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\begin{document}

\title[Elliptic Solutions of Soliton Equations]{Toward a
Characterization of Elliptic Solutions\\
of Hierarchies of Soliton Equations}

%    Information for first author
\author{F. Gesztesy}
\address{Department of Mathematics, University of Missouri, Columbia,
Missouri 65211, USA}
\email{mathfg@mizzou1.missouri.edu}

%    Information for second author
\author{R. Weikard}
\address{Department of Mathematics, University of Alabama at
Birmingham, Birmingham, Alabama 35294-1170, USA}
\email{rudi@math.uab.edu}

%    \thanks will become a 1st page footnote.
\thanks{Based upon work supported by the National Science
Foundation under Grants No. DMS-9623121 and DMS-9401816.}

%    General info
\subjclass{Primary 35Q53, 34L05; Secondary  58F07}
\date{\today}


\begin{abstract}
The current status of an explicit characterization of all elliptic
algebro-geometric solutions of hierarchies of soliton equations is
discussed and the case of the KdV hierarchy is considered in detail.
More precisely, we review our recent result that an elliptic
function
$q$ is a solution of some equation of the stationary KdV hierarchy,
if and only if the associated differential equation $\psi''(E,z)+
q(z)\psi(E,z)=E\psi(E,z)$ has a meromorphic fundamental system for
every
complex value of the spectral parameter $E$.

This result also provides an explicit condition under which a
classical theorem
of Picard holds. 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 algebro-geometric solutions of completely integrable
hierarchies of nonlinear evolution equation is the principal new
aspect of our approach.

In addition, we describe most recent attempts to extend this circle
of ideas to $n$-th-order scalar differential equations and
first-order
$n \times n$ systems of differential equations with elliptic
functions
as coefficients associated with Gelfand-Dickey and matrix-valued
hierarchies of soliton equations.
\end{abstract}

\maketitle

\section{Introduction} \label{intro}
The principal purpose of this review is to describe the basic
ideas underlying an efficient characterization of elliptic
algebro-geometric solutions of general hierarchies of soliton
equations.
Since at this time the only case worked out in all details is
that of
the KdV
hierarchy, we will focus to a large extent on this case and turn
in our
final two sections to possible extensions to the Gelfand-Dickey
and
matrix-valued hierarchies.

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 \linebreak[0]
\begin{equation}\label{1.1}
q(x)=-g(g+1)\wp(x+\omega_3), \; g\in \bb N, \; x \in\bb R
\end{equation}
in connection with the second-order ordinary differential
equation
\begin{equation}\label{1.2}
\psi''(E,x) + q(x) \psi(E,x) = E\psi(E,x), \; E\in\bb C.
\end{equation}
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,
\begin{equation}\label{1.3}
\sigma(L_2)=(-\infty, E_{2g}] \cup \bigcup^g_{m=1}
\left[ E_{2m-1},E_{2m-2}\right],
\; E_{2g}<E_{2g-1}, < \ldots < E_0.
\end{equation}

What we call the Lam\'e--Ince potential has, in fact,
a long history and many investigations of it precede Ince's work
\cite{36}. Without possibly trying to be complete we refer the
interested reader, for instance, to \cite{2b}, \cite{3},
Sect. 59,
\cite{4a}, Ch. IX, \cite{6a}, Sect. 3.6.4, \cite{13},
Sects. 135--138, \cite{13a}, \cite{13c}, \cite{13b}, \cite{29},
\cite{34}, p. 494--498, \cite{35}, p. 118--122, 266--418,
475--478,
\cite{37}, p. 378--380, \cite{39a}, \cite{41}, p. 265--275,
\cite{49b},
\cite{49a}, \cite{64}, \cite{66}, \cite{67}, Ch. XXIII as
pertinent
publications before and after Ince's fundamental paper.

Following the traditional terminology, any real-valued potential
$q$ that
gives rise to a spectrum of the type (\ref{1.3}) is called an
algebro-geometric potential. The proper extension of this notion
to general complex-valued
meromorphic
potentials $q$ and its connection with stationary solutions
of the KdV
hierarchy on the basis of elementary algebro-geometric concepts
is then
obtained as follows. Let $L_2(t)$ be the second-order
differential
expression
\begin{equation}\label{1.4}
L_2 (t)=\dfrac{d^2}{dx^2} + q(x,t), \; (x,t) \in \bb R^2,
\end{equation}
where $q$ depends on the additional (deformation) parameter
$t$.  It is
well known (see, e.g., Ohmiya \cite{49b}, Schimming \cite{56d},
Wilson \cite{68}, \cite{69}) that one
can find coefficients
$p_j(x,t)$ in
\begin{equation}\label{1.5}
P_{2g+1}(t) = \dfrac{d^{2g+1}}{dx^{2g+1}} + p_{2g}(x,t)
\dfrac{d^{2g}}{dx^{2g}} + \cdots + p_0(x,t),
\end{equation}
in such a way that $[P_{2g+1},L_2]$ is a multiplication
operator. The
coefficients $p_j$ are then certain differential polynomials
in $q$, that is, polynomials in $q$ and its $x$-derivatives.
The pair
$(P_{2g+1},L_2)$ is called a Lax pair, and the equation
\begin{equation}\label{1.6}
\dfrac{d}{dt}L_2 =\left[P_{2g+1},L_2 \right], \quad
\text{that is,}\quad
q_t=\left[P_{2g+1},L_2 \right]
\end{equation}
is a nonlinear evolution equation for $q$.  The collection of
all such
equations for all possible choices of $P_{2g+1}$, $g\in\bb N_0$
is then
called the KdV hierarchy (see Section \ref{kdv} for more
details).
Due to the commutator structure in (\ref{1.6}), solutions
$q(.,t)$ of
the nonlinear evolution equations of the KdV hierarchy
represent
isospectral deformations of $L_2 (0)$. Moreover,
a $t$-dependent
solution
$q(x,t)$ of one of the KdV equations in (\ref{1.6}), for each
fixed
$t\in \bb R$, satisfies one (and hence infinitely many)
stationary (in
general, higher-order) KdV equations. Therefore, without
loss of
generality, we mostly focus on stationary solutions in the
reminder
of this review.

Novikov \cite{48}, Dubrovin \cite{15}, Its and Matveev \cite{39},
and
McKean and van Moerbeke \cite{45} then showed that a real-valued
smooth
potential $q$ is an algebro-geometric potential if and only
if it
satisfies one of the higher-order stationary (i.e.,
$t$-independent) KdV
equations. Because of these facts it is common to call any
complex-valued meromorphic function $q$ an algebro-geometric
potential
if $q$ satisfies one (and hence infinitely many) of the
equations of the
stationary KdV hierarchy.

The stationary KdV hierarchy, characterized by $q_t=0$
or $[P_{2g+1},L_2]=0$, is intimately connected with the
question
of commutativity of ordinary differential expressions.
Thus, if
$[P_{2g+1},L]=0$, a celebrated theorem of Burchnall and
Chaundy
\cite{11}, \cite{12} implies that
$P_{2g+1}$ and $L_2$ satisfy an algebraic relation of the form
\begin{equation}\label{1.7}
P_{2g+1}^2 = \displaystyle \prod^{2g}_{m=0} (L_2 -E_m), \;
\{E_m\}^{2g}_{m=0} \subset \bb C.
\end{equation}
The locations $E_m$ of the (finite) branch points and singular
points
of the associated hyperelliptic curve
\begin{equation}\label{1.8}
F^2=\prod^{2g}_{m=0}(E-E_m)
\end{equation}
are precisely the band (gap) edges of the spectral bands of
$L_2$
(see (\ref{1.3})) whenever $q(x)$ is real-valued and smooth for
$x\in\bb R$ (with appropriate generalizations to the
complex-valued case,
see Section \ref{kdv}). It is the (possibly singular)
hyperelliptic
compact Riemann surface $K_g$ of (arithmetic) genus $g$,
obtained upon
one-point compactification of the curve (\ref{1.8}), which
signifies
that $q$ in $L_2 =d^2/dx^2+q(x)$ represents an algebro-geometric
potential.

While these considerations pertain to general solutions of the
stationary KdV hierarchy, we now concentrate on the additional
restriction that $q$ be an elliptic function (i.e., meromorphic
and
doubly periodic) and hence return to our main subject, elliptic
algebro-geometric potentials $q$ for $L_2 =d^2/dx^2+q(x)$, or,
equivalently, elliptic solutions of the stationary KdV hierarchy.
Ince's remarkable algebro-geometric result (\ref{1.3}) remained
the only
explicit elliptic algebro-geometric example until the KdV flow
$q_t=\dfrac{1}{4}q_{xxx}+
\dfrac{3}{2}qq_x$ with the initial condition
$q(x,0)=-6\wp(x)$ was
explicitly integrated by Dubrovin and Novikov \cite{17} in
1975 (see
also \cite{18}, \cite{19}, \cite{20}, \cite{38}), and found to
be of
the type
\begin{equation}\label{1.9}
q(x,t) =- 2\sum^3_{j=1} \wp(x-x_j(t))
\end{equation}
for appropriate $\{x_j(t)\}_{1\le j\le 3}$.
As observed above, all potentials $q(\cdot,t)$ in (\ref{1.9}) are
isospectral to $q(\cdot,0)=-6\wp(\cdot)$. Given these results
it was
natural to ask for a systematic account of all elliptic
solutions of
the KdV hierarchy, a problem posed, for instance, in \cite{49},
p. 152.

