\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<nm}}
c_{a,b} p^a\ell^b,
\end{equation}
and a differential expression $P$ of order $m$, such that\\
 1. $Q(P,L)=0$ and \\
 2. if $L\in \bb C[R]$ for some first order differential expression $R$
then $P\not\in \bb C[R]$.
\end{dfn}
The most trivial examples of algebro-geometric differential expressions
are given by expressions with constant coefficients when one may choose
$P=D$, the operator of taking a first derivative.

\begin{thm}
Suppose $P$ and $L$ are differential expressions of relatively prime
orders $m$ and $n$ respectively. Then $P$ and $L$ commute if and only
if there exists a polynomial of the form \eqref{BC} such that
$Q(P,L)=0$.
\end{thm}
This theorem, obtained in the early 1920s by Burchnall and Chaundy
\cite{BC1}, may serve as a characterization for algebro-geometric
differential expressions:
\begin{cor}
A differential expression $L$ of order $n\geq2$ and leading coefficient
one is algebro-geometric if and only if there exists a natural number
$m$, relatively prime with respect to $n$ and a differential expression
$P$ of order $m$, such that\\
 1. $[P,L]=0$ and \\
 2. if $L\in \bb C[R]$ for some first order differential expression $R$
then $P\not\in \bb C[R]$.
\end{cor}
The restriction to $n\geq2$ is due to the fact that $L=D+q(x)$ and
$[P,L]=0$ imply that $P\in\bb C[L]$ regardless what $q$ is. This is seen
as follows: Suppose $P$ commutes with $L$ and is of order $m$. Without
loss of generality we may assume that $P$ has leading coefficient one.
Then $P-L^m$ commutes with $L$, has order less than $m$ and a constant
leading coefficient. Induction proves the claim.

Now consider the differential expression $\hat L=D^n+\hat q_{n-1}
D^{n-1}+...+\hat q_0$ and let $E$ be the operator of multiplication by
$\exp(\int^x \hat q_{n-1} dt/n)$. Then
\begin{equation} \label{normalL}
L=E\hat LE^{-1}=D^n+q_{n-2}D^{n-2}+...+q_0
\end{equation}
for appropriate functions $q_0$, ..., $q_{n-2}$. We call $L$ the normal
form of $\hat L$. If $\hat P$ is some other differential expression and
if $P=E\hat PE^{-1}$, then $[P,L]=0$ if and only if $[\hat P,\hat L]=0$.
Therefore, to characterize the algebro-geometric differential
expressions we may restrict ourselves to those which are of the form
\eqref{normalL}.

If $L\in\bb C[R]$ for some first order differential expression
$R=D+a(x)$ and if $L$ is of the form \eqref{normalL} then $a$ must be
constant, that is $L$ has constant coefficients and it commutes with
$P=D$ and hence is algebro-geometric. On the other hand, if $L$ is in
the form \eqref{normalL} and does not have constant coefficients then
it is not in $\bb C[R]$ for any first order expression $R$ and we
obtain the following characterization:
\begin{cor} \label{C2}
The differential expression $L$ given in \eqref{normalL} is
algebro-geometric if and only if there exists a natural number $m$,
relatively prime with respect to $n$, and a differential expression $P$
of order $m$ such that $[P,L]=0$.
\end{cor}

