text:
\section{Introduction}
Graphene is a two-dimensional lattice of carbon atoms arranged on a
honeycomb structure. Due to its unusual properties it has attracted a
great deal of attention since its discovery
\cite{castro2009electronic,novoselov2004electric}.  One of the
striking facts about graphene is that the dynamics of its low-energy
excitations (the charge carriers) can be described by massless
two-dimensional Dirac operators. An interesting feature of Dirac
fermions is the lack of localization under the influence of an
external electric potential \cite{vogelsang,Kalfetal2003}. This fact, related to Klein's paradox \cite{castro2009electronic}, is
due to the peculiar cone-like gapless structure of the spectrum of  massless
free Dirac operators.

It was suggested in \cite{de2007magnetic} that it is possible to
confine such Dirac fermions in graphene by inhomogeneous magnetic
fields of the type $B=B_0+b$, where $B_0>0$ is a constant and $b$ a
perturbation with negative flux  that decays at infinity. The spectrum of the
corresponding Dirac operator in a constant magnetic field $B_0$ is given by
the (relativistic) Landau levels. The idea is that as the perturbation
$b$ is turned on eigenvalues will emerge from the Landau levels giving
rise to states localized on the bulk of the support of $b$. In this
manner a so-called (magnetic) quantum dot or artificial atom can be
created.  These type of models, also with an external electric
potential $V$, have been further studied in the physics literature, for instance,  in
\cite{martino2010spectrum,masir,wang2009magnetically,kormanyos2008bound}
for the one particle case and in \cite{hausler2009artificial,EDSS}
for the multiparticle case.  The articles
\cite{martino2010spectrum,masir,wang2009magnetically,kormanyos2008bound}
deal with specific electromagnetic fields for which the model is partly
solvable or suitable for numerical computations.

In this article we consider a large class of electromagnetic
perturbations $(b,V)$ with $V\in L_{\rm loc}^p(\R^2)$,
$p\in(2,\infty]$, and $b\in L_{\rm loc}^q(\R^2)$, $q\in(1,\infty]$,
both decaying at infinity. The essential spectrum of the corresponding
Dirac-operator $H$ describing the quantum dot is given by the Landau
levels. We show that eigenfunctions belonging to the discrete spectrum
of $H$ are superexponentially localized, i.e., they decay faster than
any exponential.  In the case when $V=0$ we verify the existence of
eigenvalues between the zeroth and the first Landau-level assuming
that $b< 0$. Assuming that a certain analyticity conditions on the
angular variable of $V$ is fulfilled and that $b$ is rotationally
symmetric we prove that those states are actually Gaussian-like
localized. These type of results on superexponential and Gaussian
localization, although new for Dirac operators, are known to hold for
spinless magnetic Schr\"odinger operators
\cite{CorneanNenciu1998,Erdos,Nakamura1996,Sordoni1998}. We benefit from this
insight to prove our statements. A precise description of our results
is given in the next section.
\section{Results}
We consider the massless two-dimensional Dirac operator with an external
magnetic field $B:\R^2\to \R$, pointing
perpendicularly to the plane, and an electric potential  $V:\R^2\to \R$.
We are interested in the  Hamiltonians
\begin{align}
  \label{eq:1}
  D_{\bf A}&:= {\boldsymbol \sigma}\cdot({\bf p}-{\bf A}),\\
\label{leila}
H&:=D_{\bf A}+V\,,
\end{align}
a priori defined on $C_0^\infty(\R^2;\C^2)\subset L^2(\R^2;\C^2)$.
Here ${\bf p}:=\tfrac{1}{\mathrm{i\,}}\nabla$ is the momentum of the  particle and  $ {\boldsymbol \sigma}:= (\sigma_1,
\sigma_2)$ is a  vector whose entries
\begin{equation*}
  \label{eq:2}
 \sigma_1=:
\left(
\begin{array}{cc}
0&1\\
1&0
\end{array}\right),\qquad
\sigma_2=:
\left(
\begin{array}{cc}
0&-\mathrm{i\,}\\
\mathrm{i\,}&0
\end{array}\right)\,,
\end{equation*}
are   Pauli matrices.
The magnetic field $B$ enters in the definitions \eqref{eq:1} and \eqref{leila} by means of the magnetic vector potential ${\bf A}=(A_1,A_2):\R^2\to \R^2$ through the relation
\begin{equation}\label{carla1}
 B=\partial_1 A_2-\partial_2 A_1=:{\rm curl} \,{\bf A}\,,
\end{equation}
which is understood in the sense of distributions.

Throughout this article we assume the following on $(B,V)$:
 \begin{itemize}
\item[(A1)] $B=B_0+b$ where $B_0>0$ is a number and  $b\in L^{q}_{\rm loc}(\R^2;\R)$ for some $q\in(1,\infty]$ and
$\lim_{n\to \infty} \| \mathbbm{1}_{\{|{\bf x}|\ge n\}}b\|_\infty=0.$
\item[(A2)]  $V\in L_{loc}^{p}(\R^2;\R)$
for some $p\in (2, \infty]$ and  $
\lim_{n\to\infty}\|\mathbbm{1}_{\{|{\bf x}|\ge n\}}V\|_\infty=0\,.$
\end{itemize}
Here $\mathbbm{1}_{I}(\cdot)$ denotes the  characteristic function on the set $I$.
Assuming that $B$ fulfills (A1) we can always find ${\bf A}\in L^t_{\rm
  loc}(\R^2;\R^2)$ for some $t\in (2,\infty]$ satisfying \eqref{carla1} (see Remark \ref{poisson}). For such
magnetic vector potentials and electric potentials $V$ satisfying (A2)
we know that the operators defined in \eqref{eq:1} and \eqref{leila}
are essentially self-adjoint (see Subsection \ref{def-of-H}). We denote their self-adjoint extensions by the same symbols and their domains by $\mathcal{D}(D_{{\bf A}})$ and $\mathcal{D}(H)$ respectively.

To the homogeneous magnetic  field $B_0$ we associate  the vector potential
\begin{align}\label{a0}
{\bf A}_0:=\frac{B_0}{2} (-x_2,x_1)\,,
\end{align}
satisfying ${\rm curl} \,{\bf A}_0=B_0$. It is well known that the spectrum of $ D_{{\bf A}_0}$ consists of infinitely degenerated eigenvalues $(l_n)_{n\in\Z}$, called Landau levels, given by
$$l_n:={\mathrm{sgn}(n)}\sqrt{2|n|B_0},\qquad n\in \Z\,,$$
where $\mathrm{sgn}(n)=n/|n|$  if $n\not= 0$ and equals one if $n=0$.

Given a   self-adjoint operator $T$ we write $\sigma_{\rm pp}(T), \sigma_{\rm d}(T)$, and $\sigma_{\rm ess}(T)$ to denote the pure point, discrete, and essential spectra of $T$ respectively.
Our first main result is as follows.
\begin{theorem}\label{Teo1}
Assume that $B$ satisfies (A1) and let ${\bf A}\in L^p_{\rm loc}(\R^2;\R^2),\,p\in(2,\infty],$ with ${\rm curl}\, {\bf A} = B$. Then, the spectrum of $D_{{\bf A}}$ is symmetric with respect to zero and
$$\sigma_{\rm ess}(D_{{\bf A}})= (l_n)_{n\in\Z} \,.$$
Moreover,
\begin{itemize}
\item[(a)] If $b\le 0$ and strictly negative on some open set,  then the discrete spectrum of $D_{{\bf A}}$ on $(0,l_1)$ is non-empty, i.e.,  $\sigma_{\rm d}(D_{{\bf A}})\cap (0,l_1)\not=\emptyset$ and
$${\rm dim(Ran}(\mathbbm{1}_{(0,l_1)}(D_{\mathbf{A}})))=\infty\,.$$
\item[(b)] If $b\ge 0$ then
$${\rm dim(Ran}(\mathbbm{1}_{(0,l_1)}(D_{\mathbf{A}})))=0\,.$$
\end{itemize}
\end{theorem}
This theorem is a consequence of lemmas \ref{Cor} and \ref{lemma}. That the spectrum of $D_{{\bf A}}$ is symmetric with respect to zero is well known; see, however,  Proposition \ref{supersymmetry}.
\begin{remark}
  A similar result to Theorem \ref{Teo1} is shown in \cite{Besch2000}
  when $b$ is replaced by $\lambda b$ and $\lambda$ is assumed to be
  sufficiently large.  Moreover, in \cite{Besch2000} stronger
  regularity assumptions on $b$ are made. In addition, the magnetic
  vector potential ${\bf a}$ associated to $b$ is assumed to decay at
  infinity. However, the results of \cite{Besch2000} hold for more
  general background magnetic fields than $B_0$. We note also that our
  proof differs from the one in \cite{Besch2000}.
\end{remark}
\begin{remark}
Assume that (A1) and (A2) are fulfilled. As a consequence of lemmas \ref{essentialda} and \ref{Cor} below,
\begin{align*}
  \sigma_{\rm ess}(H)= \sigma_{\rm ess}(D_{{\bf A}})=\sigma_{\rm ess}(D_{{\bf A}_0})\,,
\end{align*}
for any ${\bf A}\in L^p_{\rm loc}(\R^2;\R^2),\,p\in(2,\infty],$ with ${\rm curl}\, {\bf A} = B$.
\end{remark}
Our next result state that eigenfunctions corresponding to the discrete spectrum of $H$ are super-exponentially localized.
\begin{theorem}\label{Maintheorem}
Assume that $B$ and $V$ satisfy (A1) and (A2) respectively and let ${\bf A}\in L^p_{\rm loc}(\R^2;\R^2),\,p\in(2,\infty],$ with ${\rm curl}\, {\bf A} = B=B_0+b$. Then, for any eigenfunction $\Psi$ of $H=D_{{\bf A}}+V$ with $H\Psi=E\Psi$ and
$E\in \R\setminus\sigma(D_{{\bf A}_0})$ the following holds:

For every $r\in[2,\infty]$ and $\gamma>0$ there exists an $R>0$ such that
\begin{equation}\label{sexpon}
\|\mathbbm{1}_{\{|{\bf x}|\ge R\}}e^{\gamma |{\bf x}|}\Psi\|_r<\infty\,.
\end{equation}
\end{theorem}
This theorem is proven in Section \ref{superex}.
\begin{remark}
This type of results are known to hold for magnetic Schr\"odinger operators
$({\bf p}-{\bf A})^2+B$. Our proof   follows the ideas presented in \cite{CorneanNenciu1998}.
In fact, since our operator is linear in ${\bf A}$, some parts of the argument are more straightforward. For instance, we do not require that $b\in C^1(\R^2;\R)$  decays in the $C^1$-norm as  done in \cite{CorneanNenciu1998}.
\end{remark}
\begin{remark}
One  essential ingredient in the proof of Theorem \ref{Maintheorem} is
the explicit knowledge of the  Green function $G_0$   of $D_{{\bf A}_0}$. This is calculated in  Appendix~\ref{greenf}.
\end{remark}
In order to obtain  Gaussian decay we make further assumptions on $(B,V)$. Let $T=\R/(2\pi \Z)$ and let $v=v(r,\theta),(r,\theta)\in\R^+\times T$ be the potential $V$ written in
polar coordinates. We assume:
\begin{enumerate}
\item[(A3)] $B$  is radially symmetric, i.e., $b({\bf x})=b(r),\ r=|{\bf x}|$.
\item[(A4)] For any $(r,\theta)\in
  \R^+\times T$ the mapping $\R\ni a \mapsto v(r,\theta+a)=:{v}_a(r,\theta)$ has an analytic continuation $\widetilde{v}_z(r,\theta)$ to $\C$. Moreover, for any $\tau>0$ there exist a $p\in(2,\infty]$ and a real-valued function $u_\tau\in L_{\rm loc}^p(\R^+\times T,rd r\,d \theta)$ such that $\|\mathbbm{1}_{\{r>n\}}u_\tau\|_\infty\to 0$  as $n\to\infty$ and
$$|\widetilde{v}_z(r,\theta)|\le u_\tau (r,\theta)\,,$$
for any  $(r,\theta)\in
  \R^+\times T$ and $z\in S_\tau:= \{z \in \C\,:\, |\operatorname{Im}
  z|\le \tau \}$.
\item[(A5)] $v$ is differentiable with respect to $r$ and  $\R\ni a \mapsto \partial_rv(r,\theta+a)$ can be  analytically  continued to $\partial_r\widetilde{v}_z(r,\theta)$ on $\C$. Moreover, there exist a $\rho>0$ such that for any $\tau>0$ there is  $\kappa_\tau>0$ such that $|\mathbbm{1}_{\{r>\rho\}} \partial_rv_z(r,\theta)|\le \kappa_\tau $ for   any  $(r,\theta)\in
  \R^+\times T$ and $z\in S_\tau$.
\end{enumerate}
\begin{theorem}\label{Maintheorem3}
Assume that $B$ satisfies (A1) and (A3) and $V$  satisfies (A2),(A4) and (A5). Let  ${\bf A}\in L^p_{\rm loc}(\R^2;\R^2),\,p\in(2,\infty],$ with ${\rm curl}\, {\bf A} = B$.  Then, for any eigenfunction $\Psi$ of $H=D_{{\bf A}}+V$ with $H\Psi=E\Psi$ and
$E\in \R\setminus\sigma(D_{{\bf A}_0})$ the following holds:
 For every $0<\alpha<1$,  we have
\begin{equation}
\|e^{\alpha B_0/4|{\bf x}|^2}\Psi\|_2<\infty\,.
\end{equation}
\end{theorem}
This theorem is proven in Section \ref{Gauss}.
\begin{remark}
The analyticity assumption (A4)  on the angular variable of $V$  implies, by a Paley-Wiener argument, exponential decay of the
Fourier modes of the potential in its angular momentum decomposition
 (see equations \eqref{defW}, \eqref{FourierKoeff} and \eqref{decV} below). The assumption (A5) is similar to (A4) but for the radial derivative of the potential.
\end{remark}
\begin{remark}
  The first proof of Gaussian localization for magnetic Schr\"odinger
  operators using assumptions like (A4) (but not (A5)) was given in
  \cite{Erdos}. In addition, an example of a potential decaying at
  infinity for which the corresponding ground state decays slower than
  a Gaussian is also given in \cite{Erdos}.  The proof in \cite{Erdos}
  is based on a generalized Feynman-Kac formula.  An alternative proof
  using Agmon-type estimates with localizations in space and angular
  momentum was given in \cite{Nakamura1996}. A variation of the method
  in \cite{Nakamura1996} was used in \cite{Sordoni1998} to treat the
  general $n$-dimensional case, again for magnetic Schr\"odinger
  operators. Our proof follows the ideas developed in
  \cite{Nakamura1996}. However, it turns out to be more involved since
  our operator is not bounded from below.  To overcome this difficulty
  we square the Dirac operator (or parts of it). This is the reason
  why  (A5) is used in our setting.
\end{remark}
\noindent
{\it The article is organized as follows}: In Section \ref{preliminars}
we review some essentially well known facts about
magnetic Dirac operators. Sections \ref{spectrum}, \ref{superex}, and
\ref{Gauss} are devoted to the proofs of theorems \ref{Teo1},
\ref{Maintheorem}, and \ref{Maintheorem3} respectively. The article
ends with an appendix containing some useful technical results.

