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\def\D{{\bf D}}         % Dirac operator
\def\A{{\bf A}}         % vector potential
\def\P{{\bf P}}         % velocity operator
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\begin{document}

\tighten

\title{ Ground state of a spin-1/2 charged particle in a
  two-dimensional magnetic field }

\author{Masao Hirokawa}
\address{
  Department of Mathematics,\\
  Okayama University, Okayama, 700-8530, JAPAN,\\
  {\it e-mail: hirokawa@math.okayama-u.ac.jp\/}  }

\author{Osamu Ogurisu}
\address{
  Department of Computational Science,\\
  Kanazawa University, Kanazawa, 920-1192, JAPAN,\\
  {\it e-mail: ogurisu@lagendra.s.kanazawa-u.ac.jp\/} }

%\date{\today}
\date{September 4, 2000}

\maketitle  

\begin{abstract}
  It is investigated that the structure of the kernel of the
  Dirac-Weyl operator \(\D\) of a charged particle in the magnetic
  field \(B=B_0+b\), given by the sum of a strongly singular magnetic
  field \(B_0(\cdot)=\sum_j\gamma^j\delta(\cdot-a_j)\) and a magnetic
  field \(b\) with a bounded support.  Here the magnetic field \(b\)
  may have some singular points with the order of the singularity less
  than~2.
  %
  At a glance, it seems that, following ``Aharonov-Casher Theorem''
  [Phys.Rev.A, {\bf{19}}, 1979], the dimension of the kernel of
  \(\D\), \(\dim\ker\D\), is a function of one variable, the total
  magnetic flux of \(B\) (\(=\int_{\R^2}b\,dx\,dy+\sum_j\gamma_j\)).
  %
  However, since the influence of the strongly singular points occurs,
  \(\dim\ker\D\) indeed is a function of several variables, the total
  magnetic flux and each of \(\gamma_j\)'s.
\end{abstract}

%\newpage

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

\section{Introduction}
\label{sec:introduction}

In 1979, Y.~Aharonov and A.~Casher~\cite{AC79:Ground} investigated the
dimension of the kernel of the Dirac-Weyl operator \(\D\) for a
spin-1/2 charged particle in a two-dimensional magnetic field.  After
them, many
authors~\cite{Ara93:Properties,Ree88:remark,DN80:Ground,Shi91:Spectral,Iwa82:essential,HNW89:spectre,MN91:Encadrement,Ogu97:Anticommutativity,EHO99:Anomalous}
(and references therein) studied it with various situations.

Aharonov and Casher~\cite{AC79:Ground} treated the case when the
magnetic field \(B\) is a smooth function on \(\R^2\).  In this case,
\(\D\) is essentially selfadjoint operator on \(C_0^\infty(\R^2)\),
the space of smooth functions with compact supports and \(\dim\ker\D\)
is equal to the largest natural number which is less than
\(|q\Phi|/2\pi-1\), where \(q\) is the charge of the particle and the
quantity \(\Phi\) is the total magnetic flux of \(B\) defined by
\begin{math}
  \Phi = \int_{\R^2}B\,dx\,dy.
\end{math}
This result is well known as ``Aharonov-Casher Theorem.''

In 1993, A.\ Arai~\cite{Ara93:Properties} treated the another case
when \(B\) is strongly singular, more precisely,\ \(B\) is equal to 0
except on some finite points \(\a_\nu\) (\(\nu=1\), \(\dots\), \(n\))
and can be written as the sum of delta functions with coefficients
\(\gamma_\nu\) (\(\nu=1\), \(\dots\), \(n\)) and their derivatives
(See, Eq.~(\ref{eq:s-magnet})).  The total magnetic flux \(\Phi\) of
\(B\) is equal to
\begin{math}
  \sum_{\nu=1}^n\gamma_\nu.
\end{math}
He have proved that, in this case, \(\D\) is not essentially
selfadjoint on
\begin{math}
  C_0^\infty(M)
\end{math} 
with
\begin{math}
  M = \R^2\setminus\{\a_1,\dots,\a_n\}
\end{math}
and the dimension of the kernel of two selfadjoint extensions of
\(\D\) is depend on not only \(\Phi\) but also each of
\(\gamma_\nu\)'s.  (He has given an explicit
formula~\cite{Ara93:Properties}.)  This means that, in this case,
``Aharonov-Casher Theorem'' is not valid.

In this article, we treat the case when magnetic field is mixed with
the Aharonov-Casher type and the Arai type, i.e., both non-zero
magnetic field regions with non-zero area and strongly singular points
exist, which situation no one has treated yet.  The following question
arises: whether dose ``Aharonov-Casher theorem'' or ``Arai's theorem''
state \(\dim\ker\D\)?  Roughly speaking, we prove that \(\dim\ker\D\)
can be expressed by a formula similar to Arai's one.  (See
Theorem~\ref{thm:main})

The plan of this article is as follows: In Section~\ref{sec:pre}, we
define \(\D\) and prove some lemmas.  In
Section~\ref{sec:vanishingtheorem}, we prove ``Vanishing Theorem.''
In Section~\ref{sec:zerostates}, we prove our main theorem, which
states the dimension of the kernel of \(\D\).  

