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Dirac operators, threshold energies, threshold eigenfunctions, Weyl-Dirac operators, zero modes
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\documentclass[a4paper, 11pt,reqno]{amsart} 
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\usepackage{amsmath}




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




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          \@addtoreset{equation}{section}
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\begin{document}

\baselineskip=1.2\baselineskip   

\title[Dirac operators]{Eigenfunctions at the threshold energies 
of magnetic Dirac operators with positive mass
}



\author{Yoshimi Sait\={o}
and Tomio Umeda\\}

\address{
Department of Mathematics \\
University of Alabama at Birmingham, 
Birmingham, AL 35294, USA}
\email{saito@math.uab.edu}

\address{Department of Mathematical Sciences,
University of Hyogo, Himeji 671-2201, Japan
}
\email{umeda@sci.u-hyogo.ac.jp}

\date{\it March 30, 2009}



\maketitle

\vspace{10pt}

\textbf{Abstract.} We  propose a  framework for supersymmetric
Dirac operators to have threshold energies.  We apply the obtained results
to magnetic Dirac operators with positive mass in order
to investigate the asymptotic behaviors at infinity 
of the threshold eigenfunctions. 
It turns out that the results of most of the existing 
works on the zero modes of the Weyl-Dirac operators remain to hold 
for the threshold eigenfunctions of the magnetic Dirac operators
with positive mass.

\vspace{15pt}

\section{Introduction}

This paper
is concerned with eigenfunctions at the threshold energies
of Dirac operators with vector potentials
%
\begin{equation} \label{eqn:1-1}
H_{\!A}= \alpha \cdot (D - A(x) ) + m \beta , \quad D=\frac{1}{\, i \,}
\nabla_x,
\,\,\, x \in {\mathbb R}^3.
\end{equation}
%
Here $\alpha= (\alpha_1, \, \alpha_2, \, \alpha_3)$ is
the triple of  $4 \times 4$ Dirac matrices
%
\begin{equation*}\label{eqn:1-2}
\alpha_j = 
\begin{pmatrix}
 \mathbf 0 &\sigma_j \\ \sigma_j &   \mathbf 0 
\end{pmatrix}  \qquad  (j = 1, \, 2, \, 3)
\end{equation*}
with  the $2\times 2$ zero matrix $\mathbf 0$ and
the triple of  $2 \times 2$ Pauli matrices
%
\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}, \,\,\,
\sigma_3 =
\begin{pmatrix}
1&0 \\ 0&-1
\end{pmatrix},
\end{equation*}
%
and
\begin{equation*}\label{eqn:1-4}
\beta=
\begin{pmatrix}
I_2& \mathbf 0 \\ \mathbf 0 & -I_2
\end{pmatrix}.
\end{equation*}
%
The constant $m$ is assumed to be positive.


Throughout the present paper we assume that  each component
of the vector potential $A(x)=(A_1(x), \,A_2(x), \, A_3(x))$ 
is a real-valued 
measurable function. In
addition to this, we shall later impose several different sets of
 assumptions on
$A(x)$ under which the operator $-\alpha \cdot A(x)$
is relatively compact with respect to the free Dirac operator 
$H_0=\alpha \cdot D  + m \beta $. 
Therefore, under any set of assumptions to be made, the essential spectrum 
of the Dirac 
operator $H_{\!A}$ is given by the union of the intervals 
$(-\infty, \, -m]$ and $[m, \, +\infty)$:
%
\begin{equation*} \label{eqn:1-5}
\sigma_{\rm{ess}}(H_{\!A}) =  (-\infty, \, -m] \cup  [m, \, +\infty) .
\end{equation*}
%
We shall later see  that
 the discrete spectrum of $H_{\!A}$ is empty.
In other words,
there are no discrete eigenvalues with finite  multiplicity
 in the 
spectral gap $(-m, \, m)$.
Under each set of the assumptions
in the present paper, we thus have 
%
\begin{equation*}
\sigma(H_{\!A})=
\sigma_{\rm{ess}}(H_{\!A}) =  (-\infty, \, -m] \cup  [m, \, +\infty) .
\end{equation*}
%

By the threshold energies of $H_{\!A}$, we mean  the values $\pm m$, 
the edges of the
essential spectrum $\sigma_{\rm{ess}}(H_{\!A})$.
It is a common practice in quantum scattering theory 
that the edge(s) of the
essential spectrum of a quantum Hamiltonian is normally
excluded. On the other hand, 
these values of Dirac operators are
 of particular importance and of interest from
the physics point of  view. 
See Pickl and D\"urr \cite{PicklDurr} and Pickl \cite{Pickl},
where they investigate generalized eigenfunctions near
the edges $\pm m$ of the essential spectrum of $H_{\!A}$
with the emphasis on the famous relativistic effect
of pair creation. 

The aim of the present paper is twofold. 
First, we  consider 
a class of 
supersymmetric Dirac operators ( a class of abstract 
Dirac operators; see (\ref{eqn:SSDO-K2-Sec1}) below) and 
 investigate the eigenspaces
corresponding to the energies $\pm m$.
The eigenspaces
corresponding to the energies $\pm m$ of 
supersymmetric Dirac operators seem to have not been 
treated in the literature (see Remark \ref{rem:Thaller}).
In addition, we
propose a framework in which the supersymmetric
Dirac operators possess the threshold energies $\pm m$
at the edges of the essential spectrum. 
Second, we apply 
the established results  
to the Dirac operators (\ref{eqn:1-1}) in order to
describe the asymptotic behaviors at infinity
of eigenfunctions  
of the operators (\ref{eqn:1-1}) 
corresponding to the threshold eigenvalues $\pm m$,
 and  investigate the set of 
vector potentials which give rise
to eigenfunctions 
of the operators (\ref{eqn:1-1}) 
corresponding to the threshold eigenvalues $\pm m$.
Actually, it turns out that 
the asymptotic behaviors of eigenfunctions of
the operators (\ref{eqn:1-1}) at the threshold eigenvalues
are exactly the same as those of eigenfunctions of  the
Weyl-Dirac operator 
%
$$
T_{\!A}=\sigma \cdot (D - A(x)) 
$$
%
 at 
the zero eigenvalue.
In particular, the asymptotic behaviors of eigenfunctions of
the operators (\ref{eqn:1-1}) at the threshold eigenvalues
at  $\pm m$ are independent of $m$.


Explanations should be in order.
The supersymmetric Dirac operator $H$ which we shall consider
in the present paper
is defined as follows:
%
\begin{equation}   \label{eqn:SSDO-K2-Sec1}
H := 
\begin{pmatrix}
0  &  T^*  \\
T  &  0
\end{pmatrix}
 + 
m  \begin{pmatrix}
I  &  0  \\
0  &  -I
\end{pmatrix} 
\;\; \mbox{ on } \;
\mathcal K = \mathcal H_+  \oplus \mathcal H_-,
\end{equation}
%
where $T$ is a densely defined 
 operator from a Hilbert space $\mathcal H_+$ to  
another Hilbert $\mathcal H_-$,
and the identity operators in $\mathcal H_+$ and
$\mathcal H_-$ are both denoted by $I$ with an abuse
of notation.  
We should like to mention that
$T$ does not need  to be a closed operator,
and that $T^*$ does not need to be densely
defined; see Theorems \ref{thm:SSDO-main1},
\ref{thm:SSDO-main2} and Corollary \ref{cor:SSDO-C1}
in section 2.
Also, we would like to mention that
we do not need the Foldy-Wouthuysen transformation,
which is a major tool in the standard theory
of the supersymmetric Dirac operator.
 With these respects, our approach is
different from the standard theory of
the supersymmetric Dirac operators;
see  Chapter 5 of the book by 
Thaller \cite{Thaller} for the 
standard theory of the supersymmetric Dirac operator.




By the supercharge $Q$ and
the involution $\tau$, we mean 
the first  and the second term, respectively,
on the right hand side of 
(\ref{eqn:SSDO-K2-Sec1}):
%
\begin{equation}   \label{eqn:SSDO-K3-Sec1}
Q = 
\begin{pmatrix}
0  &  T^*  \\
T  &  0
\end{pmatrix}, \quad
%
\tau = 
\begin{pmatrix}
I  &  0  \\
0  &  -I
\end{pmatrix}.
\end{equation}
%
Note that the inner product of the Hilbert space $\mathcal K$
 is
defined as follows: 
%
\begin{equation}  \label{eqn:SSDO-K4-Sec1}
(f, \, g)_{\mathcal K} 
 := ( \varphi^+, \, \psi^+)_{\mathcal H_+}
 +
( \varphi^-, \, \psi^-)_{\mathcal H_-}
\end{equation}
%
for
%
\begin{equation}    \label{eqn:SSDO-K5-Sec1} 
f= 
\begin{pmatrix}
\varphi^+  \\
\varphi^-
\end{pmatrix}, \quad
g=
\begin{pmatrix}
\psi^+  \\
\psi^-
\end{pmatrix}  \in \mathcal K .
\end{equation}
%
Our main result on the supersymmetric  Dirac operators
asserts that $\mbox{Ker}(H - m)$ (resp.  $\mbox{Ker}(H + m)$),
{\it i.e.}  
the eigenspace 
of $H$ corresponding to the eigenvalue $m$ (resp. $-m$), 
is given as the direct sum of 
 $\mbox{Ker}(T)$ (resp. $\mbox{Ker}(T^*)$) and 
the zero space $\{0 \}$; see
Corollary \ref{cor:SSDO-C1}.
It straightforward from our results that
$\mbox{Ker}(H - m)$ (resp. $\mbox{Ker}(H + m)$)
is linearly isomorphic to $\mbox{Ker}(T)$ 
(resp. $\mbox{Ker}(T^*)$).
In particular,  it follows that
$\mbox{Ker}(H\pm m)$ are 
independent of $m$.

