Content-Type: multipart/mixed; boundary="-------------1309160841603"

This is a multi-part message in MIME format.

---------------1309160841603
Content-Type: text/plain; name="13-76.comments"
Content-Transfer-Encoding: 7bit
Content-Disposition: attachment; filename="13-76.comments"

2000 Mathematics Subject Classification: 35Q40, 47F05, 81Q10
---------------1309160841603
Content-Type: text/plain; name="13-76.keywords"
Content-Transfer-Encoding: 7bit
Content-Disposition: attachment; filename="13-76.keywords"

massless Dirac operators, scalar potentials, graphene, embedded eigenvalues
---------------1309160841603
Content-Type: application/x-tex; name="SchmUm-arXiv.tex"
Content-Transfer-Encoding: 7bit
Content-Disposition: inline; filename="SchmUm-arXiv.tex"

\documentclass[11pt,reqno]{article}
\usepackage{amssymb,eucal}
\usepackage{amsmath}
%\usepackage[notcite,notref]{showkeys}
\usepackage{color}



\newtheorem{thm}{Theorem}[section]
\newtheorem{prop}{Proposition}[section]
\newtheorem{lem}{Lemma}[section]
\newtheorem{rem}{Remark}[section]
\newtheorem{cor}{Corollary}[section]
\newtheorem{expl}{Example}[section]


\makeatletter
   \renewcommand{\theequation}{\thesection.\arabic{equation}}
          \@addtoreset{equation}{section}
%\makeatother


\setlength{\oddsidemargin}{0.3cm}
\setlength{\evensidemargin}{0.1cm}
\textwidth=15cm
\textheight=23cm
\topmargin =-1cm

\renewcommand{\baselinestretch}{1.1}


\begin{document}

\title{
Schnol's theorem and spectral properties of massless  Dirac
operators with scalar potentials
}


\author{Karl Michael Schmidt$^1$~~and Tomio Umeda$^2\!$
\footnote{Supported by
 the Japan Society for the Promotion of Science
     ``Grant-in-Aid for Scientific Research'' (C)
    No.  21540193.}}

\date{}
\maketitle

\begin{quote}
\begin{itemize}
\item[$^1$] School of Mathematics,  Cardiff University, Senghennydd Road,\\
Cardiff CF24 4AG, Wales, UK.\\
E-mail: {\tt schmidtkm@cardiff.ac.uk}
\item[$^2$] Department of Mathematical Sciences, University of Hyogo, Shosha,\\ Himeji 671-2201, Japan \\
E-mail: {\tt umeda@sci.u-hyogo.ac.jp}
\end{itemize}
\end{quote}


\begin{abstract}
We show that the point spectrum of massless Dirac operators with suitable scalar
potentials is confined to small sets which can easily be characterised
in terms of properties of the potentials.
Furthermore, we prove that a Schnol'-type theorem holds for massless Dirac
operators under minimal assumptions on the potential, and use this result to
conclude that the spectrum of a certain class of such operators covers the whole
real line.
\end{abstract}

\textbf{2000 Mathematics Subject Classification:} 35Q40, 47F05, 81Q10, 

\smallskip

\textbf{Keywords:} massless Dirac operators, scalar potentials, graphene, 
embedded eigenvalues

\vspace{16pt}


\section{Introduction}

\noindent


The Dirac operators we shall consider  in this paper are 
%
\begin{equation}  \label{eq:do2d}
H_2= -i \,\sigma \cdot \!\nabla + q(x)  \;\; \mbox{ in } \;
{\mathsf L}^2({\mathbb R}^2; {\mathbb C}^2)
\end{equation}
% 
and 
%
\begin{equation}   \label{eq:do3d}
H_3= -i \, \alpha  \cdot \! \nabla + q(x)  \;\; \mbox{ in } 
\;
{\mathsf L}^2({\mathbb R}^3; {\mathbb C}^4),
\end{equation}
% 
where  $\sigma=(\sigma_1, \, \sigma_2)$ 
and 
$\alpha=(\alpha_1, \, \alpha_2, \, \alpha_3)$
are given as follows:
%
\begin{equation*}\label{eqn:1-3}
\sigma_1 =
\begin{pmatrix}
0&1 \\ 1& 0
\end{pmatrix}, \,\,\,
\sigma_2 =
\begin{pmatrix}
0& -i  \\ i&0
\end{pmatrix}
\end{equation*}
%
and
%
\begin{equation*}\label{eqn:1-4}
\alpha_j =
\begin{pmatrix}
0&\sigma_j \\ \sigma_j& 0
\end{pmatrix}
\quad (\, j\in \{1, \, 2, \, 3\})
 \,\,\,
\mbox{ with } \,\,\,
\sigma_3 =
\begin{pmatrix}
1&0 \\ 0&-1
\end{pmatrix}\!.
\end{equation*}
%

\vspace{4pt}
\noindent
The dot products are to be read as
%
\begin{equation*}
\sigma \cdot \!\nabla= \sigma_1 \frac{\partial}{\partial x_1} 
+\sigma_2  \frac{\partial}{\partial x_2} 
\end{equation*}
in (\ref{eq:do2d})
and 
%
\begin{equation*}
\alpha \cdot \!\nabla= \alpha_1 \frac{\partial}{\partial x_1} 
+\alpha_2  \frac{\partial}{\partial x_2} 
+  \alpha_3  \frac{\partial}{\partial x_3} 
\end{equation*}
%
in (\ref{eq:do3d}).
The potential 
$q$ is a real-valued function on ${\mathbb R}^d$, 
where $d=2$ or $d=3$, respectively.
The operators $H_2$ and $H_3$ differ from the standard Dirac operator
in that they lack
a mass term, usually represented by an additional anti-commuting
matrix: $\sigma_3$ for the two-dimensional case and 
%
\begin{equation*}
\beta= 
\begin{pmatrix}
I & 0  \\  0 & -I 
\end{pmatrix}
\end{equation*}
%
for the three-dimensional case, where $I$ is a $2\times2$ identity matrix.

The purpose of the present paper
 is twofold.
Firstly, we establish Schnol's theorem for $H_d$, $d=2$ and $3$,
under minimal assumptions on $q$.   
Schnol's theorem
for Schr\"odinger operators is well-known (cf. 
 \cite[p.21]{CFKS}); it asserts that
an energy with polynomially bounded
eigensolution  belongs to the spectrum
of the Schr\"odinger operator. 
In  this context, 
we would like to mention some recent works 
on Schnol's theorem for 
generators of Dirichlet forms,
cf. \cite{BMLS}, \cite{HK} and \cite{LSV}.
To our knowledge, however, the present paper is the first to
establish  Schnol's theorem for Dirac operators.
Secondly, we shall
 show that
 $\sigma(H_d) = \mathbb R$
under minimal assumptions on $q$ as before.
We shall not require any restriction on the growth 
or decay of
 the potential $q$ at infinity.
\vspace{10pt}


We have two motivations for the latter purpose mentioned above. 
First,
the spectrum of the one-dimensional massless 
Dirac operator 
%
\begin{equation*}
H_1 =- i \sigma_2 \frac{d}{dx} + q(x)
\;\; \mbox{ in } \;
{\mathsf L}^2({\mathbb R}; {\mathbb C}^2)
\end{equation*}
%
covers the whole real axis and  is purely absolutely continuous
whenever $q \in {\mathsf L}^1_{loc}(\mathbb R, \, \mathbb R)$.
This surprising fact  was first pointed out 
 by one of the authors in  \cite{KMS-1}.
By separation in spherical polar coordinates, this
result also implies that 
$\sigma(H_d)= \mathbb R$
if $q$ is rotationally symmetric.
 Second,
it is believed that
the energy spectrum of graphene, 
in which electron transport is governed by a two-dimensional Dirac equation without 
a mass term, 
has no bandgap (zero bandgap); see \cite{BTB}, \cite{DSDLA},
\cite{NG}.
For these reasons, it is natural to make an attempt
to show 
that
 $\sigma(H_d) = \mathbb R$
 under minimal assumptions on $q$.

An announcement of the present paper can be found in
\cite{SU}.