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

\section{Preliminary}
\label{sec:pre}

The aim of this article is to investigate the structure of the kernel
of the Dirac-Weyl operator \(\D\) of a charged spin-1/2 particle in
the magnetic field \(B=B_0+b\), the sum of a strongly singular
magnetic field \(B_0\) (See, Eq.~(\ref{eq:s-magnet}) and
Ref.~\cite{Ara93:Properties}) and a magnetic field \(b\) with a
bounded support.  As mentioned above, the magnetic field \(b\) may
have some singular points with the order of the singularity less
than~2.


In this article, we denote the charge of the particle by
\begin{math}
  q \in \R \setminus \{0\}.
\end{math}
We assume that \(B_0\) and \(b\) may be singular at the finite
isolated points
\begin{math}
  \a_\nu = (a_{\nu 1},a_{\nu 2}) \in \R^2,
\end{math}
\(\nu=1\), \(\dots\), \(n\), and we put
\begin{displaymath}
  M = \R^2 \setminus \{\a_1,\dots,\a_n\}.
\end{displaymath}

\begin{remark}
  By the symbols indexed with 0 (resp.\ 1), we denote operators
  \(P_0\) (resp.\ \(P_1\)); vector potentials \(\A_0\) (resp.\
  \(\A_1\)); auxiliary potentials \(\phi_0\) (reps.\ \(\phi_1\)); and
  etc., which are corresponding to the strongly singular part \(B_0\)
  (resp.\ the remainder part \(b\)) of \(B\).  In the case when \(b\)
  is equal to 0, our model we shall examine is the same one treated in
  Ref.~\cite{Ara93:Properties}.
\end{remark}

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

\subsection{Strongly singular part $B_0$}
\label{sec:stronglypart}

We denote by \(B_0\) the strongly singular part of the given magnetic
field \(B\).  Let
\begin{equation}
  \label{eq:s-magnet}
  B_0(\r)
  =
  \sum_{\nu=1}^n
  \sum_{0 \le \alpha+\beta \le m}
  C_{\alpha,\beta}^{(\nu)} D_x^\alpha D_y^\beta \delta(\r-\a_\nu),
  \quad
  \r=(x,y)\in\R^2
\end{equation}
with a nonnegative integer \(m\) and real constants
\(C_{\alpha,\beta}^{(\nu)}\), where \(D_x\) and \(D_y\) denote the
distributional partial differential operators in \(x\) and \(y\),
respectively, and \(\delta(\r)\) is the Dirac delta distribution on
\(\R^2\).  A gauge potential \(\A_0(\r)\) of \(B_0\) is defined to be
on \(\R^2\)-valued function
\begin{math}
  \A_0 = (A_{01},A_{02})
\end{math}
on \(M\) such that
\begin{displaymath}
  B_0(x,y) = D_xA_{02}(x,y) - D_yA_{01}(x,y)
\end{displaymath}
in the sense of distribution on \(\R^2\).

Let
\begin{displaymath}
  \phi_0(\r) 
  =
  \sum_{\nu=1}^n 
  \sum_{0 \le \alpha+\beta \le m}
  \frac{C_{\alpha,\beta}^{(\nu)}}{2\pi} D_x^\alpha D_y^\beta \log|\r-\a_\nu|,
\end{displaymath}
which satisfies
\begin{displaymath}
  \Delta \phi_0(\r) = B_0(\r)
\end{displaymath}
in the sense of distribution on \(\R^2\).  Thus, we can choose
\begin{displaymath}
  \A_0 = (A_{01},A_{02}) = (-D_y\phi_0,D_x\phi_0)
\end{displaymath}
as a gauge potential of \(B_0\).  For later use, we define some
symbols as same as in Ref.~\cite{Ara93:Properties}.  Put
\begin{equation}
  \label{eq:constants}
  C_k^{(\nu)} = (-1)^k k! \sum_{\alpha=0}^k C_{\alpha,k-\alpha}^{(\nu)} i^{k-\alpha},
  \ 
  \gamma_\nu = C_{0,0}^{(\nu)} = C_0^{(\nu)}
  \mbox{ and }
  \Phi_0 = \sum_{\nu=1}^n \gamma_\nu.
\end{equation}

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

\subsection{The remainder part $b$}
\label{sec:lowerb}

Through this article, we suppose that the remainder part \(b\) of
\(B\) satisfies that
\begin{equation}
  \label{eq:smoothness}
  b \in C_0^\infty(M)
  = 
  \left\{
    f\in C^\infty(M); {\rm{supp\,}}{f}\text{ is bounded}
  \right\}
\end{equation}
and that for arbitrary \(\nu=1\), \(\dots\), \(n\), there exist constants
\(c_\nu>0\) and \(\epsilon_{\nu}<2\) such that
\begin{equation}
  \label{eq:sing-penetrating}
  |b(\r)| \le \frac{c_{\nu}}{|\r-\a_\nu|^{\epsilon_\nu}} \mbox{ near } \r=\a_{\nu}.
\end{equation}

Let
\begin{displaymath}
  \phi_1(\r) = \frac{1}{2\pi} \int_{\R^2} b(\r') \log|\r-\r'| d\r'
  \mbox{ for }
  \r=(x,y)\in\R^2.
\end{displaymath}
Then we have that
\begin{math}
  b(\r) = \Delta \phi_1(\r)
\end{math}
on \(M\).  Since \(b\in L^{\epsilon}(\R^2)\) with
\begin{math}
  1 < \epsilon < 2,
\end{math}
using Young's inequality, we can prove the following lemma.
\begin{Lemma}
  \label{lem:smoothness}
  Let \(b \in C_0^\infty(M)\) and suppose
  Eq.~(\ref{eq:sing-penetrating}).  Then \(\phi_1\in C(\R^2)\).
\end{Lemma}
We shall need the local boundedness of \(\phi_1\) on \(\R^2\) as a
conclusion of this lemma.