It is of special interest to consider the 
case where the Hilbert space $\mathcal H_+$ coincides with
$\mathcal H_-$ and $T$ is self-adjoint ($T^*=T$).
In this case, our result says,
   in addition to symmetry between
   $(-\infty, \, -m)$ and  $(m, \, \infty)$, 
that
the spectrum of $H$ at $\pm m$ is symmetric with respect to $0$
in the following sense: (1) $m$ is an eigenvalue $H$ if and 
only if $-m$ is an eigenvalue of $H$;
(2) the multiplicity of the eigenvalue $m$ is the
same as that of the eigenvalue $-m$.
As mentioned above,
 we propose a
framework under which 
$\sigma(H)$ is given by
the union $(-\infty, \, -m] \cup [m, \, \infty)$
while $\sigma(T) \supset (0, \, \infty)$.
Thus $\pm m$ are always threshold energies of 
the supersymmetric Dirac operator $H$ 
under the assumption $\sigma(T) \supset (0, \, \infty)$.
It will become  clear in section 2 that the involution $\tau$ causes
the gap $(-m, \, m)$.


Note that in the standard theory of supersymmetric
Dirac operators, symmetry with respect to $0$
 of the spectrum of the operator of the form 
(\ref{eqn:SSDO-K2-Sec1})  is 
only assured 
between the intervals $(-\infty, \, -m)$ and
$(m, \infty)$,
excluding $\pm m$; see
 Thaller \cite[p. 155, Corollary 5.14]{Thaller}. 
We shall recall the discussion of \cite[\S 5.6.1]{Thaller}
in our context: A supersymmetric Dirac operator $H$ 
of the form (\ref{eqn:SSDO-K2-Sec1}) is unitarily
equivalent to the operator
%
\begin{equation*}
\begin{pmatrix}
\sqrt{T^* T + m^2} & 0  \\
0 & -\sqrt{T T^* + m^2} 
\end{pmatrix}.
\end{equation*}
%
It follows
that 
$T T^* + m^2 =S (T^* T + m^2) S^*$,
where $S$ is a partial isometry with
initial subspace ${\mathcal H}\ominus\mbox{Ker}(T)$.
(This is the reason why  symmetry mentioned above has to
exclude $\pm m$.)


We should like to emphasize the broad applicability
of our results on the supersymmetric Dirac operator.
Namely, thanks to the generality of 
our results on 
the supersymmetric Dirac operator, 
we are able to utilize most of the existing works on
the zero modes of the Weyl-Dirac operator $T_{\!A}$
 (cf.
\cite{AdamMuratoriNash1,AdamMuratoriNash2,AdamMuratoriNash3,
BalinEvan1,BalinEvan2,BalinEvan3,
BalinEvanSaito,
BugliaroFeffermanGraf, Elton, 
ErdosSolovej1, ErdosSolovej2,  ErdosSolovej3,
LossYau})
for the purpose of  
investigating  the threshold
eigenfunctions of 
the Dirac operator $H_{\!A}$.  
Among these works,
 we  shall utilize ones  by Loss and Yau \cite{LossYau}, 
by Adami,
Muratori and Nash
\cite{AdamMuratoriNash1},
 by Balinsky and Evans
\cite{BalinEvan2},
and
by Elton \cite{Elton}. 
Accordingly, we shall get  deep and multifaceted understandings of the threshold
eigenfunctions of the magnetic Dirac operator  $H_{\!A}$
defined by (\ref{eqn:1-1}) .




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




\section{Supersymmetric Dirac operators}

This section is devoted to a discussion about
spectral properties of a  class
of supersymmetric Dirac operators.
We should like to emphasize that
our approach appears to be in the reverse 
direction in the sense that
 we start with two Hilbert spaces, and introduce
a supersymmetric Dirac operator on
the direct sum of the two Hilbert spaces.
We find this approach convenient for 
our purpose.



Let $\mathcal H_{\pm}$ be  Hilbert spaces, and 
 let $\mathcal K =  \mathcal H_+  \oplus \mathcal H_-$.
Note that the inner product of $\mathcal K$ is
defined as in (\ref{eqn:SSDO-K4-Sec1}),
(\ref{eqn:SSDO-K5-Sec1}).
The main object in this section is 
a supersymmetric Dirac operator $H$  in $\mathcal K$:
%
\begin{equation}   \label{eqn:SSDO-K2}
H := 
\begin{pmatrix}
0  &  T^*  \\
T  &  0
\end{pmatrix}
 + 
m  \begin{pmatrix}
I  &  0  \\
0  &  -I
\end{pmatrix},
\end{equation}
%
where $T$ is a densely defined 
  operator from $\mathcal H_+$ to $\mathcal
H_-$,
 $m$  is a positive
constant, and the identity operators in $\mathcal H_+$ and
$\mathcal H_-$ are both denoted by $I$. 


We now state the main results in
this section, which reveal the nature of 
eigenvectors of the supersymmetric Dirac
operator (\ref{eqn:SSDO-K2}) at the 
eigenvalues $\pm m$.

\vspace{4pt}

\begin{thm} \label{thm:SSDO-main1}
Suppose that $T$ is a densely defined  operator from $\mathcal H_+$ to
$\mathcal H_-$.  Let $H$  be a
supersymmetric Dirac operator defined by {\rm(\ref{eqn:SSDO-K2})}. 

\vspace{4pt}

{\rm (i)} If $f= {}^t (\varphi^+ , \, \varphi^-)
\in \mbox{\rm Ker} (H -m)$,
then $\varphi^+ \in \mbox{\rm Ker}(T)$ and $\varphi^-=0$.


\vspace{4pt}

{\rm (ii)} Conversely, if $\varphi^+ \in \mbox{\rm Ker}(T)$, then
$f= {}^t (\varphi^+ , \, 0)
\in \mbox{\rm Ker} (H -m)$.
\end{thm}


\vspace{4pt}


\begin{thm} \label{thm:SSDO-main2}
Assume that $T$ and $H$ are the same
 as in Theorem \ref{thm:SSDO-main1}.

\vspace{4pt}

{\rm (i)} If $f= {}^t (\varphi^+ , \, \varphi^-)
\in \mbox{\rm Ker} (H + m)$,
then $\varphi^+=0$ and $\varphi^- \in \mbox{\rm Ker}(T^*)$.


\vspace{4pt}

{\rm (ii)} Conversely, if $\varphi^- \in \mbox{\rm Ker}(T^*)$, then
$f= {}^t ( 0 , \, \varphi^-)
\in \mbox{\rm Ker} (H + m)$.
\end{thm}


\vspace{4pt}

As immediately consequences, we have 

\vspace{4pt}

\begin{cor}  \label{cor:SSDO-C1}
Assume that $T$ and $H$ are the same
 as in Theorem \ref{thm:SSDO-main1}. Then

\vspace{4pt}

{\rm(i)} $\mbox{\rm Ker}(H-m) = \mbox{\rm Ker}(T)\oplus \{ 0 \}$,
$\mbox{\rm dim} 
 \large( \mbox{\rm Ker} (H - m) \large) 
= 
\mbox{\rm dim} 
 \large( \mbox{\rm Ker} (T) \large)$.

