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\begin{document}

%\renewcommand{\thefootnote}{$*$}



\title[]{\huge A Criterion for Essential Self-Adjointness}

\author{\large B. Thaller}

%\address{Erwin Schr\"odinger International Institute for Mathematical Physics,

%Pasteurgasse 6/7, A-1090 Vienna, Austria (Permanent address:

%Inst. f. Mathematik, Universit\"at Graz, Heinrichstra\ss e 36, A-8010 Graz, Austria)}

%\email{thaller@@bkfug.kfunigraz.ac.at}



\maketitle



\bigskip\bigskip



\begin{center}



{\sc Erwin Schr\"odinger International Institute for Mathematical Physics,

Pasteurgasse 6/7, A-1090 Vienna, Austria}



\end{center}



\bigskip\bigskip



\begin{abstract}

We prove some simple facts on the essential self-adjointness of a symmetric operator $T$ in a Hilbert space $\frak H$.

The main tools are bounded commutators of $T$ with a suitable sequence of bounded self-adjoint operators.

A special case is a Theorem of J\"orgens and Chernoff stating that the essential self-adjointness of the

Dirac operator in an external field depends only on the local properties of the

potential.

\end{abstract}



\vfill

\noindent {Permanent address: Institut f\"ur Mathematik, Universit\"at Graz, Heinrichstra\ss e 36, A-8010 Graz, Austria}\\

{\it E-mail address}: thaller@@bkfug.kfunigraz.ac.at



%%%%%%%%%%%%%%%%%%%%%%%%%-------------> Erzwungener Seitenwechsel

\newpage

%%%%%%%%%%%%%%%%%%%%%%%%%







%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Assumptions and Main Results}\label{sec1}

%%%%%%%%%%%%%%%%%%%%%%%%%

%

\setcounter{equation}{0}

\setcounter{prop}{0}

\bigskip



\noindent Let $\frak H$ be a separable Hilbert space.

$\{A_n\}_{n=0}^\infty$ denotes a sequence of bounded and self-adjoint operators on $\frak H$.

We consider a linear operator $T$ in $\frak H$, densely defined and symmetric on

$\Dom(T)=\Dom_0 \subset \frak H$

.



\begin{nota}

If a linear operator $S$ is defined and closeable on $\Dom_0$ we denote its adjoint by $S^*$

and its closure $S^{**}$ by $S^c$.

The restriction to $\Dom_0$ of an operator $B$ defined on some larger domain

is occasionally denoted by $B|_{\Dom_0}$.

\end{nota}



\smallskip



For the operators $A_n$ and $T$ we formulate the following conditions:



\medskip



\begin{quote}

\begin{enumerate}



\item[$\bold{A_1}$:\quad]

For all $n$ there exist $k$, $m$ with

\end{enumerate}

\end{quote}

%

\vspace{-.2cm}

\begin{equation}

A_kA_n=A_n,\qquad A_mA_k=A_k.

\end{equation}



\smallskip

\begin{quote}

\begin{enumerate}

\item[$\bold{A_2}$:\quad]

For all $n$, $[T,A_n] \equiv TA_n - A_nT$ is defined on $\Dom_0$ and bounded.



\medskip



\item[$\bold{A_3}$:\quad]

For all $n$ the operator ${A_nTA_n}|_{\Dom_0}$ is essentially self-adjoint.



\medskip



\item[$\bold{A_4}$:\quad]

For all $\psi\in \Dom(T^*)$, there is a subsequence $\{A_{n_k}\}_{k=0}^\infty$ converging weakly to ${\bold 1}$ such that

\end{enumerate}

\end{quote}

\vspace{-.1cm}

\begin{equation}

\lim_{k\to \infty}(T^*A_{n_k}\psi,\psi-A_{n_k}\psi) = 0.

\end{equation}



\smallskip



\noindent One of the main results which can be proved under the conditions above (or under similar conditions discussed

below) is the following:



%%%%%%%%%%%%%%%%%%%%%-THEOREM

%

\begin{thm}\label{th1}

Assume $\bold{A_1}$--\,$\bold{A_4}$. Then $T$ is essentially self-adjoint on $\Dom_0$.

\end{thm}

%

%%%%%%%%%%%%%%%%%%%%%



\smallskip



\noindent

As an illustration consider a Hilbert space which is an infinite orthogonal sum of closed subspaces.

Define

%

\begin{equation}

\frak{H} = \bigoplus_{k=0}^\infty \frak{H}^{(k)}\, ,\qquad

\frak{H}_n = \bigoplus_{k=0}^n \frak{H}^{(k)} \, ,

\label{B1}

\end{equation}

%%

and let $A_n$ be the projection onto $\frak{H}_n$.

Hence $A_n$ converges (strongly) to $\bold 1$ and Assumption $\bold{A_1}$ is trivially satisfied.

Let $T$ be symmetric on some dense subset $\Dom_0$.

Then Theorem \ref{th1} gives conditions which assure that the essential self-adjointness of the

restrictions $A_nTA_n$ of $T$ 

is equivalent to the essential self-adjointness of $T$ (the equivalence follows with the help of Theorem~\ref{th3}

and Remark~\ref{r3} below).



