@@ -149,7 +149,7 @@ There is a coordinate-free description as well:

\[

\benbrace*{\mathd,f}\omega=\mathd(f \omega)- f(\mathd\omega)=(\mathd f)\wedge\omega+ f \mathd\omega- f \mathd\omega=(\mathd f)\wedge\omega

\]

Again via \cref{thm:diff_op_coordfree} we see that $\mathd$ is a differential operator of order one.

Again via \cref{thm:diff_op_coordfree} we see that $\mathd$ is a differential operator of order 1.

\item Let $E \to M$ be an arbitrary vector bundle with a \Index{connection}$\nabla\colon\Gamma(M,E)\to\Gamma(M, \Tang^*M \otimes E)$, i.e. $\nabla$ is linear and fulfils $\nabla(f s)=\mathd f \otimes s + f \nabla s$ for $f \in C^\infty(M)$ and $s \in\Gamma(M,E)$.

In particular

\[

...

...

@@ -298,7 +298,7 @@ Furthermore a symmetric differential operator may be \Index{selfadjoint}, i.e.\

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:

Consider the open $1$-manifold $M=(0,1)$ (with the usual Riemannian metric) and $D= i \diffd{}{x}$ (we need the factor $i$ here for $D$ to be symmetric, use integration by parts to see this).

The Hilbert space we get is $L^2[0,1]$.

...

...

@@ -310,15 +310,17 @@ As the following example shows, all three operators may be different:

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:

\begin{lemma}

\begin{lemma}\label{lem:symmetric_ess_sa}

Let $D$ be a symmetric differential operator and $u \in L^2(M,E)$\emph{compactly supported}.

Then $u$ belongs to the minimal domain of $D$ if and only if it belongs to the maximal domain.

\end{lemma}

In particular we see, that every symmetric differential operator on a closed manifold is essentially selfadjoint.

More generally every compactly supported, symmetric differential operator on an open manifold is essentially selfadjoint.

\begin{corollary}

Every symmetric differential operator on a closed manifold is essentially selfadjoint.

More generally every compactly supported, symmetric differential operator on an open manifold is essentially selfadjoint.

\end{corollary}

\begin{proof}

\begin{proof}[Proof of \cref{lem:symmetric_ess_sa}]

Let $K$ be a compact subset of $M$.

We need \emph{Friedrichs' mollifiers} to proceed:

A local variant is discussed in \cref{prop:friedrichs}, which can be globalised via the usual arguments to operators $F_t \colon L^2(K,E)\to L^2(M,E)$ such that

...

...

@@ -333,11 +335,20 @@ More generally every compactly supported, symmetric differential operator on an

\[

D F_t u = F_t D u +\benbrace*{D,F_t} u

\]

By \cref{rmk:max_domain_distribution} and (a) we know, that the first summand is uniformly bounded; the second one is uniformly bounded by (d).

By \cref{rmk:max_domain_distribution} and (a) we know, that the first summand is uniformly bounded; the second one is uniformly bounded by (d).

Considering $t=\frac{1}{n}$ we get a sequence of elements $u_n$ in the domain of $D$ converging to $u$ such that $\norm*{D u_n}$ is uniformly bounded.

Hence \cref{lem:charac_minimal_domain} implies, that $u$ is in the minimal domain of $D$.

\end{proof}

As seen in \cref{ex:max_domain_neq_min_domain} the selfadjointness of differential operators on open manifolds is a more delicate matter.

We need to look at multiplication operators.

\begin{definition}

Let $D$ be a differential operator on $M$.

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.

...

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@@ -345,8 +356,6 @@ More generally every compactly supported, symmetric differential operator on an

\begin{proof}

\todo{add proof}

\end{proof}

\todo[inline]{include wave operator stuff as well?}