\vspace{4pt}

{\rm(ii)} $\mbox{\rm Ker}(H+m) = \{ 0 \}\oplus\mbox{\rm Ker}(T^*)$,
$\mbox{\rm dim} 
 \large( \mbox{\rm Ker} (H + m) \large) 
= 
\mbox{\rm dim} 
 \large( \mbox{\rm Ker} (T^*) \large)$.
\end{cor}


\vspace{4pt}

We should like to emphasize that
comparison between Corollary \ref{cor:SSDO-C1} above
and 
\cite[p. 144, (5.23)]{Thaller} 
(i.e. $\mbox{\rm Ker}(Q) 
= \mbox{\rm Ker}(T)\oplus\mbox{\rm Ker}(T^*)$) 
indicates the importance
of the presence of the involution 
%
\begin{equation}   
\begin{pmatrix}
I  &  0  \\
0  &  -I
\end{pmatrix}.
\end{equation}
%
Also, we should like to emphasize that
Corollary \ref{cor:SSDO-C1} has a significant 
implication:
the eigenspaces of $H$ corresponding to
the eigenvalue $\mp m$ are independent of $m$.


\begin{rem}   \label{rem:Thaller}
{\rm
To our knowledge, Thaller \cite[p. 195, Theorem
7.1]{Thaller} is the first to mention the properties
{\rm(i)}, {\rm(ii)} in Theorem \ref{thm:SSDO-main1},
 but only for the
magnetic Dirac operator {\rm(\ref{eqn:1-1})}.
It should be noted that in his theorem mentioned above, 
  $\pm m$ are not necessarily
the thresholds energies of the magnetic Dirac
operator; compare this with
 Theorem \ref{thm:SSDO-main3} below.
Also, we would like to emphasize that abstractness 
is essential  
 in Theorems \ref{thm:SSDO-main1},
\ref{thm:SSDO-main2} and Corollary \ref{cor:SSDO-C1},
which enable us to apply these results to
the magnetic Dirac operator under
various assumptions on the vector potential
in the later sections: see sections 3--6.
In fact, we shall even deal with vector potentials
with local singularities in section 5.
}
\end{rem}



%
\begin{proof}[{\bf Proof of Theorem \ref{thm:SSDO-main1} }]
We first prove assertion (i).  
Let $f= {}^t (\varphi^+ , \, \varphi^-)\in \mbox{\rm Ker} (H -m)$.
We then have
%
\begin{equation}   \label{eqn:SSDO-K2-pr}
\begin{pmatrix}
0  &  T^*  \\
T  &  0
\end{pmatrix}
\begin{pmatrix}
\varphi^+   \\
\varphi_-
\end{pmatrix}
 + 
m  \begin{pmatrix}
I  &  0  \\
0  &  -I
\end{pmatrix}
\begin{pmatrix}
\varphi^+   \\
\varphi_-
\end{pmatrix}
= m
\begin{pmatrix}
\varphi^+   \\
\varphi_-
\end{pmatrix},
\end{equation}
%
hence
%
\begin{equation}   \label{eqn:SSDO-K5}
\begin{cases}
T^* \varphi^- + m \varphi^+
& =m \varphi^+  \\
\noalign{\vskip 4pt}
T \varphi^+ - m \varphi^-
& =m \varphi^-,
\end{cases}
\end{equation}
%
which immediately implies that $T^*\varphi^-=0$ and $T\varphi^+=2m\varphi^-$.
It follows that
%
\begin{gather}  \label{eqn:SSDO-K6}
\begin{split}
\Vert T \varphi^+ \Vert^2_{{\mathcal H}_-}  
&= 
(T \varphi^+, \, T \varphi^+)_{{\mathcal H}_-}   \\
&= 
(T \varphi^+, \, 2m \varphi^-)_{{\mathcal H}_-}   \\
&= 
(\varphi^+, \, 2m \,T^* \! \varphi^-)_{{\mathcal H}_+}   \\
&=0.
\end{split}
\end{gather}
%
Thus we see that $\varphi^+ \in \mbox{Ker}(T)$, and that
$\varphi^-= (2m)^{-1} \, T \varphi^+ = 0$.

We next prove assertion (ii). 
Let $\varphi^+ \in \mbox{\rm Ker}(T)$ and
put $f:= {}^t (\varphi^+ , \, 0)$. 
Then it follows that
$Hf = {}^t (m\varphi^+ , \, T\varphi^+) 
= m \, {}^t (\varphi^+ , \, 0)=mf$.
\end{proof}

\vspace{5pt}

We omit the proof of Theorem \ref{thm:SSDO-main2}, which
 is quite similar to that of 
Theorem \ref{thm:SSDO-main1}.



In the rest of this paper,
we only deal with the
case  
 where
${\mathcal H_+}={\mathcal H_-}:= \mathcal H$ and
$T$ is a self-adjoint operator in $\mathcal H$.
In this case, the supersymmetric operator $H$
becomes of the form
%
\begin{equation}   \label{eqn:SSDO-SA-1}
H = 
\begin{pmatrix}
0  &  T  \\
T  &  0
\end{pmatrix}
 + 
m  \begin{pmatrix}
I  &  0  \\
0  &  -I
\end{pmatrix}
\end{equation}
%
in the Hilbert space 
${\mathcal K}={\mathcal H}\oplus{\mathcal H}$,
and 
it follows from Theorems
\ref{thm:SSDO-main1} and  
\ref{thm:SSDO-main2} that
the operator $H$ of the form (\ref{eqn:SSDO-SA-1}) possesses of 
an important equivalence:
%
\begin{gather}  \label{eqn:SSDO-K4}
\begin{split}
T \mbox{ has a zero mode}   
&\Longleftrightarrow  
\; m \mbox{ is an eigenvalue of } H   \\
{}&\Longleftrightarrow  \; - m \mbox{ is an eigenvalue of } H. 
\end{split}
\end{gather}
%
Here we say that $T$ has a \textit{zero mode} if $T$  has 
an eigenvector corresponding to the eigenvalue $0$.
In other words, the fact that $T$ has a zero mode
is equivalent to the fact
that $0$ is an eigenvalue  of $T$.
Furthermore, Theorems
\ref{thm:SSDO-main1} and  
\ref{thm:SSDO-main2} imply the following equivalence for
a zero mode $\varphi$ of $T$:
%
\begin{gather}  \label{eqn:SSDO-K4+1}
\begin{split}
 T\varphi =0 
\; 
\Longleftrightarrow  
\;  
H
\begin{pmatrix}
\varphi  \\
0  
\end{pmatrix}
=
m
\begin{pmatrix}
\varphi  \\
0  
\end{pmatrix}
\;
\Longleftrightarrow
\;
H
\begin{pmatrix}
0  \\
\varphi    
\end{pmatrix}
=
-m
\begin{pmatrix}
0 \\
\varphi   
\end{pmatrix}
\end{split}
\end{gather}
%
% 
The equivalences (\ref{eqn:SSDO-K4}) and  (\ref{eqn:SSDO-K4+1})
are particularly
interesting in the context of the
following theorem, where $\pm m$ are the thresholds.


\vspace{4pt}
\begin{thm} \label{thm:SSDO-main3}
Let $T$ be a self-adjoint operator
in the Hilbert space $\mathcal H$. Suppose that
$\sigma(T) \supset [0, \, +\infty)$.
Then  
%
$$
\sigma(H) =(-\infty, \, -m] \cup [m, \, +\infty).
$$
%  
In particular,
$\sigma_d(H)=\emptyset$, i.e., 
the set of discrete eigenvalues of $H$ with finite 
multiplicity is empty.
\end{thm}
 

\begin{proof}
It follows from (\ref{eqn:SSDO-SA-1}) that
${\mathcal D}(H^2)=\mathcal D(T^2) \oplus \mathcal D(T^2)$
and that
%
\begin{equation}   \label{eqn:SSDO-SA-2}
H^2=
\begin{pmatrix}
 T^2 + m^2 I & 0 \\
0 & T^2 + m^2 I 
\end{pmatrix} 
\ge m^2 
\begin{pmatrix}
 I & 0 \\
0 &  I 
\end{pmatrix}. 
\end{equation}
%
This inequality implies that 
$\sigma(H) \subset (-\infty, \, -m] \cup [m, \, +\infty)$.

