Content-Type: multipart/mixed; boundary="-------------0105051231582" This is a multi-part message in MIME format. ---------------0105051231582 Content-Type: text/plain; name="01-168.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="01-168.keywords" Pauli operator, singular magnetic field ---------------0105051231582 Content-Type: application/x-tex; name="acfin.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="acfin.tex" %maj 2 %marc 7 %feb27 %jan 15 %dec 15 %nov 29 %Nov 7 %Nov 6 %oct 1, 2000 \documentstyle[12pt]{article} \setlength{\oddsidemargin}{-.1in} \setlength{\textwidth}{6.6in} \setlength{\textheight}{8.5in} \setlength{\topmargin}{-.8in} \renewcommand{\baselinestretch}{1.4} \title{Pauli operator and Aharonov Casher theorem for measure valued magnetic fields} \author{L\'aszl\'o Erd\H os \thanks{Email address: {\tt lerdos@math.gatech.edu}. Partially supported by NSF grant DMS-9970323} \\ School of Mathematics, Georgiatech, Atlanta GA 30332 \\ and \\ Vitali Vougalter \thanks{Email address: {\tt vitali@math.ubc.ca}} \\ Department of Mathematics, University of British Columbia \\ Vancouver, B.C., Canada V6T 1Z2} \date{May 3, 2001} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{counterexample}[theorem]{Counterexample} \newcommand{\rd}{{\rm d}} \newcommand{\be}{\begin{equation}} \newcommand{\ee}{\end{equation}} \newcommand{\bey}{\begin{eqnarray}} \newcommand{\eey}{\end{eqnarray}} \newcommand{\sfrac}[2]{{\textstyle \frac{#1}{#2}}} \newcommand{\triple}{ |\! |\! |} \newcommand{\bone}{{\bf 1}} \newcommand{\bk}{{\bf k}} \newcommand{\bK}{{\bf K}} \newcommand{\bX}{{\bf X}} \newcommand{\bB}{{\bf B}} \newcommand{\bA}{{\bf A}} \newcommand{\bZ}{{\bf Z}} \newcommand{\bp}{-i\nabla} \newcommand{\bu}{{\bf u}} \newcommand{\bsigma}{\mbox{\boldmath $\sigma$}} \newcommand{\bsigmac}{\mbox{\boldmath $\sigma$}\cdot} \newcommand{\bU}{{\bf U}} \newcommand{\cU}{{\cal U}} \newcommand{\cp}{{\pi}} \newcommand{\tg}{\tilde g} \newcommand{\ta}{\tilde{\bf a}} \newcommand{\tf}{\tilde f} \newcommand{\txi}{\tilde \xi} \newcommand{\tn}{\tilde{\bf n}} \newcommand{\iint}{\int \!\! \int} \newcommand{\obA}{{\bf A}^{(1)}} \newcommand{\tbA}{{\bf A}^{(2)}} \newcommand{\bR}{{\bf R}} \newcommand{\bC}{{\bf C}} \newcommand{\bE}{{\bf E}} \newcommand{\bP}{{\bf P}} \newcommand{\bn}{{\bf n}} \newcommand{\bv}{{\bf v}} \newcommand{\bN}{{\bf N}} \newcommand{\bbe}{{\bf e}} \newcommand{\ep}{\varepsilon} \newcommand{\wh}{\widehat} \newcommand{\wt}{\widetilde} \newcommand{\Av}{\mbox{Av}} \newcommand{\ppn}{\perp {\bf n}} \newcommand{\pln}{\parallel {\bf n}} \newcommand{\ppN}{\perp {\bf N}} \newcommand{\plN}{\parallel {\bf N}} \newcommand{\bxi}{\mbox{\boldmath $\xi$}} \newcommand{\bzeta}{\mbox{\boldmath $\zeta$}} \newcommand{\bpsi}{\mbox{\boldmath $\psi$}} \newcommand{\cG}{{\cal G}} \newcommand{\cS}{{\cal S}} \newcommand{\cC}{{\cal C}} \newcommand{\cB}{{\cal B}} \newcommand{\cF}{{\cal F}} \newcommand{\cA}{{\cal A}} \newcommand{\cE}{{\cal E}} \newcommand{\cP}{{\cal P}} \newcommand{\cD}{{\cal D}} \newcommand{\cV}{{\cal V}} \newcommand{\cW}{{\cal W}} \newcommand{\cK}{{\cal K}} \newcommand{\cL}{{\cal L}} \newcommand{\cM}{{\cal M}} \newcommand{\cN}{{\cal N}} \newcommand{\cR}{{\cal R}} \newcommand{\tpup}{\tilde P_+} \newcommand{\tpdo}{\tilde P_-} \newcommand{\sigmatpp}{\bsigma_{\tilde\perp}} \newcommand{\sigmatn}{\bsigma_{\tilde\bn}} \newcommand{\sigmatppc}{\bsigma_{\tilde\perp}\cdot} \newcommand{\sigmatnc}{\bsigma_{\tilde\bn}\cdot} \newcommand{\magmom}{(\bp + \bA)} \newcommand{\twocovder}{\nabla^{(2)}} \newcommand{\threecovder}{\nabla^{(3)}} \newcommand{\naone}{\nabla_{e_1}} \newcommand{\natwo}{\nabla_{e_2}} \newcommand{\nan}{\nabla_n} \newcommand{\sione}{\sigma(e_1)} \newcommand{\sitwo}{\sigma(e_2)} \newcommand{\sithree}{\sigma(e_3)} \newcommand{\sn}{\sigma(n)} \newcommand{\D}{{\cal D}} \newcommand{\bD}{{\bf D}} \newcommand{\tdo}{\tilde\D_\Omega} \newcommand{\tdoc}{\tilde\D_{\Omega, c}} \newcommand{\tce}{\tilde \cE} \newcommand{\hce}{\hat \cE} \newcommand{\tdt}{\tilde {\cal D}_3} \newcommand{\tdp}{\tilde {\cal D}_\perp} \newcommand{\Dal}{\Delta_\xi^{\alpha_c}} \newcommand{\Om}{\Omega} \newcommand{\al}{\alpha} \renewcommand{\t}{\theta} \newcommand{\th}{\widetilde h} \def\req#1{\eqno(\hbox{Requirement #1})} %\def\label#1{\qquad({\hbox{\bf #1}})} %\def\ref#1{{\bf #1}} %\def\cite#1{{\bf [#1]}} %\catcode `\@=11 %\renewcommand{\label}[1]{ %\@bsphack\if@filesw {\let\thepage\relax % \def\protect{\noexpand\noexpand\noexpand}% % \edef\@tempa{\write\@auxout{\string % \newlabel{#1}{{\@currentlabel}{\thepage}}}}% % \expandafter}\@tempa % \if@nobreak \ifvmode\nobreak\fi\fi\fi\@esphack %\qquad(\hbox{\bf #1})} %\renewcommand{\ref}[1]{{\bf #1}: %\@ifundefined{r@#1}{{\reset@font\bf ??}\@warning % {Reference `#1' on page \thepage \space % undefined}}{\edef\@tempa{\@nameuse{r@#1}}\expandafter % \@car\@tempa \@nil\null} %} %\renewcommand{\cite}[1]{{\bf [#1]}} %\catcode `\@=12 \begin{document} \maketitle \begin{abstract} We define the two dimensional Pauli operator and identify its core for magnetic fields that are regular Borel measures. The magnetic field is generated by a scalar potential hence we bypass the usual $\bA\in L^2_{loc}$ condition on the vector potential which does not allow to consider such singular fields. We extend Aharonov-Casher theorem for magnetic fields that are measures with finite total variation and we present a counterexample in case of infinite total variation. One of the key technical tools is a weighted $L^2$ estimate on a singular integral operator. \end{abstract} \medskip\noindent {\bf AMS 2000 Subject Classification:} 81Q10 \medskip\noindent {\it Running title:} Pauli Operator for Measure Valued Fields. \section{Introduction} We consider the usual Pauli operator in $d=2$ dimensions with a magnetic field $B$ $$ H = \big[\bsigma \cdot (-i\nabla + \bA) \big]^2 = (-i\nabla + \bA)^2 + \sigma_3B \qquad \mbox{on} \quad L^2(\bR^2, \bC^2), $$ $B:=\mbox{curl}(\bA) = \nabla^\perp\!\cdot\! \bA$ with $\nabla^\perp : = (-\partial_2, \partial_1)$. Here $\bsigma \cdot (-i\nabla + \bA)$ is the two dimensional Dirac operator on the trivial spinorbundle over $\bR^2$ with vector potential $\bA$ and $\bsigma = (\sigma_1, \sigma_2, \sigma_3)$ are the Pauli matrices. Precise conditions on $\bA$ and $B$ will be specified later. The Aharonov-Casher Theorem \cite{AC} states that the dimension of the kernel of $H$ is given \be \mbox{dim}\; \mbox{Ker} (H) =\lfloor |\Phi|\rfloor, \label{ACeq}\ee where $$ \Phi:= \frac{1}{2\pi}\int_{\bR^2} B(x) \rd x $$ (possibly $\pm \infty$) is the flux (divided by $2\pi$) and $\lfloor \quad \rfloor$ denotes the lower integer part ($\lfloor n \rfloor = n-1$ for $n\ge 1$ integer and $\lfloor 0\rfloor=0$). Moreover, $\sigma_3 \psi= -s\psi$ for any $\psi\in \mbox{Ker} (H)$, where $s=\mbox{sign}(\Phi)$. On a $Spin^c$-bundle over $S^2$ with a smooth magnetic field the analogous theorem is equivalent to the index theorem (for a short direct proof see \cite{ES}). From topological reasons the analogue of $\Phi$, the total curvature of a connection, is an integer (the Chern number of the determinant line bundle), and the number of zero modes of the corresponding Dirac operator is $\big| \Phi\big|$. \medskip In the present paper we investigate two related questions: \medskip (i) What is the most general class of magnetic fields for which the Pauli operator can be properly defined on $\bR^2$? \medskip (ii) What is the most general class of magnetic fields for the Aharonov-Casher theorem to hold on $\bR^2$? \medskip Pauli operators are usually defined either via the magnetic Schr\"odinger operator, $(-i\nabla +\bA)^2$, by adding the magnetic field $\sigma_3 B$ as an external potential, or directly by the quadratic form of the Dirac operator $\bsigma \cdot (-i\nabla + \bA)$ (see Section \ref{stand}). In both ways, the standard condition $\bA\in L^2_{loc}$ is necessary. On the other hand, the statement of the Aharonov-Casher theorem uses only that $B\in L^1$, and in fact $B$ can even be a measure. It is therefore a natural question to extend the Pauli operator for such magnetic fields and investigate the validity of the Aharonov-Casher theorem. However, even if $B\in L^1$, it might not be generated by an $\bA\in L^2_{loc}$. For example, any gauge $\bA$ generating the radial field $B(x)= |x|^{-2}| \log |x| \; |^{-3/2} {\bf 1}(|x|\leq \sfrac{1}{2}) \in L^1$ satisfies $\int_{|x|\leq 1/2} |\bA(x)|^2 \rd x \ge \int_0^{1/2} (r|\log r|)^{-1} \rd r =\infty$ (here ${\bf 1}$ is the characteristic function). Hence the Pauli operator cannot be defined in the usual way on $C_0^\infty$ as its core. In case of a point singularity at $p\in\bR^2$ one can study the extensions from $C_0^{\infty}(\bR^2\setminus \{ p\})$, but such approach may not be possible for $B$ with a more complicated singular set. In this paper we present an alternative method which enables us to define the Pauli operator for any magnetic field that is a regular Borel measure (Theorem \ref{bm}). \bigskip The basic idea is to define the Pauli operator via a real generating potential function $h$, satisfying \be \Delta h = B \label{poisson}\ee instead of the usual vector potential $\bA$. This potential function appears in the