of solutions which are elliptic of the second kind for second-order ordinary differential equations with elliptic coefficients associated with a common period lattice. The fundamental link between Picard's theorem and elliptic finite-gap solutions of completely integrable hierarchies of nonlinear evolution equations, as established in this paper, is without precedent in the literature. In addition, a detailed description of the singularity structure of the Green's function of the operator $H=d^2/dx^2+q$ in $L^2(\bbR)$ and its precise connection with the branch and singular points of the underlying hyperelliptic curve is given. \end{abstract} \section{Introduction} \label{intro} Hill's equation has drawn an enormous amount of consideration due to its ubiquity in applications as well as its structural richness. Of particular importance in the last 20 years is its connection with the KdV hierarchy and hence with integrable systems. We show in this paper that regarding the independent variable as a complex variable yields a breakthrough for the problem of an efficient characterization of all elliptic finite-gap potentials, a major open problem in the field. Specifically, we show that elliptic finite-gap potentials of Hill's equation are precisely those for which all solutions for all spectral parameters are meromorphic functions in the independent variable, complementing a classical theorem of Picard. The intimate connection between Picard's theorem and elliptic finite-gap solutions of completely integrable systems is established in this paper for the first time. In addition, we construct the hyperelliptic Riemann surface associated with a finite-gap potential (not necessarily elliptic), i.e., determine its branch and singular points from a comparison of the geometric and algebraic multiplicities of eigenvalues of certain operators associated with Hill's equation. These multiplicities are intimately correlated with the pole structure of the diagonal Green's function of the operator $H=d^2/dx^2+q(x)$ in $L^2(\bbR)$. Our construction is new in the present general complex-valued periodic finite-gap case. Before describing our approach in some detail, we shall give a brief account of the history of the problem involved. This theme dates back to a 1940 paper of Ince \cite{36} who studied what is presently called the Lam\'e--Ince potential $$\label{1.1} q(x)=-g(g+1)\calP(x+\omega_3), \; g\in \bbN, \; x \in\bbR$$ in connection with the second-order ordinary differential equation $$\label{1.2} \psi''(E,x) + q(x) \psi(E,x) = E\psi(E,x), \; E\in\bbC.$$ Here $\calP(x) := \calP(x;\omega_1,\omega_3)$ 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 $L=d^2/dx^2 + q(x)$ in $L^2(\bbR)$ exhibits finitely many bands (respectively gaps), i.e., \label{1.3} \sigma(L)=(-\infty, E_{2g}) \cup \bigcup^g_{m=1} \left[ E_{2m-1},\; E_{2(m-1)}\right], \; E_{2g}0$,$M_j\in\bbR.$Without loss of generality we may assume that the fundamental periods$2\omega_1$and$2\omega_3$have been chosen in such a way that the angle$2\theta$between the axes of$S_1$and$S_3$(i.e.,$\e^{i\theta}=\dfrac{\omega_3}{\omega_1}\left| \dfrac{\omega_1}{\omega_3} \right|$) satisfies$2\theta \in (0,2\pi)\backslash \{\pi\}$. 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{t3.1} then yields a proof of Theorem \ref{1.1} (see the proof of Theorem \ref{t3.7}). Finally, we close Section \ref{picard} with a series of remarks that put Theorem \ref{1.1} 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. \section{The KdV Hierarchy and Hyperelliptic Curves} \label{kdv} \setcounter{equation}{0} In this section we review basic facts on the stationary KdV hierarchy. Since this material is well known (see, e.g., \cite{4}, \cite{14}, Ch. 12, \cite{26}, \cite{28}), we confine ourselves to a brief account. Assuming$q\in C^{\infty}(\bbR)$or$q$meromorphic in$\bbC$(depending on the particular context in which one is interested) and hence either$x\in\bbR$or$x\in\bbC$, consider the recursion relation $$\label{2.1} \hat{f}'_{j+1}(x)=\dfrac{1}{4} \hat{f}^{'''}_j(x) + q(x)\hat{f}'_j(x)+ \dfrac{1}{2}q'(x)\hat{f}_j(x), \; 0\le j\le g,\; \hat{f}_0(x)=1$$ and the associated differential expressions (Lax pair) $$\label{2.2} L=\dfrac{d^2}{dx^2} + q(x),$$ $$\label{2.3} \hat{P}_{2g+1}=\sum^{g}_{j=0} \left[-\dfrac{1}{2} \hat{f}'_j(x)+ \hat{f}_j(x) \dfrac{d}{dx}\right] L^{g-j}, \; g\in\bbN_0$$ (here$\bbN_0:=\bbN\cup\{0\}).$One can show that $$\label{2.4} \left[ \hat{P}_{2g+1}, L \right]=2\hat{f}'_{g+1} ={1\over2} \hat f_g'''(x)+ 2q(x)\hat f_g'(x)+q'(x)\hat f_g(x)$$ ([$\cdot,\cdot$] the commutator symbol) and explicitly computes from (\ref{2.1}), $$\label{2.5} \hat{f}_0=1, \; \hat{f}_1=\dfrac{1}{2} q+c_1,\; \hat{f}_2=\dfrac{1}{8} q^{''} + {3\over8} q^2 +\dfrac{c_1}{2} q+c_2, \quad \text{etc.},$$ where the$c_j$are integration constants. Using the convention that the corresponding homogeneous quantities obtained by setting$c_{\ell}=0$for$\ell=1,2,\ldots$are denoted by$f_j$, i.e.,$f_j=\hat{f}_j(c_{\ell}\equiv 0)$, the (homogeneous) stationary KdV hierarchy is then defined as the sequence of equations $$\label{2.6} \kdv_g(q)=2f'_{g+1} =0, \; g\in\bbN_0.$$ Explicitly, this yields $$\label{2.7} \kdv_0(q)=q_x=0, \quad \kdv_1(q) =\dfrac{1}{4}q^{'''}+\dfrac{3}{2} qq'=0, \quad \text{etc. }$$ The corresponding inhomogeneous version of$\kdv_g(q)=0$is then defined by $$\label{2.8} \hat{f}'_{g+1}=\sum^{g}_{j=0}c_{g-j}f'_{j+1}=0,$$ where$c_0=1$and$c_1,...,c_g$are arbitrary complex constants. If one assigns to$q^{(\ell)}=d^{\ell}q/dx^{\ell}$the degree$\deg(q^{(\ell)}) = \ell+2, \; \ell\in\bbN_0$, then the homogeneous differential polynomial$f_j$with respect to$q$turns out to have degree$2j$, i.e., $$\label{2.9} \deg (f_j)=2j, \; j\in\bbN_0.