To complete the proof, 
we shall prove the fact that 
$\sigma(H) \supset (-\infty, \, -m] \cup [m, \, +\infty)$. To this end,
 suppose  $\lambda_0 \in (-\infty,  -m] \cup [m, \, +\infty)$ be given.
Since  $\sqrt{\lambda_0^2 - m^2} \ge 0$,
 we see, by the assumption of the theorem, that
$\sqrt{\lambda_0^2 - m^2} \in \sigma(T)$. 
Therefore, we can find 
a sequence $\{ \psi_n \}_{n=1}^{\infty} \subset \mathcal H$
such that
%
\begin{equation}     \label{eqn:SSDO-SA-3}
\Vert \psi_n \Vert_{\mathcal H}=1, \;\;
\psi_n \in \mbox{Ran}
\big( 
E_T (\nu - \frac{1}{\,n \,}, \,
     \nu  + \frac{1}{\,n \,})
\big),
\quad \nu_0:=\sqrt{\lambda_0^2 - m^2}  
\end{equation}
%
for each $n$, where $E_T(\cdot)$ is the spectral measure associated 
with $T$:
%
\begin{equation}    \label{eqn:SSDO-SA-4}
T = \int_{-\infty}^{\infty} \lambda \, dE_T(\lambda).
\end{equation}
%
Here we have used a basic property of the 
spectral measure: see, for example, Reed and Simon 
\cite[p. 236, Proposition]{ReedSimon}.
It is straightforward to see that
%
\begin{align}   \label{eqn:SSDO-SA-5}
\Vert (T^2  - \lambda_0^2 + m^2) \psi_n \Vert_{\mathcal H} \to 0
\mbox{ as } n \to \infty   
\end{align}
%
and that
%
\begin{gather}    \label{eqn:SSDO-SA-6}
\begin{split}
{}&\Vert (T  + \lambda_0 + m) \psi_n \Vert^2_{\mathcal H} 
\to 
(\nu_0 + \lambda_0 + m)^2
\mbox{ as } n \to \infty,   \\
\noalign{\vskip 4pt}
{}&\Vert (T  + \lambda_0 - m) \psi_n \Vert^2_{\mathcal H} 
\to 
(\nu_0 + \lambda_0 - m)^2
\mbox{ as } n \to \infty. 
\end{split}  
\end{gather}
%

We shall construct a sequence 
$\{f_n \} \subset \mathcal D(H) =\mathcal D(T)
\oplus \mathcal D(T)$ satisfying 
$\Vert f_n \Vert_{\mathcal K}=1$ and 
$\Vert  (H-\lambda_0) f_n \Vert_{\mathcal K}\to 0$
as $n \to \infty$.
To this end, we first put
%
\begin{equation}      \label{eqn:SSDO-SA-7-1}
g_n :=
\begin{pmatrix}
\psi_n \\
\psi_n 
\end{pmatrix}\!.
\end{equation}
%
We next see that
%
\begin{equation}   \label{eqn:SSDO-SA-9}
\Vert (H+\lambda_0)g_n \Vert_{\mathcal K}  \to
\sqrt{2(\nu_0 + \lambda_0)^2 + 2m^2 \,} 
\mbox{ as }  n \to  \infty
\end{equation}
%
by using (\ref{eqn:SSDO-SA-6}) and
%
\begin{equation*}   \label{eqn:SSDO-SA-8}
(H+\lambda_0)g_n
=
\begin{pmatrix}
(T  + \lambda_0 + m)\psi_n   \\
\noalign{\vskip 4pt}
 (T  + \lambda_0 - m)\psi_n
\end{pmatrix}.
\end{equation*}
%
It is clear that (\ref{eqn:SSDO-SA-9})
 enables us to define 
%
\begin{equation}      \label{eqn:SSDO-SA-7}
f_n := \frac{1}{\Vert (H + \lambda_0)g_n \Vert_{\mathcal K}}
(H + \lambda_0)g_n.
\end{equation}
%
Since we have
%
\begin{equation}   \label{eqn:SSDO-SA-10}
(H-\lambda_0)(H+\lambda_0)g_n
=
\begin{pmatrix}
(T^2  - \lambda_0^2 + m^2)\psi_n   \\
\noalign{\vskip 4pt}
 (T^2 -  \lambda_0^2 + m^2)\psi_n
\end{pmatrix},
\end{equation}
%
we can conclude from (\ref{eqn:SSDO-SA-7}),
(\ref{eqn:SSDO-SA-10}) and (\ref{eqn:SSDO-SA-5}) 
that $\Vert  (H-\lambda_0) f_n \Vert_{\mathcal K}
\to 0$ as $n \to \infty$.
Thus we have shown 
that $\lambda_0 \in \sigma(H)$.
\end{proof}




\vspace{2pt}
In all the later sections,
we shall apply the obtained results on the supersymmetric
Dirac operator to the Dirac operator $H_{\!A}$ 
of the form (\ref{eqn:1-1}) in the Hilbert space
${\mathcal K} = \big[L^2({\mathbb R}^3)\big]^4$, where we take
$T$ to be the Weyl-Dirac operator
%
\begin{equation} \label{eqn:2-WD-1}
T_{\!A}= \sigma \cdot (D - A(x) )  
\end{equation}
%.
acting in the Hilbert space 
${\mathcal H}=\big[L^2({\mathbb R}^3)\big]^2$.


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

\section{Vector potentials yielding threshold eigenfunctions}

This section is devoted to examples of vector potentials
$A(x)$ which yield eigenfunctions of
Dirac operators $H_{\!A}= \alpha \cdot (D - A(x) ) + m \beta$
 at the threshold energies $\pm m$. 
The basic idea in this section is to
exploit the equivalences (\ref{eqn:SSDO-K4}) and
(\ref{eqn:SSDO-K4+1}).
Thus we shall apply the results in section 2 
in the way described in the last paragraph  
of section 2. It turns out that
 beautiful spectral properties are
in common to all the examples of Dirac operators in this
section. See properties
(i) -- (iii) of Example \ref{expl:LY}.

\begin{expl} \label{expl:LY}
Let 
%
\begin{align}      \label{eqn:LY-1}
A_{{}_{LY}}(x) = 3 \langle x \rangle^{-4} 
  \big\{ (1-|x|^2)  w_0 + 2 (w_0 \cdot x) x + 2 w_0 \times x \big\}
\end{align}
%
where
$\langle x \rangle = \sqrt{1 + |x|^2 \,}$,
$\phi_0 = {}^t(1, \, 0)$ {\rm (}$\phi_0$ can
be any unit vector in ${\mathbb C}^2${\rm )}, 
and
%
\begin{equation}      \label{eqn:LY-1-1}
w_0=  \phi_0 \cdot (\sigma \phi_0 )
:=\big( 
(\phi_0, \, \sigma_1 \phi_0)_{\!_{{\mathbb C}^2}}, \, 
  (\phi_0, \, \sigma_2 \phi_0)_{\!_{{\mathbb C}^2}}, \,
(\phi_0, \, \sigma_3 \phi_0)_{\!_{{\mathbb C}^2}}
 \big).
\end{equation}
%
Here $w_0 \cdot x$ and $w_0 \times x$
denotes the inner product  and 
the exterior 
product of
$\,{\mathbb R}^3$ respectively,
 and
$(\phi_0, \, \sigma_1 \phi_0)_{\!_{{\mathbb C}^2}}$ 
etc. denotes the inner
product of
${\mathbb C}^2$.
Then the Dirac operator
%
$$
H_{{}_{LY}}:=H_{\!A_{LY}} = \alpha \cdot (D - A_{{}_{LY}}(x) ) + m
\beta
$$ 
%
has  the following properties:
%
\begin{itemize}
\item[(i)] $\sigma(H_{{}_{LY}})=
   \sigma_{\rm ess}(H_{{}_{LY}})=
(-\infty, \, -m] \cup [m, \,\infty)${\rm;} 
\item[(ii)] The point spectrum of $H_{{}_{LY}}$ consists only of $\pm m$,
i.e. $\sigma_p(H_{{}_{LY}})=\{ -m,\, m\}${\rm;} 
\item[(iii)] $H_{{}_{LY}}$ is absolutely continuous on 
$(-\infty, \, -m) \cup (m,\, \infty)$.
\end{itemize}
%


We shall show these properties.
It is easy to see that $-\sigma \cdot
A_{{}_{LY}}(x)$  is  relatively compact perturbation of  $\sigma \cdot D$,
hence the Weyl-Dirac operator 
%
$$
T_{{}_{LY}}:= T_{\!A_{LY}} =\sigma \cdot (D - A_{{}_{LY}}(x))
$$
% 
is a self-adjoint
operator in the Hilbert space
${\mathcal H}=\big[L^2({\mathbb R}^3)\big]^2$ with 
the domain $\big[H^1({\mathbb R}^3)\big]^2$. 
By $H^s({\mathbb R}^3)$, we mean the Sobolev space
of order $s${\rm:}
%
$$
H^s({\mathbb R}^3) = \big\{ \, u \; \big| \,
  \Vert  
\langle D \rangle^s   u 
\Vert_{L^2({\mathbb R}^3)} < + \infty
\, \big\},
$$
%
where $\langle D \rangle= \sqrt{1 -\triangle}$.
Since the spectrum of the operator $\sigma \cdot D$
equals the whole real line,
we see that $\sigma(T_{{}_{LY}})=(-\infty, \,
\infty)$. Property {\rm(i)} immediately follows from Theorem
\ref{thm:SSDO-main3}. 