original proof of the Aharonov-Casher theorem. The key identity is the following \be \int\Big|\bsigma \cdot (-i\nabla + \bA) \psi\Big|^2 = 4 \int \Big|\partial_{\bar z} (e^{-h}\psi_+)\Big|^2e^{2h} + \Big|\partial_{z} (e^{h}\psi_-)\Big|^2e^{-2h} \label{quad}\ee for regular data, with $\bA := \nabla^\perp h$ (integrals without specified domains are understood on $\bR^2$ with respect to the Lebesgue measure). We will {\it define} the Pauli quadratic form by the right hand side even for less regular data. It turns out that any magnetic field that is a regular Borel measure can be handled by an $h$-potential. The main technical tool is that for an appropriate choice of $h$, the weight function $e^{\pm 2h}$ (locally) belongs to the Muckenhoupt $A_2$ class (\cite{GR}, \cite{St}). Therefore the maximal operator and certain singular integral operators are bounded on the weighted $L^2$ spaces. This will be essential to identify the core of the operators. We point out that this approach does {\it not} apply to the magnetic Schr\"odinger operator $(-i\nabla +\bA)^2$. \bigskip The Aharonov-Casher theorem has been rigorously proven only for a restricted class of magnetic fields on $\bR^2$. The conditions involve some control on the decay at infinity and on local singularities. In fact, to our knowledge, the optimal conditions have never been investigated. The original paper \cite{AC} does not focus on conditions. The exposition \cite{CFKS} assumes compactly supported bounded magnetic field $B(x)$. The Ph.D. thesis by K. Miller \cite{Mi} assumes boundedness, and assumes that $\int |B(x)| \log |x| \rd x < \infty$. The boundedness condition is clearly too strong, and it can be easily replaced with the assumption that $B\in \cK(\bR^2)$ Kato class. Miller also observes that in case of integer $\Phi\neq0$ there could be either $|\Phi|$ or $|\Phi|-1$ zero states, but if the field is compactly supported then the number of states is always $|\Phi|-1$ \cite{CFKS}. \medskip The idea behind each proof is to construct a potential function $h$ satisfying (\ref{poisson}). {\it Locally}, $H\psi =0$ is equivalent to $\psi = (e^hg_+, e^{-h}g_-)$ with $\partial_{\bar z}g_+=0$, $\partial_{ z}g_-=0$, where we identify $\bR^2$ with $\bC$ and use the notations $x =(x_1, x_2)\in \bR^2$ and $z=x_1+ix_2 \in \bC$ simultaneously. The condition $\psi\in L^2(\bR^2,\bC^2)$ together with the explicit growth (or decay) rate of $h$ at infinity determines the {\it global} solution space by identifying the space of (anti)holomorphic functions $g_\pm$ with a controlled growth rate at infinity. For bounded magnetic fields decaying fast enough at infinity, a solution to (\ref{poisson}) is given by \be h(x) = {1\over 2\pi} \int_{\bR^2} \log |x-y| B(y) \rd y \label{trivsol} \ee and $h(x)$ behaves as $ \approx\Phi \log |x|$ for large $x$. If $\Phi\ge 0$, then $e^hg_+$ is never in $L^2$, and $e^{-h}g_-\in L^2$ if $|g_-|$ grows at most as the $\Big(\lfloor\Phi\rfloor -1\Big)$-th power of $|x|$. If $\Phi <\infty$, then $g_-$ must be a polynomial of degree at most $\lfloor\Phi\rfloor -1$. If $\Phi= \infty$, then the integral in (\ref{trivsol}) is not absolutely convergent. If the radial behavior of $B$ is regular enough, then $h$ may still be defined via (\ref{trivsol}) as a conditionally convergent integral and we then have a solution space of infinite dimension. \medskip Conditions on local regularity and decay at infinity are used to establish bounds on the auxiliary function $h$ given by (\ref{trivsol}), but they are not a priori needed for the Aharonov-Casher Theorem (\ref{ACeq}). We show that local regularity conditions are irrelevant by proving the Aharonov-Casher theorem for any measure valued magnetic fields with finite total variation (Theorem \ref{ACprec}). Many fields with infinite total variation can also be covered; some regular behavior at infinity is sufficient (Corollary \ref{reg}). However, some control is needed in general, as we present a counterexample to the Aharonov-Casher theorem for a magnetic field with infinite total variation. \begin{counterexample}\label{cont} There exists a continuous bounded magnetic field $B$ such that $\int_{\bR^2} |B| =\infty$ and \be \Phi := \lim_{r\to\infty} \Phi(r) = \lim_{r\to\infty} {1\over 2\pi} \int_{|x|\leq r} B(x) \rd x \label{impr}\ee exists and $\Phi >1$, but $\mbox{dim}\; \mbox{Ker} H = 0$. \end{counterexample} \medskip Finally, we recall a conjecture from \cite{Mi}: \begin{conjecture}\label{millconj1} Let $B(x)\ge 0$ with $\Phi := \sfrac{1}{2\pi}\int B$, which may be infinite. Then the dimension of $\mbox{Ker}(H)$ is at least $\lfloor \Phi\rfloor$. \end{conjecture} The proof in \cite{Mi} failed because it would have relied on the conjecture that for any continuous function $B\ge 0$ there exists a positive solution $h$ to (\ref{poisson}). This is false. A counterexample (even with finite $\Phi$) was given by C. Fefferman and B. Simon and it was presented in \cite{Mi}. However, the same magnetic field does {\it not} yield a counterexample to Conjecture \ref{millconj1}. Theorem \ref{ACprec} settles this conjecture for $\Phi<\infty$, but the case $\Phi =\infty$ remains open. The magnetic field in our counterexample does not have a definite sign, in fact $\Phi$ is defined only as an improper integral. \section{Definition of the Pauli operator}\label{sec:def} \subsection{Standard definition for $\bA\in L^2_{loc}$}\label{stand} The standard definition of the {\it magnetic Schr\"odinger operator}, $(-i\nabla +\bA)^2$, or the {\it Pauli operator}, $[\bsigma\cdot(-i\nabla +\bA)]^2$, as a quadratic form, requires $\bA \in L^2_{loc}$ (see e.g. \cite{LS, LL} and for the Pauli operator \cite{Sob}). We define $\Pi_k := -i\partial_k +A_k$, $Q_\pm: = \Pi_1 \pm i\Pi_2$, or with complex notation $Q_+ = -2i\partial_{\bar z} + a$, $Q_- = -2i\partial_{z} + \bar a$ with $a: = A_1 + iA_2$. These are closable operators, originally defined on $C_0^\infty(\bR^2)$. Their closures are denoted by the same letter on the minimal domains $\cD(\Pi_j)$ and $\cD(Q_\pm)$. Let $$ s_\bA(u,u) : = \| \Pi_1 u\|^2 + \| \Pi_2 u\|^2 = \int | (-i\nabla +\bA)u|^2 \; ,\qquad u\in C_0^\infty(\bR^2) $$ be the closable quadratic form associated with the magnetic Schr\"odinger operator on the minimal form domain $\cD (s_\bA)$. It is known \cite{Si} that the minimal domain coincides with the maximal domain which consists of all $u\in L^2$ such that $s_\bA(u,u)<\infty$. The closable quadratic form associated with the {\it Pauli operator} is $$ p_\bA(\psi, \psi): = \| Q_+\psi_+\|^2+ \| Q_-\psi_-\|^2 = \int | \bsigma\cdot(-i\nabla +\bA)\psi|^2 \; , \qquad \psi = \pmatrix{\psi_+\cr \psi_-} \in C_0^\infty(\bR^2, \bC^2) \; , $$ again on the minimal form domain $\cD (p_\bA) = \cD(Q_+)\otimes\cD(Q_-)$. The condition $\bA\in L^2_{loc}$ is obviously necessary. In both cases there exists a unique self-adjoint operator, $S_\bA$ and $P_\bA$, associated with these quadratic forms. A simple calculation shows that $\cD (s_\bA)\otimes\cD (s_\bA) \subset \cD (p_\bA)$. This inclusion already indicates that the Pauli operator may be defined in a more general setup. In case of $B=\nabla^\perp\!\cdot\! \bA \in L^\infty$, these two domains are equal and $P_\bA = S_\bA\otimes I_2 + \sigma_3 B$. If $B\in L^\infty_{loc}$ only, then the form domains coincide locally. For more details on these statements, see Section 2 of \cite{Sob}. \subsection{Measures and integer point fluxes} Let $\cM$ be the set of signed real Borel measures $\mu(\rd x)$ on $\bR^2$ with finite total variation, $|\mu|(\bR^2)=\int_{\bR^2} |\mu|(\rd x)<\infty$. Let $\overline\cM$ be the set of signed real regular Borel measures $\mu$ on $\bR^2$, in particular they have $\sigma$-finite total variation. If $\mu(\rd x) = B(x)\rd x$ is absolutely continuous, then $\mu\in\cM$ is equivalent to $B\in L^1$. Let $\overline{\cM^*}$ be the set of all measures $\mu\in \overline{\cM}$ such that $\mu(\{x \})\in (-2\pi,2\pi)$ for any point $x\in \bR^2$, and $\cM^*: = \cM \cap \overline{\cM^*}$. \begin{definition}\label{mustdef} Two measures $\mu, \mu'\in \overline{\cM}$ are said to be {\bf equivalent} if $\mu-\mu' = 2\pi\sum_j n_j \delta_{z_j}$, where $n_j\in \bZ$, $z_j\in\bR^2$. The equivalence class of any measure $\mu\in \overline{\cM}$ contains a unique measure, denoted by $\mu^*$, such that $\mu^*(\{x\})\in [-\pi, \pi)$ for any $x\in \bR^2$. In particular, $\mu^*\in \overline{\cM^*}$. \end{definition} The Pauli operator associated with $\mu\in \overline{\cM}$ will depend only on the equivalence class of $\mu$ up to a gauge transformation, so we can work with $\mu\in \overline{\cM^*}$. This just reflects the physical expectation that any magnetic point flux $2\pi n\delta_z$, with integer $n$, is removable by the gauge transformation $\psi(x) \to e^{in \varphi}\psi(x)$, where $\varphi = \mbox{arg}(x-z)$. In case of several point fluxes, $2\pi\sum_j n_j \delta_{z_j}$, the phase factor should be $\exp\Big( i\sum_j n_j \mbox{arg}(x-z_j)\Big)$, but it may not converge for an infinite set of points $\{ z_j \}$. However, any $\mu\in \overline{\cM}$ can be uniquely written as $\mu = \mu^* + 2\pi\sum_j n_j \delta_{z_j}$ with a set of points $z_j$ which do not