Another example is provided by the Dirac operator in the Hilbert space $L^2(\Bbb{R}^3)^4$.

The theorem contains as a special case a famous result of Chernoff \cite{1,2} and J\"orgens \cite{3} stating that

the essential self-adjointness of the Dirac operator on $\cal{C}_0^\infty(\Bbb{R}^3)^4$

depends only on the local properties of the potential.

In this case the $A_n$ are multiplication operators with suitable $\cal{C}_0^\infty$-functions.

See Section~4 for a discussion and a simplified proof of this result.



We conclude this section with some remarks on the assumptions.



%%%%%%%%%%%%%%%%

\begin{rem}

Assumption $\bold{A_1}$ is satisfied, e.g., if the $A_n$ form an increasing sequence: $A_mA_n=A_n$ for all $m>n$.

One might think of a sequence of projections, but we do not require $A_n^2=A_n$.

From $A_n^*=A_n$ for all $n$ we easily conclude

that the operators $A_n$, $A_m$, and $A_k$ in $\bold{A_1}$ commute.

\end{rem}



%%%%%%%%%%%%%%%%

\begin{rem}

The commutator $[T,A_n]$ is assumed to be defined on its natural domain. Since $A_n$ is defined on all of $\frak H$,

we have $\Dom([T,A_n]) = \Dom(TA_n) \cap \Dom(T)$.

Assumption $\bold{A_2}$ requires that $\Dom([T,A_n])=\Dom_0=\Dom(T)$ which means

%

\begin{equation}

A_n\Dom_0 \subset \Dom_0.

\label{R2}

\end{equation}

Boundedness of $[T,A_n]$ on a dense domain implies the existence of

a unique bounded extension $[T,A_n]^c$ to all of $\frak H$.

\end{rem}



%%%%%%%%%%%%%%%%%%%%%

\begin{rem}

Theorem \ref{th1} can be formulated as a perturbation theoretic result.

Let $T=H_0+V$, where $H_0$ is essentially self-adjoint and $V$ is symmetric on $\Dom_0$.

If we replace $\bold{A_3}$ by the assumption that $H_0+A_nVA_n$ be essentially self-adjoint on $\Dom_0$, then

we may conclude the essential self-adjointness of $T$. See Section~5 for details.

\end{rem}



%%%%%%%%%%%%%%%%

\begin{rem}

The expression $T^*A_{n_k}\psi$ occurring in

$\bold{A_4}$ is well defined for all $\psi \in \Dom(T^*)$, provided $\bold{A_2}$ holds

(see Lemma~\ref{Lemma2} below). If $\bold{A_4}$ holds, then

\begin{equation}

\|A_{n_k}\| \le K,

\label{R2'}

\end{equation}

where the constant $K$ is independent of $k$ (any weakly convergent sequence of operators is bounded).

If the commutators $\,[T,A_{n_k}]^c$

are also bounded uniformly in $k$, then we can replace $\bold{A_4}$ by a more convenient assumption:

\end{rem}

%%%%%%%%%%%%%%%%%%%%%-LEMMA

%

\begin{lem}\label{lem1}

Assume $\bold{A_2}$. Then $\bold{A_4}$ is implied by

\end{lem}



\begin{quote}

\begin{enumerate}

\item[$\bold{A'_4}$:\quad]

For all $\psi\in \frak H$ there is a subsequence $\{A_{n_k}\}_{k=0}^\infty$ converging strongly to ${\bold 1}$ such that

$$

\|\,[T, A_{n_k}]^c \,\psi\| \le C(\psi),

$$

where $C(\psi)>0$ is independent of $k$.

\end{enumerate}

\end{quote}

%

%%%%%%%%%%%%%%%%%%%%%



\smallskip

\noindent By the uniform boundedness principle, Assumption $\bold{A'_4}$ implies

$\|\,[T,A_{n_k}]^c\| \le K'$, uniformly in $k$.





%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Lemmas and Proofs}\label{sec2}

%%%%%%%%%%%%%%%%%%%%%%%%%

%

\setcounter{equation}{0}

\setcounter{prop}{0}

\medskip



\noindent We start by proving some elementary technical lemmas.

As always we assume that $A_n$ is self-adjoint

and bounded and that $T$ is symmetric on $\Dom_0$.



\smallskip



%%%%%%%%%%%%%%%%%%%%%-LEMMA

%

\begin{lem}\label{Lemma2}

Assume $\bold{A_2}$. Then $A_n \Dom(T^*) \subset \Dom(T^*)$ and

%

\begin{equation}

[T, A_n]^c \psi = T^*A_n\psi - A_nT^*\psi\qquad\text{for all $\psi\in\Dom(T^*)$.}

\label{E2.4}

\end{equation}

%%

\end{lem}

\smallskip

%%%%%%%%%%%%%%%%%%%%%-PROOF

%

\begin{pf}

Let $\psi\in \Dom(T^*)$. For all $\phi\in\Dom_0$

%

\begin{equation}

(A_n\psi,T\phi) = (\psi,(TA_n-[T,A_n])\phi) = ((A_nT^* - [T,A_n]^*)\psi,\phi).