\vspace{10pt}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{Embedded eigenvalues and the absolutely continuous spectrum}
It is difficult to imagine that  the spectra of 
$H_2$ and $H_3$ are always purely absolutely
continuous 
regardless of $q$. 
Actually, in the three-dimensional case,
 we have an example 
of $q$ which gives rise to a zero mode of $H_3$, 
\textit{i.e.} an example of
$q$ for which $H_3$ has the embedded eigenvalue $0$.

\begin{expl}  \label{ex:3}
Let
$q(x)= -3/ \langle x \rangle^2$, where
$\langle x \rangle= \sqrt{1 + |x|^2}$.
Then  there exists
a unique self-adjoint realization of $H_3$ in
${\mathsf L}^2({\mathbb R}^3; {\mathbb C}^4)$
with 
$\mbox{\rm Dom}(H_3)={\mathsf H}^1({\mathbb R}^3; {\mathbb C}^4)$, 
the Sobolev space
of order $1$. 
If one puts
%
\begin{equation*}  \label{eq:zeromd3}
f(x)= \langle x \rangle^{-3} (I_4 + i \alpha \cdot x) \phi_0
\end{equation*}
%
 with 
$\phi_0$ a unit vector in ${\mathbb C}^4$, 
then a direct calculation shows that $H_3f=0$.
Since $f \in {\mathsf H}^1({\mathbb R}^3; {\mathbb C}^4)$, 
this implies that $0 \in \sigma_{\rm p}(H_3)$.
Thus $H_3$ has a zero mode. 
As $\lim\limits_{|x|\rightarrow\infty} q(x) = 0$, a simple singular sequence
argument shows that $\sigma(H_3)= \mathbb R$. Hence the energy $0$ is 
an embedded eigenvalue of $H_3$.
\end{expl}


We would like to mention that
the potential $q$ and the zero mode $f$  in Example \ref{ex:3}
were motivated by \cite{LY}.

The analogous two-dimensional
construction in Example \ref{ex:2} below gives
a zero resonance of $H_2$,  not a zero mode of $H_2$.
We do not know if the potential $q$ 
in  Example \ref{ex:2} gives rise to a zero mode of  $H_2$.
However, zero modes of $H_2$ are known to occur with compactly supported
rotationally symmetric potentials, see Theorem 3 of \cite{KMS-2}.


\begin{expl}  \label{ex:2}
Let
$q(x)= -2/ \langle x \rangle^2$.
Then there exists
a unique self-adjoint realization of $H_2$ in
${\mathsf L}^2({\mathbb R}^2; {\mathbb C}^2)$
with 
$\mbox{\rm Dom}(H_2)
=
{\mathsf H}^1({\mathbb R}^2; {\mathbb C}^2)$,
and $\sigma(H_2)= \mathbb R$. 
If
% 
\begin{equation*}
\psi(x) =\langle x \rangle^{-2} (I_2 + i \sigma \cdot x) \phi_0,
\end{equation*} 
%
$\phi_0$ a unit vector in ${\mathbb C}^2$,
then one sees that $H_2 \psi =0$.
However, it is clear that
 $\psi \not \in {\mathsf L}^2({\mathbb R}^2; {\mathbb C}^2)$.
Therefore, $\psi$ is not a zero mode of $H_2$. 
On the other hand, one finds that
$\psi \in 
{\mathsf L}^{2}_{-s}({\mathbb R}^2; {\mathbb C}^2)$ 
for any $s>0$, 
where
%
\begin{equation*}
{\mathsf L}^{2}_{-s}({\mathbb R}^2; {\mathbb C}^2)
=
\big\{ \, \varphi \;  \big| \;  
 \Vert \langle x \rangle^{-s}\varphi \Vert_{L^2} < \infty \big\}.
\end{equation*}
%
This means that $\psi$ is a zero resonance of $H_2$.
\end{expl}

It is not 
an easy task  to clarify whether $H_d$ has
embedded eigenvalues for general potentials. 
However,
we have a good control of the embedded 
eigenvalues of $H_d$ if
$q(x)$ is rotationally symmetric. 
To formulate a result, 
we need to introduce the definition of 
the limit range  $\mathcal R_{\infty}(q)$ of $q$:
%
\begin{equation*}
\mathcal R_{\infty}(q) = \bigcap_{r>0} 
\overline{\,\big\{ 
\, q(x)  \, \big| \, |x|\ge r \big\} },
\end{equation*}
where $\overline{A}$ denotes the closure of a subset of 
$A \subset \mathbb R$.


%
\begin{thm}[\textbf{Schmidt\cite{KMS-2}}]  \label{th:KMS}
Let $q(x)=\eta(|x|)$ and let $\eta \in 
{\mathsf L}^1_{loc}(0, \, \infty)$.
Suppose that there exists a real number
$\lambda \in \mathbb R \setminus \mathcal R_{\infty}(q)$
such that 
%
\begin{equation} \label{eq:BV}
\frac{1}{\, r(\lambda - \eta (r) )-1 \,} \in BV(r_0, \infty) 
\end{equation}
%
for some $r_0 >0$,
where $BV(r_0, \infty)$ denotes
the set of functions of bounded variations on 
the interval $(r_0, \infty)$.
Then $\sigma_p(H_d) \subset \mathcal R_{\infty}(q)$.
\end{thm}

\vspace{3pt}

Theorem \ref{th:KMS} is a direct consequence of
\cite[Corollary 1]{KMS-2}.
One should note that under the assumption that 
$\eta \in {\mathsf L}^1_{loc}(0, \, \infty)$
there exists a distinguished  self-adjoint
realization of $H_d$, see Propositions 2 and 3 of \cite{KMS-2}.

Note that the condition (\ref{eq:BV}) just fails in the radially periodic case,
i.e. if $\eta(r+p) = \eta(r)$ $(r \ge 0)$ with period $p > 0$.
Indeed, when we take $r, s \in [0, \, p]$ and $n \in {\mathbb N}$, then for $\lambda$ not
in the (limit) range of $\eta$,
\begin{align*}
&\frac{1}{(r+np)(\lambda - \eta(r+np)) - 1} - \frac{1}{(s+np)(\lambda - \eta(s+np)) - 1}
\\
&\qquad\sim \frac{1}{(r+np)(\lambda - \eta(r))} - \frac{1}{(s+np)(\lambda - \eta(s))}
\\
&\qquad= \frac{sr}{(s+np)(r+np)} \left(\frac{1}{r\,(\lambda - \eta(r))} - \frac{1}{s\,(\lambda - \eta(s))}\right)
\\
&\qquad\qquad+ \frac{np}{(s+np)(r+np)}\left(\frac{1}{\lambda - \eta(r)} - \frac{1}{\lambda - \eta(s)}\right)
\end{align*}
as $n \rightarrow \infty$, so the total variation in the $n$th period interval is
\begin{equation*}
\mathop{\textrm{Var}}\limits_{r \in [np, \,(n+1)p]} \frac{1}{r(\lambda - \eta(r)) - 1)}
\sim \frac{1}{np}\,\mathop{\textrm{Var}}\limits_{[0, \, p]} \frac{1}{\lambda - \eta}
\qquad
(n\rightarrow\infty).
\end{equation*}
In fact, the limit range of the potential plays no role at all in the
radially periodic case, as our following result shows.

\begin{thm} \label{th:radper}
Let $q(x) = \eta(|x|)$ and let $\eta \in L^1_{loc}(0,\infty)$ be $p$-periodic.
Let $\hat\eta := \frac{1}{p} \int_0^p \eta$.
Then $H_d$ has purely absolutely continuous spectrum in ${\mathbb R}\setminus (\frac{\pi}{p}{\mathbb Z}
+ \hat\eta)$.
\end{thm}

\medskip
\noindent
{\bf Proof.}
By a suitable shift of the spectral parameter, we may assume without loss of generality
that $\hat\eta = 0$.

By separation of variables in polar coordinates (see e.g. \cite{Weidmann}, Appendix to
Section 1),
\begin{equation*}
H_d \cong \bigoplus_{k\in J_d} -i\sigma_2 \frac{d}{dr} + \eta(r) + \sigma_1 \frac{k}{r},
\end{equation*}
where the index set $J_d = {\mathbb Z}\setminus\{0\}$ if $d = 3$ and $J_d = {\mathbb Z} + \frac{1}{2}$
if $d = 2$.
Hence it is sufficient to show that each of the half-line Dirac operators with the
angular momentum term $\sigma_1 \frac{k}{r}$ has purely absolutely continuous spectrum
in ${\mathbb R}\setminus \frac{\pi}{p}{\mathbb Z}$.

It follows from Gilbert-Pearson subordinacy theory (\cite{GP}, \cite{Gilbert};
see also \cite{KMS-3}) that it is sufficient
for this purpose to show that all solutions of the eigenvalue equation
\begin{equation}\label{eq:raddir}
-i\sigma_2 \frac{d}{dr}\,u(r) + \eta(r)\,u(r) + \sigma_1 \frac{k}{r}\,u(r) = \lambda\,u(r)
\end{equation}
are bounded at infinity if $\lambda \notin \frac{\pi}{p}{\mathbb Z}$.

Let $\varepsilon > 0$; we shall prove the boundedness of all solutions of (\ref{eq:raddir})
for large $r$ and for all
$\lambda \in \Lambda := {\mathbb R} \setminus \bigcup_{n \in {\mathbb Z}} (\frac{n\pi}{p} - \varepsilon,
\frac{n\pi}{p} + \varepsilon)$ by adapting an idea of Stolz \cite{Stolz}; see also
\cite[Theorem 5.2.1]{BES}.

