\documentclass{amsart}
\begin{document}
{\noindent\small {\em Electronic Journal of Differential Equations},
Vol.~2000(2000), No.~19, pp.~1--11.\newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu \quad ejde.math.unt.edu (login: ftp)}
\thanks{\copyright 2000 Southwest Texas State University and
University of North Texas.}
\vspace{1cm}
\title[\hfilneg EJDE--2000/19\hfil On commuting differential operators]
{ On commuting differential operators }
\author[ R. Weikard \hfil EJDE--2000/19\hfilneg]
{ R. Weikard }
\address{Rudi Weikard \hfill\break
Department of Mathematics, University of Alabama at
Birmingham, Birmingham, Alabama 35294-1170, USA}
\email{rudi@math.uab.edu}
\date{}
\thanks{Submitted February 22, 2000. Published March 9, 2000.}
\subjclass{34M05, 37K10, 37K20 }
\keywords{ Meromorphic solutions,Commuting differential expressions,
\hfill\break\indent Lax pairs, KdV, Gelfand-Dikii systems }
\begin{abstract}
The theory of commuting linear differential expressions has received a
lot of attention since Lax presented his description of the KdV
hierarchy by Lax pairs $(P,L)$. Gesztesy and the present author have
established a relationship of this circle of ideas with the property
that all solutions of the differential equations
$Ly=zy$, $z\in {\mathbb C}$, are meromorphic. In this paper this
relationship is explored further by establishing its existence for
Gelfand-Dikii systems with rational and simply periodic coefficients.
\end{abstract}
\maketitle
\newtheorem{thm}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{cor}{Corollary}
\theoremstyle{definition}
\newtheorem{dfn}{Definition}
\newcommand{\bb}[1]{{\mathbb{#1}}}
\newcommand{\e}{\hbox{\rm e}}
\section{Introduction}
The theory of commuting linear differential expressions was begun by
Floquet \cite{Fl79} in 1879 and advanced significantly when Wallenberg
\cite{Wa03} and Schur \cite{Sc05} addressed it some 25 years later. An
even bigger impact had Burchnall and Chaundy with a series of papers
(\cite{BC1}, \cite{BC2}, \cite{BC3}) in the 1920s when they discovered
a relationship with algebraic geometry (see Section \ref{s:BC}). The
exploration of commuting differential expressions was again taken up in
the 1970s and 1980s because of the connection with completely
integrable partial differential equations. The ones in question here
are the Gelfand-Dikii systems which may be represented by equations of
the type $L_t=[P,L]$ where $P$ and $L$ are linear differential
expressions. The most famous such equation is the Korteweg-de Vries
(KdV) equation
$$q_t=\frac14q_{xxx}+\frac32 qq_x$$
which is obtained by choosing $L=D^2+q$ and $P=D^3+\frac32 q D +
\frac34 q_x$ when $D$ denotes the differential expression $d/dx$. The
letters $P$ and $L$ where chosen by Gelfand and Dikii in honor of Peter
Lax who first represented the KdV equation using a Lax pair \cite{Lax}.
Only a select few expressions $L$ will allow the existence of an
expression $P$ whose order is relatively prime to the order of $L$ but
which commutes with $L$ and, due to the Burchnall-Chaundy theorem, such
$L$ are also called algebro-geometric (see Section \ref{s:BC}
for precise statements and definitions). From the works of Its and
Matveev \cite{IM} and Krichever \cite{Kr77a}, \cite{Kr77b} it is clear
that the coefficients of $L$ should be given in terms of specific
differential polynomials of a Riemann theta function (i.e., a
polynomial in that function and its derivatives). However, to recognize
whether a given differential expression is algebro-geometric, this
knowledge is of little value.