In 1977, Airault, McKean and Moser, in their seminal paper
\cite{2},
presented the first systematic
study of the isospectral torus $I_{\bb R}(q_0)$ of real-valued
smooth
potentials $q_0(x)$ of the type
\begin{equation}\label{1.10}
q_0(x) =-2\sum^M_{j=1} \wp(x-x_j)
\end{equation}
with an algebro-geometric spectrum of the form (\ref{1.3}).
Among a variety
of results they proved that any element $q$ of $I_{\bb R}(q_0)$
is an
elliptic function of the type (\ref{1.10}) (with different $x_j$)
with
$M$ constant throughout $I_{\bb R}(q_0)$ and that
$\dim I_{\bb R}(q_0)\le
M$. In particular, if $q_0$ evolves according to any equation of
the KdV
hierarchy it remains an elliptic algebro-geometric potential.
The potential (\ref{1.10}) is intimately connected with
completely
integrable many-body systems of the Calogero-Moser-type
\cite{13a},
\cite{47a} (see also \cite{13c}, \cite{13b}). This
connection with
integrable particle systems was
subsequently exploited by Krichever \cite{44} in his fundamental
construction of elliptic algebro-geometric solutions of the
Kadomtsev-Petviashvili equation. In particular, he explicitly
determined
the underlying algebraic curve $\Gamma$ and characterized the
Baker-Akhiezer function associated with it in terms of elliptic
functions
as well as the corresponding theta function of $\Gamma$. The next
breakthrough occurred in 1988 when Verdier \cite{65}
published new
explicit examples of elliptic algebro-geometric potentials.
Verdier's examples
spurred a flurry of activities and inspired Belokolos and
Enol'skii
\cite{8}, Smirnov \cite{58}, and subsequently Taimanov \cite{59}
and
Kostov and Enol'skii \cite{40} to find further such examples by
combining
the reduction process of Abelian integrals to elliptic integrals
(see
\cite{5}, \cite{6}, \cite{6a}, Ch. 7, \cite{7}) with the
aforementioned
techniques of Krichever
\cite{44}, \cite{44a}. This development
finally culminated in a series of recent results of Treibich
and Verdier
\cite{61}, \cite{62}, \cite{63} where it was shown that a general
complex-valued potential of the form
\begin{equation}\label{1.11}
q(x)=-\sum^4_{j=1}d_j \; \wp(x-\omega_j)
\end{equation}
$(\omega_2 =\omega_1+\omega_3, \; \omega_4=0)$ is an
algebro-geometric potential
if and only if $d_j/2$ are triangular numbers, that is, if
and only if
\begin{equation}\label{1.12}
d_j=g_j(g_j+1) \text{ for some } g_j\in \bb Z, \; 1\le j\le 4.
\end{equation}
We shall from now on refer to potentials of the type
\begin{equation}\label{1.13}
q(x)=-\sum^4_{j=1} g_j(g_j+1)\wp(x-\omega_j),
\; g_j\in\bb Z, \; 1\le
j\le 4
\end{equation}
as Treibich-Verdier potentials. The methods of Treibich and
Verdier are
based on hyperelliptic tangent covers of the torus $\bb C/\Lambda$
($\Lambda$ being the period lattice generated by $2\omega_1$ and
$2\omega_3$). The state of the art of elliptic algebro-geometric
solutions up
to 1993 was recently reviewed in issues 1 and 2 of volume 36
of Acta
Applicandae
Math., see, for instance, \cite{8a}, \cite{20a}, \cite{44b},
\cite{58a}, \cite{59a}, \cite{60a} and also in \cite{8b},
\cite{13f},
\cite{14a},
\cite{20b}, \cite{32b}, \cite{55a}, \cite{58b}, \cite{60b}. In
addition to
these investigations on elliptic solutions of the KdV hierarchy,
the study
of other soliton hierarchies, such as the modified KdV
hierarchy, nonlinear
Schr\"odinger hierarchy, and Boussinesq hierarchy has also
begun. We refer, for instance, to \cite{13d}, \cite{17a},
\cite{29}, \cite{30}, \cite{44c}, \cite{45a}, \cite{46a},
\cite{55b},
\cite{55c}, \cite{58c}, \cite{58d}, \cite{58e}, \cite{58f}.

Despite the efforts described thus far, an efficient
characterization
of all elliptic solutions of the KdV hierarchy remained an open
problem until 1994. Around 1992 we became aware of this problem
and
started to develop our own approach
toward it's solution.  In contrast to all existing
basically algebro-geometric approaches in this area, we realized
early on that the most powerful analytic tool in this context, a
theorem of Picard (Theorem \ref{t3.1}) concerning the existence
of solutions which are elliptic of the second kind of ordinary
differential equations with elliptic coefficients, had not
been applied at all.  As we
have recently shown in \cite{32a} (see also \cite{32c}), Picard's
theorem
combined with Floquet theoretic results indeed provides a very
simple and efficient characterization of all elliptic
algebro-geometric
solutions of the KdV hierarchy. Moreover, for reflection
symmetric
elliptic algebro-geometric potentials
$q$ (i.e., $q(z)=q(2z_0-z)$ for some $z_0\in\bb C)$ including
Lam\'e-Ince and Treibich-Verdier potentials, our approach
reduces the
computation of the branch points and singular points of the
underlying
hyperelliptic curve $K_g$ to certain (constrained) linear
algebraic
eigenvalue problems as shown in \cite{29}, \cite{30}, and
\cite{31}.

Since the main hypothesis in Picard's theorem for a second-order
differential equation of the form
\begin{equation} \label{1.14}
\psi''(z)+q(z)\psi(z)=E\psi(z), \; E\in\bb C
\end{equation}
with an elliptic potential $q$ assumes the existence of a
fundamental
system of solutions meromorphic in $z$ for each value of the
spectral
parameter $E\in\bb C$, we call any
such elliptic function $q$ which gives rise to this property
a {\bf
Picard potential}. The principal result, a characterization of all
elliptic
algebro-geometric solutions of the stationary KdV hierarchy,
then reads as
follows:
\begin{theorem}\label{t1.1} (\cite{32a})
$q$ is an elliptic algebro-geometric potential if and only if
$q$ is a Picard
potential (i.e., if and only if for each $E\in\bb C$ every
solution of
$\psi''(z)+ q(z)\psi(z)=E\psi(z)$ is meromorphic with respect
to $z$).
\end{theorem}

In particular, Theorem \ref{t1.1} sheds new light on Picard's
theorem
since it identifies the elliptic coefficients $q$ for which
there
exists a meromorphic fundamental system of solutions of
(\ref{1.14})
precisely as the elliptic algebro-geometric solutions of the
stationary KdV
hierarchy. Moreover, we stress its straightforward applicability
based
on an elementary Frobenius-type analysis which decides whether
or not
(\ref{1.14}) has a meromorphic fundamental system for each
$E\in\bb C$.
In addition, we might mention the obvious connections between
this result
and the Weierstrass theory
of reduction of Abelian to elliptic integrals.

The proof of Theorem \ref{1.1} in Section \ref{picard} (Theorem
\ref{t3.7}) relies on two main ingredients:  A purely Floquet
theoretic
part to be discussed in Section \ref{floquet} and an
elliptic function part sketched in Section \ref{picard}.

The result embodied by Theorem \ref{t1.1} in the special
context of the KdV hierarchy, 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 $z$) fundamental
system
of solutions for the underlying linear Lax
differential expression (for all values of the corresponding
spectral
parameter $E$).

Having dealt with the second-order Lax differential expression
$L_2=
d^2/dx^2 +q$ underlying the KdV hierarchy, it is natural
to seek
extensions to $n$-th-order Lax differential expressions $L_n$
associated
with the Gelfand-Dickey hierarchy and more generally, to
matrix-valued hierarchies of soliton equations. At present our
results
in these directions are promising but far from being complete.
We provide a recent generalization of Picard's theorem to
first-order
$n \times n$ systems of differential equations in Section
\ref{systems}
and devote Section \ref{nthorder} to partial progress in the
context of
$n$-th-order scalar differential expressions with elliptic
coefficients.





\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{13e},
\cite{14}, Ch. 12, \cite{14b}, \cite{14c}, \cite{26}, \cite{28},
\cite{49b}, \cite{56d}, \cite{56c}, \cite{69}), we confine
ourselves to
a brief account.  Assuming $q\in C^{\infty}(\bb R)$ or $q$
meromorphic
in $\bb C$ (depending on the particular context in which one is
interested) and hence either $x\in\bb R$ or $x\in\bb C$, consider
the
recursion relation
\begin{equation}\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
\end{equation}
and the associated differential expressions (Lax pair)
\begin{equation}\label{2.2}
L_2 =\dfrac{d^2}{dx^2} + q(x),
\end{equation}
\begin{equation}\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_2^{g-j}, \; g\in\bb N_0
\end{equation}
(here $\bb N_0:=\bb N\cup\{0\}).$  One can show that
\begin{equation}\label{2.4}
\left[ \hat{P}_{2g+1}, L_2 \right]=2\hat{f}'_{g+1}
=\frac12 \hat f_g'''(x)+ 2q(x)\hat f_g'(x)+q'(x)\hat f_g(x)
\end{equation}
([$\cdot,\cdot$] the commutator symbol) and explicitly computes
from
(\ref{2.1}),
\begin{equation}\label{2.5}
\hat{f}_0=1, \; \hat{f}_1=\dfrac{1}{2} q+c_1,\;
\hat{f}_2=\dfrac{1}{8}
q^{''} + \frac38 q^2 +\dfrac{c_1}{2} q+c_2,
\quad \text{etc.},
\end{equation}
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 $f_j$, that is,
$f_j=\hat{f}_j(c_{\ell}\equiv 0)$, the (homogeneous)
stationary KdV
hierarchy is then defined as the sequence of equations
\begin{equation}\label{2.6}
\kdv_g(q)=2f'_{g+1} =0, \; g\in\bb N_0.
\end{equation}
Explicitly, this yields
\begin{equation}\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. }
\end{equation}
The corresponding non-homogeneous version of $\kdv_g(q)=0$ is
then
defined by
\begin{equation}\label{2.8}
\hat{f}'_{g+1}=\sum^{g}_{j=0}c_{g-j}f'_{j+1}=0,
\end{equation}
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
$\deg(q^{(\ell)}) = \ell+2, \; \ell\in\bb N_0$, then the
homogeneous
differential polynomial $f_j$ with respect to $q$ turns
out to have
degree $2j$, that is,
\begin{equation}\label{2.9}
\deg (f_j)=2j, \; j\in\bb N_0.
\end{equation}
Next, introduce the polynomial $\hat{F}_g(E,x)$ in $E\in\bb C$,
\begin{equation}\label{2.10}
\hat{F}_g(E,x) = \sum^{g}_{j=0} \hat{f}_{g-j}(x) E^j.
\end{equation}
Since $\hat f_0(x)=1$,
\begin{equation}\label{2.12}
\hat{R}_{2g+1}(E,x)=(E-q(x))\hat F_g(E,x)^2-\frac12
\hat F_g''(E,x)
\hat F_g(E,x)+\frac14\hat F_g'(E,x)^2
\end{equation}
is a monic polynomial in $E$ of degree $2g+1$.
However, equations (\ref{2.1}) and (\ref{2.8}) imply that
\begin{equation}\label{2.11}
\dfrac{1}{2} \hat{F}^{'''}_g-2(E-q)\hat{F}'_g+q'\hat{F}_g=0
\end{equation}
and this shows that $\hat{R}_{2g+1}(E,x)$ is in fact
independent of $x$.
Hence it can be written as
\begin{equation}\label{2.13}
\hat{R}_{2g+1}(E)=\prod^{2g}_{m=0} (E-\hat{E}_m), \;
\{\hat{E}_m\}^{2g}_{m=0} \subset \bb C.
\end{equation}