accumulate in $\bR^2 \equiv \bC$. Hence there exist analytic functions $F_\mu(x)$ (recall $x=x_1+ix_2$) that has zeros exactly at $z_j$ with multiplicity $n_j$ for $n_j>0$ and $G_{\mu}(x)$ having zeros at $z_{j}$ with multiplicity $-n_{j}$ for $ n_{j}<0$ by Weierstrass theorem. Let $L_\mu(x): =F_\mu(x) \overline{G}_{\mu}(x)$. Then the integer point fluxes can be removed by the unitary gauge transformation \be U_\mu: \psi(x) \to {L_\mu(x)\over |L_\mu(x)|} \, \psi(x) \label{umudef}\ee and clearly $L_\mu(x)/|L_\mu(x)| = \exp\Big( i n_j \mbox{arg}(x-z_j) + i H_j(x)\Big)$, where $H_j$ is a real harmonic function for $x$ near $z_j$. In particular $U_\mu^* (-i\nabla)U_\mu \psi = \Big(-i\nabla + n_j \bA_j + \nabla H_j\Big)\psi$ for $\psi$ whose support contains only $z_j$ among the flux points, and here $\nabla^\perp\cdot \bA_j = 2\pi \delta_{z_j}$. \subsection{Potential function}\label{sec:pot} The Pauli quadratic form for magnetic fields $\mu\in \overline{\cM^*}$ will be defined via the right hand side of (\ref{quad}), where $h$ is a solution to $\Delta h =\mu$. The following theorem shows that for $\mu\in \cM^*$ one can always choose a good potential function $h$. Later we will extend it for $\mu\in \overline{\cM^*}$. \begin{theorem}\label{thm:hdef} Let $\mu\in \cM^*$ and $\Phi: = {1\over 2\pi}\int \mu(\rd x)$ be the total flux (divided by $2\pi$). There is $0<\ep(\mu)\leq 1$ such that for any $0< \ep < \ep(\mu)$ there exists a real valued function $h=h_\ep \in \cap_{p<2}W^{1,p}_{loc}$ with $\Delta h= \mu$ (in distributional sense), such that (i) For any compact set $K\subset \bR^2$ and any square $Q\subset K$ \be \Big( {1\over |Q|}\int_Q e^{2h}\Big) \Big( {1\over |Q|}\int_Q e^{-2h}\Big) \leq C_1(K,\ep,\mu) \label{A2}\ee (ii) $e^{\pm 2h}\in L_{loc}^{1+\ep}$. (iii) $h$ can be split as $h = h_1 + h_2$ with the following estimates: \be \Bigg| \; { h_1(x)\over \log |x|} - \Phi \; \Bigg|\leq \ep \qquad \mbox{for} \quad |x|\ge R(\ep,\mu) \label{mainh}\ee and \be \int_{Q(u)} e^{\pm 2 h_2} \leq C_2(\ep) \langle u\rangle^{2\ep} \label{errorh} \ee with some constants $C_1(K,\ep,\mu), C_2(\ep)$ and $R(\ep,\mu)$. Here $Q(u)= [u-\sfrac{1}{2}, u+\sfrac{1}{2}]^2$ denotes the unit square about $u\in\bR^2$ and $\langle u\rangle = (u^2+1)^{1/2}$. \end{theorem} {\it Remark.} The property (i) means that $e^{2h}$ satisfies a certain reversed H\"older inequality locally; in fact the constant in (\ref{A2}) is independent of $K$ if the scale of $Q$ is big enough. If (\ref{A2}) were true for any square, then $e^{2h}$ would be in the weight-class $A_2$ used in harmonic analysis (see \cite{GR,St}). Nevertheless, this property will allow us to use weighted $L^2$-bounds on a certain singular integral operator locally (Lemma \ref{local}). We also remark that property (ii) follows from the local analog of the well-known fact that $\omega \in A_2 \Longrightarrow \omega \in A_{p}$ for some $p<2$. \begin{corollary}\label{w11} If $h\in L^1_{loc}$ satisfies $\Delta h \in \overline{\cM^*}$, then $h\in W^{1,p}_{loc}$ for all $p<2$ and $e^{\pm 2h}\in L^{1}_{loc}$. If, in addition, $\Delta h \in \cM^*$, then $e^{\pm 2h}\in L^{1+\ep}_{loc}$ with some $\ep>0$. \end{corollary} {\it Proof.} Suppose first that $\mu: = \Delta h \in \cM^*$. Choose $\ep < \ep(\mu)$ and consider $h_\ep \in \bigcap_{p<2}W^{1,p}_{loc}$ constructed in Theorem \ref{thm:hdef} with $e^{\pm 2h_\ep} \in L^{1+\ep}_{loc}$. Since $\Delta (h-h_\ep)=0$, we have $h=h_\ep + \varphi$ with a smooth function $\varphi$ so the statements follow for $h$ as well. If $\mu = \Delta h$ has infinite total variation, then Theorem \ref{thm:hdef} cannot be applied directly. But for any compact set $K$ one can find another compact set $K^*$ with $K\subset \mbox{int}( K^*)$ and then the measure $\Delta h \in \overline{\cM^*}$ restricted to $K^*$ has finite total variation. Therefore one can find a function $h^*\in \bigcap_{p<2}W^{1,p}_{loc}$, $e^{\pm h^*}\in L^2_{loc}$ with $\Delta h^* =\mu$ on $K^*$, i.e., $h-h^*$ is harmonic on $K^*$, hence it is smooth and bounded on $K$. So $h\in \bigcap_{p<2} W^{1,p}(K)$ and $e^{\pm h} \in L^{2}(K)$ follows from the same properties of $h^*$. $\,\,\Box$ \bigskip {\it Proof of Theorem \ref{thm:hdef}.} Step 1. First we write $\mu = \mu_d + \mu_c$, where $\mu_d := 2\pi \sum_j C_j\delta_{z_j}$ ($C_j\in (-1,1)$, $z_j\in \bC \equiv \bR^2$) is the discrete part of the measure $\mu$, and $\mu_c$ is continuous, i.e., $\mu_c(\{ x \})=0$ for any point $x\in \bR^2$. The summation can be infinite, finite or empty, but $\sum_j |C_j|<\infty$. We also assume that $z_j$'s are distinct. Let $$ \ep(\mu):= {1\over 10}\min_j \Big\{ 1 - |C_j| \Big\} $$ then clearly $\ep(\mu)>0$. We fix an $0<\ep <\ep(\mu)$. All objects defined below will depend on $\ep$, but we will neglect this fact in the notations. We split the measure $\mu_d = \mu_{d,1} + \mu_{d,2}$ such that $$ \mu_{d,1} := 2\pi \sum_{j=1}^N C_j\delta_{z_j} \; , \qquad \mu_{d,2} := 2\pi\sum_{j=N+1}^\infty C_j\delta_{z_j} \; , $$ where $N$ is chosen such that $2\pi\sum_{j=N+1}^\infty |C_j| < \ep/2$. In particular $|\mu_{d, 2}|(\bR^2)< \ep/2$. We define \be h_{d,j}(x) : = {1\over 2\pi}\int \log {|x-y|\over \langle y\rangle} \mu_{d,j}(\rd y)\; , \qquad j=1,2 \; , \label{hddef}\ee so that $\Delta h_{d,j} = \mu_{d,j}$. Notice that $ h_{d,j}(x)$ is well defined for a.e. $x$, moreover $ h_{d,j} \in W^{1,p}_{loc}$ for all $p<2$ by Jensen's inequality. \bigskip Step 2. We split $\mu_c = \mu_{c,1}+ \mu_{c,2}$ such that $\mu_{c,1}$ be compactly supported and $|\mu_{c, 2}|(\bR^2) < \ep/2$. We set $\mu_j:= \mu_{d, j}+ \mu_{c,j}$, $j=1,2$. Then we define \be h_{c,j}(x): = {1\over 2\pi} \int_{\bR^2} \log {|x-y|\over \langle y\rangle} \mu_{c,j}(\rd y)\; , \qquad j=1,2 \; , \label{hcdef}\ee clearly $h_{c,j} \in W^{1,p}_{loc}$ for all $p<2$, and $\Delta h_{c,j}= \mu_{c,j}$ (in distributional sense). Finally, we define \be h_1 := h_{d,1}+h_{c,1}\; ,\qquad h_2: = h_{d,2}+h_{c,2} \; ,\qquad h: = h_1 + h_2 \label{htotdef}\ee and clearly $\Delta h_j = \mu_j$. Since $\mu_{d, 1}$ and $\mu_{c,1}$ are compactly supported, the estimate (\ref{mainh}) is straightforward. We will also need the notation $\nu: =\mu_{d,2} + \mu_c = \mu_{c,1}+\mu_2$. \bigskip Step 3. For any integer $L$ we define $\Lambda_L:= (2^{-L}\bZ)^2 + (2^{-L-1}, 2^{-L-1})$ to be the shifted and rescaled integer lattice. We define the {\it dyadic squares of scale $L$} to be the squares $$ D^{(L)}_{k}: =\Big[ k_1- 2^{-L-1}, k_1+ 2^{-L-1}\Big) \times \Big[k_2- 2^{-L-1}, k_2+ 2^{-L-1}\Big) $$ of side-length $2^{-L}$ about the lattice points $k=(k_1, k_2)\in \Lambda_L$. The squares $$ \wt D^{(L)}_{k} : =\Big[ k_1- 2^{-L}, k_1+ 2^{-L}\Big) \times \Big[k_2- 2^{-L}, k_2+ 2^{-L}\Big) $$ of double side-length with the same center $k$ are called {\it doubled dyadic squares of scale $L$}. Similarly, the squares $$ \wh D^{(L)}_{k} : =\Big[ k_1- 3\cdot 2^{-L-1}, k_1+ 3\cdot 2^{-L-1}\Big) \times \Big[k_2- 3\cdot 2^{-L-1}, k_2+ 3\cdot 2^{-L-1}\Big) $$ are called {\it tripled dyadic squares of scale $L$}. For a fixed scale $L$ the collection of dyadic squares is denoted by $\cD_L$. $\wt \cD_L$ and $\wh\cD_L$ denote the set of doubled and tripled dyadic squares, respectively. The elements of $\cD_L$ partition $\bR^2$ for each $L$. Notice also that every square $Q\subset \bR^2$ can be covered by a doubled dyadic square of area not bigger than a universal constant times $|Q|$. \begin{lemma}\label{Mlemma} There exists $1\leq M=M(\mu,\ep)<\infty$ such that $|\mu|(Q) < 2\pi(1-\ep)$ for any $Q\in\wh\cD_M$. \end{lemma} {\it Proof.} We first notice that the support of $\mu_{d,1}$ consists of finitely many points, hence for large enough $L$ each element of $\wh\cD_L$ contains at most one point from this support. Second, since the measure $|\nu| = |\mu_c| + |\mu_{d,2}|$ does not charge more than $\ep/2$ to any point, we claim that there exists a positive integer $1\leq M=M(\mu,\ep)<\infty$ such that $|\nu|(D) < \ep$ for any dyadic square of scale $M$. We can choose $M(\mu,\ep)\ge L$. This statement is clear by a dyadic decomposition; we start with the partition of $\bR^2$ into dyadic squares of scale $L$. There are just finitely many squares $D\in \cD_L$ such that $|\nu|(D)\ge \ep$. We split these squares further into four identical dyadic squares. If this process stops after finitely many steps, then we have reached our $M$ as the scale of the finest decomposition. Now suppose on the contrary that this process never stops. Then we could find a strictly decreasing sequence of nested dyadic squares $D_1 \supset D_2 \supset \ldots$ such that $|\nu|(D_j)\ge \ep$, but $|\nu|$ would charge at least $\ep$ weight to their intersection which is a point. Finally, since $|\mu|= |\mu_{d,1}| + |\nu|$ and every tripled square can be covered by 9 dyadic squares of the same scale, we have $|\mu|(Q) \leq 9\ep + 2\pi\max_j |C_j| < 2\pi(1-\ep)$ for each $Q\in \wh \cD_M$ for large enough $M$. $\,\,\,\Box$. \bigskip Step 4. Now we turn to the proof of (\ref{A2}) and first we prove it for any doubled dyadic square of big scale. %Since every square $Q$ with $|Q|\leq 1$ % can be covered by a doubled %dyadic square of comparable size, it is enough to prove %(\ref{A2}) for