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%
%   The spectrum of Schr\"odinger operators 
%   with random $\delta$ magnetic fields
%
%    by Takuya Mine and Yuji Nomura
%
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\def\a{\mathbf{a}}
\def\N{\mathbf{N}}
\def\Z{\mathbf{Z}}
\def\Q{\mathbf{Q}}
\def\R{\mathbf{R}}
\def\C{\mathbf{C}}
\def\Re{\mathop{\rm Re}\nolimits}
\def\Im{\mathop{\rm Im}\nolimits}

\def\dim{\mathop{\rm dim}\nolimits}
\def\rank{\mathop{\rm rank}\nolimits}
\def\Ran{\mathop{\rm Ran}\nolimits}
\def\Ker{\mathop{\rm Ker}\nolimits}
\def\rot{\mathop{\rm curl}\nolimits}
\def\sgn{\mathop{\rm sgn}\nolimits}
\def\fra{\mathop{\rm frac}\nolimits}
\def\dist{\mathop{\rm dist}\nolimits}
\def\diam{\mathop{\rm diam}\nolimits}
\def\supp{\mathop{\rm supp}\nolimits}
\def\mult{\mathop{\rm mult}\nolimits}
\def\tr{\mathop{\rm tr}\nolimits}
\def\min{\mathop{\rm min}\limits}

\renewcommand{\labelenumi}{(\roman{enumi})}
\renewcommand{\labelenumii}{(\alph{enumii})}

\begin{document}
\begin{center}
 {
\Large\bf
The spectrum of Schr\"odinger operators
with random $\delta$ magnetic fields
}
\vspace{0.2cm}

by Takuya Mine\footnote{
Department of Comprehensive Sciences,
Kyoto Institute of Technology,
Matsugasaki,
Sakyo-ku,
Kyoto 606-8585,
Japan.\\
email: mine@kit.ac.jp}
%
and
%
Yuji Nomura\footnote{
Department of Computer Science,
Graduate School of Science and Engineering,
Ehime University,
3 Bunkyo-cho,
Matsuyama, Ehime 790-8577,
Japan.\\
email: nomura@cs.ehime-u.ac.jp
}
\vspace{0.2cm}
\end{center}

{\bf Abstract.} 
We shall consider the Schr\"odinger 
operators on $\R^2$ with the magnetic field
given by a nonnegative constant field plus 
random $\delta$ magnetic fields
of the Anderson type or 
of the Poisson-Anderson type.
We shall investigate the spectrum 
of these operators
by the method of the admissible potentials
by Kirsch--Martinelli \cite{K-M}.
Moreover, we shall prove
the lower Landau levels are infinitely degenerated eigenvalues
when the constant field is sufficiently large,
by estimating the growth order
of the eigenfunctions
using the entire function theory
by Levin \cite{L}.

\section{Introduction}

The $\delta$ magnetic fields 
are sometimes called the Aharonov-Bohm fields,
after the celebrated paper by Aharonov--Bohm \cite{A-B}.
There are numerous works which study
the Aharonov-Bohm fields;
see e.g. 
Ruijsenaars \cite{Rui},
Helffer \cite{H}
or
Nambu \cite{N}.
Especially, 
Geyler--Grishanov \cite{G-G} and
Geyler--\v S\v tov\'\i\v cek \cite{G-S} 
studied the infinite degeneracy of the zero modes
of the $2$-dimensional Pauli operator with $\delta$ magnetic fields;
Rozenblum--Shirokov \cite{Ro-Sh}
also studied the same subject 
in the case the magnetic field is a signed Borel measure.
One of the authors \cite{M} studied the structure of 
the whole spectrum of the Schr\"odinger operators with 
a constant magnetic field plus $\delta$ magnetic fields,
in the case the number of $\delta$ fields is finite,
or in the case $\delta$ fields are well-separated;
the authors \cite{M-N} also studied the same subject in the case 
$\delta$ fields vary periodically.
In the present paper,
we consider the case
there is some randomness
in the positions of $\delta$ magnetic fields
or in their intensities,
and study some fundamental 
spectral properties of the Schr\"odinger operators
with random $\delta$ magnetic fields.


Define a differential operator ${\cal L}_\omega$
on $\R^2$ by
\[
 {\cal L}_\omega 
 = \left(\frac{1}{i}\nabla + \a_\omega\right)^2,
\]
where $\omega$ is an element of 
a probability space $\Omega$,
and $\a_\omega$ is the magnetic vector potential.
The magnetic field corresponding to 
a vector potential $\a = (a_x, a_y)$
is defined by
$\rot \a = \partial_x a_y - \partial_y a_x$
in the distribution sense.
We assume the magnetic field $\rot \a_\omega$ 
is given by
\begin{equation}
\label{delta_magnetic}
 \rot \a_\omega (z)= B + \sum_{\gamma\in\Gamma_\omega}
               2\pi \alpha_\gamma(\omega) 
                   \delta(z -\gamma),
\end{equation}
where $B$ is a nonnegative constant,
$\Gamma_\omega$ 
is a discrete subset of $\R^2$,
$\alpha_\gamma(\omega)$ is a constant belonging to $[0,1)$,
and $\delta$ is the Dirac $\delta$ measure concentrated on the origin.
The assumption $\alpha_\gamma(\omega)\in [0,1)$
loses no generality,
since the integral differences of $\alpha_\gamma(\omega)$'s 
can be gauged away; see \cite[section 6]{G-S}.


We shall work on
the following two cases in the present paper:

\begin{enumerate}
 \item 
{\bf The Anderson type random $\delta$ magnetic fields.}
The set $\Gamma_\omega$ is 
a lattice $\Gamma$ of rank $2$
independent of $\omega$, that is,
there exist linearly 
independent vectors $e_1$, $e_2$ such that
$\Gamma_\omega = \Gamma = \Z e_1 \oplus \Z e_2$.
The random variables $\{\alpha_\gamma\}_{\gamma\in\Gamma}$
are independently, identically distributed (abbr.\ i.i.d.).
We denote the common distribution measure 
for $\{\alpha_\gamma\}_{\gamma\in\Gamma}$
by $\mu = \mathbf{P} \circ \alpha_\gamma^{-1}$
(independent of $\gamma$;
$\mathbf{P}$ is the probability measure on $\Omega$).
We assume
\begin{equation}
\label{nontrivial}
 \supp\mu \neq \{0\},
\end{equation}
since the case $\supp\mu = \{0\}$
is the trivial case.
We denote 
\begin{equation}
\label{mean}
 \bar{\alpha}=\mathbf{E}[\alpha_\gamma],
\quad
 p = \mathbf{P}\{\alpha_\gamma \neq 0\},
\end{equation}
where $\mathbf{E}[X]$ denotes the expectation of
the random variable $X$.
The values $\bar{\alpha}$ and $p$ are independent of
$\gamma$, since $\{\alpha_\gamma\}_{\gamma\in\Gamma}$ are i.i.d.

 \item 
{\bf The Poisson-Anderson type random $\delta$
magnetic fields.}
The set $\Gamma_\omega$ 
is the Poisson configuration 
(the support of the Poisson point process)
with intensity measure $\rho dxdy$, 
where $\rho$ is a positive constant
(for the definition of the Poisson point process, 
see e.g.\ 
Reiss \cite{Rei} or
Ando--Iwatsuka--Kaminaga--Nakano \cite{A-I-K-N}).
The random variables $\{\alpha_\gamma\}_{\gamma\in\Gamma_\omega}$
are i.i.d.\ with common distribution measure $\mu$
satisfying (\ref{nontrivial}),
which are independent of the Poisson configuration $\Gamma_\omega$
\footnote{
More precisely, we construct the random variables 
$\{\alpha_\gamma\}_{\gamma\in\Gamma_\omega}$ as follows.
Let $\Omega_1$ be the probability space on which the Poisson configuration
$\Gamma_\omega$ is defined, and number
 the elements $\{\gamma_j\}_{j=1}^\infty$ of $\Gamma_\omega$
as $0<|\gamma_1|<|\gamma_2|<\cdots$
(the probability that there exist two points of $\Gamma_\omega$
with the same absolute value is zero).
Let $\Omega_2$ be the  probability space on which 
i.i.d.\ random variables $\{\alpha_j\}_{j=1}^\infty$
are defined.
Put $\Omega=\Omega_1\times\Omega_2$,
and denote $\alpha_{\gamma_j}(\omega) = \alpha_j(\omega)$ $(j=1,2,\ldots)$.
}.
We use the same notation as (\ref{mean}).


\end{enumerate}

It is known that there exists a vector potential 
$\a_\omega$ satisfying (\ref{delta_magnetic})
(see \cite[section 4]{G-S}; see also section \ref{preliminaries} below).
We denote 
\begin{displaymath}
 L_\omega u = {\cal L}_\omega u,\quad
 D(L_\omega)
 =  C_0^\infty(\R^2 \setminus \Gamma_\omega),
\end{displaymath}
where $C_0^\infty(U)$ denotes the space of the compactly supported
smooth functions in $U$, and
$D(X)$ the operator domain of the operator $X$.
We denote the Friedrichs extension of $L_\omega$
by $H_\omega$.
By a usual ergodicity argument,
we can prove that there exists 
a closed set $\Sigma$ in $\R$ independent of $\omega$
such that
\begin{equation}
 \sigma(H_\omega) = \Sigma
\end{equation}
almost surely (see also Proposition \ref{admissible} below).
Moreover, the inequality $H_\omega \geq B$ 
(see \cite[Proposition 3.3 (iii)]{M}) implies
\begin{equation}
 \Sigma \subset [B,\infty).
\end{equation}
We denote the free operator by $H_0$, that is,
$H_0$ is the Friedrichs extension of the operator $L_0$ given by
\[
 L_0 = \left(\frac{1}{i}\nabla + \a_0\right)^2,~
\a_0 = \left(-\frac{By}{2}, \frac{Bx}{2} \right),~
 D(L_0) = C_0^\infty(\R^2).
\]
The spectrum of $H_0$ is well-known:
\[
 \sigma(H_0) = \cases{
  [0,\infty) & ($B=0$),\cr
  \bigcup_{n=1}^\infty \{E_n\} & ($B>0$),
}
\]
where $E_n = (2n-1)B$ is called the $n$-th Landau level.
When $B>0$, all the Landau levels are infinitely degenerated eigenvalues
of $H_0$.

First we exhibit our result for the Anderson type.
In the sequel, we denote
\begin{displaymath}
 \mult(\lambda;H) = \dim \Ker (H-\lambda)
\end{displaymath}
for a self-adjoint operator $H$.

\begin{theorem}
\label{ath}
Let $\a_\omega$ be the Anderson type.
Then, we have the following:
\begin{enumerate}
 \item 
Assume 
\begin{equation}
\label{touch}
 \supp\mu \cap (\{0\}\cup\{1\}) \neq \emptyset.
\end{equation}
Then, we have $\Sigma \supset \sigma(H_0)$.
In particular, if $B=0$ and (\ref{touch}) holds, 
then $\Sigma = [0,\infty)$.

 \item 
Assume
\begin{equation}
\label{nottouch}
 \supp\mu \cap (\{0\}\cup\{1\}) = \emptyset
\end{equation}
and $B=0$.
Then, we have $\inf \Sigma >0$.

 \item 
For $n\in \N =\{1,2,\ldots\}$, we have
\begin{eqnarray*}
 \mult(E_n;H_\omega) 
 = \infty && \mbox{if}~ \frac{B|{\cal D}|}{2\pi} + \bar{\alpha}
                        > np, \\
 \mult(E_1;H_\omega) = 0 && \mbox{if}~ \frac{B|{\cal D}|}{2\pi} +
  \bar{\alpha} < p
\end{eqnarray*}
almost surely, where ${\cal D}$ is the fundamental domain of $\Gamma$
 given by
\begin{displaymath}
 {\cal D}= \left\{s e_1 + t e_2~|~ 
  -\frac{1}{2}\leq s < \frac{1}{2},~
  -\frac{1}{2}\leq t < \frac{1}{2}\right\},
\end{displaymath}
and $|{\cal D}|$ is the area of ${\cal D}$.

 \item 
Assume (\ref{nottouch}) and $B>0$. Put
$R = \min_{\gamma \in \Gamma, \gamma \neq 0}|\gamma|$.
Put $\alpha_- = \inf \supp \mu$ and
$\alpha_+ = \sup \supp \mu$
($0<\alpha_- \leq \alpha_+ <1$ by (\ref{nottouch})).
Then, for any $n_0\in \N$,
there exist constants $C>0$ and $c>0$ dependent only on 
$n_0, \alpha_-, \alpha_+$ satisfying the following:
if $BR^2 \geq C$, then
the first $n_0$ Landau levels 
$E_1$, $\ldots$, $E_{n_0}$ are the isolated, 
infinitely degenerated eigenvalues of $H_\omega$
almost surely,
and
\[
 \Sigma \cap [B, E_{n_0+1})
 = \bigcup_{n=1}^{n_0}
   \{E_n\}\cup S_n,
\]
where $S_n$ is a closed subset of $\R$ satisfying
\[
 S_n \subset \bigcup_{\alpha\in \supp\mu}
               [E_n + (2\alpha -e^{-cBR^2})B,
                E_n + (2\alpha +e^{-cBR^2})B].
\]
\end{enumerate}
\end{theorem}
%
The assertion (iii) is an extension
of \cite[Theorem 1.1]{M-N},
which is in the periodic case.
The assertions (iii) and (iv) roughly mean
the lower Landau levels tend to be stable
under the perturbation by $\delta$ magnetic fields,
even if it is random.
Similar results are obtained in the case of 
(scalar) point interactions by
Ge\u\i ler \cite{G},
Avishai--Redheffer--Band \cite{A1}, 
Avishai--Redheffer \cite{A2}, 
Avishai--Azbel--Gredeskul \cite{A3} and 
Dorlas--Macris--Pul\'e \cite{D-M-P},
or in the case of $\delta$ magnetic fields
\cite{G-G, G-S,Ro-Sh, M}.
It may be interesting to compare the above results
with those in the case of regular potentials 
(see Zak \cite{Z}, 
Dinaburg--Sinai--Soshnikov \cite{D-S-S}); 
in that case,
it is widely believed that the Landau levels
are broadened and there exist some extended states
corresponding to the center of the Landau level.


Next we shall exhibit the result for the Poisson-Anderson case.
In the sequel, $[x]$ denotes the integer part of a real number $x$
(the maximal integer which does not exceed $x$), and 
$\fra(x)$ denotes the fractional part of $x$
(i.e. $\fra(x) = x-[x]$).
\begin{theorem}
\label{pth}
Let $\a_\omega$ be the Poisson-Anderson type.
Then, we have the following:
\begin{enumerate}
 \item 
$\Sigma \supset \sigma(H_0)$.
In particular, if $B=0$, then $\Sigma = [0,\infty)$.

 \item 
Assume $B>0$. Put
\begin{equation}
 F =\{ \fra(\alpha_1 + \cdots + \alpha_m)~|~
      \alpha_1,\ldots,\alpha_m\in \supp\mu,~m\in\N\}.
\end{equation}
Then, we have
\begin{equation}
 \Sigma \supset \bigcup_{n=1}^\infty (E_n + 2 B F),
\end{equation}
where $E_n + 2BF = \{E_n + 2B \alpha~|~\alpha\in F\}$.
In particular,
if $F$ is dense in $[0,1)$, then we have
\begin{equation}
 \label{pth1}
 \Sigma = [B, \infty).
\end{equation}

 \item 
We have
\begin{eqnarray*}
 \mult(E_n;H_\omega) = \infty && \mbox{if}~ 
  \frac{B}{2\pi \rho} + \bar{\alpha} > np, \\
 \mult(E_1;H_\omega) = 0 && \mbox{if}~ 
  \frac{B}{2\pi \rho} + \bar{\alpha} < p
\end{eqnarray*}
almost surely.
\end{enumerate}
\end{theorem}
%
The assumption `$F$ is dense in $[0,1)$' is satisfied
if $\supp \mu$ contains an irrational number
or $\supp\mu$ is an infinite set.
So the equation (\ref{pth1}) tells 
$\sigma(H_\omega)$ generically fills 
the whole possible energy range $[B,\infty)$;
similar results are found in the case of
the Schr\"odinger operators
with the Poisson-Anderson type random scalar potentials
\cite{A-I-K-N}.
This phenomenon is caused by 
the maximal randomness of the Poisson point process;
for any distribution of finite points, 
we can find an infinite number of its imitations
in the Poisson configuration,
so the system contains all the energy levels
of all the finite point systems (see Proposition \ref{admissible}
below).
However, we could not prove
(\ref{pth1}) when $\supp\mu$ is a finite set of rationals.
The difficulty is the treatment of the perturbation
of the singular magnetic fields;
we could only prove 
the strong resolvent continuity
when the intensities of $\delta$ fields and their positions 
are perturbed (see section \ref{perturbation_of_delta}).
The assertion (iii) corresponds to (iii) of Theorem \ref{ath},
since $1/\rho$ is the area of `the fundamental domain'
of the Poisson configuration with intensity $\rho dxdy$
(i.e. $\mathbf{E}[\#(\Gamma_\cdot\cap{\cal D})]=1$
if $|{\cal D}|=1/\rho$).