By (\ref{2.4}) the non-homogeneous KdV equation (\ref{2.8}) is
equivalent to the commutativity of $L_2$ and $\hat{P}_{2g+1}$.
This shows
that
\begin{equation}\label{2.14}
[\hat{P}_{2g+1}, L_2 ]=0,
\end{equation}
and therefore, if $L_2 \psi=E\psi$, this implies that
$\hat P^2_{2g+1}\psi
=\hat R_{2g+1}(E)\psi$. Thus $[\hat{P}_{2g+1}, L_2 ]=0$
implies
\begin{equation} \label{2.15}
\hat P^2_{2g+1} = \hat R_{2g+1}(L_2) =
\prod^{2g}_{m=0} (L_2 -\hat{E}_m),
\end{equation}
a celebrated theorem by Burchnall and Chaundy \cite{11},
\cite{12} (see,
e.g.,
\cite{14a}, \cite{25a}, \cite{32b}, \cite{55a}, \cite{69}
for more recent
accounts).

In the second part of Section \ref{floquet} we will need
the converse
of the above procedure. It is given by

\begin{lemma} \label{p2.1}
(\cite{32a})
Assume that $\hat F_g(E,x)$, given by (\ref{2.10}) with
$\hat f_0(x)=1$, is twice differentiable with
respect to
$x$, and that
\begin{equation} \label{e2.1}
(E-q(x))\hat F_g(E,x)^2-\frac12 \hat F_g''(E,x)
\hat F_g(E,x)+\frac14\hat F_g'(E,x)^2
\end{equation}
is independent of $x$. Then $q\in C^\infty(\bb R)$. 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
\begin{equation}\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,
\end{equation}
that is, the differential expression $\hat P_{2g+1}$ given
in (\ref{2.3})
commutes with the expression $L_2 =d^2/dx^2+q$.
\end{lemma}


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
\begin{equation}\label{2.16}
F^2=\hat{R}_{2g+1}(E)=\prod^{2g}_{m=0}(E-\hat{E}_m).
\end{equation}

The above formalism leads to the following standard definition.

\begin{definition} \label{d2.1}
Any solution $q$ of one of the stationary KdV
equations (\ref{2.8}) is called an {\bf algebro-geometric
potential} associated with the KdV hierarchy.
\end{definition}

Algebro-geometric 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 and Algebro-Geometric Potentials}
\label{floquet}
\setcounter{equation}{0}
In the first part of this section we discuss Floquet theory in
connection with a complex-valued non-constant periodic potential
$q$.
In the second part we recall a criterion for $q$ to be
an algebro-geometric potential in terms of Floquet solutions.

Suppose
\begin{equation}\label{2.17}
q\in L^1_{\rm loc}(\bb R), \; q(x+\Omega)=q(x), \; x\in \bb R
\end{equation}
for some $\Omega>0$ and let $\el(E)$ be the (two-dimensional)
space of
solutions of $L_2y=Ey$. Then $T(E)$, the restriction of the
operator
defined by $y\mapsto y(\cdot+\Omega)$ to $\el(E)$, commutes
with the
corresponding restriction of $L_2$ and
hence maps $\el(E)$ to itself. The eigenvalues and
eigenfunctions of
$T(E)$ are called Floquet multipliers and Floquet solutions of
$L_2y=Ey$. On $\el(E)$ we introduce the basis $c(E,x,x_0)$ and
$s(E,x,x_0)$ defined by
\begin{equation}\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.
\end{equation}
Using this basis the operator $T(E)$ is represented by the so
called
monodromy matrix
\begin{equation}
\begin{pmatrix} 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{pmatrix}.
\end{equation}
Since $\det (T(E))=1$ the Floquet multipliers $\rho_{\pm}(E)$
are given by
\begin{equation} \label{2.35a}
\rho_{\pm}(E) = \Delta(E) \pm \sqrt{\Delta(E)^2-1},
\end{equation}
where $\Delta(E)$ denotes the Floquet discriminant,
\begin{equation}
\Delta(E)=\frac{1}{2}\operatorname{tr} (T(E))
=[c(E,x_0+\Omega,x_0)+s'(E,x_0+\Omega,x_0)]/2.
\end{equation}

For each $E\in\bb C$ 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
\begin{equation}\label{2.40}
\psi_\pm(E,x,x_0) = c(E,x,x_0)
+\frac{\rho_\pm(E)-c(E,x_0+\Omega,x_0)}
{s(E,x_0+\Omega, x_0)} s(E,x,x_0),
\end{equation}
if $s(E,x_0+\Omega,x_0)\neq0$, or by
\begin{equation} \label{2.40a}
\tilde \psi_\pm(E,x,x_0) =  s(E,x,x_0) +\frac{\rho_\pm(E)-
s'(E,x_0+\Omega,x_0)}{c'(E,x_0+\Omega, x_0)} c(E,x,x_0),
\end{equation}
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.

Associated with the second-order differential expression $L_2
=d^2/dx^2+q(x)$ we consider the densely defined closed linear
operators
$H$, $H_D(x_0)$, $H(\beta,x_0)$, $\beta\in\bb C$, and $H(\theta)$,
$\theta\in\bb C$. While $H$ will be an operator in $L^2(\bb R)$,
the
others will be defined in $L^2(I(x_0))$, where
$I(x_0)=(x_0,x_0+\Omega)$
for some $x_0\in\bb R$. Specifically, the operators are given as
restrictions of the expression $L_2$ to the following domains:
\begin{gather}
\mathcal D(H)=H^{2,2}(\bb R), \label{2.18} \\
\mathcal D(H_D(x_0))=\{g\in H^{2,2}(I(x_0)):
g(x_0)=g(x_0+\Omega)=0\}, \label{2.19} \\
\mathcal D(H(\beta,x_0))=\{g\in H^{2,2}(I(x_0)):
U_1(\beta,g)(x_0)=U_1(\beta,g)(x_0+\Omega)=0\}, \label{2.19a} \\
\mathcal D(H(\theta))=\{g\in H^{2,2}(I(x_0)):
U_2(\theta,g)(x_0)=U_2(\theta,g)'(x_0)=0\}, \label{2.20}
\end{gather}
where $U_1(\beta,y)=y'+\beta y$ and $U_2(\theta,y)
=y(\cdot +\Omega)-e^{i\theta}y(\cdot)$ and where $H^{p,r}(\cdot)$
are the
usual Sobolev spaces with $r$ distributional derivatives in
$L^p(\cdot)$. Next we denote the purely discrete
spectra of $H_D(x_0)$, $H(\beta,x_0)$, and $H(\theta)$ by
$\sigma(H_D(x_0))=\{\mu_n(x_0)\}_{n\in\bb N}$,
$\sigma(H(\beta,x_0))=\{\lambda_n(\beta,x_0)\}_{n\in\bb N_0}$
and
$\sigma(H(\theta))=\{E_n(\theta)\}_{n\in\bb N_0}$,
respectively. While $H(\theta)$ depends on $x_0$ its spectrum
does not. We agree that here, 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 $H_D(x_0)$ are called Dirichlet eigenvalues with
respect to the interval $[x_0,x_0+\Omega]$. The eigenvalues of
$H(\theta)$ are precisely those values $E$ where $T(E)$ 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 $E_n$, $n\in\bb N_0$. They are
repeated
according to their multiplicity and are related to the
(anti)periodic
eigenvalues via
\begin{equation}
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)
\end{equation}
for $n\in\bb N_0$. We also introduce
\begin{equation} \label{ordd}
p(E)=\ord_E(\Delta(\cdot)^2-1),
\end{equation}
the order of $E$ as a zero of $\Delta(\cdot)^2-1$ ($p(E)=0$
if $\Delta(E)^2-1\neq0$).

Similarly, the eigenvalues of $H_D(x_0)$ and $H(\beta,x_0)$ are
the zeros
of the functions $s(\cdot,x_0+\Omega,x_0)$ and
$h(\cdot,\beta,x_0)=(\beta^2s+\beta(s'-c)-c')
(\cdot,x_0+\Omega,x_0)$,
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}
d(E,x_0)&= \ord_E (s(\cdot,x_0+\Omega,x_0)), \label{ords} \\
r(E,\beta,x_0)&= \ord_E (h(\cdot,\beta,x_0)), \label{ordc'}
\end{align}
and remark that $d(E,x_0)$ and $r(E,\beta,x_0)$ are combinations
of movable and immovable parts. Specifically, define $d_i(E)=
\min\{d(E,x_0):x_0\in\bb R\}$, $r_i(E,\beta)=\min\{r(E,\beta,x_0):
x_0\in\bb R\}$ and $d_m(E,x_0)$ and $r_m(E,x_0)$ by
\begin{align}
d(E,x_0)&=d_i(E)+d_m(E,x_0), \\
r(E,\beta,x_0)&=r_i(E,\beta)+r_m(E,\beta,x_0).
\end{align}
If $d_i(E)>0$ then $E$ is a Dirichlet eigenvalue irrespective of
the
value of $x_0$
and we will call $E$ an immovable Dirichlet eigenvalue. Otherwise,
if
$d_i(E)=0$ but $d(E,x_0)>0$ we call $E$ a movable Dirichlet
eigenvalue.
(Note that here we use a notation different from the one in
\cite{32a}, in particular, the multiplicties $d$, $d_i$, and
$d_m$
now refer to Dirichlet
eigenvalues while the multiplicities $p$ refer to periodic or
antiperiodic eigenvalues).


