$, with $=\Omega^{-1} \int^{x_0+\Omega}_{x_0} q(x) dx$, i.e., \begin{equation}\label{4.14} \sigma(H)=\left(\bigcup^{\tilde{g}}_{n=1} \tilde{\sigma}_n \right) \cup \tilde{\sigma}_{\infty}, \end{equation} where each $\tilde{\sigma}_n$ and $\tilde{\sigma}_{\infty}$ is a union of some of the spectral arcs $\sigma_n$ in (\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 $k 0$, $\Re(\omega_3) \ge 0$, $\Im(\omega_3/\omega_1)>0$. 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 \begin{equation}\label{3.11} e^{i\theta} = \dfrac{\omega_3}{\omega_1} \left| \dfrac{\omega_1}{\omega_3} \right|, \; \theta \in (0,\pi), \end{equation} and \begin{equation}\label{3.12} t_j=\omega_j/|\omega_j|, \; j=1,3 \end{equation} and define \begin{equation}\label{3.13} q_j(x):=t^2_j q (t_jx+z_0), \; j=1,3 \end{equation} 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 \begin{equation}\label{3.14} \psi''(z)+q(z)\psi(z)=E\psi(z) \end{equation} and \begin{equation}\label{3.15} w''(x) + q_j(x)w(x) = \lambda w(x), \; j=1,3, \end{equation} connected by the variable transformation \begin{equation}\label{3.16} z=t_jx+z_0, \; \psi(z) = w(x), \end{equation} one concludes that $w$ is a solution of (\ref{3.15}) if and only if $\psi$ is a solution of (\ref{3.14}) with \begin{equation}\label{3.17} \lambda=t_j^2E, \; j=1,3. \end{equation} 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 \begin{equation}\label{3.18} \tilde{c}(\lambda,0) = \tilde{s}'(\lambda,0)=1, \quad \tilde{c}'(\lambda,0) = \tilde{s}(\lambda,0)=0. \end{equation} The corresponding Floquet discriminant is then given by \begin{equation}\label{3.19} \tilde{\Delta}(\lambda)=[\tilde{c}(\lambda,\tilde{\Omega}) + \tilde{s} (\lambda,\Omega)]/2 \end{equation} and the same techniques that lead to the asymptotic expansion (\ref{2.29}) also yield \begin{equation}\label{3.20} \tilde{\Delta}(\lambda)= \cos[i\tilde{\Omega}\lambda^{1/2}(1+O(\lambda^{-1}))] \end{equation} 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 \begin{equation}\label{3.21} \left|\tilde{\lambda}_n + m^2\pi^2 \tilde{\Omega}^{-2}\right|\le \tilde{C} \end{equation} 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 \begin{equation}\label{3.22} \tilde{S}=\{\lambda\in\bbC | \left|\Im(\lambda)\right|\le \tilde{C}, \; \Re(\lambda)\le \tilde{M}\} \end{equation} 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 \begin{equation}\label{3.23} \tilde{\Omega} \tilde{\lambda}^{1/2}_n = a_n+ib_n, \quad a_n, b_n \in \bbR \end{equation} then (\ref{3.20}) implies \begin{equation}\label{3.24} m\pi + b_n-ia_n = O((a_n+i b_n)^{-1}) \end{equation} for some $m\in\bbZ$. Hence \begin{equation}\label{3.25} |m\pi+b_n| \le c_1, \; |a_n| \le c_1 \end{equation} and (multiplying (\ref{3.24}) by $a_n+ib_n$ and taking real and imaginary parts) \begin{equation}\label{3.26} |a_n(m\pi+2b_n)|\le c_2, \; |(b_nm\pi+b_n^2-a^2_n)|\le c_2 \end{equation} 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 \begin{equation}\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|} \end{equation} and \begin{equation}\label{3.28} |b_n(m\pi +b_n)| \le a^2_n+ c_2 \le c^2_1+c_2, \end{equation} i.e., \begin{equation}\label{3.29} |m\pi + b_n| \le (c_1^2+c_2) / |b_n|. \end{equation} Consequently, we infer from (\ref{3.27}), (\ref{3.29}) and from the fact that $|a_n|$ is bounded that \begin{equation}\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} \end{equation} 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}), \begin{equation}\label{3.32} \Delta_j(\lambda)=\cos [i\Omega_j\lambda^{1/2} (1+O(\lambda^{-1}))], \; j=1,3 \end{equation} 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 \begin{equation}\label{3.33} S_j=\{E\in\bbC: |\Im(t^2_jE)|\le C_j, \; \Re (t^2_jE) \le M_j\} \end{equation} 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., \begin{equation}\label{3.34} (\bbC\backslash \overline{D(0,R)}) \cap (S_1\cap S_3) = \emptyset. \end{equation} 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 \begin{equation}\label{3.35} \rho^2_j-2\Delta_j \rho_j+1=0, \; j=1,3. \end{equation} 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) \begin{equation} \sum^{g}_{k=0}c_{g-k}{df_{k+1}(q_1(x))\over dx}=0, \end{equation} 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 \begin{equation}\label{3.37} q_1^{(\ell)}(x)=t_1^{\ell+2} q^{(\ell)}(z), \end{equation} (where $z=t_1x+z_0$) we obtain \begin{equation} \sum^{g}_{k=0}c_{g-k} t_1^{2k+3} {df_{k+1}(q(z))\over dz}=0, \end{equation} 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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