$$ Next, introduce the polynomial$\hat{F}_g(E,x)$in$E\in\bbC,$$$\label{2.10} \hat{F}_g(E,x) = \sum^{g}_{j=0} \hat{f}_{g-j}(x) E^j.$$ Since$\hat f_0(x)=1$, $$\label{2.12} \hat{R}_{2g+1}(E,x)=(E-q(x))\hat F_g(E,x)^2-{1\over2} \hat F_g''(E,x) \hat F_g(E,x)+{1\over4}\hat F_g'(E,x)^2$$ is a monic polynomial in$E$of degree$2g+1$. However, equations (\ref{2.1}) and (\ref{2.8}) imply that $$\label{2.11} \dfrac{1}{2} \hat{F}^{'''}_g-2(E-q)\hat{F}'_g+q'\hat{F}_g=0$$ and this shows that$\hat{R}_{2g+1}(E,x)$is in fact independent of$x$. Hence it can be written as $$\label{2.13} \hat{R}_{2g+1}(E)=\prod^{2g}_{m=0} (E-\hat{E}_m), \; \{\hat{E}_m\}^{2g}_{m=0} \subset \bbC.$$ By (\ref{2.4}) the inhomogeneous KdV equation (\ref{2.8}) is equivalent to the commutativity of$L$and$\hat{P}_{2g+1}$. This shows that $$\label{2.14} [\hat{P}_{2g+1}, L]=0,$$ and therefore, if$L\psi=E\psi$, this implies that$\hat P^2_{2g+1}\psi =\hat R_{2g+1}(E)\psi$. Thus$[\hat{P}_{2g+1}, L]=0$implies $$\label{2.15} \hat P^2_{2g+1} = \hat R_{2g+1}(L) = \prod^{2g}_{m=0} (L-\hat{E}_m),$$ a celebrated theorem by Burchnall and Chaundy \cite{11}, \cite{12}. In Section \ref{fingap} we will need the converse of the above procedure. It is given by \begin{prop} \label{p2.1} Assume that$\hat F_g(E,x)$, given by (\ref{2.10}) with$\hat f_0(x)=1$, is twice continuously differentiable with respect to$x$, and that $$\label{e2.1} (E-q(x))\hat F_g(E,x)^2-{1\over2} \hat F_g''(E,x) \hat F_g(E,x)+{1\over4}\hat F_g'(E,x)^2$$ is independent of$x$. Then$q\in C^\infty(\bbR)$. Also the functions$\hat f_j(x)$are infinitely often differentiable and satisfy the recursion relations (\ref{2.1}) for$j=0,...,g-1$. Moreover,$\hat f_g$satisfies $$\label{e2.2} \dfrac{1}{4} \hat{f}^{'''}_g(x) + q(x)\hat{f}'_g(x)+ \dfrac{1}{2}q'(x)\hat{f}_g(x)=0,$$ i.e., the differential expression$\hat P_{2g+1}$given in (\ref{2.3}) commutes with the expression$L=d^2/dx^2+q$. \end{prop} \begin{pf} The expression given in (\ref{e2.1}) is a monic polynomial in$E$of degree$2g+1$whose coefficients are constants. We denote it by $$\label{e2.3} \hat R_{2g+1}(E)=\sum_{j=0}^{2g} \tilde c_{2g+1-j} E^j,$$ where$\tilde c_0=1$. Comparing the coefficients of$E^{2g+1-k}$in (\ref{e2.1}) and (\ref{e2.3}) yields $$\label{e2.4} q(x)=2\hat f_1(x)-\tilde c_1$$ for$k=1, and \begin{align} \label{e2.7} \hat f_{k-1}''(x)=4\hat f_k(x)-4q(x)&\hat f_{k-1}(x) +2\hat f_{k-1}(x)\hat f_1(x) -2\tilde c_1 \nonumber \\ -\sum_{j=1}^{k-2} \Bigl(&\hat f_j''(x) \hat f_{k-1-j}(x) -{1\over2} \hat f_j'(x) \hat f_{k-1-j}'(x) \nonumber \\ &+2q(x) \hat f_j(x) \hat f_{k-1-j}(x) -2 \hat f_j(x) \hat f_{k-j}(x)\Bigr) \end{align} fork=2,...,g+1$. In (\ref{e2.7}) we have introduced$\hat f_{g+1}=0$for ease of notation. By hypothesis$\hat f_1,..., \hat f_g \in C^2(\bbR)$. Equation (\ref{e2.4}) now shows that$q\in C^2(\bbR)$. Then the equations in (\ref{e2.7}) show recursively that$\hat f_k \in C^4(\bbR)$for$k=1,...,g$. By induction it follows that$q$and the functions$\hat f_1$, ...,$\hat f_g$are infinitely often differentiable. Thus we may differentiate (\ref{e2.4}) with respect to$x$to obtain (\ref{2.1}) for$j=0$. Also we may differentiate all the equations (\ref{e2.7}) with respect to$x$. Applying this procedure inductively then proves the validity of (\ref{2.1}) for$j=1,...,g-1$and of (\ref{e2.2}). The final statement then follows from (\ref{2.4}). \end{pf} The nonlinear recursion formalism for$\hat f_k$resulting from (\ref{e2.7}) can be read off from the results of Section 2.2 in \cite{GD75}. Equation (\ref{2.15}) illustrates the intimate connection between the stationary KdV equation$\hat{f}'_{g+1}=0$in (\ref{2.8}) and the compact (possibly singular) hyperelliptic curve$K_g$of (arithmetic) genus$g$obtained upon one-point compactification of the curve $$\label{2.16} F^2=\hat{R}_{2g+1}(E)=\prod^{2g}_{m=0}(E-\hat{E}_m).$$ The above formalism leads to the following standard definition. \begin{defn} \label{d2.1} Any solution$q$of one of the stationary KdV equations (\ref{2.8}) is called an {\bf (algebro-geometric) finite-gap potential} associated with the KdV hierarchy. \end{defn} Finite-gap potentials$q$can be expressed in terms of the Riemann theta function or through$\tau$-functions associated with the curve$K_g$(see, e.g., \cite{39}, \cite{57}). \section{Floquet Theory} \label{floquet} \setcounter{equation}{0} Next we turn to Floquet theory in connection with a complex-valued periodic potential$q$and assume for the rest of this section that $$\label{2.17} q\in C^0(\bbR), \; q(x+\Omega)=q(x), \; x\in \bbR$$ for some$\Omega>0$. Floquet solutions$\psi(E,x)$of$Ly=Ey$are characterized by the property $$\label{2.34} \psi(E,x+\Omega) = \rho(E)\psi(E,x) \quad \text{for all x\in\bbR,}$$ where$\rho(E)$is a so called Floquet multiplier. We introduce the fundamental system of solutions$c(E,x,x_0)$and$s(E,x,x_0)$of$Ly=Ey$defined by $$\label{2.23} c(E,x_0,x_0)=s'(E,x_0,x_0)=1, \; c'(E,x_0,x_0)= s(E,x_0,x_0)=0.$$ The functions$c(E,x,x_0)$and$s(E,x,x_0)$and their$x$-derivatives are entire functions of$E$for every choice of$x$and$x_0$. The coefficients of Floquet solutions written as linear combinations of the fundamental solutions$c(E,x,x_0)$and$s(E,x,x_0)$and the Floquet multipliers are given through eigenvectors and eigenvalues of the monodromy matrix of$L$, i.e., through $$M(E,x_0)=\left( \begin{array}{cc} c(E,x_0+\Omega,x_0) & s(E,x_0+\Omega,x_0) \\ c'(E,x_0+\Omega,x_0) & s'(E,x_0+\Omega,x_0) \end{array} \right).$$ More precisely, the Floquet multipliers are the eigenvalues of$M(E,x_0)$and hence are given by $$\label{2.35a} \rho_{\pm} = \Delta(E) \pm \sqrt{\Delta(E)^2-1},$$ where$\Delta(E)$is half the trace of$M(E,x_0)$, i.e., $$\label{2.24} \Delta(E)={1\over2} [c(E,x_0+\Omega,x_0)+s'(E,x_0+\Omega,x_0)].