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\markboth{\hfil Cesaro asymptotic equipartition \hfil EJDE--2000/70}
{EJDE--2000/70\hfil Stefan Boller \hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations},
Vol.~{\bf 2000}(2000), No.~70, 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 ftp ejde.math.unt.edu (login: ftp)}
 \vspace{\bigskipamount} \\
 %
  Cesaro asymptotic equipartition of energy in the coupled case 
 %
\thanks{ {\em Mathematics Subject Classifications:} 36G10, 47B25, 47D03, 34D05.
\hfil\break\indent
{\em Key words:} Cesaro asymptotic equipartition of energy; selfadjoint
operator matrices; \hfil\break\indent
direct sum Hilbert space; evolution equations; initial
boundary value problem; \hfil\break\indent
coupled boundary condition.
\hfil\break\indent
\copyright 2000 Southwest Texas State University. \hfil\break\indent
Submitted March 14, 2000. Published November 20, 2000.} }
\date{}
%
\author{ Stefan Boller }
\maketitle

\begin{abstract} 
    It is well known from earlier results that certain types of selfadjoint
    operators, e.g. operators allowing a representation as operator matrices,
    show equipartition of energy. In this paper we examine the question
    whether there are more selfadjoint operators showing equipartition of
    energy in the Cesaro mean. For this purpose we proof a necessary and
    sufficient criterion for equipartition of energy and use this criterion to
    show equipartition for a system of partial differential equations with a
    coupled boundary condition. 
\end{abstract}

\newtheorem{thm}{Theorem}[section]
\newtheorem{cor}[thm]{Corollary}
\newtheorem{lem}[thm]{Lemma}
\theoremstyle{definition}
\newtheorem{defn}{Definition}[section]
\theoremstyle{remark}
\newtheorem{rem}{Remark}[section]
\numberwithin{equation}{section}

 \newcommand{\thmref}[1]{Theorem~\ref{#1}}
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 \newcommand{\corref}[1]{Corollary~\ref{#1}}
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%    Enclose the argument in vert-bar delimiters:
 \newcommand{\abs}[1]{\left\lvert#1\right\rvert}
%    Enclose the argument in double-vert-bar delimiters:
 \newcommand{\norm}[1]{\left\lVert#1\right\rVert}
 \newcommand{\SProd}[2]{\bigl\langle#1\rvert#2\bigr\rangle}
%    Hilbert-R"aume
 \renewcommand{\H}{\mathcal{H}}
%    Definitionsbereich, Nullraum und Wertebereich
 \newcommand{\Dom}[1]{\mathcal{D}(#1)}
 \newcommand{\Ran}[1]{\mathcal{W}(#1)}
 \newcommand{\Lim}[1]{\lim_{#1\rightarrow\infty}}


\section{Introduction}

In this paper we examine the phenomenon of asymptotic equipartition of energy
for abstract evolution equations involving selfadjoint operators. This means
that if a Hilbert space $\H$ is the direct sum of $n$ Hilbert spaces $\H_{i}$
($i=1,\dots,n$), then, roughly speaking, each component contributes equal parts
to the conserved total energy. More precisely, let $\pi_{i}$ be the
(orthogonal) projections on the $i$-th component of $\H$ with respect to this
decomposition and let $A$ be be a selfadjoint operator on $\H$ with domain
$\Dom{A}$. Then we consider the following evolution equation:
\begin{equation}\label{Diffgl}
  \begin{split}
    D_{0}u(t)&=Au(t)\\
    u(0)&=u_0,
  \end{split}
\end{equation}
where $D_{0}=\frac{1}{2\pi i}\partial_{0}$ is the differentiation with respect
to the time $t$. The uniquely determined solution of \eqref{Diffgl} is the
unitary group $U(t):=e^{2\pi iAt}$ generated by $2\pi iA$. 

