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Quantum Hall effect, extended states
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%%%%%%%  declarations:  %%%%%%%%%%

\newtheorem{thm}{Theorem}
\newtheorem{lem}[thm]{Lemma}
\newtheorem{cor}[thm]{Corollary}
\newtheorem{exa}[thm]{Example}
\title{Extended Quantum Hall edge states. General domains}
\author{J. Fr\"ohlich, G.M. Graf, J. Walcher\\
\normalsize\it Theoretische Physik,
ETH-H\"onggerberg, CH--8093 Z\"urich}

\begin{document}


\maketitle
\vspace{0.4cm}
\begin{abstract}
We extend recent results about extended edge states in Quantum Hall systems,
which have been obtained for a half--plane, to more general geometries.
\end{abstract}
\bigskip
\section{Results}
\label{sect:r}

At first, we consider an electron of charge $e=+1$ moving in the half--plane
\begin{equation}
\Omega=\{(x_1,x_2)\in\Bbb R^2\mid x_1<0\}
\label{eq:r2}
\end{equation}
subject to a homogeneous magnetic field $B=\partial_1A_2-\partial_2A_1>0$.
The {\it guiding center} of its trajectory is 
\begin{equation}
Z=x-B^{-1}\varepsilon (p-A)\;,\qquad A=(A_1,A_2)\;,\qquad\varepsilon=
\left(\matrix 0&-1\\1& 0\endmatrix\right)\;,
\label{eq:r3}
\end{equation}
with $\varepsilon$ representing a rotation of angle $\pi/2$. Classically, the 
electron circles clockwise around the fixed guiding center as long 
as its trajectory does not hit the domain boundary. If it does, then the 
electron follows the boundary counterclockwise by repeatedly bouncing at it. 
The quantum mechanical counterpart thereof is some absolutely continuous 
spectrum which,
due to the chirality of the extended edge states, is stable under addition of 
small perturbations, like e.g. disorder (\cite{0}).

The unperturbed Hamiltonian is 
\begin{equation*}
H_0=(-\i\nabla-A)^2
\end{equation*}
on ${\rm L}^2(\Omega)$ with Dirichlet boundary conditions, i.e., with domain 
$\mathcal D(H_0)={\rm W}^2(\Omega, A)\cap{\rm W}^1_0(\Omega, A)$, where the
magnetic Sobolev spaces are defined using covariant derivatives, e.g., 
\begin{equation}
{\rm W}^1(\Omega, A)=\{\psi\in{\rm L}^2(\Omega)\mid
(-\i\partial_i-A_i)\psi\in {\rm L}^2(\Omega),\,i=1,2\}\;.
\label{eq:r5}
\end{equation}
The perturbation $V=V(x)$ is included through
$H=H_0+V$, where we assume $V$ to be bounded.

\begin{thm} Let $E/B\notin 2\mathbb N +1$. For sufficiently small
$\|V\|_\infty/B$, the spectrum of $H$ on the half--plane (\ref{eq:r2}) is 
purely absolutely continuous near $E$.
\label{thm:r1}
\end{thm}

\medskip\noindent
Here, and below, `sufficiently small' is meant depending only on quantities
mentioned before, i.e., on $E/B$, but not, e.g., on $B$. We remark that 
$(2\mathbb N +1)B$ is just the spectrum, consisting of Landau 
levels, of $H_0$ in the full plane $\mathbb R^2$. 

A discussion of various boundary conditions for quantum Hall systems is 
found in \cite{1}.
Closely related to our work is \cite{2}, where a confining
potential instead of Dirichlet boundary condition is considered, and the
spectrum is shown to be continuous at $E$. 
(see also \cite{3}). The proof of Theorem 
\ref{thm:r1} given below is basically contained in \cite{3,4};
we reproduce it below in a slightly streamlined form, thereby also using ideas
from \cite{5}, where the same result is established. However, we 
shall not rely on methods which, like Fourier 
decomposition
along the $x_2$--axis, are special to the half--plane geometry. This allows 
for a partial extension of the result to more general unbounded domains 
$\Omega\subset\mathbb R^2$. The precise {\it geometric assumption (GA)} is 
stated in Section \ref{sect:gd}. Here we just mention that it allows for the 
domains (a, b) in the following figure, but not for (c).
\begin{figure}[h]
\begin{center}