\bigskip

\noindent
{\bf Acknowledgements.}
E.S. thanks Horia Cornean for stimulating discussions in the conference `Spectral Days' in Santiago.
Both authors have been partially supported by the DFG (SFB/TR12).

\section{Preliminaries}\label{preliminars}
\subsection{Essential self-adjointness}\label{def-of-H}
Throughout this article we consider magnetic
potentials ${\bf A}\in L^p_{{\rm
   loc}}(\R^2;\R^2)$ and electric potentials $V\in L^q_{{\rm
   loc}}(\R^2),\ p,q\in(2,\infty]$.
In order to show essential self-adjointness of the operators $H$ and $D_{\bf A}$ defined in \eqref{eq:1} and \eqref{leila} it suffices
to prove that
$$ H_R\phi:= {\boldsymbol \sigma}\cdot({\bf p}-\mathbbm{1}_{\{|{\bf x}|\le R\}}{\bf A})\phi+ \mathbbm{1}_{\{|{\bf x}|\le R\}}V\phi\,,\qquad \phi\in C_0^\infty(\R^2;\C^2)\,,$$
is essentially self-adjoint  for every $R>0$ (see \cite{Chernoff77}).
Using that for $f\in L^p(\R^2;\C)$ and $2<p<\infty$
\begin{equation*}\label{Intop}
f({\bf x})({\bf p}^2+1)^{-1/2}
\end{equation*}
is a compact operator  (see \cite[Theorem 4.1]{Simon2005}) we get
that $\mathbbm{1}_{\{|{\bf x}|\le R\}}(V-{\boldsymbol \sigma}\cdot \bf A)$ is a relative compact perturbation of $D_{\bf 0}$.
This shows essential self-adjointness of $H_R$, since $D_{\bf 0}$ is essentially self-adjoint on $C_0^\infty(\R^2;\C^2)$.
\subsection{Gauge invariance}\label{SecGauge}
Let ${\bf A}, \hat{\bf A}\in L^p_{{\rm
   loc}}(\R^2;\R^2),\ 2< p<\infty,$ be two vector potentials  with
$$
{\rm curl}\, {\bf A}={\rm curl}\, \hat{\bf A}
$$
in the sense of distributions. According  to \cite{Leinfelder1983}  there is a gauge function $\hat{\Phi}\in W^{1,p}_{loc}(\R^2;\R)$ such that
$$
{\bf A}=\hat{\bf A}+\nabla \hat{\Phi}\,.
$$
It follows, for any electric potential $V\in L^q_{{\rm
   loc}}(\R^2;\R),\, q\in(2,\infty]$, that
$$ (D_{\bf A}+V) =e^{\mathrm{i\,} \hat{\Phi}} (D_{\hat{\bf A}}+V)e^{-\mathrm{i\,} \hat{\Phi}}\,.$$
In particular, $e^{\mathrm{i\,} \hat{\Phi}} (D_{\hat{\bf A}}+V)e^{-\mathrm{i\,} \hat{\Phi}}$ is essentially self-adjoint on $C_0^\infty(\R^2;\C^2)$.

\noindent
This can be seen as follows:
Note that $\mathcal{D}(e^{\mathrm{i\,} \hat{\Phi}} (D_{\hat{\bf A}}+V)e^{-\mathrm{i\,} \hat{\Phi}})=\{
f\in L^2(\R^2;\C^2)\,:\, e^{-\mathrm{i\,} \hat{\Phi}}f\in \mathcal{D}(D_{\hat{\bf A}}+V)\}$.
Pick functions $\eta,\eta'\in {C}^\infty_0(\R^2;\C^2)$ and a sequence $(\hat{\Phi}_{m})_{m\in\N}$
 in ${C}^\infty(\R^2;\R)$ with $\hat{\Phi}_{m}\to \hat{\Phi}$ in $W^{1,p}_{loc}(\R^2)$ (and hence in $W^{1,2}_{loc}(\R^2)$) as $m\to\infty$. Then,
\begin{align*}
\sps{(D_{\hat{\mathbf{A}}}+V)\eta'}{e^{-\mathrm{i\,} \hat{\Phi}}\eta}
&=\lim_{m\to\infty}\sps{(D_{\hat{\mathbf{A}}}+V)\eta'}{e^{-\mathrm{i\,} \hat{\Phi}_{m}}\eta}\\
&=\lim_{m\to\infty}\sps{e^{\mathrm{i\,} \hat{\Phi}_{m}}\eta'}{(D_{\hat{\mathbf{A}}}+V)\eta}-\lim_{m\to\infty}\sps{e^{\mathrm{i\,} \hat{\Phi}_{m}}\eta'}{{\boldsymbol \sigma}\cdot\nabla\hat{\Phi}_{m}\eta}\\
&=\sps{e^{\mathrm{i\,} \hat{\Phi}}\eta'}{(D_{\hat{\mathbf{A}}}+V)\eta}-\sps{e^{\mathrm{i\,} \hat{\Phi}}\eta'}{{\boldsymbol \sigma}\cdot\nabla\hat{\Phi}\eta}\,.
\end{align*}
Since $\eta'$ is an arbitrary element of a core of $D_{\hat{\mathbf{A}}}+V$, it follows that $e^{-\mathrm{i\,}\hat{\Phi}}\eta\in \mathcal{D}(D_{\hat{\mathbf{A}}}+V)$
and $(D_{\hat{\mathbf{A}}}+V)e^{-\mathrm{i\,}\hat{\Phi}}\eta=e^{-\mathrm{i\,} \hat{\Phi}}(D_{\hat{\mathbf{A}}}+V-{\boldsymbol \sigma}\cdot\nabla\hat{\Phi})\eta$
which implies that
\begin{align*}
e^{\mathrm{i\,} \hat{\Phi}}(D_{\hat{\mathbf{A}}}+V)e^{-\mathrm{i\,}\hat{\Phi}}\eta=(D_{{\mathbf{A}}}+V)\eta\,,\qquad \eta\in {C}^\infty_0(\R^2;\C^2)\,.
\end{align*}
Due to the essential self-adjointness of  $D_{\mathbf{A}}+V$  we deduce that
$e^{\mathrm{i\,}\hat{\Phi}}(D_{\hat{\mathbf{A}}}+V)e^{-\mathrm{i\,}\hat{\Phi}}$ is also essentially self-adjoint on ${C}^\infty_0(\R^2;\C^2)$ and that the two operators coincide.
\subsection{Supersymmetry}\label{supers}
For ${\bf A}=(A_1,A_2)$ with $A_j\in L^p_{{\rm loc}}(\R^2),
\,p\in(2,\infty]$, $j=1,2$, we define the following two operators
\begin{align*}
 d_1\phi= [(p_1-A_1)+\mathrm{i\,} (p_2-A_2)]\phi\,,\qquad \phi \in C_0^\infty(\R^2;\C)\,,\\
d_2\phi=[(p_1-A_1)-\mathrm{i\,} (p_2-A_2)]\phi\,,\qquad \phi \in C_0^\infty(\R^2;\C)\,.
\end{align*}
Clearly, we have that
\begin{align*}
 D_{\bf A}\upharpoonright_{C_0^\infty(\R^2;\C^2)}=
\left(
\begin{array}{cc}
0 & d_2\\
d_1 & 0
\end{array}\right)\,.
\end{align*}
Since $ D_{\bf A}\upharpoonright_{C_0^\infty(\R^2;\C^2)}$ is essentially self-adjoint it follows that $d_1$ and $d_2$ are closable \cite[Section 5.2.2]{Thaller}. In addition,
setting $d:=\overline{d_1}$ one finds that $d^*=\overline{d_2}$ and
\begin{equation}
  \label{eq:4}
  D_{\bf A}=
\left(
\begin{array}{cc}
0 & d^*\\
d & 0
\end{array}\right)\,\quad\mbox{on}\quad \mathcal{D}(D_{\bf A})=
\mathcal{D}(d)\oplus\mathcal{D}(d^*)\,.
\end{equation}
It is known that $dd^*$ and $d^*d$ are self-adjoint with domains $\mathcal{D}(dd^*)=\{ \phi\in \mathcal{D}(d^*)\,:\,d^*\phi\in \mathcal{D}(d)\}$ and $\mathcal{D}(d^*d)=\{ \phi\in \mathcal{D}(d)\,:\,d\phi\in \mathcal{D}(d^*)\}$. Moreover, there is a unitary map $S$ from $\rm Ker(dd^*)^\bot$ to $\rm Ker(d^*d)^\bot$, such that
\begin{equation}\label{Unitary}
 dd^*\upharpoonright_{\rm Ker(dd^*)^\bot} = S^* d^*d\upharpoonright_{\rm Ker(d^*d)^\bot}  S\,.
\end{equation}
Let us note that we can block-diagonalize  $D_{{\bf A}}$ using the
Foldy-Wouthuysen transformation. Setting
\begin{equation*}
  a_+=\left\{
\begin{array}{ll}
1/\sqrt{2} &\,\mbox{on}\,\,\,\mathrm{Ker}(D_{\bf A})^\perp\\
1 &\,\mbox{on}\,\,\,\mathrm{Ker}(D_{\bf A})
\end{array}\right.,\,\,\,\,\,\,
a_-=\left\{
\begin{array}{ll}
1/\sqrt{2} &\,\mbox{on}\,\,\,\mathrm{Ker}(D_{\bf A})^\perp\\
0 &\,\mbox{on}\,\,\,\mathrm{Ker}(D_{\bf A})
\end{array}\right.,
\end{equation*}
we define the  Foldy-Wouthuysen transformation  as
\begin{equation*}
  U=a_++\sigma_3 \mathrm{sgn}(D_{\bf A}) a_-\,,
\end{equation*}
where $\mathrm{sgn}(D_{\bf A})=D_{{\bf A}}/|D_{{\bf A}}|$ on $\mathrm{Ker}(D_{\bf A})^\perp$ and equals zero on $\mathrm{Ker}(D_{\bf A})$ and
\begin{equation*}
 \sigma_3=
\left(
\begin{array}{cc}
1&0\\
0&-1
\end{array}\right)\,.
\end{equation*}
The unitarity of the above transformation can be easily verified observing
that $\varphi\in \mathrm{Ker}(D_{\bf A}) \Leftrightarrow
\sigma_3\varphi\in \mathrm{Ker}(D_{\bf A})$ and that $\sigma_3\mathrm{sgn}(D_{\bf A})=-\mathrm{sgn}(D_{\bf A})\sigma_3$. The latter relation holds since $\sigma_3D_{\bf A}=-D_{\bf A}\sigma_3$ and $\sigma_3|D_{\bf A}|=|D_{\bf A}|\sigma_3$. A direct computation yields
\begin{equation}\label{Isom}
  U D_{\bf A} U^{*}=\left(
\begin{array}{cc}
\sqrt{d^* d} & 0\\
0 & -\sqrt{d d^*}
\end{array}\right)\,.
\end{equation}
Equation \eqref{Unitary} and \eqref{Isom} imply the following statement.
\begin{proposition}\label{supersymmetry}
Let ${\bf A}\in L^p_{{\rm
   loc}}(\R^2;\R^2)$ for some $p\in(2,\infty]$. Then, the spectrum of $D_{{\bf A}}$  is symmetric with respect to zero and
\begin{equation*}\label{www}
\sigma_{\sharp}(D_{{\bf A}})\cap (0,\infty)= \sigma_{\sharp}(\sqrt{d^*d})\setminus \{0\}\,,
\qquad \sharp\in\{\rm pp, d, ess\}\,.
\end{equation*}
\end{proposition}
\section{The spectrum of $D_{{\bf A}}$}\label{spectrum}
The aim of this section is to show Theorem \ref{Teo1}.  An important
ingredient is the study of the essential spectrum of $D_{{\bf A}}$.  In order
to do that we  modify an argument from \cite{Iwatsuka1983}
obtaining Lemma \ref{essentialda} below. We combine this  with a result
from \cite{Rozenblum2006} on the infiniteness of
zero modes for Pauli-operators (see Lemma \ref{Cor} below). The proof of the theorem is then a consequence of lemmas \ref{Cor} and \ref{lemma}.

In the following discussion we assume that $B=B_0+b$ with $B_0>0$ and
 $b\in L^1_{\rm loc}(\R^2;\R)$ such that $|b|^{1/2}$ is relative $\sqrt{{\bf p}^2+1}$- compact. Let ${\bf A}\in L^p_{\rm loc}(\R^2;\R^2),\,p\in (2,\infty],$ with ${\rm curl}\, {\bf A} = B$.
We start by  observing that, for $\phi\in C_0^\infty(\R^2;\C)$,
\begin{equation}\label{w1}
\begin{split}
\sps{d^*\phi}{d^*\phi}&= \sum_{j=1}^2 \| (p_j-A_j)\phi\|^2+ \sps{\phi}{B \phi}\\
\sps{d\phi}{d\phi}&= \sum_{j=1}^2 \| (p_j-A_j)\phi\|^2- \sps{\phi}{B \phi}
\end{split}
\end{equation}
holds. This implies the commutator relation
\begin{align}
  \label{CommutatorB}
\sps{\phi}{[d,d^*]\phi}:=
\sps{d^*\phi}{d^*\phi}-\sps{d\phi}{d\phi}=2\sps{\phi}{ B\phi}\,,\quad\phi\in C_0^\infty(\R^2;\C)\,.
\end{align}
The idea in \cite{Iwatsuka1983} is to use this commutator  to study
the essential spectrum of  $dd^*$ and $d^*d$.
In order to extend this identity we define these operators
as quadratic forms and show that
$\mathcal{Q}(b)\supset \mathcal{Q}(d^*d)=\mathcal{Q}(dd^*)$
and $|b|^{1/2}( d^*d +1)^{-1/2}$ is a compact operator. Here $\mathcal{Q}(\cdot)$ is used to denote the form domain.