Let
\begin{displaymath}
  \Phi_1 = \int_{\R^2} b(\r)\,d\r.
\end{displaymath}
Since
\begin{math}
  b\in C_0^\infty(M)
\end{math}
and Eq.~(\ref{eq:sing-penetrating}), we have 
\begin{math}
  b\in L^1(\R^2).
\end{math}
Thus, \(\Phi_1\) has a finite value.  Considering the boundedness of
the support of \(b\), we can prove the following lemma with a little
calculating.

\begin{Lemma}
  Suppose \(b \in C_0^\infty(M)\).  Then
  \begin{displaymath}
    e^{\phi_1(\r)} \sim |\r|^{\Phi_1} \mbox{ as } \r\to\infty.
  \end{displaymath}
\end{Lemma}

\begin{remark}
  We say that \(f(x)\sim g(x)\) as \(x\to\infty\) if and only if there
  exist positive constants, \(c\) and \(d\), such that for all \(x\)
  enough large,
  \begin{math}
    cf(x) \le g(x) \le df(x)
  \end{math}
  holds.
\end{remark}

Define an \(\R^2\)-valued function
\begin{math}
  \a=(a_1,a_2)
\end{math}
on \(M\) by
\begin{displaymath}
  \a(\r) = (a_1(\r), a_2(\r)) = (-\partial_y\phi_1(\r), \partial_x\phi_1(\r)).
\end{displaymath}
Here, 
\begin{math}
  \partial_x=\partial/\partial x
\end{math}
and
\begin{math}
  \partial_y=\partial/\partial y.
\end{math}
Then \(\a\) is a gauge potential of \(b\), i.e.\
\begin{displaymath}
  b(x,y) = D_xa_2(x,y)-D_ya_1(x,y)
\end{displaymath}
in the sense of distribution on \(\R^2\).

\begin{remark}
  For simplicity, we assume the smoothness of \(b\) in
  Eq~(\ref{eq:smoothness}).   P.\ Exner and the
  authors~\cite{EHO99:Anomalous} have proven that under the
  assumptions, \(b(x)={\cal O}(|x|^{-2-\delta})\) for some
  \(\delta>0\) and \(b\in L_{\rm loc}^{1+\epsilon}(\R^2)\) for some
  \(\epsilon>0\), there exists a positive \(R\) such that
  \begin{displaymath}
    |\phi_1(x) - \Phi_1\ln|x||<\epsilon\ln|x|
  \end{displaymath}
  for all \(|x|>R\) and \(\phi_1\) is continuous.  (See,  Prop.~2.2
  and Prop.~2.5 in \cite{EHO99:Anomalous}.)
  Since we do not need the compact supportness of \(b\), we can
  remove this restriction.
\end{remark}

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

\subsection{The total magnetic field \(B\)}
\label{sec:TMF}

  From now on, using the notations in the previous subsections, we
rewrite the given (total) magnetic field \(B\) as
\begin{displaymath}
  B(\r) = B_0(\r) + b(\r)
\end{displaymath}
and take
\begin{displaymath}
  \A(\r) = \A_0(\r) + \a(\r)
\end{displaymath}
as a gauge potential of \(B\).  We denote by \(\Phi\) the total
magnetic flux:
\begin{displaymath}
  \Phi = \Phi_0 + \Phi_1.
\end{displaymath}
Let
\begin{displaymath}
  \phi(\r) = \phi_0(\r) + \phi_1(\r).
\end{displaymath}
Then we have
\begin{displaymath}
  \A(\r) = (-D_y\phi(\r), D_x\phi(\r)).
\end{displaymath}

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

\subsection{Dirac-Weyl operators}
\label{sec:DWo}

In this subsection, we define the Dirac-Weyl operators with \(B_0\)
and \(B\) acting in \(\C^2\otimes{L^2(\R^2)}\).  Let \(\sigma_j\),
\(j=1\), \(2\), \(3\), be the Pauli matrices:
\begin{displaymath}
  \sigma_1=
  \left(
    \begin{array}{cc}
      0 & 1 \\ 1 & 0
    \end{array}
  \right),
  \quad
  \sigma_2=
  \left(\begin{array}{cc}
      0 & -i\\ i & 0
    \end{array}\right),
  \quad
  \sigma_3=
  \left(\begin{array}{cc}
      1 & 0 \\ 0 & -1
    \end{array}\right).
\end{displaymath}
We denote the domain of operator \(T\) by \(D(T)\).  