Since $U(\cdot)$ is unitary, the square of the norm $\norm{u(t)}^{2}$
(the energy) of $u(t)$ is conserved, i.e.
\begin{equation*}
  \sum_{i=1}^{n}\norm{\pi_{i}u(t)}^{2}=\norm{u(t)}^{2}=
  \norm{U(t)u_{0}}^{2}=\norm{u_{0}}^{2}=\mbox{const.}
\end{equation*}

Now we can give the following
\begin{defn}\label{equipartition}
  \begin{enumerate}
    \item
    The selfadjoint operator $A$ admits \textbf{asymptotic
    equipartition of energy}, if for all $u_{0}\in\H$ and
    the corresponding solution $u(t)$ the following asymptotic condition is
    true 
    \begin{equation}
      \Lim{t}\norm{\pi_i u(t)}^{2}=\frac{1}{n}\norm{u_0}^{2}.
    \end{equation}
    (See \cite{GS79})
    \item
    The selfadjoint operator $A$ admits \textbf{asymptotic
    equipartition of energy in the Cesaro sense}, if for all $u_{0}\in\H$ and
    the corresponding solution $u(t)$ the following asymptotic condition is
    true
    \begin{equation}
      \Lim{T}\frac{1}{T}\int_{0}^{T}\norm{\pi_i u(t)}^{2}dt=
      \frac{1}{n}\norm{u_0}^{2}.
    \end{equation}
  \end{enumerate}
\end{defn}

The earliest results concerning energy equipartition seem to be presented by
Brodsky (\cite{Br67}) and Lax and Phillips (\cite[Cor. 2.3,
p. 106]{LaxPhi67}). In the following there was a continuing
interest in this question. In
particular Goldstein and Sandefur contributed a lot to this area
(e.g. \cite{Go69}, \cite{Go70}, \cite{GS79}, \cite{Go86}, \cite{GS87}, they
treated also more general situations). Goldstein (\cite{Go70}) and Duffin
(\cite{Du70}) showed results concerning energy equipartition from a
finite time on. Also
Picard and Seidler (cf. e.g. \cite{PiSe87}, \cite{Pi86}, \cite{Pi91}) examined
equipartition results, where they choose matrices of operators as an ansatz for
the operator $A$. For further contributions consult the references. A
recent paper is written by Goldstein, de~Laubenfels and Sandefur
(\cite{GLS93}).

In the known results on equipartition the operator $A$ is assumed to be
decomposable in operators with special properties, e.g. as a matrix of
closed operators or generators of (regularized) semigroups. This is
typically the case for a partial differential equation with decoupled
boundary conditions, i.e. when the boundary condition restricts the components
separately.

In this paper we want to examine, if there are more operators not
belonging to this class, which nevertheless show equipartition of energy. For
simplicity we restrict our considerations to the case of energy equipartition
in the Cesaro sense for 2-component systems ($n=2$ in \defnref{equipartition}).

After giving some prerequisites in section \ref{sec3} we proof a necessary and
sufficient condition for asymptotic energy equipartition in section
\ref{sec3:2}. This condition contains some previous results. Further we apply
this condition in section \ref{sec2:2} to a system of partial differential
equations with a coupled boundary condition. 

In the last section we give a short outlook on open questions in this area. 

\section{Prerequisites}
\label{sec3}

As the starting point for our considerations we use the following ansatz given
by Picard and Seidler in \cite{PiSe87} essentially equivalent to that examined
by Goldstein in \cite{Go69}, \cite{Go70}. They considered $2\times
2$-operator-matrices in $\H=\H_{1}\oplus\H_{2}$ of the form
\begin{equation*}
  A=\begin{pmatrix}0&B^{*}\\B&0\end{pmatrix}
\end{equation*}
with $B:\Dom{B}\subset\H_{1}\to\H_{2}$ a densely defined, closed operator
and the corresponding initial value problem
\begin{equation}\label{AnfsProbl}
  \begin{split}
    D_{0}u&=Au\\
    u(0)&=u_{0}\in\H.
  \end{split}
\end{equation}
They proved the following theorem concerning equipartition in the Cesaro sense:
\begin{thm}\label{satzPic2}
  For every initial value $u_{0}\in\H$ the following asymptotic relations hold
  \begin{equation*}
    \begin{split}
      \Lim{T}\frac{1}{T}\int_{0}^{T}\norm{\pi_{1}u(t)}^{2}dt
      &=\tfrac{1}{2}\norm{Qu_{0}}^{2}+\norm{\pi_{1}Pu_{0}}^{2}\\
      \Lim{T}\frac{1}{T}\int_{0}^{T}\norm{\pi_{2}u(t)}^{2}dt
      &=\tfrac{1}{2}\norm{Qu_{0}}^{2}+\norm{\pi_{2}Pu_{0}}^{2},
    \end{split}
  \end{equation*}
  where $P$, $Q=I-P$ are the projections to the kernel and the
   closure of the range of $A$, respectively.
\end{thm}