Let us  define
\begin{align*}
q_1(\phi,\phi)={q}_1[\phi]:= \|d\phi\|^2,
\qquad q_2(\phi,\phi)={q}_2[\phi]:=\|d^*\phi\|^2,
\end{align*}
with form domains $\mathcal{Q}(q_1)=\mathcal{D}(d)$ and $\mathcal{Q}(q_2)=\mathcal{D}(d^*)$. Since $d$ and $d^*$ are closed (see Subsection \ref{supers}) we have that $q_1$ and $q_2$ are closed and positive. Thus, associated to $q_j$, $j=1,2$, there is a unique self-adjoint operator $T_j$ characterized as follows:
\begin{equation}
  \label{eq:3}
\begin{split}
\sps{\psi}{T_j \varphi}=q_j(\psi,\varphi),\quad \psi\in \mathcal{Q}(q_j), \varphi\in\mathcal{D}(T_j)\,,\\
\mathcal{D}(T_j)=\{\varphi\in Q(q_j)\,|\, \exists \eta\in L^2(\R^2;\C), \forall \psi\in \mathcal{C}, q_j(\psi,\varphi)=\sps{\psi}{\eta}\}\,,
\end{split}
\end{equation}
where $\mathcal{C}$ is any form core of $q_j$. It is easy to check using \eqref{eq:3} that in fact $T_1=d^* d$ and $T_2=d d^*$.
Note that since the restrictions of $d$ and $d^*$ to $C_0^\infty(\R^2,\C)$ are closable $C_0^\infty(\R^2,\C)$ is a form core for $q_1$ and $q_2$.
We define yet another quadratic form. For  $\phi\in C_0^\infty(\R^2,\C)$ we set
\begin{align*}
\tilde{q}_3[\phi]:=\sum_{j=1}^2\|(p_j-A_j)\phi\|^2\,.
\end{align*}
It is known \cite{Simon1979} that $\tilde{q}_3$ is closable
and we denote its closure by $q_3$. Its associated self-adjoint operator $H_S=:({\bf p}-{\bf A})^2$ is the usual magnetic Schr\"odinger operator. Recall that $|b|^{1/2}$ is relative $\sqrt{{\bf p}^2+1}$-compact. Using the
diamagnetic inequality for $|{\bf p} -{\bf A}|$ (see e.g. \cite{Franketal2007}) and arguing as in \cite[Theorem 2.6]{AHS-I} we conclude that $\mathcal{Q}(q_3)=\mathcal{D}(H_S^{1/2})\subset\mathcal{D}(|b|^{1/2})$ and that
$|b|^{1/2}$ is relative $H_S^{1/2}$-compact. Thus, the quadratic form
\begin{align*}
  \beta[\varphi]:=B_0\|\varphi\|^2+
\sps{{\rm sgn}(b)|b|^{1/2}\varphi}{|b|^{1/2}\varphi}
\end{align*}
is in absolute value bounded with respect to $q_3$ with bound $0$. In particular,
\begin{align*}
  q_3^{\pm}[\varphi]:=q_3[\varphi]\pm\beta[\varphi]\,,\quad \varphi\in\mathcal{Q}(q_3)\,,
\end{align*}
 is closed. Observing that by \eqref{w1} we have that $q_2\upharpoonright_{C_0^\infty}=q_3^+\upharpoonright_{C_0^\infty}$
and $q_1\upharpoonright_{C_0^\infty}=q_3^-\upharpoonright_{C_0^\infty}$ and
using that $C_0^\infty(\R^2;\C)$ is a form core for $q_1, q_2, q_3 $ and $ q_3^{\pm}$ we conclude that $\mathcal{Q}(q_1)=\mathcal{Q}(q_2)=\mathcal{Q}(q_3)\equiv \mathcal{Q}$
and $q_1=q_3^-$ and $q_2=q_3^+$. Moreover,
\begin{equation}\label{dddd}
\begin{split}
  d d^*&= ({\bf p}-{\bf A})^2+B\,\\
  d^* d&=({\bf p}-{\bf A})^2-B
\end{split}
\end{equation}
in the sense of quadratic forms on $\mathcal{Q}$ and hence the commutator formula \eqref{CommutatorB} extends to $\mathcal{Q}$.

\begin{lemma}\label{essentialda}
Let $B=B_0+b$ with $B_0>0$ and $|b|^{1/2}\in L^2_{\rm loc}(\R^2;\R)$ be relative $\sqrt{{\bf p}^2+1}$-compact.
Let ${\bf A}\in L^p_{\rm loc}(\R^2;\R^2), p\in (2,\infty]$, with
${\rm curl} \,{\bf A} =B.$  Then,  either one of the following statements holds
\begin{itemize}
\item [i)] $\sigma_{ess}(d^*d)=\emptyset\qquad$
\item [ii)] $\sigma_{ess}(d^*d)=\{ 2B_0 n\,:\, n\in \N_0\}$ and $\sigma_{ess}(dd^*)=\{ 2B_0 n\,:\, n\in \N\}$\,.
\end{itemize}
In addition, if $V$ satisfies (A2) then $V$ is relative $D_{{\bf A}}$- compact
and in particular $\sigma_{\rm ess}(D_{{\bf A}})=\sigma_{\rm ess}(D_{{\bf A}}+V)$.
\end{lemma}
\begin{remark}\label{poisson}
Note that our assumption on $B$ are satisfied if $B$  fulfills (A1). Indeed, in this case  $|b|^{1/2}({\bf p}^2+1)^{-1/2}$ is compact by Lemma \ref{compsi} in Appendix \ref{family}.

Moreover, note that if $B\in L_{\rm loc}^{q}(\R^2;\R)$ for some $q>1$ we can always find  ${\bf A}\in L^p_{\rm loc}(\R^2;\R^2)$ for some  $p\in (2,\infty]$. In order to see this define
$h$ to be a solution of
\begin{equation}\label{newton}
\Delta h=B\,.
\end{equation}
A local solution to this equation is given by the Newton potential $h_N$ of $B$. We know that $h_N\in W^{2,q}(\Omega)$ by the Calderon-Zygmund inequality, where $\Omega\subset \R^2$ is a bounded domain (see e.g. \cite[Sec.9.4]{GT1983}). This property extends to any solution $h$ of \eqref{newton} since  $h-h_N$ is harmonic on $\Omega$. Therefore, $h\in W_{\rm loc}^{2,q}(\R^2)$. Now one can define ${\bf A}:=(-\partial_2 h,\partial_1 h)$. Clearly, $A_j\in  W_{\rm loc}^{1,q}(\R^2)$. By standard Sobolev inequalities one obtains that $A_j\in L_{\rm loc}^t(\R^2;\R)$ for some $2<t<\infty$
if $q\in (1,2]$   and $A_j\in L_{\rm loc}^\infty(\R^2;\R)$ if $q> 2$.
\end{remark}
\begin{proof}
First note that for any $\lambda\ge 0$ the
operator $(d^*d+2B_0+\lambda)^{-1/2}$ maps $L^2(\R^2;\C)$ onto $\mathcal{D}(\sqrt{d^*d})$ which equals $\mathcal{Q}$ and $\mathcal{D}(H_S^{1/2})$. Thus, by the closed graph theorem, the operator
$(H_S+1)^{1/2}(d^*d+2B_0+\lambda)^{-1/2}$ is bounded. In particular,
\begin{align*}
|b|^{1/2}(d^*d+2B_0+\lambda)^{-1/2}=|b|^{1/2}(H_S+1)^{-1/2}(H_S+1)^{1/2}(d^*d+2B_0+\lambda)^{-1/2}
\end{align*}
is compact. Hence, the operator
\begin{align*}
  T(\lambda):=\overline{(d^*d+2B_0+\lambda)^{-1/2}{\rm sgn}(b)|b|^{1/2}} |b|^{1/2}(d^*d+2B_0+\lambda)^{-1/2}\,
\end{align*}
is also compact. It is easy to see that $\lambda>0$ can be chosen so large that $\|T(\lambda)\|<1$. For such $\lambda$'s we have, according to the resolvent formula for operators defined as quadratic forms (see \cite{Simon1971}),  that
\begin{align*}
  (d^*d+2B_0+2b+\lambda)^{-1}=
(d^*d+2B_0+\lambda)^{-1/2}(1+T(\lambda))^{-1}(d^*d+2B_0+\lambda)^{-1/2}\,.
\end{align*}
Note that the inverse of $1+T(\lambda)$ is well defined as a geometric expansion. Since $(1+T(\lambda))^{-1}-1$ is compact, we conclude that the resolvent difference between $d^*d+2B_0+2b+\lambda$ and $d^*d+2B_0+\lambda$
is also compact. Therefore, by Weyl's theorem, the two operators have the same essential spectrum. Using this and \eqref{dddd} we deduce that
\begin{align}\label{eess}
  \sigma_{\rm ess}(dd^*)=\sigma_{\rm ess}(d^*d+2B_0+2b)=\sigma_{\rm ess}(d^*d+2B_0)\,.
\end{align}
The latter equality and Equation \eqref{Unitary} imply (here we follow \cite{Iwatsuka1983})
\begin{equation}
  \label{iwa1}
\begin{split}
  &S:=\sigma_{\rm ess}(d^*d)\,,\qquad S\subset[0,\infty)\,,\\
&S\setminus\{0\}=S+2B_0\,.
\end{split}
\end{equation}
Assume now that $S\not=\emptyset$, then it is easy  to see from \eqref{iwa1}  that $0\in S$ and hence $2B_0n\in S, \,n\in\N_0$. Note also that no other points can belong to $S$. Hence,
using \eqref{eess} we get that  $ \sigma_{\rm ess}(dd^*)=2B_0n,\, n\in\N$.

Now, assume that $V$ fulfills (A2). Then, $V$ is relative
$\sqrt{{\bf p}^2+1}$- compact (see Lemma \ref{compsi} in Appendix
\ref{family}). It follows by the diamagnetic inequality that $V$ is
relative $H_S^{1/2}$- compact and consequently (arguing as before for
$b$) $\mathcal{D}(V)\supset \mathcal{Q}$ and the operators
$V(dd^*+\lambda^2)^{-1/2}$ and $V(d^*d+\lambda^2)^{-1/2}$ are compact
for any $\lambda\not=0$. From these considerations follow that
$V(D_{\bf A}-\mathrm{i\,} \lambda)^{-1}$ is compact, since the identity
  \begin{align*}
    (D_{\bf A}-\mathrm{i\,} \lambda)^{-1}&=(D_{{\bf A}}^2+\lambda^2)^{-1/2}[(D_{{\bf A}}^2+\lambda^2)^{-1/2}(D_{{\bf A}}+\mathrm{i\,}\lambda)]\\
&=\left(\begin{array}{cc}
(d^*d+\lambda^2)^{-1/2} & 0\\
0 &(dd^*+\lambda^2)^{-1/2}
\end{array}\right)\cdot[(D_{{\bf A}}^2+\lambda^2)^{-1/2}(D_{{\bf A}}+\mathrm{i\,}\lambda)]
  \end{align*}
holds and the operator in $[\dots]$ is bounded. Therefore,
$$ \sigma_{ess}(D_{{\bf A}}+V)=\sigma_{ess}(D_{{\bf A}}).$$
\end{proof}
We note that if $b$ satisfies (A1) then ${\rm Ker }(d^*d)$ is infinitely degenerated. Indeed, this follows from the fact that
\begin{equation}
\int_{\R^2}[B]_+d^2x =\infty\,,\qquad  \int_{\R^2}[B]_-d^2x <\infty\,,
\end{equation}
(where $[f]_+$  and $[f]_-$ are the positive and negative parts of $f$) which
shows that $B=B_0+b$ fulfills the conditions of \cite[Corollary 3.4]{Rozenblum2006}. In particular, we know that
\begin{align*}
  {\rm Ker }(d^*d)= \{ \omega e^{-h}\,|\, \omega e^{-h}\in L^{2}(\R^2;\C)\,,\, \omega \,\,\mbox{is analytic in}\,\,x_1+\mathrm{i\,} x_2\}\,,
\end{align*}
where $h$ is a solution of the equation $\Delta h=B$ \cite{Rozenblum2006}.
Therefore, we get:
\begin{lemma}\label{Cor}
Assume that $B$ satisfies (A1) and let ${\bf A}\in L^p_{\rm loc}(\R^2;\R^2),\,p\in(2,\infty],$  with ${\rm curl}\, {\bf A} = B$. Then,
\begin{gather}\label{finaless}
\sigma_{\rm ess}(d^*d)=\{ 2B_0 n\,|\, n\in \N_0\}\,,
\qquad \sigma_{\rm ess}(dd^*)=\{ 2B_0 n\,|\, n\in \N\}\,.
\end{gather}
In particular,
\begin{align*}
 \sigma_{\rm ess}(D_{{\bf A}})= \sigma_{\rm ess}(D_{{\bf A}_0})=\{ l_n\,|\,n\in \Z\}\,.
\end{align*}
Moreover, $0$ is an isolated point of $\sigma(D_{{\bf A}})$ and $\sigma(d^*d)$.
\end{lemma}
\begin{proof}
Due to our previous discussion we see that $0\in \sigma_{\rm ess}(d^*d)$. This combined with Lemma \ref{essentialda} imply \eqref{finaless}.  That
$0$ is an isolated point of $\sigma(d^*d)$
follows  by noting that, since  $0\not \in \sigma_{ess}(dd^*)$,  $0$  is neither an accumulation point  of $\sigma(dd^*)$ nor of $\sigma(d^*d)$. The statements on $\sigma(D_{{\bf A}})$ are now a consequence of Proposition \ref{supersymmetry}.
\end{proof}
\begin{lemma}\label{lemma}
Assume that $B$ satisfies (A1) and let ${\bf A}\in L^p_{\rm loc}(\R^2;\R^2),\,p\in(2,\infty],$  with ${\rm curl}\, {\bf A} = B$. Then, we have:
\begin{itemize}
\item[(a)] If $b\le 0$ and strictly negative on some open set,  then
$${\rm dim(Ran}(\mathbbm{1}_{(0,\sqrt{2B_0})}(D_{\mathbf{A}})))={\rm dim(Ran}(\mathbbm{1}_{(-\sqrt{2B_0},0)}(D_{\mathbf{A}})))=\infty\,.$$
\item[(b)] If $b\ge 0$ then
$${\rm dim(Ran}(\mathbbm{1}_{(0,\sqrt{2B_0})}(D_{\mathbf{A}})))={\rm dim(Ran}(\mathbbm{1}_{(-\sqrt{2B_0},0)}(D_{\mathbf{A}})))=0\,.$$
\end{itemize}
\end{lemma}
\begin{proof}
We may choose ${\bf A}:=(-\partial_2 h,\partial_1 h)$ where $h$ is a solution of
$\Delta h=B$. Due to Remark \ref{poisson} we know that ${\bf A}\in L^p_{\rm loc}(\R^2,\R^2)$
for some $p>2$.
\noindent
Part (a): Let $\Omega$ be an open set with $b\upharpoonright \Omega<0$.
Recall that there are infinitely many functions $\omega$, analytic in $x_1+\mathrm{i\,} x_2$, with $\psi:=\omega e^{-h }\in {\rm Ker}(d^*d)$.
For such $\psi$ we have, using
  \eqref{dddd},
\begin{align}
  \label{eq:11}
 \sps{\psi}{d d^* \psi}=2\sps{\psi}{B \psi}\le 2 B_0 \|\psi\|^2+2\int_{\Omega} b({\bf x})|\psi({\bf x})|^2 d {\bf x} <2B_0 \|\psi\|^2\,,
\end{align}
where in the last inequality we use the fact that $\psi$ can not
vanish on $\Omega$.  Let $(\psi_{n})_{n\in\N}$ be an orthonormal
system such that $\psi_n:= e^{-h}\omega_n\in {\rm Ker}\,d^*d$ with
$\omega_n$ analytic  in $x_1+\mathrm{i\,} x_2$. For $N\in \N$ define the self-adjoint matrix $M_N :=
( \sps{ \psi_n}{dd^*\psi_m})_{1\le n,m\le N}$.
It follows from \eqref{eq:11} that
$M_N<2B_0$. The Rayleigh-Ritz principle implies
$$0\le  \mu_n(dd^*)\le \mu_n(M_N)<2B_0,\ n=1,\ldots,N\,,$$
where we write $$\mu_n(T):= \sup_{\phi_1,\ldots,\phi_{n-1}}\inf_{\psi\in {\rm span}\{ \phi_1,\ldots,\phi_{n-1}\}^\perp \atop \|\psi\|=1,\psi\in Q(T)} \sps{\psi}{T\psi}$$
 for some self-adjoint operator T. Since $N$ is arbitrary the mini-max principle implies that
${\rm dim(Ran}(\mathbbm{1}_{[0,\sqrt{2B_0})}(dd^*)))=\infty$. It follows that
${\rm dim(Ran}(\mathbbm{1}_{(0,\sqrt{2B_0})}(dd^*)))=\infty$, for $0\notin \sigma_{\rm ess}(d d^*)$ by Lemma \ref{Cor}.
The claim is now a consequence of Proposition \ref{supersymmetry} and \eqref{Unitary}.
\\
\noindent
Part (b): In this case we have that  $dd^*\ge 2B_0$, since $dd^*-d^*d=2B\ge 2B_0$. Thus, the claim follows now from Proposition \ref{supersymmetry} and \eqref{Unitary}.
\end{proof}
\section{Super-exponential localization}\label{superex}
The proof of Theorem \ref{Maintheorem} follows the ideas developed in \cite{CorneanNenciu1998}. An essential ingredient is that,
by means of suitable local gauge transformations on certain regions
outside a big ball of radius $n$ centered at the origin, one can
replace the operator $D_{\bf A}$ by a Dirac operator $D_{{\bf A}_n}$
with ${\bf A}_n={\bf A}_0 + {\bf a}^n$, where ${\bf a}^n$ is a
magnetic vector potential of a magnetic field $b_n$ satisfying
$\lim_{n\to\infty} \|b_n\|_{\infty}=0$. The advantage of this is that we can
obtain explicit $L^p$ estimates (see Lemma~\ref{expansion} below) for
the resolvents of $D_{{\bf A}_n}$, conjugated with exponential
weights. These estimates can be derived using a certain resolvent
expansion (see \eqref{eq:21}) in combination with an explicit
expression for the Green kernel of $D_{{\bf A}_0}$ that can be found
in Appendix \ref{greenf} below.

