Commit bd382af5 authored by Jannes Bantje's avatar Jannes Bantje

split into another section

parent 834eb326
Pipeline #8032 passed with stages
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......@@ -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}
......
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