$$ That$\Delta(E)$is independent of$x_0$follows from equations (\ref{cx0}) and (\ref{sx0}) below. The functions$c(E,x,x_0)$and$s(E,x,x_0)$satisfy certain Volterra integral equations which imply that$\partial s(E,x,x_0)/\partial x_0=-c(E,x,x_0)$and$\partial c(E,x,x_0)/\partial x_0= (q(x_0)-E) s(E,x,x_0). Hence \begin{align} {d\over dx_0} c^{(k)}(E,x_0+\Omega,x_0) &=c^{(k+1)}(E,x_0+\Omega,x_0)+(q(x_0)-E)s^{(k)}(E,x_0+\Omega,x_0), \label{cx0} \displaybreak[0]\\ {d\over dx_0} s^{(k)}(E,x_0+\Omega,x_0) &=s^{(k+1)}(E,x_0+\Omega,x_0)-c^{(k)}(E,x_0+\Omega,x_0) \label{sx0} \end{align} fork=0,1$. For each$E\in\bbC$there exists at least one nontrivial Floquet solution. In fact, since together with$\rho(E)$,$1/\rho(E)$is also a Floquet multiplier, there are two linearly independent Floquet solutions for a given$E$provided$\rho(E)^2\neq1$. Floquet solutions can be expressed in terms of the fundamental system$c(E,x,x_0)$and$s(E,x,x_0)$by $$\label{2.40} \psi_\pm(E,x,x_0) = c(E,x,x_0) + {\rho_\pm(E) - c(E,x_0+\Omega,x_0) \over s(E,x_0+\Omega, x_0)} s(E,x,x_0),$$ if$s(E,x_0+\Omega,x_0)\neq0$, or by $$\label{2.40a} \tilde \psi_\pm(E,x,x_0) = s(E,x,x_0) + {\rho_\pm(E)- s'(E,x_0+\Omega,x_0)\over c'(E,x_0+\Omega, x_0)} c(E,x,x_0),$$ if$c'(E,x_0+\Omega,x_0)\neq0$. If both$s(E,x_0+\Omega,x_0)$and$c'(E,x_0+\Omega,x_0)$are equal to zero, then$s(E,x,x_0)$and$c(E,x,x_0)$are linearly independent Floquet solutions and hence every solution of$Ly=Ey$is Floquet. Associated with the second-order differential expression$L=d^2/dx^2+q(x)$we consider densely defined closed linear operators$H$,$H_D(x_0)$,$H_N(x_0)$, and$H_\theta$in$L^2(\bbR)$and$L^2((x_0,\; x_0+\Omega))$,$x_0\in\bbR$, respectively. Let \begin{gather} Hf=Lf, \quad f\in H^{2,2}(\bbR), \label{2.18} \\ H_D(x_0)f=Lf, \quad f\in\{g\in H^{2,2}((x_0, x_0+\Omega)): g(x_0)=0=g(x_0+\Omega)\}, \label{2.19} \\ H_N(x_0)f=Lf, \quad f\in\{g\in H^{2,2}((x_0, x_0+\Omega)): g'(x_0)=0=g'(x_0+\Omega)\}, \label{2.19a} \end{gather} and for$\theta\in \bbC$$$\label{2.20} H(\theta)f=Lf, \; f\in\{g\in H^{2,2}((x_0,x_0+\Omega)): g^{(k)} (x_0+\Omega) = e^{i\theta}g^{(k)}(x_0), \; k=0,1\}$$$(H^{p,r}(\cdot)$being the usual Sobolev spaces with$r$distributional derivatives in$L^p(\cdot)$). Next we denote the purely discrete spectra of$H_D(x_0)$,$H_N(x_0)$, and$H(\theta)$by $$\label{2.21} \sigma(H_D(x_0))=\{\mu_n(x_0)\}_{n\in\bbN}, \quad \sigma(H_N(x_0))=\{\nu_n(x_0)\}_{n\in\bbN_0}, \quad \sigma(H(\theta))=\{E_n(\theta)\}_{n\in\bbN_0},$$ respectively. Note that, while$H(\theta)$depends on$x_0$its spectrum does not. We agree that in (\ref{2.21}) as well as in the rest of the paper all point spectra (i.e., sets of eigenvalues) are recorded in such a way that all eigenvalues are consistently repeated according to their algebraic multiplicity unless explicitly stated otherwise. The eigenvalues of$H_D(x_0)$($H_N(x_0)$) are called Dirichlet (Neumann) eigenvalues with respect to the interval$[x_0,x_0+\Omega]$. The eigenvalues of$H(\theta)$are precisely those values$E$where the monodromy matrix$M(E,x_0)$of$L$has eigenvalues$\rho=e^{\pm i\theta}$. The eigenvalues$E_n(0)$($E_n(\pi)$) of$H(0)$($H(\pi)$) are called the periodic (antiperiodic) eigenvalues associated with$q$. Note that the (anti)periodic eigenvalues$E_n(0)$($E_n(\pi)$) are the zeros of$\Delta(\cdot)-1$($\Delta(\cdot)+1$) and that their algebraic multiplicities coincide with the orders of the respective zeros (see, e.g., \cite{32}). In the following we denote the zeros of$\Delta(E)^2-1$by$E_n$,$n\in\bbN_0$. They are repeated according to their multiplicity and are related to the (anti)periodic eigenvalues via $$E_{4n}=E_{2n}(0), \quad E_{4n+1}=E_{2n}(\pi), \quad E_{4n+2}=E_{2n+1}(\pi), \quad E_{4n+3}=E_{2n+1}(0)$$ for$n\in\bbN_0$. We also introduce $$\label{ordd} d(E)=\ord_E(\Delta(\cdot)^2-1),$$ the order of$E$as a zero of$\Delta(\cdot)^2-1$($d(E)=0$if$\Delta(E)^2-1\neq0$). Similarly the Dirichlet eigenvalues$\mu_n(x_0)$and the Neumann eigenvalues$\nu_n(x_0)$of$H_D(x_0)$and$H_N(x_0)$are the zeros of the functions$s(\cdot,x_0+\Omega,x_0)$and$c'(\cdot,x_0+\Omega,x_0)$, respectively. Again their algebraic multiplicities coincide precisely with the multplicities of the respective zeros (see, e.g., \cite{32}). These multiplicities depend in general on$x_0. We introduce the notation \begin{align} p(E,x_0)&= \ord_E (s(\cdot,x_0+\Omega,x_0)), \label{ords} \\ r(E,x_0)&= \ord_E (c'(\cdot,x_0+\Omega,x_0)), \label{ordc'} \end{align} and remark thatp(E,x_0)$and$r(E,x_0)are a combination of a movable and an immovable part, i.e., \begin{align} p(E,x_0)&=p_i(E)+p_m(E,x_0), \displaybreak[0]\\ r(E,x_0)&=r_i(E)+r_m(E,x_0), \end{align} wherep_m(E,\cdot)$and$r_m(E,\cdot)$vanish for some$x_0 \in\bbR$. If$p_i(E)>0$($r_i(E)>0$) we will call$E$an immovable Dirichlet (Neumann) eigenvalue. The asymptotic behavior of$c'(E,x_0+\Omega,x_0)$,$s(E,x_0+\Omega,x_0)$and$\Delta(E)$as$|E|tends to infinity, i.e., \begin{align} s(E,x_0+\Omega,x_0) &= (-E)^{-1/2} \sin[(-E)^{1/2} \Omega] +O(|E|^{-1}e^{|\Im(-E)^{1/2}|\Omega}), \label{2.28}\\ c'(E,x_0+\Omega,x_0) &= (-E)^{-1/2} \sin[(-E)^{1/2} \Omega] +O(|E|^{-1}e^{|\Im(-E)^{1/2}|\Omega}), \label{2.28a}\\ \Delta(E) &= \cos[(-E)^{1/2}\Omega]+O(|E|^{-1/2} e^{|\Im(-E)^{1/2}|\Omega}), \label{2.29} \end{align} obtained by a standard iteration of Volterra integral equations together with Rouch\'e's theorem, then proves the following facts: \begin{enumerate} \item The zeros\mu_n(x_0)$of$s(E,x_0+\Omega,x_0)$and the zeros$\nu_n(x_0)$of$c'(\cdot,x_0+\Omega,x_0)$are simple for$n\in\bbN$sufficiently large. \item The