Before stating these $L^p$ estimates let us fix some notation. For $p,q\in[1,\infty]$ we denote by
$\mathcal{B}(p,q)$ the space of bounded operators from
$L^p(\R^2;\C^2)$ to $L^q(\R^2;\C^2)$ and write, for $T\in
\mathcal{B}(p,q) $, %
\begin{align}
  \label{eq:23}
  \|T\|_{p,q}:=\|T\|_{\mathcal{B}(p,q)}\,.
\end{align}
Let $\gamma\ge 0$ and ${\bf u}\in\R^2$ with $|{\bf u}|=1$.
We define the exponential weight function as
\begin{equation*}
F({\bf x}):=\gamma\, {\bf u}\cdot {\bf x}\,,\quad {\bf x}\in\R^2\,.
\end{equation*}
Let $b_n$ be a magnetic field with $\lim_{n\to\infty}
\|b_n\|_{\infty}=0$ and  ${\bf a}^n$ be the associated vector potential in the transversal gauge, i.e.,
\begin{equation}
  \label{eq:15}
 {\bf a}^n({\bf x}):= \int_0^1  b_n(s {\bf x})\wedge {\bf x} \,s ds\,,
\end{equation}
where we write $a\wedge {\bf v}:= a(-v_2,v_1)$ for $a\in\R$ and ${\bf v}\in \R^2$.
The proof of the Lemma below can be found at the end of this section.
\begin{lemma}\label{expansion}
  Let $V_n \in L^\infty(\R^2;\R)$, $n\in\N$, be a family of electric
  potentials satisfying
\begin{equation*}\label{decayPotential}
\lim_{n\to\infty}\|V_n\|_\infty=0\,.
\end{equation*}
For any $n\in\N$ define the family of self-adjoint operators $D_{{\bf A}_n}+V_n$,
where ${\bf A}_n:={\bf A}_0+{\bf a}^n$ and ${\bf a}^n$ is given in \eqref{eq:15}.  Let $z\in \R\setminus\sigma(D_{{\bf
    A}_0})$ and $q,r\in[1,\infty]$ be such that
$1+\tfrac{1}{r}-\tfrac{1}{q}=\tfrac{1}{p}$ for some $p\in[1,2)$.
Then, there exists $N>0$ such that, for all $n>N$, $z\notin\sigma(D_{{\bf
    A}_n}+V_n)$ and
\begin{equation}
  \label{eq:28}
  e^{F}(D_{{\bf A}_n}+V_n-z)^{-1} e^{-F}\in \mathcal{B}(q,r)\,.
\end{equation}
\end{lemma}
In what follows we  apply the above result  to show Theorem \ref{Maintheorem}.
\begin{proof}[Proof of Theorem \ref{Maintheorem}]
For $n\in\N$ and ${\bf u}\in\R^2$ with $|{\bf u}|=1$ set
\begin{equation}
\Omega_{n}=\{ {\bf x}\in \R^2\,:\, {\bf u}\cdot {\bf x} > n\}\,.
\end{equation}
For $j\in\{1,2,3\}$ define $\chi_{j}\in C^\infty(\R^2;[0,1])$ with $\chi_{j}=0$ on $\R^2\setminus\Omega_{jn}$ and $\chi_{j}=1$ on $\Omega_{(j+1)n}$. We choose
$n$ so large that
$$\|b\|_{L^\infty(\Omega_n)}<\infty\,.$$
Since $b\in L_{\rm loc}^q(\R^2),\,q>1,$ we find a vector potential ${\bf a}\in L^p_{\rm loc}(\R^2;\R^2),\, p>2,$ with ${\rm curl}\,{\bf a}=b$ (see Remark \ref{poisson}).
Define, for ${\bf x} \in \R^2$,
\begin{align*}
   {\bf a}^n({\bf x})= \int_0^1  b_n(s {\bf x})\wedge {\bf x} \,s ds\,,
\end{align*}
where $b_n:=\mathbbm{1}_{\Omega_{n}}b\in L^\infty(\R^2)$. Observe that
\begin{align*}
  {\rm curl}\,{\bf a}={\rm curl}\,{\bf a}^n\quad\mbox{on}\quad\Omega_{n}\,,
\end{align*}
that $\Omega_n$ is simply connected, and that ${\bf a}^n, {\bf a}\in L^p_{\rm loc}(\R^2;\R^2)$ for some $p>2$. Therefore, there exists a gauge function
$\tilde{\Phi}_n\in W^{1,p}_{\rm loc}(\Omega_n)$ such that
 (see \cite[Lemma 1.1]{Leinfelder1983})
\begin{align}\label{total}
  \nabla \tilde{\Phi}_n= {\bf a}-{\bf a}^n\quad\mbox{on}\quad\Omega_{n}\,.
\end{align}
By multiplying $\tilde{\Phi}_n$ with a $C^\infty$- cutoff function we may define a
$\Phi_n\in W^{1,p}_{loc}(\R^2)$ that coincides with $\tilde{\Phi}_n$ on $\Omega_{2n}$. In particular, we find that
\begin{align}
  \label{total1}
   \nabla {\Phi}_n= {\bf a}-{\bf a}^n\quad\mbox{on}\quad\Omega_{2n}\,.
\end{align}
Define now $V_n:=\chi_1V$ and observe that $\|V_n\|_\infty\to 0$ as $n\to\infty$.
Then we get, for any $\eta\in \mathcal{C}^\infty_0(\R^2;\C^2)$ and  $j\in\{2,3\}$, using \eqref{total1} and the identity $\chi_j=\chi_1\chi_j$,
\begin{align*}
\chi_j(D_{\mathbf{A}}+V)\eta&=\chi_j(D_{\mathbf{A}_0}-{\boldsymbol \sigma}\cdot{\bf a}+V_n)\eta\\
&=\chi_j (D_{\mathbf{A}_0}- {\boldsymbol \sigma}\cdot\nabla\Phi_n-{\boldsymbol \sigma}\cdot{\bf a}^{n}+V_n)\eta\\
&=\chi_j e^{\mathrm{i\,} \Phi_n}(D_{\mathbf{A}_0}-{\boldsymbol \sigma}\cdot {\bf a}^{n}+V_n)e^{-\mathrm{i\,} \Phi_n}\eta\,.
\end{align*}

Set ${\bf A}_n:={\bf A}_0+{\bf a}^n$ and let $\Psi$ be an eigenfunction of $D_{\bf A}$ with eigenvalue $E\notin \sigma(D_{\mathbf{A}_0})$. By the previous computation we obtain, for any $\eta\in \mathcal{C}^\infty_0(\R^2;\C^2)$ and  $j\in\{2,3\}$,
\begin{align*}
\langle\, e^{\mathrm{i\,} \Phi_n}(D_{\mathbf{A}_n}+V_n-E)e^{-\mathrm{i\,} \Phi_n}\eta\,|\,\chi_j\,\Psi\,\rangle
&=\langle \,(D_{\mathbf{A}}+V-E)\eta\,|\,\chi_j\,\Psi\,\rangle\\
&=\langle\, \mathrm{i\,} {\boldsymbol \sigma}\cdot \nabla \chi_j \eta\,|\,\Psi\,\rangle.
\end{align*}
This equality extends to any $\eta$ in the domain of $e^{\mathrm{i\,} \Phi_n}(D_{\mathbf{A}_n}+V_n-E)e^{-\mathrm{i\,} \Phi_n}$,
since $ \mathcal{C}^\infty_0(\R^2;\C^2)$ is a core for $e^{\mathrm{i\,} \Phi_n}(D_{\mathbf{A}_n}+V_n-E)e^{-\mathrm{i\,} \Phi_n}$  (see Subsection \ref{SecGauge}). Clearly, ${\bf A}_n$ and $V_n$ satisfy the assumptions of Lemma~\ref{expansion}.
 Thus, for $n$ sufficiently large, $E\notin \sigma(D_{\mathbf{A}_n})$ and
we may replace $\eta$ by $e^{\mathrm{i\,} \Phi_n}(D_{\mathbf{A}_n}+V_n-E)^{-1}e^{-\mathrm{i\,} \Phi_n}\eta$ obtaining that
\begin{equation}\label{eq:23a}
\chi_j \Psi
=-\mathrm{i\,} e^{\mathrm{i\,} \Phi_n}(D_{\mathbf{A}_n}+V_n-E)^{-1}e^{-\mathrm{i\,} \Phi_n}
 ({\boldsymbol \sigma}\cdot\nabla \chi_j)\Psi,\quad j\in \{2,3\}\,.
\end{equation}
Observe that using \eqref{eq:23a} for $j=2$
in combination with Lemma~\ref{expansion} (with $q=2, r=3$, and $F=0$) we obtain  that
\begin{equation}\label{eq:25a}
\chi_2 \Psi \in L^3(\R^2;\C^2)\,.
\end{equation}
We use again \eqref{eq:23a}, for large $n$,  to get in addition that
\begin{equation}\label{eq:27}
e^{F}\chi_3 \Psi
=-\mathrm{i\,} e^{\mathrm{i\,} \Phi_n}\big(e^{F}(D_{\mathbf{A}_n}+V_n-E)^{-1}e^{-F}\Big)\big(e^{-\mathrm{i\,} \Phi_n}
 e^{F}({\boldsymbol \sigma}\nabla \chi_3) \chi_2\Psi\big)\,.
\end{equation}
Since $\operatorname{supp}(\nabla\chi_3)\subset
\Omega_{3n}\setminus\Omega_{4n}$ we find thanks to \eqref{eq:25a} that
$e^{-\mathrm{i\,} \Phi_n} e^{F}({\boldsymbol \sigma}\nabla \chi_3) \chi_2\Psi
\in L^2(\R^2;\C^2)\cap
L^3(\R^2;\C^2)$.  Thus, we may apply
Lemma~\ref{expansion} with $q=3, r=\infty$ and $q=2, r=2$ to obtain
the decay in the $L^\infty$ and $L^2$ norms respectively for $n\ge
n_0$ sufficiently large.  We obtain the desired bound \eqref{sexpon} from \eqref{eq:27}
by varying $F$ over finitely many vectors ${\bf u}$.
\end{proof}
\begin{proof}[Proof of Lemma \ref{expansion}]
Recall that the magnetic vector potential is given by ${\bf A}_n={\bf A}_0+{\bf a}^{n}$ where ${\bf a}^{n}$ is defined in  \eqref{eq:15}.