The rest of the present paper is organized as follows.
In section \ref{preliminaries}, 
we shall prepare some notations 
used in the present paper.
In section \ref{admissible_potentials},
we shall define the admissible potentials for our models.
In section \ref{eigenfunction_for_landau_levels},
we shall construct explicit eigenfunctions 
for Landau levels.
In section \ref{proof_of_ath},
we shall prove Theorem \ref{ath}.
In section \ref{proof_of_pth},
we shall prove Theorem \ref{pth}.
In section \ref{perturbation_of_delta},
we shall prove the strong resolvent continuity of our operators
when the intensities of $\delta$ fields
and their positions are perturbed.
In section \ref{appendix},
we shall outline 
a proof of a generalization of the entire function theory by 
Levin \cite{L},
and give a characterization of
the domain of the Friedrichs extension.



\section{Preliminaries}
\label{preliminaries}
In the sequel,
we identify a vector $z=(x,y)\in \R^2$
with a complex number $z=x+iy\in\C$.
So
$L^2(\R^2) = L^2(\R^2;dxdy)$ is identified
with $L^2(\C) = L^2(\C;dxdy)$, etc.

For a nonnegative constant $B$
and a meromorphic function $\psi$ on $\C$
having at most 1-st order poles
and real residues,
put 
\begin{equation}
 \label{phidef}
 \phi (z) = \frac{B\bar{z}}{2} + \psi(z).
\end{equation}
We denote
\[
 {\cal L}_\phi = \left({\frac{1}{i}\nabla} + \a_\phi\right)^2,
\]
where
\[
  \a_\phi(z) = (\Im \phi(z), \Re \phi(z)).
\]
Let $\Gamma$ be the set of the (1-st order) poles of $\psi$.
Let $\alpha_\gamma$ be the (real) residue of $\psi$ at $z=\gamma$,
and put $\alpha = (\alpha_\gamma)_{\gamma\in\Gamma}$.
Then we have
\[
 \rot \a_\phi (z)= B + \sum_{\gamma\in \Gamma} 
 2\pi \alpha_\gamma \delta(z -\gamma)
\]
(see \cite[section 4]{G-S}).
Define a linear operator $L_\phi$ by
\[
 L_\phi u = {\cal L}_\phi u,\quad 
 D(L_\phi) = C_0^\infty(\C\setminus \Gamma).
\]
We denote
the Friedrichs extension of $L_\phi$ by $H_\phi$.


For $r>0$ and $z\in \C$, 
we denote 
\begin{displaymath}
 B_r(z)=\{w\in \C~|~|w-z|\leq r\}.
\end{displaymath}
When $\#(\Gamma\cap B_r(0))$ grows not extremely fast,
we can prove
\begin{eqnarray}
 D(H_\phi) 
&=& \{u\in L^2(\C)~|~
 {\cal L}u \in L^2(\C),~\nonumber\\
&&\quad \limsup_{z\rightarrow\gamma} |u(z)| < \infty
 \mbox{ for any }\gamma\in \Gamma
\}
 \label{domain}
\end{eqnarray}
(see Proposition \ref{friedrichs_domain}).

Define
\begin{displaymath}
 D_0(H_\phi) = \{
 u\in D(H_\phi)~|~\supp u~\mbox{is~bounded}
\}.
\end{displaymath}
We see that $D_0(H_\phi)$ is an operator core for
$H_\phi$ by cut-off argument.

Define differential operators ${\cal A}_\phi$ and ${\cal A}_\phi^\dag$ by
\begin{equation}
\label{ccrop}
 {\cal A}_\phi = 2 \partial_z + \phi(z),\quad
 {\cal A}_\phi^\dag = - 2 \partial_{\bar{z}} + \overline{\phi(z)},
\end{equation}
where $\partial{z}=(\partial_x - i\partial_y)/2$,
$\partial{\bar{z}}=(\partial_x + i\partial_y)/2$.
These operators satisfy the canonical commutation relation
\begin{displaymath}
  {\cal L}_\phi = {\cal A}_\phi^\dag {\cal A}_\phi + B
          = {\cal A}_\phi {\cal A}_\phi^\dag - B
\end{displaymath}
as an operator on ${\cal D}'(\C\setminus \Gamma)$.


Although the results are independent of the choice of the gauge,
we choose the following gauge in the sequel,
to clarify the argument.
Put
\begin{equation}
\label{phi_omega} 
 \phi_\omega (z) 
  = \frac{B\bar{z}}{2} 
   + \frac{\alpha_0(\omega)}{z} 
   + \sum_{\gamma \in \Gamma_\omega\setminus\{0\}}
     \alpha_\gamma(\omega)\left(\frac{1}{z-\gamma} 
                + \frac{1}{\gamma}
                + \frac{z}{\gamma^2} \right),
\end{equation}
where $\alpha_0(\omega) = 0$ if $0\not\in\Gamma_\omega$.
We can verify the convergence of the right hand side of (\ref{phi_omega})
both in the Anderson case and in the Poisson-Anderson case
(see Proposition \ref{canonical_product} and 
(ii) of Lemma (\ref{explicit_solution})).
We denote $H_\omega=H_{\phi_\omega}$.

\section{Admissible potentials}
\label{admissible_potentials}
The spectra of the Schr\"odinger operators with random potentials
are determined by those with some special potentials,
called the admissible potentials.
The method of this type is introduced by
Kirsch--Martinelli \cite{K-M},
and also used by \cite{A-I-K-N} et al.
In our cases, their method can be formulated as follows.


\begin{definition}
{\bf (Admissible sequences for the Anderson type fields)}
Let $\Gamma$ be the period lattice in the definition of the Anderson type
$\delta$ magnetic fields.
Let $\alpha = (\alpha_\gamma)_{\gamma\in \Gamma}$ be a 
$[0,1)$-valued sequence. 
We say $\alpha$ is periodic if
there exists a rank 2-sublattice $\Gamma'$ of $\Gamma$
such that
$
 \alpha_{\gamma+\gamma'} = \alpha_\gamma
$
for every $\gamma\in \Gamma$, $\gamma'\in \Gamma'$.
We say a sequence $\alpha$ 
is admissible for the Anderson type fields if 
$\alpha$ is a $\supp\mu$-valued periodic sequence.
We denote the set of all the admissible sequences
by ${\cal P}_A$.

For a periodic sequence $\alpha$,
take a complete system of representatives 
$\{\gamma_1,\ldots, \gamma_K\}$ 
of $\Gamma / \Gamma'$
($K=\#(\Gamma / \Gamma')$),
and define
\begin{displaymath}
 \phi_\alpha(z) = \frac{B\bar{z}}{2} +
 \sum_{k=1}^K \alpha_{\gamma_k} \zeta_{\Gamma'}(z-\gamma_k), 
\end{displaymath}
where $\zeta_{\Gamma'}$ is the Weierstrass $\zeta$-function
corresponding to the lattice $\Gamma'$, that is,
\[
 \zeta_{\Gamma'}(z) 
 = \frac{1}{z} + \sum_{\gamma'\in \Gamma'\setminus\{0\}}
   \left(
   \frac{1}{z-\gamma'} + \frac{1}{\gamma'} + \frac{z}{{\gamma'}^2}
\right).
\]
We denote $H_\alpha = H_{\phi_\alpha}$.
\end{definition}

\begin{definition}{\bf (Admissible pairs for the Poisson-Anderson
 type
fields)}
We say a pair $(\Gamma,\alpha)$ is admissible 
for the Poisson-Anderson type fields
if $\Gamma$ is a finite subset of $\C$ (maybe the empty set)
and $\alpha=(\alpha_\gamma)_{\gamma\in\Gamma}$ 
is a $\supp\mu$-valued sequence.
We denote the set of all the admissible pairs
by ${\cal F}_A$.
For an admissible pair $(\Gamma,\alpha)$,
we define
\begin{displaymath}
 \phi_{\Gamma,\alpha}(z) = \frac{B\bar{z}}{2} + 
                \sum_{\gamma\in\Gamma}\frac{\alpha_\gamma}{z-\gamma}.
\end{displaymath}
We denote $H_{\Gamma,\alpha} = H_{\phi_{\Gamma,\alpha}}$.
\end{definition}

\begin{proposition}
\label{admissible}
\begin{enumerate}
 \item 
Let $\a_\omega$ be the Anderson type. Then, we have
\[
   \sigma(H_\omega) = \sigma_{\rm ess}(H_\omega) 
  =\overline{\bigcup_{\alpha \in{{\cal P}_A}} \sigma(H_\alpha)}
\]
almost surely.

 \item 
Let $\a_\omega$ be the Poisson-Anderson type. Then, we have
\[
   \sigma(H_\omega) = \sigma_{\rm ess}(H_\omega) 
  =\overline{\bigcup_{(\Gamma,\alpha)\in {\cal F}_A} \sigma(H_{\Gamma,\alpha})}
\]
almost surely.
\end{enumerate}
\end{proposition}
%
%
%
Though the proof is similar to those of known results 
\cite{K-M, A-I-K-N},
we shall give it here
to show the singularity of $\a_\omega$ does not
violate the argument.
%
%
%
\Proof
We prove only the assertion (ii).
The proof of (i) is similar.

Take a countable dense subset $X$ of $\C$,
a countable dense subset $S$ of $\supp \mu$ and put
\begin{displaymath}
 \widetilde{{\cal F}_A}
 = \{(\Gamma, \alpha) \in {{\cal F}_A}~|~
   \gamma\in X\mbox{ and }
   \alpha_\gamma\in S~\mbox{for~every}~\gamma\in\Gamma  \}.
\end{displaymath}
Notice that the set $\widetilde{{\cal F}_A}$ is countable.
We put
\begin{eqnarray*}
   \Sigma = \overline{\bigcup_{(\Gamma,\alpha)\in{{\cal F}_A}} 
                      \sigma(H_{\Gamma,\alpha})},\quad
   \widetilde{\Sigma} 
   = \overline{\bigcup_{(\Gamma, \alpha)\in\widetilde{{\cal F}_A}} 
                      \sigma(H_{\Gamma,\alpha})}.
\end{eqnarray*}

We shall divide the proof into $4$ steps.

\noindent{\bf Step 1.} 
$\sigma(H_\omega)\supset \sigma_{\rm ess}(H_\omega)$
clearly holds.

\noindent{\bf Step 2.} 
$\Sigma \supset \sigma(H_\omega)$.
\Proof
Let $\lambda \in \sigma(H_\omega)$.
Then,
for any $\epsilon>0$,
 we can take an $\epsilon$-approximating normalized eigenfunction
$u_\epsilon$ of $H_\omega$ for 
$\lambda$ (i.e. $\|u_\epsilon\|=1$, 
$\|(H_\omega-\lambda)u_\epsilon\| \leq \epsilon$)
from $D_0(H_\omega)$.
Then, using a gauge transform
in an open neighborhood of $\supp u_\epsilon$,
we can construct an $\epsilon$-approximating normalized eigenfunction
of $H_{\Gamma,\alpha}$ for $\lambda$ 
for some $(\Gamma,\alpha)\in {\cal F}_A$.
This implies $\dist(\lambda,\Sigma) \leq \epsilon$,
so the conclusion holds.
\QED

\noindent{\bf Step 3.}
$\widetilde{\Sigma}\supset \Sigma$
immediately follows from 
Corollary \ref{contspec}.

\noindent{\bf Step 4.}
$\sigma_{\rm ess}(H_\omega) \supset \widetilde{\Sigma}$
almost surely.
\Proof
Since $\widetilde{{\cal F}_A}$ is countable
and $\sigma_{\rm ess}(H_\omega)$ is closed, it suffices to show
\begin{displaymath}
 \sigma_{\rm ess}(H_\omega) \supset \sigma(H_{\Gamma,\alpha})
\end{displaymath}
almost surely, for every $(\Gamma,\alpha)\in\widetilde{{\cal F}_A}$. 
Moreover, since 
$\sigma_{\rm ess}(H_\omega)$ is closed, 
it suffices to show that, if
\begin{equation}
\label{ad2}
(r,s)\cap \sigma(H_{\Gamma,\alpha})\neq \emptyset,
 ~~r,s\in\Q, ~~r<s,~~
 (\Gamma,\alpha) \in \widetilde{{\cal F}_A},
\end{equation}
then we have
\begin{equation}
\label{ad1}
(r,s) \cap \sigma_{\rm ess}(H_\omega)\neq \emptyset
\end{equation}
almost surely.
Take $r,s,(\Gamma,\alpha)$ satisfying (\ref{ad2}) and 
take $\lambda \in (r,s)\cap \sigma(H_{\Gamma,\alpha})$.
Take $\epsilon>0$ so that 
$(\lambda -2\epsilon, \lambda + 2\epsilon)\subset (r,s)$.
In the sequel, we use the notation in section \ref{section_scont}
below.
Take a bounded open set $O \supset \Gamma$,
and let 
the subspaces $\{D_{\Gamma',\alpha'}\}$ and
the operators $\{T_{\Gamma',\alpha'}\}$
as in Lemma \ref{strong_continuity_delta_fields}.
Take an $\epsilon$-approximating normalized eigenfunction $u$ of $H_{\Gamma,\alpha}$
for $\lambda$ from $D_{\Gamma,\alpha}$,
and put $u_{\Gamma',\alpha'} = T_{\Gamma',\alpha'} u/  \| T_{\Gamma',\alpha'} u\|$.
Take a bounded open set $O' \supset O \cup \supp u$.
By Lemma \ref{strong_continuity_delta_fields}, 
there exists a constant $\delta>0$ such that
\begin{equation}
\label{ad3}
 \|(H_{\Gamma',\alpha'} - \lambda) u_{\Gamma',\alpha'}\| \leq
 \epsilon,\quad
 \supp u_{\Gamma',\alpha'}\subset O'
\end{equation}
for any $(\Gamma',\alpha')\in {\cal F}$ with 
$d((\Gamma',\alpha'), (\Gamma,\alpha))\leq\delta$.


Take a sequence $\{z_n\}_{n=1}^\infty\subset \C$
such that $\{O' + z_n\}_{n=1}^\infty$ are disjoint.
Put $K=\#\Gamma$, 
$\Gamma=(\gamma_k)_{k=1}^K$ and
$\alpha=(\alpha_k)_{k=1}^K$.
For $n\in\N$,
consider the event $A_n$ which consists of all $\omega$ satisfying
\begin{eqnarray}
 && \Gamma_\omega\cap B_{\delta/\sqrt{2K}}(\gamma_k + z_n) 
   = \{\gamma_{kn}(\omega)\}\quad\mbox{(1 point set)},\quad
  \label{ad41}\\
 && |\alpha_{\gamma_{kn}(\omega)}(\omega) - \alpha_k | \leq
  \frac{\delta}{\sqrt{2K}}
  \label{ad42}
\end{eqnarray}
for $k=1,\ldots,K$, and
\begin{equation}
  \Gamma_\omega\cap \left(
(O' + z_n) \setminus \bigcup_{k=1}^K B_{\delta/\sqrt{2K}}(\gamma_k + z_n) \right)
   = \emptyset.
 \label{ad43}
\end{equation}
The  events $\{A_n\}_{n\in\N}$
are independent and have the same positive probability.
Thus, for almost sure $\omega$,
we can take a subsequence $\{n_l\}_{l=1}^\infty$ 
such that (\ref{ad41}), (\ref{ad42}) and (\ref{ad43})
hold for $n = n_l$.
By (\ref{ad3}), (\ref{ad41}),
(\ref{ad42}), (\ref{ad43}) and a gauge transform,
we can construct a sequence 
$\{v_l\}_{l=1}^\infty\subset D_0(H_\omega)$
satisfying
\begin{displaymath}
 \|(H_\omega - \lambda) v_l \| \leq \epsilon,\quad
 \|v_l\|=1
\end{displaymath}
and $\supp v_l \subset O' + z_{n_l}$, almost surely.
This implies 
$\dist (\lambda, \sigma_{\rm ess}(H_\omega))\leq \epsilon $,
so (\ref{ad1}) holds.
\QED
Thus the proof of (ii) of Proposition \ref{admissible}
is completed.
\QED

\section{Eigenfunctions for Landau levels}
\label{eigenfunction_for_landau_levels}
In this section, we assume $B>0$ and
construct eigenfunctions
for Landau Levels.
Similar solutions are found in 
\cite{G-G, G-S, Ro-Sh, M-N}.