\label{E2.5}

\end{equation}

%%

Hence, by the definition of the adjoint operator, $A_n\psi\in \Dom(T^*)$ and

%

\begin{equation}

T^*A_n\psi=(A_nT^* - [T,A_n]^*)\psi.

\label{E2.6}

\end{equation}

%%

Since $\rmi[T,A_n]$ is obviously symmetric on $\Dom_0$, its bounded extension is self-adjoint, i.e.,

%

\begin{equation}

[T,A_n]^* = - [T,A_n]^c.

\label{E2.7}

\end{equation}

%%

Combining this with Eq.~(\ref{E2.6}) completes the proof of the Lemma \ref{Lemma2}.

\end{pf}



\medskip



%%%%%%%%%%%%%%%%%%%%%-LEMMA

%

\begin{lem}\label{Lemma3}

Assume $\bold{A_1}$--\,$\bold{A_3}$. Then $A_n \Dom(T^*) \subset \Dom(T^c)$ and

%

\begin{equation}

T^cA_n\psi = (A_mTA_m)^cA_n\psi + [T,A_m]^cA_n\psi\qquad\text{for all $\psi\in\Dom(T^*)$,}

\label{E2.8}

\end{equation}

%%

where $m$ is chosen according to Assumption $\bold{A_1}$.

\end{lem}

\smallskip

%%%%%%%%%%%%%%%%%%%%%-PROOF

%

\begin{pf}

Let $\psi\in \Dom(T^*)$.

By Lemma \ref{Lemma2}, $A_n\psi\in\Dom(T^*)$.

Choose $m$, $k$ according to $\bold{A_1}$ and note that $A_mA_n = A_mA_kA_n = A_kA_n = A_n$.

Hence for $\phi\in\Dom_0$,

%

\begin{equation}

(A_n\psi,A_mTA_m\phi)=(A_mT^*A_n\psi,\phi).

\label{E2.9}

\end{equation}

%%

Since by Assumption $\bold{A_3}$

%

\begin{equation}

(A_mTA_m|_{\Dom_0})^* = (A_mTA_m|_{\Dom_0})^c,

\label{E2.11}

\end{equation}

%%

Eq.~(\ref{E2.9}) implies

%

\begin{equation}

A_n\psi \in \Dom\bigl( (A_mTA_m|_{\Dom_0})^c \bigr).

\label{E2.12}

\end{equation}

%%

Hence there is a sequence $\{\chi_j\}_{j=0}^\infty$ with

$\chi_j \in \Dom_0$ for all $j$, $\lim\chi_j=A_n\psi$, such that

$\{A_mTA_m\chi_j\}_{j=0}^\infty$ is convergent.

But then the sequence $\{\xi_j\}_{j=0}^\infty$ with $\xi_j=A_k\chi_j$ has the same

properties:

$\xi_j \in \Dom_0$ and (by continuity of $A_n$) $\lim\xi_j$ exists. Furthermore

%

\begin{equation}

A_mTA_m\xi_j = A_kA_mTA_m\chi_j + A_m[T,A_k]A_m\chi_j,

\label{E2.13}

\end{equation}

%%

which converges, as $j\to\infty$. Therefore

%

\begin{equation}

T\xi_j = TA_m\xi_j = A_mTA_m\xi_j + [T,A_m]\xi_j

\label{E2.14}

\end{equation}

%%

again converges in $\frak H$, as $j\to\infty$. Hence $A_n\psi=\lim\xi_j\in\Dom(T^c)$

and

%

\begin{equation}

T^cA_n\psi = \lim_{j\to\infty} T \xi_j,

\label{E2.15}

\end{equation}

%%

which is just what we wanted to prove.

\end{pf}



\medskip



%%%%%%%%%%%%%%%%%%%%%-PROOF

%

\noindent {\it{Proof of Theorem \ref{th1}}}.

Let $\psi\in\Dom(T^*)$. By Lemma \ref{Lemma3}, $A_n\psi\in\Dom(T^c)$, and

%

\begin{equation}

(\psi, T^*\psi) = (T^cA_{n_k}\psi, A_{n_k}\psi)

+ (T^cA_{n_k}\psi,\psi-A_{n_k}\psi) +

(\psi-A_{n_k}\psi,T^*\psi).

\label{E2.16}

\end{equation}

%%

Using $\bold{A_4}$ we find

%

\begin{equation}

|(T^cA_{n_k}\psi,\psi-A_{n_k}\psi)| + |(\psi-A_{n_k}\psi,T^*\psi)| @>>{n\to\infty}> 0.

\label{E2.18}

\end{equation}

%%

Note that

%

\begin{equation}

(T^cA_{n_k}\psi, A_{n_k}\psi)=(A_{n_k}\psi, T^cA_{n_k}\psi)

\label{E2.17}

\end{equation}

%%

is real, because the closure of a symmetric operator is always symmetric.

