Content-Type: multipart/mixed; boundary="-------------0705300258935" This is a multi-part message in MIME format. ---------------0705300258935 Content-Type: application/x-tex; name="mine_nomura.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="mine_nomura.tex" % % The spectrum of Schr\"odinger operators % with random $\delta$ magnetic fields % % by Takuya Mine and Yuji Nomura % \documentclass{article} \usepackage{amsfonts} \newtheorem{theorem}{\indent Theorem}[section] \newtheorem{proposition}[theorem]{\indent Proposition} \newtheorem{lemma}[theorem]{\indent Lemma} \newtheorem{corollary}[theorem]{\indent Corollary} \newtheorem{definition}[theorem]{\indent Definition} \setcounter{section}{0} \newenvironment{demo}[1]{\begin{trivlist}% \item[]\hspace{\parindent}{\it #1}\ }{\end{trivlist}} % \newcommand{\Proof}{\begin{demo}{{\bf Proof.}}} \newcommand{\qbd}[1]{\begin{demo}{#1}} \newcommand{\qed}{\end{demo}} \newsavebox{\toy} \savebox{\toy}{\framebox[0.65em]{\rule{0cm}{1ex}}} \newcommand{\QED}{\usebox{\toy}\end{demo}} \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, ~~r0$ 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}~|~00$ 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} 00$ 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 ${\cal F}' = \{(\Gamma',\alpha') \in {\cal F} ~|~ d((\Gamma',\alpha'),(\Gamma,\alpha) )0$, 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 $00$ 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 $n \leq a 0$ 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. 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