Let
\begin{displaymath}
  p_1 = -iD_x, \quad p_2 = -iD_y
\end{displaymath}
be distributional differential operators acting in \(L^2(\R^2)\).
The velocity operator
\begin{math}
  \P_0 = (P_{01}, P_{02})
\end{math}
with the gauge potential \(\A_0\) is given by
\begin{displaymath}
  P_{0j} = p_j -qA_{0j} \mbox{ with } D(P_{0j})=C_0^\infty(M).
\end{displaymath}
Define the Dirac-Weyl operator \(\Qm\) with the strongly singular
magnetic field \(B_0\) by
\begin{displaymath}
  \Qm = \sigma_1P_{01} + \sigma_2P_{02} 
  \mbox{ with }
  D(\Qm)= \C^2\otimes{C_0^\infty(M)}.
\end{displaymath}
As the above equation, in this article, we often omit the tensorial
sign, $\otimes$, between two operators.  The velocity operator
\begin{math}
  \P = (P_{1},P_{2})
\end{math}
with the gauge potential \(\A\) is given by
\begin{displaymath}
  P_{j} = p_j -q(A_{0j} + a_j) \mbox{ with } D(P_{j})=C_0^\infty(M).
\end{displaymath}
Define the Dirac-Weyl operator \(\Dm\) with the total magnetic field
\(B\) by
\begin{displaymath}
  \Dm = \sigma_1P_{1} + \sigma_2P_{2}
  \mbox{ with }
  D(\Dm)= \C^2\otimes{C_0^\infty(M)}.
\end{displaymath}

Note that \(\Qm\) is written in the form
\begin{displaymath}
  \Qm = 
  \left(\begin{array}{cc}
    0 & Q_- \\ Q_+ & 0
  \end{array}\right),
\end{displaymath}
where
\begin{displaymath}
  Q_\pm = P_{01} \pm i P_{02} \mbox{ with } D(Q_\pm)=C_0^\infty(M),
\end{displaymath}
and \(\Dm\) is written in
\begin{displaymath}
  \Dm = 
  \left(\begin{array}{cc}
    0 & D_- \\ D_+ & 0
  \end{array}\right),
\end{displaymath}
where
\begin{displaymath}
  D_\pm = P_{1} \pm i P_{2} \mbox{ with } D(D_\pm)=C_0^\infty(M).
\end{displaymath}
The operators \(\Qm\) and \(\Dm\) are symmetric, and thus, \(\Qm\) and
\(\Dm\) are closable.  We denote by \(\bar \Qm\) and \(\bar\Dm\) the
closure of \(\Qm\) and \(\Dm\), respectively.


Let
\begin{displaymath}
  \D_1 = 
  \left(\begin{array}{cc}
    0 & D_+^* \\ \bar D_+ & 0
  \end{array}\right)
  \mbox{ and }
  \D_2 = 
  \left(\begin{array}{cc}
    0 & \bar D_- \\ D_-^* & 0
  \end{array}\right)
\end{displaymath}
be Dirac-Weyl operators with \(B\) acting in
\(\C^2\otimes{L^2(\R^2)}\).  These are selfadjoint extensions of
\(\Dm\).  In Section~\ref{sec:zerostates}, it is proven that these do
not coincide under some conditions.

\begin{remark}
  In Ref.~\cite{Ara93:Properties}, for the symmetric operator \(\Qm\),
  two selfadjoint extensions \(\Qm^{(1)}\) and \(\Qm^{(2)}\) of it
  have been constructed in the same fashion as above.  Under some
  conditions, these selfadjoint extensions are not coincided with
  each other.
\end{remark}

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

\subsection{Generalized kernel}
\label{sec:genkernel}

In this subsection, we preliminarily examine the generalized kernel
(the space of generalized eigenvectors with eigenvalue 0) of \(\Dm\)
to identify the kernels of \(\D_1\) and \(\D_2\).  We use the
following natural identification of \(\R^2\) and \(\C\): each
\((x,y)\in\R^2\) corresponds to \(x+iy\in\C\) with the the imaginary
unit \(i\).  We set that
\begin{math}
  \partial = (\partial/\partial x -i \partial/\partial y)/2,
\end{math}
\begin{math}
  \bar\partial = (\partial/\partial x +i \partial/\partial y)/2,
\end{math}
and
\begin{math}
  a_\nu = a_{\nu_1}+ia_{\nu_2}
\end{math}
for
\begin{math}
  \a_\nu = (a_{\nu 1},a_{\nu 2}).
\end{math}


The following proposition shall be needed later.

\begin{Proposition}
  \label{prop:DandQ}
  The following identity as differential operators
  \begin{displaymath}
    D_\pm = e^{\mp q\phi_1}Q_\pm e^{\pm q\phi_1}
  \end{displaymath}
  holds on \(C^\infty(M)\).
\end{Proposition}

\begin{proof}
  By direct computations, we have
  \begin{eqnarray*}
    &&{} Q_+ = - 2ie^{- q\phi_0}\bar\partial e^{+ q\phi_0}, 
    \\
    &&{} Q_- = - 2ie^{+ q\phi_0}    \partial e^{- q\phi_0},
    \\
    &&{} D_+ = - 2ie^{- q\phi}\bar\partial e^{+ q\phi},
    \\
    &&{} D_+ = - 2ie^{+ q\phi}    \partial e^{- q\phi}
  \end{eqnarray*}
  on \(C^\infty(M)\).  Thus, we obtain the desired result.
\end{proof}