The functions $c(\cdot,x,x_0)$ and $s(\cdot,x,x_0)$ and their
$x$-derivatives are entire functions of order $1/2$ for every
choice of
$x$ and $x_0$. This and their asymptotic behavior as $|E|$ tends
to infinity is obtained via Volterra integral equations. Invoking
Rouch\'e's theorem then yields the following facts:
\begin{enumerate}
\item The zeros $\mu_n(x_0)$ of $s(E,x_0+\Omega,x_0)$ and
the zeros $\lambda_n(\beta,x_0)$ of $h(\cdot,\beta,x_0)$ are
simple
for $n\in\bb N$ sufficiently large.
\item The zeros $E_n$ of $\Delta(E)^2-1$ are at most double
for $n\in\bb N$ large enough.
\item $\mu_n(x_0)$, $\lambda_n(\beta,x_0)$, and $E_n$ can be
arranged (and
will be subsequently) such that they have the following
asymptotic
behavior as $n$ tends to infinity:
\begin{align}
\mu_n(x_0) &= -\frac{n^2\pi^2}{\Omega^2} + O(1), \label{2.32}\\
\lambda_n(\beta,x_0)&=-\frac{n^2\pi^2}{\Omega^2}+O(1),
\label{2.32a} \\
E_{2n-1}, E_{2n} &= -\frac{n^2\pi^2}{\Omega^2}
+ O(1).\label{2.33}
\end{align}
\end{enumerate}

The Hadamard factorization of $s(E, x_0+\Omega, x_0)$ therefore
reads
\begin{equation} \label{2.25}
s(E,x_0+\Omega,x_0) = c_1(x_0) \prod_{n=1}^\infty
\left(1-\frac{E}{\mu_n(x_0)}\right)
=F_D(E,x_0) D(E),
\end{equation}
where all those factors which do not depend on $x_0$ are
collected in $D(E)$. Here we assume that none of the eigenvalues
is
equal to zero; otherwise, obvious modifications have to be used.

For more details on algebraic versus geometric multiplicities of
eigenvalue problems of the type of $H_D(x_0)$ and $H(\theta)$ see,
for instance, \cite{32}.

It was shown by Rofe-Beketov \cite{56} that the spectrum of $H$ is
equal
to the conditional stability set of $L_2$, that is, the set of all
spectral parameters $E$ for which a nontrivial bounded solution of
$L_2 \psi=E\psi$ exists. Hence
\begin{equation}\label{2.22}
\sigma(H)= \bigcup\sb{\theta \in[0,2\pi]} \sigma(T(\theta))
=\bigcup_{n\in\bb N_0} \sigma_n,\; \text{where}\;
\sigma_n= \bigcup_{\theta\in[0,\pi]}E_n(\theta).
\end{equation}
We note that in the general case where $q$ is complex-valued
some of the spectral arcs $\sigma_n$ may cross each other, see,
for instance, \cite{32} and \cite{52} for explicit examples.

The Green's function $G(E,x,x')$ of $H$, that is, the integral
kernel
of the resolvent of $H$,
\begin{equation}\label{2.36}
G(E,x,x') = (H-E)^{-1}(x,x'), \; E\in\bb C\backslash
\sigma(H), \; x,x'\in\bb R,
\end{equation}
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}\hspace{-4mm}.
\end{align}
Here $f_{\pm}(E,\cdot)$ solve $L_2 f=Ef$ and are chosen such
that
\begin{equation}\label{2.38}
f_{\pm}(E,\cdot)\in L^2((R,\pm \infty)), \;
E\in\bb C\backslash \sigma(H), \, R\in\bb R,
\end{equation}
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 differentiable and satisfies the nonlinear second-order
differential equation (see, e.g., \cite{25b}, \cite{47})
\begin{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
\end{equation}
(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
\begin{equation}\label{2.42}
W(f_{-}(E,\cdot), f_{+}(E,\cdot))
=\frac{e^{i\theta}-e^{-i\theta}}{ s(E,x_0+\Omega, x_0)}
=-2\frac{[\Delta(E)^2-1]^{1/2}}{ s(E,x_0+\Omega, x_0)}.
\end{equation}
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})
\begin{equation}\label{2.43}
G(E,x_0,x_0)
= -\frac{s(E,x_0+\Omega, x_0)}{2[\Delta(E)^2-1]^{1/2}}.
\end{equation}

\begin{theorem} \label{t2.1b} (\cite{32a}, \cite{w3})
If $q$ is a locally integrable periodic function on $\bb R$ then
$p(E)-2d_i(E)\geq 0$ for all $E\in\bb C$.
\end{theorem}
\begin{proof}
Equations \eqref{2.25}, \eqref{2.68}, and \eqref{2.43} show that
\begin{equation} \label{371}
4(E-q(x))F_D(E,x)^2-2F_D(E,x)F_D''(E,x)+F_D'(E,x)^2
=\frac{4(\Delta(E)^2-1)}{D(E)^2}.
\end{equation}
Since the left hand side is entire the claim follows immediately
from
the definitions of the numbers $p(E)$ and $d_i(E)$.
\end{proof}

A somewhat bigger effort allows one to prove also
\begin{theorem} \label{t2.1a} (\cite{32a}, \cite{w3})
If $q$ is a locally integrable periodic function on $\bb R$ then
$d_i(E)=r_i(E,\beta)$ unless $q$ is a constant and $E=q+\beta^2$.
Moreover, if $d_i(E)>0$ then there exist two linearly independent
Floquet solutions of $L_2y=Ey$. Finally, $p(E)-2d_i(E)>0$ if and
only if
there exists an $x_0\in\bb R$ such that $W(z,x_0)$, the Wronskian
of the
Floquet solutions $\psi_\pm$ given by \eqref{2.40}, tends to zero
as
$z$ tends to $E$.
\end{theorem}
Hence, if there are not two linearly independent Floquet solutions
for
$L_2y=Ey$ then $\rho^2=1$ and $p(E)>0$ but $d_i(E)=0$ and thus
$p(E)-2d_i(E)>0$ at all such points.

Nowhere in this section did we use thus far that $q$ is an
algebro-geometric potential. Next we give necessary and sufficient
conditions for this in terms of properties of multiplicities of
eigenvalues of (anti)periodic boundary value problems on one hand
and
the Dirichlet problem on the other hand. We begin with
\begin{definition}
The number $\operatorname{def}(L_2)=\sum_{E\in\bb C}
(p(E)-2d_i(E))$ is
called the Floquet defect. The number $\sum_{E\in\bb C}
d_m(E,x_0)$ will be called the number of movable Dirichlet
eigenvalues; similarly, $\sum_{E\in\bb C}
r_m(E,\beta,x_0)$ denotes the number of movable eigenvalues of
$H(\beta,x_0)$.
\end{definition}
Note that by Theorem \ref{t2.1b}, $\operatorname{def}(L_2)$ is
either infinite or else a nonnegative integer. If it is finite then
$\operatorname{def}(L_2)=\deg(4(\Delta^2-1)/D^2)$. Both,
$\operatorname{def}(L_2)$ and the number of
movable Dirichlet eigenvalues are in general infinite.

\begin{theorem}\label{t2.2a} (\cite{32a}, \cite{w3})
Assume that $q$ is a locally integrable, periodic function of
period
$\Omega>0$ on $\bb R$. Then the following statements are
equivalent:\\
1. The Floquet defect $\operatorname{def}(L_2)$ equals
$2g+1$.
\\
2. The number of movable Dirichlet eigenvalues equals $g$.\\
3. There exists a monic differential expression $\hat P_{2g+1}$
of order $2g+1$ which commutes with $L_2$ but none of smaller odd
order,
i.e., $q$ is an algebro-geometric potential.
\end{theorem}
In particular, $\operatorname{def}(L_2)$ is either odd or
infinite.

\begin{proof}[Sketch of proof]
If $\operatorname{def}(L_2)$ is finite, asymptotic considerations
show
that only finitely many Dirichlet eigenvalues can be movable.
Hence
$F_D(\cdot,x_0)$ is a polynomial, say of degree $\hat g$. By
equation
\eqref{371} $4(\Delta^2-1)/D^2$ is a polynomial of degree
$2\hat g+1$.
Hence $\hat g=g$. This shows the equivalence of the first two
statements.

Next one shows that the leading coefficent of $F_D(\cdot,x_0)$ is
independent of $x_0$. The third statement follows then from the
second
using Lemma \ref{p2.1}. To prove that the third  statement
implies the
other two one has to show that the zeros of the function $\hat
F_g(\cdot,x_0)$ in \eqref{2.10} are precisely the movable
Dirichlet
eigenvalues. This follows from applying $\hat P_{2g+1}$ as given
in
\eqref{2.3} succesively to the generalized Dirichlet
eigenfunctions.
\end{proof}

\begin{theorem}\label{t2.2b} (\cite{32a}, \cite{w3})
Assume that $q$ is a non-constant, locally integrable, periodic
function of period $\Omega>0$ on $\bb R$ and that any (and hence
all) of
the three statements in Theorem \ref{t2.2a} is satisfied.
Then the following statements hold.\\
1. The number of movable eigenvalues of $H(\beta,x_0)$ equals
$g+1$,
i.e.,
\begin{equation}
\sum_{E\in\bb C} r_m(E,\beta,x_0)=g+1.
\end{equation}
2. $q\in C^\infty(\bb R)$.\\
3. The differential expression $\hat P_{2g+1}$ satisfies the
Burchnall-Chaundy relation
\begin{equation}\label{4.13}
\hat P^2_{2g+1}=\prod_{z\in\bb C}(L-z)^{p(z)-2d_i(z)}.
\end{equation}
4. The diagonal Green's function $G(\cdot,x,x)$ of $H$ is
continuous
on $\bb C\backslash \{z: p(z)-2d_i(z)>0\}$ and is of
the type
\begin{equation}\label{4.6}
G(E,x,x) = -\frac12\frac{\prod_{z\in\bb C}(E-z)^{d_m(z,x)}}
{\prod_{z\in\bb C}(E-z)^{p(z)-2d_i(z)}}.
\end{equation}
5. The spectrum of $H$ consists of finitely many bounded spectral
arcs $\tilde{\sigma}_n$, $1\le n\le \tilde{g}$ for some
$\tilde{g}\leq
g$ and one unbounded (semi-infinite) arc
$\tilde{\sigma}_{\infty}$ which
tends to $-\infty+<q>$, with
$<q>=\Omega^{-1} \int^{x_0+\Omega}_{x_0}
q(x) dx$, that is,
\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{theorem}