zeros$E_n$of$\Delta(E)^2-1$are at most double for$n\in\bbN$large enough. \item$\mu_n(x_0)$,$\nu_n(x_0)$, and$E_n$can be arranged such that they have the following asymptotic behavior as$ntends to infinity (we assume in the following that they are actually arranged in this way): \begin{align} \mu_n(x_0) &= -{n^2\pi^2\over\Omega^2} + O(1), \label{2.32} \displaybreak[0] \\ \nu_n(x_0) &= -{n^2\pi^2\over\Omega^2} + O(1), \label{2.32a} \\ E_{2n-1}, E_{2n} &= -{n^2\pi^2\over\Omega^2} + O(1). \label{2.33} \end{align} \end{enumerate} The Hadamard factorizations ofs(E, x_0+\Omega, x_0)$,$c'(E,x_0+\Omega,x_0)$, and$\Delta(E)^2-1therefore read \begin{align} s(E,x_0+\Omega,x_0) &= c_1(x_0) \prod_{n=1}^\infty \left(1-{E\over\mu_n(x_0)}\right), \label{2.25} \\ c'(E,x_0+\Omega,x_0) &= c_2(x_0) \prod_{n=0}^\infty \left(1-{E\over\nu_n(x_0)}\right), \label{2.25a} \\ \Delta(E)^2-1 &= c_3^2 \prod_{n=0}^\infty \left(1-{E\over E_n}\right) \label{2.26} \end{align} for suitableE$-independent and nonvanishing$c_1(x_0)$,$c_2(x_0)$, and$c_3$. Equations (\ref{2.25}) -- (\ref{2.26}) assume that none of the eigenvalues is equal to zero. If this were to happen these equations have to be replaced by obvious modifications. For more details on algebraic versus geometric multiplicities of eigenvalue problems of the type of$H_D(x_0)$and$H(\theta)$see, e.g., \cite{32}. We then have the following \begin{prop} \label{t2.1} (i) The number$\lambda$is an immovable Dirichlet eigenvalue if and only if it is also an immovable Neumann eigenvalue. In particular, in this case there are two linearly independent (anti)periodic solutions of$Ly=Ey$.\\ (ii) If$\lambda$is an (anti)periodic eigenvalue and also both a Dirichlet and a Neumann eigenvalue with respect to the interval$[x_0,x_0+\Omega]$, then$\lambda$is an immovable Dirichlet and Neumann eigenvalue. \end{prop} \begin{pf} (i) Assume that$\lambda$is an immovable Dirichlet eigenvalue. Then$s(\lambda,x_0+\Omega,x_0)=0$for every$x_0\in\bbR$. By (\ref{sx0}),$s'(\lambda,x_0+\Omega,x_0)$is continuous as a function of$x_0$. On the other hand$s'(\lambda,x_0+\Omega,x_0)$is always equal to one of the two Floquet multipliers. Therefore, we infer that$s'(\lambda,x_0+\Omega,x_0)$is in fact equal to some constant$\rho$. In particular we have$s(\lambda,x+\Omega,x_0)=\rho s(\lambda,x,x_0)$for all$x_0\in\bbR$. Now consider the function$s(\lambda,x,\tilde x_0). Then we obtain \begin{align} s(\lambda,x+\Omega,\tilde x_0) &= \alpha s(\lambda,x+\Omega,x_0)+\beta c(\lambda,x+\Omega,x_0) \nonumber\\ &= \alpha \rho s(\lambda,x,x_0)+\beta c(\lambda,x+\Omega,x_0) \end{align} and $$s(\lambda,x+\Omega,\tilde x_0) = \rho s(\lambda,x,\tilde x_0) = \rho( \alpha s(\lambda,x,x_0)+\beta c(\lambda,x,x_0) ).$$ Since the left hand sides are equal, comparing the right hand sides yields $$c(\lambda,x+\Omega,x_0)=\rho c(\lambda,x,x_0)$$ if\tilde x_0$is chosen such that$\beta=s(\lambda,\tilde x_0,x_0) \neq0$. Hence$c(\lambda,x,x_0)$is a solution satisfying Neumann boundary conditions, i.e.,$\lambda$is among the Neumann eigenvalues$\{\nu_n(x_0)\}_{n\in\bbN_0}$. Since$x_0$is arbitrary we find that$\lambda$is, in fact, an immovable Neumann eigenvalue. Moreover,$c(\lambda,x,x_0)$and$s(\lambda,x,x_0)$are linearly independent Floquet solutions of$Ly=\lambda y$with the same Floquet multiplier$\rho$implying$\rho^2=1$. Thus$c(\lambda,x,x_0)$and$s(\lambda,x,x_0)$are indeed both periodic or both antiperiodic. (ii) Suppose for brevity that$\lambda$is a periodic eigenvalue. Then$M(E,x_0)$is the identity matrix and therefore every solution of$L\psi =\lambda\psi$is periodic. Thus we obtain for any$x_1\in\bbR$that$s(\lambda,x_1+\Omega,x_1)=s(\lambda,x_1,x_1)=0$and$c'(\lambda,x_1+\Omega,x_1)=c'(\lambda,x_1,x_1)=0$implying that$\lambda$is an immovable Dirichlet and Neumann eigenvalue. \end{pf} It was shown by Rofe-Beketov \cite{56} that the spectrum of$H$is equal to the conditional stability set of$L$, i.e., the set of all spectral parameters$E$for which a nontrivial bounded solution of$L\psi=E\psi$exists. Hence $$\label{2.22} \sigma(H)= \bigcup\sb{\theta \in[0,2\pi]} \sigma(H(\theta)) =\bigcup_{n\in\bbN_0} \sigma_n,\; \text{where}\; \sigma_n= \bigcup_{\theta\in[0,\pi]}E_n(\theta).$$ We note that in the general case where$q$is complex-valued some of the spectral arcs$\sigma_n$may cross each other, see, e.g., \cite{32} and \cite{52} for explicit examples. The Green's function$G(E,x,x')$of$H$, i.e., the integral kernel of the resolvent of$H$$\label{2.36} G(E,x,x') = (H-E)^{-1}(x,x'), \; E\in\bbC\backslash \sigma(H), \; x,x'\in\bbR,$$ is explicitly given by \begin{align}\label{2.37} G(E,x,x') = W(f_{-}(E,x), f_{+}(E,x))^{-1} \begin{cases} f_{+}(E,x) f_{-}(E,x'),\; x \ge x'\\ f_{-}(E,x) f_{+}(E,x'), x\le x'. \end{cases} \end{align} Heref_{\pm}(E,\cdot)$solve$Lf=Ef$and are chosen such that $$\label{2.38} f_{\pm}(E,\cdot)\in L^2((R,\pm \infty)), \; E\in\bbC\backslash \sigma(H), R\in\bbR$$ with$W(f,g)=fg'-f'g$the Wronskian of$f$and$g$. Equation (\ref{2.37}) implies that the diagonal Green's function is twice continuously differentiable and satisfies the nonlinear second-order differential equation $$\label{2.68} 4(E-q(x))G(E,x,x)^2-2G(E,x,x)G''(E,x,x)+G'(E,x,x)^2=1$$ (the primes denoting derivatives with respect to$x$). It follows from (\ref{2.22}) that$|\rho(E)|\neq1$unless$E\in\sigma(H)$. Therefore, if$E\not\in\sigma(H)$there is precisely one Floquet solution in$L^2((-\infty,R))$and one in$L^2((R,\infty))$. Letting$\rho_{\pm}(E)=e^{\pm i\theta}$with$\Im(\theta)>0$we obtain$|\rho_+(E)|<1<|\rho_-(E)|$. Hence$f_+(E,x)=\psi_+(E,x,x_0)$and$f_-(E,x)=\psi_-(E,x,x_0)$. Since$\psi_\pm(E,x_0,x_0)=1$, equations (\ref{2.35a}) and (\ref{2.40}) imply $$\label{2.42} W(f_{-}(E,\cdot), f_{+}(E,\cdot)) ={e^{i\theta}-e^{-i\theta}\over s(E,x_0+\Omega, x_0)} =-2{[\Delta(E)^2-1]^{1/2}\over s(E,x_0+\Omega, x_0)}.