We will now give a sufficient condition for $L$ to be algebro-geometric
in terms of the solutions of the differential equations $Ly=zy$.
\begin{thm} \label{bcidea}
Let $L$ be a differential expression of the form \eqref{normalL} and
suppose that, for every $z\in\bb C$, the equation $Ly=zy$ has a
solution of the form
$$\psi(\lambda,x)=(\lambda^g+r_{g-1}(t(x))\lambda^{g-1}+...+r_0(t(x)))
 \exp(\lambda x)$$
where $r_0$, ..., $r_{g-1}$ are rational functions, $t$ is a
meromorphic function, and
$$\lambda^n+\rho_{n-2}\lambda^{n-2}+...+\rho_0=z$$
for certain complex numbers $\rho_0$, ..., $\rho_{n-2}$. Then there
exists a differential expression $P$ whose order $m$ is relatively
prime with respect to $n$ such that $[P,L]=0$. In particular, $L$ is
algebro-geometric.
\end{thm}
\begin{proof} Define
$$U=D^g+r_{g-1}(t(x))D^{g-1}+...+r_0(t(x))$$
and
$$L_0=D^n+\rho_{n-2}D^{n-2}+...+\rho_0.$$
Then consider the differential expressions $V=LU-UL_0$. Since
$L_0(\exp(\lambda x))=z\exp(\lambda x)$ and $U(\exp(\lambda
x))=\psi(\lambda,x)$ we obtain
$$V(\exp(\lambda x))=(L-z)U(\exp(\lambda x))=(L-z)\psi(\lambda,x)=0$$
for every $\lambda\in\bb C$. Since the functions $\exp(\lambda x)$ are
linearly independent for distinct $\lambda$ we obtain that $V$ is the
zero expression, that is,
$$LU=UL_0.$$
Let $\{y_1,...,y_g\}$ be a basis of $\ker U$. To each element $y_\ell$
of this basis we may associate a differential expression $H_\ell$ with
constant coefficients in the following way. Since $y_\ell\in\ker U$, so
is $L_0y_\ell$ and, in fact, $L_0^jy_\ell$ for every $j\in\bb N$. Since
$\ker U$ is finite-dimensional there exists a $k\in\bb N$ and complex
numbers $\beta_0,...,\beta_k$ such that $\beta_0=1$ and
$$\sum_{j=0}^k \beta_{k-j} L_0^jy_\ell=0.$$
Then define $H_\ell=\sum_{j=0}^k \beta_{k-j} L_0^j$. Since the
expressions $H_\ell$ commute among themselves we obtain that $\ker U
\subset \ker(D^j\prod_{\ell=1}^g H_\ell)$ for any nonnegative integer
$j$. Let $S_\mu$ be the set of all differential expressions $H$ of
order $\mu$ with constant coefficients such that $\ker U\subset \ker
H$. We have just shown that $S_\mu$ is not empty provided $\mu$ is
sufficiently large. Hence there exists the number
$$m=\min\{\mu\in\bb N: \gcd(\mu,n)=1, S_\mu\neq\emptyset\}.$$
Let $P_0$ be an element of $S_m$. Since $\ker U\subset \ker P_0$, we
obtain that there exists an expression $P$ such that $PU=UP_0$. Hence
$[P,L]U=PLU-LPU=UP_0L_0-UL_0P_0=U[P_0,L_0]=0$ and thus $[P,L]=0$. 
In
view of Corollary \ref{C2} this proves that $L$ is algebro-geometric.
\end{proof}




\section{The theorems of Picard, Floquet, and Halphen} \label{pfh}
As mentioned in the introduction the proof of Theorem \ref{main} relies
on classical theorems by Floquet and Halphen concerning the linear
differential equation
\begin{equation} \label{lde}
y^{(n)}+q_{n-1}y^{(n-1)}+...+q_0y=0
\end{equation}
with simply periodic and rational coefficients, respectively. These
theorems, in turn, where inspired by a theorem of Picard which is
concerned with the elliptic case.

While Picard's theorem \cite{Picard} will not be used I state it for
the sake of its historic significance.
\begin{thm} \label{picard}
Assume that the coefficients $q_0,...,q_{n-1}$ in \eqref{lde} are
elliptic functions with common fundamental periods $2\omega_1$ and
$2\omega_2$. If the differential equation \eqref{lde} has only
meromorphic solutions then it has a solution which is elliptic of the
second kind\footnote{A function $f$ is called elliptic of the second
kind, if it is meromorphic and if there exist two numbers $a_1$ and
$a_2$, independent over the real numbers, and two numbers $\rho_1$ and
$\rho_2$ such that $f(x+a_1)=\rho_1 f(x)$ and $f(x+a_2)=\rho_2 f(x)$
for all $x$.}.
\end{thm}