Note that the set $B$ of
values of $E$ where $p(E)-2d_i(E)>0$ contains $B_1$, the set of
all those
points where only less than two linearly independent Floquet
solutions
exist. For $B\backslash B_1$ to be nonempty, it is necessary that
$p(z)\geq3$ for some (anti)periodic eigenvalue $z$. While it seems
difficult to construct an explicit example where
$B\backslash B_1\neq
\emptyset$, the very existence of this phenomenon has first been
noted
in \cite{32a}. References
\cite{24}, \cite{25}, \cite{33}, \cite{50}, \cite{51} treat
potentials
with $p(E)\leq 2$ and references \cite{9}, \cite{10} require that
algebraic and geometric multiplicities of all (anti)periodic
eigenvalues
coincide and hence also that $p(E) \leq 2$. Generically one has
$p(E)-2d_i(E)=1$ if this is positive at all and $B=B_1$
(cf. \cite{60}).

\begin{remark}[Singularity structure of the Green's function]
As Theorem \ref{t2.2b} shows, it is precisely the multiplicity
$p-2d_i$
of the branch and singular points in the Burchnall-Chaundy
polynomial
(\ref{4.13}) which determines the singularity structure of the
diagonal
Green's function $G(E,x,x)$ of $H$. Moreover, since (see, e.g.,
\cite{47})
\begin{equation}\label{2.79}
G(E,x,x')=[G(E,x,x)G(E,x',x')]^{1/2} \exp [-\dfrac{1}{2}
\int^{\max (x,x')}_{\min(x,x')} G(E,s,s)^{-1}ds],
\end{equation}
this observation extends to the off-diagonal Green's function
$G(E,x,x')$ of $H$ as well.
\end{remark}

\begin{remark}[Inverse square singularities]
The case of the Lam\'e-Ince potential, where $q$ has
singularities of the form $-g(g+1)/x^2$, indicates the necessity to
consider also potentials with such singularities. This is
possible by
modifying the usual approach via Volterra integral equations which
are used to obtain the asymptotic properties
(\ref{2.32})--(\ref{2.33})
of the
corresponding eigenvalue
distributions. One obtains essentially the same results as in the
present review, the only
difference being that the conditional stability set cannot be
interpreted as the spectrum of an operator in $L^2(\bb R)$. This
approach has been worked out in detail in \cite{w2} and \cite{w3}.
\end{remark}

\begin{remark}[Finite-band potentials]
For real-valued potentials Novikov \cite{48} and Dubrovin \cite{15}
showed that $q$ is an algebro-geometric potential if and only
if the
spectrum of the operator $H$ consists of only finitely many bands.
This
is no longer true for complex-valued potentials. In fact, for
$q=\e^{ix}$
one infers $\sigma(H)=(-\infty,0]$ but every Dirichlet eigenvalue
is movable
(see \cite{w3}).
\end{remark}





\section{A Characterization of Elliptic Solutions of the
KdV Hierarchy}
\label{picard}
\setcounter{equation}{0}
In this section we discuss the principal result in
\cite{32a}, an
explicit
characterization of all elliptic algebro-geometric
solutions of the
KdV hierarchy.  One
of the
two key ingredients
in our main Theorem \ref{t3.7} (the other being Theorem
\ref{t2.2a})
is a systematic use of a powerful theorem of Picard (see
Theorem \ref{t3.1} below) concerning the existence of
solutions which
are elliptic functions of the second kind of ordinary
differential
equations with elliptic coefficients.

We start with Picard's theorem.

\begin{theorem}\label{t3.1}
(\cite{63}, \cite{64}, \cite{65}, see, e.g., \cite{3},
p. 182--187,
\cite{37}, p. 375--376)  Let $q_m$, $1\leq m \leq n$ be
elliptic functions with a common period lattice spanned
by the
fundamental periods $2\omega_1$ and
$2\omega_3$. Consider the differential equation
\begin{equation}\label{3.1}
\sum^n_{m=0} q_m(z)\psi^{(m)}(z)=0, \; q_n(z)=1, \; z\in\bb C
\end{equation}
and assume that (\ref{3.1}) has a meromorphic fundamental
system of
solutions. Then there exists at least one solution $\psi_0$
which is
elliptic of the second kind, that is, $\psi_0$ is
meromorphic and
\begin{equation}\label{3.2}
\psi_0(z+2\omega_j)=\rho_j\psi_0(z), \; j=1,3
\end{equation}
for some constants $\rho_1, \rho_3\in\bb C$. If in addition,
the
characteristic equation corresponding to the substitution
$z\to
z+2\omega_1$ or $z\to z+2\omega_3$ (see \cite{37}, p. 358,
376) has
distinct roots then there exists a fundamental system of
solutions of
(\ref{3.1}) which are elliptic functions of the second kind.
\end{theorem}

The characteristic equation associated with the substitution
$z\to z+2\omega_j$ alluded to in Theorem \ref{t3.1} is given by
\begin{equation}\label{3.3}
\det [A-\rho I]=0,
\end{equation}
where
\begin{equation}\label{3.4}
\phi_{\ell}(z+2\omega_j) = \sum^n_{m=1} a_{\ell,m}\phi_m(z), \;
A=(a_{\ell,m})_{1\le \ell, m \le n}
\end{equation}
and $\phi_1,....,\phi_n$ is any fundamental system of solutions of
(\ref{3.1}).

What we call Picard's theorem following the usual convention in
\cite{3}, p. 182--185, \cite{13}, p. 338--343, \cite{34},
p. 536--539,
\cite{41}, p. 181--189, appears, however, to have a longer
history.  In
fact, Picard's investigations \cite{53}, \cite{54}, \cite{55}
were
inspired by earlier work of Hermite in the special case of
Lam\'e's
equation \cite{35}, p. 118--122, 266--418, 475--478 (see also
\cite{6a},
Sect. 3.6.4 and \cite{67}, p. 570--576).  Further contributions
were
made by Mittag-Leffler \cite{46}, and Floquet \cite{21},
\cite{22},
\cite{23}. Detailed accounts on Picard's differential equation
can be
found in \cite{34}, p. 532--574, \cite{41}, p. 198--288.

In this context it seems appropriate to recall the well-known
fact (see,
e.g., \cite{3}, p. 185--186) that $\psi_0$ is elliptic of the
second
kind if and only if it is of the form
\begin{equation}\label{3.5}
\psi_0(z)=Ce^{\lambda z}
\prod^m_{j=1}[\sigma(z-a_j)/\sigma(z-b_j)]
\end{equation}
for suitable $m\in\bb N$ and $C,\lambda,a_j,b_j
\in\bb C$, $1\le j\le
m$. Here $\sigma(z)$ is the Weierstrass sigma function associated
with
the period lattice $\Lambda$ spanned by $2\omega_1, 2\omega_3$
(see
\cite{1}, Ch. 18).

Picard's Theorem \ref{t3.1}, restricted to the second-order case
\begin{equation}\label{3.6}
\psi''(z) + q(z)\psi(z)=E\psi(z),
\end{equation}
motivates the following definition.

\begin{definition} Let $q$ be an elliptic function.  Then $q$
is called a
{\bf Picard potential} if and only if the differential
equation
(\ref{3.6}) has a meromorphic fundamental system of
solutions (with
respect to $z$) for each value of the spectral parameter
$E\in\bb C$.
\end{definition}

For completeness we recall the following result.

\begin{theorem}(\cite{31}) \label{p3.3}
(i)  Any non-constant Picard potential $q$ has a representation
of the
form
\begin{equation} \label{3.7}
q(z)=C-\sum^m_{j=1} s_j(s_j+1) \wp(z-b_j)
\end{equation}
for suitable $m, s_j\in\bb N$ and $C, b_j \in
\bb C$, $1\le j\le m$,
where the $b_j$ are pairwise distinct $\operatorname{mod}
(\Lambda)$ and
$\wp(z)$ denotes the Weierstrass $\wp$-function associated with
the
period lattice $\Lambda$ (\cite{1}, Ch. 18).\\
(ii) Let $q(z)$ be given as in (\ref{3.7}). If
$\psi^{''}+q\psi=E\psi$
has a meromorphic fundamental system of solutions for a number of
distinct values of $E$ which exceeds $\max\{s_1,\ldots,s_m\},$
then
$q$ is a Picard potential.
\end{theorem}

We emphasize that while any Picard potential is necessarily
of the form
(\ref{3.7}), a potential $q$ of the type (\ref{3.7}) is a
Picard
potential only
 if
the constants $b_j$ satisfy a series of additional intricate
constraints,
see, for instance, Section 3.2 in \cite{31}.

The following result indicates the connection between
Picard potentials
and elliptic algebro-geometric potentials.

\begin{theorem}\label{t3.4}
(Its and Matveev \cite{39}, Krichever \cite{42}, \cite{43},
Segal and Wilson \cite{57})  Every elliptic algebro-geometric
potential $q$ is a Picard potential.
\end{theorem}

\begin{proof}[Sketch of Proof]
For nonsingular curves
$K_g:F^2=\prod^{2g}_{j=0}(E-\hat{E}_j)$ associated with
$q$ (see (\ref{2.16})), where
$\hat{E}_{\ell}\ne \hat{E}_{\ell'}$ for $\ell\ne \ell'$,
Theorem \ref{t3.4} is obvious from the
standard
representation of the Baker-Akhiezer function in terms of the
Riemann theta function of $K_g$ (\cite{16}, \cite{39},
\cite{42},
\cite{43}). For singular curves $K_g$ the result follows from the
$\tau$-function representation of the Floquet solutions
$\psi_{\pm}(E,x)$ associated with $q$
\begin{equation}\label{3.8}
\psi_{\pm}(E,x)=e^{\pm k(E)x} \tau_{\pm}(E,x) /\tau(x),
\end{equation}
where
\begin{equation}\label{3.9}
q(x)=C+2\{\ln[\tau(x)]\}''
\end{equation}
and from the fact that $\tau(x)$ and $\tau_{\pm}(E,x)$ are
entire with
respect to $x$ (cf. \cite{57}).
\end{proof}

Naturally, one is tempted to conjecture that the converse of
Theorem
\ref{t3.4} is true as well.  The rest of this section will
explain our
proof of this conjecture in \cite{32a}.