A simple calculation shows that the vector potential
\begin{equation}
  \label{eq:12}
  {\bf a}^n_{\bf x'}({\bf x})= \int_0^1  b_n({\bf x}'+s ({\bf x}-{\bf x}'))\wedge ({\bf x}-{\bf x}') \,s ds\,,\quad {\bf x}'\in\R^2\,,
\end{equation}
is also a vector potential of  the magnetic field $b_n$.
A crucial property of ${\bf a}^n_{{\bf x}'}$ is that
\begin{align}
  \label{eq:25}
  |{\bf a}^n_{{\bf x}'}({\bf x})|\le \|b_n\|_\infty\cdot |{\bf x}-{\bf x}'|, \quad {\bf x},{\bf x}'\in\R^2\,.
\end{align}
Since
$\rm{curl}\, {\bf a}^n_{\bf x'}= \rm{curl}\, {\bf a}^n$ there exists a function
$\varphi_n:\R^2\to\R$ with
\begin{equation}
  \label{eq:13}
  \nabla_{{\bf x}} \varphi_n({\bf x},{\bf x}')={\bf a}^n({\bf x})-{\bf a}^n_{{\bf x}'}({\bf x})\,.
\end{equation}
We may further require that
\begin{equation}
  \label{eq:14}
  \varphi_n({\bf x},{\bf x})=0.
\end{equation}
The proof of Lemma~\ref{expansion} is based upon $L^p$ estimates for the  resolvent expansion \eqref{eq:21} below. We start by defining the relevant objects and list their $L^p$ properties.  For $z\in\R\setminus\sigma(D_{{\bf A}_0})$ let   $G_0({\bf x},{\bf x}',z)$ be a representation of the Green kernel
  of $(D_{{\bf A}_0}-z)^{-1}$ as $2\times 2$-matrix. Due to \eqref{eq:17} from Appendix \ref{greenf} and the triangular inequality we obtain that
\begin{align*}
  \big\|e^{F({\bf x})}G_0({\bf x},{\bf x}';z)e^{-F({\bf x}')}\big\|_{\C^2\otimes\C^2}\le e^{-\theta({\bf x}-{\bf x}')+\gamma|{\bf x}-{\bf x}'|}\omega({\bf x}-{\bf x}';z):=g_{F}({\bf x}-{\bf x}')\,.
\end{align*}
We observe that thanks to \eqref{eq:18} we have that
\begin{equation}
  \label{eq:16}
g_{F} \in L^t(\R^2),\quad t\in[1,2)\,\quad\mbox{and} \quad|{\bf x}|g_{F}\in  L^t(\R^2), \quad t\in[1,\infty]\,.
\end{equation}
We introduce, for $n\in \N$, the integral operators $ S_n(z),T_n(z) :
L^2(\R^2;\C^2)\to L^2(\R^2;\C^2)$ with
\begin{align}
  \label{eq:20}
  \big(S_n(z)f\big)({\bf x})&:=
\int_{\R^2}e^{\mathrm{i\,}\varphi_n({\bf x},{\bf x}')}G_0({\bf x},{\bf x}';z) f({\bf x}')  d{\bf x}',\\
  \big(T_n(z)f\big)({\bf x})&:=
\int_{\R^2}{\boldsymbol \sigma}\cdot a^n_{{\bf x}'}({\bf x})\,e^{\mathrm{i\,}\varphi_n({\bf x},{\bf x}')}G_0({\bf x},{\bf x}';z)f({\bf x}')  d{\bf x}'\,,
\end{align}
where $\varphi_n$ is determined by \eqref{eq:13} and \eqref{eq:14}.
Notice that in view of \eqref{eq:16}, \eqref{eq:25},
and Young's inequality (see \cite[Section 4.2]{LL2001})  both operators are well defined and bounded. In fact, since
$$
\|(e^{F} T_n(z) e^{-F})({\bf x},{\bf x}')\|_{\C^2\otimes\C^2}\le \|b_n\|_\infty |{\bf x}-{\bf x}'|
g_{F}({\bf x}-{\bf x}')\,,
$$
we find by \eqref{eq:16} and Young's inequality that, for $q\in[1,\infty]$,
\begin{equation}
    \label{eq:24}
\lim_{n\to\infty} \|e^F T_n(z) e^{-F}\|_{q,q}=0\,.
  \end{equation}
Furthermore, a similar argument implies that, for $q,r\in[1,\infty]$ and
$t\in [1,2)$ with  $\tfrac{1}{t}=1+\tfrac{1}{r}-\tfrac{1}{q}$,
\begin{align} \label{eq:24-b}
\sup_{n\in \N}\|e^{F}S_n(z)e^{-F}\|_{q,r}<\infty\,.
\end{align}
Our next task is to show the following resolvent
formula in $L^2(\R^2;\C^2)$, for $n\in\N$ so large that $\|T_n(z)\|_{2,2}<1$ (see \eqref{eq:24}),
\begin{equation}
  \label{eq:21}
  (D_{{\bf A}_n}-z)^{-1}= S_n(z)\sum_{k=0}^\infty T_n(z)^k\,.
\end{equation}
Pick functions $f\in L^2(\R^2;\C^2)$ and $g\in C^\infty_0(\R^2;\C^2)$ and, we find
\begin{align*}
\label{eq:22}
&\SP{(D_{{\bf A}_n}-z)g}{S_n(z) f}\\
&\quad=\int_{\R^2}\SP{\big[(D_{{\bf A}_n}-z)g]({\bf x})}{\int_{\R^2} e^{\mathrm{i\,}\varphi_n({\bf x},{\bf x}')} G_0({\bf x},{\bf x}';z)f({\bf x}')\,d{\bf x}'}_{\C^2}\,d{\bf x}\\
&\quad=\int_{\R^2}\int_{\R^2}\SP{e^{-\mathrm{i\,} \varphi_n({\bf x},{\bf x}')}
\big[(D_{{\bf A}_n}-z)g]({\bf x})}{  G_0({\bf x},{\bf x}';z)
f({\bf x}')}_{\C^2}\,d{\bf x}\,d{\bf x}'\,,
\end{align*}
where Young's inequality together with Lemma \ref{GreenKernel} from Appendix \ref{greenf} enabled us to
use Fubini's theorem in the last equality.
Observe that due to \eqref{eq:13}
\begin{align*}
  e^{-\mathrm{i\,} \varphi_n({\bf x},{\bf x}')}
  \big[(D_{{\bf A}_n}-z)g]({\bf x})
  =\big[(D_{{\bf A}_0}-{\boldsymbol \sigma}\cdot {\bf a}^n_{{\bf x}'}-z)
  e^{-\mathrm{i\,} \varphi_n(\cdot,{\bf x}')}g]({\bf x})\,.
\end{align*}
Hence, using again Fubini's theorem,
\begin{equation}\label{ide}
\begin{split}
  &\SP{(D_{{\bf A}_n}-z)g}{S_n(z) f}=\\
&\quad\int_{\R^2}\SP{ \int_{\R^2} G_0({\bf x}',{\bf x};z)
\big[(D_{{\bf A}_0}-z)e^{-\mathrm{i\,} \varphi_n(\cdot,{\bf x}')}g]({\bf x})\, d{\bf x}}{ f({\bf x}')}_{\C^2}\,d{\bf x}'\\
&-\int_{\R^2}\int_{\R^2}\SP{ {\boldsymbol \sigma}\cdot{\bf a}^n_{{\bf x}'}({\bf x})
  e^{-\mathrm{i\,} \varphi_n({\bf x},{\bf x}')}g({\bf x})}{  G_0({\bf x},{\bf x}';z)f({\bf x}')}_{\C^2}\,d{\bf x}'\,d{\bf x}\\
&=\SP{g}{ f}- \SP{g}{T_n(z) f}\,.
\end{split}
\end{equation}
For the first integral after the first equality
above we used \eqref{eq:14} together with the fact
that $G_0$ is the Green kernel of $D_{{\bf A}_0}$
and thus, for any $\tilde{g}\in C^\infty_0(\R^2;\C^2)$,
\begin{align*}
  \int_{\R^2} G_0({\bf x}',{\bf x};z)\big[(D_{{\bf A}_0}-z)\tilde{g}]({\bf x})\,d{\bf x}
  =\tilde{g}({\bf x}')\quad \mbox{a.e.}\,.
\end{align*}
Now,  since  $D_{{\bf A}_n}$ is essentially self-adjoint on $C^\infty_0(\R^2;\C^2)$
we can extend the identity \eqref{ide} for all $g\in \mathcal{D}(D_{{\bf A}_n})$.
From this extension  follows that $S_n(z)$ maps $L^2(\R^2;\C^2)$ in
$\mathcal{D}(D_{{\bf A}_n})$ and
\begin{equation*}
  \label{eq:26}
  (D_{{\bf A}_n}-z)S_n(z)f=f- T_n(z)f\,,\quad f\in L^2(\R^2;\C^2)\,.
\end{equation*}
This yields, for $n$ large enough, the operator identity
\begin{align*}
  S_n(z)= (D_{{\bf A}_n}-z)^{-1}\big(1- T_n(z) \big)\,,
\end{align*}
from which follows \eqref{eq:21}.

We observe that  \eqref{eq:28}, for $V_n=0$, is a consequence of
\eqref{eq:24},  \eqref{eq:24-b}, and
\begin{align*}
  \Big\| e^{F}(D_{{\bf A}_n}-z)^{-1} e^{-F} \Big\|_{q,r}\le
  \Big\| e^{F}S_n (z)e^{-F} \Big\|_{q,r}\cdot \sum_{k=0}^\infty
  \Big( \big\|e^F T_n(z)e^{-F} \big\|_{q,q}\Big)^k\,.
\end{align*}
Note that the last sum above converges for $n$ large enough due to \eqref{eq:24}.

In order to show \eqref{eq:28} for $V_n\not=0$  we  note that by  H\"older's inequality
\begin{equation}\label{eqeq}
\begin{split}
\|V_n e^F (D_{{\bf A}_n}-z)^{-1}e^{-F}\|_{q,q}\le&
\|V_n\|_\infty\,\|e^F (D_{{\bf A}_n}-z)^{-1}e^{-F}\|_{q,q}\to 0\,,
\end{split}
\end{equation}
 as $n\to\infty$. Therefore, the following computation is meaningful for $n$ large enough
\begin{align*}
\|e^F&(D_{\mathbf{A}_n}+ V_n-z)^{-1}e^{-F}\|_{q,r}\\ \nonumber
&=\|e^F(D_{\mathbf{A}_n}-z)^{-1}(1+ V_n (D_{\mathbf{A}_n}-z)^{-1})^{-1}e^{-F}\|_{q,r}\\  \nonumber
&\le\|e^F(D_{\mathbf{A}_n}-z)^{-1}e^{-F}\|_{q,r}\sum_{m=0}^\infty
\|\{V_n e^F(D_{\mathbf{A}_n}-z)^{-1}e^{-F}\}^m\|_{q,q}\,.
\end{align*}
This finishes the proof.
\end{proof}
\section{Gaussian-localization}\label{Gauss}
In this section we show  Theorem \ref{Maintheorem3} on Gaussian localization of eigenfunctions with energies in the
discrete spectrum of
\begin{equation}
H= D_{\mathbf A} + V\,,
\end{equation}
under the assumptions (A1)-(A5) stated in the introduction. We choose the magnetic vector potential to be given by
\begin{equation}\label{DefArad1}
{\mathbf A}({\bf x}):= r^{-1}A(r)\left (\begin{array}{c}-x_2\\ x_1
\end{array}\right),\quad A(r)= r^{-1} \int_0^r B(s)s\,ds\,.
\end{equation}
If $B\in L_{\rm loc}^q(\R^2,\R)$ it is easy to see, using H\"older's
inequality, that if $q\in(1,2]$ then ${\mathbf A}\in L_{\rm
  loc}^p(\R^2;\R^2)$, for some $p\in(2,\infty)$, and that ${\mathbf
  A}\in L_{\rm loc}^\infty(\R^2;\R^2)$ whenever $q\in(2,\infty]$ .

The proof of Theorem \ref{Maintheorem3}, given in Subsection \ref{agmon}, follows the ideas of \cite{Nakamura1996} consisting in Agmon-type estimates with localizations in space and in the angular momentum variable. Of course, we have to adapt the method of \cite{Nakamura1996} since our Hamiltonian is not bounded form below.

In Subsection \ref{vale4} we transform the operator $H$ to polar
coordinates and we decompose it in the angular momentum variable
$m_j$.  The analyticity condition  (A4) on $V$ permits us to obtain
exponential decay in $|m_j|$ of eigenfunctions of $H$ with eigenvalues
$E\in\sigma_{\rm d}(H)$ (see Lemma \ref{RotAnalytic} in Subsection
\ref{vale}). In order to obtain the Agmon estimates, in Subsection
\ref{agmon}, we square the transformed free Dirac operator $K_0^{(2)}$ (see
\eqref{DefK02} for its definition). The Gaussian decay is essentially
due to a positive term in $(K_0^{(2)})^2$ that goes like $r^2$. This
term is in competition with a term that behaves like $m_j$ when $m_j\ge 0$. The
Gaussian weights in the Agmon estimates are localized in the region
where $m_j \lesssim r^2$. The complementary region, on the other hand, is controlled by the exponential decay in $|m_j|$.