doubled dyadic squares of nonnegative scale. %Moreover, it is enough to prove it for doubled %dyadic squares of scale at least $M$ %at the expense of increasing the constant $C_1(K,\ep,\mu)$ by %a factor $16^{M(\mu,\ep)}$. Let $Q = \wt D^{(K)}_k\in \wt\cD_K$ be a doubled dyadic square with $K\ge M$ and let $\wh Q = \wh D^{(K)}_k$ be the corresponding tripled square with the same center $k\in \Lambda_K$. We split the measure $\mu$ as $$ \mu = \mu^{int} + \mu^{ext} : = {\bf 1}_{\wh Q} \mu + {\bf 1}_{\wh Q^c} \mu $$ with $|\mu| = |\mu^{int}| + |\mu^{ext}|$ and $h$ is decomposed accordingly as $h = h^{int}+ h^{ext}$ with $$ h^{\#}(x) := {1\over 2\pi} \int_{\bR^2} \log {|x-y|\over \langle y \rangle}\mu^\# (\rd y) \; , $$ where $\# = \mbox{int, ext}$. We also define $$ \wt h^{int}(x) := {1\over 2\pi} \int_{\bR^2} \log |x-y|\mu^{int} (\rd y) = h^{int}(x) + {1\over 2\pi} \int_{\bR^2} \log \langle y \rangle\mu^{int} (\rd y)\; . $$ Let $\Av_{Q}h^{ext} : = |Q|^{-1}\int_{Q} h^{ext}$ be the average of $ h^{ext}$ on $Q$. A simple calculation shows that \be \Big| h^{ext}(x) -\Av_{Q} h^{ext}\Big| \leq C |\mu|(\bR^2)\; , \qquad \forall x\in Q \label{av} \ee with a universal constant $C$ using that $\mu^{ext}$ is supported outside of the tripled square. Therefore $$ \Big( {1\over | Q|}\int_{Q} e^{2h}\Big) \Big( {1\over | Q|}\int_{ Q} e^{-2h}\Big) \leq e^{4C|\mu|(\bR^2)}\Big( {1\over |Q|}\int_{ Q} e^{2\wt h^{int}}\Big) \Big( {1\over | Q|}\int_{Q} e^{-2\wt h^{int}}\Big) $$ We split $\mu^{int}$ into its positive and negative parts: $\mu^{int} = \mu^{int}_+ - \mu^{int}_-$, we let $\phi_\pm: = {1\over 2\pi} \int_{\wh Q} \mu^{int}_\pm\ge 0$. By Lemma \ref{Mlemma} and $K\ge M$ we have $\phi:=\phi_++\phi_- <(1-\ep)$. Now we apply Jensen's inequality for the probability measures $(2\pi\phi_\pm)^{-1} \mu^{int}_\pm$ (if $\phi_\pm\neq0$): $$ \int_{ Q} e^{2\wt h^{int}} = \int_{ Q} \exp\Big( {1\over2\pi\phi_+} \int_{\wh Q}\log |x-y|^{2\phi_+} \mu^{int}_+(\rd y)\Big) \exp\Big( {1\over2\pi\phi_-} \int_{\wh Q}\log |x-y'|^{-2\phi_-} \mu^{int}_-(\rd y')\Big) \rd x $$ \be \leq \int_{Q} \rd x {1\over 2\pi\phi_+} \int_{\wh Q} \mu^{int}_+(\rd y) {1\over 2\pi\phi_-}\int_{\wh Q} \mu^{int}_-(\rd y') |x-y|^{2\phi_+}|x-y'|^{-2\phi_-} \leq C(\ep)|Q|^{1+\phi_+-\phi_-} \label{one}\ee with an $\ep$-dependent constant. When performing the $\rd x$ integration, we used the fact that $\phi_- < 1-\ep$, hence the singularity is integrable. Similarly, we have $$ \int_{Q} e^{-2\wt h^{int}} \leq C(\ep)| Q|^{1-\phi_++\phi_-} $$ which completes the proof of (\ref{A2}) for doubled dyadic squares of scale at least $M$ with a $K$-independent constant. \bigskip Step 5. Next, we prove $e^{\pm 2h}\in L_{loc}^{1+\ep}$. We can follow the argument in Step 4. On any square $Q\in \wt\cD_M$ we can use that $h^{ext}$ is bounded by (\ref{av}) and we can focus on $\exp (\pm 2 \wt h_{int})$. Then we use Jensen's inequality (\ref{one}) and use the fact that $x\mapsto |x-y|^{- 2(1+\ep)\phi_\pm}$ is locally integrable since $\phi_\pm < (1-\ep)$. \bigskip Step 6. Now we complete the proof of (\ref{A2}) for all squares $Q\subset K$. Since every square can be covered by a doubled dyadic square of comparable size, we can assume that $Q$ is such a square. If the scale of $Q$ is smaller than $M$, then $|Q|^{-1}\leq 4^{M(\mu,\ep)}$ and we can simply use $e^{\pm 2h}\in L_{loc}^1$ to estimate the integrals. \bigskip Step 7. Finally, we prove (\ref{errorh}). Let $\wh Q(u):= [u-1, u+1]^2$ and we split the measure $\mu_2$ as $$ \mu_2 = \mu_2^{int} + \mu_2^{ext} := {\bf 1}_{\wh Q(u)} \mu_2 + {\bf 1}_{\wh Q(u)^c} \mu_2 $$ and the function $h_2= h_2^{int} + h_2^{ext}$, where $$ h_2^\# (x) = {1\over 2\pi}\int_{\bR^2} \log {|x-y|\over \langle y\rangle } \mu_2^\#(\rd y) \; , \qquad \# = \mbox{int, ext} \; . $$ Similarly to the estimates (\ref{av}) and (\ref{one}) in Step 4, we obtain $$ \int_{Q(u)} e^{\pm2 h_2} \leq C(\ep) \exp{ \Big(\pm 2\Av_{Q(u)} h_2^{ext}\Big)} \exp{ \Big(2 |\mu_2|(\wh Q(u)) \log \langle u \rangle\Big)} \; , $$ and a simple calculation shows $$ \Big| \Av_{Q(u)} h_2^{ext}\Big| \leq \int_{Q(u)} \int_{\wh Q(u)^c} \Big| \log {|x-y|\over \langle y\rangle}\Big| \; |\mu_2^{ext}|(\rd y) \rd x \leq |\mu_2|(\wh Q^c(u))\log \langle u \rangle + C(\ep) \; . $$ From these estimates (\ref{errorh}) follows using that $ |\mu_2|(\bR^2)\leq \ep$. $\,\,\,\Box$ \bigskip \subsection{Definition of the Pauli operator for measure valued fields}\label{sec:core} For any real valued function $h\in L^1_{loc}(\bR^2)$ we define the following symmetric quadratic form: $$ \cp^h(\psi, \xi): = \cp^h_+(\psi_+, \xi_+) + \cp^h_-(\psi_-, \xi_-) $$ with $$ \cp^h_+(\psi_+, \xi_+):= 4\int \overline{\partial_{\bar z} (e^{-h}\psi_+)} \partial_{\bar z} (e^{-h}\xi_+) e^{2h}\; , \qquad \cp^h_-(\psi_-, \xi_-):= 4\int \overline{\partial_{ z} (e^{h}\psi_-)} \partial_{z} (e^{h}\xi_-) e^{-2h} $$ on the natural maximal domains $$ \cD(\cp^h_\pm) = \Big\{ \psi_\pm \in L^2(\bR^2)\; : \; \cp^h_\pm(\psi_\pm, \psi_\pm)<\infty \Big\}\; , $$ $$ \cD(\cp^h) =\cD(\cp^h_+)\otimes\cD(\cp^h_-)= \Big\{ \psi = \pmatrix{\psi_+\cr \psi_-} \in L^2(\bR^2, \bC^2)\; : \; \cp^h(\psi, \psi)<\infty \Big\} $$ We use $\| \cdot \|$ to denote the usual $L^2(\bR^2, \rd x)$ or $L^2(\bR^2, \bC^2,\rd x)$ norms. We define the following norms on functions $$ \triple f \triple_{h,+} := \Big[ \| f \|^2 + \| \partial_{\bar z} (e^{-h}f) e^{h}\|^2 \Big]^{1/2}, \qquad \triple f \triple_{h,-}:= \Big[ \| f \|^2 + \| \partial_{z} (e^{h}f) e^{-h}\|^2\Big]^{1/2} $$ and for a spinor $\psi$ we let \be \triple \psi \triple_h: = \triple \psi_+ \triple_{h,+} + \triple \psi_- \triple_{h,-} \; . \label{triplenorm}\ee For any real function $h \in L^1_{loc}$ with $\Delta h \in \overline{\cM}$, we define the set \be \cC_h: = \Big\{ \psi = \pmatrix{g_+ e^{h}\cr g_- e^{-h}} \; : \; g_\pm \in C_0^\infty(\bR^2)\Big\} \; . \label{ccdef}\ee Notice that this set depends only on $\mu =\Delta h$: if $h, h'$ are two functions such that $\Delta h = \Delta h' = \mu$ in distributional sense, then $h-h'$ is harmonic, i.e., smooth. Therefore $e^h$ and $e^{h'}$ differ by a smooth multiplicative factor, i.e. $\cC_h =\cC_{h'}$, hence we can denote this set by $\cC_\mu$. Moreover, by Theorem \ref{thm:hdef}, for any $\mu\in \overline{\cM^*}$ and any compact set $K$, there exists an $h \in L^1_{loc}$ with $\Delta h =\mu$ on $K$, and $h$ is unique up to a smooth additive factor. Since the support of $g_\pm$ is compact, the following set is well-defined for all $\mu \in \overline{\cM^*}$ \be \cC_\mu : = \Big\{ \psi = \pmatrix{g_+ e^{h}\cr g_- e^{-h}} \; : \; g_\pm \in C_0^\infty(\bR^2), \; \Delta h = \mu \quad \mbox{on} \; \mbox{supp}(g_-)\cup\mbox{supp}(g_+) \Big\} \; . \label{cmudef}\ee \medskip \begin{theorem}\label{Hdef} Let $h\in L^1_{loc}(\bR^2)$ be a real valued function such that $\mu:=\Delta h \in \overline{\cM^*}$. Then (i) The quadratic form $\cp^h$ is nonnegative, symmetric and closed, hence it defines a unique selfadjoint operator $H_h$ $$ (H_h\psi, \xi) := \cp^h(\psi, \xi)\; , \qquad \psi\in \cD(H_h), \; \xi\in \cD(\cp^h) $$ with domain $$ \cD(H_h) : = \{ \psi\in \cD(\cp^h)\; : \; \cp^h(\psi, \cdot)\in L^2(\bR^2, \bC^2)'\} $$ (ii) The set $\cC_\mu$ is dense in $\cD(\cp^h)$ with respect to $\triple \cdot \triple_h$, i.e., it is a form core of $H_h$. (iii) For any $L^1_{loc}$-functions $h$ and $h'$ with $\Delta h = \Delta h' \in \overline{\cM^*}$, the operators $H_h$ and $H_{h'}$ are unitarily equivalent by a $U(1)$-gauge transformation. In particular, the spectral properties of $H_h$ depend only on $\mu=\Delta h$. \end{theorem} \begin{definition}\label{deff} For any real function $h\in L^1_{loc}$ with $\mu=\Delta h \in \overline{\cM^*}$ the operator $H_h$ will be called the {\bf Pauli operator with generating potential $h$}. For any $\mu\in \overline{\cM^*}$ the unitarily equivalent operators $\{ H_h\; : \; \Delta h =\mu\}$ are called the {\bf Pauli operators with a magnetic field $\mu$}. The Pauli operators for any $\mu\in \overline{\cM}$ are defined as $U_\mu^*HU_\mu$ on the core $U_\mu^*C_\mu$, where $H$ is a Pauli operator with field $\mu^*\in\overline{\cM^*}$ (see Definition \ref{mustdef}) and $U_\mu$ is defined in (\ref{umudef}). \end{definition} To complete the definition of the Pauli operator for any magnetic field $\mu\in \overline{\cM}$, we need \begin{theorem}\label{bm} For any $\mu\in \overline{\cM^*}$, there exists $h\in L^1_{loc}$ with $\Delta h = \mu$. Hence the above definition of $H_h$ actually defines the Pauli operators for any measure valued magnetic field $\mu\in \overline{\cM}$. \end{theorem} \medskip {\it Proof of Theorem \ref{Hdef}.} From Corollary \ref{w11} we know that $e^{\pm h}\in L^2_{loc}$, and we show below that for any doubled dyadic square $Q_0$ the estimate \be \Big( {1\over |Q|}\int_Q e^{2h}\Big) \Big( {1\over |Q|}\int_Q e^{-2h}\Big) \leq C_3(h,Q_0) \label{A2loc}\ee analogous to (\ref{A2}) is valid on any square $Q\subset Q_0$, with a $(h, Q_0)$-dependent constant. These are the two properties of $h$ which we use below. For any $Q_0$ one can find a compact set $K$ such that $Q_0\subset \mbox{int}(K)$ and $\mu = \Delta h$ restricted to $K$, $\mu|_K$, has finite total variation. Let $\ep =\ep(\mu|_K)/2$ and we consider $h_\ep$ defined in Theorem \ref{thm:hdef}. Since $\Delta h = \Delta h_\ep$ on $K$, we can write $h= h_\ep + \varphi$ with a smooth real function $\varphi$ depending on $h$. In particular, for any doubled dyadic square $Q_0$ the estimate (\ref{A2}) for $h_\ep$ implies that (\ref{A2loc}) is valid for $h= h_\ep + \varphi$ on any square $Q\subset Q_0$. \medskip {\it Part (i).