\subsection{Multi-valued canonical product}
There is a beautiful theory by B.\  Ja.\  Levin
about the relation between the growth 
order of the canonical product
and the distribution of its zeros \cite{L}.
His theory also holds for the multi-valued canonical product,
with the modification as follows.

Let $\Gamma$ be a discrete subset of $\C$ and
$\alpha=(\alpha_\gamma)_{\gamma\in\Gamma}$ 
be a sequence of non-negative real numbers.
For $r>0$ and $\theta_1, \theta_2\in \R$ with
$0 \leq \theta_2 - \theta_1 \leq 2\pi$, 
put
\begin{equation}
\label{lev1}
 n(r, \theta_1, \theta_2) = \sum_{0<|\gamma|\leq r,
                            \theta_1 \leq \arg \gamma < \theta_2} \alpha_\gamma
\end{equation}
(we abbreviate `$\gamma\in\Gamma$' in the sum, as in the sequel).
Put  $n(r)=n(r,0,2\pi)$.
We assume 
\begin{equation}
 \label{lev2}
 n(r) = O(r^2)\quad\mbox{as }r\rightarrow\infty.
\end{equation}
Define 
a sum $\zeta_{\Gamma, \alpha}$ and
a product $\sigma_{\Gamma, \alpha}$ by
\begin{eqnarray}
\label{lev3}
 \zeta_{\Gamma,\alpha}(z) &=& \frac{\alpha_0}{z} 
 + \sum_{\gamma\neq 0}
   \alpha_\gamma
   \left(
   \frac{1}{z-\gamma} + \frac{1}{\gamma} + \frac{z}{\gamma^2}
   \right),\\
\label{lev4} 
 \sigma_{\Gamma,\alpha}(z) 
&=& z^{\alpha_0} \prod_{\gamma\neq 0}
%   E\left(\frac{z}{\gamma},2\right)^{\alpha_\gamma}.
 \left(1 - \frac{z}{\gamma}\right)^{\alpha_\gamma}
 e^{\alpha_\gamma\left(
    \frac{z}{\gamma} + \frac{z^2}{2\gamma^2}\right)}
\end{eqnarray}
(we put $\alpha_0=0$ when $0\not\in\Gamma$).
When $\Gamma$ is a lattice of rank $2$ and
$\alpha_\gamma \equiv 1$,
then $\zeta_{\Gamma,\alpha}$ is the Weierstrass $\zeta$ function,
and  $\sigma_{\Gamma,\alpha}$ is the Weierstrass $\sigma$ function.

Let ${\cal C}=\{C_j\}_{j=1}^\infty$ be a system of disks,
where $C_j = B_{r_j}(z_j)$. We say ${\cal C}$ has the upper linear 
density $\bar{\rho}^*({\cal C})$ if
\begin{displaymath}
  \bar{\rho}^*({\cal C}) 
 = \limsup_{r\rightarrow\infty}
 \frac{1}{r}\sum_{|z_j|\leq r} r_j.
\end{displaymath}
We say ${\cal C}$ is a $C^0$-set if $\bar\rho^*({\cal C})=0$.
We often identify ${\cal C}$ with 
the union set of the disks belonging to ${\cal C}$.



\begin{proposition}
\label{canonical_product}
Assume (\ref{lev2}) holds.
Then the following holds:
\begin{enumerate}
 \item 
The sum (\ref{lev3}) converges uniformly in a compact subset of
 $\C\setminus\Gamma$.
If we take the branches of the functions
$\{(1-\frac{z}{\gamma})^{\alpha_\gamma}\}_{\gamma\in\Gamma\setminus\{0\}}$ appropriately,
then the right hand side of (\ref{lev4}) converges uniformly
in a simply connected compact subset of $\C\setminus\Gamma$.
For $k=0,1,2,\ldots$,
the function $|\left(\frac{d}{dz}\right)^k\sigma_{\Gamma,\alpha}(z)|$
is independent of the choice of the branches.
Moreover, we have
\begin{equation}
 \label{sigma_formula}
 \frac{d}{dz}\sigma_{\Gamma,\alpha}(z) = \sigma_{\Gamma,\alpha}(z)\zeta_{\Gamma,\alpha}(z).
\end{equation}

 \item 
Assume additionally that
\begin{enumerate}
 \item 
there exists $I_0\subset [0,2\pi)$ such that
$[0,2\pi)\setminus I_0$ is countable and
the limit
\begin{equation}
 \label{lev5}
 \Delta(\theta_1,\theta_2) 
 = \lim_{r\rightarrow\infty} 
   \frac{n(r,\theta_1,\theta_2)}{r^2}
\end{equation}
exists for any $\theta_1, \theta_2\in I_0 + 2\pi \Z$
with $0 \leq \theta_2-\theta_1 \leq 2\pi$,
and

\item 
the limit
\begin{equation}
\label{lev6}
 \delta_{\Gamma,\alpha} = \frac{1}{2} \lim_{r\rightarrow \infty} 
          \sum_{0<|\gamma|\leq r}\frac{\alpha_\gamma}{\gamma^2}
\end{equation}
exists and finite.
\end{enumerate}
%
Let $d\Delta$ be the Lebesgue-Stieltjes measure 
given by the relation 
$\int_{[\theta_1, \theta_2)}d \Delta(\psi) 
= \Delta(\theta_1, \theta_2)$.
Then, 
there exists a 
$C^0$-set ${\cal C}$ such that
\begin{equation}
 \label{lev7}
 \lim_{r\rightarrow \infty, re^{i\theta}\not\in {\cal C}}
 \frac{ \log|\sigma_{\Gamma,\alpha}(re^{i\theta})| }{r^2}
       = H(\theta),
\end{equation}
where the function $H(\theta)$ is defined by the Stieltjes integral
\begin{displaymath}
 H(\theta) 
 = -\int_{\theta-2\pi}^{\theta} (\psi-\theta)  
                   \sin 2(\psi-\theta) ~d \Delta(\psi) 
  + \Re (e^{2i\theta}\delta_{\Gamma,\alpha} ).
\end{displaymath}
The convergence (\ref{lev7})
is uniform with respect to $\theta\in [0,2\pi)$.
\end{enumerate}
\end{proposition}

\indent{\bf Remark.} There is a misprint in the first edition of \cite{L};
there must be the minus sign before the integral in \cite[(2.06)]{L}.

%
The second assertion of the above lemma
is a generalization of \cite[Theorem 2 in Chap. II, Sec. 1]{L},
and its proof is also similar.
We shall outline a proof in the appendix.


\begin{corollary}
 \label{uniform_decay}
In addition to the assumption of (ii) of Proposition \ref{canonical_product},
assume that
\begin{displaymath}
 \Delta(\theta_1, \theta_2) = c(\theta_2 - \theta_1)
\end{displaymath}
for some positive constant $c$.
Put 
\begin{displaymath}
 \widetilde{\sigma}_{\Gamma,\alpha}(z) 
 = e^{-\delta_{\Gamma,\alpha}z^2} \sigma_{\Gamma,\alpha}(z).
\end{displaymath}
Then, there exists  some $C^0$-set ${\cal C}$ satisfying the following;
for any $\epsilon >0$, we have
\begin{equation}
 \label{uni2}
  \left|\widetilde{\sigma}_{\Gamma,\alpha}(z)\right|
 \leq e^{(c\pi + \epsilon)|z|^2 }
\end{equation}
for sufficiently large $z$,
and
\begin{equation}
 \label{uni3}
  \left|\widetilde{\sigma}_{\Gamma,\alpha}(z)\right|
 \geq e^{(c\pi - \epsilon)|z|^2 } 
\end{equation}
for sufficiently large $z$ outside ${\cal C}$.
\end{corollary}
\Proof
By Proposition \ref{canonical_product} and the equality
\begin{displaymath}
 - c \int_{\theta -2\pi}^\theta (\psi-\theta) \sin 2(\psi-\theta) d\psi
 = c \pi,
\end{displaymath}
we see that there exists  some $C^0$-set ${\cal C}$
such that both (\ref{uni2}) and (\ref{uni3}) hold
for sufficiently large $z$ outside ${\cal C}$.
Since ${\cal C}$ is a $C^0$-set,
the limitation $z\in \C\setminus{\cal C}$ on (\ref{uni2})
can be eliminated by using the maximum modulus principle
(see the argument after the proof of \cite[Lemma 5 in Chap. II, Sec. 3]{L}).
%we mimic the argument after the proof of \cite[Chap. II, Sec. 3, Lemma 5]{L}.
%To eliminate the limitation $z\in \C\setminus{\cal C}$ from (\ref{uni2}),
%we mimic the argument after the proof of \cite[Chap. II, Sec. 3, Lemma 5]{L}.
%Let $z\in \C \setminus {\cal C }$,
%and $U_z$ the connected component of $\C \setminus {\cal C }$
%containing $z$.
%Since $\cal C$ is a $C^0$-set,
%we have 
%\begin{displaymath}
% \lim_{|z|\rightarrow \infty,~z \in \C\setminus{\cal C}} 
%\frac{\diam U_z}{|z|} =0.
%\end{displaymath}
%Take $z\in \C \setminus {\cal C}$ sufficiently large 
%so that $\diam U_z \leq \epsilon |z|$.
%Since (\ref{uni2}) holds on $\partial U_z$,
%we have by the maximum modulus principle
%\begin{eqnarray*}
% |\widetilde{\sigma}_{\Gamma,\alpha} (z)|
% &\leq& \max_{z\in \partial U_z} |\widetilde{\sigma}_{\Gamma,\alpha} (z)|\\
% &\leq& e^{(c\pi + \epsilon)(|z| + \diam U_z)^2}
% \leq e^{(c\pi + \epsilon)(1+\epsilon)^2|z|^2}.
%\end{eqnarray*}
%Replacing $\epsilon$ by smaller one, we see that (\ref{uni2}) 
%holds for all $z$ sufficiently large.
\QED


For an entire function $f$,
it is well-known that
$f$ and its derivatives $\frac{d^kf}{dz^k}$
have the same exponential growth order (see \cite[Chap 1., Sec.\  2]{L}).
For a multi-valued holomorphic function $f$, 
we have the following.
\begin{lemma}
 \label{est_dif}
 Let $f$ be a multi-valued holomorphic function on $\C$
and $n_0$ a nonnegative integer.
Let $\Gamma$ be the set of the branch points of $f$.
Assume the following conditions hold:
\begin{enumerate}
 \item 
 In a neighborhood $U_\gamma$ of each $\gamma\in \Gamma$, $f$ is written as
\begin{displaymath}
 f(z) = (z-\gamma)^{\alpha_\gamma}g_\gamma(z),
\end{displaymath}
where $\alpha_\gamma > n_0$ 
and $g_\gamma$ is a function holomorphic in $U_\gamma$.

\item
 $\#\{\gamma\in\Gamma~|~\left|\gamma\right| \leq r\}
=O(r^2)$ as $r\rightarrow \infty$.

\item
 There exists a constant $a>0$ such that
\begin{displaymath}
 |f(z)| \leq   e^{a|z|^2}
\end{displaymath}
for sufficiently large $z$.
\end{enumerate}
Then, for any $\epsilon>0$,
we have for any $k=0,1,\ldots,n_0$ 
\begin{equation}
 \label{exp2}
 \left|\frac{d^k f}{dz^k}(z)\right| \leq e^{(a+\epsilon)|z|^2}
\end{equation}
for sufficiently large $z\in \C\setminus \Gamma$.
\end{lemma}

\indent{\bf Remark.}
By (i), the function $|\frac{d^kf}{dz^k}(z)|$
is single-valued.
\Proof
By (i), we have
\begin{displaymath}
 \lim_{z\rightarrow \gamma}\left|\frac{d^kf}{dz^k} (z)\right| =0
\end{displaymath}
for $k=0,\ldots,n_0$. 
Thus the function 
$M_k(r)=\max_{|z|=r}\left|\frac{d^k f}{dz^k}(z)\right|$ 
is monotone nondecreasing,
by the maximum modulus principle.
By (ii), we can take $A\in \N$
such that
\begin{displaymath}
 \#\{\gamma\in\Gamma~|~\left|\gamma\right| \leq r\} \leq A r^2 - 1.
\end{displaymath}
Take $l\in \N$.
Dividing the ring $\{l-1 < |z| \leq l\}$ into $Al^2$ subrings,
we find a subring 
$\{r_l- \frac{1}{2Al^2} < |z| \leq r_l + \frac{1}{2Al^2}\}$
which contains no point of $\Gamma$. 
Then, for $|z|= r_l$, 
we have by the Cauchy integral formula
\[
  \frac{d^k f}{dz^k} (z)
 = \frac{k!}{2\pi i} 
   \int_{|w-z|= \frac{1}{3Al^2}} \frac{f(w)}{(w-z)^{k+1}}dw.
\]
Thus we have
\[
 M_k(l-1)
 \leq (3Al^2)^k k! M_0\left(l \right).
\]
Therefore (\ref{exp2}) follows from 
the assumption (iii).
\QED



\subsection{Explicit solution}
Let us construct the eigenfunctions for Landau levels
for our random models
and estimate them using the results in the previous subsection.
Note that 
a similar argument is found in 
Chistyakov--Lyubarskii--Pastur \cite{C-L-P}
(they give an estimate
for entire functions with randomly distributed zeros).




Put 
$\alpha(\omega) = (\alpha_\gamma(\omega))_{\gamma\in\Gamma_\omega}$,
$\zeta_\omega = \zeta_{\Gamma_\omega,\alpha(\omega)}$
and $\sigma_\omega = \sigma_{\Gamma_\omega,\alpha(\omega)}$.
Then, the operators 
${\cal A}_\omega={\cal A}_{\phi_\omega}$,
${\cal A}_\omega^\dag={\cal A}_{\phi_\omega}^\dag$
defined in (\ref{ccrop}) are written as
 \begin{equation}
 {\cal A}_\omega
 = 
   2\partial_z + \frac{B\bar{z}}{2} + \zeta_\omega(z),\quad
 {\cal A}_\omega^\dag 
 = 
   -2\partial_{\bar{z}} + \frac{Bz}{2}+ \overline{\zeta_\omega(z)}.
\label{ccra}
\end{equation}
Put
\begin{eqnarray*}
 \widetilde{\alpha}_\gamma(\omega)
 = \cases{ 1 & ($0<\alpha_\gamma(\omega)<1$), \cr
            0 & ($\alpha_\gamma(\omega)=0$), \cr
           }\quad
 \widetilde{\sigma}_\omega = \sigma_{\Gamma_\omega, \widetilde{\alpha}(\omega)},
\end{eqnarray*}
where $\widetilde{\alpha}(\omega) 
= (\widetilde{\alpha}_\gamma(\omega))_{\gamma\in\Gamma_\omega}$.
Notice that $\widetilde{\sigma}_\omega$ is an entire function.

\begin{lemma}
 \label{explicit_solution}
Let $\a_\omega$ be the Anderson type or 
the Poisson-Anderson type,
and $n$ a positive integer.
\begin{enumerate}
 \item 
Let $f$ be an entire function.
Put
\begin{equation}
\label{sol1}
 u(z) = {{\cal A}_\omega^\dag}^{n-1} \left(e^{-\frac{B}{4}|z|^2}
       |\sigma_\omega(z)|^{-1}
       \overline{\widetilde{\sigma}_\omega(z)^n f(z)}\right).
\end{equation}
If $u\in L^2(\C)$, then $u\in D(H_\omega)$ and 
$H_\omega u = E_n u$.
Moreover, if $u\in D(H_\omega)$ satisfies
$H_\omega u = B u$, 
then there exists an entire function $f$ such that
(\ref{sol1}) holds with $n=1$.