We start with a bit of notation. Let $q(z)$ be an elliptic
function
with fundamental periods $2\omega_1, 2\omega_3$ and assume,
without
loss of generality, that $\re(\omega_1)>0$,
$\re(\omega_3) \ge 0$,
$\im(\omega_3/\omega_1)>0$. The fundamental period
parallelogram then consists 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\phi} = \dfrac{\omega_3}{\omega_1} \left|
\dfrac{\omega_1}{\omega_3} \right|, \; \phi \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 $z_0\in\bb C$ 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 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 $\tilde{q}\in C^0(\bb R)$ 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 now given by
\begin{equation}\label{3.19}
\tilde{\Delta}(\lambda)=[\tilde{c}(\lambda,\tilde{\Omega})
+ \tilde{s}
(\lambda,\Omega)]/2
\end{equation}
and Rouch\'e's theorem then yields
\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{lemma}\label{p3.5}
Let $\tilde{\lambda}_n$ be a periodic or antiperiodic
eigenvalue of
$\tilde{q}$. Then there exists an $m\in\bb Z$ 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 $n\in\bb N_0$.  In
particular, all
periodic and antiperiodic eigenvalues
$\tilde{\lambda}_n$, $n\in\bb N_0$
of $\tilde{q}$ are contained in a half-strip $\tilde{S}$
given by
\begin{equation}\label{3.22}
\tilde{S}=\{\lambda\in\bb C | \left|\im(\lambda)\right|\le
\tilde{C}, \;
\re(\lambda)\le \tilde{M}\}
\end{equation}
for some $\tilde{M}\in\bb R.$
\end{lemma}

In order to apply Lemma \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 Lemma \ref{p3.5} immediately yields the
following result.

\begin{lemma}\label{p3.6}
Let $j=1$ or $3$. Then all $2\omega_j$-(anti)periodic
eigenvalues
$E_{j,n}$, $n\in\bb N_0$ associated with $q$ lie in the
half-strip
$S_j$ given by
\begin{equation}\label{3.33}
S_j=\{E\in\bb C: |\im(t^2_jE)|\le C_j, \;
\re (t^2_jE) \le M_j\}
\end{equation}
for suitable constants $C_j>0, M_j\in\bb R.$  The angle
between the axes
of the strips $S_1$ and $S_3$ equals $2\phi \in (0,2\pi).$
\end{lemma}

Lemmas \ref{p3.5} and \ref{p3.6} apply to any elliptic
potential
whether or not they are algebro-geometric. In our final step
we shall now
invoke Picard's Theorem \ref{t3.1} to obtain our
characterization of
elliptic algebro-geometric potentials.

\begin{theorem}\label{t3.7}
$q$ is an elliptic algebro-geometric potential if and only if
$q$ is a Picard
potential (i.e., if and only if for each $E\in\bb C$ every
solution of
$\psi''(z)+ q(z)\psi(z)=E\psi(z)$ is meromorphic with
respect to $z$).
\end{theorem}

\begin{proof}
By Theorem \ref{t3.4} it remains to prove that a Picard
potential is
algebro-geometric. 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})),
that is,
\begin{equation}\label{3.34}
(\bb C\backslash \overline{D(0,R)}) \cap (S_1\cap S_3) =
\emptyset.
\end{equation}
Let $\rho_{j,\pm}(\lambda)$ be the Floquet multipliers
of $q_j(x),$
that is, 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 $E\in\bb C\backslash
\overline{D(0,R)}$, 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 $E\in
\bb C\backslash
\overline{D(0,R)}$ 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{t2.2a} then proves
that both $q_1$
and $q_3$ are algebro-geometric potentials.

By (\ref{2.8}) (in slight abuse of notation)
\begin{equation}
\sum^{g}_{k=0}c_{g-k}\frac{df_{k+1}(q_1(x))}{ dx}=0,
\end{equation}
where $g\in\bb N_0$, $f_{k+1}$, $k=0,...,g$, 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} \frac{df_{k+1}(q(z))}{ dz}=0,
\end{equation}
that is, $q$ is an algebro-geometric 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{proof}

We add a series of remarks further illustrating the
significance of
Theorem \ref{t3.7}.

\begin{remark} [Complementing Picard's theorem]
First we note that Theorem \ref{t3.7} extends and complements
Picard's
Theorem \ref{t3.1} in the sense that it determines the elliptic
functions which satisfy the hypothesis of the theorem precisely as
(elliptic) algebro-geometric solutions of the stationary KdV
hierarchy.
\end{remark}

\begin{remark}[Characterizing elliptic algebro-geometric
potentials]
While an explicit proof of the algebro-geometric property of
$q$ 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 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 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}). Theorem \ref{t3.7} appears to be the only
effective tool to identify general elliptic algebro-geometric
solutions
of the KdV hierarchy.
\end{remark}

\begin{remark} [Reduction of Abelian integrals]
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, for instance, \cite{5},
\cite{6},
\cite{6a}, Ch. 7, \cite{7}, \cite{8}, \cite{13d}, \cite{18},
\cite{19},
\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 (\ref{2.15}), associated with Picard potentials,
reduce to
one-dimensional theta functions.
\end{remark}

\begin{remark} [Computing genus and branch points] \label{r3.12}
Even though Theorem \ref{t3.7} characterizes all elliptic
algebro-geometric 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 $g$ previously had 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 reflection symmetric elliptic
algebro-geometric
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
reflection
symmetric elliptic
algebro-geometric potential to the solution of constraint
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\wp(z-\omega_j)-12\wp(z-\omega_k),
\displaybreak[0] \\
\hat{q}_4(z)=-20\wp(z-\omega_j)-6\wp(z-\omega_k)
-6\wp(z-\omega_{\ell}), \\
q_5(z)=-30\wp(z-\omega_j)-2\wp(z-\omega_k), \\
\hat{q}_5(z)=
-12\wp(z-\omega_j)-12\wp(a-\omega_k)-6\wp
(z-\omega_{\ell})-2\wp(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 -- $2\wp(x-b_n)$ (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{remark}





\section{Picard's Theorem for First-Order Systems}
\label{systems}
\setcounter{equation}{0}
Having characterized all elliptic algebro-geometric solutions of
the KdV
hierarchy which are related to the second-order expression $L_2 =
d^2/dz^2+q$, it is natural to try to extend Theorem \ref{t3.7} to
$n$-th order expressions $L_n$ (connected with the
Gel'fand-Dickey hierarchy). Actually, a more general extension to
integrable systems related to general first-order $n\times n$
matrix-valued differential expressions seems very desirable in
order to
include AKNS systems (see, e.g., \cite{27b}) and the matrix
hierarchies
of integrable equations described in detail, for instance, in
\cite{14}, Sects.
9, 13--16, \cite{15a}, \cite{16a}. Picard's Theorem \ref{t3.1}
generalizes in a straightforward manner to first-order systems,
that is, pairwise
distinct Floquet multipliers in one of the fundamental directions
and a
meromorphic fundamental system of solutions guarantee the
existence of a
fundamental system of solutions which are elliptic of the second
kind
(see (\ref{5.5})). Moreover, it is possible to obtain the explicit
Floquet-type structure of these solutions (cf. Theorem \ref{t5.3}).

Denote by $M(n)$ the set of $n\times n$ matrices with entries
in $\bb
C$ and consider the linear homogeneous system
\begin{equation}\label{5.1}
  \Psi'(z) = Q(z)\Psi(z),   \quad x\in\bb C,
\end{equation}
where $Q(z)\in M(n)$ and where the entries of $Q(z)$ are elliptic
functions with a common period lattice $\Lambda$ spanned by
$2\omega_1$ and $2\omega_3$ which satisfy the same conditions
as before.

Assuming without loss of generality that no pole of $Q(z)$ lies on
the line containing the segments $[0,2\omega_j]$, Floquet theory
with
respect to these directions yields the existence of fundamental
matrices
$\Phi_j(z)$ of the type
\begin{equation}\label{5.3}
\Phi_{j}(z) = P_{j}(z)\exp\lp z K_j\rp,
\end{equation}
where $P_{j}(z)$ is a periodic matrix with period $2\omega_j$
and $K_j$ is
a constant matrix. The monodromy matrix is given by
$M_j=\exp(2\omega_j K_j)$. We want to establish the existence
of a
Floquet representation {\it simultaneously} for both directions
$2\omega_1$ and $2\omega_3$.  More precisely, we intend to find
solutions $\underline\phi$ of $\Psi'(z) = Q(z)\Psi(z)$ satisfying
\begin{equation}\label{5.5}
\underline\phi(z+2\omega_j) = \rho_j\underline\phi, \quad  j=1,3,
\end{equation}
where $\rho_j\in\cmz$. Solutions $\underline\phi(z)$ of
(\ref{5.1})
satisfying (\ref{5.5}) are again called elliptic of the second
kind.

Even though Picard did mention certain extensions of his result
to
first-order systems (see, e.g., \cite{gray}, p. 248--249),
apparently he
did not seek a Floquet representation for systems in the elliptic
case. The first to study such a representation seems to have been
Fedoryuk who proved the following result.
\begin{theorem}\label{t5.1}(\cite{fedor})
Let $Q(z)$ be an $n\times n$ matrix whose entries are elliptic
functions
with fundamental periods $2\omega_1$ and $2\omega_3$ and suppose
that
(\ref{5.1}) has a single-valued fundamental matrix of solutions.
Then (\ref{5.1}) admits a fundamental matrix $\Phi(z)$ of the
type
\begin{equation}\label{5.6}
  \Phi(z) = D(z)\exp\lp zS+\zeta(z)T\rp,
  \quad z\in\bb C,
\end{equation}
where $S,T\in M(n)$, $D(z)$ is invertible and doubly periodic and
\begin{equation}\label{5.8}
S = \frac{1}{\pi i} \lb2\omega_3\zeta\lp\omega_1\rp K_3
     - 2\omega_1\zeta\lp\omega_3\rp K_1\rb,
  \quad
  T = -\frac{2\omega_1 \omega_3}{\pi i}\lp K_3- K_1\rp.
\end{equation}
Moreover, $K_1$ and  $K_3$, and hence $S$ and $T$ commute.
\end{theorem}

Fedoryuk's representation (\ref{5.6}) has the peculiar feature
that it
seems to stress an apparent essential singularity structure of
solutions
at $z=0$. Indeed, since $\zeta(z)$ has a first-order pole at
$z=0$, the
term $\exp\lp \zeta(z)T\rp$ in (\ref{5.6}) exhibits an essential
singularity unless $T$ is nilpotent.  Hence, the doubly periodic
matrix
$D(z)$, in general, will cancel the essential singularity of
$\exp\lp
\zeta(z)T\rp$ and therefore cannot be meromorphic and hence not
elliptic. Thus Fedoryuk's result cannot be considered the natural
extension of Picard's Theorem \ref{t3.1}.