For the proof of this theorem the following lemma is essential. We
will use this lemma later, so we cite it here.
\begin{lem}\label{lemPic}
  Let $E_{\lambda}$ be the spectral measure of $A$ and
  $T:=\begin{pmatrix}1&0\\0&-1\end{pmatrix}$. Then
  \begin{equation*}
    TE_{I}=E_{-I}T
  \end{equation*}
  for all intervals $I\subset\mathbb{R}$.
\end{lem}

As a corollary of \thmref{satzPic2} we get the following result originally
showed by Goldstein in \cite[Theorem 4]{Go70}:
\begin{cor}\label{corPic}
  We have energy equipartition in the Cesaro sense for every initial value
  $u_{0}\in\H$ of \eqref{AnfsProbl} if and only if $0$ is not an eigenvalue of
  $A$.
\end{cor} 

If looking for a criterion for equipartition of energy in the Cesaro
sense we must examine the asymptotic behaviour of 
\begin{equation}\label{produkt}
  S_{u_{0}}(t):=\SProd{u(t)}{T u(t)}=
  \norm{\pi_{1}u(t)}^{2}-\norm{\pi_{2}u(t)}^{2},
\end{equation}
where $u(t)$ is the solution of \eqref{Diffgl} for the initial value $u_{0}$
and $T$ is the operator defined in \lemref{lemPic}.
We see that we have energy equipartition if and only if
\begin{equation}\label{eqpart}
  \Lim{T}\frac{1}{T}\int_{0}^{T}S_{u_{0}}(t)dt=
  \Lim{T}\frac{1}{T}\int_{0}^{T}\norm{\pi_{1}u(t)}^{2}dt
  -\Lim{T}\frac{1}{T}\int_{0}^{T}\norm{\pi_{2}u(t)}^{2}dt=
  0.
\end{equation}
Inserting the unitary group $e^{2\pi iAt}$ generated by $2\pi iA$
in \eqref{produkt} we get:
\begin{equation*}
  S_{u_{0}}(t)=\SProd{\exp(2\pi iAt)u_{0}}{T\exp(2\pi iAt)u_{0}}.
\end{equation*}
For the examination of this expression the following theorem about a functional
calculus for scalar products of non-commuting operator functions is quite
useful. 
\begin{thm}\label{prodsatz}
  Let $A,B$ be selfadjoint (unbounded) operators on a Hilbert
  space $\H$, $T$ a bounded operator on $\H$. Let further
  $f,g:\mathbb{R}\to\mathbb{C}$ be measurable  functions (measurable with
  respect to the Borel-$\sigma$-algebras on $\mathbb{R}$ and
  $\mathbb{C}$, respective),  and $x\in\Dom{f(A)}$, $y\in\Dom{g(B)}$.  
  Further let $\overline{f}\otimes g$ be integrable with respect to
  $\mu_{x,y}$, where $\mu_{x,y}$ is the uniquely determined (complex) measure
  on the Borel-$\sigma$-algebra $\Sigma^{2}$ in $\mathbb{R}^{2}$, such that
  for every measurable rectangle $R=M_{1}\times M_{2}$
  ($M_{1},M_{2}\in\Sigma$, $\Sigma$ the Borel-$\sigma$-algebra on $\mathbb{R}$)
  \begin{equation*}
    \mu_{x,y}(R)=\SProd{E_{M_{1}}x}{TF_{M_{2}}y}
  \end{equation*}
  is valid. Here $E_{M_{1}}$ and $F_{M_{2}}$ are the spectral
  measures with respect to the two operators $A$ and
  $B$, resp. 
  For this measure we have the following result:
  \begin{equation}\label{prodgleich}
    \SProd{f(A)x}{Tg(B)y}=
    \int_{\mathbb{R}\times\mathbb{R}}\overline{f}\otimes gd^{2}\mu_{x,y}.
  \end{equation}
\end{thm}

\begin{proof}
  The proof is standard measure theory, so we omit the details
  (cf. \cite{Bol}). 
\end{proof}

\begin{cor}\label{cor31}
  The assertion of \thmref{prodsatz} applies in particular to bounded,
  measurable functions $f$, $g$.
\end{cor} 