 \item 
For almost sure $\omega$,
the assumptions (a) and (b) in (ii) of Proposition \ref{canonical_product}
are satisfied with $\Gamma=\Gamma_\omega$, 
$\alpha=\beta(\omega)
=(n\widetilde{\alpha}_\gamma(\omega)-\alpha_\gamma(\omega))_{\gamma\in\Gamma_\omega}$ and
\begin{displaymath}
 \Delta (\theta_1, \theta_2) 
 = \cases{(\theta_2 - \theta_1) (np-\bar{\alpha})/
         2|{\cal D}| & (Anderson type),\cr
           \rho (\theta_2 - \theta_1) (np-\bar{\alpha})/2
                      & (Poisson-Anderson type). \cr
}
\end{displaymath}

 \item 
Let $\omega\in\Omega$ satisfy the conclusion of (ii).
Let $\delta_\omega=\delta_{\Gamma_\omega,\beta(\omega)}$
be the constant defined by (\ref{lev6})
for $\Gamma=\Gamma_\omega$ and $\alpha=\beta(\omega)$.
For a polynomial $g\not \equiv 0$, 
let $u_{n,g}$ be the function $u$ defined by (\ref{sol1})
with $f(z) = e^{-\delta_\omega z^2} g(z)$.
Then, there exists a $C^0$-set ${\cal C}$ such that
for any $\epsilon>0$
\begin{equation}
 \label{sol12}
 \cases{
 |u_{n,g} (z)| \leq \exp\left(
 \left(-\frac{B}{4} + \frac{\pi (np- \bar{\alpha})}{2|{\cal D}|} +
 \epsilon\right)|z|^2\right) &
 (Anderson type),\cr
 |u_{n,g} (z)| \leq \exp\left(
 \left(-\frac{B}{4}+ \frac{\pi\rho (np- \bar{\alpha})}{2} +
 \epsilon\right)|z|^2\right) &
 (Poisson-Anderson type) \cr
}
\end{equation}
for sufficiently large $z$, and
\begin{equation}
 \label{sol13}
 \cases{
 |u_{1,g} (z)| \geq \exp\left(
 \left(-\frac{B}{4} + \frac{\pi (p-\bar{\alpha})}{2|{\cal D}|} -
 \epsilon\right)|z|^2\right) &
 (Anderson type),\cr
 |u_{1,g} (z)| \geq \exp\left(
 \left(-\frac{B}{4} + \frac{\pi\rho (p-\bar{\alpha})}{2} -
 \epsilon\right)|z|^2\right) &
 (Poisson-Anderson type) \cr
}
\end{equation}
for sufficiently large $z$ outside ${\cal C}$.
\end{enumerate}
\end{lemma}

\Proof
(i) 
The proof is almost the same as \cite[Lemma 4.1]{M-N}.
The only difference is that
the function $u$ does not necessarily vanish at 
the point $\gamma$ with $\alpha_\gamma(\omega)=0$.
So we change the $\sigma^n$ in \cite[(4.1)]{M-N}
into $\widetilde{\sigma}_\omega^n$.

(ii)
We prove the statement only for the Poisson-Anderson type
(the Anderson type can be treated similarly).

First we prove the assumption (a) is satisfied.
For $N =m+ni\in \Z\oplus\Z i$, define a square $Q_N$ by
\begin{displaymath}
 Q_N = \left\{s + ti ~|~m-\frac{1}{2} \leq s < m + \frac{1}{2},~
                   n-\frac{1}{2} \leq t < n + \frac{1}{2}\right\}
\end{displaymath}
and put
\begin{displaymath}
 X_N(\omega) = \sum_{\gamma\in \Gamma_\omega \cap Q_N} \beta_\gamma(\omega).
\end{displaymath}
Then the random variables $\{X_N\}_{N\in \Z \oplus \Z i}$
are independent and
\begin{displaymath}
 \mathbf{E}[X_N] 
= \mathbf{E}_{\Omega_1}[\#(\Gamma_\cdot \cap Q_N)]
 \mathbf{E}_{\Omega_2}[n \widetilde{\alpha}_\gamma - \alpha_\gamma]
=\rho(np-\bar{\alpha}),
\end{displaymath}
where we used $\mathbf{E}[\#(\Gamma_\cdot \cap U)] = \rho |U|$
(for the probability spaces $\Omega_1$ and $\Omega_2$,
see the footnote about the definition of
the Poisson-Anderson fields).
For $r>0$ and $\theta_1, \theta_2\in\R$ with 
$0\leq \theta_2-\theta_1 \leq 2\pi$, put
\begin{eqnarray*}
 &&S(r, \theta_1, \theta_2)
 = \{s e^{i\theta}~|~0<s\leq r,~\theta_1 \leq \theta < \theta_2\},\\
 && N(r,\theta_1,\theta_2)
    = \{N\in \Z\oplus \Z i~|~Q_N \subset S(r, \theta_1,\theta_2) \},\\
 && \widetilde{n}(r,\theta_1,\theta_2)
 = \sum_{N\in N(r,\theta_1,\theta_2)} X_N.
\end{eqnarray*}
Then we have
\begin{eqnarray*}
 && \frac{\widetilde{n}(r,\theta_1,\theta_2)}{r^2}
 = \frac{\sum_{N\in N(r, \theta_1, \theta_2)} X_N}
        {\# N(r,\theta_1,\theta_2)}
        \frac{\# N(r,\theta_1,\theta_2)}{r^2}
 \rightarrow \frac{\rho(np-\bar{\alpha})(\theta_2-\theta_1)}{2}
\end{eqnarray*}
almost surely, by the law of large numbers.
Moreover, we readily have 
\begin{eqnarray*}
 \lim_{r\rightarrow\infty}
 \frac{\widetilde{n}(r,\theta_1,\theta_2)
       - n(r,\theta_1,\theta_2)}{r^2}=0
\end{eqnarray*}
almost surely.
Thus we have
\begin{equation}
 \label{sol14}
 \lim_{r\rightarrow\infty} \frac{n(r,\theta_1,\theta_2)}{r^2}
  = \frac{\rho(np-\bar{\alpha})(\theta_2-\theta_1)}{2}
\end{equation}
almost surely,
for each $\theta_1, \theta_2\in \Q$ with $0\leq \theta_2-\theta_1 \leq
2\pi$.
By the monotonicity of the function $n(r,\theta_1,\theta_2)$
with respect to $\theta_1$ or $\theta_2$,
we see that $(\ref{sol14})$ holds for every $\theta_1,\theta_2 \in \R$,
almost surely.

Next we show the assumption (b) holds.
Put
\begin{displaymath}
 \delta(r) = \sum_{1 < |\gamma| \leq r} \frac{\beta_\gamma}{\gamma^2}.
\end{displaymath}
We shall prove $\delta(r)$ converges as $r\rightarrow\infty$, almost surely.

For $m=1,2,\ldots$ and $k=0,\ldots,4m-1$,
put
\[
 U_{m,k} = \left\{ re^{i\theta}~\biggl|~ m^2 < r \leq (m+1)^2,
  \frac{k\pi}{2m} \leq \theta < \frac{(k+1)\pi}{2m}\right\},
\]
\[
 c_{m,k} = m^2 e^{i\frac{k\pi}{2m}},\quad
 \Gamma_{m,k}=\Gamma\cap U_{m,k},\quad
 \delta_{m,k}=\sum_{\Gamma_{m,k}}\frac{\beta_\gamma}{\gamma^2}.
\]
In the sequel, we denote the general constants
independent of $m,k,\omega$ by $C$.
For $\gamma\in U_{m,k}$,  we have
\begin{equation}
\label{sol15}
 \left|
 \frac{1}{\gamma^2} - \frac{1}{c_{m,k}^2}
\right|
=
 \left|
 \frac{(\gamma+c_{m,k})(\gamma-c_{m,k})}{\gamma^2 c_{m,k}^2}
\right|
\leq 
C m^{-5}.
\end{equation}
Put $\bar{\beta}= \mathbf{E}[\beta_\gamma]=np-\bar{\alpha}$.
Then we have
\begin{eqnarray}
&&\left|
  \delta_{m,k}
  +
  \delta_{m,k+m}
\right|\nonumber\\
&\leq&
 m^{-4}\left|
  \sum_{\Gamma_{m,k}} \beta_{\gamma} -
  \sum_{\Gamma_{m,k+m}} \beta_{\gamma}
\right|
+ Cm^{-5} \left(\# \Gamma_{m,k} + \# \Gamma_{m,k+m}\right)\nonumber\\
&\leq& m^{-4}
  \Biggl(\left| \sum_{\Gamma_{m,k}} (\beta_\gamma - \bar{\beta})\right|
      + \left| \sum_{\Gamma_{m,k+m}} (\beta_\gamma - \bar{\beta})\right|
      + |\# \Gamma_{m,k} - \# \Gamma_{m,k+m}|\bar{\beta} \Biggr)
\nonumber\\
&&\quad
+ Cm^{-5} \left(\# \Gamma_{m,k} + \# \Gamma_{m,k+m}\right),
\label{sol16}
\end{eqnarray}
where we used (\ref{sol15}) and 
$c_{m,k+m}^2=-c_{m,k}^2$ in the first inequality.
By the Schwarz inequality and 
the independentness of $\{\beta_\gamma\}$, we have
\begin{eqnarray}
 &&\mathbf{E}\left[
\left|\sum_{\Gamma_{m,k}}(\beta_\gamma -\bar{\beta})\right|\right]
=\mathbf{E}_{\Omega_1}\left[
 \mathbf{E}_{\Omega_2}\left[
\left|\sum_{\Gamma_{m,k}}(\beta_\gamma -\bar{\beta})\right|\right]\right]\nonumber\\
&\leq& \mathbf{E}_{\Omega_1}\left[
      \left( \mathbf{V}_{\Omega_2} \left[\sum_{\Gamma_{m,k}}\beta_\gamma \right]
 \right)^{1/2}\right]
= \mathbf{E}_{\Omega_1} \left[
  \left( \# \Gamma_{m,k} 
 \mathbf{V}_{\Omega_2} \left[\beta_\gamma\right]
 \right)^{1/2}\right]
\nonumber \\
&\leq&
 \left(
 \mathbf{E}_{\Omega_1} [\# \Gamma_{m,k}]\right)^{1/2}
 \left(\mathbf{V}_{\Omega_2} \left[\beta_\gamma\right]
 \right)^{1/2}
 \leq   C m,
 \label{sol17}
\end{eqnarray}
where $V[X]$ denotes the variance of a random variable $X$.
$\mathbf{E}\left[
\left|\sum_{\Gamma_{m,k+m}}(\beta_\gamma -\bar{\beta})\right|
\right]$
is estimated in the same way. Moreover, we have
\begin{eqnarray}
       \mathbf{E}[|\# \Gamma_{m,k} - \# \Gamma_{m,k+m}|]
&\leq&  2   \mathbf{E}[\left|\# \Gamma_{m,k} - \rho|U_{m,k}|\right|]\nonumber\\
&\leq&  2   \mathbf{V}[\# \Gamma_{m,k}]^{1/2}
\leq C m,\label{sol18}\\
       \mathbf{E}[\# \Gamma_{m,k} + \# \Gamma_{m,k+m}]
&=&  2   \rho|U_{m,k}|\leq  C m^2, \label{sol19}
\end{eqnarray}
where we used
$\mathbf{V} [\# \Gamma_{m,k}] = \rho |\Gamma_{m,k}| \leq C m^2$.
By (\ref{sol16}), (\ref{sol17}), (\ref{sol18}) and (\ref{sol19}), 
we have
\begin{displaymath}
  \mathbf{E} \left[|\delta_{m,k} +\delta_{m,k+m}|\right] \leq C m^{-3},
\end{displaymath}
so
\begin{eqnarray*}
 \sum_{m=1}^\infty
 \sum_{k=0}^{m-1}
 \mathbf{E} \left[
|\delta_{m,k} +\delta_{m,k+m} +\delta_{m,k+2m} +\delta_{m,k+3m}| 
\right]< \infty.
\end{eqnarray*}
Therefore we conclude the sequence $\{\delta(m^2)\}_{m=1}^\infty$
converges almost surely.

Now it is sufficient to show that 
\begin{eqnarray}
\label{sol6}
 \sup_{m^2 < r < (m+1)^2} |\delta(r)-\delta(m^2)| \rightarrow 0
\quad \mbox{ as } m\rightarrow \infty
\end{eqnarray}
almost surely.
As in the proof of (a), we can prove
\begin{displaymath}
 \frac{\# \{\gamma\in \Gamma_\omega ~|~m^2 < |\gamma| <(m+1)^2 \}}
      {\pi(m+1)^4 - \pi m^4} \rightarrow \rho
\end{displaymath}
almost surely.
This implies 
\begin{displaymath}
 \# \{\gamma\in \Gamma_\omega ~|~m^2 < |\gamma| <(m+1)^2 \}
 \leq C m^3
\end{displaymath}
almost surely. Thus we have
\[
  |\delta(r)-\delta(m^2)| \leq 
  \# \{\gamma\in \Gamma_\omega ~|~m^2 < |\gamma| <(m+1)^2 \}
  m^{-4} 
\leq Cm^{-1}
\]
for $m^2 < r < (m+1)^2$, which implies (\ref{sol6}).

(iii) 
By (\ref{sigma_formula}) and (\ref{ccra}),
we have
\begin{eqnarray*}
 {\cal A}_\omega^\dag 
& =& \sgn \sigma_\omega(z)^{-1}
   \left(-2\partial_{\bar{z}} + \frac{Bz}{2}\right)
   \sgn \sigma_\omega(z),
\end{eqnarray*}
where $\sgn(z) = \frac{z}{|z|} = \frac{|z|}{\bar{z}}$.
Thus we have
\begin{eqnarray*}
 &&u_{n,g}(z) \\
 &=& e^{-\frac{B}{4}|z|^2}
     \sgn \sigma_\omega(z)^{-1}
     \left(-2\partial_{\bar{z}} + Bz\right)^{n-1}
    \overline{\sigma_\omega(z)^{-1}\widetilde{\sigma}_\omega(z)^n 
    e^{-\delta_\omega z^2}g(z)}\\
 &=& e^{-\frac{B}{4}|z|^2}
     \sgn \sigma_\omega(z)^{-1}
     \left(-2\partial_{\bar{z}} + Bz\right)^{n-1}
    \overline{\widetilde{\sigma}_{\Gamma_\omega, \beta(\omega)}(z)g(z)}.
\end{eqnarray*}
Since
\begin{displaymath}
 \frac{ \# (\Gamma_\omega \cap B_r(0)) }
      {r^2} \rightarrow \pi\rho
\end{displaymath}
almost surely, we have
\begin{equation}
 \label{sol7}
  \# (\Gamma_\omega \cap B_r(0)) = O(r^2)
 \quad\mbox{as } r\rightarrow \infty
\end{equation}
almost surely.
So the conclusion follows from 
(ii) of this lemma,
Corollary \ref{uniform_decay},
Lemma \ref{est_dif}
and the Leibniz rule.
\QED




\section{Proof of Theorem \ref{ath}}
\label{proof_of_ath}
Let us once postpone the argument about the domains
of self-adjoint extensions, and begin the proof of main theorems.

\qbd{Proof of Theorem \ref{ath}} 
(i)
Let us identify a real value $\lambda$ with the constant sequence 
$(\lambda)_{\gamma\in\Gamma}$.
When $\supp\mu \ni 0$ or $1$,
the constant sequence $0$ or $1$ is admissible.
Thus the assertion follows from (i) of Proposition \ref{admissible},
since $H_0$ is the free operator
and $H_1$ is unitarily equivalent to $H_0$ by \cite[section 6]{G-S}.


(ii)
The same statement in the case $\alpha$ is periodic
is already proved by
Melgaard--Ouhabaz--Rozenblum \cite{M-O-R},
and the proof of this assertion is also similar.
We use the Hardy type inequality 
(see Laptev--Weidl \cite{L-W},
\cite{M-O-R}).
Similarly to \cite[Proposition 7.7]{M-O-R},
we see that there exists a constant $C>0$
dependent only on $\Gamma$ such that
\begin{displaymath}
 \int_{\mathbf{R}^2} \left|(\nabla + i \a_\omega) u\right|^2 dxdy
 \geq C \rho(\alpha)^2 \int_{\mathbf{R}^2} W(z)|u(z)|^2 dxdy
\end{displaymath}
for every $u\in C_0^\infty(\C\setminus\Gamma)$,
where $\rho(\alpha) = \min(\alpha_-, 1-\alpha_+)$
and $W(z) = \dist(z, \Gamma)^{-2}$.
Since $\inf W(z)>0$, we have the conclusion.