\input{fig1.pstex_t}
 
\caption{Examples}
\label{figure:fig1}
\end{center}
\end{figure}

At each collision the guiding 
center moves forward in the direction of the tangent vector to 
$\partial\Omega$ at the collision point. Yet, this may be a backward motion
w.r.t. the tangent vector at the next collision point, with the result that a 
classical edge 
trajectory can get trapped. No trapping is possible however if the
cyclotron radius of the electron is small compared 
to the radius of curvature of $\partial\Omega$ (more precisely: compared to 
the radius of injectivity of the tubular map associated with 
$\partial\Omega$). Then this
bouncing motion will result in an effective counterclockwise progression along
$\partial\Omega$. 

\begin{thm} Assume $\Omega$ satisfies (GA). Fix $E/B\notin 2\mathbb N +1$ and 
let $\|V\|_\infty/B$ be small 
enough. Then, the spectrum of $H$ is purely absolutely continuous near $E$,
provided $B$ is large enough. 
\label{thm:r2}
\end{thm}

\medskip\noindent
Finally, no trapping is possible if $\Omega$ lies on one side of the graph 
of a function $f$,
\begin{equation}
\Omega=\{(x_1,x_2)\in\Bbb R^2\mid x_1<f(x_2)\}\;,
\label{eq:r10}
\end{equation}
with $f'\in{\rm C}^2_{\rm unif}(\mathbb R)$. 
In fact, 
as for the half--plane, $Z_2$ is increasing at each reflection.

\begin{thm} Let $E/B\notin 2\mathbb N +1$, and $B_0>0$. For sufficiently small
$\|V\|_\infty/B$, the spectrum of $H$ on the domain (\ref{eq:r10}) is 
purely absolutely continuous near $E$, provided $B\ge B_0$.
\label{thm:r3}
\end{thm}

\medskip\noindent
Under the scaling $x\to\lambda x,\,B\to \lambda^{-2}B,\,V\to \lambda^{-2}V$
we have $H\to\lambda^{-2}H$. Without loss of generality we may thus set 
\begin{equation*}
B=1
\end{equation*}
in Theorem \ref{thm:r1}, and the same is true for 
Theorem~\ref{thm:r2} if its last sentence is replaced by: {\it Then, the 
spectrum of 
$H$ on $\lambda\Omega$ is purely absolutely continuous near $E$,
provided $\lambda>0$ is large enough}. Similarly the conclusion of Theorem 
\ref{thm:r3} is {\it ... the spectrum of $H$ on the domain defined by 
$f_\lambda(x)=\lambda f(x/\lambda)$ is  is 
purely absolutely continuous near $E$, provided 
$\lambda\ge\lambda_0=B_0^{1/2}$.} The theorems will be proved in this
form.


Theorems 1--3 do not rule out the possibility that the spectrum of $H$ is 
empty near
$E$. However, one can show that, if $E/B>1$, then $E\in\sigma(H_0)$ under the 
assumptions of any of the theorems. This statement typically extends from 
$\sigma(H_0)$ to
$\sigma(H)$, e.g., for $V$ a disorder potential (see \cite{5a} for details).


\bigskip
\section{The half--plane}
\label{sect:hs}

We begin the proof of Theorem \ref{thm:r1} with some preliminaries (1.--3.):

\smallskip

1. We shall work in the Landau gauge
\begin{equation}
A=(0, x_1)\;.
\label{eq:hs15}
\end{equation}
\smallskip

2. Let $p=-\i\nabla$. The component
$p_1=-\i\partial_1$ can not be defined self-adjointly on ${\rm L}^2(\Omega)$,
a fact related to the integration by parts identity 
\begin{equation}
(p_1\omega,\rho)-(\omega,p_1\rho)=\i\langle\omega,\rho\rangle\;,
\qquad(\omega,\rho\in {\rm W}^1(\Omega, A))\;,
\label{eq:hs17}
\end{equation}
where $\langle\cdot,\cdot\rangle$
denotes the inner product on ${\rm L}^2(\partial\Omega)$. Strictly speaking 
one should write $\langle L\omega,L\rho\rangle$ instead of 
$\langle\omega,\rho\rangle$, where $L$ is the boundary trace operator, which
is bounded as a map
$L:\,{\rm W}^1(\Omega, A)\to{\rm L}^2(\partial\Omega)$.
Below we shall need another, similar statement (see e.g. \cite{6}, Theorem 8.3
or \cite{7}, Section 7.50)

\smallskip\noindent 
{\bf Theorem A} {\it The map
\begin{equation*}
L:\,{\rm W}^2(\Omega, A)\to
{\rm W}^{3/2}(\partial\Omega)\times{\rm W}^{1/2}(\partial\Omega)\;,\qquad
\psi\mapsto(\psi\restriction\partial\Omega,\partial_1\psi\restriction\partial
\Omega)
\end{equation*}
is continuous, has $\ker L={\rm W}^2_0(\Omega, A)$, and admits a continuous
right inverse $R$: $LR=1$.}