Let $Q(r) := \int_0^r \eta$ $(r \ge 0)$; then $Q$ is $p$-periodic and $Q(0) = 0$.
For $j \in {\mathbb N}$, the matrix-valued function 
$\Phi_j(r) := \exp\big[-i\sigma_2 \big\{ Q(r) - \lambda (r-(j-1)p)\big\}\big]$ $(r \ge 0)$ satisfies
the unperturbed, periodic differential equation
\begin{equation*}
-i\sigma_2 \frac{d}{dr}\,\Phi_j(r) + \eta(r)\,\Phi_j(r) = \lambda\,\Phi_j(r),
\end{equation*}
and $\Phi_j((j-1)p) = I$.
$M(\lambda) := \Phi_j(jp) = I \cos \lambda p + i \sigma_2 \sin \lambda p$ is the monodromy
matrix  (cf. \cite[p.5]{BES}) of the periodic equation and $D(\lambda) := \mathop{\textrm{tr}} M(\lambda) = 2 \cos \lambda p$
its discriminant (cf. \cite[p.9]{BES}).
Clearly there exists $\delta > 0$ such that $|D(\lambda)| \le 2 - 2\delta$ $(\lambda \in
\Lambda)$.

By the variation of constants method (cf. \cite[p.3]{BES}), 
we can find an integral equation for the matrix-valued
solution $\Psi_j$ of (\ref{eq:raddir}) such that $\Psi_j((j-1)p) = I$,
\begin{equation} \label{eq:inteq}
\Psi_j(r) = \Phi_j(r) - \Phi_j(r) \int_{(j-1)p}^r \Phi_j(s)^{-1} \sigma_3 \frac{k}{s}\,
  \Psi_j(s)\,ds
\qquad
(r \ge (j-1)p).
\end{equation}
Using the fact that $\Phi_j$ is always unitary, we hence obtain the estimate for the
matrix operator norm
\begin{equation*}
|\Psi_j(r)| \le 1 + \int_{(j-1)p}^r \frac{|k|}{s}\,|\Psi_j(s)|\,ds
\qquad
(r \ge (j-1)p).
\end{equation*}
By Gronwall's inequality, it follows that
\begin{equation*}
|\Psi_j(r)| \le \exp\left(|k| \log\frac{r}{(j-1)p}\right)
= \frac{r^{|k|}}{(j-1)^{|k|} p^{|k|}},
\end{equation*}
and hence by (\ref{eq:inteq}), for $(j-1)p \le r \le jp$,
\begin{align}
|\Psi_j(r) - \Phi_j(r)| &\le \int_{(j-1)p}^r \frac{|k|}{s}\,|\Psi_j(s)|\,ds
\le \frac{r^{|k|} - (j-1)^{|k|} p^{|k|}}{(j-1)^{|k|} p^{|k|}} \nonumber \\
&\le \frac{j^{|k|} - (j-1)^{|k|}}{(j-1)^{|k|}}
= \big( 1 + \frac{1}{j-1} \big)^{|k|} - 1
\rightarrow 0 \qquad (j\rightarrow\infty). \label{eq:psiphi}
\end{align}
In particular, the matrices $M_j(\lambda) := \Psi_j(jp)$ satisfy
\begin{equation}\label{eq:monocon}
\lim\limits_{j\rightarrow\infty} |M_j(\lambda) - M(\lambda)| = 0
\end{equation}
uniformly in
$\lambda \in {\mathbb R}$.
This implies that $D_j(\lambda) := \mathop{\textrm{tr}} M_j(\lambda) \rightarrow D(\lambda)$ uniformly
in $\lambda$ as $j\rightarrow\infty$.
Thus there is $J \in {\mathbb N}$ such that $|D_j(\lambda)| \le 2 - \delta$ for all $j \ge J$
and $\lambda \in \Lambda$.

For such $j$ and $\lambda$, the matrices $M_j(\lambda)$ have complex conjugate eigenvalues
$\mu_j(\lambda), \overline{\mu_j(\lambda)}$, $|\mu_j(\lambda)| = 1$.
(See \cite[{\it Case 3} \, in p. 10]{BES} in conjunction with the fact that
 $\text{det}\Psi_j(r)=1$, which is obtained from
 \cite[Liouville's formula in p.3]{BES}).
A suitable matrix of eigenvectors can be written as
\begin{equation*}
E_j(\lambda) := \left(\begin{matrix}
 \mu_j(\lambda) - (M_j(\lambda))_{22} & \overline{\mu_j(\lambda)} - (M_j(\lambda))_{22}\\
 (M_j(\lambda))_{21} & (M_j(\lambda))_{21}
  \end{matrix}\right);
\end{equation*}
in the limit $j \rightarrow \infty$, this converges uniformly in $\lambda \in \Lambda$
to a corresponding matrix
$E(\lambda)$ of eigenvectors of $M(\lambda)$ in view of (\ref{eq:monocon}).

Now consider the matrix-valued solution $\Psi_J$. For $n \ge J$ (omitting the variable
$\lambda$ for brevity),
\begin{align*}
\Psi_J(np) &= M_n M_{n-1} \cdots M_J \\
 &= E_n {\textstyle\left(\begin{matrix}\mu_n & 0 \\ 0 & \overline{\mu_n}\end{matrix}\right)} E_n^{-1}
 E_{n-1} {\textstyle \left(\begin{matrix}\mu_{n-1} & 0 \\ 0 & \overline{\mu_{n-1}}\end{matrix}\right)} E_{n-1}^{-1} \cdots
 E_J {\textstyle \left(\begin{matrix}\mu_J & 0 \\ 0 & \overline{\mu_J}\end{matrix}\right)} E_J^{-1}.
\end{align*}
Hence the matrix operator norm can be estimated as
\begin{equation*}
|\Psi_J(np)| \le |E_n|\,|E_n^{-1} E_{n-1}|\,|E_{n-1}^{-1} E_{n-2}| \cdots |E_{J+1}^{-1} E_J|\, |E_J^{-1}|.
\end{equation*}

In order to estimate $|E_j^{-1} E_{j-1}|$, we again solve (\ref{eq:raddir}) on the
interval $[(j-1)p, jp]$ by variation of constants, but this time using $\Psi_{j-1}(r-p)$
as a reference instead of $\Phi_j(r)$; this gives
\begin{equation*}
\Psi_j(r) = \Psi_{j-1}(r-p) + \Psi_{j-1}(r-p) \int_{(j-1)p}^r \Psi_{j-1}(s-p)^{-1}
  \sigma_3\, \frac{kp}{s(s-p)}\,\Psi_j(s)\,ds
\end{equation*}
$(r \ge (j-1)p)$.
Consequently,
\begin{align*}
|M_j(\lambda) - M_{j-1}(\lambda)| &\le |M_{j-1}(\lambda)| \int_{(j-1)p}^{jp}
|\Psi_{j-1}(s-p)^{-1}|\, \frac{|k|p}{s(s-p)}\,|\Psi_j(s)|\,ds \\
&\le C^3 |k|p\int_{(j-1)p}^{jp} \frac{ds}{s(s-p)}
\end{align*}
with a constant $C$ which is independent of $\lambda$ due to the uniform bound
(\ref{eq:psiphi}).
This also implies such an estimate for $|D_j(\lambda) - D_{j-1}(\lambda)|$ and,
since the $D_j$ are Lipschitz continuous on $\Lambda$, for
$|\mu_j(\lambda) - \mu_{j-1}(\lambda)|$ $(\lambda \in \Lambda)$.
Hence
\begin{equation*}
|E_j(\lambda) - E_{j-1}(\lambda)| \le C' \int_{(j-1)p}^{jp} \frac{ds}{s(s-p)}
\end{equation*}
with some other uniform constant $C'$.
Now we can estimate
\begin{align*}
|\Psi_J(np)| &\le |E_n|\,|E_J^{-1}| \prod_{j=J+1}^n |E_j^{-1} E_{j-1}|
\le C'' \prod_{j=J+1}^n (1 + |E_j^{-1}|\,|E_j - E_{j-1}|) \\
&\le C'' \exp \bigg(\sum_{j=J+1}^n |E_j^{-1}|\,|E_j - E_{j-1}|\bigg)
\le C'' \exp \left(C''' \int_{Jp}^{np} \frac{ds}{s(s-p)}\right),
\end{align*}
with uniform constants $C'', C'''$; this is bounded as $n \rightarrow \infty$.
Hence $\Psi_J(r)$ is bounded at infinity,
since $\Psi_J(r)= \Psi_n(r) \Psi_J((n-1)p)$ and, by (\ref{eq:psiphi}),
%
\begin{equation*}
|\Psi_n(r)| \le \big( 1 + \frac{1}{n-1} \big)^{|k|} \le 2^{|k|}  
\end{equation*}
for $(n-1)p \le r \le np$ and $n > J$.
%

This concludes the proof of Theorem \ref{th:radper}, since every solution of
(\ref{eq:raddir}) is a linear combination of the columns of $\Psi_J$.
$\;\blacksquare$

\medskip
The above method of proof does not work at the points
$\lambda\in\frac{\pi}{p}{\mathbb Z} + \hat\eta$;
these points are potential candidates for embedded eigenvalues.
However, it seems to be a rather delicate question to decide whether such embedded
eigenvalues actually occur.
%%%%%%%%%%%%%%%%%%%%%%%%%
\medskip

If $q$ is not assumed to be rotationally symmetric, we can prove 
the following. 