} Let $\psi_n =(\psi_{n+}, \psi_{n-})$ be a Cauchy sequence in the norm $\triple \cdot \triple_h$, i.e., $\psi_n\to \psi$ in $L^2(\rd x)$, $ \partial_{\bar z} (e^{-h}\psi_{n+}) \to u_+$ in $L^2(e^{2h}\rd x)$ and $ \partial_{ z} (e^{h}\psi_{n-}) \to u_-$ in $L^2(e^{-2h}\rd x)$. We have to show that $ \partial_{\bar z} (e^{-h} \psi_+) = u_+$, $ \partial_{z} (e^{h} \psi_-) = u_-$. For any $\phi \in C_0^\infty (\bR^2)$ $$ \int \overline{\phi} u_+ = \lim_{n\to\infty} \int \overline{\phi} \partial_{\bar z} (e^{-h}\psi_{n+}) = - \lim_{n\to\infty} \int \partial_{\bar z}\overline{\phi} \, e^{-h}\psi_{n+} = -\int \partial_{\bar z}\overline{\phi} \, e^{-h}\psi_+ $$ hence $ \partial_{\bar z} (e^{-h}\psi_+) = u_+$ in distributional sense. Here we used that $$ \Big|\int \overline{\phi} \Big( u_+ - \partial_{\bar z} (e^{-h}\psi_{n+})\Big)\Big| \leq \| \overline{\phi} e^{-h}\| \Big\| u_+ - \partial_{\bar z} (e^{-h}\psi_{n+})\Big\|_{L^2(e^{2h})} \to 0 $$ and $$ \Big|\int \partial_{\bar z}\overline{\phi}\, e^{-h}(\psi_+-\psi_{n+}) \Big|\leq \Big\| \partial_{\bar z}\overline{\phi} e^{-h}\Big\| \| \psi_+-\psi_{n+}\|\to 0\; , $$ which follows from $e^{-h} \in L^2_{loc}$. The proof of the spin-down component is similar. This shows that the form $\pi^h$ is closed. The rest of the argument is standard (see, e.g., Lemma 1 in \cite{LS}). \bigskip {\it Part (ii).} The spin-up and spin-down parts can be treated separately and analogously, so we focus only on the spin-up part. \medskip Step 1. We first show that the set $$ \cC_0: = \{ f\in \cD(\pi^h_+), \;\mbox{supp}(f) \; \mbox{compact}\} $$ is dense in $\cD(\pi^h_+)$ with respect to $\triple \cdot \triple_{h,+}$. This is standard: let $\chi(x)$ be a compactly supported smooth cutoff function, $0\leq \chi \leq 1$, $\chi(x) \equiv 1$ for $|x|\leq 1$, and let $\chi_n(x): =\chi(x/n)$. For any $f\in \cD(\pi^h_+)$ we consider $f_n=\chi_n f$, then clearly $\triple f-f_n\triple_{h,+}\to 0$. \bigskip Step 2. We need the following \begin{lemma}\label{nabla} Let $f\in\cC_0$ then $\nabla (fe^{-h})\in L^2(e^{2h})$. \end{lemma} {\it Proof of Lemma \ref{nabla}.} Let $g:=fe^{-h}$. Let $Q_1$ be a doubled dyadic square that contains a neighborhood of $K:= \mbox{supp}\, (g)$, and let $Q_0$ be a doubled dyadic square that strictly contains $Q_1$ and $|Q_0|=4|Q_1|$. We define \be \omega (x): = \left\{ \begin{array}{cr} e^{2h(x)}& \mbox{for}\; x\in Q_1 \cr 1 & \mbox{for} \; x\in Q_1^c \; . \end{array} \right. \label{omgive}\ee \begin{lemma}\label{local} The function $\omega (x)$ satisfies the inequality \be \Big( {1\over |Q|}\int_Q \omega\Big) \Big( {1\over |Q|}\int_Q \omega^{-1}\Big) \leq C_4(h, Q_0) \label{hold}\ee for any square $Q\subset \bR^2$, i.e., $\omega$ is an $A_2$-weight (see \cite{GR, St}). \end{lemma} {\it Proof of Lemma \ref{local}.} It is sufficient to prove (\ref{hold}) for all doubled dyadic squares $Q$. It is easy to see that one of the following cases occurs: (i) $Q$ is disjoint from $Q_1$, (ii) $Q\subset Q_0$, (iii) $|Q_1| \leq 9 |Q|$. In the first case (\ref{hold}) is trivial, in case (ii) it follows from (\ref{A2loc}). Finally, in case (iii) we have $$ \Big( {1\over |Q|}\int_Q \omega\Big) \Big( {1\over |Q|}\int_Q \omega^{-1}\Big) \leq 36^2 \Big( 1 + {1\over |Q_0|}\int_{Q_0} e^{2h}\Big) \Big( 1 + {1\over |Q_0|}\int_{Q_0} e^{-2h}\Big) $$ hence (\ref{hold}) holds with an appropriate constant. $\,\,\Box$. \medskip Since $|\nabla g|^2 = 2(|\partial_z g|^2 + |\partial_{\bar z} g|^2)$ and $\omega = e^{2h}$ on $\mbox{supp}\, (g)$, Lemma \ref{nabla} follows immediately from \be \int_{\bR^2} |\partial_{z}g|^2\omega \leq C_5(h, Q_0) \int_{\bR^2} |\partial_{\bar z}g|^2\omega \, . \label{key1} \ee Notice that $$ \wh{\partial_{ z}g} (\xi) = m(\xi)\wh{\partial_{ \bar z}g} (\xi) \quad \mbox{with} \quad m(\xi):= {(\xi_1 -i\xi_2)^2\over |\xi|^2}\; , $$ where hat stands for Fourier transform, $\xi\in \bR^2$, and $m(\xi)$ is a homogeneous multiplier of degree 0. Hence (\ref{key1}) is just the weighted $L^2$-inequality for the regular singular integral operator $T_m$ with Fourier multiplier $m(\xi)$ and with weight $\omega\in A_2$ \cite{GR, St}. $\,\,\Box$ \bigskip Step 3. To conclude that $\cC_0\cap e^h C_0^\infty$ is dense in $\cC_0$ with respect to $\triple \, \cdot\, \triple_{h,+}$, we use the fact that $C_0^\infty$ is dense in the weighted Sobolev space $W^{1, 2}(\omega)$ with the $A_2$-weight $\omega$ (see e.g. \cite{K}). Here we only recall the key point of the proof. Let $g\in W^{1, 2}(\omega)$ compactly supported and $g_\ep := J_\ep \ast g \in C_0^\infty$ where $J_\ep(x): = \ep^{-2} J(x/\ep)$ is a standard mollifier: $0\leq J \leq 1$, $\int J =1$, $J$ smooth, compactly supported. Then the functions $|\nabla g_\ep| \leq J_\ep \ast |\nabla g|$, have an $L^2$-integrable majorant by the weighted maximal inequality \cite{St} applied to $|\nabla g| \in L^2(\omega)$, hence $g_\ep\to g$ in $W^{1, 2}(\omega)$ as $\ep\to0$. Notice that every $g_\ep$ is supported on a common compact neighborhood of the support of $g$. \bigskip {\it Part (iii).} Since $\Delta h = \Delta h'$, we can write $h' = h +\varphi$ with a smooth real function $\varphi$. We define $\lambda$ as the harmonic conjugate of $\varphi$, $\nabla \lambda = \nabla^\perp\varphi$, which exists and is smooth by $\Delta \varphi =0$. By $\partial_{\bar z} (\varphi + i\lambda)=0$ we have $$ \pi^h(\psi, \psi) = \pi^{h'}\Big( e^{-i\lambda}\psi, e^{-i\lambda}\psi\Big) \; ,\qquad \psi \in \cC_\mu \; , $$ and then by the density of $\cC_\mu$ we obtain the same relation for all $\psi\in \cD(\pi^h)$. $\,\,\,\Box$ {\it Proof of Theorem \ref{bm}.} Since $|\mu|$ is finite on every bounded set, we can find a sequence of disjoint rings, $R_j := \{ x\; : \; r_j \leq |x| \leq r_j+2\delta_j\}$, $j=1, 2,\ldots$, with appropriate widths $2\delta_j>0$ and radii $r_j\to\infty$ (as $j\to\infty$), such that $\sum_j |\mu|(R_j) < \infty$. For $j=0$ we set $r_j=\delta_j=0$. Let $0\leq \chi_j\leq 1$ ($j=0,1,\ldots$) be smooth functions such that $\chi_j(x)\equiv 1$ for $r_j +2\delta_j \leq |x|\leq r_{j+1}$ and $\chi_j(x) \equiv 0$ for $|x|\leq r_j +\delta_j$ or $|x|\ge r_{j+1}+\delta_{j+1}$. Notice that the supports of $\chi_j$ are disjoint. We define $\mu_j:=\mu \cdot {\bf 1}\{ x\; : r_j+2\delta_j \leq |x|\leq r_{j+1}\}$, By Theorem \ref{thm:hdef} there exist $h_j \in \bigcap_{p<2}W^{1,p}_{loc}$, $e^{\pm h_j}\in L^2_{loc}$ with $\Delta h_j = \mu_j$. We notice that $$ \Delta \Big(\sum_j \chi_j h_j) = \nu+\sum_j \chi_j \mu_j $$ where $\nu$ is absolutely continuous, $\nu = N(x)\rd x$ with $$ N = \sum_j \Big[ 2\nabla \chi_j\cdot \nabla h_j + h_j \Delta \chi_j\Big] \in L^1_{loc} $$ We can find a decomposition $N= N_1 + N_2$, $N_1\in L^\infty_{loc}$ and $N_2\in L^1(\bR^2)$. Let $\kappa : = \mu - \sum_j \chi_j \mu_j - N_2(x) \rd x$, then $\kappa\in \cM$ since $N_2\in L^1(\bR^2)$ and the measure $\mu - \sum_j \chi_j \mu_j$ belongs to $\cM$ since it vanishes on the complement of $\bigcup_j R_j$, it has a total variation smaller than $|\mu|$ on each $R_j$ and $\sum_j |\mu|(R_j)<\infty$. It is also clear that $\kappa$ does not charge more to any point than $\mu$ does since $0\leq\chi_j\leq 1$ and they have disjoint supports, hence $\kappa\in \cM^*$. By Theorem \ref{thm:hdef} there is $k \in \bigcap_{p<2}W^{1,p}_{loc}$, $e^{\pm k}\in L^2_{loc}$ such that $\Delta k =\kappa$. We define $h^*: = k + \sum_j \chi_j h_j$, clearly $h^* \in \bigcap_{p<2}W^{1,p}_{loc}$ and $\mu = \Delta h^* -N_1(x)\rd x$. %Since $\mu\in \overline{\cM^*}$ and $N_1\in L^1_{loc}$, %we have $\Delta h \in \overline{\cM^*}$, hence % $e^{\pm h^*}\in L^2_{loc}$ by Corollary \ref{w11}. By the Poincar\'e formula there exists $\bA\in L^\infty_{loc}$ with $\nabla^\perp\!\cdot \! \bA = - N_1$. For any fixed $p<\infty$, one can find $\wt\bA\in L^p_{loc}$ with $\nabla^\perp\!\cdot \! \wt\bA = - N_1$, $\nabla\cdot \wt\bA =0$ by Lemma 1.1. (ii) \cite{L}. But then $\wt\bA^\perp = (-\wt A_2, \wt A_1)$ is curl-free, hence $\wt\bA^\perp = \nabla \wt h$ for some $\wt h \in W^{1,p}_{loc}$ by Lemma 1.1. (i) \cite{L}. Then $\Delta \wt h = N_1$, hence $h:= h^* - \wt h \in L^1_{loc}$ satisfies $\Delta h =\mu$. $\;\;\Box$ \medskip Finally, we have to verify that the Pauli operator $H_h$ defined in this Section coincides with the standard Pauli operator if $\bA\in L^2_{loc}$, modulo a gauge transformation. \begin{proposition}\label{coinc} Let $\bA\in L^2_{loc}$ and let $P_\bA$ be the operator defined in Section \ref{stand}. We assume that $\nabla^\perp\!\cdot\! \bA$ (in distributional sense) is a measure and that $\mu:= \nabla^\perp\!\cdot\! \bA\in \overline{\cM}$. Then $\mu\in \overline{\cM^*}$, in fact $\mu$ has no discrete component. Moreover, if $\Delta h =\mu$ with some $h\in L^1_{loc}$, then the operator $H_h$ defined in Theorem \ref{Hdef} is unitarily equivalent to $P_\bA$. \end{proposition} {\it Remark:} $\bA\in L^2_{loc}$ does not imply that $\nabla^\perp\!