In the remainder of this section we shall focus on an alternative
to
Theorem \ref{t5.1} and describe a generalization of Picard's
theorem
in the context of
first-order systems with elliptic coefficients.

\begin{theorem}\label{t5.3} (\cite{27a})
Let $Q(z)$ be an elliptic $n\times n$ matrix with fundamental
periods $2\omega_1$ and $2\omega_3$ and suppose that (\ref{5.1})
has a
meromorphic fundamental matrix $\Psi(z)$ of solutions.
Then (\ref{5.1}) admits a fundamental matrix of the type
\begin{align}\label{5.16}
\Phi(z)=&E(z)\sigma(z)^{-1} \sigma\lp zI_n
-\frac{2\omega_1 \omega_3}{\pi i}\lp  K_3- K_1\rp\rp\\
&\times \exp\lbr\frac{z}{\pi i}
\lb2\omega_3\zeta\lp\omega_1\rp K_3-
2\omega_1\zeta\lp\omega_3\rp K_1\rb\rbr, \quad z\in\bb C, \notag
\end{align}
where $E(z)$ is an elliptic matrix with periods $2\omega_j$ and
$K_1$,
$K_3$ (and hence $M_j=\exp\lp2\omega_j K_j\rp$, $ j=1,3$) are
commuting matrices. Moreover, linearly independent solutions
$\underline\phi_m(z)\in\bb C^n$, $ 1\leq m\leq n$ of
(\ref{5.1}), that is, column vectors of
(\ref{5.16}), are of the type
\begin{equation}\label{5.19}
\underline\phi_m(z)=\sum_{k_{1}=0}^{n_{1}}\sum_{k_{2}=0}^{n_{2}}
\underline{e}_{m,k_1,k_2}(z)\exp\lp z \mu_{m,k_1,k_2}\rp
z^{k_{1}}\zeta(z)^{k_{2}},
\end{equation}
where the vectors $\underline{e}_{m,k_1,k_2}(z)$ are elliptic,
the numbers
$\mu_{m,k_1,k_2}$ denote the (not necessarily distinct)
eigenvalues of
\begin{equation}
\lp 1/\pi i\rp\lb2\omega_3\zeta\lp2\omega_1/2\rp K_3
 -2\omega_1\zeta\lp2\omega_3/2\rp K_1\rb,
\end{equation}
and, most notably, the upper limits of the sums in (\ref{5.19})
satisfy
\begin{equation}\label{5.20}
n_{1}+n_{2}\leq n-1.
\end{equation}
In particular, there exists at least one solution
$\underline\phi_{m_0}(z)$ of (\ref{5.1}) which is elliptic of
the second
kind, that is, $\underline\phi_{m_0}(z)$ is meromorphic on
$\bb C$ and
\begin{equation}\label{5.21}
\underline\phi_{m_0}(z+2\omega_j)=\rho_{m_{0},j}
\underline\phi_{m_0}(z),
\quad  j=1,3, \quad z\in\bb C
\end{equation}
for some $\rho_{m_{0},j}
=\exp\lp2\omega_j\mu_{m_{0},0,0}\rp\in\cmz$,
$ j=1,3$. In addition, if all eigenvalues of $M_1$ or $M_3$ are
distinct, then there exists a fundamental system of solutions
$\{\underline\phi_m(z)\}_{ 1\leq m\leq n}$ of (\ref{5.1})
with all $\underline\phi_{m}(z)$ elliptic of the second kind.
\end{theorem}

For the proof one considers a meromorphic fundamental matrix
$\tilde\Psi(z)$ of \ref{5.1} and defines
\begin{align}\label{5.23}
E(z)=&\tilde\Psi(z)\exp\lp -\frac{z}{\pi i}
\lb2\omega_3\zeta\lp\omega_1\rp K_3
-2\omega_1\zeta\lp\omega_3\rp K_1 \rb\rp \\
&\times \sigma\lp z I_n
-\frac{2\omega_1 \omega_3}{\pi i}\lp K_3- K_1\rp\rp^{-1}\sigma(z).
\notag
\end{align}
By hypothesis, $E(z)$ is meromorphic and, applying the
addition theorem
\begin{equation}\label{5.13}
\sigma(z+2\omega_j)
=-\sigma(z)\exp\left\{2\zeta\lp\omega_j\rp\lb z
+\lp\omega_j\rp\rb\right\},  \quad  1\leq j\leq 3
\end{equation}
(more precisely, a matrix-valued generalization thereof), one
verifies that
\begin{equation}\label{5.24}
E(z+2\omega_j) =  E(z), \quad  j=1,3,
\end{equation}
that is, $E(z)$ is elliptic. The remaining assertions in Theorem
\ref{t5.3} follow by transforming $(K_3 -K_1)$ and, say, $K_1$
(separately) into their Jordan normal forms (cf. \cite{27a}).

\begin{remark}\label{r5.4}
If $M_1$ or $M_3$ has distinct eigenvalues, one can show that
equation
(\ref{5.1}) has a fundamental system of solutions
$\underline\phi_m(z)$ of the form,
\begin{align}\label{5.25}
\underline\phi_m(z)=&\underline{e}_m(z)\sigma(z)^{-1}
\sigma\lp z-\frac{2\omega_1\omega_3}{\pi i}k_{3,1,m}\rp \\
&\times\exp\lp\frac{2\omega_3 z}{\pi i}\zeta\lp\omega_1\rp
k_{3,1,m}\rp\exp\lp z k_{1,m}\rp,\quad  1\leq m\leq n,
\quad z\in\bb C,
\notag
\end{align}
where $\underline{e}_m(z)$ are elliptic with period
lattice $\Lambda$ and $\lbr k_{3,1,m}\rbr_{ 1\leq m\leq n}$ and
$\lbr k_{1,m}\rbr_{ 1\leq m\leq n}$ are the eigenvalues of
$( K_3- K_1)$
and $K_1$, respectively.
\end{remark}

\begin{remark}\label{r5.5}
In the special scalar case (\ref{3.1}), the bound (\ref{5.20}) was
proved
in \cite{37}, p. 377--378 for $n=2$ and stated (without proof) for
$n\geq 3$.
\end{remark}





\section{The Higher-Order Scalar Case}\label{nthorder}
\setcounter{equation}{0}
In our final section we consider a differential expression
$L_n$ of
the form
\begin{equation}\label{6.1}
L_n y=y^{(n)}+q_{n-2}y^{(n-2)}+...+q_0y,
\end{equation}
where initially the coefficients $q_0,...,q_{n-2}$ are continuous
complex-valued functions of a real variable periodic with period
$\Omega>0$. Again we denote the ($n$-dimensional) vector space of
solutions of the differential equation $L_ny=Ey$ by $\el(E)$ and
the
operator which shifts the argument of a function in $\el(E)$ by
a period
$\Omega$ by $T(E)$. As before $T(E)$ and $L_n$ commute which
implies
that $T(E)$ maps $\el(E)$ to itself. Floquet multipliers, that is,
eigenvalues of $T(E)$ are given as zeros of the polynomial
\begin{equation}\label{6.2}
\F(E,\rho)=(-1)^n \rho^n + (-1)^{n-1} a_{1}(E) \rho^{n-1}+ ...
-a_{n-1}(E)\rho+1=0,
\end{equation}
where the functions $a_1,...,a_{n-1}$ are entire. This is obvious
after choosing the basis $\phi_1(E,x),...,\phi_n(E,x)$ of $\el(E)$
satisfying the
initial conditions $\phi_j^{k-1}(E,x_0)=\delta_{j,k}$.

Note that $\F(E,\cdot)$ has $n$ distinct zeros unless the
discriminant
of $\F(E,\cdot)$, which is an entire function of $E$, is equal to
zero. Thus, this happens at most at countably many points.
Denote by
$m_g(E,\rho)$ and $m_f(E,\rho)$ the geometric and algebraic
multiplicity, respectively, of the eigenvalue $\rho$ of $T(E)$.
Then the
number $m_f(E,\rho)-m_g(E,\rho)\in\{0,1,...,n-1\}$ counts
the ``missing'' Floquet solutions of $L_n y=Ey$ with multiplier
$\rho$.
We will be interested in the case where this number is positive
only for
finitely many points $E$.

For $\theta\in\bb C$, consider the operator $H(\theta)$ associated
with the differential expression $L_n$ in $L^2([x_0,x_0+\Omega])$
with
domain
\begin{equation}\label{6.3}
D(H(\theta))=\{g\in H^{2,n}([x_0,x_0 +\Omega]):
g^{(k)}(x_0 +\Omega)=e^{i\theta} g^{(k)}(x_0), \, k=0,...,n-1\}.
\end{equation}

$H(\theta)$ has discrete spectrum. In fact, its eigenvalues, which
will be called Floquet eigenvalues, are given as the zeros of
$\F(\cdot,\rho)$. Moreover, the algebraic multiplicity
$m_a(E,\rho)$ of $E$ as an eigenvalue of $H(\theta)$ is given as
the order
of $E$ as a zero of $\F(\cdot,\rho)$ (see, e.g., \cite{32}).

With any differential expression $L_n$ given by (\ref{6.1}) with
continuous complex-valued periodic coefficients $q_0,...,q_{n-2}$
of a real variable we associate the corresponding closed operator
$H: H^{2,n}\to L^2(\bb R)$, $Hy=L_n y$.