(iii)
Suppose ${B|{\cal D}|}/{2\pi} + \bar{\alpha}> np$ holds.
Then, there exists $\epsilon > 0$ such that
$-\frac{B}{4} + \pi (np- \bar{\alpha})/2|{\cal D}| +
 \epsilon<0$.
For any polynomial $g$, 
the function $u_{n,g}$ is an eigenfunction of $H_\omega$
corresponding to the eigenvalue $E_n$,
by (\ref{sol12}).
Thus we have
$\mult(E_n;H_\omega) = \infty$.

Next, suppose ${B|{\cal D}|}/{2\pi} +  \bar{\alpha} < p$
holds.
Then, there exists $\epsilon > 0$ such that
$-B/4 + \pi (p- \bar{\alpha})/2|{\cal D}| -
 \epsilon >0$.
By (\ref{sol13}), we have
\begin{equation}
\label{p_ath0}
 |u_{1,1}(z)| \geq 1
\end{equation}
for sufficiently large $z$ outside some $C^0$-set ${\cal C}$.
Adding some disk centered at the origin to ${\cal C}$,
we may assume (\ref{p_ath0}) holds for every $z\in \C\setminus{\cal C}$.
Let 
\begin{displaymath}
 S_0 = \{r>0 ~|~ \{|z|=r\}\cap {\cal C} = \emptyset \}.
\end{displaymath}
Suppose some $u\in D(H)$ satisfies $Hu = Eu$.
By (i) of Lemma \ref{explicit_solution},
$u$ is written as $u = u_{1,1}\bar{f}$ for some entire function 
$f=\sum_{n=0}^\infty a_n z^n$.
Then we have
\begin{equation}
 \label{p_ath1}
 \int_{\C} |u|^2 dxdy
 \geq 2\pi\sum_{n=0}^\infty \int_{S_0}  |a_n|^2 r^{2n+1} dr.
\end{equation}
Since ${\cal C}$ is a $C^0$-set, we have
\begin{eqnarray*}
 \int_{(0,R)\cap S_0} r^{2n+1} dr
 \geq |(1,R)\cap S_0| \rightarrow \infty
\end{eqnarray*}
as $R\rightarrow \infty$, where $|S|$ denotes the Lebesgue measure of $S$.
Thus the right hand side of (\ref{p_ath1})
diverges if some $a_n$ is not zero.
This implies $u=0$, so we have $\mult(E_1;H_\omega)=0$.

(iv) 
By the scaling $z' = \sqrt{B}z$, 
we can reduce the proof into the case $B=1$.
Then, the assertion is an immediate corollary of
\cite[Theorem 1.2 (ii)]{M-N} and (i) of Proposition \ref{admissible}
(notice that the constants $R_0$ and $c$ in 
\cite[Theorem 1.2 (ii)]{M-N}
depend only on $n_0$, $B$, $\alpha_-$, $\alpha_+$).
\QED





\section{Proof of Theorem \ref{pth}}
\label{proof_of_pth}

\qbd{Proof of Theorem \ref{pth}}
(i) 
This is an immediate corollary of (ii) of Proposition \ref{admissible},
since the empty set $\emptyset$ is admissible and
$H_{\emptyset,\emptyset} = H_0$.

(ii)
By (ii) of Proposition \ref{admissible} and 
Lemma \ref{gathering} (proved later), we have
for any $n\in \N$ and any admissible pair $(\Gamma,\alpha)$
\begin{displaymath}
 \Sigma \supset
 \overline{\bigcup_{\epsilon>0}\sigma(H_{\epsilon\Gamma,\alpha})}
 \ni E_n + 2B \fra(\alpha_1+\cdots+\alpha_K),
\end{displaymath}
where $K = \# \Gamma$
and $\alpha=(\alpha_k)_{k=1}^K$.
Thus we have $\Sigma \supset E_n + 2B F$.

(iii)
Similar to the proof of (iii) of Theorem \ref{ath}.
\QED

\section{Perturbation of $\delta$ magnetic fields}
\label{perturbation_of_delta}
In this section, we prove the strong resolvent continuity
of $H_{\Gamma,\alpha}$
with respect to $(\Gamma,\alpha)$
(we have already used it in the proof of Proposition \ref{admissible}).
Since our magnetic potential has strong singularity,
a careful analysis of the domain is necessary.

\subsection{Self-adjoint extensions of minimal operators}
\label{domain_properties}
In this subsection,
we review some properties about the domain
of the self-adjoint extension of $D(L_\phi)$.

We prepare some notation 
for the case $\#\Gamma = 1$.
Let $B\geq 0$, $0 \leq \alpha \leq 1$,
and $\gamma\in \C$
(the case $\alpha$=1 is contained for convenience).
Put
\[
 \phi_\alpha^{\gamma,1}(z)
= \frac{B\overline{(z-\gamma)}}{2} + \frac{\alpha}{z-\gamma}.
\]
We denote 
${\cal L}_\alpha^{\gamma,1}= {\cal L}_{\phi_\alpha^{\gamma,1}}$,
$L_\alpha^{\gamma,1}= L_{\phi_\alpha^{\gamma,1}}$ and
$H_\alpha^{\gamma,1}= H_{\phi_\alpha^{\gamma,1}}$.

Let $R>0$.
Let $\chi\in C_0^\infty(\C)$ 
such that $0 \leq \chi \leq 1$
and 
\begin{displaymath}
 \chi(z) = \cases{
            0 & ($|z|\geq \frac{R}{2}$),\cr
            1 & ($|z|\leq \frac{R}{3}$).\cr
}
\end{displaymath}
For $\gamma\in\Gamma$, put
$\chi_\gamma(z)= \chi(z-\gamma)$,
$r_\gamma=|z-\gamma|$, $\theta_\gamma = \arg(z-\gamma)$
and
\begin{eqnarray*}
 f_\alpha^{\gamma,1}(z) = \chi_\gamma(z)
   {r_\gamma}^{\alpha_\gamma} ,&&
 g_\alpha^{\gamma,1}(z) = \chi_\gamma(z)
  e^{-i\theta_\gamma}{r_\gamma}^{1- \alpha_\gamma},\\
 h_\alpha^{\gamma,1}(z) = \chi_\gamma(z)
  {r_\gamma}^{-\alpha_\gamma},&&
 i_\alpha^{\gamma,1}(z) =\chi_\gamma(z)
  e^{-i\theta_\gamma}{r_\gamma}^{\alpha_\gamma -1},\\
 j_\alpha^{\gamma,1}(z) =\chi_\gamma(z)
  \log r_\gamma,&&
 k_\alpha^{\gamma,1}(z) = \chi_\gamma(z)
  e^{-i\theta_\gamma} \log r_\gamma.
\end{eqnarray*}


\begin{lemma}
\label{fundamental}
Let $\phi$ be a function given by (\ref{phidef}).
Assume $0\leq \alpha_\gamma \leq 1$ for every $\gamma\in\Gamma$.
Then, the following holds:

\begin{enumerate}
 \item 
We regard ${\cal L}_\phi$ as an operator on ${\cal D}'(\C\setminus
 \Gamma)$. Then,
\begin{eqnarray*}
 D(L_\phi^*) &=& \{u\in L^2(\C)~|~{\cal L}_\phi u\in L^2(\C)\},\\
 D(H_\phi) &=& D(L_\phi^*)\cap \overline{D(L_\phi)}^Q,
\end{eqnarray*}
where $\overline{\phantom{M} }^Q$ denotes the closure with respect to
the form norm. In particular, for $u\in D(H_\phi)$ we have
\begin{displaymath}
 \left(\frac{1}{i}\nabla +\a_\phi\right)u \in L^2(\C)^2.
\end{displaymath}

 \item 
Let $u\in D(L_\phi^*)$ and $\chi\in C_0^\infty(\C)$
with $\chi(z)=1$ in a neighborhood of $\gamma$
for every $\gamma\in \Gamma \cap \supp\chi$.
Then, $\chi u \in D(L_\phi^*)$.

 \item 
Let $u\in D(H_\phi)$. Then, for any $\chi\in C_0^\infty(\C)$,
we have $\chi u \in D(H_\phi)$.

 \item 
Suppose that there exists a constant $R$ satisfying
\begin{displaymath}
 0<R\leq \inf_{\gamma\neq \gamma'}|\gamma-\gamma'|.
\end{displaymath}
Suppose also that
there exist functions $\{\Phi^\gamma\}_{\gamma\in\Gamma}$ satisfying
$ \Phi^\gamma\in C^\infty(B_{R/2}(\gamma))$,
$|\Phi^\gamma(z)|=1$ and
\begin{displaymath}
 {\cal L}_\phi \Phi^\gamma = \Phi^\gamma {\cal L}_{\alpha_\gamma}^{\gamma,1}
\end{displaymath}
in $B_{R/2}(\gamma)$.
Put
\begin{eqnarray*}
 f_\alpha^\gamma(z) = \Phi^\gamma f_{\alpha_\gamma}^{\gamma,1},\quad
 g_\alpha^\gamma(z) = \Phi^\gamma g_{\alpha_\gamma}^{\gamma,1},\quad \ldots,
\end{eqnarray*}
where the $R$ in the definition
of $f_{\alpha_\gamma}^{\gamma,1}$, $\ldots$
is the present one.
Put
\begin{eqnarray*}
 \Gamma_0 &=& \{\gamma\in\Gamma~|~\alpha_\gamma=0\},\\
 \Gamma_1 &=& \{\gamma\in\Gamma~|~\alpha_\gamma=1\},\\
 \Gamma_2 &=& \{\gamma\in\Gamma~|~0<\alpha_\gamma<1\}.
\end{eqnarray*}
Then, when $\Gamma$ is a finite set, we have
\begin{eqnarray}
 D(L_\phi^*) &=& D(\overline{L_\phi}) \oplus
 {\rm L.h.}\Biggl(
 \bigcup_{\gamma\in\Gamma_0} \{f_\alpha^\gamma, j_\alpha^\gamma\}
  \cup
 \bigcup_{\gamma\in\Gamma_1} \{g_\alpha^\gamma, k_\alpha^\gamma\}\nonumber\\
  &&\qquad\qquad\qquad\cup
 \bigcup_{\gamma\in\Gamma_2} 
  \{f_\alpha^\gamma, g_\alpha^\gamma, h_\alpha^\gamma, i_\alpha^\gamma\}\Biggr),
\label{domainL}\\
 D(H_\phi) &=& D(\overline{L_\phi}) \oplus {\rm L.h.}\Biggl(
 \bigcup_{\gamma\in\Gamma_0} \{f_\alpha^\gamma\}
 \cup
 \bigcup_{\gamma\in\Gamma_1} \{g_\alpha^\gamma\}
 \cup
 \bigcup_{\gamma\in\Gamma_2} 
 \{f_\alpha^\gamma, g_\alpha^\gamma\}\Biggr),\label{domainH}
\end{eqnarray}
where ${\rm L.h.}X$ denotes the finite linear combinations
of the vectors in $X$.
When $\Gamma$ is an infinite set, the right hand side
of (\ref{domainL}) (or (\ref{domainH}))
is dense in the left hand side
with respect to the graph norm.
\end{enumerate}
\end{lemma}
The proof of the above proposition is similar to those of
\cite[Proposition 5.4, Corollary 5.10, Lemma 5.15]{M}, so we shall omit it.
To prove (iv), roughly speaking, we construct
an isomorphism from $D(L_\phi^*)/D(\overline{L_\phi})$
to $\bigoplus_{\gamma\in\Gamma} 
  D({L_{\alpha_\gamma}^{\gamma,1}}^*/
  \overline{L_{\alpha_\gamma}^{\gamma,1}})$
by cut-off and gauge transform,
and use the results in 
Albeverio--Gesztesy--H{\o}egh-Krohn--Holden \cite{A-G-H-H} and
Exner--{\v S}t'ov\'{\i}{\v c}ek--Vyt{\v r}as \cite{E-S-V}.

\subsection{Criterion for the strong resolvent convergence}
We shall prove the strong resolvent continuity
via the following elementary lemma.
\begin{lemma}
 Let ${\cal H}$ be a Hilbert space.
 Let $A_n$ ($n=1,2,\ldots$), $A$ be self-adjoint operators
on ${\cal H}$.
Then, the following two conditions are equivalent:
\begin{enumerate}
 \item 
$A_n \rightarrow A$ in the strong resolvent sense.

 \item 
There exist linear operators $T_n$ and 
an operator core $D$ for $A$ such that $T_n D \subset D(A_n)$
and for every $u\in D$
\begin{displaymath}
 T_n u \rightarrow u,\quad
 A_n T_n u \rightarrow A u
\end{displaymath}
in ${\cal H}$.
\end{enumerate}
\end{lemma}
\indent{\bf Remark.}
The assertion (ii)$\Rightarrow$(i) in the case $T_n=Id$
is well-known (see Reed-Simon \cite[Theorem VIII.25]{Re-Si1}).
\Proof
{\bf (i) $\Rightarrow$ (ii)}
Put $T_n = (A_n + i)^{-1}(A+i)$, $D=D(A)$.

{\bf (ii) $\Rightarrow$ (i)}
Let $u\in D$ and put $v = (A+i)u$. Then,
\begin{eqnarray*}
  &&  (A_n + i)^{-1}v - (A+i)^{-1}v\\
  &=& (A_n + i)^{-1}\left((A+i)u - (A_n+i)T_n u  \right)
    + T_nu- u \rightarrow 0.
\end{eqnarray*}
Since $(A_n + i)^{-1} - (A+i)^{-1}$ is uniformly bounded
and $(A+i)D$ is dense in ${\cal H}$, we have the assertion.
\QED