\vspace{3pt}

%
\begin{thm}  \label{th:virial}
Let $q \in C^1(\mathbb R^d; \mathbb R)$, and suppose
that both $q$ and $(x \cdot \nabla)q$ are 
bounded functions. 
Then  
$\sigma_p (H_d) \subset [m_q, \, M_q]$, where
%
\begin{equation*}
m_q= \inf_x \{ q(x) + (x\cdot \nabla)q(x) \}, 
\quad 
M_q= \sup_x \{ q(x) + (x\cdot \nabla)q(x) \}.
\end{equation*}
%
\end{thm}
%

\vspace{5pt}
To prove Theorem \ref{th:virial}, we shall apply  the following simple abstract version
of the virial theorem.

\begin{lem}[\textbf{Balinsky and Evans\cite{BE}}, \cite{BE-1}]
Let $U(a)$, $a>0$, be a one-parameter family of unitary operators
on a Hilbert space $\mathcal H$, which converges strongly to 
the identity as $a \to 1$. Let $T$ be
a self-adjoint operator in $\mathcal H$ and
$T_a:= aU(a)TU(a)^{-1}$.
If $f$ belongs to
$\text{\rm Dom}(T) \cap \text{\rm Dom}(T_a)$
and is an eigenvector of $T$ corresponding to 
an eigenvalue $\lambda$,
then
%
\begin{equation*}
\lim_{a \to 1} \Big(
f_a, 
\Big[
\frac{T_a - T}{a-1}
\Big] f
\Big)_{\mathcal H} 
=
\lambda \, \Vert f \Vert_{\mathcal H}^2,
\end{equation*}
%
where $f_a = U(a)f$.
\end{lem}


\noindent
{\bf Proof of Theorem \ref{th:virial}.}
We only give the proof for $H_3$, because the proof for $H_2$ is exactly the same.

Let $\lambda \in \sigma_p(H_3)$, and let
$f$ be a corresponding eigenfunction with $\Vert f \Vert =1.$
In particular, 
$f \in \mbox{Dom}(H_3)={\mathsf H}^1({\mathbb R}^3; {\mathbb C}^4)$
and 
 $H_3 f= \lambda f$. With the dilation group
$\{U(a)\}_{a>0}$, defined by 
$U(a)g (x):=a^{3/2}g(ax)$, $g \in {\mathsf L}^2({\mathbb R}^3; {\mathbb C}^4)$, 
we introduce a family of self-adjoint operators
$\{ H(a)\}_{a>0}$ by 
$H(a):= a \, U(a) H U(a)^{-1}$. 
We then see that
%
\begin{equation*}
\big( U(a)f, \, H(a)f \big) 
=  \big( H(a)U(a)f, \, f \big)
=  \big( aU(a) Hf, \, f \big)
=   \lambda a \big( U(a) f, \, f \big),
\end{equation*}
% 
which implies that
%
\begin{equation}  \label{eq:vrl1}
\big( U(a)f, \, H(a)f - Hf \big)  = \lambda (a -1) \big( U(a) f, \, f \big).
\end{equation}
%

On the other hand, we find that
%
\begin{equation*}
[H(a) g](x)= -i \alpha \cdot \nabla g(x)+ a \,q (a x) g(x), 
\quad \forall g \in {\mathsf H}^1({\mathbb R}^3; {\mathbb C}^4),
\end{equation*}
%
hence we have
%
\begin{equation} \label{eq:vrl2}
 [H(a)f](x) - [Hf](x)  = \{a \,q (a x)  - q(x)\}f(x).
\end{equation}
%
Combining (\ref{eq:vrl1}) with (\ref{eq:vrl2}) 
yields
%
\begin{equation} \label{eq:vrl3}
\big( U(a)f, \, \frac{a \, q(a\,\cdot) - q( \,\cdot \,)}{a-1}f  \,\big)  
= \lambda  \big( U(a) f, \,   f\, \big).
\end{equation}
%
Since $\mbox{s\,-}\lim_{a \to 1} U(a)=I$, (\ref{eq:vrl3}) implies, by
the Lebesgue dominated convergence theorem, that
%
\begin{equation}    \label{eq:vrl4}
\int_{{\mathbb R}^3}
 \big( 
q(x) +  (x \cdot \nabla q)(x) 
\big)
|f(x)|^2 \, dx = \lambda.
\end{equation}
%
The conclusion of the theorem follows from (\ref{eq:vrl4}).
$\;\blacksquare$


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



\section{Schnol's theorem}
In this section, we state and prove Schnol's theorem for $H_d$.
The idea of our proof is based on that
of \cite[p. 21, Theorem 2.9]{CFKS},
where Schnol's theorem for Schr\"odinger operators 
is established.
In the three-dimensional case,
our Schnol's theorem can be stated as follows:

%
\begin{thm}  \label{schnol3}
Let $q\in {\mathsf L}^2_{loc}({\mathbb R}^3, {\mathbb R})$,
and let $\lambda$ be a real number.
Suppose $f$ is a polynomially bounded measurable function 
on ${\mathbb R}^3$, not identically $0$, and satisfies
the equation 
%
\begin{equation}  \label{eq:eveq3}
(- i \alpha \cdot \nabla + q) f = \lambda f
\end{equation}
%
in the distribution sense. 
Then $\lambda \in \sigma(H_3)$ 
for any
self-adjoint realization $H_3$ such that
$\mbox{\rm Dom}(H_3) 
\supset {\mathsf H}^1({\mathbb R}^3; {\mathbb C}^4)\cap \mbox{\rm Dom}(q)$.
\end{thm}
%


\vspace{2pt}

\noindent
{\bf Proof.} It is sufficient to prove the assertion for $\lambda =0$,
because any $\lambda\not=0$ can be absorbed in $q$. 
The proof will be devided into two steps.

\noindent
{\it Step 1. The case of $f \in {\mathsf L}^2({\mathbb R}^3; {\mathbb C}^4)$.} 

Let $\varphi \in C_0^{\infty}({\mathbb R}^3, {\mathbb C})$. Then
it follows that
%
\begin{align}
(- i\alpha \cdot \nabla )(\varphi f)
&=   
(- i\alpha \cdot \nabla \varphi) f 
+ 
\varphi (- i\alpha \cdot \nabla ) f   
 \nonumber  \\
&=  
(- i\alpha \cdot \nabla \varphi) f - \varphi \, q f,
\label{eqn:sa3-3}
\end{align}
%
where we have used (\ref{eq:eveq3})  in the second equality.
Since $\varphi q\in {\mathsf L}^2({\mathbb R}^3; {\mathbb C})$
and $f$ is locally bounded, 
we see  that 
$\varphi \, q f\in {\mathsf L}^2({\mathbb R}^3; {\mathbb C}^4)$, 
hence by (\ref{eqn:sa3-3}) that 
$(- i\alpha \cdot \nabla )(\varphi f)\in {\mathsf L}^2({\mathbb R}^3; {\mathbb C}^4)$.
This implies that 
$(- i\alpha \cdot \nabla )^2(\varphi f) \in 
{\mathsf H}^{-1}({\mathbb R}^3; {\mathbb C}^4)$, the Sobolev
space of order $-1$.
On the other hand, 
$(- i\alpha \cdot \nabla )^2(\varphi f)= -\Delta(\varphi f)$. 
Hence we find that $\varphi f \in 
{\mathsf L}^2({\mathbb R}^3; {\mathbb C}^4)
\subset 
{\mathsf H}^{-1}({\mathbb R}^3; {\mathbb C}^4)$
and $ -\Delta(\varphi f) \in  {\mathsf H}^{-1}({\mathbb R}^3; {\mathbb C}^4)$.
We now apply the ellipticity argument, 
and we get  
%
\begin{equation}  \label{eqn:sa2}
\varphi f \in {\mathsf H}^{1}({\mathbb R}^3; {\mathbb C}^4)
 \ \  \mbox{ for } \ 
\forall  \varphi \in C_0^{\infty}({\mathbb R}^3; {\mathbb C}).
\end{equation}
%
By the ellipticity argument, we mean the following:
\lq\lq 
$\, u \in {\mathsf H}^{\ell}({\mathbb R}^3; {\mathbb C})$ and 
$\Delta u \in {\mathsf H}^{\ell}({\mathbb R}^3; {\mathbb C})$ 
for some $\ell \in \mathbb R
\Longrightarrow  u \in 
{\mathsf H}^{\ell+2}({\mathbb R}^3; {\mathbb C}).$\rq\rq

