split into another section

contents/diffop.tex

@@ -313,7 +313,7 @@ Using distribution theory\footnote{see \url{https://en.wikipedia.org/wiki/Distri
 	To make sense of the expression $D u$ here, one has to use distribution theory, i.e. the \Index{distribution} $D u$ is required to be an actual Element of $L^2(M,E)$.
 \end{remark}
 
-Furthermore a symmetric differential operator may be \Index{selfadjoint}, i.e.\ fullfil $\overline{D}=D^*$.
+Furthermore a symmetric differential operator may be \Index{essentially selfadjoint} (or just \Index{selfadjoint}), i.e.\ fullfil $\overline{D}=D^*$.
 Note that the closure $\overline{D}$ is an extension of $D$ in general and that $D^*$ is an extension of $\overline{D}$ in the symmetric case.
 As the following example shows, all three operators may be different:
 
@@ -327,7 +327,18 @@ As the following example shows, all three operators may be different:
 \end{example}
 
 This example already suggests, that non-selfadjointness is in some way connected to \enquote{boundary conditions} on a non-compact manifold $M$.
-The same phenomenon shows up in the following statement:
+We investigate this phenomenon in the next section and conclude the present section with the following statement about elliptic operators.
+
+\begin{lemma}\label{lem:adjoint_elliptic}
+	If $D$ is a symmetric, essentially selfadjoint elliptic operator, then its adjoint is elliptic as well.
+\end{lemma}
+\begin{proof}
+	\todo{add proof}
+\end{proof}
+% section hilbert_space_techniques_and_the_formal_adjoint (end)
+
+\section{Selfadjointness of Differential Operators} % (fold)
+\label{sec:self_adjointness_of_differential_operators}
 
 \begin{lemma}\label{lem:symmetric_ess_sa}
 	Let $D$ be a symmetric differential operator and $u \in L^2(M,E)$ \emph{compactly supported}.
@@ -367,15 +378,9 @@ We need to look at multiplication operators.
 	Then $M$ is called \Index{complete} for $D$ if there is a smooth proper function $g \colon M \to \mathbb{R}$ such that $\benbrace*{D,\rho(g)}$ is a bounded Hilbert space operator.
 \end{definition}
 
-\todo[inline]{include wave operator stuff as well?}
 
-\begin{lemma}\label{lem:adjoint_elliptic}
-	If $D$ is a symmetric elliptic operator, then its adjoint is elliptic as well.
-\end{lemma}
-\begin{proof}
-	\todo{add proof}
-\end{proof}
-% section hilbert_space_techniques_and_the_formal_adjoint (end)
+\todo[inline]{include wave operator stuff as well?}
+% section self_adjointness_of_differential_operators (end)
 
 \section{Analysis of Elliptic Differential Operators} % (fold)
 \label{sec:analysis_of_elliptic_differential_operators}