Let \(D(M)=C_0^\infty(M)\) and \(D'(M)\) be the space of distributions
on \(M\). We say that \(f\in D'(M)\) is a generalized eigenvector with
eigenvalue \(0\) of the operators \(T\) on \(C_0^\infty(M)\) if
\begin{displaymath}
  \langle f, (Tg)^*\rangle = 0 \mbox{ for all } g\in C_0^\infty(M),
\end{displaymath}
where \(\langle\cdot,\cdot\rangle\) denotes the canonical bilinear
form on \(D'(M)\times C_0^\infty(M)\).  We denote by \(S(T)\) the
space of all the generalized eigenvectors with eigenvalue 0 of \(T\).
Note that
\begin{math}
  \ker\bar{D}_\pm \subset S(D_\pm).
\end{math}

For a function \(\psi\) on \(M\), let
\begin{eqnarray*}
  &{}H_{\psi}(M) = \{e^{-q\psi}f| f\mbox{ is holomorphic on }M\},
  \\
  &{} H_{\psi}^a(M) = \{e^{q\psi}f| f\mbox{ is antiholomorphic on }M\}.
\end{eqnarray*}
The following lemma is Lemma~4.4 in Ref.~\cite{Ara93:Properties}.
\begin{Lemma}
  We have
  \begin{displaymath}
    S(Q_+) = H_{\phi_0}(M) \mbox{ and } S(Q_-) = H_{\phi_0}^a(M).
  \end{displaymath}
\end{Lemma}

The following lemma can be proven in the same way as in the proof of
Lemma~4.4 in Ref.~\cite{Ara93:Properties}.
\begin{Lemma}
  \label{lem:holomorphic}
  We have
  \begin{displaymath}
    S(D_+) = H_\phi(M) \mbox{ and } S(D_-) = H_\phi^a(M).
  \end{displaymath}
\end{Lemma}
Note that
\begin{math}
  S(\Qm) = S(Q_+)\oplus S(Q_-)
\end{math}
and
\begin{math}
  S(\Dm) = S(D_+)\oplus S(D_-).
\end{math}

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

\section{Vanishing Theorem}
\label{sec:vanishingtheorem}

In this section, we examine the vanishing theorem for Dirac-Weyl
operators \(\D_1\) and \(\D_2\).  Suppose that
\begin{math}
  b \in C_0^\infty(M)
\end{math}
and that Eq.~(\ref{eq:sing-penetrating}) holds.  We need the following
lemma proved by Arai.
\begin{Lemma}[Arai, Lemma~4.3 in Ref.~\cite{Ara93:Properties}]
  \label{lem:araisvanishing}
  We have
  \begin{math}
    \ker\bar Q_\pm = \{0\}
  \end{math}
  i.e.\
  \begin{math}
    \ker\bar \Qm = \{0\}.
  \end{math}
\end{Lemma}

The following is the main theorem in this section.

\begin{Theorem}[Vanishing Theorem]
  \label{thm:vanishingtheorem}  
  Suppose that the multiplication operator \(e^{q\phi_1}\) (resp.\ 
  \(e^{-q\phi_1}\)) acting in \(L^2(\R^2)\) is bounded.  Then
  \begin{displaymath}
    \ker\bar D_+ = \{0\}.\ (\mbox{resp.\ \(\ker\bar D_- = \{0\}.\)})
  \end{displaymath}
\end{Theorem}

\begin{remark}
  The following three statements are equivalent.
  %%
  (i) The operator \(e^{q\phi_1}\) is bounded;
  %% 
  (ii)
  \begin{math}
    \limsup_{|\r|\to\infty} q\phi_1(\r) < \infty;
  \end{math}
  %% 
  (iii)
  \begin{math}
    q\Phi_1 < 0.
  \end{math}
\end{remark}