We now choose $\chi \in C_0^{\infty}({\mathbb R}^3; {\mathbb C})$ 
so that
$\chi(x)=1 \ (|x| \le 1)$ and $=0 \ (|x| \ge 2)$,
and we set
%
\begin{equation}  \label{eqn:sa2-1}
\chi_n(x) = \chi(x/n)  \quad (n =1, \, 2, \, 3, \, \cdots).
\end{equation}
%
It follows from (\ref{eqn:sa2}) that 
$\chi_n f \in  {\mathsf H}^{1}({\mathbb R}^3; {\mathbb C}^4)$. 
It is evident that $\chi_n f \in \text{Dom}(Q)$.
Hence $\chi_n f \in \text{Dom}(H_3)$ for
$n =1, \, 2, \, 3, \, \cdots$.
To construct a singular sequence, we define
%
\begin{equation}  \label{eqn:sa3-5}
f_n = \frac{1}{\, \Vert \chi_n f \Vert_{{\mathsf L}^{2}}}\,
\chi_n f.
\end{equation}
%
It is obvious that 
$f_n \in \text{Dom}(H_3)$
and $\Vert f_n \Vert_{ {\mathsf L}^{2} }=1$.
We now only have to show that $\Vert H f_n \Vert_{{\mathsf L}^{2}}\to 0$
as $n \to \infty$.
In fact, we see that
%
\begin{gather}  \label{eqn:sa3-6}
\begin{split}
Hf_n
&=
 \frac{1}{\, \Vert \chi_n f \Vert_{{\mathsf L}^{2}}  }\,
\big[
\{ (-i \alpha \cdot \nabla )\chi_n \}f + 
\chi_n ( -i \alpha \cdot \nabla + q)f \big] \\
&=
\frac{1}{\, \Vert \chi_n f \Vert_{{\mathsf L}^{2}}  }\,
\Big[
\frac{1}{\, n \,}
\Big\{ 
(-i \alpha \cdot \nabla \chi) \big(\frac{x}{n} \big) 
\Big\} f  
\Big],
\end{split}
\end{gather}
%
where we have used the hypothesis   that 
$(-i \alpha \cdot \nabla + q) f =0$.
Noting the fact that 
$\lim_{n\to \infty}\Vert \chi_n f \Vert_{{\mathsf L}^{2}}
=\Vert f \Vert_{{\mathsf L}^{2}}\not= 0$,
we can deduce 
from (\ref{eqn:sa3-6})
that $\Vert H f_n \Vert_{{\mathsf L}^{2}}\to 0$.
Hence we can conclude that $0 \in \sigma(H_3)$.


\vspace{3pt}

\noindent
{\it Step 2. The case of $f \not\in {\mathsf L}^2({\mathbb R}^3; {\mathbb C}^4)$.} 


We may assume that $f$ satisfies the estimate
%
\begin{equation}  \label{eqn:arara-1}
|f(x)| \le C (1+|x|)^N 
\end{equation}
%
for some $N \in \mathbb N$.
Let $f_n$ be defined in the same way as (\ref{eqn:sa3-5}). 
As was shown in Step 1, 
it follows that
$ f_n \in \mbox{\rm Dom}(H_3)$ and that (\ref{eqn:sa3-6}) is still valid.
Then we have
%
\begin{equation}    \label{eqn:sa3-8}
\Vert H f_n \Vert^2_{{\mathsf L}^2}
\le
\frac{1}{n^2 \Vert \chi_n f \Vert^2_{{\mathsf L}^2}} \,
 \Big[ \sup_{1\le |x| \le 2} |\nabla \chi (x)|  \Big]^2 
\int_{n \le |x| \le 2n} |f(x)|^2 \,dx.
\end{equation}
%

We now introduce a sequence
$(M(n))_{n \in {\mathbb N}}$ by
%
\begin{equation}   \label{eqn:sa3-9}
M(n):= \int_{|x| \le n} |f(x)|^2 \,dx
\quad (n\in {\mathbb N}),
\end{equation}
%
which is diverging and monotonically increasing.
It follows from (\ref{eqn:sa3-8}) and (\ref{eqn:sa3-9})
that
%
\begin{equation}  \label{eqn:arere-2}
\Vert H f_n \Vert^2_{{\mathsf L}^2}
\le
C \frac{M(2n) - M(n)}{n^2 M(n)},
\end{equation}
%
where $C$ is a positive constant, independent of $n$.

For the sake of contradiction, suppose that
%
\begin{equation}   \label{eqn:sa3-10}
\liminf_{n \to \infty}  \,\frac{M(2n) - M(n)}{n^2 M(n)} >0.
\end{equation}
%
Then there would be a large integer $\nu_0$ and
a positive constant $\alpha_0$ such that
%
\begin{equation}   \label{eqn:sa3-11}
\frac{M(2n) - M(n)}{n^2 M(n)} \ge \alpha_0
\quad \mbox{for } \forall n \ge \nu_0.
\end{equation}
%
This implies that
%
\begin{equation}   \label{eqn:sa3-12}
M(2n) \ge (1 + \alpha_0 n^2) M(n)
\quad \mbox{for } \forall n \ge \nu_0.
\end{equation}
%
By repeated use of (\ref{eqn:sa3-12}), we obtain
%
\begin{equation}   \label{eqn:sa3-13}
M(2^n \nu_0) \ge 
\Big\{
\prod_{j=0}^{n-1} (1 + \alpha_0 \nu_0^2 2^{2j}) 
\Big\}
M(\nu_0)
\quad (n\in{\mathbb N}).
\end{equation}
%
We now write $n=2\ell$. It follows from
(\ref{eqn:sa3-13}) that
%
\begin{gather}  \label{eqn:sa3-14}
\begin{split}
M(4^{\ell} \nu_0) 
&
\ge 
\Big\{
\prod_{j=0}^{2\ell-1} (1 + \alpha_0 \nu_0^2 4^j) 
\Big\}
M(\nu_0)  \\
&
\ge
\Big\{
\prod_{j=\ell}^{2\ell-1} (1+ \alpha_0 \nu_0^2 4^j) 
\Big\}
M(\nu_0)  \\
&
\ge
\alpha_0^{\ell} \nu_0^{2\ell} 4^{\ell^2}
M(\nu_0).
\end{split}
\end{gather}
%
On the other hand, it follows from (\ref{eqn:arara-1}) and
(\ref{eqn:sa3-9}) that
%
\begin{gather}   \label{eqn:sa3-15}
\begin{split}
M(4^{\ell} \nu_0) 
&
\le 
C \int_{|x| \le 4^{\ell} \nu_0} (1+|x|)^{2N} \,dx   \\
&
=
C^{\prime}
 \int_0^{4^{\ell} \nu_0} (1+r)^{2N} r^2 \, dr   \\ &
\le
C^{\prime\prime}_{\nu_0} 4^{(2N+3)\ell}.
\end{split}
\end{gather}
% 
It follows from (\ref{eqn:sa3-14}) and (\ref{eqn:sa3-15})
that
%
\begin{equation}  \label{eqn:sa3-16}
C^{\prime\prime}_{\nu_0} 4^{(2N+3)\ell}  \ge 
\alpha_0^{\ell} \nu_0^{2\ell} 4^{\ell^2} M(\nu_0)
\quad (\ell \in{\mathbb N}).
\end{equation}
%
Taking the logarithm of both sides of (\ref{eqn:sa3-16}), 
one gets
%
\begin{gather}  \label{eqn:sa3-17}
\begin{split}
{}& \log C^{\prime\prime}_{\nu_0}
 + {(2N+3)\ell} \,\log 4  \ge  \\
{}& \qquad \qquad
\ell \,\log (\alpha_0 \nu_0^{2})
+  {\ell^2} \,\log4 + \log M(\nu_0) 
\qquad (\ell \in {\mathbb N}).
\end{split}
\end{gather}
%
Since the right hand side of (\ref{eqn:sa3-17})
grows faster than the left hand side of (\ref{eqn:sa3-17})
as $\ell$ goes to infinity,
the inequality (\ref{eqn:sa3-17})
 is a contradiction. Hence we can deduce
that
%
\begin{equation}   \label{eqn:sa3-18}
\liminf_{n \to \infty}  \,\frac{M(2n) - M(n)}{n^2 M(n)} =0,
\end{equation}
%
which yields that 
there is a subsequence $\{ M(n_k) \}$
 such that
%
\begin{equation}   \label{eqn:sa3-19}
\lim_{k \to \infty}  \,\frac{M(2n_k) - M(n_k)}{n_k^2 M(n_k)} =0.
\end{equation}
%
This fact, together with (\ref{eqn:arere-2}), implies that
$\Vert H f_{n_k} \Vert_{{\mathsf L}^2} \to 0$ 
as $k \to \infty$.
Thus we can conclude that $0 \in \sigma(H_3)$.
$\blacksquare$


\vspace{10pt}

In the two dimensional case,
Schnol's theorem is as follows:

%
 \begin{thm}  \label{schnol2}
 Let $q\in {\mathsf L}^2_{loc}({\mathbb R}^2, {\mathbb R})$  
and $\lambda$ be a real number.
Suppose $\psi$ is a polynomially bounded measurable function 
 on ${\mathbb R}^2$, not identically $0$, and satisfies
 the equation
%
 \begin{equation*}
 (- i \sigma \cdot \nabla + q) \psi = \lambda\psi
 \end{equation*}
%
 in the distribution sense. 
 Then $\lambda \in \sigma(H_2)$ 
 for any
 self-adjoint realization $H_2$ such that
 $\mbox{\rm Dom}(H_2) 
 \supset {\mathsf H}^1({\mathbb R}^2; {\mathbb C}^2)\cap \mbox{\rm Dom}(q)$.
 \end{thm}

The proof of Theorem \ref{schnol2} is similar to
 that of Thorem \ref{schnol3}, and is omitted.