\smallskip\noindent 
The statement in \cite{6,7} refers to $A=0$; to
extend it to the present situation, pick a function 
$j\in{\rm C}_0^\infty(-\infty,0]$ with $j(x_1)=1$ for $x_1$ near $0$. Then 
$L$ and $R$ can be replaced by $Lj$, resp. $jR$. The claim now follows,
since the map $\psi\mapsto j\psi$ is bounded as a map 
${\rm W}^2(\Omega, A)\to {\rm W}^2(\Omega, 0)$, resp. 
${\rm W}^2(\Omega, 0)\to {\rm W}^2(\Omega, A)$, for the special gauge
(\ref{eq:hs15}).

\smallskip

3. The proof of Theorem \ref{thm:r1} is based on Mourre theory
(\cite{9,10,11}), which we reproduce from \cite{12} using the 
following pieces of
notation: $\mathcal B(\mathcal H)$ is the algebra of bounded operators on the
Hilbert space $\mathcal H$; $\rho(H)\ni z$ is the resolvent set of $H$, where
$R(z):=(H-z)^{-1}$; $E_\Delta(H)$ is the spectral projection for $H$ onto 
$\Delta\subset\mathbb R$.

\goodbreak
\smallskip\noindent 
{\bf Theorem B} {\it Let $H,\,C$ be self--adjoint operators on $\mathcal H$, 
and $E\in\mathbb R$.
Assume:
\begin{itemize} 
\item[(i)]
There is $z\in\rho(H)$ such that the map
\begin{equation}
g:\,\mathbb R\to\mathcal B(\mathcal H)\;,\qquad 
s\mapsto\e^{\i sC}R(z)\e^{-\i sC}
\label{eq:hs18}
\end{equation}
is of class ${\rm C}^{1+\varepsilon}$ with $0+\le\varepsilon\le 1$ in the 
norm topology (with $\varepsilon=0+$ meaning that $g'(s)$ is 
Dini continuous);
\item[(ii)]
There is an open interval $\Delta\ni E$ and $\alpha>0$ such that
\begin{equation}
E_\Delta(H)\i[H,C]E_\Delta(H)\ge\alpha E_\Delta(H)\;.
\label{eq:hs20}
\end{equation}
\end{itemize} 
Then the spectrum of $H$ is purely absolutely continuous near $E$.
}

\smallskip\noindent

 
According to Theorem 6.2.10 in \cite{12}, $g\in{\rm C}^1$ implies that 
$[H,C]$, defined 
as a quadratic form on $\mathcal D(H)\cap\mathcal D(C)$, extends to a 
bounded operator $[H,C]:\mathcal D(H)\to\mathcal D(H)^*$. In particular, the 
l.h.s. of (\ref{eq:hs20}) is a bounded operator on $\mathcal H$. 


\medskip
An observable classically increasing in time is 
\begin{equation}
K=Z_2=x_2-p_1\;,
\label{eq:hs21}
\end{equation}
which like $p_1$ can not be defined self--adjointly on ${\rm L}^2(\Omega)$.
Nevertheless we consider its commutator with $H_0$ in the sense of quadratic
forms:

\begin{lem} For $\varphi,\psi\in\mathcal D(H_0)\cap \mathcal D(x_2)$ we have 
\begin{equation}
\i[(H_0\varphi,K\psi)-(K\varphi,H_0\psi)]=
\langle p_1\varphi,p_1\psi\rangle\;.
\label{eq:hs23}
\end{equation}
\end{lem}
  