We conclude that

%

\begin{equation}

(\psi, T^*\psi) = \lim_{k\to\infty}(A_{n_k}\psi, T^cA_{n_k}\psi)

\label{E2.19}

\end{equation}

%%

is real as a limit of real numbers. Hence (by the polarization identity)

$T^*$ is symmetric which proves that $T$ is essentially self-adjoint.\qed

%

%%%%%%%%%%%%%%%%%%  END PROOF



\medskip



%%%%%%%%%%%%%%%%%%%%%-PROOF

%

\noindent {\it{Proof of Lemma \ref{lem1}}}.

Assuming $\bold{A_2}$ we find with the help of Lemma \ref{Lemma2}

that for all $\psi\in\Dom(T^*)$

%

\begin{equation}

\|T^*A_{n_k}\psi\|  \le  \|A_{n_k}\|\,\|T^*\psi\| + \|\,[T,A_{n_k}]^c\,\psi\|.

\label{E2.20}

\end{equation}

%%

Using Eq~(\ref{R2'}) and

$\bold{A'_4}$ we find

%

\begin{equation}

\|T^*A_{n_k}\psi\| \le K \|T^*\psi\| + C(\psi) \equiv C_1(\psi).

\label{E2.21}

\end{equation}

%%

Hence we obtain

%

\begin{equation}

|(T^cA_{n_k}\psi,\psi-A_{n_k}\psi)| \le C_1(\psi)\|\psi-A_{n_k}\psi\|,

\label{E2.21a}

\end{equation}

%%

which tends to zero because the sequence $A_{n_k}$ is assumed to converge strongly to $\bold 1$.

This proves Lemma \ref{lem1}.\qed

%

%%%%%%%%%%%%%%%%%%  END PROOF



%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Some Remarks and Further Results}\label{sec3}

%%%%%%%%%%%%%%%%%%%%%%%%%

%

\setcounter{equation}{0}

\setcounter{prop}{0}

\medskip



\noindent %

\begin{rem}

A slight modification of the proof of Theorem \ref{th1} allows to replace the condition

$\bold{A_4}$ resp. $\bold{A_4'}$ with, e.g.,

\end{rem}

\begin{quote}

\begin{enumerate}



\item[$\bold{B_4}$:\quad]

For some subsequence $\{A_{n_k}\}_{k=0}^\infty$ and for all $\psi\in \frak H$,

%

                   $$\sum_{k=1}^\infty A_{n_k}={\bold 1},$$

where the series converges in the strong sense, and

%

                   $$\|\sum_{k=1}^j [T, A_{n_k}]^c\psi\|\le C(\psi),$$

where $C(\psi)>0$ is independent of $j$.



\end{enumerate}

\end{quote}



\smallskip



\noindent Repeat the calculation Eqs.~(\ref{E2.16})--(\ref{E2.21a}) with

$A_n$ replaced by

$\tilde{A}_j \equiv \sum_{k=1}^j A_{n_k}$

to arrive at the conclusion of Theorem \ref{th1}.

\medskip



\begin{rem}

An immediate consequence of the essential self-adjointness of $T$ on $\Dom_0$

and the boundedness of the commutator is the

following.

\end{rem}

\medskip



%%%%%%%%%%%%%%%%%%%%%-LEMMA

%

\begin{cor}\label{Lemma4}

Assume $\bold{A_2}$ and let $T$ be essentially self-adjoint on $\Dom_0$.

Denote

%

\begin{equation}

\psi(t) \equiv \exp(- {\rom{\rmi}} t T^c)\,\psi, \quad\text{for all $\psi\in\frak H$.}

\label{E2.22}

\end{equation}

%%

Then for all $\psi$, $\phi\in \frak H$ the function $t\to(\psi(t),A_n\phi(t))$ is continuously differentiable with



\vspace{-.5cm}



%

\begin{equation}

\frac{d}{dt} (\psi(t),A_n\phi(t)) = (\psi(t),{\rom{\rmi}} [T,A_n]^c\phi(t)).

\label{E2.23}

\end{equation}

%%

\end{cor}

\smallskip

%%%%%%%%%%%%%%%%%%%%%-PROOF

%

\begin{pf} For convenience we give a proof of this simple fact:

For $\psi$, $\phi\in \frak H$ choose $\{\psi_j\}$, $\{\phi_j\}$ in $\Dom_0$ with

$\psi=\lim\psi_j$, $\phi=\lim\phi_j$. By the strong continuity of the unitary group,

$f_j(t) \equiv (\psi_j(t),A_n\phi_j(t))$

is continuous, and by $\bold{A_2}$ even continuously differentiable with

$f_j'(t) \equiv (\psi_j(t),\rmi [T,A_n]\phi_j(t))$.

Define $g(t)\equiv (\psi(t), \rmi [T,A_n]^c\phi(t))$. Then

%

\begin{equation}

|f_j'(t)-g(t)| \le

\|[T,A_n]^c\|( \|\phi_j\|\|\psi_j-\psi\|+\|\psi\|\|\phi_j-\phi\|) < \varepsilon,

\label{E2.25}

\end{equation}

%%

where $\varepsilon$ can be chosen independently of $t$ and is arbitrarily small

for $j$ large. Hence we can exchange the differentiation and the limit to conclude

that $f(t)=\lim f_j(t)$ is differentiable with $f'(t)=\lim f_j'(t)$.