When applying either of Theorems \ref{schnol3} and  \ref{schnol2},
one needs to construct a polynomially bounded eigensolution for
a given energy  of the Dirac operator $H_d$ 
with $q$ being locally square integrable. 
However, it is not easy to construct such an eigensolution
unless $q$ decays at infinity in an appropriate sense.
If $q$ decays rapidly, it is well-known that 
one can construct bounded eigensolutions (generalized eigenfunctions)
of $H_d$ by exploiting the limiting absorption principle. 
In Example \ref{schnlapp} below, 
we shall construct a bounded eigensolution (cf. (\ref{eq:sa-4-}))
 for a given energy of $H_d$ with potential $q$  of a specific form. 
We would like to stress that 
we do not require any decay assumption of $q$ at infinity.



%
\begin{expl} \label{schnlapp}
Let $\eta$ be a real-valued continuous function on $\mathbb R$ and define
$q(x):= \eta(x \cdot k)$ on ${\mathbb R}^d$, $d\in \!\{2, \, 3\}$,
where $k \in {\mathbb R}^d$ is a unit vector.
One can show, by the standard technique 
{\rm (}{\it cf.} {\rm \cite[{\it p.257, Corollary}\,]{RS-I}},
{\rm \cite[{\it p.113,  Theorem 4.3}\,]{Thaller}\rm )} ,
that $H_d$ is essentially self-adjoint on 
$H^1({\mathbb R}^d; {\mathbb C}^{2^{d-1}})\cap \mbox{\rm Dom}(q)$.
Let $H_d$ be the unique self-adjoint realization.
Then $\sigma(H_d)= \mathbb R$.

For $d=3$, this fact is proved in the following manner\,{\rm :} 
Put
%
\begin{equation*}
\xi(t) = \int_0^t \eta (\tau) \, d \tau.
\end{equation*}
%
As the eigenvalues of the matrix $\alpha\cdot k$ are $\pm 1$
 {\rm(}each with geometric multiplicity {\rm 2)}
we can choose a spinor  $\phi_0 \in {\mathbb C}^4$ 
so that $|\phi_0|=1$, $(\alpha \cdot k) \phi_0= \phi_0$.
For a given $\lambda \in \mathbb R$, define
%
\begin{equation}   \label{eq:sa-4-}
f(x)= e^{-i(\alpha\cdot k) \, \xi (x \cdot k)} 
 e^{i\lambda x\cdot k} \phi_0.
\end{equation}
%
Then $f$ is in $ C^1({\mathbb R}^3; {\mathbb C}^4)$, and
 satisfies the equation (\ref{eq:eveq3}).
In fact, one can see that
%
\begin{equation*}
-i \alpha_j \frac{\partial f}{\partial x_j}
=
-i \alpha_j  k_j  \, (-i \alpha \cdot k) \, q(x)   f  + \lambda \alpha_j k_j f
\qquad
(\, j\in \{1, \, 2, \, 3\}),
\end{equation*}
%
hence that 
%
\begin{align}
(-i \alpha\cdot \nabla )f
&=
 (-i \alpha \cdot k)^2 \, q(x)  f  + \lambda (\alpha \cdot k) f  \nonumber\\
&=
-q(x)  f  + \lambda  \, e^{-i(\alpha\cdot k) \, \xi (x \cdot k)} 
 e^{i\lambda x\cdot k} (\alpha \cdot k) \phi_0     \label{eq:sa-4}\\
&=
 -q(x)  f  + \lambda f,  \nonumber
\end{align}
%
where we have used the fact that $(\alpha \cdot k) \phi_0= \phi_0$
and the  facts that $(\alpha \cdot k)^2=I_4$ and
 that  $\alpha\cdot k$ commutes with the exponential 
$e^{-i(\alpha\cdot k) \, \xi (x \cdot k)}$.
It is obvious that 
 $| f(x) |_{{\mathbb C}^4} =1$ 
for all $x \in {\mathbb R}^3$.
Hence, it follows from 
Theorem \ref{schnol3} that
$\lambda \in \sigma(H_3)$.
Since $\lambda$ is an arbitrary real number, one can conclude that 
$\sigma(H_3)=\mathbb R$.

For $d=2$, the proof is similar to that for $d=3$ and is omitted.
\end{expl}
%






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


\section{Spectra of \boldmath $H_d$}


In this section, we shall prove that $\sigma(H_3)= \mathbb R$
under the assumption that
the potential $q$ is sufficiently close, in
an ${\mathsf L}^2$ sense, to a function which is constant on planes perpendicular to
a particular direction in each of a sequence of growing balls.
We would like to 
emphasize  that we don't need any growth 
or decay property of the potential $q$ at infinity.  


%
\begin{thm}  \label{th:singseq3}
Let $q \in {\mathsf L}^2_{loc}({\mathbb R}^3; {\mathbb R})$. 
Suppose that there is a sequence $(k_n)_{n\in{\mathbb N}}$ of
unit vectors in ${\mathbb R}^3$,
a sequence $(B_{r_n}(a_n))_{n\in{\mathbb N}}$ of balls with centre
$a_n \in {\mathbb R}^3$ and radius $r_n \to \infty$ $(n\rightarrow\infty)$,
and a sequence of square-integrable functions
$\eta_n: (-r_n,r_n)\rightarrow{\mathbb R}$ $(n\in{\mathbb N})$ such  that
%
\begin{equation*} 
r_n^{-3}
\int_{B_{r_n}(a_n)} 
 \big|q(x) - \eta_n \big( (x - a_n)\cdot k_n) \big) \big|^2
\, dx \to 0 
\end{equation*}
%
as $n \to \infty$. Then $\sigma(H_3) = {\mathbb R}\;$   
for
any self-adjoint extension $H_3$ of
%
\begin{equation*}  
(-i\alpha \cdot \nabla + q) 
\big|_{C_0^\infty({\mathbb R}^3;  {\mathbb C}^4)}.
\end{equation*}
% 
\end{thm}



\vspace{6pt}

\begin{rem}
The two dimensional analogue of Theorem \ref{th:singseq3} holds true.
\end{rem}



\noindent
{\bf Proof of Theorem 4.1.}
Let $\lambda\in {\mathbb R}$; we shall show that $\lambda$ belongs to the spectrum
of $H_3$ by constructing a Weyl singular sequence.

In a similar way to Example \ref{schnlapp}, we can choose a sequence of
spinors $(\phi_n)_{n\in{\mathbb N}}$ in ${\mathbb C}^4$ such that
%
\begin{equation} \label{eq:phi-n}
|\phi_n|=1, \quad (\alpha \cdot k_n) \phi_n = \phi_n.
\end{equation}
%
Since $C^\infty(-r_n, r_n)$ is dense in ${\mathsf L}^2(-r_n, r_n)$,
there are functions $\tilde \eta_n\in C^\infty(-r_n, r_n)$ such that
%
\begin{equation}
\frac{1}{2 r_n}\int_{-r_n}^{r_n} |\eta_n(\tau) - \tilde \eta_n(\tau)|^2  d\tau \to 0
\end{equation}
%
 as
 $n\to\infty$.
Let $\xi_n(t) := \int_0^t \tilde \eta_n (\tau) d\tau$ \ 
$(t\in(-r_n, r_n); n\in{\mathbb N})$, and
define
%
\begin{equation}  \label{eq:Fn}
F_n(x) := e^{-i(\alpha \cdot k_n)
\xi_n((x-a_n)\cdot k_n)}
e^{i \lambda x \cdot k_n} \phi_n
\, : \, B_{r_n}(a_n) \to {\mathbb C}^4.
\end{equation}
%
Since $e^{-i(\alpha \cdot k_n)\xi_n((x-a_n)\cdot k_n)}$ is 
a unitary matrix, it follows
from (\ref{eq:phi-n}) that 
$|F_n(x)|_{{\mathbb C}^4}=1$ for all $n\in{\mathbb N}$ and all $x \in B_{r_n}(a_n)$.
Furthermore, we see that $F_n \in C^\infty(B_{r_n}(a_n))^4$, and
we make the same computation as in (\ref{eq:sa-4}) to get
%
\begin{gather} \label{eq:DOFn}
\begin{split}
(-i \alpha \cdot \nabla) F_n(x)
=
\big\{ \! 
- \tilde \eta_n\big((x - a_n) \cdot k_n\big) + \lambda  \,
\big\}\,F_n(x).
\end{split}
\end{gather}
%
Here we have used (\ref{eq:phi-n})
and the facts that $(\alpha\cdot k_n)^2 = I_4$ and that $\alpha\cdot k_n$
commutes with the exponentials in (\ref{eq:Fn}).