\subsection{Unitary transform}\label{vale4}
In the following we  derive an equivalent representation of $H$.
We denote by $U$ the unitary map that represents $H$ in polar coordinates
(see e.g.  \cite[Section 7.3.3]{Thaller})
\begin{align*}
UHU^*&=H^{(1)}= K^{(1)}_0+v(r,\theta),\\
 K^{(1)}_0&:=S_\theta\big\{ -\mathrm{i\,} \partial_r+i r^{-1}
\sigma_3 J_3-\mathrm{i\,} \sigma_3 A(r)\big\},\\
\end{align*}
acting on $\mathcal{H}^{(1)}:=L^2(\R^+)\otimes L^2(T;\C^2)^2$, where
\begin{align*}
  J_3&:= -\mathrm{i\,} \partial_{\theta}+1/2\sigma_3, \quad
S_\theta:=\begin{pmatrix}
 0& e^{-\mathrm{i\,} \theta}\\
e^{\mathrm{i\,} \theta}&0
\end{pmatrix}\,.
\end{align*}
Next we identify $L^2(T;\C^2)$ with $\ell^2(\Z)^2$
by means of  the transformation $\mathcal{F}:L^2(T;\C^2)\to \ell^2(\Z)^2$  given by
\begin{align*}
  \mathcal{F}[f](j):=\frac{1}{\sqrt{2\pi}}\int_0^{2\pi}M_\theta e^{-\mathrm{i\,} m_j \theta}f(\theta) \,d \theta\,,
\end{align*}
for $f\in L^2(T;d\theta)^2$, where  $m_j = (2j+1)/2,\ j\in \Z$, and
\begin{align*}
  M_\theta:=\left(\begin{array}{cc}
e^{\mathrm{i\,}\theta/2}&0\\
0&\mathrm{i\,} e^{-\mathrm{i\,} \theta/2}\,
\end{array}
\right)\,.
\end{align*}
Under these transformations we find  the decomposition
\begin{align*}
& L^2(\R^2;\C^2) \cong \mathcal{H}^{(2)}:=\bigoplus_{j\in \Z}L^2(\mathbb{R}^+ ;dr)^2\,
\end{align*}
and the corresponding  operator
\begin{align*}
& H \cong H^{(2)}=K^{(2)}_0+W\,,
\end{align*}
which is essentially self-adjoint on $\mathcal{D}^{(2)}:=\mathcal{F}U C^\infty_0(\R^2;\C^2)$. For $h\in \mathcal{D}^{(2)}$,  $K^{(2)}_0=\mathcal{F}U D_{\mathbf A} U^* \mathcal{F}^*$ acts as
\begin{equation}\label{DefK02}
(K^{(2)}_0h)(r,j)= \big(-\mathrm{i\,} \sigma_2 \partial_r + \sigma_1(-m_j r^{-1}+A(r))\big)h(r,j)\,,
\end{equation}
where we used that $\mathcal{F} S_\theta \mathcal{F}^*=\sigma_2$, $\mathcal{F}S_\theta \sigma_3\mathcal{F}^*=\mathrm{i\,} \sigma_1$, and that $\mathcal{F} J_3 \mathcal{F}^*$ is the multiplication operator by $m_j$.
The electric potential
$W=\mathcal{F}v \mathcal{F}^*$ acts as
\begin{equation}\label{defW}
(Wh)(r,l):= \sum_{j \in \Z}
\hat{v}(r,l-j)
h(r,j),
\end{equation}
where
\begin{equation}\label{FourierKoeff}
\hat{v}(r,n) =\frac{1}{2\pi}\int_0^{2\pi}e^{-\mathrm{i\,} n\theta}v(r,\theta) d \theta\,,\quad n\in\Z\,.
\end{equation}
Two other quantities play an important role in our analysis, namely $W_1:=\mathcal{F}\partial_r v \mathcal{F}^*$ given by
\begin{equation}
  \label{eq:19}
 (W_1 h)(r,l):= \sum_{j \in \Z}
\partial_r \hat{v}(r,l-j)
h(r,j),
\end{equation}
and $W_2:=\mathcal{F}\partial_\theta v \mathcal{F}^*$ that acts as
\begin{align}
  \label{eq:29}
  (W_2h)(r,l):= \sum_{j \in \Z}
\mathrm{i\,}(j-l)\hat{v}(r,l-j)
h(r,j)\,.
\end{align}
\subsection{Rotation-analyticity}\label{vale}
For $f\in\mathcal{H}^{(1)}$ and  $a\in \R$ we set
\begin{equation}
\big(U_af \big)(r,\theta):= (e^{\mathrm{i\,} J_3 a} f)(r,\theta)
=e^{\mathrm{i\,} \sigma_3 a/2}f(r, \theta + a)\,.
\end{equation}
We call a vector $f\in \mathcal{H}^{(1)}$
rotation-analytic, if and only if the series
$$
\sum_{n\in\N} \frac{\|J_3^n f\|}{n!} \rho^n\,,\quad \rho>0\,,
$$
has an infinite radius of convergence.
We start by presenting a lemma that gives us some a priori decay of some
eigenfunctions of $H^{(2)}$ in the angular momentum variable.
\begin{lemma}\label{RotAnalytic}
Assume that (A1)-(A4) hold. Let $\Psi\in
\mathcal{H}^{(2)}$ be an eigenfunction of $H^{(2)}$ to the eigenvalue  $E\in \sigma_d(H^{(2)})$.
Then, for every $\gamma>0$, we have
\begin{align}
\sum_{j\in\Z}&
\int_0^\infty
e^{2\gamma |m_j|}|\Psi(r,j)|^2 d r<\infty.
\end{align}
\end{lemma}
\begin{proof}The proof is analog to the one given in \cite[Section 3]{Nakamura1996}. We sketch it here for the reader's convenience.
Due to Lemma \ref{family2} (in Appendix~\ref{family}) $\{H^{(1)}(z)\}_{z\in\C}$ defined on $\mathcal{D}(K_0^{(1)})$ through
\begin{align*}
  H^{(1)}(z)=K_0^{(1)}+\widetilde{v}_z\,,
\end{align*}
is an analytic family of type (A)  (see \cite{ReedSimon1978}). Note that when $a\in\R$ the identity $H^{(1)}(a)=U_a H^{(1)}U_a^*$ holds. Moreover, by
Lemma~\ref{family1} (in Appendix~\ref{family}) we have that
\begin{align*}
\widetilde{v}_z (K_0^{(1)}-\mathrm{i\,})^{-1}
\end{align*}
is a compact operator in $\mathcal{H}^{(1)}$ for any $z\in \C$.
Therefore,
$\sigma_{\mathrm{ess}}(H^{(1)}(z))=\sigma_{\mathrm{ess}}(K^{(1)}_0)$
by Weyl's theorem. Arguing with analytic perturbation theory and using the fact that the spectrum of $H^{(1)}(a)$ and $H^{(1)}$ is the same for $a$ real (see
e.g. the proof of Theorem XIII.36 in \cite{ReedSimon1978} for a similar argument)  we find
that $E\in \sigma_{\mathrm{d}}(H^{(1)})$ of multiplicity $N\in\N$ is also an eigenvalue of
$H^{(1)}(z)$ of the same multiplicity.

Let   $P_z$ be the $N$-dimensional $E$-eigenprojection of $H^{(1)}(z)$.
Since rotation-analytic vectors are dense in $\mathcal{H}^{(1)}$ (see e.g. \cite{Nelson1959}) we find some rotation-analytic vectors $f_1,\dots, f_N$ such that
$\mathrm{Ran} P_0=\mathrm{Span}\{P_0f_1,\dots, P_0f_N\}\,.$
Observing that,
for $a\in\R$ and $j\in\{1,\dots,N\}$,
$$U_aP_0 f_j=P_a U_af_j\,, $$
we find an analytic continuation of $a\mapsto U_aP_0f_j\in \mathcal{H}^{(1)}$ to the complex plane. In particular,  $e^{\mathrm{i\,} J_3 z}P_0f_j$ belongs to $\mathcal{H}^{(1)}$ for any $z\in \C$.
Let $\Psi_1\in \mathrm{Ran}P_0$ be such that $\mathcal{F}\Psi_1=\Psi$. By the discussion above we get that
$$\mathcal{F} e^{ J_3 \gamma}\Psi_1 \in \mathcal{H}^{(2)}\,,\quad \gamma\in\R\,.$$
This ends the proof since $(\mathcal{F}e^{ J_3 \gamma}\Psi_1)(r,j)=e^{m_j\gamma} \Psi(r,j)$ and
\begin{align*}
  \sum_{j\in\Z}&
\int_0^\infty
e^{2\gamma |m_j|}|\Psi(r,j)|^2 d r \\
&\le\sum_{j\in\Z}
\int_0^\infty
e^{-2\gamma m_j}|\Psi(r,j)|^2 d r + \sum_{j\in\Z}
\int_0^\infty
e^{2\gamma m_j}|\Psi(r,j)|^2 d r <\infty\,.
\end{align*}
\end{proof}

\subsection{Agmon-type Estimate}\label{agmon}
In this section we deduce the Agmon estimates needed in the proof of Theorem \ref{Maintheorem3}.
They were obtained  in \cite{Nakamura1996} for
the case of a magnetic Schr\"odinger operator. These estimates uses
heavily the fact that the Schr\"odinger  operator is bounded
from below. As we commented before we will obtain these estimates for the square of  the  Dirac operator $K_0^{(2)}$.

Fix a number $\tilde{B}>B_0$ and note that, due to  (A2), there exists $R_0>0$ so large that the estimate \eqref{aeqq:1} is fulfilled and  moreover
\begin{equation}\label{r0}
\|\mathbbm{1}_{\{r>R_0\}} B\|<\tilde{B}\,,\quad r>R_0\,.
\end{equation}
We set, for $0<q_2<q_1<1$,
\begin{align}
r(j)&:=\begin{cases}\sqrt{ \frac{4\tilde{B}}{(q_1^2-q_2^2)B_0^2}m_j },& m_j\ge 0\\
0,&m_j<0\,,\\
\end{cases}\\
\Omega_{q_1,q_2}
&:=\{ (r,j)\in \R^+\times \Z\,|\,r\ge r(j)\}\,.
\end{align}
Moreover, we define
\begin{align}
\rho(r,j)&:=
\begin{cases}
q_2 B_0/4(r^2-r(j)^2),& m_j\ge 0,\ r\ge r(j)\\
q_2 B_0 r^2/4& m_j<0\\
0& m_j\ge 0,\ r< r(j)\,.
\end{cases}
\end{align}
Eventually we will  choose $q_2$ to be sufficiently close to $1$.
A direct  calculation shows that
\begin{equation}
|\rho(r,j_1)-\rho(r,j_2)|\le \frac{q_2\tilde{B}}{(q_1^2-q_2^2)B_0}|j_1-j_2|.
\end{equation}
Let $\rho_\epsilon:= \rho(1+ \epsilon\rho)^{-1}$.  It is easy to see
that
\begin{equation}\label{Diffrho}
|\rho_\epsilon(r,j_1)-\rho_\epsilon(r,j_2)|\le \frac{q_2\tilde{B}}{(q_1^2-q_2^2)B_0}|j_1-j_2|.
\end{equation}
Finally, for $R>R_0$, we fix a smooth function $f_R$ in $r$ with bounded derivatives in $\R^+\times\Z$ satisfying
\begin{equation}
f_R(r,j)=
\begin{cases}
1&   r\ge 2R \textrm{ and } (r,j)\in\Omega_{q_1,q_2}\\
0&  r \le R \textrm{ or } (r,j)\not \in\Omega_{q_1,\mu q_2}\,
\end{cases}
\end{equation}
where $\mu\in (0,1)$ is a fixed number that  will be chosen sufficiently close to $1$. Note that $\Omega_{q_1,q_2}\subset \Omega_{q_1,\mu q_2}$.

Let $\Psi$ be the eigenfunction from Theorem~\ref{Maintheorem3} and  $\hat{\Psi}:=\mathcal{F}U \Psi$ be a normalized eigenfunction of $H^{(2)}$ with
corresponding energy $E\in \sigma_d(H^{(2)})$. We set, for $R>R_0$ and $\delta\in(0,\mu)$,
\begin{equation}\label{Defg}
g:=e^{\delta \rho_\epsilon}f_R \hat{\Psi}\,.
\end{equation}
Observe that $\delta$ can be chosen arbitrarily close to $1$.
\begin{lemma}\label{KonjugiertW}
We
  find constants $R_1>R_0$ and $c>0$ such that, for any $\delta\in(-1,1), \rho>R_1$, and
  $j\in\{0,1,2\}$,
\begin{align}\label{aeqq:1}
\sup_{\epsilon>0}\|\,\theta_\rho e^{ \delta\,\rho_\epsilon}W_j
e^{- \delta\,\rho_\epsilon} \theta_\rho \|< c \,,
\end{align}
where $W_0:=W$ and $\theta_\rho:=\mathbbm{1}_{\{r>\rho\}}$.  In particular,
the commutator
\begin{align}
  \label{eq:32}
  [K_0^{(2)},W_0]=-\mathrm{i\,}\sigma_2 W_1+\frac{\mathrm{i\,} \sigma_1}{r}W_2
\end{align}
satisfies the estimate
\begin{align}
  \label{eq:33}
  \sup_{\epsilon>0}\|\, e^{ \delta\,\rho_\epsilon}f_R[K_0^{(2)},W_0]e^{- \delta\,\rho_\epsilon}\|<2c\,,\quad R>R_1\,.
\end{align}
\end{lemma}
\begin{proof}
We show \eqref{aeqq:1} only for $j=2$, for the other cases follow analogously.
For any $m\in \Z$ and $r>0$ we define the analytic function $\C\ni z\mapsto h_m(r,z):=e^{-\mathrm{i\,} m z} \widetilde{v}_z(r,0)$.
 Using  \eqref{FourierKoeff} and the decay and  analyticity assumptions on $v$ (A5) we find for any $r>\rho$ (sufficiently large), $m\in\Z$,
 and  $\gamma\in\R$  that there is a constant $C>0$ such that
\begin{equation}\label{decV}
\begin{split}
  |\hat{v}(r, m)|=&\frac{1}{2\pi}\big| \int_{0}^{2\pi} h_m(r,\theta) d \theta\big|
=\frac{1}{2\pi} \big|\int_{0}^{2\pi} h_m(r,\theta-\mathrm{i\,}\gamma) d \theta\big|\\
\le&
\frac{e^{-m\gamma}}{2\pi} \int_{0}^{2\pi} u_{2|\gamma|}(r,\theta)  d \theta
\le \|\theta_\rho u_{2|\gamma|}\|_\infty e^{-m\gamma}\le C e^{-m\gamma}\,.
\end{split}
\end{equation}
Here we also used Cauchy's integral theorem and the fact that
$h_m(r,z)$ is $2\pi$-periodic with respect to ${\rm Re}(z)$. In particular, replacing $\gamma$ by $-\gamma$ in the above estimate we see  that, for $\gamma>0$ and  $m\in\Z$, the bound
$ \theta_\rho|\hat{v}(r, m)|\le C e^{-|m|\gamma}$ holds.
Therefore, using \eqref{Diffrho}, \eqref{eq:29}, and Young's inequality for $\ell^2(\Z^2;\C^2)$
in combination with the Cauchy-Schwarz inequality for $L^2((0,\infty);\C)$,
we get, for $\gamma$ sufficiently large and every $f\in \mathcal{H}^{(2)}$, that $|\sps{f}{\theta_\rho e^{ \delta\,\rho_\epsilon}W_2
e^{- \delta\,\rho_\epsilon} \theta_\rho f}|$ is bounded by
\begin{align*}
\int_0^\infty\sum_{l\in\Z}|f(r,l)|  \sum_{j\in\Z} |\theta_\rho \hat{v}(r,l-j)||l-j| e^{\delta \tau |l-j|} |f(r,j)|d r \le \tilde{C} \|f\|^2\,,
\end{align*}
for some constant $\tilde{C}>0$, where $\tau:=\tfrac{q_2\tilde{B}}{(q_1^2-q_2^2)B_0}$.