\section{A Criterion for Energy Equipartition for Selfadjoint Operators}
\label{sec3:2}

We want to use \corref{cor31} for the examination of \eqref{produkt}. Let
$T$ be the following bounded operator on $\H$:
\begin{equation*}
    (u,v)\mapsto T(u,v)=(u,-v).
\end{equation*}
According to \corref{cor31} we have
\begin{equation}\label{eq3:2:1}
    S_{u_{0}}(t)=\int_{\mathbb{R}\times\mathbb{R}}\exp(-2\pi i(\lambda-\mu) t)
          d^{2}\nu_{u_{0}}(\lambda,\mu),
\end{equation}
where $\nu_{u_{0}}$ is the measure of \thmref{prodsatz} for
    $\SProd{E_{M_{1}}u_{0}}{TE_{M_{2}}u_{0}}$, with
$E_{\lambda}$ the spectral measure of $A$. Now we
will prove the following theorem giving a necessary and sufficient
criterion for energy equipartition.

\begin{thm}\label{satz3:2:1}
  If $u_{0}\in\H$, $S_{u_{0}}$ as in \eqref{eq3:2:1}, then
  \begin{equation*}
    \Lim{T}\frac{1}{T}\int_{0}^{T}{S_{u_{0}}(t)dt}=
    \int_{\left\{\lambda=\mu\right\}}{d^{2}\nu_{u_{0}}(\lambda,\mu)}
    .
  \end{equation*}
\end{thm}

\begin{proof}
  Applying Fubini's theorem and the dominated convergence theorem we get
  \begin{equation*}
    \begin{split}
      \Lim{T}\frac{1}{T}\int_{0}^{T}{S_{u_{0}}(t)dt}
      &=\Lim{T}\frac{1}{T}\int_{0}^{T}\left({\int_{\mathbb{R}\times\mathbb{R}}
          \exp(-2\pi i(\lambda-\mu) t)d^{2}\nu_{u_{0}}(\lambda,\mu)}\right)dt\\
      &=\Lim{T}\frac{1}{T}\int_{\left\{\lambda\not=\mu\right\}}
        {\left(\int_{0}^{T}{\exp(-2\pi i(\lambda-\mu) t)dt}\right)
        d^{2}\nu_{u_{0}}(\lambda,\mu)}\\
      &\quad+\Lim{T}\frac{1}{T}\int_{\left\{\lambda=\mu\right\}}
        {\left(\int_{0}^{T}{\exp(-2\pi i(\lambda-\mu) t)dt}\right)
        d^{2}\nu_{u_{0}}(\lambda,\mu)}\\
      &=\int_{\left\{\lambda\not=\mu\right\}}
        {\Lim{T}\frac{\exp(-2\pi i(\lambda-\mu)T)-1}{-2\pi i(\lambda-\mu)T}
        d^{2}\nu_{u_{0}}(\lambda,\mu)}\\
      &\quad+\int_{\left\{\lambda=\mu\right\}}{d^{2}\nu_{u_{0}}(\lambda,\mu)}\\
      &=\int_{\left\{\lambda=\mu\right\}}{d^{2}\nu_{u_{0}}(\lambda,\mu)},
    \end{split}
  \end{equation*}
  which proves the assertion.
\end{proof}

As a consequence of this theorem we can formulate

\begin{thm}\label{satz3:2:2}
  Asymptotic energy equipartition in the Cesaro sense for the operator
  $A$ holds, if and only if 
  \begin{equation}\label{int:beh}
    \int_{\left\{-K<\lambda=\mu\leq K\right\}}
         {d^{2}\nu_{u_{0}}(\lambda,\mu)}=0
  \end{equation}
  for every $u_{0}\in\H$ and every $K\in\mathbb{R}^{+}$.
\end{thm}

\begin{proof}
  It follows from \eqref{eqpart} with the help of \thmref{satz3:2:1}.
\end{proof}