We shall show property {\rm(ii)}. 
According to Loss and Yau \cite[section II]{LossYau}, 
the Weyl-Dirac operator 
$T_{{}_{LY}}$
has a zero mode $\varphi_{{}_{LY}}$ defined by
%
\begin{equation}    \label{eqn:LY-2}
\varphi_{{}_{LY}}(x) = \langle x \rangle^{-3}
 \big( I_2 + i \sigma \cdot x \big) \phi_0. 
\end{equation}
%
It follows from  {\rm(\ref{eqn:SSDO-K4})} and
{\rm(\ref{eqn:SSDO-K4+1})}
 that ${}^t(\varphi_{{}_{LY}}, \, 0 )$ 
{\rm(}resp. ${}^t(0, \,\varphi_{{}_{LY}}) )$ is 
an eigenfunction of the Dirac operator $H_{{}_{LY}}$
corresponding to the threshold eigenvalue $m$ 
{\rm(}resp. $-m${\rm)}. 
Hence $\sigma_p(H_{{}_{LY}}) \supset \{-m, \, m\}$.
On the other hand, it follows from 
Yamada \cite{Yamada-1} 
that $H_{{}_{LY}}$ has no eigenvalue in 
$(-\infty, -m) \cup (m, \,\infty)$.
{\rm(}Note that the vector potential $A_{{}_{LY}}$ 
satisfies the assumption of 
\cite[Proposition 2.5]{Yamada-1}.{\rm)}
This fact, together with property {\rm(i)}, implies that
 $\sigma_p(H_{{}_{LY}}) \subset \{-m, \, m\}$. Summing up, 
we get property {\rm(ii)}.

Property {\rm(iii)} is a direct consequence of 
Yamada \cite[Corollary 4.2]{Yamada-1}.
As for absolutely continuity and 
limiting absorption principle for
Dirac operators, see also Yamada \cite{Yamada-2},
Balslev and Helffer \cite{BalslevHellfer}, and
Pladdy, Sait{\=o} and Umeda \cite{PladdySaitoUmeda}.
\end{expl}


\begin{expl} \label{expl:AMN}
In the same spirit as in Example \ref{expl:LY},
we can show the existence of countably infinite
number of vector potentials with which the Dirac
operators have the properties {\rm (i) -- (iii)} in
Example \ref{expl:LY}.

In fact,
we shall exploit a result on the Weyl-Dirac 
operator by Adam, Muratori and Nash \cite{AdamMuratoriNash1},
where they construct a series 
 of 
vector potentials $A^{(\ell)}$
$(\ell = 0$, $1$, $2$, $\cdots)$, each of  which gives rise a zero
mode $\psi^{(\ell)}$ of the Weyl-Dirac operator 
$T^{(\ell)}:= \sigma \cdot (D - A^{(\ell)}(x) )$.
The idea of  \cite{AdamMuratoriNash1} is an extension  of 
that of  Loss and Yau \,\cite[section II]{LossYau}{\rm;} Indeed 
$A^{(0)}$ and $\psi^{(0)}$ give the same
vector potential and zero mode as in
{\rm(\ref{eqn:LY-1})} and  {\rm(\ref{eqn:LY-2})}.
For $\ell \ge 1$, the construction of the zero mode
$\psi^{(\ell)}(x)$ is based on an anzatz {\rm(}see {\rm(7)} in section II of
{\rm \cite{AdamMuratoriNash1})} and 
the definition of $A^{(\ell)}$  is given by
%
\begin{align}      
A^{(\ell)}(x) =
\frac{h^{(\ell)}(x)}{| \psi^{(\ell)}(x)|^2}
\{ \psi^{(\ell)}(x) \cdot (\sigma \psi^{(\ell)}(x)) \},  \label{eqn:AMN-2}
\end{align}
%
where $h^{(\ell)}(x)$ is a real valued function and 
$\psi^{(\ell)}(x) \cdot (\sigma \psi^{(\ell)}(x))$ is 
defined in the same way as in {\rm(\ref{eqn:LY-1-1})}.
For $\ell = 1$,  the zero mode is given by
%
\begin{align}      
\psi^{(1)}(x) =\langle x \rangle^{-5}   
\big\{  ( 1 -\frac{5}{3} |x|^2) I_2  
  + (\frac{5}{3} - |x|^2) i\sigma \cdot x \big\} \phi_0.   \label{eqn:AMN-3}
\end{align}
%

By the same arguments as in 
Example \ref{expl:LY}, we can deduce that
 the Dirac operator 
$H^{(\ell)}:= \alpha \cdot (D - A^{(\ell)}(x) ) + m
\beta$, $\;\ell = 0$, $1$, $2$, $\cdots$, has 
 the properties  {\rm (i) -- (iii)}
of
Example \ref{expl:LY}.
\end{expl}

\vspace{10pt}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Asymptotic limits of threshold eigenfunctions}

In the previous section, we have seen there exists infinitely many
 $A$'s  such that the corresponding magnetic Dirac 
operators $H_{\! A}$ have the threshold  eigenvalues $\pm m$.
In this section,
we consider a class of  magnetic Dirac operators
$H_{\! A}$ under Assumption(SU) below,
and  will focus  on the asymptotic behaviors at infinity of 
 eigenfunctions of $H_{\! A}$  corresponding 
to $\pm m$ provided that
these two energies are eigenvalues.


\vspace{15pt}

\noindent
\textbf{Assumption(SU).} 

\noindent
Each element $A_j(x)$ 
($j=1, \, 2, \, 3$) of $A(x)$ is 
a measurable function satisfying
%
\begin{equation} \label{eqn:2-1}
| A_j(x) | \le C \langle x \rangle^{-\rho} 
\quad   ( \rho >1 ),
\end{equation}
%
where $C$ is a positive constant.

\vspace{15pt}

It is easy to see that under Assumption(SU)
the Dirac operator $H_{\! A}$
is a self-adjoint operator in the Hilbert space
 ${\mathcal K}=\big[L^2({\mathbb
R}^3)\big]^4$ with  
$\mbox{Dom}(H_{\! A}) = [H^1({\mathbb R}^3)]^4$.
Also it is easy to see that under Assumption(SU)
the Weyl-Dirac operator $T_{\! A}$
is a self-adjoint operator in the Hilbert space
 ${\mathcal H}=\big[L^2({\mathbb R}^3)\big]^2$ with  
$\mbox{Dom}(T_{\! A}) = \big[H^1({\mathbb R}^3)\big]^2$.
Since the operator $-\sigma \cdot A(x)$
is relatively compact with respect to the operator 
$T_0:=\sigma \cdot D$, and since $\sigma(T_0)=(-\infty, \, \infty)$,
it follows that  $\sigma(T_A)=(-\infty, \, \infty)$. 
Recalling that
%
\begin{equation}   \label{eqn:MDO}
H_{\! A} = 
\begin{pmatrix}
0  &  T_A  \\
T_A  &  0
\end{pmatrix}
 + 
m  \begin{pmatrix}
I  &  0  \\
0  &  -I
\end{pmatrix},
\end{equation}
 we apply Theorem \ref{thm:SSDO-main3}
to  $H_{\! A}$, and get
%
$$
\sigma(H_{\!A}) = 
\sigma_{\rm ess}(H_{\!A}) = (-\infty, \, -m] \cup  [m, \,+\infty).
$$
%
Hence $\pm m$ are the threshold energies of 
the operator $H_{\! A}$. 
Assuming that $\pm m$ are the
eigenvalues of
$H_{\! A}$,
we should emphasize here that
the eigenspaces corresponding to the eigenvalues
$\pm m$ of $H_{\! A}$ are given as 
the direct sum of $\mbox{Ker}(T_A)$ and 
the zero space $\{ 0 \}$ 
(cf. Corollary \ref{cor:SSDO-C1}), and that these two eigenspaces themselves
as well as their dimensions
are independent of $m$.


\begin{thm} \label{thm:ABEF-main}
Suppose that Assumption{\rm(SU)} is verified, and 
that $m$ {\rm(}resp. $-m${\rm)} is an eigenvalue
 of $H_{\! A}$. 
Let $f$ be an eigenfunction of $H_{\! A}$ corresponding 
to $m$ {\rm(}resp. $-m${\rm)}. Then there exists a zero
mode $\varphi^+$  {\rm(}resp. $\varphi^-${\rm)} of 
$T_A$ such that for any 
$\omega \in {\mathbb S}^2$
%
\begin{equation}    \label{eqn:limit-1}
\lim_{r \to \infty} r^2 f(r\omega) 
= 
\begin{pmatrix}
u^+(\omega)  \\
0
\end{pmatrix} 
\quad
\large( \mbox{resp.}
\begin{pmatrix}
0  \\
u^-(\omega) 
\end{pmatrix} 
\large),
\end{equation}
%
where 
%
\begin{equation}   \label{eqn:lim-1-WD}
u^{\pm}(\omega)  
=
\frac{i}{\, 4\pi \,}
\int_{{\mathbb R}^3}  
\big\{ 
  \big( \omega\cdot A(y) \big)  I_2
    + i\sigma \cdot 
    \big( \omega\times A(y) \big) 
   \big\}  \varphi^{\pm}(y) \, dy,
\end{equation}
%
and the convergence is uniform with respect to  $\omega$.
\end{thm}

Theorem \ref{thm:ABEF-main}  is a direct consequence
of Corollary \ref{cor:SSDO-C1}, together with 
Sait\={o} and Umeda \cite[Theorem 1.2]{SaitoUmeda1}.
Note that under Assumption(SU) every 
eigenfunction of $H_{\! A}$ corresponding to
either one of eigenvalues $\pm m$ is a continuous   
function of $x$ (cf. Sait\={o} and Umeda \cite[Theorem 2.1]{SaitoUmeda2}),
therefore the expression $f(r\omega)$ in (\ref{eqn:limit-1}) makes sense 
for each $\omega$.