Halphen's theorem is concerned with the rational case. A proof is given
by Ince \cite{Ince} and this proof can be used to state the following
version which is different from Ince's version.
\begin{thm} \label{halphen}
Let the coefficients $q_0,...,q_{n-1}$ in \eqref{lde} be rational
functions which are bounded at infinity and define
$\rho_j=\lim_{x\to\infty} q_j$. If the differential equation
\eqref{lde} has only meromorphic solutions then there is a solution
$R(x)\exp(\lambda x)$ where $R$ is a rational function and $\lambda$
satisfies
$$\lambda^n+\rho_1\lambda^{n-1}+... +\rho_n=0.$$
\end{thm}

Floquet's famous theorem (see e.g. Eastham \cite{Eastham} or Magnus and
Winkler \cite{MW}) on periodic differential equation, though inspired
by Picard's results, has a broader scope but also gives less
information on the structure of solutions when compared with the
theorems by Picard and Halphen.  We will therefore provide a below an
analogue of Picard's or Halphen's theorem for the simply periodic case.

Let us first remember a few basic facts from the theory of meromorphic,
simply periodic functions (for more information see, e.g., Markushevich
\cite{Markushevich}, Chapter III.4). If $f$ is a meromorphic periodic
function with period $2\pi$ then
$$f^*(t)=f(-i\log(t))$$
is meromorphic on $\bb C-\{0\}$. If $f$ is entire then $f^*$ is
analytic on $\bb C-\{0\}$.

A meromorphic simply periodic function $q$ with period $p$ which has
only finitely many poles in the period strip $\{x\in\bb C:
0\leq\Re(x/p)<1\}$ and which is bounded as $|\Im(x/p)|$ tends to
infinity is of the form
$$q(x)=\frac{a_0+a_1\e^{2\pi ix/p}+...+a_m \e^{2\pi imx/p}}
 {b_0+b_1\e^{2\pi ix/p}+...+b_m \e^{2\pi imx/p}}.$$
We will call such functions bounded at the ends of the period strip.
Note that
$$\lim_{\Im(x/p)\to\infty} q(x)=\frac{a_0}{b_0}=q^*(0)$$
and
$$\lim_{\Im(x/p)\to-\infty} q(x)=\frac{a_m}{b_m}=q^*(\infty).$$

\begin{thm} \label{floquetplus}
Let the coefficients $q_0,...,q_{n-1}$ in \eqref{lde} be meromorphic,
simply periodic with period $p$, and bounded at the ends of the period
strip. If the differential equation \eqref{lde} has only meromorphic
solutions then there is a solution $R(\e^{2\pi ix/p})\exp(i\lambda x)$
where $R$ is a rational function and $\lambda$ satisfies
$$(i\lambda)^n+q_{n-1}^*(0)(i\lambda)^{n-1}+...+q_0^*(0)=0.$$
\end{thm}
Since a proof of this theorem does not seem to be readily available I
will give an outline below. But first I will present a few lemmas which
will be needed.
\begin{lemma} \label{lemma.f1}
Let $v$ be a polynomial with $v(0)\neq0$, abbreviate $\e^{ix}$ by $t$,
and suppose that
$$y(x)=\frac{u(t)}{v(t)} t^\lambda$$
is meromorphic with respect to $x$. Then
$$y^{(k)}(x)=\frac{t^\lambda}{v(t)^{k+1}}
 \sum_{j=0}^k t^j f_{j,k}(\lambda,t) u^{(j)}(t)$$
where the $f_{j,k}$ are polynomials in both of their variables and
$u^{(j)}$ denotes the $j$-th derivative of $u$ with respect to $t$. In
particular, $f_{j,j}(\lambda,t) =(iv(t))^j$. Moreover, $\deg
f_{j,k}(\lambda,\cdot)\leq k \deg v$.
\end{lemma}
\begin{proof}
The first statement follows immediately from an induction over $k$. In
fact
$$f_{j,k+1}=iv ((\lambda+j)f_{j,k}+f_{j-1,k})
 +it(vf_{j,k}'-(k+1)v'f_{j,k})$$
where primes denote derivatives with respect to $t$ and $f_{j,k}=0$
unless $j\in\{0,...,k\}$.