For calculation purposes the next lemma is useful.
\begin{lem}\label{satz3:2:3}
  We have
  \begin{equation*}
    \int_{\left\{-K<\lambda=\mu\leq K\right\}}{d^{2}\nu_{u_{0}}(\lambda,\mu)}=
    \Lim{n}\sum_{k=-n}^{n}{\SProd{E_{I_{k}^{(n)}}u_{0}}
                                 {TE_{I_{k}^{(n)}}u_{0}}},
  \end{equation*}
  where 
  \begin{equation*}
    I_{k}^{(n)}:=\left\{\lambda\in\mathbb{R}\vert
         (k-\tfrac{1}{2})\epsilon_{n}<\lambda\leq(k+\tfrac{1}{2})\epsilon_{n}
       \right\}
    \quad(k=-n,\dots,n)
  \end{equation*}
  for $n\in\mathbb{N}$ and $\epsilon_{n}=\frac{2K}{2n+1}$. 
\end{lem}

\begin{proof}
  By the dominated convergence theorem we get  
  \begin{equation*}
     \int{\chi_{\left\{-K<\lambda=\mu\leq K\right\}}
       d^{2}\nu_{u_{0}}(\lambda,\mu)}
     =\Lim{n}\sum_{k=-n}^{n}
        \int{\chi_{(I_{k}^{(n)}\times I_{k}^{(n)})}(\lambda,\mu)
      d^{2}\nu_{u_{0}}(\lambda,\mu)},
  \end{equation*}
  because
  \begin{equation*}
    \chi_{\left\{-K<\lambda=\mu\leq K\right\}}=
    \Lim{n}\chi_{\bigcup_{k=-n}^{k=n}(I_{k}^{(n)}\times I_{k}^{(n)})},
  \end{equation*}
  where $\chi_{M}$ is the characteristic function of the set $M$.
  Furthermore we get
  \begin{equation*}
    \begin{split}
      \int{\chi_{(I_{k}^{(n)}\times I_{k}^{(n)})}(\lambda,\mu)
        d^{2}\nu_{u_{0}}(\lambda,\mu)}
      &=\int{\chi_{I_{k}^{(n)}}(\lambda)\chi_{I_{k}^{(n)}}(\mu)
        d^{2}\nu_{u_{0}}(\lambda,\mu)}\\
      &=\SProd{E_{I_{k}^{(n)}}u_{0}}
                                 {TE_{I_{k}^{(n)}}u_{0}}.
    \end{split}
  \end{equation*}
\end{proof}

\begin{rem}
  From the last result we can extract the condition given by Picard and Seidler
in \cite{PiSe87} using \lemref{lemPic}. 
\end{rem} 

In the case of operators with compact resolvent we get the following simple
criterion. 
\begin{cor}\label{cor3:1}
    Let $A$ be an operator with compact resolvent. Then we have energy
      equipartition in the Cesaro sense if and only if
      \begin{equation*}
        \norm{\pi_1 P_j u_{0}}^{2}=\norm{\pi_2 P_j u_{0}}^{2}
        \text{ for all $u_{0}\in\H$.}
      \end{equation*}
\end{cor}

\begin{proof}
  If $A$ has a compact resolvent there exists an at most countably infinite
  set of eigenvalues of $A$ with no (finite) accumulation point. Hence there
  exists an $N\in\mathbb{N}$, such that in every interval $I_{k}^{(N)}$ of
  \lemref{satz3:2:3} lies at most one eigenvalue, i.e. $E_{I_{k}^{(n)}}=0$ or
  $E_{I_{k}^{(n)}}=P_j$, where $P_j$ is the (orthogonal) projection on the
  eigenspace for the eigenvalue, which lies in the interval
  $I_{k}^{(n)}$. Using \lemref{satz3:2:3} and \thmref{satz3:2:2} we get the
  condition:  
  \begin{equation*}
    0=\Lim{n}\sum_{k=-n}^{n}{\SProd{E_{I_{k}^{(n)}}u_{0}}
      {T E_{I_{k}^{(n)}}u_{0}}}
     =\sum_{j}{\SProd{P_j u_{0}}{T P_j u_{0}}}
    \text{ for all $u_{0}\in\H$.}
  \end{equation*}
  Here the summation extends over all $j$, that lie in the interval
  $(-K,K\rbrack$. Since the condition is true for every $u_{0}$ and
  every $K\in\mathbb{R}^{+}$, the assertion follows.
\end{proof}