%\subsection{Perturbation of the intensity}
%\label{section_pint}
%Let $\Gamma$ be a lattice of rank $2$,
%$\Gamma'$ a sublattice of $\Gamma$
%which is also of rank $2$.
%Let $K=\#(\Gamma/\Gamma')$ and let 
%$\{\gamma_1,\ldots,\gamma_K\}$ be a complete system of
%representatives of $\Gamma/\Gamma'$.
%Let ${\cal P}$ be the space of all the $[0,1]$-valued 
%$\Gamma'$-periodic sequences. 
%Define a metric $d$ on ${\cal P}$ by
%\[
%  d(\alpha,\beta)
% = \left(\sum_{k=1}^K \left|\alpha_k - \beta_k \right|^2
%\right)^{1/2}
%\]
%for $\alpha$, $\beta\in {\cal P}$,
%where $\alpha_k = \alpha_{\gamma_k}$, $\beta_k = \beta_{\gamma_k}$.
%We denote
%$L_\alpha = L_{\phi_\alpha}$,
%$H_\alpha = H_{\phi_\alpha}$ for $\alpha \in {\cal P}$,
%where $\phi_\alpha$ is the function given by (\ref{periodic_phi}).
%
%\begin{lemma}
%\label{perturbation_intensity}
%Let $\alpha\in {\cal P}$.
%Then, 
%there exist a neighborhood ${\cal P}'$ of $\alpha
%$,
%subspaces $\{D_\beta\}_{\beta\in {\cal P}'}$ of $L^2(\C)$ 
%and linear operators $\{T_\beta\}_{\beta\in {\cal P}'}$
%satisfying the following conditions:
%
%{\rm (a)} $D_\beta$ is an operator core for $H_\beta$, 
%for any $\beta\in {\cal P}'$.
%
%{\rm (b)} $T_\beta$ is a linear operator 
%from $D_{\alpha}$ to $D_\beta$,
%for any $\beta\in {\cal P}'$.
%Moreover, if $\supp u$ is contained in the open set
%\begin{equation}
%\label{omegaj} 
% \Omega_j = \left\{s\omega_1 + t \omega_2~\biggl|~
%            -j+\frac{1}{2} < s <  j-\frac{1}{2},~
%            -j+\frac{1}{2} < t <  j-\frac{1}{2}
%    \right\}
%\end{equation}
%for some positive integer $j$, then $\supp T_\beta u \subset \Omega_j$.
%
%{\rm (c)} When $\beta \rightarrow \alpha$ in ${\cal P}'$,
%we have
%\begin{equation}
%\label{int0}
% T_\beta u \rightarrow u,\quad
% H_\beta T_\beta u \rightarrow H_{\alpha} u 
%\end{equation}
%in $L^2(\C)$, for any $u\in D_{\alpha}$.
%\end{lemma}
%\Proof
%Let $R = \dist(0, \partial\Omega_1)$.
%For $\alpha\in {\cal P}$, define
%\begin{eqnarray*}
% \Phi_\alpha^\gamma (z)
% &=& \exp \Im \int_\gamma^z \left(-\frac{B\bar{\gamma}}{2}
%   + \frac{\alpha_\gamma}{z-\gamma}
%   - \sum_{k=1}^K \alpha_k\zeta_{\Gamma'}(z-\gamma_k)
%\right)dz .
%\end{eqnarray*}
%Since the singularity of the integrand at $z=\gamma$ is removable,
%the function $\Phi_\alpha^\gamma$ is single-valued, smooth and satisfies
%\begin{equation}
%\label{intertwine1}
% {\cal L}_\alpha  \Phi_\alpha^\gamma 
%=  \Phi_\alpha^\gamma  {\cal L}_\alpha^{\gamma,1},
%\end{equation}
%in $\{\left|z- \gamma\right| < \frac{R}{2}\}$.
%The function $\Phi_\alpha^\gamma$ is also smooth with respect to
%$K$-variables $\alpha_1,\cdots,\alpha_K$.
%
%Let us apply  Lemma \ref{fundamental} 
%with the above $R$ and $\Phi^\gamma = \Phi_\alpha^\gamma$.
%According to (iv) of Lemma \ref{fundamental},
%the following subspace $D_\alpha$ 
%is a core of $D(H_\alpha)$:
%\begin{equation}
% \label{dalpha_def}
% D_\alpha = {\rm L.h.}\left(
%            \bigcup_{\gamma\in \Gamma_0}\{ f_\alpha^\gamma\}
%       \cup \bigcup_{\gamma\in \Gamma_1}\{ g_\alpha^\gamma\}
%       \cup \bigcup_{\gamma\in \Gamma_2}\{ f_\alpha^\gamma\}
%            \cup\{g_\alpha^\gamma\}\right)
%       \oplus C_0^\infty(\C\setminus \Gamma).
%\end{equation}
%We shall construct the operator $T_\beta$.
%Let $\Gamma_{0}$, $\Gamma_{1}$ and $\Gamma_{2}$ be the sets
%defined by (\ref{gam0}), (\ref{gam1}) and (\ref{gam2}) for $\alpha$,
%respectively.
%If $\Gamma_{2} \neq \emptyset$, put
%\[
% r = \min_{\gamma\in \Gamma_{2}}\min(\alpha_\gamma,
% 1-\alpha_\gamma).
%\]
%If $\Gamma_{2}=\emptyset$, put $r=1$.
%Let ${\cal P}' = \{\beta\in {\cal P} ~|~ d(\alpha, \beta)< r\}$.
%For $\beta \in {\cal P}'$, 
%define a linear operator $T_\beta$ from $D_\alpha$
%to $D_\beta$ by
%\begin{eqnarray*}
%  &T_\beta&\left(
% \sum_{\gamma\in \Gamma_{0}} c_\gamma f_\alpha^\gamma 
%+\sum_{\gamma\in \Gamma_{1}} d_\gamma g_\alpha^\gamma 
%+\sum_{\gamma\in \Gamma_{2}} (c_\gamma f_\alpha^\gamma 
%                           + d_\gamma g_\alpha^\gamma ) + \xi
%\right)\\
%  &=&
% \sum_{\gamma\in \Gamma_{0}} c_\gamma f_\beta^\gamma 
%+\sum_{\gamma\in \Gamma_{1}} d_\gamma g_\beta^\gamma 
%+\sum_{\gamma\in \Gamma_{2}} (c_\gamma f_\beta^\gamma 
%                           + d_\gamma g_\beta^\gamma ) + \xi,
%\end{eqnarray*}
%where $c_\gamma$, $d_\gamma$ are constants and 
%$\xi \in C_0^\infty(\C\setminus \Gamma)$.
%
%We shall check that the operators $\{T_\beta\}$
%have the desired properties.
%If $\supp u \subset \Omega_j$ for some $j$, then
%$c_\gamma = d_\gamma=0$ for $\gamma\not\in\Omega_j \cap \Gamma$
%and $\supp\xi\subset \Omega_j$. Thus we have 
%$\supp T_\beta u \subset \Omega_j$.
%Since the functions $\Phi_\beta^\gamma$, 
%$f_\beta^\gamma$, $g_\beta^\gamma$ 
%are smooth with respect to 
%$\beta_1,\ldots,\beta_K$, clearly the convergence
%(\ref{int0}) holds pointwise.
%By the dominated convergence theorem,
%it suffices to show that the functions $T_\beta u$ and 
%$H_\beta T_\beta u$ are bounded uniformly 
%with respect to $\beta_1,\ldots,\beta_K$.
%By (\ref{intertwine1}),
%it suffices to show 
%$f_{\beta_\gamma}^{\gamma,1}$,
%$g_{\beta_\gamma}^{\gamma,1}$,
%$H_{\beta_\gamma}^{\gamma,1} f_{\beta_\gamma}^{\gamma,1}$ and
%$H_{\beta_\gamma}^{\gamma,1} g_{\beta_\gamma}^{\gamma,1}$ 
%are uniformly bounded.
%This can be done by a straightforward calculation
%using the polar coordinate.
%Put $r=|z-\gamma|$, $\theta=\arg(z-\gamma)$,
%$b = \beta_\gamma$, $\chi = \chi_\gamma$.
%Then we have
%\begin{equation}
% \label{polar}
% H_{\beta_\gamma}^{\gamma,1}
% = - \frac{1}{r}\partial_r r \partial_r
%        + \frac{1}{r^2}
%   \left(
%     \frac{1}{i}\partial_\theta + b + \frac{Br^2}{2}
%   \right)^2
%\end{equation}
%and
%\begin{eqnarray*}
% |f_{\beta_\gamma}^{\gamma,1}| &=& \chi r^b
%\leq M,\\
%  \left|H_{\beta_\gamma}^{\gamma,1} f_{\beta_\gamma}^{\gamma,1}\right|
% &\leq& (2b+1)|\partial_r \chi| r^{b-1}
% +  (|\partial_r^2\chi| + Bb)r^b 
% + \frac{B^2}{4}\chi r^{b+2} \\
% &\leq& 3 m^{-1} C_1 + (C_2 + B M) + \frac{B^2}{4} M^3,
%\end{eqnarray*}
%where $M=\max\left(\frac{R}{2}, 1\right)$,
%$m = \min \left(\frac{R}{3}, 1\right)$,
%$C_1 = \sup|\partial_\chi|$, $C_2 = \sup|\partial_r^2\chi|$.
%The estimates for
%$g_{\beta_\gamma}^{\gamma,1}$ and
%$H_{\beta_\gamma}^{\gamma,1} g_{\beta_\gamma}^{\gamma,1}$ 
%can be done similarly.
%\QED
%
%\begin{corollary}
%$H_\beta \rightarrow H_\alpha$ in the strong resolvent sense
%as $\beta \rightarrow \alpha$ in ${\cal P}$.
%\end{corollary}
%
%\begin{corollary}
%\label{acontspec}
%For any $\lambda \in \sigma(H_\alpha)$,
%$\lim_{\beta\rightarrow\alpha} \dist(\sigma(H_\beta),\lambda)=0$.
%\end{corollary}
%\Proof
%This can be easily proved by constructing
%approximating eigenfunctions
% using the operator $T_\beta$.
%\QED








\subsection{Strong resolvent continuity
for the perturbation of $\delta$ magnetic fields}
\label{section_scont}
Let $K$ be a positive integer.
Let ${\cal F}$ be the space of all the pairs 
$(\Gamma,\alpha)$, where 
$\Gamma = (\gamma_k)_{k=1}^K$ 
is a sequence of $K$ distinct complex numbers,
and $\alpha=(\alpha_k)_{k=1}^K$
is a $[0,1]$-valued sequence.
Define a metric $d$ on ${\cal F}$ by
\[
   d((\Gamma,\alpha), (\Gamma',\alpha'))
  =\left(\sum_{k=1}^K(|\gamma_k-\gamma_k'|^2 
                    + |\alpha_k-\alpha_k'|^2)\right)^{1/2}
\]
for $(\Gamma,\alpha), (\Gamma',\alpha')\in{\cal F}$,
where $\Gamma'=(\gamma_k')_{k=1}^K$,
$\alpha'=(\alpha_k')_{k=1}^K$.
We often regard a sequence $\Gamma$
as a $K$-point subset of $\C$.
We denote
${\cal L}_{\Gamma,\alpha} = {\cal L}_{\phi_{\Gamma,\alpha}}$,
$L_{\Gamma,\alpha} = L_{\phi_{\Gamma,\alpha}}$ and
$H_{\Gamma,\alpha} = H_{\phi_{\Gamma,\alpha}}$ 
for $(\Gamma,\alpha) \in {\cal F}$,
where 
\begin{displaymath}
 \phi_{\Gamma,\alpha}(z)
 = \frac{B\bar{z}}{2}  + 
  \sum_{k=1}^K \frac{\alpha_k}{z-\gamma_k}.
\end{displaymath}





\begin{lemma}
\label{strong_continuity_delta_fields}
Let $O$ be an open set and 
$(\Gamma,\alpha)\in {\cal F}$ with $\Gamma \subset O$.
Then, there exist an open neighborhood ${\cal F}'$ of $(\Gamma,\alpha)$,
subspaces $\{D_{\Gamma',\alpha'}\}_{(\Gamma',\alpha')\in {\cal F}'}$ 
of $L^2(\C)$ 
and linear operators $\{T_{\Gamma',\alpha'}\}_{(\Gamma',\alpha')\in {\cal F}'}$
satisfying the following conditions:
\begin{enumerate}
 \item 
$D_{\Gamma',\alpha'}$ is an operator core 
of $H_{\Gamma',\alpha'}$
and 
$D_{\Gamma',\alpha'}\subset D_0(H_{\Gamma',\alpha'})$, 
for any $(\Gamma',\alpha')\in {\cal F}'$.

 \item 
 $T_{\Gamma',\alpha'}$ is a linear operator 
from $D_{\Gamma,\alpha}$ to $D_{\Gamma',\alpha'}$
for any $(\Gamma',\alpha')\in {\cal F}'$.
Moreover, $\supp T_{\Gamma',\alpha'} u \subset O\cup \supp u$.

 \item 
When $(\Gamma',\alpha') \rightarrow 
    (\Gamma,\alpha)$ in ${\cal F}'$,
we have
\begin{equation}
\label{pos0}
 T_{\Gamma',\alpha'} u \rightarrow u,\quad
 H_{\Gamma',\alpha'} T_{\Gamma',\alpha'} u \rightarrow H_{\Gamma,\alpha} u 
\end{equation}
in $L^2(\C)$, for any $u\in D_{\Gamma,\alpha}$.
\end{enumerate}
\end{lemma}
\Proof
Take $(\Gamma,\alpha)\in {\cal F}$.
Put
\begin{eqnarray*}
 {\cal K}_0 &=& \{k=1,\ldots,K~|~\alpha_k=0\},\\
 {\cal K}_1 &=& \{k=1,\ldots,K~|~\alpha_k=1\},\\
 {\cal K}_2 &=& \{k=1,\ldots,K~|~0<\alpha_k<1\}.
\end{eqnarray*}
If ${\cal K}_2 \neq \emptyset$, put
\[
 A = \min_{k \in \Gamma_{2}}\min(\alpha_k, 1-\alpha_k).
\]
If ${\cal K}_2=\emptyset$, put $A=1$.
Take  
$R'>0$ so that
$B_{3R'}(\gamma_k)\subset O$ for $k=1,\ldots,K$
and $\{B_{3R'}(\gamma_k)\}_{k=1}^K$ are disjoint.
Take a small positive number $R<\min(A, R')$ (determined later)
and put ${\cal F}' = \{(\Gamma',\alpha') \in {\cal F} 
~|~ d((\Gamma',\alpha'),(\Gamma,\alpha) )<R\}$.

We shall construct diffeomorphisms 
$\{F_{\Gamma',\alpha'}\}_{(\Gamma',\alpha')\in {\cal F}'}$ on $\C$
satisfying
\begin{equation}
 \label{pos1}
  F_{\Gamma',\alpha'} (z) = z\quad\mbox{for}~
  z\in \C\setminus \bigcup_{k=1}^K B_{3R'}(\gamma_k),
\end{equation}
\begin{equation}
 \label{pos2}
 F_{\Gamma',\alpha'} (z) = z- \gamma_k + \gamma_k' \quad\mbox{for}~
  z\in B_{R'}(\gamma_k),
\end{equation}
\begin{equation}
 \label{pos3}
 \|\partial_x^l\partial_y^m(F_{\Gamma',\alpha'}(z) - z)\|_\infty
 \leq C_{lm} d((\Gamma,\alpha),(\Gamma',\alpha'))
\end{equation}
for any $(\Gamma',\alpha')\in {\cal F}'$ and $l,m=0,1,2,\ldots$, 
where $C_{lm}$ is a positive constant independent of $(\Gamma',\alpha')$.
For this purpose, take a smooth function $\eta$ on $\R^1$
such that $0\leq \eta(r)\leq 1$ and
\[
 \eta(r) = \cases{
  1 & ($ r \leq R'$), \cr
  0 & ($r  \geq 2R'$).\cr
}
\]
Define $F_{\Gamma',\alpha'}$ by (\ref{pos1}) and
\[
 F_{\Gamma',\alpha'}(z) = z + \eta(|z-\gamma_k|)(\gamma_k' - \gamma_k)
\]
for $z\in B_{3R'}(\gamma_k)$, $k=1,\ldots,K$.
Then, $F_{\Gamma',\alpha'}$ clearly satisfies 
(\ref{pos2}) and (\ref{pos3}).
If we take $R$ sufficiently small, we see that
$F_{\Gamma',\alpha'}$ is a diffeomorphism by the Hadamard
inverse function theorem.

For $(\Gamma',\alpha') \in {\cal F}'$ and $k=1,\ldots,K$, define
\begin{eqnarray*}
 \Phi_{\Gamma',\alpha'}^k (z)
 &=& \exp i \Im\int_{\gamma_k'}^z \left(-\frac{B\bar{\gamma_k'}}{2}
   - \sum_{l\neq k} \frac{\alpha_l'}{w-\gamma_l'}
\right)dw.
\end{eqnarray*}
The function $\Phi_{\Gamma',\alpha'}^k$ is single-valued, 
smooth and satisfies
\begin{equation}
\label{intertwine1}
 {\cal L}_{\Gamma',\alpha'}  \Phi_{\Gamma',\alpha'}^k
=  \Phi_{\Gamma',\alpha'}^k  {\cal L}_{\alpha_k'}^{\gamma_k',1}
\end{equation}
in $B_{3R'}(\gamma_k')$.

Let us apply Lemma \ref{fundamental} with 
the above $R$ and $\{\Phi_{\Gamma',\alpha'}^k\}_{k=1}^K$.
Define a core $D_{\Gamma',\alpha'}$ for $H_{\Gamma',\alpha'}$
by
\[
 D_{\Gamma',\alpha'} =
{\rm L.h.}\left(
            \bigcup_{k\in {\cal K}_0}\{ f_{\alpha_k'}^{\gamma_k'}\}
       \cup \bigcup_{k\in {\cal K}_1}\{ g_{\alpha_k'}^{\gamma_k'}\}
       \cup \bigcup_{k\in {\cal K}_2}\{ f_{\alpha_k'}^{\gamma_k'}\}
            \cup\{g_{\alpha_k'}^{\gamma_k'}\}\right)
       \oplus C_0^\infty(\C\setminus \Gamma').
\]
For $(\Gamma',\alpha') \in {\cal F}'$, 
define a linear operator $T_{\Gamma',\alpha'}$
from $D_{\Gamma,\alpha}$ to $D_{\Gamma',\alpha'}$ by
\begin{eqnarray*}
  &&T_{\Gamma',\alpha'} \left(
  \sum_{k\in \Gamma_0} c_k f_{\alpha_k}^{\gamma_k}
 +\sum_{k\in \Gamma_1} d_k g_{\alpha_k}^{\gamma_k}
 +\sum_{k\in \Gamma_2} (c_k f_{\alpha_k}^{\gamma_k}
                           + d_k g_{\alpha_k}^{\gamma_k} )
  + \xi \right)\\
  &=&
  \sum_{k\in \Gamma_0} c_k f_{\alpha_k'}^{\gamma_k'}
 +\sum_{k\in \Gamma_1} d_k g_{\alpha_k'}^{\gamma_k'}
 +\sum_{k\in \Gamma_2} (c_k f_{\alpha_k'}^{\gamma_k'}
                           + d_k g_{\alpha_k'}^{\gamma_k'} ) 
              + \xi\circ F_{\Gamma',\alpha'}^{-1},
\end{eqnarray*}
where $c_k$ and $d_k$ are constants and 
$\xi \in C_0^\infty(\C\setminus \Gamma)$.