\begin{proof}
  Consider the case when \(e^{q\phi_1}\) is bounded.  By
  Proposition~\ref{prop:DandQ}, we have
  \begin{equation}
    \label{eq:generalkernel}
    e^{q\phi_1}S(D_+) = S(Q_+).
  \end{equation}
  Let \(\Phi\in\ker\bar D_+\).  Then there exists a sequence
  \begin{math}
    \{\Phi_n; n\in\N\} \subset C_0^\infty(M)
  \end{math} 
  such that
  \begin{math}
    \Phi_n \to \Phi
  \end{math}
  and
  \begin{math}
    D_+\Phi_n\to 0
  \end{math}
  in \(L^2(\R^2)\) as 
  \begin{math}
    n\to\infty.
  \end{math}
  Since
  \begin{math}
    \Phi\in S(D_+),
  \end{math}
  by Eq.~(\ref{eq:generalkernel}) and the assumption,
  \begin{displaymath}
    e^{q\phi_1}\Phi \in S(Q_+) \cap L^2(\R^2).
  \end{displaymath}
  We have
  \begin{math}
    e^{q\phi_1}\Phi_n\in C_0^\infty(M)
  \end{math}
  for all \(n\in\N\) and, since
  \begin{math}
    e^{q\phi_1}
  \end{math}
  is bounded, we have
  \begin{displaymath}
    e^{q\phi_1}\Phi_n\to e^{q\phi_1}\Phi
    \quad\text{in}\quad L^2(\R^2)
  \end{displaymath}
  as \(n\to\infty\).  Moreover,
  \begin{displaymath}
    \|Q_+ e^{q\phi_1}\Phi_n\|
    \le
    \|e^{q\phi_1}\|\, \|e^{-q\phi_1}Q_+ e^{q\phi_1}\Phi_n\|
    =
    \|e^{q\phi_1}\|\, \|D_+\Phi_n\|
    \to 0
  \end{displaymath}
  as \(n\to\infty\).  Thus,
  \begin{math}
    e^{q\phi_1}\Phi \in D(\bar Q_+)
  \end{math}
  and
  \begin{math}
    \bar Q_+e^{q\phi_1}\Phi = 0.
  \end{math}
  Therefore, by Lemma~\ref{lem:araisvanishing}, 
  \begin{math}
    e^{q\phi_1}\Phi=0.
  \end{math}
  Thus, we have
  \begin{math}
    \Phi=0,
  \end{math}
  which means
  \begin{math}
    \ker\bar D_+ =\{0\}.
  \end{math}
  %%
  In the case when \(e^{-q\phi_1}\) is bounded, we can obtain the
  desired result in the same way as above.
\end{proof}

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

\section{Kernel of Dirac-Weyl operator}
\label{sec:zerostates}

In the previous section, in the case when \(e^{q\phi_1}\) (resp.\
\(e^{-q\phi_1}\)) is bounded, we proved that
\begin{math}
  \ker\bar D_+=\{0\} \mbox{ (resp.\ \(\ker\bar D_-=\{0\}\)).}
\end{math}
In this section, we examine \(\ker\bar D_-\) and \(\ker\bar D_+\), and
we identify the kernels of the two selfadjoint extensions \(\D_1\) and
\(\D_2\) in restricted cases, respectively.

Let \(\Z_+\) be the set of nonnegative integers. We introduce the set
\begin{displaymath}
  W_\pm(\Phi)
  =
  \left\{
    (p,k_1,\dots,k_n) \in \Z_+ \times \Z^n
    ;
    p+\sum_{\nu=1}^n k_\nu < \pm \frac{q\Phi}{2\pi}-1,
    \pm\frac{q\gamma_\nu}{2\pi}-1 < k_\nu,
    \nu=1,\dots,n
  \right\},
\end{displaymath}
and put
\begin{displaymath}
  N_\pm(\Phi;n;q) = \# W_\pm(\Phi),
\end{displaymath}
the number of the elements of \(W_\pm(\Phi)\).  Note that
\begin{math}
  N_-(\Phi;n;q) = N_+(\Phi;n;-q).
\end{math}
Let
\begin{displaymath}
  F(z) = - \frac{1}{2\pi}\sum_{\nu=1}^{n} \sum_{k=1}^m \frac{C_k^{(\nu)}}{k(z-a_\nu)^k}.
\end{displaymath}
Here, the constants \(C_k^{(\nu)}\) were defined by
Eq.~(\ref{eq:constants}).  For
\begin{math}
  (p_\pm, k_1,\dots, k_n) \in \Z_+\times\Z^n ,
\end{math}
define functions on \(M\) by
\begin{displaymath}
  \Omega^+_{p_+, k_1,\dots,k_n} (\r)
  =
  \left(
    \prod_{\nu=1}^n |\r-\a_\nu|^{-q\gamma_\nu/2\pi}(z-a_\nu)^{k_\nu}
  \right)
  P_+(z) e^{iq\Im F(z)},
\end{displaymath}
\begin{displaymath}
  \Omega^-_{p_-, k_1,\dots,k_n} (\r) 
  =
  \left(
    \prod_{\nu=1}^n |\r-\a_\nu|^{q\gamma_\nu/2\pi}(\bar z-\bar a_\nu)^{k_\nu}
  \right)
  P_-(\bar z) e^{-iq\Im \bar F(z)},
\end{displaymath}
with \(P_\pm\) a polynomial of order \(p_\pm\) such that
\begin{math}
  P_+(a_\nu)\ne 0,
\end{math}
\begin{math}
  P_-(\bar a_\nu)\ne 0,
\end{math}
\(\nu=1\), \(\dots\), \(n\).  

\begin{remark}
  The sets \(W_\pm(\Phi_0)\) and the numbers \(N_\pm(\Phi_0;n;q)\) are
  equal to \(W_\pm\) and \(N_\pm(n;q)\) given in
  Ref.~\cite{Ara93:Properties} respectively, and these functions
  \(\Omega^\pm_{p_\pm,k_1,\dots,k_n}\) are the same ones given in
  Ref.~\cite{Ara93:Properties}.
\end{remark}

First of all, we solve the differential equations
\begin{math}
  D_\pm \Omega = 0
\end{math}
on \(M\) and examine when the solutions \(\Omega\) of it are in
\(L^2(M)\).

\begin{Lemma}
  \label{lem:pde}
  (i) The functions
  \begin{math}
    e^{\mp q\phi_1}\Omega^\pm_{p_\pm,k_1,\dots,k_n}
  \end{math}
  satisfy the partial differential equations
  \begin{displaymath}
    D_\pm e^{\mp q\phi_1}\Omega^\pm_{p_\pm,k_1,\dots,k_n} = 0 \mbox{ on } M,
  \end{displaymath}
  respectively.