Equation \eqref{eq:33} follows from \eqref{aeqq:1} and \eqref{eq:32}. Equation \eqref{eq:32} is a consequence of
\begin{align}
  \label{eq:34}
  [K_0^{(2)},W]=&\mathcal{F}[K_0^{(1)},v]\mathcal{F}^*\,\\
=&\mathcal{F} (-\mathrm{i\,} S_\theta \partial_r v+\frac{S_\theta \sigma_3}{r} \partial_\theta v)\mathcal{F}^*\,,
\end{align}
and the fact that $\mathcal{F} S_\theta \mathcal{F}^*=\sigma_2$ and $\mathcal{F}S_\theta \sigma_3\mathcal{F}^*=\mathrm{i\,} \sigma_1$.
\end{proof}
Before continuing let us state a simple technical result.
\begin{lemma}\label{wdef}
For any $\gamma\in\R$ we have that $e^{\gamma \rho_\epsilon}f_R \hat{\Psi} \in \mathcal{D}(K_0^{(2)})$.
\end{lemma}
\begin{proof}
Let  $\lambda>0$ and $\eta\in \mathcal{F}U C^\infty_0(\R^2;\C^2)$. First observe that a simple computation shows that
$$(\partial_r e^{\gamma \rho_\epsilon}f_R)e^{-\lambda r}$$ extends to a bounded operator on $\mathcal{H}^{(2)}$.
 In addition,  $e^{\lambda r}\hat{\Psi}\in\mathcal{H}^{(2)}$ by Theorem \ref{Maintheorem}. Therefore, we get by   explicit calculation on $\mathcal{F}U C^\infty_0(\R^2;\C^2)$, that
\begin{align*}
  \sps{K_0^{(2)}\eta}{e^{\gamma \rho_\epsilon}f_R \hat{\Psi}}&=\sps{f_Re^{\gamma \rho_\epsilon}K_0^{(2)}\eta}{\hat{\Psi}}\\
&= \sps{K_0^{(2)}f_Re^{\gamma \rho_\epsilon}\eta}{\hat{\Psi}}+\sps{\mathrm{i\,}\sigma_2(\partial_r e^{\gamma \rho_\epsilon}f_R)\eta}{\hat{\Psi}}\\
&= \sps{\eta}{e^{\gamma \rho_\epsilon}f_R K_0^{(2)}\hat{\Psi}}-\mathrm{i\,}\sps{\eta}{\sigma_2(\partial_r e^{\gamma \rho_\epsilon}f_R)e^{-\lambda r} \,(e^{\lambda r}\hat{\Psi})}\,.
\end{align*}
Since $\eta$ can be chosen arbitrarily  from the domain of essential self-adjointness of $K_0^{(2)}$ we get the desired result.
\end{proof}
An important role in our analysis is played by the quantity
\begin{equation} \label{defQ}
Q:= \operatorname{Re} \langle K^{(2)}_0e^{\delta\,\rho_\epsilon}g\,|\,K^{(2)}_0e^{-\delta\,\rho_\epsilon}g
 \rangle\,,
\end{equation}
which is well defined due to Lemma \ref{wdef}.
Before we show Theorem~\ref{Maintheorem3} we state two preparatory lemmata whose proofs are given in the   next subsection.
\begin{lemma}\label{LowerboundQ}
There are $R,\epsilon $-independent constants $C_1,C_2>0$ such that, for $R>R_1$ sufficiently large,
\begin{equation}
Q\ge (C_1R^2-C_2)\|g\|^2\,.
\end{equation}
\end{lemma}
\begin{lemma}\label{UpperboundQ}
There is an $R,\epsilon $-independent constant $C_3$ and an $\epsilon$-independent constant $C(R)$ such that, for $R>R_1$ sufficiently large,
\begin{equation}
Q\le C_3\|g\|^2+ C(R)\|g\|\,.
\end{equation}
\end{lemma}
\begin{proof}[Proof of Theorem \ref{Maintheorem3}]
Fix $\delta,q_1, q_2\in(0,1)$.
Combining Lemma \ref{UpperboundQ} and \ref{LowerboundQ} we find,
for  $R>R_1$ sufficiently large,
\begin{equation}\label{eq31a}
\|g\|\le (C_1R^2- C_2-C_3)^{-1} C(R)
\end{equation}
Since the right hand side of \eqref{eq31a} is
independent of $\epsilon$
we obtain, by the monotone convergence theorem,
\begin{equation}
\|e^{\delta\rho}\hat{\Psi}\|^2=\lim_{\epsilon\to 0} \|e^{\delta\rho_\epsilon}\hat{\Psi}\|^2\le ( \sup_{\epsilon>0}\|g\|+\|e^{\delta \rho}(1-f_R)\|)^2 <\infty\,.
\end{equation}
For  $M>1$ define
\begin{equation}
\widetilde{\Omega}_{q_1,q_2,M}= \Big\{ (r,j)\in \R^+\times \Z\,|\,r^2\ge M r(j)^2\Big\}\,.
\end{equation}
We have that $\widetilde{\Omega}_{q_1,q_2,M}\subset \Omega_{q_1,q_2}$. Thus,  for any $(r,j)\in \widetilde{\Omega}_{q_1,q_2,M}$, we get
\begin{equation}
\rho(r,j)=  \frac{q_2B_0}{4}(r^2-r(j)^2)\ge \frac{q_2B_0}{4} \big(1-\frac{1}{M}\big) r^2\,.
\end{equation}
Therefore, setting $\alpha:= \delta q_2 (1-M^{-1})$, we obtain
\begin{equation}\label{wwww}
\|e^{\alpha B_0/4 r^2} \mathbbm{1}_{\widetilde{\Omega}_{q_1,q_2,M}}\hat{\Psi}\|<\infty\,.
\end{equation}
 If
$(r,j)\not \in\widetilde{\Omega}_{q_1,q_2,M}$ then
\begin{equation}
m_j\ge \frac{(q_1^2-q_2^2)B_0^2r^2}{4M\tilde{B}}=:\beta r^2.
\end{equation}
Thus, thanks to Lemma \ref{RotAnalytic} we deduce, for any $\gamma>0$, that
\begin{equation}\label{deca}
\|e^{\beta \gamma r^2} \mathbbm{1}_{\widetilde{\Omega}^c_{q_1,q_2,M}}\hat{\Psi}\| <\infty\,.
\end{equation}
Choosing $\gamma= \alpha/\beta \cdot B_0/4$ and combining \eqref{deca} with \eqref{wwww} we conclude that $\|e^{\alpha B_0/4 r^2} \hat{\Psi}\| <\infty$. The latter holds for $\alpha>0$  arbitrarily close to 1, since $\delta$ and $q_2$
can be chosen arbitrarily close to 1 and $M>1$ can be as large as we want. This proves the theorem.
\end{proof}
\subsection{Proof of lemmas \ref{LowerboundQ} and  \ref{UpperboundQ} }
Before we give the proof of lemmas  \ref{LowerboundQ}  and  \ref{UpperboundQ} we need a preparatory result.
\begin{lemma}\label{LowerboundK02}
For $R>R_1$ sufficiently large we have that
\begin{equation}
\|K^{(2)}_0 g\|^2_{\mathcal{H}^{(2)}}
\ge \mu^2 q_2^2B_0^2 \|r\,g\|^2/4-\|r^{-1}\,g\|^2/4
-\tilde{B} \,\|g\|^2\,.
\end{equation}
\end{lemma}
\begin{proof}
Let us write $g=(g^+,g^-)^{\mathrm{T}}$ and $g_j^\pm:= g^\pm(\cdot,j)$.
By Equation \eqref{DefK02} we have
\begin{align*}
\|K^{(2)}_0 g\|^2_{\mathcal{H}^{(2)}}&
=\sum_{j\in  \Z} \Big(\|(\partial_r - m_jr^{-1}+A(r))g_j^+\|^2\\
&\phantom{=\sum_{j\in  \Z} \Big(\|(\partial_r}
\quad +\|(-\partial_r - m_jr^{-1}+A(r))g_j^-\|^2\Big)\,.
\end{align*}
Furthermore, dropping the term $-\partial_r^2$, we get
\begin{align*}
\|(\pm \partial_r - m_jr^{-1}+A(r))g_j^\pm\|^2
&\ge \big \langle g^\pm_j\,|\,\big((m^2_j\mp m_j)r^{-2}+A(r)^2 \big)g_j^\pm\big \rangle\\
&\quad+ \big \langle g^\pm_j\,|\,\mp \partial_r A(r)-2m_j r^{-1}A(r)\big)g_j^\pm\big \rangle\,.
\end{align*}
Observe that  (A2) implies that
\begin{align}
  \label{eq:7}
  \frac{1}{r^2}\int_{0}^r b(s)s ds={\rm o}(1)\,,\quad\mbox{as}\quad r\to\infty\,.
\end{align}
This can be seen by splitting the integral above in the regions where $b(s)s$ is integrable and the  one where $b$ decays in the $L^\infty$- norm. Hence, given $q_3\in(q_1,1)$ we find, using \eqref{eq:7}, a constant $R_2>R_1$ such that, for all $r>R_2$,
\begin{equation}
  \label{eq:8}
\begin{split}
  &B(r)\ge q_3 B_0\,,\qquad A(r)\ge q_1 B_0r/2\,,\\
&|\partial_r A(r)|\le \tilde{B}\,,\qquad
A(r)\le \tilde{B} r/2\,.
\end{split}
\end{equation}
Therefore, for all $r>R>R_2$, we get
\begin{align*}
  &\|(\pm \partial_r - m_jr^{-1}+A(r))g_j^\pm\|^2\\&\qquad
\ge \sps{g_j^\pm}{(-r^{-2}/4+q_1^2B_0^2 r^2/4-2m_jr^{-1}A(r) -\tilde{B})g_j^\pm}\,,
\end{align*}
where we also use that $(m^2_j\pm m_j)\ge -1/4$.

Assume that $m_j<0$. Since  $q_1>q_2$ and   $A(r)>0$, for $r>R_2$, we find that
\begin{equation}
\|(\pm \partial_r - m_jr^{-1}+A(r))g_j^\pm\|^2
\ge \big \langle g^\pm_j\,|\,\big(q_2^2B_0^2 r^2/4-r^{-2}/4-\tilde{B} \big)g_j^\pm\big \rangle\,.
\end{equation}
Assume now that $m_j\ge 0$.  Recall that $A(r)\le \tilde{B} r/2$, for $r>R_2$. Using  that $m_j\le r^2(q_1^2-\mu^2q_2^2)B_0^2/(4\tilde{B})$ on $\operatorname{supp}g \subset \Omega_{q_1,\mu q_2}$ we get
\begin{align*}
\|&(\pm \partial_r - m_jr^{-1}+A(r))g_j^\pm\|^2\\
&\ge \big \langle g^\pm_j\,|\,\big(q_1^2B_0^2 r^2/4-r^{-2}/4-m_j \tilde{B}-\tilde{B} \big)g_j^\pm\big \rangle\\
&\ge \big \langle g^\pm_j\,|\,\big( \mu^2 q_2^2 B_0^2 r^2/4-r^{-2}/4
-\tilde{B}\big)g_j^\pm\big \rangle\,.
\end{align*}
This finishes the proof.
\end{proof}
\begin{proof}[Proof of Lemma \ref{LowerboundQ}]
Notice that
\begin{align*}
e^{\pm \delta\,\rho_\epsilon}&K^{(2)}_0 e^{\mp \delta\,\rho_\epsilon}
= K^{(2)}_0+ Z^{\pm\rho_\epsilon},\qquad
Z^{\pm\rho_\epsilon}:= \pm \mathrm{i\,} \delta\partial_r \rho_\epsilon \sigma_2\,.
\end{align*}
Thus, we have
\begin{align*}
Q&=\operatorname{Re} \langle (K^{(2)}_0
+Z^{-\rho_\epsilon})g\,|\,(K^{(2)}_0+Z^{\rho_\epsilon})g \rangle\\
&= \| K^{(2)}_0 g \|^2 -\delta^2\|\partial_r\rho_\epsilon g\|^2
\end{align*}
Since $|\partial_r\rho_\epsilon|\le|\partial_r\rho|\le q_2 B_0 r/2$  we find
\begin{equation}
Q\ge \| K^{(2)}_0 g \|^2-(1/4)\delta^2q_2^2B_0^2\| r g\|^2\,.
\end{equation}
Combining this with Lemma \ref{LowerboundK02} and that $\operatorname{supp}g\subset \{ (r,j)\,|\, r\ge R\}$ we obtain
(recall that $0<\delta<\mu<1$)
\begin{equation}
Q\ge \Big((\mu^2-\delta^2)q_2^2B_0^2 R^2/4-R^{-2}/4-\tilde{B}\Big)\|g\|^2\,.
\end{equation}
This concludes the proof.
\end{proof}
\begin{proof}[Proof of Lemma \ref{UpperboundQ}]
We clearly have
\begin{align}\label{qq}
Q&\le |\sps{K_0^{(2)} e^{\delta \rho_\epsilon}g}{f_R (E-W)\hat{\Psi}}| +
|\sps{K_0^{(2)} e^{\delta \rho_\epsilon}g}{\sigma_2 (\partial_r f_R) \hat{\Psi}}|\,.
\end{align}
We analyze each of the above terms separately. Using that $(K_0^{(2)}+W)\hat{\Psi}=E\hat{\Psi}$ and noting that $W f_R$ extends  trivially  to a bounded operator (for $R>R_1$ large enough), we have, for any  $\eta\in\mathcal{F}UC_0^\infty (\R^2;\C^2)$,
\begin{align*}
  \sps{K_0^{(2)} \eta}{f_R (E-W)\hat{\Psi}}&= \sps{(E-W)f_R K_0^{(2)} \eta}{\hat{\Psi}}\\
&=\sps{K_0^{(2)}f_R (E-W)\eta}{\hat{\Psi}}+
\sps{[(E-W)f_R,K_0^{(2)} ]\eta}{\hat{\Psi}}\\
&=\sps{\eta}{(E-W)^2f_R\hat{\Psi}}+\sps{\eta}{[W,K^{(2)}_0]f_R\hat{\Psi}}\\
&\quad+\sps{\eta}{\mathrm{i\,} \sigma_2 (\partial_r f_R)(W-E)\hat{\Psi}}\,.
\end{align*}
This identity extends to any $\eta\in \mathcal{D}(K^{(2)}_0)$, in particular, we may choose  $\eta=e^{\delta \rho_\epsilon}g$ (see Lemma~\ref{wdef}). Thus,  using Lemma~\ref{KonjugiertW}, we find  a constant $C>0$, independent of  $R$ and $\epsilon$, such that
\begin{align*}
&|\sps{K_0^{(2)} e^{\delta \rho_\epsilon}g}{f_R (E-W)\hat{\Psi}}|\\
&\qquad\le \|g\|\,
\|e^{\delta \rho_\epsilon} \big[ (E-W)^2f_R+[W,K^{(2)}_0]f_R+\mathrm{i\,} \sigma_2 (\partial_r f_R)(W-E)\big]\hat{\Psi}\|\\
&\qquad\le C \|g\| \, \|e^{\delta \rho_\epsilon}\hat{\Psi}\|\le
C\|g\|(\|g\|+\|e^{\delta \rho}(1-f_R)\|)\,.
\end{align*}
We now treat the second term in \eqref{qq}.
We define the operators $\Upsilon$ and $L$ acting, for any
$h\in\mathcal{H}^{(2)}$ and $ (r,j)\in\R^+\times \Z$, as
\begin{align*}
 & (\Upsilon h)(r,j)=e^{-|m_j|}h(r,j)\,,\\
&(Lh)(r,j)= (2\sigma_1\sigma_2(m_j r^{-1}+A(r))(\partial_rf_R\Upsilon h)(r,j)\,.
\end{align*}
Clearly, since $A(r)$ is bounded on the support of $\partial_rf_R$ (for $R>R_1$ large enough; see \eqref{eq:8}) $L$ is an anti-symmetric  bounded operator on $\mathcal{H}^{(2)}$.
With these definitions we have, using again the eigenvalue equation, that for any $\eta\in\mathcal{F}UC_0^\infty (\R^2;\C^2)$
\begin{align*}
\sps{&K_0^{(2)} \eta}{\sigma_2 (\partial_r f_R) \hat{\Psi}}\\ \nonumber
&=\sps{K_0^{(2)}\sigma_2 (\partial_r f_R)\eta}{  \hat{\Psi}}
+ \sps{\eta}{  \mathbbm{1}_{\operatorname{supp}\partial_rf_R}(\mathrm{i\,} \partial^2_r f_R\hat{\Psi}- L \Upsilon^{-1}\hat{\Psi} )}\\ \nonumber
&= \sps{\eta}{\mathbbm{1}_{\operatorname{supp}\partial_rf_R}\big(\sigma_2 (\partial_r f_R)(E-W)\hat{\Psi}+  \mathrm{i\,} \partial^2_r f_R\hat{\Psi}- L \Upsilon^{-1}\hat{\Psi} \big)}\,.
\end{align*}
Note that $\Upsilon^{-1}\hat{\Psi}\in \mathcal{H}^{(2)}$ by Lemma \ref{RotAnalytic}.
Next, we extend this identity to $\eta\in \mathcal{D}(K_0^{(2)})$ and replace $\eta$ by $e^{\delta \rho_\epsilon}g$.
Using that $e^{\delta \rho_\epsilon}\mathbbm{1}_{\operatorname{supp}\partial_rf_R}$ is bounded uniformly in $\epsilon>0$, we find  $\epsilon$-independent constants $C(R),C'(R)>0$ such that
\begin{align*}\label{com1}
  & |\sps{K_0^{(2)}e^{\delta \rho_\epsilon}g}{\sigma_2 (\partial_r
    f_R) \hat{\Psi}}| \\
&\quad\le C'(R)\|g\| \,\|e^{\delta
    \rho}\mathbbm{1}_{\operatorname{supp}\partial_rf_R}\| \big(\|
  \Upsilon^{-1} \hat{\Psi}\|+\|\mathbbm{1}_{\operatorname{supp}\partial_rf_R}W\hat{\Psi}\|\big)\\
&\quad\le
  C(R)\|g\|\,,
\end{align*}
where in the last inequality we use again Lemma \ref{RotAnalytic}.
Therefore, we obtain from \eqref{qq} and the above bounds that $$Q\le \|g\|\big(C\|g\|+C\|e^{\delta \rho}(1-f_R)\|+C(R)\big)\,,$$
which concludes the proof.
\end{proof}