\begin{rem}
  The condition of \corref{cor3:1} can be formulated also in terms of the
  eigenvectors of $A$. Let $B_k:=\left\{x_j^{(k)}\right\}$ be a (finite)
  orthonormal basis of the eigenspace of the $k$-th eigenvalue $\lambda_k$. 
  Then the condition of \corref{cor3:1} becomes
  \begin{equation*}
    \SProd{\pi_i x_j^{(k)}}{\pi_i x_l^{(k)}}=\frac{1}{2}\delta_{jl} 
    \quad i=1,2,\forall j,k,l.
  \end{equation*}
\end{rem}

\section{Energy Equipartition for a System with Coupled Boundary Condition}\label{sec2:2}

  In this section we examine as an example for the theorem just proven an
  operator which can not be regarded as a matrix of operators like in the
  articles of Picard and Seidler (\cite{PiSe87}, \cite{Pi91}).

  We consider on the Hilbert space $\H:=L_{2,p}(I)\oplus L_{2,q}(I)$, with the
  scalar product 
    \begin{equation*}
      \SProd{(u,v)}{(u',v')}_{\H}:=\SProd{u}{u'}_{p}+\SProd{v}{v'}_{q}:=
      \SProd{p^{-1}u}{u'}_{0}+\SProd{q^{-1}v}{v'}_{0},
    \end{equation*} 
  where $p,q\in L_{\infty}(I)$ with $p(x),q(x)\geq c_{0}>0$ for almost every
  $x\in I$ ($I=(a,b)$ with $a,b\in\mathbb{R}$), the following operator:

  \begin{subequations}\label{Op:Def}
    \begin{equation}
      A=
      \begin{pmatrix}
        0 &pD\\
        qD&0
      \end{pmatrix}
    \end{equation}
    with domain
    \begin{equation}
      \Dom{A}:=\{(u,v)\in\H_{1}(I)\oplus\H_{1}(I)|
                             u(a)+\alpha v(a)=0\land u(b)+\beta v(b)=0\},
    \end{equation}
    where $\H_{1}(I)=W^{1}_{2}(I)$ is the Sobolev space of once differentiable
    $L_{2}$-functions ($\H_{1}(I)\subset\mathbf{C}(\overline{I})$!).
  \end{subequations}
  By a simple calculation we see that this operator is selfadjoint for
  $\alpha,\beta$ being purely imaginary but can not
  be regarded as an operator matrix (with respect to the given decomposition
  of $\H$), if $\alpha\not=0$ or $\beta\not=0$. Also one can see that
  $\lambda=0$ is an eigenvalue if and only if $\alpha=\beta$.

  With standard arguments we can now show that $A$
  is an operator with compact resolvent (cf. e.g. \cite[ch.7.4,
  p.142ff]{Le86} or \cite{Bol}). We get the following
  \begin{thm}\label{komp}
    \begin{enumerate}
      \item $\Ran{A}$ is closed, where $\Ran{A}$ is the range of $A$.
      \item $A^{-1}: \Ran{A}\rightarrow\Dom{A}\cap\Ran{A}$ exists,
      \item $A^{-1}$ is compact.
    \end{enumerate}
  \end{thm}


  If we restrict $A$ to the range $\Ran{A}$ we get a selfadjoint operator
  with compact resolvent, which will be denoted again with $A$.
  