  (ii) The functions
  \begin{math}
    e^{\mp q\phi_1}\Omega^\pm_{p_\pm,k_1,\dots,k_n}
  \end{math}
  are in \(L^2(\R^2)\) if and only if
  \begin{math}
    (p_\pm,k_1,\dots,k_n)\in W_\pm(\Phi),
  \end{math}
  respectively.
\end{Lemma}

\begin{proof}
  By Lemma~4.6~(i) in Ref.~\cite{Ara93:Properties}, we have
  \begin{displaymath}
    Q_\pm \Omega^\pm_{p_\pm,k_1,\dots,k_n} = 0 \mbox{ on } M.
  \end{displaymath}
  Thus, with Proposition~\ref{prop:DandQ}, we obtain part~(i).  We
  prove part~(ii).  We have
  \begin{displaymath}
    |e^{-q\phi_1}\Omega^+_{p_+,k_1,\dots,k_n}(\r)|
    \sim
    {\rm{const.}}|\r|^{ -(q\Phi/2\pi) + p_+ +\sum_{\nu=1}^n k_\nu }
  \end{displaymath}
  as \(|\r|\to \infty\), and
  \begin{displaymath}
    |e^{-q\phi_1}\Omega^+_{p_+,k_1,\dots,k_n}(\r)|
    \sim
    {\rm{const.}}|\r-\a_\nu|^{ -(q\gamma_\nu/2\pi) + p_+ }
  \end{displaymath}
  as
  \begin{math}
    \r\to \a_\nu.
  \end{math}
  Therefore, the desired assertion on
  \(e^{-q\phi_1}\Omega^+_{p_+,k_1,\dots,k_n}\) follows.  Similarly we
  can prove the assertion on
  \(e^{q\phi_1}\Omega^-_{p_-,k_1,\dots,k_n}\).
\end{proof}

Using this lemma, we can identify the kernels of \(D_\pm^*\).

\begin{Theorem}
  \label{thm:kerofminmalsadj}
  We have
  \begin{equation}
    \label{eq:kerDminus*}
    \ker D_-^*
    =
    \left\{
      e^{-q\phi_1}\Omega^+_{p,k_1,\dots,k_n}; (p,k_1,\dots,k_n)\in W_+(\Phi)
    \right\},
  \end{equation}
  where
  \begin{math}
    \ker D_-^*=\{0\}
  \end{math}
  if 
  \begin{math}
    W_+(\Phi) = \emptyset
  \end{math}, and
  \begin{equation}
    \label{eq:kerDplus*}
    \ker D_+^*
    =
    \left\{
      e^{q\phi_1}\Omega^-_{p,k_1,\dots,k_n}; (p,k_1,\dots,k_n)\in W_-(\Phi)
    \right\}
  \end{equation}
  where
  \begin{math}
    \ker D_+^*=\{0\}
  \end{math}
  if
  \begin{math}
    W_-(\Phi) = \emptyset.
  \end{math}
\end{Theorem}

\begin{proof}
  By Lemma~\ref{lem:pde}, the sets on the RHS of
  Eq.~(\ref{eq:kerDminus*}) and Eq.~(\ref{eq:kerDplus*}) are included
  in \(\ker D_-^*\) and \(\ker D_+^*\), respectively.
  %%
  To prove the converse inclusion relations, let
  \begin{displaymath}
    \Omega = (\Omega_+, \Omega_-) \in \ker D^*_- \oplus \ker D^*_+.
  \end{displaymath}
  Note that
  \begin{displaymath}
    \ker D^*_- \oplus \ker D^*_+ 
    = \ker\Dm^* 
    = \C^2\otimes L^2(\R^2) \cap S(\Dm).
  \end{displaymath}
  Hence \(\Omega\in S(D)\) and \(\Omega_\pm \in L^2(\R^2)\).
  Therefore, by Lemma~\ref{lem:holomorphic}, there exists a
  holomorphic function \(f\) and an anti-holomorphic function \(g\) on
  \(M\) such that
  \begin{displaymath}
    \Omega_+ = e^{-q(\phi_1+\phi_0)}f,
    \quad
    \Omega_- = e^{q(\phi_1+\phi_0)}g.
  \end{displaymath}
  We note that
  \begin{displaymath}
    \phi_0(\r) = \Re F(z) + \sum_{\nu=1}^n \frac{\gamma_\nu}{2\pi}\log|\r-\a_\nu|
  \end{displaymath}
  and
  \begin{math}
    \phi_1 \in C(\R^2).
  \end{math}
  Thus, using the condition, \(\Omega_+\in L^2(\R^2)\), we see that
  \(f\) must be of the form
  \begin{displaymath}
    f(z) = e^{q F(z)} h(z)
  \end{displaymath}
  with a meromorphic function \(h\) on \(\C\cup\{\infty\}\) with
  possible poles at \(z=a_\nu\), \(\nu=1\), \(\dots\), \(n\).  Thus,
  \(\Omega_+\) has to take the form
  \begin{displaymath}
    \Omega_+ = e^{-q\phi_1}\Omega^+_{p,k_1,\dots,k_n}
  \end{displaymath}
  with some
  \begin{math}
    (p,k_1,\dots,k_n) \in \Z_+ \times \Z^n
  \end{math}
  and a polynomial \(P_+\) of degree \(p\) such that 
  \begin{math}
    P_+(a_\nu) \ne 0,
  \end{math}
  \(\nu=1\), \(\dots\), \(n\).  By Lemma~\ref{lem:pde}~(ii),
  \((p,k_1,\dots,k_n)\) must be in \(W_+(\Phi)\).  Thus, \(\Omega_+\)
  is in the set on the RHS of Eq.~(\ref{eq:kerDminus*}).  In the same
  way, we can prove that \(\Omega_-\) is in the set on the RHS of
  Eq.~(\ref{eq:kerDplus*}).
\end{proof}