This implies that $f_{j,j}(\lambda,t) =(iv(t))^j$. The statement about
the degree of $f_{j,k}(\lambda,\cdot)$ follows now, for fixed $j$, by
another induction over $k$.
\end{proof}

\begin{lemma} \label{lemma.f2}
The polynomials $f_{j,k}$ in Lemma \ref{lemma.f1} have the following
property:
$$f_{j,k}(\lambda,0)=(iv(0))^k g_{j,k}(\lambda)$$
where $g_{j,k}$ is a polynomial of degree $k-j$ with leading
coefficient $\binom{k}{j}$. Moreover, $g_{0,k}(\lambda)=\lambda^k$.
\end{lemma}
\begin{proof}
These statements are also proven by induction.
\end{proof}

\begin{proof}[Proof of Theorem \ref{floquetplus}]
Without loss of generality we assume that $p=2\pi$. For convenience we
also introduce $q_n=1$.

Each of the coefficients $q_j$ has at most finitely many poles in the
period strip $\{x\in\bb C:0\leq \Re(x)<2\pi\}$. These poles will be
denoted by $x_1$, ..., $x_m$. From Floquet's theorem we know that there
is a solutions of $Ly=zy$ of the form
$$\psi(x)=\phi(x)\e^{i\lambda x}$$
where $\phi$ is a periodic function with period $2\pi$ and $\lambda$ is
a suitable complex number which is determined up to addition of an
arbitrary integer. By hypothesis $\phi$ is a meromorphic function and
its poles may occur only at the points $x_1$, ..., $x_m$ and their
translates. Therefore there exist positive integers $s_j$ and a
polynomial
$$v(t)=\prod_{j=1}^m (t-\e^{ix_j})^{s_j}$$
such that $v(\e^{ix})\phi(x)$ is an entire meromorphic function which
is periodic with period $2\pi$. This implies that there is a function
$u_0$ which is analytic on $\bb C-\{0\}$ such that $u_0(\e^{ix})
=v(\e^{ix})\phi(x)$. We want to show that $u_0$ is a rational function.

Now multiply \eqref{lde} by $v(\e^{ix})^{n+1}$ and perform the
substitution $y(x)=t^\lambda u(t)/v(t)$ with $t=\e^{ix}$.

With the aid of Lemma \ref{lemma.f1} equation \eqref{lde} turns into
\begin{equation} \label{031298.1}
\sum_{j=0}^n t^j p_j(t) u^{(j)}(t)=0
\end{equation}
where
$$p_j(t)=\sum_{k=j}^n v(t)^{n-k}q_k^*(t) f_{j,k}(\lambda,t)$$
and where, of course, $q_k^*(t)=q_k(-i\log(t))$.

Because all solutions of \eqref{lde} are meromorphic any pole of any of
the coefficients must be a regular singular point of the differential
equation. Therefore the poles of $q_j$ have order $n-j$ at worst and
the functions $v(t) q_{n-1}^*(t)$, ..., $v(t)^n q_0^*(t)$ are
polynomials. This implies that the coefficients $p_j$ in
\eqref{031298.1} are polynomials. Zero and infinity are singular points
of the equation \eqref{031298.1} and, since $p_n(0)=(iv(0))^n\neq0$, we
obtain that zero is in fact a regular singular point. This, in turn,
implies that the isolated singularity $t=0$ of $u_0$ can not be an
essential singularity, i.e., $u_0$ is analytic in $\bb C$ with the
exception of a possible pole at zero.