\end{pf}

%

%%%%%%%%%%%%%%%%%%%% END PROOF



\medskip\noindent

The following theorem is a converse of Theorem~\ref{th1}.

We give conditions for the essential self-adjointness of

$A_nTA_n$ as a consequence of the self-adjointness of $T^c$.

First we state a lemma similar to Lemma \ref{Lemma2}.

\smallskip



%

%%%%%%%%%%%%%%%%%%%%%-LEMMA

%

\begin{lem}\label{Lemma1}

Assume $\bold{A_2}$. Then $A_n \Dom(T^c) \subset \Dom(T^c)$ and

%

\begin{equation}

[T, A_n]^c \psi = T^cA_n\psi - A_nT^c\psi\qquad\text{for all $\psi\in\Dom(T^c)$.}

\label{E2.1}

\end{equation}

%%

\end{lem}

\smallskip

%%%%%%%%%%%%%%%%%%%%%-PROOF

%

\begin{pf}

By the definition of closure, for any $\psi\in\Dom(T^c)$ there is a sequence

$\{\psi_k\}_{k=0}^\infty$ with $\psi_k\in\Dom_0$, $\lim \psi_k =\psi$, and $\lim T\psi_k=T^c\psi$.

Using the continuity of $A_n$ and $[T,A_n]$ we find

%

\begin{equation}

TA_n\psi_k = A_nT\psi_k + [T,A_n]\psi_k  @>>{k\to\infty}>

A_nT^c\psi + [T,A_n]^c\psi

\label{E2.2}

\end{equation}

%%

Hence the sequences $\{A_n\psi_k\}$ and $\{TA_n\psi_k\}$ both converge. This implies

%

\begin{equation}

\lim_{k\to\infty}A_n\psi_k=A_n\psi \in \Dom(T^c),\qquad

T^c A_n\psi = \lim_{k\to\infty}TA_n\psi_k.

\label{E2.3}

\end{equation}

%%

Now Eq.~(\ref{E2.1}) follows immediately.

\end{pf}

%



\medskip



%%%%%%%%%%%%%%%%%%%%%-THEOREM

%

\begin{thm}\label{th3}

Assume $\bold{A_2}$. Let $T$ be essentially self-adjoint on $\Dom_0$.

In addition we assume for all $\psi$ that

$A_n^2\psi\in \Dom(T^c)$ implies $A_n\psi\in\Dom(T^c)$. 

Then $A_nTA_n|_{\Dom_0}$ is essentially self-adjoint.

\end{thm}

%

%%%%%%%%%%%%%%%%%%%%%-END THEOREM





%%%%%%%%%%%%%%%%%%%%%-PROOF

%

\begin{pf}

Denote $B=A_nTA_n|_{\Dom_0}$. Let $\psi\in \Dom(B^*)$, $\phi\in\Dom_0$. Then

%

\begin{equation}

(B^*\psi,\phi) = (\psi,A_nTA_n\phi) = (A_n^2\psi,T\phi) + ([T,A_n]^*A_n\psi,\phi)

\label{E2.26}

\end{equation}

%%

shows that $A_n^2\psi\in \Dom(T^*)=\Dom(T^c)$ and

%

\begin{equation}

T^cA_n^2\psi = B^*\psi - [T,A_n]^*A_n\psi.

\label{E2.27}

\end{equation}

%%

The additional assumption implies $A_n\psi\in\Dom(T^c)$ and

with Lemma \ref{Lemma1} we obtain

%

\begin{equation}

T^cA_n^2\psi = A_nT^cA_n\psi + [T,A_n]^cA_n\psi.

\label{E2.28}

\end{equation}

%%

Using Eq.~(\ref{E2.7}) we find that 

%

\begin{equation}

B^*\psi = A_nT^cA_n\psi\quad\text{for all $\psi\in \Dom(B^*)$.}

\label{E2.29}

\end{equation}

%%

Hence $B^*$ is symmetric which is equivalent to the essential self-adjointness of $B$.

\end{pf}



\begin{rem}\label{r3}

The additional assumption in Theorem~\ref{th3} is trivially satisfied if the operators $A_n$ are projections.

\end{rem}



%%%%%%%%%%%%%%%%%%%%%%%%%

\section{An Example: Dirac Operators}\label{sec4}

%%%%%%%%%%%%%%%%%%%%%%%%%

%

\setcounter{equation}{0}

\setcounter{prop}{0}

\medskip



\noindent

In the Hilbert space ${\frak H}= L^2({\Bbb R}^3)^4$ of ${\Bbb C}^4$-valued square integrable functions on ${\Bbb R}^3$

we define the free Dirac operator as the closure of the operator

%

\begin{equation}

H_0 = -\rmi \alpha\cdot\nabla + \beta \quad\text{on}\quad

\Dom_0 = {\cal C}_0^\infty({\Bbb R}^3\setminus \{0\})^4.

\label{E3.1}

\end{equation}

%%

Here $\alpha = (\alpha_1,\alpha_2,\alpha_3)$ and $\beta$ are the Hermitian $4\times 4$ Dirac matrices (see \cite{4} for details).