$$ The sign of the square root was chosen such that$[\Delta(E)^2-1]^{1/2}$is asymptotically equal to$\rho_-(E)/2$for large positive$E$. Equation (\ref{2.42}) implies (see also \cite{27}) $$\label{2.43} G(E,x_0,x_0) = -{s(E,x_0+\Omega, x_0)\over2[\Delta(E)^2-1]^{1/2}}.$$ Closely related to$G(E,x,x)$is the function $$H(E,x,x')={\partial^2 G(E,x,x') \over \partial x \partial x'}.$$ To evaluate it for$x=x'=x_0$we use (\ref{2.40a}) and obtain $$\label{2.43a} H(E,x_0,x_0) = {c'(E,x_0+\Omega, x_0)\over2[\Delta(E)^2-1]^{1/2}}.$$ When$q\in C^1(\bbR)$the function$Hsatisfies the nonlinear second-order differential equation \begin{align}\label{2.68a} &4(E-q(x))^2H(E,x,x)^2-2q'(x) H(E,x,x)H'(E,x,x) \nonumber\\ &-2(E-q(x))H(E,x,x)H''(E,x,x)+(E-q(x))H'(E,x,x)^2 \nonumber\\ =&(E-q(x))^3. \end{align} We emphasize that both (\ref{2.68}) and (\ref{2.68a}) hold universally for anyq\in C^0(\bbR)$or$q\in C^1(\bbR)$, respectively, i.e., they do not at all depend on periodicity of$q$. While (\ref{2.68}) is a standard result (see, e.g., \cite{GD75}) the differential equation (\ref{2.68a}) appears to be new. Equations (\ref{2.68}) and (\ref{2.68a}) are the main ingredients for the following \begin{thm} \label{t2.1a} Let$q$be a differentiable periodic function of period$\Omega>0$on$\bbR$. Then for every$E\in\bbC$\begin{gather} p_i(E)=r_i(E), \label{piisri}\\ d(E)-p_i(E)-r_i(E) \geq 0 \label{pi+riFrom (\ref{2.25neu}) and (\ref{2.25aneu}) we get for all$E\in\bbC$and all$x\in\bbR$\begin{gather} {\Delta(E)^2-1 \over D(E)^2} -{1\over4} F_D'(E,x_0)^2 ={N(E)\over D(E)}F_D(E,x_0) F_N(E,x_0), \label{pklr} \\ (q(x_0)-E)^2 {\Delta(E)^2-1 \over N(E)^2}-{1\over4} F_N'(E,x_0)^2 =(q(x_0)-E)^2 {D(E)\over N(E)}F_D(E,x_0) F_N(E,x_0). \label{rklp} \end{gather} Since, according to the first part of the proof, the left hand sides of (\ref{pklr}) and (\ref{rklp}) are entire functions with respect to$E$for every$x_0 \in\bbR$so must be the right hand sides. Suppose$\lambda$is a zero of$D(E)$or$N(E)$and assume there is an$x_0$such that $$\label{conx0} (q(x_0)-\lambda)^2 F_D(\lambda,x_0) F_N(\lambda,x_0)\neq0.$$ Then (\ref{pklr}) shows that$p_i(\lambda)\leq r_i(\lambda)$while (\ref{rklp}) shows the converse inequality and hence (\ref{piisri}). To show that there is an$x_0 \in\bbR$satisfying (\ref{conx0}) observe that equation (\ref{2.68}) implies that zeros of$F_D(E,\cdot)$are isolated since$F_D'(E,x_0)\neq0$if$F_D(E,x_0)=0$and similarly, (\ref{2.68a}) implies that a zero$x_0$of$F_N(E,\cdot)$is isolated provided$q(x_0)\neq E$. \end{pf} \section{Floquet Theory and Finite-Gap Potentials} \label{fingap} \setcounter{equation}{0} In this section we prove that$q$is a finite-gap potential if the equation$\psi''+q\psi=E\psi$has two linearly independent Floquet solutions for all but finitely many values of the spectral parameter$E$. The proof reveals a number of other properties, notably about the Green's function of$H$. \begin{thm}\label{t2.2} Assume that$q(x)$is a continuous periodic function of period$\Omega>0$on$\bbR$and that$Ly=y''+q(x)y=Ey$has two linearly independent Floquet solutions for all but finitely many values of$E\in\bbC$. Then the following statements hold.\\ (i) Suppose$\{\hat E_j\}_{j=1}^{\hat M}$for some$\hat M\in \bbN$is the set where two linearly independent Floquet solutions do not exist. Then the function$\Delta(E)^2-1$has a zero at each point$\hat{E}_j$, i.e.,$d(\hat E_j)>0$for$1\leq j\leq \hat M$. Moreover, none of the$\hat E_j$,$j=1,...,\hat M$is an immovable Dirichlet (or Neumann) eigenvalue.\\ (ii) The inequality $$\label{4.2} d(E)-p_i(E)-r_i(E)\geq0$$ is strict only on a finite set$\{\hat E_j\}_{j=1}^M$,$M\geq \hat M$which includes the numbers$\hat E_1,..., \hat E_{\hat M}$.\\ (iii) The number of movable Dirichlet eigenvalues and the number of movable Neumann eigenvalues are finite. More precisely, there exists an integer$g \in\bbN_0$such that for any given$x_0 \in\bbR\begin{align} \sum_{E\in\bbC} p_m(E,x_0) &=g, \label{4.3} \displaybreak[0]\\ \sum_{E\in\bbC} r_m(E,x_0) &=g+1. \label{4.4} \end{align} This numberg$satisfies $$\label{4.5} 2g+1= \sum_{j=1}^{M} (d(\hat E_j)-p_i(\hat E_j)-r_i(\hat E_j)) =\sum_{j=1}^M \hat q_j,$$ where $$\label{4.9} \hat q_j = d(\hat E_j)-p(\hat E_j)-r(\hat E_j)\; \text{for j=1,...,M.}$$ (iv)$q\in C^\infty(\bbR)$.\\ (v)$q$is an algebro-geometric finite-gap potential associated with the compact (possibly singular) hyperelliptic curve$K_g$of (arithmetic) genus$g$obtained upon one-point compactification of the curve $$\label{4.10} F^2=\hat{R}_{2g+1}(E)=\prod_{j=1}^{M}(E-\hat{E}_j)^{\hat q_j}.$$ Equivalently, there exists a monic ordinary differential expression$\hat{P}_{2g+1}$of order$2g+1$, i.e., $$\label{4.11} \hat{P}_{2g+1} = \sum^{2g+1}_{\ell=0} p_{\ell}(x) \dfrac{d^{\ell}}{dx^{\ell}}, \; p_{2g+1}(x) =1,$$ which commutes with$L$, i.e., $$\label{4.12} [\hat{P}_{2g+1}, L]=0$$ and satisfies the Burchnall-Chaundy relation $$\label{4.13} \hat{P}^2_{2g+1} = \hat{R}_{2g+1}(L) =\prod_{j=1}^{M}(L-\hat{E}_j)^{\hat q_j}.$$ (vi) The diagonal Green's function$G(E,x,x)$of$H$is defined and continuous for all$E\in\bbC\backslash \{\hat{E}_j\}^{M}_{j=1}$and is of the type $$\label{4.6} G(E,x,x) = -{1\over2} \hat{F}_g(E,x)/\hat{R}_{2g+1}(E)^{1/2},$$ where $$\label{4.7} \hat{F}_g(E,x) = \prod_{j\in J_g}[E-\mu_j(x)]$$ and where$J_g$(of cardinality$g$) is the set of indices$j\in\bbN$such that$p_m(\mu_j(x),x)>0$(we set$\hat{F}_g(E,x)=1$for$g=0$).