Moreover, at least one of the indices of the singular point $t=0$ of
equation \eqref{031298.1} must be an integer because zero is an
isolated singularity for the solution $u_0$. Remember that $\lambda$ is
only determined up to the addition of an integer. Therefore and because
$u_0(t) t^\lambda =(t^{-m} u_0(t)) t^{\lambda+m}$ we can and will
choose the smallest integer index to be zero. Having made this
convention $u_0$ is now analytic at zero, i.e., $u_0$ is an entire
function. The product of all the indices, which equals the constant
term of the indicial equation, must now be zero, too. Hence
$$0=\sum_{k=0}^n i^k q_k^*(0) g_{0,k}(\lambda)
 =\sum_{k=0}^n q_k^*(0) (i\lambda)^k$$
which is the desired relationship for $\lambda$. The theorem will now
be proved once we show that $u_0$ is a polynomial. To see this we have
to study its behavior at infinity.

Let $s=1/t$ and define integers $a_{j,k}$ by the equality
$$u^{(j)}(t)=\sum_{k=1}^j a_{j,k} s^{j+k} w^{(k)}$$
where $u(t)=w(s)$. This yields, in particular, $a_{j,j}=(-1)^j$. Also
define $a_{0,0}=1$ and $a_{j,0}=0$ for any $j\in\bb N$.

Introducing $s$ as the independent variable we find
\begin{equation} \label{060499}
0=v(t)^{-n} \sum_{j=0}^n t^j p_j(t) u^{(j)}
  =\sum_{k=0}^n s^k \tilde p_k(s) w^{(k)}
\end{equation}
where
$$\tilde p_k(s)=v(1/s)^{-n}\sum_{j=k}^n a_{j,k} p_j(1/s)
 =\sum_{j=k}^n \sum_{m=j}^n
 a_{j,k} q_m^*(1/s) \frac{f_{j,m}(\lambda,1/s)}{v(1/s)^m}.$$
Recall from Lemma \ref{lemma.f1} that the degree of
$f_{j,m}(\lambda,\cdot)$ is not larger than the degree of $v^m$. Hence
the functions $\tilde p_k$ are bounded at zero. In particular, $\tilde
p_n(s)=(-i)^n$ is bounded but also different from zero. It now follows
that $s=0$ is a regular singular point of equation \eqref{060499}.
Therefore, and since $s=0$ must be an isolated singularity of
$w_0(s)=u_0(1/s)$, the function $w_0$ behaves like an integer power
near zero. This, in turn implies that the entire function $u_0$ behaves
like an integer power at infinity, i.e., $u_0$ is a polynomial.
\end{proof}









\section{The structure of solutions} \label{structure}
\subsection{The rational case}
\begin{thm} \label{ratstrct}
Consider the differential expression
$$L=D^n+q_{n-2}D^{n-2}+...+q_0$$
where $q_{n-2}$, ..., $q_0$ are rational functions which have
respectively the limits $\rho_{n-2}$, ..., $\rho_0$ at infinity. Assume
that, for all $z\in\bb C$ all solutions of the equation $Ly=zy$ are
meromorphic. Then $Ly=zy$ has a solution of the form
$$\psi(\lambda,x)=(\lambda^g+r_{g-1}(x)\lambda^{g-
1}+...+r_0(x))\exp(\lambda x)$$
where $r_0,...,r_{g-1}$ are rational functions and
$$z=\lambda^n+\rho_{n-2}\lambda^{n-2}+...+\rho_0.$$
\end{thm}
\begin{proof}
If $L$ has constant coefficients the theorem is trivially true with
$g=0$. Hence assume that $L$ does not have constant coefficients.

By Halphen's theorem the equation $Ly=zy$ has a solution of the form
$\psi(\lambda,x)=R_\lambda(x)\exp(\lambda x)$ where $R_\lambda$ is a
rational function and $z=\lambda^n+\rho_{n-2}\lambda^{n-2}+...+\rho_0$.
The only thing left to investigate is the behavior of $R_\lambda$ in
terms of $\lambda$.