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

\section{Sparseness of vector potentials yielding 
threshold eigenfunctions}

In this section, we shall discuss
 the sparseness of the set
of vector potentials $A$ which give rise to 
threshold eigenfunctions of magnetic Dirac
operators $H_{\! A}$,
in the sprit of Balinsky and Evans \cite{BalinEvan1},
\cite{BalinEvan2},
where they investigated Pauli operators
and Weyl-Dirac operators respectively.


We shall make the following assumption:

\vspace{15pt}

\noindent
\textbf{Assumption(BE).}

\noindent 
$A_j\in L^3({\mathbb R}^3)$  for $j=1$, $2$, $3$. 

\vspace{15pt}

Under Assumption(BE) Balinsky and Evans \cite[Lemma 2]{BalinEvan2}
showed that $-\sigma\cdot A$ is infinitesimally small with respect
to $T_0=\sigma\cdot D$ with 
$\mbox{Dom}(T_0)=\big[H^1({\mathbb R}^3)\big]^2$
(see (\ref{eqn:RangeSA-1}) below).
This fact enables us to define the self-adjoint 
realization
$T_A$ in the Hilbert space 
${\mathcal H}=\big[L^2({\mathbb R}^3)\big]^2$
as the operator sum of $T_0$
and $-\sigma\cdot A$, thus  
$\mbox{Dom}(T_A)=\big[H^1({\mathbb R}^3)\big]^2$.
It turns out that
under Assumption(BE) 
 $-\alpha\cdot A$ is infinitesimally small with respect
to $H_0:= \alpha \cdot D + m \beta$, 
and hence we can define the
self-adjoint  realization $H_{\! A}$
in the Hilbert space ${\mathcal K}=\big[L^2({\mathbb R}^3)\big]^4$ 
as the
operator sum of $H_0$ and $-\alpha\cdot A$, thus  
$\mbox{Dom}(H_{\! A})=\big[H^1({\mathbb R}^3)\big]^4$.
Therefore we can regard $H_{\! A}$ as
a supersymmetric Dirac operator, and 
shall apply the results in section 2 to $H_{\! A}$.
(Recall (\ref{eqn:MDO}) again.)
  

\vspace{5pt}

\begin{prop}    \label{prop:WD-spec}
Let Assumption\,{\rm(BE)} be satisfied. Then 
$\sigma(T_A)=(-\infty, \, \infty)$.
\end{prop}

We shall prepare a few lemmas for the proof of 
Proposition \ref{prop:WD-spec}.


\begin{lem}   \label{lem:WD-T0}
Let $z \in {\mathbb C} \setminus (-\infty, \, \infty)$.
Then
$\langle D \rangle^{1/2} (T_0 - z )^{-1}$ is a bounded
operator in ${\mathcal H}$. 
Moreover we have
% 
\begin{equation}    \label{eqn:RangeT0}
\mbox{\rm Ran}  \big( \langle D \rangle^{1/2}
 (T_0 - z)^{-1} \big) 
\subset
\big[ H^{1/2}({\mathbb R}^3) \big]^2.
\end{equation}
%
\end{lem}

\begin{proof}
It is sufficient to show the conclusions of the lemma
for $z=-i$. Let $\varphi \in \mbox{Dom}(T_0)$. 
Then we have
%
\begin{gather}  \label{eqn:RangeT0-1}
\begin{split}
\Vert (T_0 + i) \varphi \Vert^2_{{\mathcal H}}  
&= 
\int_{{\mathbb R}^3} 
\big| 
\big( (\sigma \cdot \xi) + i I_2 \big)
   \widehat \varphi (\xi)
\big|_{{\mathbb C}^2}^2 \, d\xi   \\
&= 
\int_{{\mathbb R}^3} 
( |\xi|^2 + 1 )
 \big|
\widehat \varphi (\xi)
\big|_{{\mathbb C}^2}^2 \, d\xi    \\
&= 
\Vert \langle D \rangle \varphi \Vert^2_{{\mathcal H}},
\end{split}
\end{gather}
%
where we have used the anti-commutation relation
$\sigma_j \sigma_k + \sigma_k \sigma_j
 = 2 \delta_{jk}I_2$ in the second equality.
It follows from (\ref{eqn:RangeT0-1})
that
%
\begin{equation}   \label{eqn:RangeT0-2}
\Vert \varphi \Vert_{{\mathcal H}}  
=\Vert \langle D \rangle (T_0 + i)^{-1}
 \varphi \Vert_{{\mathcal H}} 
\end{equation}
%
for all $\varphi \in {\mathcal H}$. 
Furthermore, we see that
\begin{gather}  \label{eqn:RangeT0-3}
\begin{split}
\Vert \langle D \rangle^{1/2}
 (T_0 + i)^{-1} \varphi
\Vert_{{\mathcal H}}   
&\le 
 \Vert 
\langle D \rangle^{1/2}  
(T_0 + i)^{-1} \varphi 
     \Vert_{[H^{1/2}({\mathbb R^3})]^2}    \\
&=
\Vert \langle D \rangle  
(T_0 + i)^{-1} \varphi \Vert_{{\mathcal H}}    \\
&= 
\Vert \varphi \Vert_{{\mathcal H}}.
\end{split}
\end{gather}
%
It is evident that (\ref{eqn:RangeT0-3}) proves
the conclusions of the lemma for $z=-i$.
\end{proof}


\begin{lem}   \label{lem:WD-sigma}
If $\varphi \in \big[ H^{1/2}({\mathbb R}^3) \big]^2$,
then
$(\sigma\cdot A)\langle D \rangle^{-1/2} \varphi \in {\mathcal H}$.
\end{lem}

\begin{proof}
By Balinsky and Evans \cite[Lemma 2]{BalinEvan2}, we see
that for any $\epsilon >0$, there exists
a constant $k_{\epsilon}>0$ such that
for all $\varphi \in \mbox{Dom}(T_0)$ 
%
\begin{equation}    \label{eqn:RangeSA-1}
\Vert (\sigma \cdot A) \varphi \Vert_{\mathcal H}
\le 
\epsilon  \Vert T_0 \varphi \Vert_{\mathcal H}
 + 
k_{\epsilon }
\Vert \varphi \Vert_{\mathcal H}.
\end{equation}
%
By virtue of the fact that 
$\langle D \rangle^{-1/2}\varphi \in \mbox{Dom}(T_0)$
for $\varphi \in \big[ H^{1/2}({\mathbb R}^3) \big]^2$,
it follows from (\ref{eqn:RangeSA-1})
that
%
\begin{align*}    
\Vert (\sigma \cdot A) 
\langle D \rangle^{-1/2} \varphi
\Vert_{\mathcal H}
&\le 
\epsilon  \Vert (T_0 + i) \langle D \rangle^{-1/2}
\varphi \Vert_{\mathcal H}
 + 
k_{\epsilon }
\Vert \langle D \rangle^{-1/2} \varphi 
\Vert_{\mathcal H}                \nonumber\\ 
&\le
\epsilon  \Vert \langle D \rangle^{1/2}
\varphi \Vert_{\mathcal H}
 + 
k_{\epsilon }
\Vert  \varphi \Vert_{\mathcal H} 
                 < + \infty,  %%%%% \label{eqn:RangeSA-2}
\end{align*}
%
where we have used (\ref{eqn:RangeT0-1}) 
 and the fact that
%
$
\Vert \langle D \rangle^{1/2} \varphi \Vert_{\mathcal H}
=
\Vert  \varphi \Vert_{[ H^{1/2}({\mathbb R}^3) ]^2}.
$
%
\end{proof}



\begin{lem}   \label{lem:WD-sigma1}
$\langle D \rangle^{-1} (\sigma\cdot A)\langle D \rangle^{-1/2}$ 
is a compact operator in ${\mathcal H}$.
\end{lem}

\begin{proof}
One can make a factorization
%
\begin{gather}   
\begin{split}   \label{eqn:factorization} 
{}&\langle D \rangle^{-1} (\sigma\cdot A)
                   \langle D \rangle^{-1/2}   \\
=&
\Big(
\frac{|D|^{1/2}}{\langle D \rangle} 
\Big)
\Big(
\frac{1}{|D|^{1/2}} (\sigma \cdot A ) \frac{1}{|D|^{1/2}} 
\Big)
\Big(
\frac{|D|^{1/2}}{\langle D \rangle^{1/2}} 
\Big).
\end{split}
\end{gather}
%
It is obvious that the first term and the last term
on the right hand side of  (\ref{eqn:factorization})
are bounded operators in ${\mathcal H}$.
Then it follows from (\ref{eqn:factorization}) and
Balinsky and Evans \cite[Lemma 1]{BalinEvan2}
that the conclusion of the lemma holds true.
\end{proof}