\\ (vii) Let$\lambda\in\bbC$,$B(\lambda; \varepsilon)=\{E: |E-\lambda|<\varepsilon\}$and$f_\pm(E,x)$two Floquet solutions of$L\psi=E\psi$which are linearly independent for each$E\in B(\lambda; \varepsilon)\backslash\{\lambda\}$and which, together with their$x$-derivatives, are continuous as functions of$E$in$B(\lambda; \varepsilon)$. Then the Wronskian$W(f_-,f_+)$vanishes at$\lambda$if and only if$\lambda\in\{\hat E_1,...,\hat E_M\}$.\\ (viii) The spectrum of$H$consists of finitely many bounded spectral arcs$\tilde{\sigma}_n$,$1\le n\le \tilde{g}$for some$\tilde{g}\le g$and one unbounded (semi-infinite) arc$\tilde{\sigma}_{\infty}$which tends to$-\infty+$, with$=\Omega^{-1} \int^{x_0+\Omega}_{x_0} q(x) dx$, i.e., $$\label{4.14} \sigma(H)=\left(\bigcup^{\tilde{g}}_{n=1} \tilde{\sigma}_n \right) \cup \tilde{\sigma}_{\infty},$$ where each$\tilde{\sigma}_n$and$\tilde{\sigma}_{\infty}$is a union of some of the spectral arcs$\sigma_nin (\ref{2.22}). \end{thm} \begin{pf} We introduce the following sets \begin{align} D_m(x_0)&=\{E: p_m(E,x_0)>0\},\displaybreak[0]\\ N_m(x_0)&=\{E: r_m(E,x_0)>0\},\displaybreak[0]\\ B_1&=\{E:0=p_i(E)=r_i(E); \; 00 such that for $|E|\geq\Lambda$ the inequalities $p(E,x_0),r(E,x_0)\leq1$ and $d(E)\leq2$ hold. Assume also that $\max\{|\hat E_1|,..., |\hat E_{\hat M}|\} < \Lambda$. Then we have $d(E)=2$ for all (anti)periodic eigenvalues $E$ for which $|E|\geq\Lambda$, since otherwise there would be only one linearly independent Floquet solution for $L\psi=E\psi$. Assume now that $E\in B_2$ and that $|E|\geq\Lambda$. Then $p_i(E), r_i(E)\geq1$ and $p(E,x_0), \linebreak[0] r(E,x_0) \leq1$ and hence $p_i(E)=r_i(E)=1$. Also $d(E)=2=p_i(E)+r_i(E)$ which contradicts the definition of $B_2$. Therefore no $E$ whose absolute value is larger than $\Lambda$ can be in $B_2$, i.e., $B_2$ is a finite set. Next we show that $D_m(x_0)$ and $N_m(x_0)$ are also finite sets. Consider the periodic or antiperiodic eigenvalue $E_{2n}$ where $n$ is such that $|E_{2n}|\geq\Lambda$. Then $M(E_{2n},x_0)$ has a double eigenvalue $\pm1$ with geometric multiplicity two. This forces the diagonal elements of $M(E_{2n},x_0)$ to be zero, i.e., $E_{2n}$ is both a Dirichlet and a Neumann eigenvalue for every $x_0\in\bbR$. The asymptotic behavior of these eigenvalues shows that $E_{2n}=\mu_n(x_0) =\nu_n(x_0)$ for every $x_0\in\bbR$. The same argument shows that $\mu_k(x_0)$ and $\nu_k(x_0)$ are immovable for all $k>n$. Thus, if $\mu_k(x_0)$ or $\nu_k(x_0)$ actually depend on $x_0$, then $k0$, $\Re(\omega_3) \ge 0$, $\Im(\omega_3/\omega_1)>0$. The fundamental period parallelogram $\Delta$ consists then of the points $z=2\omega_1 s +2\omega_3 t$ where $0\leq s,t<1$. We introduce $$\label{3.11} e^{i\theta} = \dfrac{\omega_3}{\omega_1} \left| \dfrac{\omega_1}{\omega_3} \right|, \; \theta \in (0,\pi),$$ and $$\label{3.12} t_j=\omega_j/|\omega_j|, \; j=1,3$$ and define $$\label{3.13} q_j(x):=t^2_j q (t_jx+z_0), \; j=1,3$$ for a $z_0\in\bbC$ which we choose 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 $z_0$ and $z_0+2\omega_1$ or on the line through $z_0$ and $z_0+2\omega_3.$ Since $q$ has only finitely many poles in the fundamental period parallelogram $\Delta$ this can always be achieved.) For such a choice of $z_0$ we infer that $q_j(x)$ are real-analytic and periodic of period $\Omega_j=2|\omega_j|, \; j=1,3.$ Comparing the differential equations $$\label{3.14} \psi''(z)+q(z)\psi(z)=E\psi(z)$$ and $$\label{3.15} w''(x) + q_j(x)w(x) = \lambda w(x), \; j=1,3,$$ connected by the variable transformation $$\label{3.16} z=t_jx+z_0, \; \psi(z) = w(x),$$ one concludes that $w$ is a solution of (\ref{3.15}) if and only if $\psi$ is a solution of (\ref{3.14}) with $$\label{3.17} \lambda=t_j^2E, \; j=1,3.$$ Next, consider $\tilde{q}\in C^0(\bbR)$ of period $\tilde{\Omega} >0$ and let $\tilde{c}(\lambda,x), \tilde{s}(\lambda,x)$ be the corresponding fundamental system of solutions of $\tilde{w}^{''}+\tilde{q}\tilde{w}= \lambda\tilde{w}$ defined by $$\label{3.18} \tilde{c}(\lambda,0) = \tilde{s}'(\lambda,0)=1, \quad \tilde{c}'(\lambda,0) = \tilde{s}(\lambda,0)=0.$$ The corresponding Floquet discriminant is then given by $$\label{3.19} \tilde{\Delta}(\lambda)=[\tilde{c}(\lambda,\tilde{\Omega}) + \tilde{s} (\lambda,\Omega)]/2$$ and the same techniques that lead to the asymptotic expansion (\ref{2.29}) also yield $$\label{3.20} \tilde{\Delta}(\lambda)= \cos[i\tilde{\Omega}\lambda^{1/2}(1+O(\lambda^{-1}))]$$ as $|\lambda|$ tends to infinity. \begin{prop}\label{p3.5} Let $\tilde{\lambda}_n$ be a periodic or antiperiodic eigenvalue of $\tilde{q}$. Then there exists an $m\in\bbZ$ such that $$\label{3.21} \left|\tilde{\lambda}_n + m^2\pi^2 \tilde{\Omega}^{-2}\right|\le \tilde{C}$$ for some $\tilde{C}>0$ independent of $n\in\bbN_0$. In particular, all periodic and antiperiodic eigenvalues $\tilde{\lambda}_n$, $n\in\bbN_0$ of $\tilde{q}$ are contained in a half-strip $\tilde{S}$ given by $$\label{3.22} \tilde{S}=\{\lambda\in\bbC | \left|\Im(\lambda)\right|\le \tilde{C}, \; \Re(\lambda)\le \tilde{M}\}$$ for some $\tilde{M}\in\bbR.$ \end{prop} \begin{pf} The periodic and antiperiodic eigenvalues are precisely the points $\lambda_0$ where $\tilde{\Delta}(\lambda_0) =\pm 1.$ Let $$\label{3.23} \tilde{\Omega} \tilde{\lambda}^{1/2}_n = a_n+ib_n, \quad a_n, b_n \in \bbR$$ then (\ref{3.20}) implies $$\label{3.24} m\pi + b_n-ia_n = O((a_n+i b_n)^{-1})$$ for some $m\in\bbZ$. Hence $$\label{3.25} |m\pi+b_n| \le c_1, \; |a_n| \le c_1$$ and (multiplying (\ref{3.24}) by $a_n+ib_n$ and taking real and imaginary parts) $$\label{3.26} |a_n(m\pi+2b_n)|\le c_2, \; |(b_nm\pi+b_n^2-a^2_n)|\le c_2$$ for some constants $c_1, c_2 > 0$. For $|\tilde{\lambda}_n|=(a^2_n+b^2_n) \tilde{\Omega}^{-2}$ sufficiently large we conclude that $|b_n|$ and hence $|m\pi|$ and $|m\pi+2b_n|$ are also large since $|a_n|$ stays bounded. By (\ref{3.25}) one obtains that $|m\pi|\le |b_n|+c_1$ and hence (\ref{3.26}) and $|c_1|\le|b_n|/2$ imply $$\label{3.27} |a_n|\le {c_2\over |m\pi+2 b_n|} \le {c_2 \over 2|b_n|-|m\pi|} \le {c_2 \over |b_n|-c_1} \le {2c_2 \over |b_n|}$$ and $$\label{3.28} |b_n(m\pi +b_n)| \le a^2_n+ c_2 \le c^2_1+c_2,$$ i.e., $$\label{3.29} |m\pi + b_n| \le (c_1^2+c_2) / |b_n|.