All finite singular points of the equation $Ly=zy$ must be regular
singular points. Therefore $q_j$ has no poles of order larger than
$n-j$. Also the poles of $R_\lambda$ must be located at the poles of
the coefficients  $q_j$. Hence there exist positive integers $s_j$ and
a polynomial
$$v(x)=\prod_{j=1}^m (x-x_j)^{s_j}.$$
such that the function $v\psi(\lambda,\cdot)$ is entire and the
functions $v^2 q_{n-2}$, ..., $v^n q_0$ are polynomials. Therefore
$$\psi(\lambda,x)=\frac{p(\lambda,x)}{v(x)} \exp(\lambda x)$$
where
$$p(\lambda,x)=\sum_{j=0}^N c_j(\lambda) x^j.$$
Hence
$$0=\psi^{(n)}+q_{n-2}\psi^{(n-2)}+...+(q_0-\lambda^n-...-\rho_0)\psi
 =\frac{\exp(\lambda x)}{v(x)^{n+1}}F(\lambda,x).$$
Since $v^j q_{n-j}$ are polynomials the function $F(\lambda,\cdot)$ is a
polynomial whose coefficients are polynomials in $c_0,..., c_n$, and
$\lambda$. In fact, as polynomials in $c_0,...,c_n$ these coefficients
are homogeneous of degree one. Each of these coefficients must be zero
and therefore the coefficients $c_j$ satisfy a system of linear
homogeneous algebraic equations with coefficients in $\bb C[\lambda]$. A
nontrivial solution exists and its components (the coefficients $c_j$)
are rational functions of $\lambda$. Hence
$$p(\lambda,x)=\frac1{h(\lambda)}\sum_{j=0}^N \tilde c_j(\lambda) x^j$$
where $\tilde c_0,...,\tilde c_n$, and $h$ are polynomials. Without
loss of generality we may assume that $h$ is a constant. Hence, for
some integer $g$,
$$p(\lambda,x)=v_g(x) \lambda^g + v_{g-1}(x) \lambda^{g-1} +...+ v_0(x)$$
and
$$\psi(\lambda,x)=(r_g(x)\lambda^g+r_{g-1}(x)\lambda^{g-1}+...+r_0(x))
 \exp(\lambda x)$$
where $r_j=v_j/v$ for $j=0,...,g$. Since $z^{1/n}=\lambda
+O(\lambda^{-1})$ as $\lambda$ tends to infinity, asymptotic
considerations along the lines of Wasow \cite{Wasow} prove that
$r_g(x)=1$. For the sake of completeness Wasow's technique is outlined
in the appendix.
\end{proof}

\subsection{The simply periodic case}
\begin{thm} \label{perstrct}
Consider the differential expression
$$L=D^n+q_{n-2}D^{n-2}+...+q_0$$
where $q_{n-2}$, ..., $q_0$ are simply periodic meromorphic functions
which are bounded at the ends of the period strip. Let the period be
$p$ and define $\rho_k=\lim_{\Im(x/p)\to\infty}q_k(x)$ for
$k=0,...,n-2$. Assume that, for all $z\in\bb C$, all solutions of the
equation $Ly=zy$ are meromorphic. Then $Ly=zy$ has a solution of the
form
$$\psi(\lambda,x)=(\lambda^g+r_{g-1}(t(x))\lambda^{g-1}+...+r_0(t(x)))
 \exp(\lambda x),$$
where $t(x)=\exp(2\pi ix/p)$, $r_0,...,r_{g-1}$ are rational functions,
and
$$z=\lambda^n+\rho_{n-2}\lambda^{n-2}+...+\rho_0.$$
\end{thm}
\begin{proof}
If $L$ has constant coefficients the theorem is trivially true with
$g=0$. Hence assume that $L$ does not have constant coefficients.