It is well known that $H_0$ is essentially self-adjoint on $\Dom_0$, its closure $H_0^c$ is

self-adjoint on the first Sobolev space

%

\begin{equation}

\Dom(H_0^c) = H^1({\Bbb R}^3)^4 \equiv

\left\{\psi\in L^2({\Bbb R}^3)^4\mid \alpha\cdot\nabla \psi\in L^2({\Bbb R}^3)^4\right\}.

\label{E3.2}

\end{equation}

%%

Here $\nabla$ denotes the distributional derivative.



We denote by $g$ the bounded operator of multiplication by a function $g\in {\cal C}_0^\infty({\Bbb R}^3)$,

%

\begin{equation}

\psi\to g\psi,\quad(g\psi)(x) = g(x)\psi(x),\quad \text{all $\psi\in{\frak{H}}$, all $x\in{\Bbb R}^3$.}

\label{E.3.3}

\end{equation}

%%

It is easy to see that

%

\begin{equation}

[H_0,g]^c = -\rmi\alpha\cdot(\nabla g)

\label{E3.4}

\end{equation}

%%

is a bounded operator of multiplication with a Hermitian matrix-valued ${\cal C}_0^\infty$-function.



\smallskip

%

%%%%%%%%%%%%%%%%%%%%%-LEMMA

%

\begin{lem}\label{Lemma3.1}

The operator $gH_0g$ is essentially self-adjoint on $\Dom_0$.

%%

\end{lem}

%%%%%%%%%%%%%%%%%%%%%-PROOF

%

\begin{pf}

Since $\beta$ is a bounded operator in ${\frak{H}}$,

it is sufficient to consider the operator

%

\begin{equation}

T_0 \equiv -\rmi \alpha\cdot\nabla

\label{E3.2a}

\end{equation}

%%

when investigating the self-adjointness-properties. Again, $T_0$ is essentially self-adjoint on $\Dom(T_0) = \Dom_0$,

self-adjoint on $\Dom(T_0^c) = \Dom (H_0^c)$, and $[T_0,g]^c=[H_0,g]^c$.

Our proof uses the self-adjointness of $T_0^c$ on $H^1({\Bbb R}^3)^4$ and the

boundedness of $[T_0,g]^c$. Denote $B=gT_0g|_{\Dom_0}$.

As in the calculation leading to Eq.~(\ref{E2.27}) we conclude for all $\psi\in\Dom(B^*)$

that $g^2\psi\in \Dom(T_0^*)=\Dom(T_0^c)$

and

%

\begin{equation}

B^*\psi = -\rmi\alpha\cdot\nabla g^2\psi + \rmi g\alpha\cdot(\nabla g) \psi.

\label{E3.5}

\end{equation}

%%

By the Leibniz rule which holds for the product of a distribution with a smooth function we obtain

%

\begin{equation}

-\rmi\alpha\cdot\nabla g^2\psi = -2\rmi g\alpha\cdot(\nabla g)\psi-\rmi g^2 \alpha\cdot\nabla \psi.

\label{E3.6}

\end{equation}

%%

This shows that for all $\psi\in \Dom(B^*)$ the distribution

$\alpha\cdot\nabla\psi\in H^{-1}({\Bbb R}^3)^4$ satisfies

$g^2\alpha\cdot\nabla\psi\in L^2({\Bbb R}^3)^4$.

Therefore we can perform the following calculation for arbitrary $\psi,\phi\in\Dom(B^*)$

%

\begin{align}

(\phi, B^*\psi) &= (\phi, -\rmi\alpha\cdot\nabla g^2\psi + \rmi g\alpha\cdot(\nabla g) \psi)\\

                &= (-\rmi g^2\alpha\cdot\nabla \phi, \psi) + (-\rmi g\alpha\cdot(\nabla g)\phi, \psi)\\

                &= (-\rmi \alpha\cdot\nabla g^2\phi + \rmi g\alpha\cdot(\nabla g) \phi,\psi) = (B^*\phi, \psi)

\label{E3.7}

\end{align}

%%

Hence $B^*$ is symmetric, i.e., $B$ is essentially self-adjoint.

\end{pf}



\medskip\noindent

Now, let consider a function $f\in {\cal C}_0^\infty({\Bbb R})$ with the properties $f(0)=1$, and $f(r) = 0$ if $r\ge 1$. Define

the sequence of multiplication operators $A_n$, $n=1,2,\ldots$ by

%

\begin{equation}

\bigl(A_n\psi\bigr)(x) = \begin{cases}

                                \psi(x) & \text{if $|x|\le n$,}\\

                                f(|x|-n)\psi(x) & \text{if $|x| \ge n$.}

                         \end{cases}

\label{E3.8}

\end{equation}

%%

The sequence $\{A_n\}$ satisfies $\bold{A_1}$ and $\bold{A_4'}$,

the commutators $[H_0,A_n]^c$ are bounded uniformly in $n$ by  sup$|f'(r)|$.

By Lemma~\ref{Lemma3.1} all the operators $A_nH_0A_n$ are essentially self-adjoint on $\Dom_0$.