  Using \corref{cor3:1} we prove now the following 
  \begin{thm}\label{randbed}
    Energy equipartition in the Cesaro sense is true for the operator
    $A$, if and only if the components $v_{i}$, $w_{i}$ of
    the (orthonormal) eigenvectors $u_{i}$ corresponding to the eigenvalue
    $\lambda_{i}$ fulfill the following condition
    \begin{equation}\label{rand}
      \left[\overline{w_{i}}v_{j}\right]_{a}^{b}=0\quad\text{for every $i,j$.}
    \end{equation}
  \end{thm}

  \begin{proof}
    With the orthonormality of the eigenvectors we get
    \begin{equation*}
      \begin{split}
        \lambda\delta_{ij} &=\lambda \SProd{u_{i}}{u_{j}}_{\H}
                            =\SProd{\lambda u_{i}}{u_{j}}_{\H}
                            =\SProd{A u_{i}}{u_{j}}_{\H}=\\
                           &=\SProd{pD w_{i}}{v_{j}}_{p}+
                             \SProd{qD v_{i}}{w_{j}}_{q}
                            =\SProd{D w_{i}}{v_{j}}_{0}+
                             \SProd{D v_{i}}{w_{j}}_{0}.
      \end{split}
    \end{equation*}
    Integrating by parts and inserting the differential equation we get
    furthermore 
    \begin{equation*}
      \begin{split}
        \SProd{D w_{i}}{v_{j}}_{0}+\SProd{D v_{i}}{w_{j}}_{0}&=
             \SProd{w_{i}}{D v_{j}}_{0}+\SProd{D v_{i}}{w_{j}}_{0}
             -\frac{1}{2\pi i}\left[\overline{w_{i}}v_{j}\right]_{a}^{b}=\\
             &=\SProd{w_{i}}{q^{-1}\lambda w_{j}}_{0}
               +\SProd{q^{-1}\lambda w_{i}}{w_{j}}_{0}
               -\frac{1}{2\pi i}\left[\overline{w_{i}}v_{j}\right]_{a}^{b}=\\
             &=2\lambda\SProd{w_{i}}{w_{j}}_{q}
               -\frac{1}{2\pi i}\left[\overline{w_{i}}v_{j}\right]_{a}^{b}.
      \end{split}
    \end{equation*}
    Hence
    \begin{equation*}
      \SProd{w_{i}}{w_{j}}_{q}=\frac{1}{2}\delta_{ij}
           +\frac{1}{4\pi i\lambda}\left[\overline{w_{i}}v_{j}\right]_{a}^{b}.
    \end{equation*}
    Similarly we get
    \begin{equation*}
      \SProd{v_{i}}{v_{j}}_{q}=\frac{1}{2}\delta_{ij}
           -\frac{1}{4\pi i\lambda}\left[\overline{w_{i}}v_{j}\right]_{a}^{b},
    \end{equation*}
    which shows the assertion with the help of \corref{cor3:1}.
  \end{proof}

  As an example for operators satisfying the condition of the last theorem but
  not belonging to the class considered in \cite{PiSe87},
  i.e. $\alpha\not=0$ or $\beta\not=0$, we examine now the special case 
  $p=q$ with $p\in\mathbf{C}_{1}(\overline{I})$. Then the components $v,w$ of
  the eigenvectors are continuously differentiable and the eigenvalue equation
  for the operator $A$ can be formulated classically. So we have simple
  spectrum by the Picard-Lindel\"of theorem.
  
  Further we get the following conservation law:
  \begin{lem}\label{erhaltung}
    Let $A$ be as described above and let $u,v$ be the components of the
    eigenvectors of $A$, then 
    \begin{equation*}
      \abs{v}^{2}+\abs{w}^{2}=const.
    \end{equation*}
  \end{lem}

  \begin{proof}
    By multiplication of the differential equation by $\overline{(pD)v}$ and
    $(pD)w$, resp., we get 
    \begin{equation*}
      \begin{split}
        \overline{(pD)v}(pD)w&=\overline{(pD)v}\lambda v\\
        (pD)w\overline{(pD)v}&=(pD)w\overline{\lambda w}.
      \end{split}
    \end{equation*}
    Hence ($0\not=\lambda\in\mathbb{R}$):
    \begin{equation*}
      -(D\overline{v})v=(Dw)\overline{w}
    \end{equation*}
    and
    \begin{equation*}
      D\abs{v}^{2}=(Dv)\overline{v}+(D\overline{v})v=
      -(D\overline{w})w-(Dw)\overline{w}=-D\abs{w}^{2}.
    \end{equation*}
  \end{proof}