$$ Consequently, we infer from (\ref{3.27}), (\ref{3.29}) and from the fact that $|a_n|$ is bounded that $$\label{3.30} |\tilde{\Omega} \tilde{\lambda}^{1/2}_n+i\pi m| =|a_n+ib_n+i\pi m| \le|a_n|+|b_n+\pi m| \le c|b_n|^{-1} \le c'|\tilde{\lambda}_n|^{-1/2}$$ for some constants $c$ and $c'$. Multiplying (\ref{3.30}) by $|\tilde{\Omega}\tilde{\lambda}^{1/2}_n - i\pi m|$ finally results in \begin{eqnarray}\label{3.31} |\tilde{\Omega}^2 \tilde{\lambda}_n+\pi^2m^2|& \le & c'|\tilde{\lambda}_n|^{-1/2} |\tilde{\Omega}\tilde{\lambda}_n^{1/2} - i\pi m|\\ & \le & c'|\tilde{\lambda}_n|^{-1/2} [2\tilde{\Omega} |\tilde{\lambda}_n|^{1/2} +|-\tilde{\Omega}\tilde{\lambda}_n^{1/2}-i\pi m|]\\ & \le & c'|\tilde{\lambda}_n|^{-1/2} [2\tilde{\Omega}|\tilde{\lambda}_n|^{1/2}+c'|\tilde{\lambda}_n|^{-1/2}] \le \tilde C. \end{eqnarray} Hence $\tilde{\lambda}_n$ is in a disk around $-m^2\pi^2 \tilde{\Omega}^{-2}$ whose radius is independent of $n$. \end{pf} In order to apply Proposition \ref{p3.5} to $q_1$ and $q_3$ we note that according to (\ref{3.20}), $$\label{3.32} \Delta_j(\lambda)=\cos [i\Omega_j\lambda^{1/2} (1+O(\lambda^{-1}))], \; j=1,3$$ as $|\lambda|$ tends to infinity, where, in obvious notation, $\Delta_j(\lambda)$ denotes the discriminant of $q_j(x)$, $j=1,3$. Next, denote by $\lambda_{j,n}$ an $\Omega_j$-(anti)periodic eigenvalue of $w^{''}+q_jw=\lambda w.$ Then $E_{j,n} = t_j^{-2}\lambda_{j,n}$ is a $2\omega_j$-(anti)periodic eigenvalue of $\psi^{''}+ q \psi = E\psi$ and vice versa. Hence Proposition \ref{p3.5} immediately yields the following result. \begin{prop}\label{p3.6} Let $j=1$ or $3$. Then all $2\omega_j$-(anti)periodic eigenvalues $E_{j,n}$, $n\in\bbN_0$ associated with $q$ lie in the half-strip $S_j$ given by $$\label{3.33} S_j=\{E\in\bbC: |\Im(t^2_jE)|\le C_j, \; \Re (t^2_jE) \le M_j\}$$ for suitable constants $C_j>0, M_j\in\bbR.$ The angle between the axes of the strips $S_1$ and $S_3$ equals $2\theta \in (0,2\pi).$ \end{prop} Propositions \ref{p3.5} and \ref{p3.6} apply to any elliptic potential whether or not they are finite-gap. In our final step we shall now invoke Picard's Theorem \ref{t3.1} to obtain our characterization of elliptic finite-gap potentials. \begin{thm}\label{t3.7} $q$ is an elliptic finite-gap potential if and only if $q$ is a Picard potential (i.e., if and only if for each $E\in\bbC$ every solution of $\psi''(z)+ q(z)\psi(z)=E\psi(z)$ is meromorphic with respect to $z$). \end{thm} \begin{pf} By Theorem \ref{t3.4} it remains to prove that a Picard potential is finite-gap. Hence we assume in the following that $q$ is a Picard potential. Since all $2\omega_j$-(anti)periodic eigenvalues $E_{j,n}$ of $q$ yield zeros $\lambda_{j,n} = t^2_jE_{j,n}$ of the entire functions $\Delta_j(\lambda)^2-1$, the $E_{j,n}$ have no finite limit point. Next we choose $R>0$ large enough such that the exterior of the closed disk $\overline{D(0,R)}$ centered at the origin of radius $R>0$ contains no intersection of $S_1$ and $S_3$ (defined in (\ref{3.33})), i.e., $$\label{3.34} (\bbC\backslash \overline{D(0,R)}) \cap (S_1\cap S_3) = \emptyset.$$ In (\ref{3.34}) it is assumed that $\theta \ne \pi/2$ such that $S_1$ and $S_3$ are not parallel. This can always be achieved by replacing the original pair of fundamental periods $2\omega_1, 2\omega_3$ by $2\omega_1$ and a suitable linear combination of $2\omega_1$ and $2\omega_3$. Let $\rho_{j,\pm}(\lambda)$ be the Floquet multipliers of $q_j(x),$ i.e., the solutions of $$\label{3.35} \rho^2_j-2\Delta_j \rho_j+1=0, \; j=1,3.$$ Then (\ref{3.34}) implies that for $E\in\bbC\backslash \overline{D(0,R)}$, at most one of the numbers $\rho_1(t_1E)$ and $\rho_3(t_3E)$ can be in $\{-1,1\}$. In particular, at least one of the characteristic equations corresponding to the substitution $z\to z+2\omega_1$ or $z\to 2\omega_3$ (cf. (\ref{3.3}) and (\ref{3.4})) has two distinct roots. Since by hypothesis $q$ is a Picard potential, Picard's Theorem \ref{t3.1} applies and guarantees for all $E\in \bbC\backslash \overline{D(0,R)}$ the existence of two linearly independent solutions $\psi_1(E,z)$ and $\psi_2(E,z)$ of $\psi^{''} + q\psi=E\psi$ which are elliptic of the second kind. Then $w_{j,k}(x)=\psi_k(t_j x+z_0)$, $k=1,2$ are linearly independent Floquet solutions associated with $q_j$. Therefore the points $\lambda$ for which $w''+q_jw=\lambda w$ 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{t2.2} then proves that both $q_1$ and $q_3$ are finite-gap potentials. By (\ref{2.8}) (in slight abuse of notation) $$\sum^{g}_{k=0}c_{g-k}{df_{k+1}(q_1(x))\over dx}=0,$$ where $g\in\bbN_0$, $f_{k+1}$, $k=0,...,g$, are differential polynomials in $q_1$ homogeneous of degree $2k+2$ (cf. (\ref{2.9})), and $c_{k}$, $k=0,...,g$ are complex constants. Since $$\label{3.37} q_1^{(\ell)}(x)=t_1^{\ell+2} q^{(\ell)}(z),$$ (where $z=t_1x+z_0$) we obtain $$\sum^{g}_{k=0}c_{g-k} t_1^{2k+3} {df_{k+1}(q(z))\over dz}=0,$$ i.e., $q$ is a finite-gap potential as well. A similar argument would have worked using the relationship between $q_3$ and $q$. In particular, the order of the operators commuting with $d^2/dz^2+q(z)$, $d^2/dx^2+q_1(x)$, and $d^2/dx^2+q_3(x)$, respectively, is the same in all cases, namely $2g+1$. \end{pf} We add a series of remarks further illustrating the significance of Theorem \ref{t3.7}. \begin{rem}{\bf (Complementing Picard's theorem.)