We shall check that $T_{\Gamma',\alpha'}$ satisfies the desired
properties.
The inclusion $\supp T_{\Gamma',\alpha'}u \subset O\cup \supp u$
holds by definition.
Since the functions
$f_{\alpha_k'}^{\gamma_k'}$,
$g_{\alpha_k'}^{\gamma_k'}$ and
$F_{\Gamma',\alpha'}$
are smooth with respect to 
$(\Gamma',\alpha')$
except $\gamma_k'\neq z$, 
the convergence (\ref{pos0}) clearly holds pointwise almost everywhere.
By the dominated convergence theorem,
it suffices to show that the functions 
$T_{\Gamma',\alpha'} u$ and 
$H_{\Gamma',\alpha'} T_{\Gamma',\alpha'} u$ are bounded uniformly 
with respect to $(\Gamma',\alpha')$.
By (\ref{intertwine1}),
it suffices to show 
$f_{\alpha_k'}^{\gamma_k',1}$,
$g_{\alpha_k'}^{\gamma_k',1}$,
$H_{\alpha_k'}^{\gamma_k',1} f_{\alpha_k'}^{\gamma_k',1}$ and
$H_{\alpha_k'}^{\gamma_k',1} g_{\alpha_k'}^{\gamma_k',1}$ 
are uniformly bounded.
This can be done by a straightforward calculation
using the polar coordinate.
Put $r=|z-\gamma_k'|$, $\theta=\arg(z-\gamma_k')$,
$a = \alpha_{\gamma_k'}$, $\chi = \chi_{\gamma_k'}$.
According to \cite[section II]{E-S-V}, we have
\begin{equation}
 \label{polar}
 H_{\alpha_k'}^{\gamma_k',1}
 = - \frac{1}{r}\partial_r r \partial_r
        + \frac{1}{r^2}
   \left(
     \frac{1}{i}\partial_\theta + a + \frac{Br^2}{2}
   \right)^2.
\end{equation}
Thus we have
\begin{eqnarray*}
 |f_{\alpha_k'}^{\gamma_k',1}| &=& \chi r^a
\leq M,\\
  \left|H_{\alpha_k'}^{\gamma_k',1} f_{\alpha_k'}^{\gamma_k',1}\right|
 &\leq& (2a+1)|\partial_r \chi| r^{a-1}
 +  (|\partial_r^2\chi| + Ba)r^a 
 + \frac{B^2}{4}\chi r^{a+2} \\
 &\leq& 3 m^{-1} C_1 + (C_2 + B)M + \frac{B^2}{4} M^3,
\end{eqnarray*}
where $M=\max\left(\frac{R}{2}, 1\right)$,
$m = \min \left(\frac{R}{3}, 1\right)$,
$C_1 = \sup|\partial_r \chi|$, $C_2 = \sup|\partial_r^2\chi|$.
The estimates for
$g_{\alpha_k'}^{\gamma_k',1}$ and
$H_{\alpha_k'}^{\gamma_k',1} g_{\alpha_k'}^{\gamma_k',1}$ 
can be done similarly.
\QED

\begin{corollary}
$H_{\Gamma',\alpha'} \rightarrow H_{\Gamma,\alpha}$ 
in the strong resolvent sense
as $(\Gamma',\alpha') \rightarrow (\Gamma,\alpha)$ in ${\cal F}$.
\end{corollary}


\begin{corollary}
\label{contspec}
For any $\lambda \in \sigma(H_{\Gamma,\alpha})$,
$\lim_{(\Gamma',\alpha')\rightarrow(\Gamma,\alpha)} 
\dist(\sigma(H_{\Gamma',\alpha'}), \lambda)=0$.
\end{corollary}
\Proof
This can be easily proved by constructing
approximating eigenfunctions
 using the operator $T_{\Gamma',\alpha'}$.
\QED


\subsection{Gathering to a point}
Next we consider the case some points are gathering to a point.
For a sequence $\Gamma=(\gamma_k)_{k=1}^K$ and $\epsilon >0$, we denote
\[
 \epsilon \Gamma = (\epsilon \gamma_k)_{k=1}^K.
\]
\begin{lemma}
 \label{gathering}
 Let $(\Gamma,\alpha) \in {\cal F}$. Then, we have
\begin{equation}
\label{gat0}
   \lim_{\epsilon \downarrow 0}
 \dist(\sigma(H_{\epsilon \Gamma,\alpha}), 
  \{E_n + 2B \fra(\alpha_1 + \cdots + \alpha_K)\})=0
\end{equation}
for any $n=1,2,\ldots$.
\end{lemma}
\Proof
Let $R= \max_{k=1,\ldots,K}|\gamma_k|$
and $\beta = \fra(\alpha_1 + \cdots + \alpha_K)$.
If $\beta=0$, then the assertion is trivial 
since $\sigma(H_{\epsilon\Gamma,\alpha})$ contains all the Landau levels
by \cite[Theorem 1.1 (i)]{M}.

Assume $0<\beta<1$.
For $\epsilon>0$ and $|z|>\epsilon R$, put\begin{displaymath}
 \Phi_\epsilon (z)
 = \exp i \Im \int_{\epsilon(R+1)}^z 
    \left(\frac{\beta}{w}
   - \sum_{k=1}^K \frac{\alpha_k}{w-\epsilon\gamma_k}
\right)dw ,
\end{displaymath}
where the integral is done along a smooth curve 
from $\epsilon(R+1)$ to $z$
contained in the region $\{|z| > \epsilon R\}$.
The function $\Phi_\epsilon$ is single-valued, smooth and satisfies
\begin{equation}
\label{intertwine3}
 {\cal L}_{\epsilon\Gamma,\alpha}  \Phi_\epsilon
=  \Phi_\epsilon  {\cal L}_\beta^{0,1}
\end{equation}
in $\{|z|>\epsilon R\}$.

Put 
\[
  u_n(z) = |z|^\beta z^{n-1} e^{-\frac{B}{4}|z|^2}.
\]
The function $u_n$ is an eigenfunction of $H_\beta^{0,1}$
for the eigenvalue $E_n + 2\beta B$.
Take a smooth function $\chi=\chi(r)$ on $\R$ satisfying
$0 \leq \chi(z)\leq 1$ and
\[
 \chi(r) = \cases{ 0 & ($0\leq r \leq R$),\cr
                   1 & ($2R\leq r $).
}
\]
Put $\chi_\epsilon(z) = \chi(\frac{|z|}{\epsilon})$ and
put
$u_\epsilon = \Phi_\epsilon \chi_\epsilon u_n 
/ \|\Phi_\epsilon \chi_\epsilon u_n \|$.
Using (\ref{intertwine3})
and the polar coordinate expression (\ref{polar}),
we can show
\begin{equation}
\label{gat1}
\|(H_{\epsilon\Gamma,\alpha} - (E_n + 2\beta B))u_\epsilon\|^2
 \leq C \epsilon^{2\beta + 2 n -4},
\end{equation}
where $C$ is a positive constant 
independent of $\epsilon$.
When $n \geq 2$, 
the inequality (\ref{gat1}) implies (\ref{gat0}).

To treat the case $n = 1$, we introduce an auxiliary operator
$H_{\epsilon\Gamma,\alpha}^-$ as in \cite[Proposition 3.3]{M}
(notice that the Friedrichs extension is denoted by $H_N^{AB}$ in
\cite{M}).
The operator $H_{\epsilon\Gamma,\alpha}^-$ is a self-adjoint extension of
$L_{\epsilon\Gamma,\alpha}$ satisfying
\begin{displaymath}
 H_{\epsilon\Gamma,\alpha} + 2B \simeq H_{\epsilon\Gamma,\alpha}^-
 |_{\Ker(H_{\epsilon\Gamma,\alpha}^- - B)^\bot}
\end{displaymath}
(see \cite[(8)]{M}).
Thus we have
\begin{equation}
 \label{gat2}
 \dist(\sigma(H_{\epsilon\Gamma,\alpha}), E_1 + 2\beta B)
 = \dist (\sigma(H_{\epsilon\Gamma,\alpha}^-), E_2 + 2\beta B).
\end{equation}
Since 
$\supp u_\epsilon \subset \{|z|\geq \epsilon R\}$, 
we have $u_\epsilon\in D(H_{\epsilon\Gamma,\alpha}^-)$.
Thus (\ref{gat1}) for $n=2$ holds even if we replace 
$H_{\epsilon\Gamma,\alpha}$ by $H_{\epsilon\Gamma,\alpha}^-$.
Combining this fact with (\ref{gat2}),
we conclude (\ref{gat0}) also holds for $n=1$.
\QED










\section{Appendix}
\label{appendix}
\subsection{Proof of Proposition \ref{canonical_product}}
\qbd{Proof of Proposition \ref{canonical_product}.}
(i)
Since $|\frac{1}{z-\gamma} + \frac{1}{\gamma} + \frac{z}{\gamma^2}| 
= O(|\gamma|^{-3})$ locally uniformly 
with respect to $z$
in $\C\setminus\Gamma$,
we have
\begin{eqnarray*}
 &&\sum_{\gamma\neq 0}
 \alpha_\gamma \left|\frac{1}{z-\gamma} + \frac{1}{\gamma} 
  + \frac{z}{\gamma^2}\right|
 \leq C \int_0^\infty r^{-3} dn(r)\\
 &=& C \left(\left[r^{-3}n(r)\right]_0^\infty
  +3 \int_0^\infty r^{-4}n(r) dr
\right)< \infty,
\end{eqnarray*}
where we used (\ref{lev2}).
Thus the sum (\ref{lev3}) converges.
Then we can define $\sigma_{\Gamma,\alpha}$
via the following formula:
\begin{equation}
\label{lev9} 
 \sigma_{\Gamma,\alpha}(z) 
 = z^{\alpha_0}
   \exp\left(\int_0^z 
\left(\zeta_{\Gamma,\alpha}(w) - \frac{\alpha_0}{w}\right)dw
\right).
\end{equation}
The right hand side of (\ref{lev9}) can be rewritten
in the form (\ref{lev4}), and then the product
converges.
The formula (\ref{sigma_formula}) follows from
(\ref{lev9}).
If we change the path of integration from $0$ to $z$, 
then $\sigma_{\Gamma,\alpha}$ is multiplied
by some $e^{2\pi i \alpha_\gamma}$'s.
Thus $|(\frac{d}{dz})^k \sigma_{\Gamma,\alpha}|$ 
is independent of the choice of the branches.


(ii) (Outline)
This assertion can be proved
in a similar way to the proof of
\cite[Theorem 2 in Chap.\ II, Sec.\ 1]{L}.
Only we have to do is to replace the definition of
the function $n(r,\theta_1,\theta_2)$
by (\ref{lev1}).
Below we shall exhibit the outline of the proof,
and show how the lemmas in \cite{L}
should be modified by this change.
Without loss of generality, we assume 
$0\notin\Gamma$, so
$\alpha_0 = 0$.

\begin{lemma}
 \label{levin_cartan}
 For any positive number $H$, 
 any finite set $\Gamma\subset\C$
 and any sequence $\alpha = (\alpha_\gamma)_{\gamma\in\Gamma}$
 of positive numbers,
 there is a system of disks in $\C$, 
 with the sum of the radii equal to $2H$,
 such that for each point $z$ outside these disks 
 we have
\begin{displaymath}
  \prod_{\gamma\in \Gamma} \left|z-\gamma\right|^{\alpha_\gamma}
  > \left(\frac{H}{e}\right)^n,
\end{displaymath}
where $n = \sum_{\gamma\in\Gamma} \alpha_\gamma$.
\end{lemma}
\qbd{Outline of Proof.}
This is a generalization of the Cartan estimate
\cite[Theorem 10 in Chap. 1, Sec. 7]{L}.
For $X\subset \C$, put
\begin{displaymath}
 n(X) = \sum_{\gamma \in \Gamma\cap X}\alpha_\gamma.
\end{displaymath}
Put $\Gamma_0 = \Gamma$, $C_0=\emptyset$.
For $j=1,2,\ldots$, define disks $C_j = B_{r_j}(z_j)$
by the following inductive procedure:
Put $\Gamma_j = \Gamma_{j-1} \setminus C_{j-1}$.
If $\Gamma_j = \emptyset$, the procedure finishes.
If $\Gamma_j\neq \emptyset$, let $C_j$ be a disk
having the largest radius among the closed disks 
$B_r(z)$ satisfying
\begin{displaymath}
 r = \frac{H}{n}n\left(B_r(z)\cap \Gamma_j \right).
\end{displaymath}
Since $\Gamma$ is a finite set, this procedure 
must finish within finite steps, and we obtain
disks $\{C_j\}_{j=1}^J$.
Put $D_j = B_{2r_j}(z_j)$.
Then the disks $\{D_j\}_{j=1}^J$ have the desired properties.
The equality
$\sum_{j=1}^J 2r_j =2H$ holds by the construction of $\{C_j\}$.
For $z\in \left(\bigcup_{j=1}^J D_j\right)^c$,
number the elements of $\Gamma$ and $\alpha$ as
$|z-\gamma_1| \leq \cdots \leq |z-\gamma_K|$
and $\alpha_k = \alpha_{\gamma_k}$.
By a similar argument as in \cite{L}
\footnote{
We show `every disk $C=B_r(z)$ with $r \geq r_j$ satisfies 
$r \geq \frac{H}{n}n(C \cap \Gamma_j)$',
and apply this fact to the disk $B_{|z-\gamma_k|}(z)$.
}, we have
\begin{displaymath}
 |z-\gamma_k| \geq \frac{H}{n}\sum_{j=1}^k \alpha_j.
\end{displaymath}
Thus we have
\begin{eqnarray*}
 \sum_{k=1}^K \alpha_k \log|z-\gamma_k|
 &\geq&  \sum_{k=1}^K \alpha_k\left(
  \log H -\log n + \log \sum_{j=1}^k\alpha_j \right)\\
 &>& n(\log H -\log n)  + \int_0^n \log x ~dx\\
 &=& n\log\frac{H}{e},
\end{eqnarray*}
where we used the concavity of $\log x$
in the second inequality.
\QED

We introduce the Weierstrass primary factors
\begin{displaymath}
 G(u; 1) = (1-u)e^{u},\quad
 G(u; 2) = (1-u)e^{u+ \frac{u^2}{2}}.
\end{displaymath}
When we consider the function $\log G (u;p)$ ($p=1,2$) in the sequel,
we make a cut $[1,\infty)$ in the complex $u$-plane,
and take the branch $\log G(0;p)=0$.
So when we consider the function 
$\log \sigma_{\Gamma,\alpha}(z) 
= \sum_{\gamma\in\Gamma}\alpha_\gamma \log G(\frac{z}{\gamma},2)$,
the variable $z$ belongs to the star region
\begin{displaymath}
 \C \setminus \bigcup_{\gamma\in\Gamma} \{t\gamma~|~t \geq 1\}.
\end{displaymath}
We denote $r=|z|$.

\begin{lemma}
\label{levin_lemma7}
 Assume (\ref{lev2}) holds.
 For $0<s<1$, put
\begin{displaymath}
  f_s (z) = \prod_{|\gamma|< s r} 
                   G\left(\frac{z}{\gamma};1\right)^{\alpha_\gamma}.
\end{displaymath}
Then, there exist $C_1>0$ and $r_1 = r_1(s)>0$ such that
\begin{displaymath}
 \left|\log f_s(z) \right| \leq C_1 s r^2
\end{displaymath}
for $r \geq r_1$, where $C_1$ is independent of $s$, $r$.
\end{lemma}
The proof is similar to that of 
\cite[Lemma 7 in Chap. 1, Sec. 17]{L}, in the case $\rho(r)=\rho=2$,
$p=1$.

\begin{lemma}
\label{levin_lemma8}
 Assume (\ref{lev2}) holds.
 For $t>2$, put
\begin{displaymath}
 {}_t f (z) = \prod_{|\gamma|> t r} 
                   G\left(\frac{z}{\gamma};2\right)^{\alpha_\gamma}.
\end{displaymath}
Then, there exist $C_2>0$ and $r_2 = r_2(t)>0$ such that
\begin{displaymath}
 \left|\log {}_t f(z) \right| \leq C_2 t^{-1} r^2
\end{displaymath}
for $r \geq r_2$, where $C_2$ is independent of $t$, $r$.
\end{lemma}
The proof is similar to that of 
\cite[Lemma 8 in Chap. 1, Sec. 17]{L}, in the case $\rho(r)=\rho=2$,
$p=2$.