The aim of the present paper is to give an easily verifiable sufficient
condition to ensure that a given differential expression $L$ (with
rational or simply periodic coefficients) is algebro-geometric (see
Theorem \ref{main}). A few years ago such a characterization was
obtained by Gesztesy and myself for expressions of the form $L=D^2+q$
with an elliptic potential $q$ (see \cite{GW96}). In fact, we found
that $L$ is algebro-geometric if and only if the equation $Ly=zy$ has
only meromorphic solution regardless what $z$ is. The corresponding
relationship exists also for rational and simply periodic potentials of
$D^2+q$ (see \cite{W}) and for the more general AKNS system (at least
in the case of elliptic coefficients, see \cite{GW98}). The clue in
\cite{GW96} was to consider the independent variable of the equation
$y''+qy=zy$ as a complex variable and use a classical theorem of Picard
treating equations with elliptic coefficients. For a survey of this and
related approaches to integrable systems see \cite{GW-bull}.
In retrospect it is clear from the work of Its and Matveev \cite{IM}
and of Segal and Wilson \cite{SW} that the solutions of $Ly=zy$ are
necessarily meromorphic if $L$ is algebro-geometric. However, it seems
that nobody thought that this was peculiar.
The following theorem, which establishes sufficient conditions for a
differential expression to be algebro-geometric, will be proven in this
paper:
\begin{thm} \label{main}
Suppose that the coefficients of the differential expression
$$L=D^n+q_{n-2}D^{n-2}+...+q_0$$
are either
\begin{itemize}
\item rational functions, which are bounded at infinity, or else
\item meromorphic, simply periodic functions with period $p$, which
remain bounded as $|\Im(x/p)|$ tends to infinity.
\end{itemize}
If, regardless of $z\in\bb C$, all solutions of the differential
equation $Ly=zy$ are meromorphic then $L$ is algebro-geometric.
\end{thm}
Therefore, given a differential expression $L$ in one of the classes
indicated, it suffices to examine the behavior of the solutions of
$Ly=zy$ near the finitely many singular points of the equation. This is
a routine, if lengthy, task.
\begin{proof}[Proof of Theorem \ref{main}]
Theorem \ref{bcidea} gives a sufficient condition for $L$ to be
algebro-geometric provided the equation $Ly=zy$ has a solution of a
certain form. That this is indeed so is guaranteed by Theorem
\ref{ratstrct} in the rational case (choose $t(x)=x$) and by Theorem
\ref{perstrct} in the simply periodic case (choose $t(x)=\exp(2\pi
ix/p)$).
\end{proof}
The proofs of Theorems \ref{ratstrct} and \ref{perstrct} rely on
results by Halphen and Floquet (concerned with the rational and simply
periodic case, respectively). These, in turn, are modeled after the
above mentioned theorem of Picard. The proof of Theorem~\ref{bcidea} is
suggested by the work of Burchnall and Chaundy \cite{BC3}.
While in the case of the KdV hierarchy the corresponding theorem was
first proven for elliptic potentials the current methods are not easily
adaptable to elliptic coefficients of $L$ when $n>2$. The reason is
that the known proofs for the KdV hierarchy rely on the recursion
relation through which the hierarchy may be defined. An analogous
representation is unknown for general $n$ (see however \cite{DGU} for
$n=3$). The current proof, on the other hand, does not extend to the
elliptic case because the relationship between $\lambda$ and $z$, which
is algebraic for rational and simply periodic coefficients, is
transcendental in the case of elliptic coefficients.
In Section \ref{s:BC} we will review the theory of Burchnall and
Chaundy and prove a characterization of algebro-geometric potentials.
Section \ref{pfh} presents the Halphen theorem and an analogous version
of the Floquet theorem. Section \ref{structure} establishes that in the
cases considered certain solutions are of the form required by Theorem
\ref{bcidea}. An important ingredient for this part is the asymptotic
behavior of the solutions as the spectral parameter tends to infinity.
This is, of course, a well researched subject and the reader is
reminded of the basic facts, following Wasow \cite{Wasow}, in the
appendix.
\section{Burchnall-Chaundy theory} \label{s:BC}
\begin{dfn} \label{d1}
A differential expression $L$ of order $n\geq2$ and leading coefficient
one is called algebro-geometric if there exists a natural number $m$,
relatively prime with respect to $n$, a polynomial $\mathcal Q$ of the
form
\begin{equation} \label{BC}
\mathcal Q(p,\ell)=p^n-\ell^m + \sum_{\substack{a,b\geq0\\ am+bn