\medskip

Consequently, we obtain our main theorem.
\begin{Theorem}
  \label{thm:main}
  (i) If \(e^{q\phi_1}\) is bounded, we have
  \begin{equation}
    \label{eq:mainplus}
    \ker D_1
    =
    \left\{
      \left(\begin{array}{c}
        0 \\ e^{q\phi_1}\Omega^-_{p,k_1,\dots,k_n}
      \end{array}\right);
      (p,k_1,\dots,k_n) \in W_-(\Phi)
    \right\},
  \end{equation}
  where
  \begin{math}
    \ker D_1=\{0\}
  \end{math}
  if 
  \begin{math}
    W_-(\Phi) = \emptyset.
  \end{math}
  In particular,
  \begin{equation}
    \label{eq:mainplusnumber}
    \dim \ker D_1 = N_-(\Phi;n;q).
  \end{equation}

  (ii) If \(e^{-q\phi_1}\) is bounded, we have
  \begin{equation}
    \label{eq:mainminus}
    \ker D_2
    =
    \left\{
      \left(\begin{array}{c}
        e^{-q\phi_1}\Omega^+_{p,k_1,\dots,k_n} \\ 0
      \end{array}\right);
      (p,k_1,\dots,k_n) \in W_+(\Phi)
    \right\},
  \end{equation}
  where
  \begin{math}
    \ker D_2 = \{0\}
  \end{math}
  if
  \begin{math}
    W_+(\Phi) = \emptyset.
  \end{math}
  In particular,
  \begin{equation}
    \label{eq:mainminusnumber}
    \dim\ker D_2 = N_+(\Phi;n;q).
  \end{equation}
\end{Theorem}

\begin{proof}
  Consider the part~(i).  By Theorem~\ref{thm:vanishingtheorem}, we
  have
  \begin{displaymath}
    \ker D_1
    =
    \left\{
      \left(\begin{array}{c}
        0 \\ \Omega
      \end{array}\right);
      \Omega \in \ker D_+^*
    \right\}.
  \end{displaymath}
  Hence, by Lemma~\ref{lem:pde}, the set on the RHS of
  Eq.~(\ref{eq:mainplus}) is included in \(\ker D_1\).  By
  Theorem~\ref{thm:kerofminmalsadj}, we obtain the converse inclusion
  relation.  Therefore, we obtain Eq.~(\ref{eq:mainplus}).  In the
  same way, we can obtain Eq.~(\ref{eq:mainminus}).  We write
  \begin{math}
    (p,k_1,\dots,k_n)= (p,\k).
  \end{math}
  Thus, \(\{e^{\mp{q\phi_1}}\Omega^\pm_{p_j,\k_j}\}^l_{j=1}\) are
  linearly independent if and only if
  \begin{math}
    (p_j,\k_j) \ne (p_i,\k_i),
  \end{math}
  \(i\ne j\), \(i\), \(j=1\), \(\dots\), \(l\).  Thus,
  Eq.~(\ref{eq:mainplusnumber}) and Eq.~(\ref{eq:mainminusnumber})
  follow.
\end{proof}

As a corollary of Theorem~\ref{thm:main}, we have the following.
\begin{Corollary}
  Let \(n\ge 2\) and suppose that 
  \begin{math}
    N_+(\Phi;n;q) + N_+(\Phi;n;q) \ge 1.
  \end{math}
  Then \(\Dm\) is not essentially selfadjoint.
\end{Corollary}

\begin{proof}
  In this case, the two selfadjoint extensions \(\D_1\) and \(\D_2\)
  are not coincided with each other.  Thus, \(\Dm\) is not
  essentially selfadjoint.
\end{proof}

\begin{remark}
  In the same way as in the proof of Corollary 4.9 in
  Ref.~\cite{Ara93:Properties}, we can prove under some conditions that
  \begin{math}
    N_\pm(\Phi_1;n;q)\ge 1
  \end{math}
  and, thus, \(\Dm\) is not essentially selfadjoint.
\end{remark}

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

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\end{document}

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