We now choose $\chi \in C_0^{\infty}({\mathbb R}^3; {\mathbb R})$ 
so that
$\mbox{supp} (\chi) \subset B_1(0)$ and
$\Vert \chi \Vert_{\mathsf L^2}=1$, and 
define
%
\begin{equation}  \label{eq:cut}
\chi_n(x):= r_n^{-3/2} \, 
\chi\big( r_n^{-1}(x - a_n)\big).
\end{equation}
%
We shall show that the sequence $(f_n)_{n\in{\mathbb N}}$ defined
by $f_n:=\chi_n F_n$ $(n\in{\mathbb N})$ is a Weyl singular sequence for
$H-\lambda$.
First, we note that $\Vert f_n \Vert_{{\mathsf L}^2}=1$
and $f_n \in C_0^\infty({\mathbb R}^3;{\mathbb C}^4)$.
Next, we see that
%
\begin{gather}  \label{eq:DOfn}
\begin{split}
(-i \alpha \cdot \nabla) f_n(x) 
&=
\big\{(-i \alpha \cdot \nabla)\chi_n(x) \big\}\,F_n(x) 
+
\chi_n(x)(-i \alpha \cdot \nabla) F_n(x)   \\
&=  r_n^{-5/2} \, 
\big[
(-i \alpha \cdot \nabla)\chi
\big]
\big( r_n^{-1}(x - a_n)\big) F_n(x) \\
&\qquad 
-  \tilde \eta_n \big((x - a_n) \cdot k_n \big)\,f_n(x)
+ \lambda f_n(x),
\end{split}
\end{gather}
%
where we have used (\ref{eq:DOFn}) and (\ref{eq:cut}).
Finally, it follows from (\ref{eq:DOfn})
that
%
\begin{gather}  \label{eq:DOfn1}
\begin{split}
(H_3-\lambda)f_n(x)
&=  
r_n^{-5/2} \, \big[(-i \alpha \cdot \nabla)\chi\big]
\big( r_n^{-1}(x - a_n)\big) \, F_n(x)  \\
{}&\;\; +
\big\{
q(x)- \tilde \eta_n\big((x - a_n) \cdot k_n\big) 
\big\}
 \, r_n^{-3/2} \, 
\chi\big(r_n^{-1}(x - a_n) \big) \, F_n(x) ,
\end{split}
\end{gather}
%
which --- adding and subtracting 
$\eta_n\big((x - a_n)\cdot k_n\big)$ --- implies that
%
\begin{gather} \label{eq:DOfn2}
\begin{split}
\Vert (H_3-\lambda)f_n \Vert
&\le 
r_n^{-1} \, \left(
\int_{|x|\le 1}
 \big| (\alpha \cdot \nabla)\chi(x) \big|^2
\, dx
\right)^{1/2}  \\
\noalign{\vskip 4pt}
&\;\;+
\Vert \chi \Vert_\infty \,
\left(
 r_n^{-3} 
\int_{B_{r_n}(a_n)}
\big| q(x)- \eta_n\big( (x - a_n) \cdot k_n \big) \big|^2
 \,  dx  \right)^{1/2}   \\
&\;\;+
\Vert \chi \Vert_\infty \,
\left(
{\pi\over r_n} 
\int_{-r_n}^{r_n}
\big|\eta_n(\tau)-\tilde \eta_n(\tau) \big|^2 d\tau \right)^{1/2}   \\
\noalign{\vskip 4pt}
&\quad \to 0
\ \  \mbox{ as } \  n \to \infty.
\end{split}
\end{gather}
%
Hence we can conclude that $\lambda \in \sigma(H_3)$. 
$\quad\blacksquare$

\vspace{8pt}


In the following theorem,
 we shall show that the result of Theorem \ref{th:singseq3}
extends to potentials which are
close to  constants on a local foliation of curved surfaces, which could
be fattened to sets of positive measure, 
provided  that 
their curvature becomes
asymptotically small. 
We assume that the potential is
approximated by  $C^\infty$ smooth functions,
which in the light of the proof of the preceding theorem is no serious
restriction of generality.

%
\vspace{10pt}

\begin{thm}  \label{th:distorted}
Let $q \in {\mathsf L}^2_{loc}({\mathbb R}^3; {\mathbb R})$. 
Suppose that there is a sequence $(k_n)_{n\in{\mathbb N}}$ of
unit vectors in ${\mathbb R}^3$,
a sequence $(B_{r_n}(a_n))_{n\in{\mathbb N}}$ of balls with centre
$a_n \in {\mathbb R}^3$ and radius $r_n \to \infty$ $(n\to\infty)$,
and sequences of functions
$\varphi_n \in C^\infty(B_{r_n}(a_n); {\mathbb R})$
and
$\eta_n \in C^\infty({\mathbb R};{\mathbb R})$ $(n\in{\mathbb N})$
such  that
%
\begin{equation} \label{eq:phin}
r_n^{-3}
\int_{B_{r_n}(a_n)} 
 \big|q(x) - \eta_n \big(x\cdot k_n + \varphi_n(x) \big) \big|^2
\, dx \to 0  
\quad (n \to \infty)
\end{equation}
%
and
%
\begin{equation} \label{eq:phin1}
r_n^{-3}
\int_{B_{r_n}(a_n)} 
\big|\nabla \phi_n(x) \big|^2 \,
 \big|\eta_n \big(x\cdot k_n + \varphi_n(x) \big) \big|^2
\, dx \to 0  
\quad (n \to \infty).
\end{equation}
%
Then $\sigma(H_3) = {\mathbb R}$   
for any self-adjoint extension $H$ of
%
\begin{equation}  \label{eq:domain2}
(-i\alpha \cdot \nabla + q) 
\big|_{C_0^\infty({\mathbb R}^3;  {\mathbb C}^4)}
\end{equation}
% 
\end{thm}

\vspace{6pt}

\begin{rem}
The two dimensional analogue of Theorem \ref{th:distorted} holds true.
\end{rem}