Hence the following corollary is an immediate consequence of Theorem~\ref{th1}



%

%%%%%%%%%%%%%%%%%%%%%-Corollary

%

\begin{cor}\label{Cor4.1}

Let $A_n$ be defined as above and let $V$ be a symmetric operator on $\Dom_0$ such that $\bold{A_4}$ holds with $T=V$,

and assume that $A_n (H_0 +V) A_n$ is essentially self-adjoint on $\Dom_0$.

Then $H_0 + V$ is essentially self-adjoint on $\Dom_0$.

%%

\end{cor}

%

\medskip\noindent

The assumptions of the corollary are trivially satisfied,

if $V$ is multiplication by a locally bounded Hermitian matrix-valued function (no matter how fast it grows at infinity),

because in this case $A_nVA_n$ is a bounded perturbation of the essentially self-adjoint operator $A_nH_0A_n$.

As noted by Chernoff \cite{2} this result is in marked contrast to the situation for the second-order Schr\"odinger operator

and is related to the existence of a limiting velocity for the propagation of wavepackets according to the Dirac equation.



Because of the Kato-Rellich theorem only the relative boundedness of $A_nVA_n$

with respect to $A_nH_0A_n$ is needed and one could also consider potentials

with local singularities. Other examples include nonlocal potentials. See

\cite{3,4} for details.



A variant of the preceeding proof

is obtained by using a partition of unity $\{f_n\}$ on ${\Bbb R}^3$ with $\sup_{x,n}|\nabla f_n(x)|\le M<\infty$

to define operators $A_n$ satisfying $\bold{B_4}$.



%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Perturbation Theory}\label{sec5}

%%%%%%%%%%%%%%%%%%%%%%%%%

%

\setcounter{equation}{0}

\setcounter{prop}{0}

\medskip



\noindent

Let $H_0$ and $V$ be symmetric operators defined on a dense subset $\Dom_0$ of some Hilbert space $\frak H$.

Let $\{A_n\}_{n=1}^\infty$ be a sequence of bounded self-adjoint operators satisfying $\bold{A_1}$.

Assume that $\bold{A_2}$ holds with $T$ replaced by $H_0$ and $V$, respectively.

Instead of $\bold{A_3}$ let us now assume



\begin{quote}

\begin{enumerate}





\item[$\bold{C_3}$:\quad]

$H_0$ is essentially self-adjoint on $\Dom_0$, and for all $n$ the operator $H_0 + {A_nVA_n}$

is essentially self-adjoint on $\Dom_0$ with

\end{enumerate}

\end{quote}

%

\vspace{-.1cm}

\begin{equation}

\Dom\bigl( (H_0 + A_nVA_n)^c \bigr) = \Dom(H_0^c).

\label{E4.1}

\end{equation}

%%

The operator $T\equiv H_0 + V$ is well defined and symmetric on $\Dom_0$ and we assume that it satisfies $\bold{A_4}$.



%%%%%%%%%%%%%%%%%%%%%-THEOREM

%

\begin{thm}\label{th4}

Under the above assumptions, $T$ is essentially self-adjoint on $\Dom_0$.

\end{thm}

%

%%%%%%%%%%%%%%%%%%%%%



\smallskip



With the help of the following lemma, the proof of Theorem~\ref{th4} is an easy modification of the proof of Theorem~\ref{th1}.



%%%%%%%%%%%%%%%%%%%%%-LEMMA

%

\begin{lem}\label{lemma4.1}

Assume $\bold{A_1}$, $\bold{C_3}$, and $\bold{A_2}$ with $H_0$ and $V$.

Then $\psi\in \Dom\bigl(T^*\bigr)$ implies that $A_n\psi\in\Dom\bigl(T^c\bigr)$ and

%

\begin{equation}

T^cA_n\psi = H_0^cA_n\psi + V^cA_n\psi,

\quad\text{for all $\psi\in \Dom\bigl(T^*\bigr)$.}

\label{E4.2}

\end{equation}

%%

\end{lem}

%

%%%%%%%%%%%%%%%%%%%%%



\begin{pf}

Let $\psi\in \Dom\bigl(T^*\bigr)$. Since $T$ satisfies $\bold{A_2}$ we find with

Lemma~\ref{Lemma2} that $A_n\psi\in\Dom\bigl(T^*\bigr)$. Hence for $\phi\in \Dom_0$ we obtain using $A_mA_n=A_n$

%

\begin{equation}

(T^* A_n\psi,\phi) = (A_n\psi, \{ T_m - [V,A_m] \}\phi),

\qquad T_m\equiv H_0 + A_mVA_m.

\label{E4.3}

\end{equation}

%%

This shows that $A_n\psi\in \Dom(T_m^*) = \Dom(T_m^c) = \Dom (H_0^c)$, where we have used $\bold{C_3}$.

By definition of closure, there is a sequence $\chi_j\in \Dom_0$ with $\chi_j\to A_n\psi$ and $H_0^c\chi_j \to H_0^cA_n\psi$,

i.e., $\{\chi_j\}$ converges in the Hilbert space $\Dom(H_0)$ equipped with the graph norm

$\|\psi\|_0^2 = \|H_0^c\psi\|^2 + \|\psi\|^2$. Since $T_m^c$ is closed on $(\Dom(H_0),\|\cdot\|_0)$,

it is bounded and hence the sequence $\{T_m^c\chi_j\}$ is again convergent.