} \label{r3.8} Theorem \ref{t3.7} extends and complements Picard's Theorem \ref{t3.1} in the sense that it determines the elliptic functions $q(z)$ which satisfy the hypothesis of the theorem precisely as (elliptic) finite-gap solutions of the stationary KdV hierarchy. \end{rem} \begin{rem}{\bf (Characterization of elliptic finite-gap potentials.)} \label{r3.9} While an explicit proof of the finite-gap property of $q$ in general is highly nontrivial (see, e.g., the references cited in connection with special cases such as the Lam\'e-Ince and Treibich-Verdier potentials in Remark \ref{r3.12} below), the fact of whether or not $\psi^{''}(z)+q(z)\psi(z)=E\psi(z)$ has a fundamental system of solutions meromorphic in $z$ for a finite (but sufficiently large) number of the spectral parameter $E\in\bbC$ can be decided by means of an elementary Frobenius-type analysis (see, e.g., \cite{29} and \cite{30}). To date Theorem \ref{t3.7} appears to be the only effective tool to identify general elliptic finite-gap solutions of the KdV hierarchy. Thus Theorem \ref{t3.7} provides an explicit characterization of all elliptic finite-gap solutions of the stationary KdV hierarchy, a problem posed, e.g., by Novikov et al. in \cite{49}, p.152. \end{rem} \begin{rem}{\bf (Reduction of Abelian integrals.)} \label{r3.11} Theorem \ref{t3.7} is also relevant in the context of the Weierstrass theory of reduction of Abelian to elliptic integrals, a subject that attracted considerable interest, see, e.g., \cite{5}, \cite{6}, \cite{6a}, Ch. 7, \cite{7}, \cite{8}, \cite{18}, \cite{19}, \cite{20}, \cite{38}, \cite{40}, \cite{44}, \cite{58}, \cite{59}. In particular, the theta functions corresponding to the hyperelliptic curves derived from the Burchnall-Chaundy polynomials (\ref{2.15}), associated with Picard potentials, reduce to one-dimensional theta functions. \end{rem} \begin{rem}{\bf (Computation of genus and branch points.)} \label{r3.12} While Theorem \ref{t3.7} characterizes all elliptic finite-gap potentials as Picard potentials, it does not yield an effective way to compute the underlying hyperelliptic curve $K_g$; in particular, its proof provides no means to compute the branch and singular points nor the (arithmetic) genus $g$ of $K_g$. To the best of our knowledge $K_g$ has been computed only for Lam\'e-Ince potentials and certain Treibich-Verdier potentials (see, e.g., \cite{4a}, \cite{8}, \cite{40}, \cite{49}, \cite{58}, \cite{59}, \cite{64}, \cite{66}, \cite{67}). Even the far simpler task of computing $g$ has only been achieved in the case of Lam\'e-Ince potentials (see \cite{36} and \cite{61} for the real and complex-valued case, respectively). In \cite{29}, \cite{30}, and \cite{31} we have treated these problems for Lam\'e-Ince, Treibich-Verdier, and even elliptic finite-gap potentials, respectively. In particular, in \cite{30} we computed $g$ for all Treibich-Verdier potentials and in \cite{31} we reduced the computation of the branch and singular points of $K_g$ for any even elliptic finite-gap potential to the solution of linear algebraic eigenvalue problems. We refrain from reproducing a detailed discussion of this matter here, instead we just recall an example taken from \cite{30} which indicates some of the subtleties involved: Consider the potentials \begin{gather}\label{3.45} q_4(z)=-20\calP(z-\omega_j)-12\calP(z-\omega_k), \displaybreak[0] \\ \hat{q}_4(z)=-20\calP(z-\omega_j)-6\calP(z-\omega_k) -6\calP(z-\omega_{\ell}), \\ q_5(z)=-30\calP(z-\omega_j)-2\calP(z-\omega_k), \\ \hat{q}_5(z)=-12\calP(z-\omega_j)-12\calP(a-\omega_k)-6\calP (z-\omega_{\ell})-2\calP(z-\omega_m), \end{gather} where $j,k,\ell,m\in\{1,2,3,4\}$ ($\omega_2=\omega_1+\omega_3$, $\omega_4=0$) are mutually distinct. Then $q_4$ and $\hat{q}_4$ correspond to (arithmetic) genus $g=4$ while $q_5$ and $\hat{q}_5$ correspond to $g=5$. However, we emphasize that all four potentials contain precisely 16 summands of the type -- $2\calP(x-b_n)$ (cf. the discussion following (\ref{1.10})). $q_5$ and $\hat{q}_5$ are isospectral (i.e., correspond to the same curve $K_5$) while $q_4$ and $\hat{q}_4$ are not. \end{rem} \begin{rem}{\bf (Generalizations.)} \label{r3.13} Finally, we remark that Theorems \ref{t3.1} and \ref{t3.4} and Propositions \ref{p3.5} and \ref{p3.6} extend to $n$-th order operators $L_n$. We have decided to restrict this paper to the second-order case $L_2=d^2/dx^2+q(x)$ since the corresponding generalization of Theorem \ref{t2.2} to algebro-geometric finite-gap solutions of the stationary Gelfand-Dickey (GD) hierarchy is beyond the scope of this paper. (Even though a recursion relation formalism for the GD hierarchy analogous to the KdV case in (\ref{2.1}) -- (\ref{2.16}) exists in principle, the explicit construction of a monic differential expression $P_r$ of order $r$ ($r$ and $n$ relatively prime) commuting with $L_n$, along the lines of our proof of Theorem \ref{t2.2}, is a formidable task which obscures the remarkable simplicity of our argument displayed in the proof of Proposition \ref{p3.5}.) Here we just mention the fact that if $L_n$ is a Picard differential expression (in the sense that $L_n\psi(z)=E\psi(z)$ has a fundamental system of solutions meromorphic in $z$ for each $E\in\bbC$) then the number of $E$-values where there exist less then $n$ Floquet solutions for $L_n\psi=E\psi$ is finite in number. \end{rem} {\bf Acknowledgments.} R. W. gratefully acknowledges support by the National Science Foundation. \begin{thebibliography}{99} \bibitem{1} M. Abramowitz and I. A. 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