\noindent
{\bf Proof of Theorem 4.2.}
We follow the general line of the proof of 
Theorem \ref{th:singseq3}. Let $\lambda \in {\mathbb R}$ be arbitrary, and
$\phi_n$ as in (\ref{eq:phi-n}).
Define $\xi_n(t) := \int_0^t \eta_n (\tau) \, d\tau$ 
$(t\in{\mathbb R})$ and
%
\begin{equation}  \label{eq:Fnphi}
F_n(x) := 
e^{-i(\alpha \cdot k_n)\xi_n(x\cdot k_n + \varphi_n(x))}
e^{i\lambda\, x \cdot k_n} \phi_n
\qquad
(x\in B_{r_n}(a_n));
\end{equation}
%
Then $F_n \in C^\infty(B_{r_n}(a_n), {\mathbb C}^4)$ and
$|F_n(x)|_{{\mathbb C}^4}=1$ $(n\in{\mathbb N}, x \in B_{r_n}(a_n))$.
Moreover, abbreviating $q_n(x) := \eta_n(x\cdot k_n + \varphi_n(x))$
we get
%
\begin{gather}
\begin{split}
-i\alpha\cdot\nabla F_n(x)
&=
-i\alpha\cdot\big(
-i(\alpha\cdot k_n)\,
q_n(x)\,
 \{k_n + \nabla\varphi_n(x) \} + i \lambda\,k_n
\big)\,F_n(x) \\
&=
-q_n(x)\,F_n(x)
 -\big(  \alpha\cdot\nabla\varphi_n(x)   \big)
\,(\alpha\cdot k_n)\,q_n(x)\,F_n(x)
 + \lambda\,F_n(x)
\end{split}
\end{gather}
%
$(x\in B_{r_n}(a_n))$.
To construct a singular sequence, let $\chi_n$ be as in (\ref{eq:cut}), and 
define $f_n:= \chi_n F_n$.
Then $\Vert f_n \Vert_{{\mathsf L}^2}=1$
and $f_n \in C_0^\infty({\mathbb R}^3; {\mathbb C}^4)$.
Furthermore, we have
%
\begin{gather}  \label{eq:DOfnphi}
\begin{split}
-i \alpha \cdot \nabla f_n(x) 
&=
\big( -i \alpha \cdot \nabla\chi_n(x)  \big) F_n(x) 
+
\chi_n(x) \big(  -i \alpha \cdot \nabla F_n(x)  \big)   \\
&=  r_n^{-5/2}\, \big[(-i \alpha \cdot \nabla)
\chi\big]
(r_n^{-1} \big(x - a_n) \big) \,F_n(x) \\
&\quad - \chi_n(x) \, q_n(x)\,F_n(x) \\
&\quad
-\chi_n(x)
\big(
\alpha \cdot \nabla \varphi_n(x)
\big) 
(\alpha \cdot k_n)
q_n(x)\,F_n(x) 
+ \lambda f_n(x),
\end{split}
\end{gather}
%
from which we obtain
%
\begin{gather}   \label{eq:singsq}
\begin{split}
(H-\lambda)f_n(x)
&=
r_n^{-5/2} \, 
\big[
(-i \alpha \cdot \nabla)\chi
\big]
\big( r_n^{-1}(x - a_n) \big)\, F_n(x) \\
&\;\; +
\chi_n(x) \big(q(x) - q_n(x) \big)\,F_n(x)  \\
&\;\;
-\chi_n(x)
\big( \alpha \cdot \nabla \varphi_n(x) \big) (\alpha \cdot k_n) \,
q_n(x) F_n(x).
\end{split}
\end{gather}
%
Hence
%
\begin{gather}   \label{eq:singsq1}
\begin{split}
\Vert  (H-\lambda)f_n \Vert_{{\mathsf L}^2}
&\le 
r_n^{-1} \, \Big(
\int_{|x|\le 1}
 \big| (\alpha \cdot \nabla)\chi(x) \big|^2
\, dx
\Big)^{1/2}  \\
\noalign{\vskip 4pt}
&+
\Vert \chi \Vert_{{\mathsf L}^{\infty}} \,
\Big(
 r_n^{-3} 
\int_{B_{r_n}(a_n)}
\big| q(x)- \eta_n\big( (x - a_n) \cdot k_n \big) \big|^2
 \,  dx  \Big)^{1/2}   \\
\noalign{\vskip 4pt}
&+
\Vert \chi \Vert_{{\mathsf L}^{\infty}} \,
\Big(
 r_n^{-3} 
\int_{B_{r_n}(a_n)}
\big| \nabla \varphi_n (x) \big|^2 \, 
\big| \eta_n\big( (x - a_n) \cdot k_n \big) \big|^2
 \,  dx  \Big)^{1/2}   \\
\noalign{\vskip 4pt}
&\quad \to 0
\ \  \mbox{ as } \  n \to \infty, 
\end{split}
\end{gather}
by (\ref{eq:phin}) and  (\ref{eq:phin1}).
Here we twice used the fact that 
$|({\alpha}\cdot v)u|=|v||u|$ for 
any $v \in {\mathbb R}^3$ and $u \in {\mathbb C}^4$. 
Thus we can conclude that $\lambda \in \sigma(H_3)$. 
$\quad\blacksquare$

\vspace{6pt}


\begin{thebibliography}{00}

\bibitem{BE}
A.A. Balinsky and W.D. Evans,
\emph{On the virial theorem for the relativistic operator
of Brown and Ravenhall, and the absence of embedding eigenvalues},
Lett. Math. Phys. {\bf 44} (1989), 233--248.

\bibitem{BE-1}
A.A. Balinsky and W.D. Evans,
\emph{Spectral Analysis of Relativistic Operators},
Imperial College Press, London, 2011.

\bibitem{BTB}
J.H. Bardarson, M. Titov and P.W. Brouwer,
\emph{Electrostatic confinement of electrons in an integrable graphene
quantum dot},
Phys. Rev. Lett. {\bf 102} (2009), 226803.


\bibitem{BMLS}
A. Boutet de Monvel, D. Lenz and P. Stollmann,
\emph{Sch'nol's theorem for strongly local forms},
Israel J. Math. {\bf 173} (2009), 189--211.

\bibitem{BES}
B.M. Brown, M.S.P. Eastham and K.M. Schmidt,
\emph{Periodic Differential Operators},
Operator Theory: Advances and Applications {\bf 230},
Birkh\"auser, Basel, 2013.

\bibitem{CFKS}
H.L. Cycon, R.G. Froese, W. Kirsch and B. Simon,
\emph{Schr\"odinger Operators},
Texts and Monographs in Physics, Springer-Verlag, Berlin-Heidelberg,
1987.

\bibitem{DSDLA}
X. Du, I. Skachko, F. Duerr, A. Luican and E.Y. Andrei,
\emph{Fractional quantum Hall effect and insulating phase of
Dirac electrons in graphene},
Nature {\bf 462} no. 8522 (2009), 192--195.

\bibitem{GP}
D. J. Gilbert and D. B. Pearson,
\emph{On subordinacy and analysis of the spectrum of one-
dimensional Schrodinger operators},
J. Math. Anal. Appl. {\bf 128} (1987), 30--56.

\bibitem{Gilbert}
D. J. Gilbert,
\emph{On subordinacy and analysis of the spectrum of Schrodinger operators with
two singular endpoints},
Proc. Royal Soc. Edinburgh {\bf 112A} (1989), 213--229.

\bibitem{HK}
S. Haeseler and M. Keller,
\emph{Generalized solutions and spectrum for
Dirichlet forms on graphs},
Random Walks, Boundaries and Spectra, 
eds. D. Lenz, F. Sobieczky and W. Woess,
Progress in Probability  {\bf 64} (2011) 181--199,
Springer Basel AG.



\bibitem{KOY}
H. Kalf, T. \={O}kaji and O. Yamada,
\emph{Absense of eigenvalues of Dirac operators with
potentials diverging at infinity},
Math. Nachr. {\bf 259} (2003), 19--41.


\bibitem{J}
K. J\"orgens,
\emph{Perturbations of the Dirac operator},
Conference on the theory of ordinary and partial
differential equation, eds. W.N. Everitt and B.D. Sleeman,
Lecture Notes in Mathematics {\bf 280} (1972), 87--102. 

\bibitem{LSV}
D. Lenz, P. Stollmann and I. Veselic',
\emph{Generalized eigenfunctions and spectral theory for strongly local
Dirichlet forms},
Spectral Theory and Analysis,
Operator Theory: Advances and Applications
{\bf 214} (2011) 83--106,
Springer Basel AG.



\bibitem{LY} M. Loss and H.T. Yau,
\emph{Stability of Coulomb systems with magnetic fields. III.  
      Zero energy bound states of the Pauli operators}, Commun. 
       Math. Phys. \textbf{104} (1986), 283--290.

 
\bibitem{NG}
K.S. Novoselov, A.K. Geim, S.V. Morozov, D. Jiang, M.I. Katsnelson,
I.V. Grigorieva, S.V. Dubonos and A.A. Firsov,
\emph{Two-dimensional gas of massless Dirac fermions in 
graphene}, Nature {\bf 438} no. 4233 (2005), 197--200.

\bibitem{RS-I}
M. Reed and B. Simon,
\emph{Methods of Modern Mathematical Physics I},
Functional Analysis, Revised and enlarged edition,
Academic Press, Orlando, 1980. 

\bibitem{KMS-3}
K.M. Schmidt,
\emph{Absolutely continuous spectrum of Dirac systems with potentials
infinite at infinity},
Math. Proc. Camb. Phil. Soc. {\bf 122} (1997), 377--384.

\bibitem{KMS-1}
K.M. Schmidt,
\emph{Dense point spectrum for the one-dimensional
Dirac operator with an electrostatic potential},
Proc. R. Soc. Edinburgh
{\bf 126A} (1996), 1087--1096.


\bibitem{KMS-2}
K.M. Schmidt,
\emph{Spectral properties of rotationally symmetric
massless Dirac operators},
Lett. Math. Phys.
{\bf 92} (2010), 231--241.

\bibitem{SU}
K.M. Schmidt and T. Umeda,
\emph{Spectral properties of massless Dirac operators
with real-valued potentials},
To appear in RIMS K\^{o}ky\^{u}roku Bessatsu,
Research Institute of Mathematical Sciences,
Kyoto University.

\bibitem{Stolz}
G. Stolz,
\emph{On the absolutely continuous spectrum of perturbed Sturm-Liouville operators},
J. Reine Angew. Math. {\bf 416} (1991), 1--23.

%
\bibitem{Thaller} B. Thaller,
     {\em The Dirac equation}, Springer-Verlag, Berlin,
      1992.
%
\bibitem{Weidmann} J. Weidmann,
     {\em Spectral Theory of Ordinary Differential Operators},
     Lect. Notes in Math. 1258, Springer-Verlag, Berlin,
      1987.

\end{thebibliography}

\end{document}
---------------1309160841603--