Theorem \ref{floquetplus} applies and gives us a solution
$$\frac{u_\lambda(t(x))}{v(t(x))}\exp(i\lambda x)$$
where $u_\lambda$ and $v$ are polynomials. Again, we only have to study
the behavior of $u_\lambda$ with respect to $\lambda$. A similar proof
as above, another call on Wasow's theorem, and a replacing $i\lambda$
by $\lambda$ shows the validity of the present claim.
\end{proof}






\appendix
\section{Asymptotic behavior} \label{asymptotics}
Suppose the functions $q_{n-2}$, ..., $q_0$ are analytic in some open
set $\Omega$ containing $x_0$. We want to study the behavior of
solutions of the differential equation
$$Ly=y^{(n)}+q_{n-2} y^{(n-2)}+...+q_0y=\mu^n y$$
as $\mu$ tends to infinity.

Let
$$T=
\begin{pmatrix}
 1 & ... & 1 \\
 \mu\sigma_1 & ... & \mu\sigma_n \\
 \vdots &  & \vdots \\
 (\mu\sigma_1)^{n-1} & ... & (\mu\sigma_n)^{n-1}
\end{pmatrix}$$
where $\sigma_1$, ..., $\sigma_n$ denote the different $n$-th roots of
one. The substitution
$$(y(x),...,y^{(n-1)}(x))^t=Tu(x-x_0)$$
and letting $\varepsilon=1/\mu$ transforms the equation $Ly=\mu^n y$
into a system $\varepsilon u'=A(\varepsilon,\cdot)u$ where
$$A(\varepsilon,t)=\sum_{j=0}^n A_j(t) \varepsilon^j$$
with $A_0=\operatorname{diag}(\sigma_1,...,\sigma_n)$, $A_1=0$, and
$A_j$ analytic in a vicinity of zero.

When $r$ is a positive number and $I$ a real open interval we denote by
$S(r,I)$ the set $\{z\in\bb C: |z|<r, \arg(z)\in I\}$ and by $K(r)$ the
set $\{z\in\bb C: |z|<r\}$.

Then, by a repeated application of Theorem 26.2 of Wasow \cite{Wasow}
and its proof (in particular, the formulas 25.19 -- 25.22) and because
of the absence of a term $\varepsilon A_1(t)$, there exist numbers
$\rho$ and $\delta$, an interval $I$, and matrix-valued functions
$P:K(\rho)\times S(\delta,I)\to\bb C$ and $B:K(\rho)\times
S(\delta,I)\to\bb C$ such that
\begin{enumerate}
\item $P$ is holomorphic in both variables.
\item Asymptotically, as $\varepsilon$ tends to zero in $S(\delta,I)$,
$$P(t,\varepsilon)\sim I+\sum_{j=2}^\infty P_{j}(t)\varepsilon^j.$$
\item The transformation $u=Pw$ takes the equation $\varepsilon u'
=A(\varepsilon,\cdot)u$ into the completely decoupled system
$$\varepsilon w'=B(\varepsilon,\cdot) w$$
where $B$ is diagonal and has the asymptotic expansion
$$B(\varepsilon,t)\sim A_0+\sum_{j=2}^\infty B_{j}(t)\varepsilon^j$$
as $\varepsilon$ tends to zero in $S(\delta,I)$.
\end{enumerate}

A fundamental matrix of $\varepsilon w'=B(\varepsilon,\cdot) w$ is
$$w(\varepsilon,t)=\exp(\varepsilon^{-1}\int_0^t B(\varepsilon,s)ds)
 =\exp(A_0\mu t) \exp(\varepsilon C(t))$$
for a suitable diagonal matrix $C(t)$. Since $P_0(t)=I$ and
$\exp(\varepsilon C(t))=I+O(\varepsilon)$ we obtain for the asymptotic
behavior of $u$
$$u(\varepsilon,t)=(I+O(\varepsilon))\exp(A_0\mu x).$$
Linear independent solutions of $Ly=\mu^ny$ are given by
\begin{align*}
y_j(\mu,x)&=\sum_{k=1}^n u_{k,j}(\varepsilon,x-x_0)
 =\sum_{k=1}^n (\delta_{j,k}+O(\varepsilon))
 \exp(\mu \sigma_j (x-x_0))\\
 &=(1+O(\mu^{-1})) \exp(\mu \sigma_j (x-x_0)).
\end{align*}













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