\cdot\! \bA$ is even locally a measure of finite variation. One example is the radial gauge $\bA (x): = \Phi(|x|)|x|^{-2} x^\perp$, $x^\perp:=(-x_2, x_1)$, that generates the radial field $$ B(x): = \sum_{n=1}^\infty {(-4)^{n}\over n}\cdot {\bf 1}( 2^{-n} \leq |x| < 2^{-n+1}). $$ with flux $\Phi(r): = \int_{|x|\leq r} B(x) \rd x$. One can easily check that $\int_{|x|\leq 1} |B(x)| \rd x = \infty$ but $\bA\in L^2_{loc}$. However, if $\nabla^\perp\!\cdot\! \bA\ge 0$ as a distribution, then it is a (positive) Borel measure $\mu \in \overline{\cM^*}$ (see \cite{LL}). \bigskip {\it Proof.} First we show that $\mu = \nabla^\perp\!\cdot\! \bA$ has no discrete component. Suppose, on the contrary, that $\mu(\{ x\})\neq 0$ for some $x$, and we can assume $x=0$, $\mu(\{ 0 \})>0$. Let $\chi$ be a radially symmetric smooth function on $\bR^2$, $0\leq \chi \leq 1$, $\mbox{supp}\chi \subset \{ |x|\leq 2\}$, $\chi (x)\equiv 1$ for $|x|\leq 1$, $|\nabla \chi|\leq 2$, and let $\chi_n(x) : = \chi(2^n x)$. Clearly $-\int \bA\cdot\nabla^\perp \chi_n =\int \chi_n \rd \mu \to \mu(\{ 0\})$ as $n\to\infty$. Using polar coordinates, we have, for large enough $n$, $$ {1\over 2}\mu(\{ 0 \}) \leq - \int \bA\cdot \nabla^\perp \chi_n \leq 4\int_{2^{-n}}^{2^{-n+1}} \int_0^{2\pi}|\bA(s, \theta)| \rd \theta\, \rd s $$ $$ \leq 4\sqrt{2\pi} \Bigg(\int_{2^{-n}}^{2^{-n+1}} \int_0^{2\pi}|\bA(s, \theta)|^2 \rd \theta\, s \, \rd s\Bigg)^{1/2} \leq 4\sqrt{2\pi} \Bigg(\int |\bA(x)|^2 \cdot {\bf 1}(2^{-n}\leq |x|\leq 2^{-n+1}) \rd x\Bigg)^{1/2} $$ hence $\int_{|x|\leq 1} |\bA|^2=\infty$. The proof also works if we assume only $\mu \in \overline{\cM}$ instead of $\mu \in {\cM}$. \medskip Now we prove the unitarity. Without loss of generality we can assume that $\nabla\cdot \bA =0$ by part (ii) Lemma 1.1. of \cite{L}. Let $\bA_h: = \nabla^\perp h$, then $\nabla^\perp\!\cdot\! \bA_h = \mu$, $\nabla\cdot \bA_h =0$ and $\bA_h\in L^1_{loc}$ by Corollary \ref{w11}. Since $\nabla^\perp\!\cdot\! (\bA- \bA_h) = 0$, there exists $\lambda \in W^{1,1}_{loc}$ such that $\bA = \bA_h + \nabla \lambda$ by part (i) Lemma 1.1. of \cite{L}. Taking the divergence, we see that $\Delta \lambda =0$. Let $\varphi$ be a smooth harmonic conjugate of $\lambda$, $\nabla\lambda = \nabla^\perp \varphi$. A simple calculation shows that $$ \int |\bsigma\cdot (-i\nabla +\bA)\psi|^2 = \pi^{h+\varphi}(\psi, \psi) =\pi^h \Big( e^{i\lambda}\psi, e^{i\lambda}\psi\Big) $$ for all $\psi\in \cD(p_\bA) = \cD(\pi^{h+\varphi})$, i.e., the quadratic forms $p_\bA$ and $\pi^h$ are unitarily equivalent. $\;\;\Box$ \subsection{Pauli operator generated by both potentials}\label{sec:inf} Theorem \ref{bm} showed that every measure $\mu\in \overline{\cM^*}$ can be generated by an $h$-potential, $\Delta h =\mu$, and we defined the Pauli operators. However, it may be useful to combine the scalar potential with the usual vector potential $\bA\in L^2_{loc}$ to generate the given magnetic field. In this way one has more freedom in choosing the potentials. Typically, the singularities can be easier handled by the $h$-potential, and the standard $h=\sfrac{1}{2\pi} \log |\, \cdot \, |\ast\mu$ formula is (locally) available. But this formula exhibits a strong non-locality of $h$, and the truncation method of the proof of Theorem \ref{bm} is not particularly convenient in practice. Large distance behavior of the bulk magnetic field is better described by a vector potential. In this section we give such a unified definition of the Pauli operator. \bigskip For any $h\in L^1_{loc}$, $\bA\in L^2_{loc}$ we define the quadratic form $$ \pi^{h,\bA}(\psi, \psi) : = \int \Big| (-2i\partial_{\bar z} + a)(e^{-h}\psi_+)\Big|^2 e^{2h} + \int \Big| (-2i\partial_{z} + \bar{a})(e^{h}\psi_+)\Big|^2e^{-2h} $$ on the maximal domain $$ \cD(\pi^{h, \bA}) : = \Big\{ \psi \in L^2(\bR^2, \bC^2) \; : \; \triple \psi \triple_{h,\bA} <\infty\Big\}\; , $$ where $a= A_1+ iA_2$ and $$ \triple \psi \triple_{h,\bA}: = \Big[ \| \psi \|^2 + \pi^{h,\bA}(\psi, \psi)\Big]^{1/2} \; . $$ Let $$ \cP^*: = \Big\{ (h, \bA) \; : \; h\in L^1_{loc}, \; e^{\pm h}\bA \in L^2_{loc}, \; \Delta h \in \overline{\cM^*}, \nabla^\perp\!\cdot\! \bA \in \overline{\cM}\Big\} $$ be the set of admissible potential pairs. The measure $\mu : = \Delta h + \nabla^\perp\!\cdot\! \bA \in \overline{\cM}$ is called the magnetic field generated by $ (h, \bA)$. We recall from Corollary \ref{w11} that $(h, \bA)\in\cP^*$ implies $h\in \bigcap_{p<2} W^{1,p}_{loc}$ and $e^{\pm h} \in L^{2}_{loc}$, moreover, $e^{\pm h} \bA\in L^2_{loc}$ implies $\bA\in L^2_{loc}$. Since $\nabla^\perp\!\cdot\! \bA $ has no discrete component (Proposition \ref{coinc}), the measure $\mu$ generated by $ (h, \bA)\in \cP^*$ is in $\overline{\cM^*}$. In particular, the set of measures generated by a potential pair from $\cP^*$ is the same as the set of measures generated by only $L^1_{loc}$ $h$-potentials (Theorem \ref{bm}). \medskip \begin{theorem}\label{thm:inf} (i) (Self-adjointness). Assume that $ (h, \bA)\in \cP^*$ and let $\mu: =\Delta h + \nabla^\perp\cdot \bA$. Then $\pi^{\bA, h}$ is a nonnegative symmetric closed form, hence it defines a unique self-adjoint operator $H_{h,\bA}$. (ii) (Core). The set $\cC_{\mu}$ (see (\ref{cmudef})) is dense in $\cD(\pi^{h,\bA})$ with respect to $ \triple \, \cdot \, \triple_{h,\bA}$, i.e., it is a form core for $H_{h,\bA}$. (iii) (Consistency). If $ (h, \bA)\in \cP^*$ and $\wt h\in L^1_{loc}$ such that $\Delta h + \nabla^\perp\!\cdot\!\bA = \Delta \wt h$, then $H_{h,\bA}$ is unitary equivalent to $H_{\wt h}$ defined in Theorem \ref{Hdef}. \end{theorem} \begin{definition} For any $(h,\bA)\in \cP^*$ the operator $H_{h,\bA}$ is called the {\bf Pauli operator with a potential pair $(h, \bA)$}. \end{definition} Notice that Proposition \ref{coinc} and (iii) of Theorem \ref{thm:inf} guarantees that the Pauli operators with the same magnetic field are unitarily equivalent, irrespectively which definition we use. {\it Proof of Theorem \ref{thm:inf}.} {\it Part (i).} The proof that $\pi^{h, \bA}$ is closed is very similar to the proof of part (i) of Theorem \ref{Hdef}. The operators $\partial_{\bar z}$ and $\partial_{z}$ should be replaced by $\partial_{\bar z}+ia$ and $\partial_{z}-i\overline{a}$, but the extra terms with $a$ can always be estimated by the local $L^2$ norm of $e^{\pm h}\bA$. {\it Part (ii).} Step 1. We need the following preliminary observation. Since $\bA\in L^2_{loc}$, we can consider the decomposition $\bA= \wt\bA + \nabla\wt\lambda$, $\nabla\cdot\wt\bA=0$, $\wt\bA\in L^2_{loc}$, $\wt\lambda\in W^{1,2}_{loc}$ (see Lemma 1.1 \cite{L}). $\wt\bA$ and $\wt\lambda$ are called the divergence-free and the gradient component of $\bA\in L^2_{loc}$, and notice that $\wt \bA$ is unique up to a smooth gradient, since if $\wt \bA + \nabla\wt\lambda =\wt \bA' + \nabla\wt\lambda'$ then $0= \nabla\cdot(\bA-\bA')= \Delta(\wt\lambda' - \wt\lambda)$, i.e., $\wt\lambda' - \wt\lambda$ is smooth. Moreover, if $\bA e^{\pm h}\in L^2_{loc}$, then $\wt\bA e^{\pm h}\in L^2_{loc}$ as well. To see this, we fix a compact set $K$ and a compact set $K^*$ whose interior contains $K$, then we choose a cutoff function $0\leq\varphi_K\leq 1$ with $\varphi_K\equiv 1$ on $K$ and $\mbox{supp}\, \varphi_K \subset K^*$. We let $\bA_K: = \varphi_K\bA \in L^2$ and let $\lambda$ be defined via its Fourier transform $$ \wh \lambda (\xi): = {\xi \cdot \wh\bA_K(\xi)\over |\xi|^2}\; , \qquad\xi\in \bR^2 $$ i.e., $-\Delta \lambda = \nabla\cdot \bA$. Then $\nabla \lambda$ is obtained from $\bA_K$ by the action of a singular integral operator whose multiplier is $\xi\otimes \xi/|\xi|^2$. Choose $\omega$ as in (\ref{omgive}), where $Q_0$ is a dyadic square containing $K^*$, then $\omega\in A_2$. Hence, by the weighted $L^2$-inequality we have $$ \int |\nabla \lambda|^2 \omega \leq C(\omega) \int |\bA_K|^2 \omega = C(\omega)\int |\bA_K|^2 e^{2h} $$ with some $\omega$-dependent constant. In particular $\nabla\lambda \in L^2_{loc}(e^{ 2h})$. The proof of $\nabla\lambda \in L^2_{loc}(e^{ -2h})$ is identical. Now $\wt\lambda$ satisfies $\Delta \wt \lambda = -\Delta \lambda$ on $K$, i.e. $\wt\lambda$ and $\lambda$ differ by an additive smooth function, hence $\nabla\wt\lambda \in L^2_{loc}(e^{\pm 2h})$, which means that $\wt\bA e^{\pm h}\in L^2_{loc}$. \medskip Step 2. We show that $\cC_\mu \subset\cD(\pi^{h, \bA})$ if $(h, \bA)\in \cP^*$, $\mu=\Delta h +\nabla^\perp\cdot \bA$. Let $\psi \in \cC_\mu$ be compactly supported on $K$ and let $K^*$ be a compact set whose interior contains $K$. Since $\mu$ restricted to $K^*$ has finite total variation, we can apply Theorem \ref{thm:hdef} for the restricted measure to construct a function $h^*\in L^1_{loc}$ such that $\Delta h^* = \mu = \Delta h + \nabla^\perp\cdot\wt\bA$ on $K^*$. Then there is a real function $\chi$ such that $\wt\bA = \nabla^\perp (h^*-h) + \nabla\chi$ (Lemma 1.1. of \cite{L}). After taking the divergence, we see that $\chi$ is harmonic on $K^*$. Let $\varphi\in C^\infty$ be its harmonic conjugate, $\nabla\chi=\nabla^\perp\varphi$. We have the identity \be \pi^{h, \bA}(e^{-i\wt\lambda}\psi, e^{-i\wt\lambda} \psi) = \pi^{h, \wt\bA}(\psi, \psi) =\pi^{h^*+\varphi}\Big( \psi, \psi\Big) \label{gaid1}\ee for any $\psi$ supported on $K^*$. Since $\psi \in \cC_\mu$, we can write $\psi_\pm = g_\pm e^{\pm h^*}$ and we see that the right hand side of (\ref{gaid1}) is finite, hence $e^{-i\wt\lambda}\psi \in \cD(\pi^{h, \bA})$. But by Schwarz inequality $$ \pi^{h, \bA}(e^{-i\wt\lambda}\psi, e^{-i\wt\lambda} \psi) \ge {1\over 2} \pi^{h, \bA}(\psi, \psi) - 4 \sum_\pm \|g_\pm\|_\infty^2\int_{K^*} |\nabla\wt\lambda|^2 e^{\pm 2h} $$ hence $\psi \in \cD(\pi^{h, \bA})$ by Step 1. \medskip Step 3. We now show that $\cC_\mu$ is dense in $\cD(\pi^{h, \bA})$ with respect to $\triple \, \cdot \,\triple_{h, \bA}$ if $(h, \bA)\in \cP^*$, $\mu=\Delta h +\nabla^\perp\cdot \bA$. We first notice that it is sufficient to show that $\cC_{\mu}$ is dense in the set $$ \overline{\cC}_0:=\{ \psi \in \cD(\pi^{h, \bA}), \; : \;\mbox{supp}(\psi) \;\mbox{compact}\} $$ similarly to Step 1 of the proof of Theorem \ref{Hdef} (ii). So let $\psi\in \cD(\pi^{h, \bA})$ be supported on a compact set $K$. As in Step 2, we let $h^*\in L^1_{loc}$ be a function such that $\Delta h^* = \mu$ on a compact neighborhood $K^*$ of $K$, $K\subset \mbox{int}(K^*)$. As before, we have $\wt\bA = \nabla^\perp (h^*-h) + \nabla\chi$ with a harmonic $\chi$ and let $\varphi\in C^\infty(K^*)$ be its harmonic conjugate, $\nabla\chi=\nabla^\perp\varphi$. The identity (\ref{gaid1}) is now written as \be \pi^{h, \bA}(\psi, \psi) =\pi^{h^*+\varphi}\Big( e^{i\wt\lambda}\psi, e^{i\wt\lambda}\psi\Big) \label{gaid}\ee for any $\psi$ supported on $K^*$. In particular, $\psi\in \cD(\pi^{h, \bA})$ implies $e^{i\wt\lambda}\psi\in\cD(\pi^{h^* +\varphi})$. We define the set $$ \wt\cC_\mu: = \Bigg\{ \psi =\pmatrix{g_+ e^h\cr g_-e^{-h}} \; : \; g_\pm \in L^{\infty}_0, \; \Delta h = \mu \quad \mbox{on}\; \mbox{supp} (g_-) \cup \mbox{supp} (g_+) \Bigg\} $$ where $L_0^\infty$ denotes the set of bounded, compactly supported functions. The set $\wt\cC_\mu$ is well defined, see the remark before the definition (\ref{cmudef}). Since $\cC_\mu$ is dense in $\cD(\pi^{h^*+\varphi})$ with respect to $\triple\, \cdot \, \triple_{h^*+\varphi}$ by part (ii) of Theorem \ref{Hdef}, we can find a sequence of spinors $\xi_n \in \cC_{\mu}$ such that $\triple \xi_n - e^{i\wt\lambda}\psi \triple_{h^*+\varphi}\to0$. We can assume that all $\xi_n$ are supported in $K^*$ (see remark at the end of Step 3 of the proof Theorem \ref{thm:hdef} (ii)). But then $\triple e^{-i\wt\lambda} \xi_n - \psi \triple_{h,\bA}\to0$ again by (\ref{gaid}), in particular the set $\overline{\cC}_1 := \cD(\pi^{h, \bA})\bigcap \wt\cC_\mu$ is dense in $\Big( \cD(\pi^{h, \bA}), \triple\, \cdot \, \triple_{h, \bA}\Big)$. Finally, we show that $\cC_\mu$ is dense in $\Big( \overline{\cC}_1, \triple\, \cdot \, \triple_{h, \bA}\Big)$. Let $\chi \in \overline{\cC}_1$, i.e., $\chi_\pm = g_\pm e^{\pm h}$ with some compactly supported bounded functions $g_\pm$. Notice that if $g$ is a bounded function, then $ge^{\pm h}\bA \in L^2_{loc}$ since $e^{\pm h}\bA\in L^2_{loc}$. In particular $(\partial_{\bar z} + ia)g_+ \in L^2(e^{2h})$ implies $\partial_{\bar z}g_+ \in L^2(e^{2h})$. But then $\nabla g_+ \in L^2(e^{2h})$ by Lemma \ref{nabla} and similarly for $g_-$. We focus only on the spin-up part, the spin-down part is similar. Let $g^{(\ep)} : = J_\ep\ast g_+$, where $J_\ep$ is a standard mollifier (see Step 3. of the proof of Theorem \ref{Hdef} (ii)). Recall that $\|g^{(\ep)} \|_\infty \leq \|g_+\|_\infty$ and the functions $|\nabla g^{(\ep)}|\leq J_\ep\ast |\nabla g_+|$ have an $L^2(e^{2h})$-integrable majorant using the weighted maximal inequality. By passing to a subsequence $g^{(\ep)}\to g_+$, $\nabla g^{(\ep)} \to\nabla g_+$ in $L^2(e^{2h})$ and $g^{(\ep)}\to g_+$ a.e. as $\ep\to0$. Therefore $$ \int \Big|(\partial_{\bar z}+ia) (g^{(\ep)} - g_+)\Big|^2 e^{2h} + \Big\| (g^{(\ep)} - g_+)e^{h}\Big\|^2 $$ $$ \leq 2\int \Big|\nabla (g^{(\ep)} - g_+)\Big|^2 e^{2h} + 2\int |\bA|^2 |g^{(\ep)} - g_+|^2 e^{2h} + \Big\| (g^{(\ep)} - g_+)e^{h}\Big\|^2 \to 0 $$ as $\ep\to0$ since $e^{ h}\in L^2_{loc}$ and $e^{h}\bA\in L^2_{loc}$. \medskip {\it Part (iii).} Since $\Delta \wt h \in \overline{\cM^*}$, we know that $\wt h\in L^2_{loc}$ (Corollary \ref{w11}). Since $\nabla^\perp\cdot (\nabla^\perp h - \nabla^\perp\wt h + \bA)=0$, there exists $\lambda\in W^{1,2}_{loc}$ with $\nabla^\perp(h-\wt h) + \bA = \nabla\lambda$. A simple calculation shows that $$ \pi^{h, \bA}(\psi, \psi) = \pi^{\wt h}( e^{i\lambda}\psi, e^{i\lambda}\psi) \; . \qquad \qquad \Box $$ \section{Aharonov-Casher theorem} We prove the following extension of the Aharonov-Casher theorem: \begin{theorem}\label{ACprec} Let $\mu\in \overline{\cM}$ and we assume that $\mu^*\in \cM^*$. Let $\Phi: = \sfrac{1}{2\pi} \int \mu^*$. The dimension of the kernel of any Pauli operator $H$ with magnetic field $\mu$ is given $$ \mbox{dim} \mbox{Ker} \, H = \left\{ \begin{array}{ccl} [ |\Phi| ] & \mbox{if} & \Phi\not\in\bZ \;\; \mbox{or} \;\; \Phi=0\cr [ |\Phi| ] \; \mbox{or} \; [ |\Phi| ]-1 & \mbox{if} & \Phi\in\bZ \setminus\{ 0\} \end{array} \right. $$ (here $[a]$ denotes the integer part of $a$). In case of nonzero integer $\Phi$ both cases can occur, but if, additionally, $\mu^*$ has a compact support or has a definite sign, then always $\mbox{dim} \mbox{Ker} \, H = [ |\Phi| ]-1$. In all cases the kernel is in the eigenspace of $\sigma_3$: $\mbox{Ker}(H)\subset\{ \psi\; : \; \sigma_3 \psi = -s\psi\}$ with $s=\mbox{sign}(\Phi)$. \end{theorem} \bigskip {\it Proof.} We can assume that $\mu\in \cM^*$ and we choose $h:= h_\ep$ with some $\ep < \ep(\mu)$. By (iii) of Theorem (\ref{Hdef}) it is sufficient to consider the operator $H_h$. We can also assume that $\Phi\ge 0$. Suppose first that $\Phi$ is not integer, let $\{ \Phi \} = \Phi - [\Phi]$ be its fractional part. Choose $\ep <\min\Big\{ \ep (\mu), \{ \Phi \}/3, (1-\{ \Phi \})/3 \Big\}$. Any normalized eigenspinor $\psi$ with $\pi^h (\psi, \psi)=0$ must be in the form $\psi = ( e^{h} g_+, e^{-h} g_-)$ where $g_+$ is holomorphic and $g_-$ is antiholomorphic. First we show that $g_+ =0$. Let $u\in\bR^2$ with $|u|\ge R(\ep, \mu)+1$, $$ 1 = \|\psi\|^2 \ge \int_{Q(u)} e^{2h}|g_+|^2 \ge \Big( \int_{Q(u)} e^{h_1}|g_+| \Big)^2 \Big( \int_{Q(u)} e^{-2h_2} \Big)^{-1} $$ then by Theorem \ref{thm:hdef} and subharmonicity of $|g_+|$ we see that $|g_+(u)|\leq C\int_{Q(u)}|g_+| \leq C\langle u \rangle^{-\Phi+2\ep} \to 0$ as $u\to\infty$, hence $g_+=0$. A similar calculation shows that $|g_-(u)| \leq C\langle u \rangle^{\Phi+2\ep}$, i.e., $g_-$ must be a polynomial of degree at most $[\Phi]$ since $\Phi + 2\ep < [\Phi]+1$. However, a polynomial of degree $[\Phi]$ would give $$ \int e^{-2h} |g_-|^2 \ge C\sum_{k\in \Lambda_0, |k|\ge R} |k|^{2[\Phi]} \int_{Q(k)} e^{-2h} \ge C\sum_{k\in \Lambda_0, |k|\ge R} |k|^{2[\Phi]} \Big(\int_{Q(k)} e^{2h}\Big)^{-1} $$ $$ \ge C\sum_{k\in \Lambda_0, |k|\ge R} |k|^{2([\Phi]-\Phi -2\ep)} =\infty $$ for some large enough $R$ and various constants $C$. On the other hand, the functions $g_-(z) = 1, z, \ldots , z^{[\Phi]-1}$ all give normalizable spinors since for these choices \be \int e^{-2h}|g_-|^2 \leq C\sum_{k\in \Lambda_0} |k|^{2([\Phi]-\Phi-1)} \int_{Q(k)} e^{-2h_2} \leq C\sum_{k\in \Lambda_0} |k|^{2([\Phi]-\Phi-1+\ep)} <\infty \; . \label{thres}\ee If $\Phi$ is integer, then the same arguments work except (\ref{thres}); in fact $g_-(z)=z^{\Phi-1}$ may or may not give normalizable solutions. If $\mu$ is compactly supported and $\Phi\ge 0$ is integer then from the definition of $h$ (\ref{hddef})-(\ref{htotdef}) we see that $\Big|h(x) - \Phi \log|x|\Big|$ is bounded for all large enough $|x|$. Similarly one can easily verify that if $\mu\ge 0$, then $h(x) \leq \Phi \log \langle x \rangle + C$ since $\log |x-y|\leq \log \langle x \rangle +\log \langle y \rangle +C$. In both cases $z^{\Phi-1}e^{-h}$ is not $L^2$-normalizable at infinity. However, if $\mu$ can change sign and not compactly supported then there could be $\Phi\in \bZ$ zero energy states. For example the radial field (with $\beta >0$, $N\in {\bf N}$) $$ B(x)= \left\{ \begin{array}{ccc} 2(N+\beta) e^{-2} & \mbox{for} & |x|\leq e \cr -\beta (|x|\log |x|)^{-2} & \mbox{for} & e <|x| \end{array} \right. \; , $$ with $\Phi = \sfrac{1}{2\pi} \int B =N$, is generated by a radial potential $h(x)$ such that $h(x) = N\log |x| + \beta \log\log |x|$ for large $x$ and is regular for small $x$. The threshold state $z^{\Phi-1}e^{-h}$ is normalizable only for $\beta >\sfrac{1}{2}$, so in this case the dimension of the kernel is $\Phi$, otherwise $\Phi -1$. $\,\,\,\,\Box$ \bigskip This theorem requires $\mu\in\cM$. We will see in Section \ref{count} that the Aharonov-Casher theorem need not be true for magnetic fields with infinite total variation. However, the proof above still works for magnetic fields that can be decomposed into the sum of a component in $\cM$ and a component with a regularly behaving generating potential. We just remark one possible extension: \begin{corollary}\label{reg} Suppose that $\mu\in \overline{\cM}$ can be written as $\mu = \mu_{rad} + \wt\mu$ such that $\wt \mu\in \cM$ and $\mu_{rad}$ is a rotationally symmetric Borel measure (i.e., $\mu_{rad} = \mu_{rad}\circ R$ for any rotation $R$ in $\bR^2$ around the origin). We can assume that $\mu_{rad}(\{ 0 \}) =0$ by including the possible delta function at the origin into $\wt\mu$. We assume that that $\Phi_{rad}: =\lim_{R\to\infty} \Phi(R):= \lim_{R\to\infty}\sfrac{1}{2\pi} \int_{|x|\leq R} \mu_{rad}(\rd x)$ exists (possibly infinite) and $h_{rad}(x): = \Phi(|x|) \log |x|$ satisfies $\nabla h_{rad}\in L^\infty_{loc}$. Then all statements of Theorem \ref{ACprec} for the Pauli operator with magnetic field $\mu$ are valid with $\Phi : = \Phi_{rad}+ \sfrac{1}{2\pi}\int \wt\mu^*(\rd x)$. \end{corollary} {\it Proof.