As in the proof of Lemma~\ref{Lemma3} we can replace $\chi_j$ by the sequence $\xi_j=A_k\chi_j$,

which has the same properties, if $A_k$ is chosen according to $\bold{A_1}$.

In particular, the sequences $\{T_m\xi_j\}$ and $\{H_0\xi_j\}$ are convergent.

But then

%

\begin{equation}

V \xi_j  = V A_m \xi_j = A_mVA_m \xi_j + [V,A_m] \xi_j = (T_m-H_0) \xi_j + [V,A_m]^c \xi_j

\label{E4.4}

\end{equation}

%%

is convergent, i.e., $\lim\xi_j = A_n\psi\in\Dom(V^c)$. Finally,

%

\begin{equation}

T\xi_j = H_0\xi_j + V\xi_j \to H_0^c A_n\psi + V^cA_n\psi,

\label{E4.5}

\end{equation}

%%

which implies $A_n\psi\in \Dom(T^c)$ together with Eq.~(\ref{E4.2}).

\end{pf}

\smallskip

%%%%%%%%%%%%%%%%%%%%%-THEOREM

%

\begin{thm}\label{th5}

Let $H_0$ be essentially self-adjoint on $\Dom_0$ and $X$ be self-adjoint on $\Dom(X)$, such that

$[H_0^c,X]$ is well defined on $\Dom_0$ and bounded.

Let $V$ be a real-valued function on $\Bbb R$, which is locally bounded.

Define $V(X) = \int V(\lambda)\,dE_X(\lambda)$ and assume $\Dom_0\subset \Dom(V(X))$.

Then $H_0+V(X)$ is essentially self-adjoint on $\Dom_0$.

\end{thm}

%

%%%%%%%%%%%%%%%%%%%%%





\begin{pf}

Let $g$ be a real-valued function, which can be written as the Fourier-transform of a function $\tilde g$, such

that ${(1+|\xi|)\tilde g(\xi)}$ is integrable:

%

\begin{equation}

g(\lambda) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty e^{\rmi \lambda \xi}\,\tilde g(\xi)\,d\xi.

\label{E4.A}

\end{equation}

%%

Define $g(X)$ by the weak integral

%

\begin{equation}

g(X)\psi = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty e^{\rmi X \xi}\psi\,\tilde g(\xi)\,d\xi,\quad\text{for all $\psi\in\frak H$}.

\label{E4.B}

\end{equation}

%%

Then

%

\begin{equation}

\|[H_0^c,g(X)]^c\| \le \frac{1}{\sqrt{2\pi}}\,\|[H_0^c,X]^c\|\int_{-\infty}^\infty |\xi|\,|\tilde g(\xi)|\,d\xi.

\label{E4.C}

\end{equation}

%%

Let $f\in {\cal C}_0^\infty(\Bbb R)$ be real-valued, with $f(\lambda)=1$, if $|\lambda|\le 1/2$ and $f(\lambda) = 0$, if $|\lambda|\ge 1$.

Define $A_n:= f(X/n)=\int_{-n}^n f(\lambda/n)\,dE_X(\lambda)$, where $E_X$ is the spectral measure of $X$.



Now it is easy to see that $T=H_0$ and $A_n$ satisfy the assumptions $\bold{A_1}$--$\bold{A_4}$, and even $\bold{A_4'}$.

By our assumptions,

$V(X) = \int V(\lambda)\,dE_X(\lambda)$ is a densely defined self-adjoint operator

which commutes with all $A_n$. Moreover, $A_nV(X)A_n$ is bounded and symmetric on $\Dom_0$.

Hence $H_0 +A_nV(X)A_n$ is essentially self-adjoint on $\Dom_0$.

Essential self-adjointness of $H_0 + V(X)$ now follows immediately from Theorem~\ref{th4}. 

\end{pf}



\medskip

\begin{ack}

I would like to thank T. Hoffmann-Ostenhof for his kind invitation to the Erwin Schr\"odinger Institute,

where part of this work was done.

\end{ack}



\medskip

\begin{thebibliography}{XXXX}



\bibitem{1} P. R. Chernoff,

{\em Essential self-adjointness of powers of generators of hyperbolic equations}. J. Func. Anal. {\bf 12}, 401--414 (1973).

\bibitem{2} P. R. Chernoff,

{\em Schr\"odinger and Dirac operators with singular potentials and hyperbolic equations}. Pacific J. Math. {\bf 72}, 361--382 (1977).

\bibitem{3} K. J\"orgens,

{\em Perturbations of the Dirac operator}. In ``Proceedings of the conference on the theory of ordinary and partial differential equations,

Dundee (Scotland)'',

W. N. Everitt and B. D. Sleeman (eds.),

Lecture Notes in Mathematics {\bf 280}, 87--102, Springer Verlag, Berlin (1972).

\bibitem{4} B. Thaller, {\em The Dirac Equation}, Texts and Monographs in Physics, Springer Verlag, Berlin (1992).



\end{thebibliography}



\end{document}