Rofe-Beketov's result \cite{56}, referred to in Section
\ref{floquet},
originally was proved for an $n$-th order operator. Hence the
spectrum $\sigma(H)$ of $H$ equals the conditional stability set
${\mathcal S}(L_n)$ of $L_n$, that is, the set of all complex
numbers $E$
for which the differential equation $L_n y=Ey$ has a nontrivial
bounded
solution. For $E$ to be in ${\mathcal S}(L_n)$ it is necessary
and
sufficient that $L_n y=Ey$ has a Floquet multiplier of modulus
one.
Hence
\begin{equation}\label{6.4}
{\mathcal S}(L_n)=\{E\in\bb C: {\F}(E,e^{it})=0 \text{ for some
$t\in\bb R$}\},
\end{equation}
where $\F$ is given by (\ref{6.2}). Since $\F$ is entire in both
its
variables, it follows that $\sigma(H)={\mathcal S}(L_n)$ consists
of
(generally) infinitely many regular analytic arcs (i.e., spectral
bands). They
end at a point where the arc fails to be regular analytic or
extends to
infinity. Finite endpoints of the spectral bands are called band
edges.

\begin{definition}\label{d6.1}
(i) $H$ is called a {\bf finite-band operator} if and only if
$\sigma(H)$ consists of a finite number of regular analytic
arcs.\\
(ii) $L_n$ is called a {\bf Picard differential expression} if
and
only if all its coefficients are elliptic functions associated
with
a common period lattice and if $L_n y=Ey$ has a meromorphic
fundamental system (with respect to the independent variable)
for
any
value of the spectral parameter $E \in \bb C$.
\end{definition}

Next, let $\Phi(E,x)$ be the fundamental matrix of $L_n y=Ey$
satisfying
the initial condition $\Phi(E,x_0)=I_n$ where $I_n$ is the
$n\times n$
identity matrix. The Floquet multipliers of the differential
equation
$L_n y=Ey$ are then the eigenvalues of the monodromy matrix
$\Phi(E,x_0+\Omega)$.

Our aim is to determine multiplicities of Floquet eigenvalues and
multipliers for large values of the spectral parameter $E$. For
large
values of $E$ the equation $L_n y=Ey$ can be treated as a
perturbation
of $y^{(n)} = Ey$. In this case there exist $n$ linearly
independent
Floquet solutions $\exp(\lambda \sigma_k x)$ with associated
Floquet
multipliers $\exp(\lambda \sigma_k)$, where $\lambda$ is such that
$\lambda^n=-E$ and the $\sigma_k$ are the different $n$-th roots
of
$-1$. The characteristic polynomial of the associated monodromy
matrix
is therefore given by
\begin{equation}\label{6.5}
\F_0(E,\rho)=(-1)^n \rho^n +(-1)^{n-1} a_{F,1}(E)\rho^{n-1}+ ...
-a_{F,n-1}(E) \rho + 1,
\end{equation}
where the $a_{F,j}$ are the elementary symmetric polynomials in
the
variables $\exp(\lambda \sigma_1)$, ..., $\exp(\lambda \sigma_n)$.
Perturbation theory now yields that the coefficients $a_j$ in
\eqref{6.2} are related to the coefficients $a_{F,j}$ in
\eqref{6.5} by
\begin{equation} \label{6.11}
a_j(E)=a_{F,j}(E)+b_j(E), \quad j=1,...n-1,
\end{equation}
where, for some suitable positive constant $M$,
\begin{equation}\label{6.12}
|b_j| \leq \frac{M}{|\lambda|} |\exp(\lambda\sigma_{n+1-j})...
\exp(\lambda\sigma_n)|, \quad j=1,...,n-1
\end{equation}
having ordered the roots in such a way that
$|\exp(\lambda\sigma_j)|
\leq |\exp(\lambda\sigma_{j+1})|$ for $j=1,...,n-1$. This allows
one to show that asymptotically, in a certain small disk about
$\exp(\lambda\sigma_k)$, there are either one or two Floquet
multpliers
of $L_ny=Ey$ depending on whether or not the inequality
\begin{equation}\label{6.14}
\left|\exp(\lambda\sigma_k) -
\exp(\lambda\sigma_j)\right| \geq
\gamma \max\{|\exp(\lambda\sigma_k)|,
|\exp(\lambda\sigma_j)|\}
\text{ for all $j\neq k$}
\end{equation}
holds. From this one may prove the following theorem concerning
algebraic multplicities of Floquet multipliers.
\begin{theorem}\label{t6.4} (\cite{66a})
Let $L_n$ be defined as in (\ref{6.1}). For every $\varepsilon>0$
there
exists a disk $B(\varepsilon)\subset \bb C$ with the following two
properties.\\
1. All values of $E$, where at least two Floquet multipliers of the
differential equation $L_n y=Ey$ coincide, lie in $B(\varepsilon)$
or in
the cone $\{E:|\im(E)|/|\re(E)|\leq \varepsilon\}$.\\
2. Every degenerate Floquet multiplier outside $B(\varepsilon)$ has
multiplicity two.
\end{theorem}
This result has been obtained earlier by McKean \cite{44d} for
$n=3$ and
by da Silva Menezes \cite{13g} for general $n$.

Moreover the above observations may be used to obtain information
concerning algebraic multiplicities of Floquet eigenvalues.
\begin{theorem}(\cite{66a})\label{t6.5}
Let $\theta_0\in\bb C$. Then there exists an $R>0$ such
that every eigenvalue $E$ of the Floquet operator $H(\theta_0)$,
which
satisfies $|E|>R$, has at most algebraic multiplicity two.
\end{theorem}

Now suppose that $\mathcal F(E,\rho)=0$ and that
$E\in\mathcal S(L_n)$
is suitably large. Since $1\leq m_f(E,\rho), m_a(E,\rho)\leq2$ we
have
to distinguish four cases and Weierstrass's preparation theorem
provides us with the following information:\\
1. If $m_f(E,\rho)=m_a(E,\rho)=1$ then one spectral band passes
through
$E$.\\
2. If $m_f(E,\rho)=2$ and $m_a(E,\rho)=1$ then two (possibly
coinciding)
spectral bands end in $E$. \\
3. If $m_f(E,\rho)=1$ and $m_a(E,\rho)=2$ then two spectral bands
intersect in $E$ forming a right angle.\\
4. If $m_f(E,\rho)=m_a(E,\rho)=2$ then two (possibly coinciding)
spectral bands pass through $E$.

This shows that a necessary condition for a suitably large $E$
to be a
band edge is that $m_a(E,\rho)=1$ and $m_f(E,\rho)=2$ for some
$\rho$.
In particular, such an $E$ is necessarily a point where strictly
less than
$n$ linearly independent Floquet solutions exist. In addition,
when
$n$ is odd then $\sigma(H)$ is ultimately in a cone with the
imaginary
axis as symmetry axis while the possible band edges (where
$m_f(E,\rho)=2$)
are in a cone whose axis is the real axis. Therefore we have the
following result.
\begin{theorem} \label{t6.7a} (\cite{66a})
The operator $H$ associated with the differential expression
$L_n$
introduced after (\ref{6.3}) is a finite-band operator
whenever $n$, the order of $L_n$, is odd.
\end{theorem}

Finally we turn to the case where $L_n$ is a Picard differential
expression. The principal result of this section,
Theorem \ref{t6.8}
below, then shows that algebraic and geometric multiplicities of
Floquet multipliers of $L_n y=Ey$ can be different only when
$E$ is one
of finitely many points.

\begin{theorem}(\cite{66a})\label{t6.8}
Suppose the differential expression $L_n$ is Picard. Then there
exist
$n$ linearly independent solutions of $L_ny=Ey$ which are
elliptic of
the second kind for all but finitely many values of the spectral
parameter $E$.
\end{theorem}
\begin{proof}[Sketch of proof]
The proof is modeled closely after the one of Theorem \ref{3.7}.
Again,
inside a compact set there can be only a finite number of values
of $E$
where Floquet multipliers associated with a fundamental period of
the coefficients of $L_n$ are degenerate. On the other hand when
$|E|$ becomes large we only have to prove that for one of the
fundamental periods of the coefficients of $L_n$ all Floquet
multipliers
of $L_n y=Ey$ are distinct according to Picard's
Theorem \ref{t3.1}.

Assume that the fundamental periods $2\omega_1$ and $2\omega_3$
are
such that the angle $\phi$ between them is less than $\pi/n$ and
assume that
$z_0$ is such that no singularity of $q_0,..., q_{n-2}$ lies on
the line
through $z_0$ and $z_0+2\omega_1$ or on the line through $z_0$
and
$z_0+2\omega_3$.

Substituting $w(x)=y(2\omega_1 x+z_0)$ and defining $p_k(x)
=(2\omega_1)^{n-k} q_k(2\omega_1 x+z_0)$ transforms $L_ny=Ey$
into
\begin{equation}\label{6.15}
w^{(n)} + p_{n-2}(x) w^{(n-2)} + ... + p_0(x) y
= (2\omega_1)^n E w.
\end{equation}
Therefore, Theorem \ref{t6.4} implies that all Floquet multipliers
associated with the periods $2\omega_1$ ($2\omega_3$) are pairwise
distinct provided the spectral parameter $(2\omega_j)^n E$
lies outside the set $S_j$, $j=1,3$, where
\begin{equation}\label{6.16}
S_j =\left\{z: \left|\frac{\im(z)}{\re(z)}\right|
\leq \frac{\phi}{3} \right\} \cup \left\{z: |z| \leq R_j\right\},
\end{equation}
with $R_j$, $j=1,3$ being suitable positive constants.

The two sets $S_1$ and $S_3$ do not intersect outside a
sufficiently
large disk $D$.
Hence, for each value of $E$ outside $D$, Picard's theorem
guarantees
the existence of $n$ linearly independent solutions of
$L_n y=Ey$ which
are elliptic functions of the second kind.
\end{proof}

In particular, when $|E|$ is large and $L_n$ is Picard, we infer
that
$m_f(E,\rho) =m_g(E,\rho) \leq m_a(E,\rho)$. Moreover, we have
shown
earlier that necessarily $m_a(E,\rho)=1$
and $m_f(E,\rho)=2$ for band edges $E$ with $|E|$ sufficiently
large.
Thus there are no band edges with sufficiently large absolute
values for Picard expressions. One may also show that at most two
bands extend to infinity. Hence we have the following final
theorem which, in view of Theorem \ref{t6.7a}, has significance
only
when $n$ is even.
\begin{theorem}(\cite{66a})\label{t6.9}
Let $L_n$ be a Picard differential expression and $H$ the
associated
operator. Then, if $\sigma(H)$ does not contain closed regular
analytic
arcs, $\sigma(H)$ consists of finitely many analytic arcs which
are
regular in their interior.
\end{theorem}




{\bf Acknowledgments.} F.G. would like to thank all organizers
of the
AMS meeting at LSU for creating a most stimulating atmosphere
and especially for the extraordinary hospitality extended to him.

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\end{document}