\begin{lem}   \label{lem:WD-T1}
Let $z \in {\mathbb C} \setminus (-\infty, \, \infty)$.
Then
$(T_A - z)^{-1}\langle D \rangle
\Big|_{ [ H^1({\mathbb R}^3) ]^2}$ 
can be
extended to a bounded operator $\widetilde R_A(z)$ in  ${\mathcal H}$.
Moreover
%
\begin{equation}    \label{eqn:R-T0}
(T_A - z )^{-1} \varphi = \widetilde R_A(z) 
\langle D \rangle^{-1}
\varphi
\quad {\mbox for \ } \forall \varphi \in {\mathcal H}.
\end{equation}
%
\end{lem}

\begin{proof}
We first show that $\langle D \rangle(T -z)^{-1}$ is
a closed operator in $\mathcal H$. 
To this end, suppose that $\{ \varphi_j \}$ is a 
sequence in $\mathcal H$ such that
$\varphi_j \to 0$ in $\mathcal H$ and 
$\langle D \rangle (T - z)^{-1}\varphi_j \to \psi$ in 
$\mathcal H$ as $j \to \infty$. 
Then $\{ (T - z)^{-1}\varphi_j \}$ is a Cauchy sequence
in $\big[ H^1({\mathbb R}^3) \big]^2$, hence
there exists a $\widetilde \psi \in \big[ H^1({\mathbb R}^3) \big]^2$
such that
%
\begin{equation}   \label{eqn:R-T1}
(T - z)^{-1}\varphi_j \to \widetilde \psi
\;\mbox{ in } \big[ H^1({\mathbb R}^3) \big]^2 
\;\mbox{ as } j \to \infty.
\end{equation}
%
Since the topology of $\big[ H^1({\mathbb R}^3) \big]^2$
is stronger than that of $\mathcal H$,
 (\ref{eqn:R-T1})
implies that
%
\begin{equation}   \label{eqn:R-T2}
(T - z)^{-1}\varphi_j \to \widetilde \psi
\;\mbox{ in } {\mathcal H}
\;\mbox{ as } j \to \infty.
\end{equation}
%
On the other hand, since $\varphi_j \to 0$ in $\mathcal H$, and
since $(T - z)^{-1}$ is a bounded operator in $\mathcal H$,
we see that
%
\begin{equation}   \label{eqn:R-T3}
(T - z)^{-1} \varphi_j \to  0  \; \mbox{ in }  \mathcal H
\end{equation}
%
as $j \to \infty$. Combining (\ref{eqn:R-T2}) and 
 (\ref{eqn:R-T3}), we see that $\widetilde\psi = 0$.
This fact, together with (\ref{eqn:R-T1}), 
$\langle D \rangle (T - z)^{-1}\varphi_j \to 0$ in 
$\mathcal H$ as $j \to \infty$. Hence $\psi=0$.
We have thus shown that $\langle D \rangle (T - z)^{-1}$
is a close operator. Noting that 
$\mbox{Dom}(\langle D \rangle (T - z)^{-1})=\mathcal H$,
we can conclude from the Banach closed graph theorem
that  
$\langle D \rangle (T - z)^{-1}$  is a bounded
operator in $\mathcal H$, which will be  
denoted by $Q_A(z)$.

We now put  $\widetilde R_A(z):= Q_A(\overline z)^*$, where 
$Q_A(\overline z)^*$ denotes the adjoint operator of $Q_A(\overline z)$.
Then for any $\varphi \in \mathcal H$ and any
$\psi \in \big[ H^1({\mathbb R}^3) \big]^2$,
we have
%
\begin{gather}
\begin{split}     \label{eqn:R-T4}
(\varphi, \, \widetilde R_A(z)\psi)_{\mathcal H}
&= 
( Q_A(\overline z) \varphi, \, \psi)_{\mathcal H}     \\
&=
(\langle D \rangle (T - {\overline z})^{-1} \varphi, \, \psi)_{\mathcal H}  
\\ &=
( \varphi, \,  (T - z)^{-1}\langle D \rangle \psi)_{\mathcal H}. 
\end{split}
\end{gather}
%
It follows from (\ref{eqn:R-T4}) that
%
\begin{equation}    \label{eqn:R-T5}
\widetilde R_A(z)\psi
=(T - z)^{-1}\langle D \rangle\psi
\end{equation}
%
for all $\psi \in \big[ H^1({\mathbb R}^3) \big]^2$.
Replacing $\psi$ in (\ref{eqn:R-T5})
with $\langle D \rangle^{-1} \varphi$, 
$\varphi \in \mathcal H$, we get
(\ref{eqn:R-T0}).
\end{proof}



\vspace{5pt}

\begin{proof}[{\bf Proof of Proposition \ref{prop:WD-spec} }]
Since $\sigma(T_0)=\sigma_{\rm ess}(T_0)=(-\infty, \, \infty)$,
it is sufficient to show that 
%
\begin{equation}   \label{eqn:WD-spec1}
\sigma_{\rm ess}(T_A)=\sigma_{\rm ess}(T_0).
\end{equation}
%
To this end, we shall prove that the difference
$(T_A +i )^{-1} -  (T_0 +i )^{-1}$
is a compact operator in $\mathcal H$. Then, this fact
implies (\ref{eqn:WD-spec1}); 
see Reed and Simon \cite[p.113, Corollary 1]{ReedSimon4}.

We see that
%
\begin{align}   
&(T_A +i )^{-1} -  (T_0 +i )^{-1}    \nonumber\\
=& (T_A +i )^{-1} 
    (\sigma \cdot A) (T_0 +i )^{-1} \nonumber \\ 
=& \widetilde R_A(-i)
\{ \langle D \rangle^{-1} (\sigma\cdot A)   
\langle D\rangle^{-1/2}  \}  
\{ \langle D\rangle^{1/2}  (T_0 +i )^{-1} \},   \label{eqn:WD-spec2}
\end{align}
%
where we have used Lemma \ref{lem:WD-T1} in  (\ref{eqn:WD-spec2}).
It follows from Lemmas \ref{lem:WD-T0}--\ref{lem:WD-T1}
that (\ref{eqn:WD-spec2}) makes sense as a product of
three bounded operators in ${\mathcal H}$ and that 
the product is a compact operator in ${\mathcal H}$. 
\end{proof}

Proposition \ref{prop:WD-spec}, together with Theorem
\ref{thm:SSDO-main3}, gives the following result 
on the spectrum of the magnetic Dirac operator
$H_{\! A}$.

\begin{thm}   \label{thm:BE-1}
Let Assumption\,{\rm(BE)} be satisfied. Then 
%
$$
\sigma(H_{\! A})=
\sigma_{\rm ess}(H_{\! A})=(-\infty,\, -m] \cup[m, \, \infty).
$$
%
\end{thm}


\vspace{5pt}

We now state the main results in this section,
which are concerned with the eigenspaces 
corresponding to the threshold eigenvalues 
of magnetic Dirac operators $H_{\! A}$.


\begin{thm}   \label{thm:BE-2}
Let Assumption\,{\rm(BE)} be satisfied. Then
\vspace{4pt}

{\rm (i)} $\mbox{\rm Ker} (H_{\! A} -m)$ is non-trivial
if and only if 
$\mbox{\rm Ker} (H_{\! A} +m)$ is non-trivial; in other
words,
%
\begin{gather*}
\begin{split}
{}&\big\{ \,
A \in \big[ L^3({\mathbb R}^3) \big]^3 
\; \big|  \;
\mbox{\rm Ker} (H_{\! A} -m) \not= \{ 0 \} \, \big\}   \\
{}&\quad=
\big\{ \,
A \in \big[ L^3({\mathbb R}^3) \big]^3 
\; \big|  \;
\mbox{\rm Ker} (H_{\! A} + m) \not= \{ 0 \} \, \big\}.
\end{split}
\end{gather*}

\vspace{4pt}

{\rm (ii)} There exists a constant $c$ such that
%
\begin{equation}  \label{eqn:Daubechies}
\mbox{\rm dim}
\big( \mbox{\rm Ker} (H_{\! A} -m)\big)
=
\mbox{\rm dim}
\big( \mbox{\rm Ker} (H_{\! A} + m)\big)
\le
c \int_{{\mathbb R}^3} |A(x)|^3 \, dx.
\end{equation}
%
Moreover, the dimension of 
$\mbox{\rm Ker} (H_{\! A} \mp m)$ 
is independent of
$m$. 