} Let $\bA_{rad}:= \nabla^\perp h_{rad} \in L^\infty_{loc}$ and let $\wt h$ be the generating function of $\wt\mu$ given in Theorem \ref{thm:hdef} for some $\ep<\ep(\wt\mu)$. Then $(\wt h, \bA_{rad})\in \cP^*$ with a magnetic field $\mu$, hence the Pauli operators are well defined and unitarily equivalent. Clearly $\pi^{\wt h,\bA_{rad}}(\psi,\psi) = \pi^h(\psi,\psi)$ with $h=h_{rad}+\wt h$, hence any zero energy state $\psi$ must be in the form $\psi = (e^h g_+, e^{-h}g_-)$ where $g_\pm$ are (anti)holomorphic. Now we can follow the proof of Theorem \ref{ACprec}. We use the estimates (\ref{A2}), (\ref{errorh}) for the $\wt h$ part of the generating potential and we estimate $h_{rad}(x): = \Phi(|x|) \log |x|$ by $\lim_{R\to\infty}\Phi(R)$ for large $R$, $\lim_{R\to0}|\Phi(R)|=0$ for small $R$ and by $\Phi \in L^\infty_{loc}(\bR)$ for intermediate $R$. $\;\;\Box$ \section{Counterexample}\label{count} In this section we present the construction of the Counterexample \ref{cont}. For simplicity, the magnetic field will be only bounded and not continuous, but it will be easy to see that a small mollification does not modify the estimates. Let $\delta < 1/10$ be a fixed small number and $N_k = 10k$ for $k=1, 2, \ldots$. We denote the $N_k$-th roots of unity by $\zeta_{k,j}: = \exp(2\pi i j/N_k)$, $j=1, 2, \ldots N_k$. Let $D_{k,n,j}: =\{ x \; : \; |x-n\zeta_{k,j}|\leq\delta\}$ be the disk of radius $\delta$ about $n\zeta_{k,j}$, let $\overline{D}_{k,n,j}: =\{ x \; : \; |x-n\zeta_{k,j}|\leq 2\delta\}$ be the twice bigger disk. Let $0<\ep< 1/4$ be fixed. The magnetic field $B$ is given as $B: = B_0 + \wt B - \wh B$ with $$ B_0(x) : = 2(1+\ep)\delta^{-2} {\bf 1}( |x|\leq \delta) $$ $$ \wt B := \sum_{k=1}^\infty \wt B_k, \qquad \wt B_k: =\sum_{n=4^k+1}^{4^k+2^k}\wt B_{k,n}, \qquad\wt B_{k,n}(x): = \sum_{j=1}^{N_k} 2 \delta^{-2} {\bf 1}(x\in D_{k,n,j}) $$ and $$ \wh B : = \sum_{k=1}^\infty \wh B_k, \qquad \wh B_k: =\sum_{n=4^k+1}^{ 4^k+2^k} \wh B_{k,n}, \qquad \wh B_{k,n}(x): = {1\over 2\pi}\int_0^{2\pi} \wt B_{k,n}(|x|e^{i\theta})\rd \theta $$ The field $\wt B$ consists of uniform field "bumps" with flux $2\pi$ localized on the disks $D_{k,n,j}$ around points $n\zeta_{k,j}$ that are located on concentric circles of radius $n$. The field $\wh B$ is the radial average of $\wt B$. The field $B_k=\wt B_k - \wh B_k$ is called the $k$-th {\it band}. The relation (\ref{impr}) is straightforward by construction. We define the potential function $h : = h_0 + \wt h - \wh h$, with $$ h_0(x):= {1\over 2\pi} \int_{\bR^2} \log|x-y|B_0(y)\rd y $$ $$ \wt h := \sum_{k=1}^\infty \wt h_k, \qquad \wt h_k: =\sum_{n=4^k+1}^{4^k+2^k}\wt h_{k,n}, \qquad\wt h_{k,n}(x): = {1\over 2\pi} \int_{\bR^2} \log|x-y|\wt B_{k,n}(y)\rd y -N_k \log n $$ and $$ \wh h := \sum_{k=1}^\infty \wh h_k, \qquad \wh h_k: =\sum_{n=4^k+1}^{ 4^k+2^k}\wh h_{k,n}, \qquad\wh h_{k,n}(x): = {1\over 2\pi} \int_0^{|x|} {\rd r\over r} \int_{|y|\leq r} \wh B_{k,n}(y)\rd y \; . $$ Clearly $\Delta \wt h_k=\wt B_k$, $\Delta \wh h_k=\wh B_k$. Easy computations yield the following relations: $$ h_0(x) = (1+\ep) \log |x| \qquad \mbox{for} \quad |x|\ge \delta\; , $$ $$ \wt h_{k,n}(x) = \sum_{j=1}^{N_k}\log \Big| \zeta_{k,j}- {x_1+ix_2\over n}\Big| = \log\Big| 1- \Big( {x_1+ix_2\over n}\Big)^{N_k}\Big| \qquad \mbox{for} \quad x \not\in \bigcup_{j=1}^{N_k} D_{k,n,j}\; , $$ $$ \wh h_{k,n}(x) = \left\{ \begin{array}{ccc} 0 & \mbox{for} & |x|\leq n-\delta \cr N_k \Big[ \log {|x|\over n} + O(n^{-1}) \Big] & \mbox{for} & |x|\ge n-\delta \; .\cr \end{array} \right. \; $$ The infinite sums in the definition of $\wt h$ and $\wh h$ are absolutely convergent, hence $h\in L^\infty_{loc}$. The sum of the $\wt h_{k}(x)$'s converges since $$ \sum_{k=1}^\infty \sum_{n=4^k+1}^{4^k+2^k} \Big| {x\over n}\Big|^{N_k} <\infty $$ for each fixed $x$ and $\wh h_k(x)$ is actually zero for all but finite $k$. Therefore we know that $\Delta h = B$ in distributional sense. Moreover, we can rearrnge the sums and write $$ h = h_0 + \sum_{k=1}^\infty h_k , \qquad h_k: = \sum_{n=4^k+1}^{4^k+2^k} h_{k,n} \; , \qquad h_{k,n} := \wt h_{k,n} - \wh h_{k,n} \; . $$ A short calculation shows that for each $k_0$ \be \sum_{k=1\atop k\neq k_0}^\infty |h_k(x)| =O(1) \qquad \mbox{for} \quad 3\cdot 4^{k_0-1}-1 \leq |x|\leq 3\cdot 4^{k_0} +1 \; . \label{out}\ee %$$ % \sum_{k> k_0} % \sum_{n=4^k+1}^{ 4^k+2^k} \Big| {x\over n}\Big|^{N_k} \leq C %$$ %if $|x|\leq 3\cdot 4^{k_0}+1$ and %$$ % \sum_{k m \; . \end{array} \right. \; $$ Writing $x_1+ix_2= m\zeta_{k,\ell}+ (\varrho_1+i\varrho_2)$, $\varrho= (\varrho_1, \varrho_2)\in \bR^2$, $\delta \leq |\varrho|\leq 2\delta$ and expanding $h_{k,n}(x)$ around $m\zeta_{k,\ell}$ up to second order in $\varrho$ we easily obtain that $h_{k,n}(x)\leq N_k O(n^{-1})$ for each $n\neq m$ if $\delta$ is small enough, hence $h_k(x)\leq h_{k,m}(x) + O(1)$. Moreover, $h_{k,m}(x) = \log \Big| 1- [(x_1+ix_2)/m]^{N_k}\Big| + O(1)$ since $|\wh h_{k,m}(x)|=O(1)$ for any $m-2\delta \leq |x|\leq m+2\delta$. Hence $$ \int e^{-2h} \ge C \sum_{k=1}^\infty \sum_{m=4^k+1}^{ 4^k+2^k} {1\over m^{2(1+\ep)}} \sum_{\ell=1}^{N_k} \int_{ \overline{D}_{k,m,\ell}\setminus D_{k,m,\ell}} \Big| 1 - \Big( {x\over m}\Big)^{N_k}\Big|^{-2} \rd x $$ $$ = C \sum_{k=1}^\infty \sum_{m=4^k+1}^{ 4^k+2^k} {N_k \over m^{2(1+\ep)}} \int_{\delta \leq |\varrho|\leq 2\delta} \Big| 1 - \Big( 1 + {\varrho_1+i\varrho_2\over m}\Big)^{N_k}\Big|^{-2} \rd \varrho \ge C \sum_{k=1}^\infty {1\over N_k}\, 2^{k(1-4\ep)} =\infty\; . $$ \medskip Finally, we have to show that $e^{-h}{\bar f} \not\in L^2(\bR^2)$ for any entire function $f$. First we show that $f$ cannot have zeros. Suppose that $a$ is (one of) its zero closest to the origin, i.e., $f(z) = (z-a)^m g(z)$, $g$ is entire, $g(0)\neq 0$, $m\ge 1$. Let $A_k:=\{ x \; : \; 3\cdot 4^k -1\leq |x|\leq 3\cdot 4^k +1\}$, then $h(x)=h_0(x) + O(1)$ for all $x\in A_k$ by (\ref{out}). Hence for a large enough $K$ $$ \int e^{-2h}|f|^2 \ge C\sum_{k=K}^\infty \int_{A_k} e^{-2h_0(x)} |x-a|^{2m} |g(x)|^2 \rd x $$ $$ \ge C\sum_{k=K}^\infty 4^{2k(m-1-\ep)} \int_{A_k}|g(x)|^2 \rd x \ge C|g(0)|^2\sum_{k=K}^\infty 4^{2k(m-1-\ep)}\cdot 4^k =\infty $$ using that $|g|^2$ is subharmonic and the area of $A_k$ is of order $4^k$. Now, since $f$ has no zeros, we can write $f=e^\varphi$ and we would like to show that $\varphi$ is constant. It is enough to show that $R:= \mbox{Re}\, \varphi$ is constant and we can assume $R(a)=0$. Suppose that $\nabla R(a)\neq 0$ for some $a\in \bC$. Let $z_k$ be the point where the maximum of $R$ over the closed disk $D_k:= \{ |x|\leq 3\cdot 4^k\}$ is attained. Since $R$ is harmonic, $|z_k| = 3\cdot 4^k$. Using (\ref{out}) and the subharmonicity of $|e^{2\varphi}|$, we have \be \int e^{-2h}|f|^2 \ge C\sum_{k=1}^\infty 4^{-2k(1+\ep)} \int_{|x-z_k|\leq 1} |e^{2\varphi(x)}|\rd x \ge C\sum_{k=1}^\infty 4^{-2k(1+\ep)} e^{2 R(z_k)}\; . \label{pois}\ee From the Poisson formula we easily obtain $|\nabla R(a)|\leq 4^{-k} \max_{D_k} R = 4^{-k} R(z_k)$ for large enough $k$. Hence $R(z_k) \ge 4^k|\nabla R(a)|$ and the integral in (\ref{pois}) is infinite. $\,\,\,\Box$. \medskip {\it Acknowledgements.} This work started during the first author's visit at the Erwin Schr\"odinger Institute, Vienna. Valuable discussions with T. Hoffmann-Ostenhof and M. Loss are gratefully acknowledged. \thebibliography{ccccccc} \bibitem[A-C]{AC} Aharonov, Y., Casher, A.: Ground state of spin-1/2 charged particle in a two-dimensional magnetic field. Phys. Rev. {\bf A19}, 2461-2462 (1979) \bibitem[CFKS]{CFKS} Cycon, H. L., Froese, R. 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Soc., 1997 \bibitem[Mi]{Mi} Miller, K., {\em Bound states of Quantum Mechanical Particles in Magnetic Fields. \/} Ph.D. Thesis, Princeton University, 1982 \bibitem[Si]{Si} Simon, B.: Maximal and minimal Schr\"odinger forms. J. Operator Theory. {\bf 1}, 37-47 (1979) \bibitem[So]{Sob} Sobolev, A.: On the Lieb-Thirring estimates for the Pauli operator. Duke J. Math. {\bf 82} no. 3, 607--635 (1996) \bibitem[St]{St} Stein, E.: {\em Harmonic Analysis. \/} Princeton University Press, 1993 \bigskip L\'aszl\'o ERD\H OS \vskip-10pt School of Mathematics \vskip-10pt Georgia Institute of Technology \vskip-10pt Atlanta, GA 30332, USA \vskip-10pt E-mail: {\tt lerdos@math.gatech.edu} \bigskip Vitali VOUGALTER \vskip-10pt Department of Mathematics \vskip-10pt University of British Columbia \vskip-10pt Vancouver, B.C. Canada V6T 1Z2 \vskip-10pt E-mail: {\tt vitali@math.ubc.ca} \end{document} ---------------0105051231582--