  Now we can derive the following
  \begin{thm}
    Let $A$ be as in \lemref{erhaltung}. We have energy equipartition in
    the Cesaro sense for $A$ iff $\alpha=\beta$ or $\alpha=-1/\beta$.
  \end{thm}

  \begin{proof}
    Inserting the boundary condition in \lemref{erhaltung} and
    \thmref{randbed} we get 
    \begin{equation*}
      \abs{w(b)}^{2}=\frac{-\alpha^{2}+1}{-\beta^{2}+1}\abs{w(a)}^{2}.
    \end{equation*}
    and
    \begin{equation*}
      \begin{split}
        \overline{w(b)}v(b)-\overline{w(a)}v(a)&=
         \overline{w(b)}(-\beta)w(b)-\overline{w(a)}(-\alpha)w(a)=\\
         &=\left(\beta\frac{\alpha^{2}-1}{-\beta^{2}+1}
                  +\alpha\right)\abs{w(a)}^{2}
         .
      \end{split}
    \end{equation*}
    The last expression is $0$ if and only if the following quadratic equation
    is fulfilled:
    \begin{equation*}
      \beta(\alpha^{2}-1)=\alpha(\beta^{2}-1).
    \end{equation*}
    (if $\abs{v(a)}^{2}=0$ we get, by \lemref{erhaltung}, that $v=0$
    which is not an eigensolution.)  
\end{proof}

\begin{rem}
  It can also be shown that this result remains true, when the interval $I$ is
  unbounded, e.g. $I=(0,\infty)$, and $p=q=1$ (see \cite{Bol}).
\end{rem}

\section{Outlook}\label{sec5}

Beginning with the result presented here there a some further questions to
solve. First there is the question, when do we have equipartition in the strong
sense, i.e. when do the components converge pointwise to the ratio
$\frac{1}{n}$ of the total energy, not only in the Cesaro mean. Here in
analogy to \thmref{satz3:2:1} the behaviour of $\Lim{t}S_{u_{0}}(t)$ has to be
examined. Here we expect, that we must state another condition like the 
{\em Riemann-Lebesgue-Condition} or the absolute continuity of the spectrum in
the decoupled case. It could be, that the absolute continuity of the
spectral measure implies also the absolute continuity of the measure
$\nu_{u_{0}}$ of \thmref{prodsatz} on the subsets
$D_{1}:=\{(\lambda,\mu)\in\mathbb{R}^{2}\vert\lambda<\mu\}$ and
$D_{2}:=\{(\lambda,\mu)\in\mathbb{R}^{2}\vert\lambda>\mu\}$, resp. So for
absolute continuous spectrum we would have the same condition as in the Cesaro
case. 

Further we could extend the considerations to bigger systems than $n=2$. Now
it can easily be seen that you can extend the condition of
\thmref{satz3:2:2} for $n>2$ to
  \begin{equation}
    \int_{\left\{-K<\lambda=\mu\leq K\right\}}
         {d^{2}\nu_{u_{0},k}(\lambda,\mu)}=0
    \text{ for all $u_{0}\in \H$ and for every $k=1,\dots,n$.,}
  \end{equation}
where the measures $\nu_{u_{0},k}$ are constructed like the measure
$\nu_{u_{0}}$ with the operators
$T_{k}=(\delta_{i,j}-n\delta_{i,j}\delta_{k,i})_{i,j}$ ($k=1,\dots,n$).

Last you could omit the restriction to selfadjoint operators and examine
equations as Goldstein et al. did for instance in \cite{Go86}, \cite{GS87}, or
\cite{GLS93} where they considered the
ratio of the components of the total energy, which is asymptotically constant
in certain cases. \smallskip

\noindent \textbf{Acknowledgment.}
The author thanks professor R. Picard for supporting this diploma thesis by
discussing and useful hints. Also he wishes to thank professors Voigt and
Timmermann, and the referee for his/her comments on the first version of this
paper. 
      
\bibliographystyle{plain}

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\noindent{\sc Stefan Boller}\\
Graduiertenkolleg Quantenfeldtheorie\\
Universit\"at Leipzig, Augustusplatz 10/11\\
04109 Leipzig, Germany\\
e-mail: Stefan.Boller@itp.uni-leipzig.de


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