\vspace{4pt}

{\rm (iii)} The set 
%
$\big\{ \,
A \in \big[ L^3({\mathbb R}^3) \big]^3 
\; \big|  \;
\mbox{\rm Ker} (H_{\! A} \mp m) = \{ 0 \} \, \big\}$
contains  an open dense subset of 
$\big[ L^3({\mathbb R}^3) \big]^3$.
%
\end{thm}

\begin{proof}
By Corollary \ref{cor:SSDO-C1}, we see that
%
\begin{gather}  \label{eqn:SSDOMagnetic}
\begin{split}
\mbox{Ker}(T_A) \mbox{ is trivial}   
&\Longleftrightarrow  
\; \mbox{\rm Ker} (H_{\! A} - m) \mbox{ is trivial}   \\
{}&\Longleftrightarrow  
\; \mbox{\rm Ker} (H_{\! A} + m) \mbox{ is trivial. } 
\end{split}
\end{gather}
%
Assertion (i) is equivalent to  (\ref{eqn:SSDOMagnetic}).
Assertion (ii) follows from  Corollary \ref{cor:SSDO-C1}
and Balinsky and Evans \cite[Theorem 3]{BalinEvan2}.
Assertion (iii) follows from  Corollary \ref{cor:SSDO-C1}
and Balinsky and Evans \cite[Theorem 2]{BalinEvan2}.
\end{proof}

\vspace{5pt}

\begin{rem} {\rm
Assertions {\rm(i)} and {\rm(ii)} of Theorem
\ref{thm:BE-2} mean the following facts{\rm:} 
The threshold energy $m$  
is an eigenvalue  of $H_{\! A}$ 
if and only if
the threshold energy $-m$ 
is an eigenvalue of $H_{\! A}$.
If this is the case, their multiplicity are the same.
}
\end{rem}

\vspace{5pt}

\begin{rem}   {\rm
As for the best constant in {\rm(\ref{eqn:Daubechies})},
see Balinsky and Evans \cite[Theorem 3]{BalinEvan2}.
}
\end{rem}

\vspace{10pt}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{The structure of the set of vector potentials 
yielding threshold eigenfunctions}

In this section, we shall discuss
a property of nonlocality of magnetic vector
potentials as well as
 the sparseness of the set
of vector potentials $A$ which give rise to 
threshold eigenfunctions of $H_{\! A}$
in the same sprit as in section 5.


We make the following assumption:

\vspace{15pt}

\noindent
\textbf{Assumption(E).}

\noindent 
Each $A_j$ ($j=1$, $2$, $3$) is a real-valued 
continuous function such that $A_j(x) = o(|x|^{-1})$
as $|x| \to \infty$.

\vspace{10pt}
It is straightforward to see that under Assumption(E), 
$-\sigma\cdot A$ is a bounded self-adjoint operator
in the Hilbert space 
$\mathcal H=\big[L^2({\mathbb R}^3)\big]^2$.
Hence we can define the self-adjoint
realization $T_A$ 
with $\mbox{Dom}(T_A)=\big[H^1({\mathbb R}^3)\big]^2$
as the
operator sum of $T_0$ and $-\sigma\cdot A$.



Also, it is straightforward to see that
 $-\alpha\cdot A$ is 
a bounded self-adjoint operator
in the Hilbert space 
${\mathcal K}=\big[L^2({\mathbb R}^3)\big]^4$,
hence we can define the
self-adjoint  realization $H_{\! A}$
with 
$\mbox{Dom}(H_{\! A})=\big[H^1({\mathbb R}^3)\big]^4$ in
${\mathcal K}$ as the operator sum of $H_0$ and
$-\alpha\cdot A$. Therefore, in the same way as in
section 5,
 we can regard $H_{\! A}$ as
a supersymmetric Dirac operator, and 
 apply the results in section 2
to $H_{\! A}$.


We note that under Assumption(E)
$(-\sigma\cdot A)(T_0 + i)^{-1}$ is a compact
operator in  
$\mathcal H$.
Hence 
in the same way as in the proof
of Proposition \ref{prop:WD-spec} 
we can show that
$\sigma(T_A) = (-\infty, \, \infty)$. 
This fact, together with Theorem \ref{thm:SSDO-main3},
implies the following result.


\begin{thm}    \label{thm:decay-1}
Let Assumption\,{\rm(E)} be satisfied. Then 
%
$$
\sigma(H_{\! A})=
\sigma_{\rm ess}(H_{\! A})=(-\infty,\, -m] \cup[m, \, \infty).
$$
%
\end{thm}

To state the main results in this section,
we need to introduce the following notation:
%
\begin{equation}    \label{eqn:eltonAssump}
{\mathcal A}:=
\{ \, A \, | \, A \mbox{ satisfies Assumption(E)} \, \}.
\end{equation}
%
We regard $\mathcal A$ as a Banach space
with the norm
%
$$
\Vert A \Vert_{\mathcal A}
= \sup_{x} \{ \langle x \rangle |A(x)| \}
$$
%
 
\vspace{1pt}

\begin{thm} \label{thm:elton-2}
Let Assumption{\rm(E)} be satisfied. Define
%
$$
{\mathcal Z}_k^{\pm} = 
\{ \, A \in \mathcal A \, | \,
  \mbox{\rm dim} (\mbox{\rm Ker}(H\mp m) ) = k
\, \}
$$
%
for $k=0$, $1$, $2$, $\cdots$. Then

\vspace{4pt}

{\rm (i)} ${\mathcal Z}_k^{+}={\mathcal Z}_k^{-}$ for all $k$.


\vspace{4pt}

{\rm (ii)} ${\mathcal Z}_0^{\pm}$ is an open and dense 
subset of $\mathcal A$.

\vspace{4pt}

{\rm(iii)} For any $k$ and any open subset
$\Omega (\not= \emptyset)$ of ${\mathbb R}^3$
there exists an  $A\in {\mathcal Z}_k^{\pm}$ 
such that $A \in 
\big[ C_0^{\infty}(\Omega) \big]^3$.
\end{thm}


\begin{proof}
Assertion (i) is a direct consequence of 
Corollary \ref{cor:SSDO-C1}.
Assertions (ii) and (iii) follows from
Corollary \ref{cor:SSDO-C1} and 
Elton \cite[Theorem 1]{Elton}.
\end{proof}

It is of some interest to point out a conclusion
following from Theorem \ref{thm:ABEF-main} and 
Assertion (iii) of Theorem \ref{thm:elton-2}.
Namely, there are (at least) countably infinite number of
vector potentials $A$ with compact support
such that the corresponding Dirac operators
$H_{\! A}$ have the threshold eigenfunctions
$f$ 
with  the property (\ref{eqn:lim-1-WD}).
The eigenfunction $f$ behave like 
$|f(x)| \sim |x|^{-2}$ for $|x| \to \infty$,
in spite of the fact that the vector potentials
and the corresponding magnetic fields vanish outside
 bounded regions.  It is obvious that
this phenomenon describes a
certain kind of non-locality.



\vspace{10pt}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{A concluding remark}

Section 4 is based upon our results on supersymmetric
Dirac operators in section 2 of the present paper
 and those of
Sait\=o and Umeda 
\cite{SaitoUmeda2}.
Section 5 is based upon our results on supersymmetric
Dirac operators in section 2 
 and those of Balinsky and Evans \cite{BalinEvan2}.
Section 6 is based upon our results on supersymmetric
Dirac operators 
 and those of Elton \cite{Elton}.
The combinations of our abstract results and
those existing works broaden the understandings
of the threshold eigenfunctions of the magnetic
Dirac operators. 

In each section from section 4 to section 6, 
we have made a different assumption on 
the vector potential. It is important to
compare these assumptions to each other.
To this end, mimicking (\ref{eqn:eltonAssump}),
we introduce
the following notation
%
\begin{align*}   
{\mathcal A}_{SU}:=&
\{ \, A \, | \, A \mbox{ satisfies Assumption(SU)} \, \},  \\
{\mathcal A}_{BE}:=&
\{ \, A \, | \, A \mbox{ satisfies Assumption(BE)} \, \}.
\end{align*}
%
We then have
%
\begin{align*}
&{\mathcal A}_{SU} \subset {\mathcal A}_{BE}, 
\quad
{\mathcal A}_{BE} \setminus {\mathcal A}_{SU}\not=\emptyset,   
 \\
%
&{\mathcal A}\setminus{\mathcal A}_{SU}\not=\emptyset,
\quad
{\mathcal A}_{SU}\setminus{\mathcal A}\not=\emptyset,
 \\ 
%
&{\mathcal A}\setminus{\mathcal A}_{BE}\not=\emptyset,
\quad
{\mathcal A}_{BE} \setminus {\mathcal A} \not=\emptyset.
\end{align*}
%

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



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\end{thebibliography}
\end{document}
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