\begin{lemma}
\label{levin_lemma9}
 Assume that $\Gamma \subset (0,\infty)$
and the limit $\Delta = \Delta(0,2\pi)$
(defined by (\ref{lev5})) exists.
Put 
\begin{equation}
 \label{lev18}
  V_r (z) = \prod_{|\gamma|\leq r} 
                   G\left(\frac{z}{\gamma};1\right)^{\alpha_\gamma}
          \prod_{|\gamma|> r} 
                   G\left(\frac{z}{\gamma};2\right)^{\alpha_\gamma}.
\end{equation}
Then, for $0<\theta<2\pi$, we have
\begin{displaymath}
 \lim_{r\rightarrow\infty}
 \frac{\log V_r(r e^{i\theta})}{r^2}
 = - \Delta\left(\frac{1}{2} - i(\theta - \pi)
  \right)e^{2i\theta}.
\end{displaymath}
The limit is uniform with respect to $\theta \in [\eta, 2\pi-\eta]$,
for any $0 < \eta< \pi$.
\end{lemma}
The proof is similar to that of 
\cite[Lemma 9 in Chap. 1, Sec. 17]{L}, in the case $\rho(r)=\rho=2$.

\begin{lemma}
\label{levin_lemma4}
 Suppose a discrete set $\Gamma = \{\gamma_k\}_{k=1}^\infty$
  and a sequence $\alpha = (\alpha_{\gamma_k})_{k=1}^\infty$
satisfy  (\ref{lev2}).
 Assume $\widetilde{\Gamma} 
         = \{\widetilde{\gamma}_k\}_{k=1}^\infty$ satisfies
\begin{displaymath}
 |\gamma_k| = |\widetilde{\gamma}_k|,\quad
 |\arg \gamma_k - \arg \widetilde{\gamma}_k| < \delta
\end{displaymath}
for some $\delta>0$ independent of $k$.
Let $V_r(z)$ as in (\ref{lev18}),
and $\widetilde{V}_r(z)$ is (\ref{lev18}) 
with $\Gamma$ replaced by $\widetilde{\Gamma}$
and $\alpha_{\widetilde{\gamma}_k}=\alpha_{\gamma_k}$.
Then, for any $\eta>0$ and $\epsilon>0$,
there exists $\delta_0>0$ dependent only on $\eta$, $\epsilon$
such that if $\delta<\delta_0$ we have
\begin{displaymath}
 \left|
 \log|V_r(z)| - \log |\widetilde{V}_r(z)|
\right| < \epsilon r^2
\end{displaymath}
for all $z$ not in the union of some disks
with upper linear density less than $\eta$.
\end{lemma}
The proof is similar to that of 
\cite[Lemma 4 in Chap. 2, Sec. 3]{L}, in the case $\rho(r)=\rho=2$.
In the proof,
 we use Lemma \ref{levin_cartan},
Lemma \ref{levin_lemma7}
and Lemma \ref{levin_lemma8}.

\begin{lemma}
\label{levin_lemma5}
 Let $\Gamma$, $\alpha$ satisfying
the assumption (a) in (ii) of Proposition \ref{canonical_product}.
Let $V_r$ as in (\ref{lev18}).
Then, 
there exists a $C^0$-set ${\cal C}$ such that
we have
\begin{equation}
 \label{lev22}
 \lim_{r\rightarrow\infty, r e^{i\theta} \not\in {\cal C}} 
  \frac{\log|V_r(r e^{i\theta})|}{r^2}
 = -\int_{\theta-2\pi}^{\theta} (\psi-\theta)  
                   \sin 2(\psi-\theta) ~d \Delta(\psi).
\end{equation}
The convergence in (\ref{lev22})
is uniform with respect to $\theta\in [0,2\pi)$.
\end{lemma}
The proof is similar to that of 
\cite[Lemma 5 in Chap.2, Sec. 3]{L}
and the subsequent argument, with $\rho(r)=\rho=2$.
Roughly speaking, we approximate $\Gamma$
by another set $\widetilde{\Gamma}$
contained in a finite number of semi-infinite lines.
The asymptotics of the function $\widetilde{V}_r$
(the function $V_r$ corresponding to $\widetilde{\Gamma}$) is 
obtained by Lemma \ref{levin_lemma9},
which leads to the conclusion 
combined with the approximating argument
using Lemma \ref{levin_lemma4}.

Using the above lemmas,
we shall prove (ii) of Proposition \ref{canonical_product}.
Notice that 
\begin{equation}
 \label{lev23}
 \frac{\log|\sigma_{\Gamma,\alpha}|}{r^2}
 = \Re \left(\sum_{|\gamma|\leq r} \frac{\alpha_\gamma}{2\gamma^2}
	e^{2i\theta} \right)
   + \frac{\log |V_r(z)|}{r^2}.
\end{equation}
The first term in the right hand side of (\ref{lev23})
converges to $\Re (\delta_{\Gamma,\alpha} e^{2i\theta})$
by assumption (b).
So the conclusion follows from Lemma \ref{levin_lemma5}.
\QED





\subsection{Domain of the Friedrichs extension}
\label{domain_of_friedrichs}
Using (\ref{domainH}),
we can verify (\ref{domain})
if $\#\Gamma<\infty$ (see the next proposition).
However, it is not trivial in the case $\#\Gamma = \infty$,
especially in the case $\Gamma$ is not uniformly separated
(in fact, the Poisson configuration
$\Gamma_\omega$ is not uniformly separated almost surely).
Here we shall prove (\ref{domain})
under some rather mild condition.
By gauge transform, we can assume $0 \leq \alpha_\gamma <1$
without loss of generality.

\begin{proposition}
 \label{friedrichs_domain}
Put $n(r) = \# (\Gamma\cap B_r(0))$ for $r>0$.
Assume
\begin{equation}
\label{fdom0}
 \liminf_{r\rightarrow \infty}
 \frac{\log \log n (r)}{r}  < \log 2.
\end{equation}
Then, (\ref{domain}) holds.
\end{proposition}
Of course, we regard the case $\#\Gamma\leq 1$ satisfies 
(\ref{fdom0}).
Not only the lattice $\Gamma$, 
but also the Poisson configuration
$\Gamma_\omega$ also satisfies (\ref{fdom0}),
by (\ref{sol7}).
The difficulty of the proof is that
$u\in D(L_\phi^*)$ does not imply $(\nabla + i\a_\phi)u \in (L^2(\C))^2$
in general.
\Proof
In the sequel, we omit the subscript $\phi$.
We denote the right hand side of (\ref{domain}) by $\widetilde{D}$,
that is, $\widetilde{D}$ consists of the elements of $D(L^*)$
satisfying the boundary conditions
\begin{equation}
 \label{bdry}
 \limsup_{z\rightarrow\gamma}|u(z)|<\infty
\end{equation}
for every $\gamma\in \Gamma$.
We shall divide the proof into 2 steps.


{\bf Step 1.} The case $\#\Gamma<\infty$.
\Proof
First we show $D(H) \subset \widetilde{D}$.
Since $\#\Gamma<\infty$, $D(H)$ is given by (\ref{domainH}).
The functions $f_\alpha^\gamma$, $g_\alpha^\gamma$
satisfy (\ref{bdry}). We shall show 
$u\in D(\overline{L})$ also satisfies (\ref{bdry}).
Take a sequence $u_n \in D(L)$ which approximates $u$
in the graph norm. Then, $v_n=(H+1)u_n \rightarrow v =(H+1)u$
in $L^2$.
By the diamagnetic inequality \cite{M-O-R}, we have
\begin{equation}
 \label{fdom1}
 |(u_n-u)(z)| = |(H+1)^{-1}(v_n - v)| (z)
              \leq (-\Delta + 1)^{-1}|v_n -v| (z)
 \quad \mbox{a.e.}
\end{equation}
The function $(-\Delta + 1)^{-1}|v_n -v|\rightarrow 0$ in 
the Sobolev space $H^2$.
By the Sobolev inequality,
the right hand  side of (\ref{fdom1}) converges
to $0$ uniformly in $z\in \C$,
so is the left hand side.
Thus $u$ is continuous and $u(\gamma)=0$.

Conversely, 
by (\ref{domainL}), (\ref{domainH}),
the function in $D(L^*)\setminus D(H)$ must have
a singularity at some $\gamma$.
So $\widetilde{D}\subset D(H)$.
\QED

\noindent{\bf Step 2.} The case $\# \Gamma = \infty$.
\Proof
For constants $a, b$ with $1\leq a < b$, we can find
 $\chi_{a,b}\in C_0^\infty(\C )$
so that
$0 \leq \chi_{a,b} \leq 1$ and
\begin{displaymath}
 \chi_{a,b}(z) = \cases{ 1 & $|z|\leq a$, \cr 
                       0 & $|z|\geq b$, \cr 
}
\end{displaymath}
\begin{displaymath}
\|\nabla \chi_{a,b}\|_\infty \leq C (b-a)^{-1},\quad
\|\Delta \chi_{a,b}\|_\infty \leq C (b-a)^{-2},
\end{displaymath}
where $C$ is a constant independent of $a$, $b$ and $z$.

First we show $D(H)\subset \widetilde{D}$.
Let $u\in D(H)$.
For $n\in \N$,
take $a,b$ with $n \leq a <b$
and put $\chi = \chi_{a,b}$.
Then, we have $\chi u \in D(H)$
by (iii) of Lemma \ref{fundamental}.
By reducing into the finite point case 
using a gauge transform, we see that
(\ref{bdry}) holds for any $\gamma \in B_n(0) \cap \Gamma$.
Taking $n\rightarrow \infty$, we have $u\in \widetilde{D}$.

Next we show $D(H)\supset \widetilde{D}$.
Let $u\in D(L^*)$ satisfying (\ref{bdry}).
Take $\eta = \chi_{a,b}$ so that $\supp\nabla\eta \cap \Gamma = \emptyset$.
By (ii) of Lemma \ref{fundamental}, we have $\eta u \in D(L^*)$,
and $\eta u$ also satisfies (\ref{bdry}). 
Thus, by reducing into the finite point case
using a gauge transform,
we have $\eta u \in D(H)$.
By (iii) of Lemma \ref{fundamental}, 
we have $\chi u\in D(H)$ for any $\chi = \chi_{a,b}$.
Thus $(\nabla + i \a)\chi u\in L^2$ and
\begin{displaymath}
 \|(\nabla + i\a)\chi u\|^2 = (H \chi u, \chi u).
\end{displaymath}
By the Leibniz rule, we have
\begin{eqnarray*}
 \|(\nabla + i\a)\chi u\|^2
 \leq
 \|\chi H u\| \| u\|
+ 2 \|\nabla \chi (\nabla + i\a)u\|\|u\|
+ \|(\Delta \chi) u\|\|u\|.
\end{eqnarray*}
Thus we have
\begin{eqnarray}
 &&\int_{|z|\leq a} \left|(\nabla + i\a)u\right|^2dxdy\nonumber\\
 &\leq& C\left(1+ (b-a)^{-2}\right) + C(b-a)^{-1}
     \left(
      \int_{a\leq |z|\leq b} \left|(\nabla + i\a)u\right|^2dxdy
     \right)^{1/2} \label{fdom2}
\end{eqnarray}
for any $1\leq a < b$,
where $C$ is  a constant independent of $a$, $b$.


We shall prove $(\nabla + i\a)u\in L^2$.
Put
\begin{displaymath}
 c_k = \int_{k-1 \leq |z| \leq k} \left|(\nabla + i\a)u\right|^2dxdy.
\end{displaymath}
Substituting $a=k$, $b=k+1$ into (\ref{fdom2}), we have
\begin{equation}
\label{fdom3}
 c_1 + \ldots + c_k \leq C ( 1 + c_{k+1}^{1/2}),
\end{equation}
where $C$ is independent of $k$.
Thus, if $c_k$ is bounded, then we have $(\nabla + i\a)u\in L^2$.
Suppose $c_k$ is unbounded.
By (\ref{fdom3}),
we see that
there exist $k_0\in \N$
and $h>0$ such that $d_k = c_{k_0+k}/h$
satisfies $d_1 \geq 1$ and
\begin{displaymath}
 d_1 + \cdots + d_k \leq d_{k+1}^{1/2}
\end{displaymath}
for any $k=1,2,\ldots$.
This inequality implies $d_3\geq 4>e$ and $d_k^2\leq d_{k+1}$, 
so we have
\begin{displaymath}
 \log \log d_k \geq (k-3) \log 2
\end{displaymath}
for $k\geq 3$,
therefore
\begin{equation}
\label{fdom4}
 \liminf_{k\rightarrow\infty}\frac{\log \log c_k}{k} \geq \log 2.
\end{equation}

Dividing the ring $\{k-1 < |z| \leq k\}$ into $2n(k)$ subrings,
we find a subring $S_k = \{r_k - l_k < |z| \leq r_k + l_k\}$
with $l_k = 1/(2n(k))$ and $S_k\cap \Gamma = \emptyset$.
Put $S_k'=\{r_k - l_k/4 < |z| \leq r_k + l_k /4\}$.
Take a covering of $S_k'$ by open balls $\{B_j'\}_{j=1}^J$,
where $B_j' = B_{l_k/2}(z_j)$, $|z_j|=r_j$. 
Put  $B_j = B_{l_k}(z_j)$.
We may assume
\begin{displaymath}
 \max_{z\in \mathbf{C}} \#\{j~|~B_j \ni z\}\leq C,
\end{displaymath}
where $C$ is independent of $k$, $j$.
Take $\Phi_j\in C^\infty(B_j)$ such that
$|\Phi_j(z)|=1$ and
\begin{displaymath}
 {\cal L} \Phi_j = \Phi_j {\cal L}_{z_j}^{0,1}
\end{displaymath}
in $B_j$.
By this equality and the elliptic estimate, we have
\begin{eqnarray*}
 &&\int_{B_j'}|(\nabla + i\a)u|^2 dxdy
 = \int_{B_j'}|(\nabla + i\a_{z_j}^{0,1})\Phi_j^{-1}u|^2 dxdy
\\
& =& \int_{B_{1/2}(0)}
    \left| (\nabla_{z'} + i l_k^2 \a_0(z'))\Phi_j^{-1} u (l_k z' + z_j)
    \right|^2 dx' dy'\\
&\leq& C \int_{B_1(0)}\left(
    | (\nabla_{z'} + i l_k^2 \a_0(z'))^2 \Phi_j^{-1} u (l_k z' + z_j)|^2
+   | \Phi_j^{-1} u (l_k z' + z_j)|^2\right)
 dx' dy'\\
&\leq & C l_k^{-2} \int_{B_j}(|H u |^2 + |u|^2) dxdy,
\end{eqnarray*}
where 
$C$ denotes general constants independent of $k$, $j$
and $z'=x'+iy'$.
Summing up this inequality with respect to $j$, we have
\begin{displaymath}
 \int_{S_k'}|(\nabla + i\a)u|^2 dxdy 
 \leq C l_k^{-2}.
\end{displaymath}
Thus, substituting $a= r_k - l_k/4$, $b=r_k + l_k/4$
 into (\ref{fdom2}), 
we have
\begin{displaymath}
 c_1 + \cdots + c_{k-1} \leq C( 1 + n(k)^2).
\end{displaymath}
So we have
\begin{displaymath}
 \liminf_{k\rightarrow \infty} \frac{\log \log c_k}{k} 
 \leq \liminf_{k\rightarrow \infty} \frac{\log \log n(k)}{k} 
 < \log 2.
\end{displaymath}
This contradicts (\ref{fdom4}).
Thus we have $(\nabla + i\a)u\in L^2(\C)$.

Put $\chi_n = \chi_{n,2n}$.
Then, by the Leibniz formula
and the fact $u$, $(\nabla + i\a)u$, 
$Hu$ are square integrable, we see that
$\chi_n u\rightarrow u$ in $D(L^*)$.
Since $\chi_n u\in D(H)$ and
$D(H)$ is a closed subspace of $D(L^*)$,
we have $u\in D(H)$.
\QED

Thus the proposition is proved.
\QED

{\bf Acknowledgments.}
The authors thank to Prof. L. Pastur 
for introducing us the paper \cite{C-L-P}.
The work of T. M. is partially supported by
JSPS grant Kiban B-18340049,
JSPS grant Kiban C-18540215 and
JSPS grant Kiban C-18540218.
The work of Y. N. is partially supported by
JSPS grant Kiban C-16540097 and
JSPS grant Kiban C-17540148.

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