redpajama_set_name: RedPajamaArXiv


Is Educational Fundraising Out of Control?
Activities range widely, from old standbys like pizza lunches, bake sales, chocolate-covered-almond and wrapping-paper drives to big moneymakers like black-tie galas and "direct-ask" donation campaigns. And this is all added to community fundraisers. "Every direction they turn, parents are saturated with fundraising," says Naomi Kruse, administrative officer with the Manitoba Association of Parent Councils, and mother of Brandon, 12, and Kirsten, 7. "It can sap a parent, not only financially but also physically and emotionally." With all the work that volunteers put in versus the amount reaped, Puddifoot says that they would get more monetary benefit if parents simply found part-time jobs and contributed that salary to the school.
Ungerleider agrees. "It's pretty easy to pass the hat in West Vancouver, one of the most advantaged communities in the country. But if you're talking about the downtown east side, it's much more difficult, because the community is less affluent and can't support it." The numbers bear this out: PFE found that the top 10 percent of fundraising schools raised more than the bottom 80 percent combined.
There are signs of a growing movement against educational fundraising. In 2006, concerned about the development of a two-tier system, a school board trustee in Victoria sued the B.C. government over school fees, arguing that they contravene the province's School Act — and its responsibility to provide free education for all children. The B.C. Supreme Court agreed, banning all fees for materials and equipment in courses that lead to graduation. In Ontario, the province's New Democratic Party introduced a private member's bill in the spring of last year that sought to prohibit fundraising for essentials. Cleverly called the "No More Tiers Act," the bill — which was not passed into law by the ruling Liberals — would have banned the use of privately raised funds for things like textbooks, teacher salaries and school repairs.
Contributing editor Tim Johnson is getting his wallet ready to buy chocolate-covered almonds when his niece, Brooke, hits school age.
Keep reading for ways to make sure your child's next school fundraiser is a hit.


When sixty-eight year old Sarah Miller moves into the Cunningham Village retirement community, she is mourning the loss of her husband, her young grandson, and the place that has been home for forty-two years. But Sarah is a survivor. As she reaches out into the retirement community that is to become home, she finds friends, activities, new hobbies, and a possible love interest. In the eighth installment of this fun, friendly series of cozy mysteries, a friend has gone missing, and Sarah is told to leave it to the professionals—but Sarah refuses to accept their assumptions about her friend's identity.


And so on, until you get to the end of the stem outline, where you don't bring your needle back up ahead of thread but just complete the stitch you just formed.
Notice your last satin stitch won't be all that small, it's maybe half as wide as the first one, the outlining here is kind of subtle. Try it, you'll see!
Simple, right? It does take a bit of practice. One thing I'd say is make sure not to pull your stitches too tight. I find that embroidering with a hoop helps with this, even if, like me, you leave your fabric a bit loose so you can more easily embroider with sewing stitches – that is, bring your needle back up to the right side of the embroidery at the same time as you stitched down to the wrong side. But the old ladies who make a lot of everything that's still embroidered in Hungary almost never use hoops… And their stitches have the right tension anyway.
This entry was posted in Embroidery, Learn, Recommendations and tagged backstitch, boszorkanyos, buzsak, colette, hungarian sampler, satin stitch, sew-along, sorbetto top, tutorials, witchy stitching. Bookmark the permalink.


CoComment is a site that looks very promising. It will allow you to track where you leave comments, put a copy of the comments you leave here and there on your own website for your readers. It brings everything full circle. Sadly it is a popularity contest apparently as you need an invitation code and nobody sent me one. Maybe I can trade a gmail invite (or wait that was then) or a Newsvine tip or two?


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text:
CROWN STREET FILMS
What Makes Crown Street Different
In the world of marketing, there are the "five P's." Product. Price, Promotion, Place, and People. At Crown Street Films, taken a little creative license (after all, that's what we do) and created our own five P's. People, Process, Personality, Product, and Passion.
There isn't a more true statement than "If you love what you do, you will never work again". In today's vernacular the word "passion" or "passionate" can feel forced, but at Crown Street Films, it's what drives us every day. We are passionate about creating films that can help deliver a message.
Even an Oscar-winning script will end up like Ishtar if you don't have the right people working on the project.
Creating compelling, educational, and compelling content is a process. We've mastered the process involved in creating your content. However, just like the United States Constitution states, to strive for a more perfect union, we always look to perfect and improve our process to serve you better.
McCann famously coined the phrase that advertising is "Truth Well Told." And that is still the case, but with the infinite amount of content disseminated daily, making your content and story stand out and resonate with your audience needs personality. We work with you to either bring your personality to your message or craft a message with personality.
That's what it's all about. It's getting your product, service, or message out into the universe.
You will never get a "surprise" bill at the end. We will be upfront and transparent about all expenses and costs associated with creating killer content for you. We've all heard the cliche about you can have it Great, Fast, and affordable, but you can only pick two.
At Crown Street Films, it's our mission to provide you with all three choices. We may not be the cheapest, but we are confident that our price to value ratio will exceed your expectations.
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"THAT is not true. That is a complete lie," said former St Pius X, Adamstown teacher Edward "Ted" Hall in Newcastle Court on Tuesday to an allegation he indecently assaulted a student at the school in 1978 while holding him up against a wall with one hand. Mr Hall strongly rejected allegations he indecently assaulted 11 male students while on school excursions, at his Merewether unit, at the school, in the surf and on overnight shooting trips during his more than 13 years at St Pius between 1973 and June 1986. During his first appearance in the third week of his trial Mr Hall answered "That never happened", "No such excursion ever occurred", "That is not true" and "Absolutely not" as his barrister Colin Heazlewood asked him to respond to specific allegations. Mr Hall pleaded not guilty to 31 charges including sexual intercourse, assault, acts of indecency and attempting to detain a boy for sexual gratification in a judge-alone trial before Judge Tim Gartelmann.  "That is a complete fabrication," he said after one former student alleged Mr Hall forced his penis into the student's mouth at the school in the early 1980s. The student alleged Mr Hall indecently assaulted him in a number of different ways between 1980 and 1981. "Did you do that at all?" Mr Heazlewood asked. "Not at all," Mr Hall said. He denied owning a selection of canes and keeping them in a cupboard near his school desk. He denied caning boys in the classroom, and particularly in the corners of the classroom where a student alleged he was also indecently assaulted. "No. Never, never, never," Mr Hall told Judge Gartelmann. He told the court he took students to stay at his parents' Junee property comprising "1500 acres plus a homestead on separate 21 acres", based on an "open invitation to students in year 8 or year 9". He took year 10 students on trips to Brisbane between 1980 and 1985 after "the kids requested a trip north" and later expanded it to include a group of year 9 students. He denied indecently assaulting a student at a YMCA camp because he had never attended the camp and denied indecently assaulting students during shooting trips to the Gloucester area. "Nothing happened," Mr Hall said after one former student alleged he was indecently assaulted at Limeburners Creek during one of the Gloucester trips. Mr Hall denied that anything of a sexual nature occurred during five shooting trips to the Gloucester area with a St Pius student in 1982 and 1983, after the student alleged he was sexually abused at a Stroud hotel during one trip where the two shared a room. During two of the trips bad weather forced them to camp overnight at Craven and Stratford railway stations and on the fifth trip bad weather forced them to stay at the hotel, Mr Hall said. The court was told the student spent the night at Mr Hall's unit after the teacher provided dinner for the student and his parents, and later offered to have the student stay the night so that he could drop him at sport the next morning. The student alleged Mr Hall showed him pornography and gave him alcohol. Mr Hall denied having a television or a video machine at the time. "Did you at any time indecently or sexually assault him?" Mr Heazlewood asked. "No I did not." Mr Hall denied ever giving students alcohol after a number alleged the teacher regularly provided alcohol and food for students at his unit. "You supplied them with alcohol?" Mr Heazlewood asked. "Never," Mr Hall said. "Never?" Mr Heazlewood repeated. "Never," Mr Hall said. He denied shaking hands with students so aggressively that one alleged a nerve in his shoulder pinched, but confirmed he engaged in a "good clutch" with students during a game called "Man or mouse" which he said predated him at the school. "I always shook their hand but never pinched a nerve," Mr Hall said. "It was a friendly sort of thing." Mr Hall left St Pius in June, 1986 shortly after a boy and his parents reported sexual abuse allegations to police, and later taught at Sydney private boys' schools including Newington College and Trinity Grammar. He was known as Tony Hall at Newington College. On Tuesday Judge Gartelmann directed that Mr Hall be found not guilty of four of the charges.  The trial continues.
https://nnimgt-a.akamaihd.net/transform/v1/crop/frm/pBvqKxHgUnSrdaeEwDT49F/f537aab2-52f0-4879-8ec1-d81d2a3bec95.jpg/r9_378_4023_2646_w1200_h678_fmax.jpg
September 4 2018 - 6:00PM
Edward 'Ted' Hall denies indecently assaulting 11 former students
Joanne McCarthy
"THAT is not true. That is a complete lie," said former St Pius X, Adamstown teacher Edward "Ted" Hall in Newcastle Court on Tuesday to an allegation he indecently assaulted a student at the school in 1978 while holding him up against a wall with one hand.
Mr Hall strongly rejected allegations he indecently assaulted 11 male students while on school excursions, at his Merewether unit, at the school, in the surf and on overnight shooting trips during his more than 13 years at St Pius between 1973 and June 1986.
During his first appearance in the third week of his trial Mr Hall answered "That never happened", "No such excursion ever occurred", "That is not true" and "Absolutely not" as his barrister Colin Heazlewood asked him to respond to specific allegations.
Mr Hall pleaded not guilty to 31 charges including sexual intercourse, assault, acts of indecency and attempting to detain a boy for sexual gratification in a judge-alone trial before Judge Tim Gartelmann.
"That is a complete fabrication," he said after one former student alleged Mr Hall forced his penis into the student's mouth at the school in the early 1980s. The student alleged Mr Hall indecently assaulted him in a number of different ways between 1980 and 1981.
"Did you do that at all?" Mr Heazlewood asked.
"Not at all," Mr Hall said.
He denied owning a selection of canes and keeping them in a cupboard near his school desk. He denied caning boys in the classroom, and particularly in the corners of the classroom where a student alleged he was also indecently assaulted.
History: Former St Pius X, Adamstown teacher Ted Hall during his time at the school. He has pleaded not guilty to 31 charges of sexually abusing former students between 1973 and 1986.
"No. Never, never, never," Mr Hall told Judge Gartelmann.
He told the court he took students to stay at his parents' Junee property comprising "1500 acres plus a homestead on separate 21 acres", based on an "open invitation to students in year 8 or year 9".
He took year 10 students on trips to Brisbane between 1980 and 1985 after "the kids requested a trip north" and later expanded it to include a group of year 9 students. He denied indecently assaulting a student at a YMCA camp because he had never attended the camp and denied indecently assaulting students during shooting trips to the Gloucester area.
"Nothing happened," Mr Hall said after one former student alleged he was indecently assaulted at Limeburners Creek during one of the Gloucester trips.
Mr Hall denied that anything of a sexual nature occurred during five shooting trips to the Gloucester area with a St Pius student in 1982 and 1983, after the student alleged he was sexually abused at a Stroud hotel during one trip where the two shared a room.
During two of the trips bad weather forced them to camp overnight at Craven and Stratford railway stations and on the fifth trip bad weather forced them to stay at the hotel, Mr Hall said.
The court was told the student spent the night at Mr Hall's unit after the teacher provided dinner for the student and his parents, and later offered to have the student stay the night so that he could drop him at sport the next morning.
The student alleged Mr Hall showed him pornography and gave him alcohol.
Mr Hall denied having a television or a video machine at the time.
"Did you at any time indecently or sexually assault him?" Mr Heazlewood asked.
"No I did not."
Mr Hall denied ever giving students alcohol after a number alleged the teacher regularly provided alcohol and food for students at his unit.
"You supplied them with alcohol?" Mr Heazlewood asked.
"Never," Mr Hall said.
"Never?" Mr Heazlewood repeated.
He denied shaking hands with students so aggressively that one alleged a nerve in his shoulder pinched, but confirmed he engaged in a "good clutch" with students during a game called "Man or mouse" which he said predated him at the school.
"I always shook their hand but never pinched a nerve," Mr Hall said.
"It was a friendly sort of thing."
Mr Hall left St Pius in June, 1986 shortly after a boy and his parents reported sexual abuse allegations to police, and later taught at Sydney private boys' schools including Newington College and Trinity Grammar. He was known as Tony Hall at Newington College.
On Tuesday Judge Gartelmann directed that Mr Hall be found not guilty of four of the charges.
This story 'That never happened': former St Pius teacher denies sexual abuse first appeared on Newcastle Herald.


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World AIDS Day, observed on December 1st every year, is dedicated to raising awareness of the AIDS pandemic caused by the spread of HIV infection.
Government and health officials observe the day, often with speeches or forums on the AIDS topics. Since 1995, the President of the United States has made an official proclamation on World AIDS Day. Governments of other nations have followed suit and issued similar announcements.
Positive Living North commemorates this day with the World AIDS Day Candle Light Vigil. This year's vigil will take place on Friday December 1st, 2017 at the Fire Pit Cultural Drop-In Centre. Please join us for a moment of silence and rememberance to honour those living with HIV and those we have lost to AIDS. The formal portion of the agenda will be followed by a dinner. If you would like further information, please contact PLN.


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