\smallskip\noindent 
{\bf Proof.} By density we may assume that $\varphi,\psi$ are of compact
support. The separate contributions from the two terms in 
$H_0=p_1^2+(p_2-x_1)^2$ are
\begin{eqnarray}
\i[(p_1^2\varphi,K\psi)-(K\varphi,p_1^2\psi)]&=&
\langle p_1\varphi,p_1\psi\rangle\;,
\label{eq:hs25}\\
\i[((p_2-x_1)^2\varphi,K\psi)-(K\varphi,(p_2-x_1)^2\psi)]&=&0\;.
\label{eq:hs26}
\end{eqnarray}
In fact, as (\ref{eq:hs25}) is concerned, the contribution of $x_2$ to $K$ 
is, using 
(\ref{eq:hs17}),
\begin{equation*}
\i[(p_1\varphi,p_1x_2\psi)-(p_1x_2\varphi,p_1\psi)]=0\;,
\end{equation*}
whereas that of $-p_1$ is
\begin{equation*}
-\i[(p_1^2\varphi,p_1\psi)-(p_1\varphi,p_1\psi)]=
\langle p_1\varphi,p_1\psi\rangle\;.
\end{equation*}
To verify (\ref{eq:hs26}), we note that 
$[p_2-x_1,x_2-p_1]\psi=0$. Hence
\begin{eqnarray*}
((p_2-x_1)^2\varphi,(x_2-p_1)\psi)&\!=\!&
((p_2-x_1)\varphi,(p_2-x_1)(x_2-p_1)\psi)\\
&&\hskip -2.2cm=((p_2-x_1)\varphi,(x_2-p_1)(p_2-x_1)\psi)=
((x_2-p_1)(p_2-x_1)\varphi,(p_2-x_1)\psi)\\
&&\hskip -2.2cm=((p_2-x_1)(x_2-p_1)\varphi,(p_2-x_1)\psi)=
((x_2-p_1)\varphi,(p_2-x_1)^2\psi)\;,
\end{eqnarray*}
where, in the third equality, we used (\ref{eq:hs17}) with 
$(p_2-x_1)\varphi\restriction\partial\Omega=0$.\hfill$\blacksquare$

\goodbreak
\smallskip\noindent 
At energies away from the Landau levels, the quadratic form (\ref{eq:hs23}) is 
strictly positive.

\begin{lem} 
Fix $E\notin 2\mathbb N +1$. Then there is an open interval $\Delta\ni E$
and $\alpha>0$ such that
\begin{equation}
\dn p_1E_\Delta(H_0)\psi\dn^2\ge\alpha\|E_\Delta(H_0)\psi\|^2\;,
\label{eq:hs30}
\end{equation}
where we set 
$\dn \cdot\dn^2=\langle\cdot,\cdot\rangle$.
\label{lem:hs4}
\end{lem} 
  
\smallskip\noindent 
{\bf Proof.} We note that
$RL\mathcal D(H_0)\subset\mathcal D(H_0)$ and write for 
$\psi\in\mathcal D(H_0)$
\begin{equation*}
(H_0-E)\psi=(H_0-E)(\psi-RL\psi)+(H_0-E)RL\psi\;.
\end{equation*}
Since $L(\psi-RL\psi)=0$, the function $\psi-RL\psi\in{\rm W}^2_0(\Omega, A)$
can be extended to $\mathbb R^2$ by $0$, and the Dirichlet boundary condition
dropped. Hence 
$\|(H_0-E)(\psi-RL\psi)\|\ge\dist(E,2\mathbb N +1)\|\psi-RL\psi\|$.
Using 
$\|(H_0-E)RL\psi\|\le\const\|RL\psi\|_2
\le\const(\dn \psi\dn_{3/2}+\dn \partial_1\psi\dn_{1/2})$ (with
indices denoting Sobolev norms) and $\psi\restriction\partial\Omega=0$ we 
obtain
\begin{equation}
\|(H_0-E)\psi\|\ge\dist(E,2\mathbb N +1)\|\psi\|-\const
\dn \partial_1\psi\dn_{1/2}\;.
\label{eq:hs32}
\end{equation}
For any $\varepsilon>0$ we have
\begin{equation*}
\dn \partial_1\psi\dn_{1/2}\le\const(\varepsilon^{-1}\dn \partial_1\psi\dn_0+
\varepsilon\dn \partial_1\psi\dn_1)
\end{equation*}
with 
$\dn \partial_1\psi\dn_1\le\const\|\psi\|_{5/2}\le\const\|H_0^{5/4}\psi\|$.
Here we used the boundedness of the map 
${\rm W}^{5/2}(\Omega,A)\to{\rm W}^1(\partial\Omega),\,
\psi\mapsto\partial_1\psi\restriction\partial\Omega $ (see \cite{6}, 
Theorem 9.4 or \cite{7}, Theorem 7.58).
We insert these estimates in eq. (\ref{eq:hs32}) and use 
$\|H_0^{5/4}E_\Delta(H_0)\psi\|\le(|E|+|\Delta|)^{5/4}\|\psi\|$ and 
$\|(H_0-E)E_\Delta(H_0)\psi\|\le|\Delta|\|\psi\|$. For sufficiently small
$\varepsilon,\,|\Delta|$ we obtain the claim.\hfill$\blacksquare$


\smallskip\noindent 

Following \cite{5} we choose $C=x_2$ as the conjugate operator to $H$, which 
in contrast to
$K$ is self--adjoint on its natural domain. The r.h.s. in
\begin{equation*}
H(\tau)=\e^{\i\tau x_2}H\e^{-\i\tau x_2}=p_1^2+(p_2-\tau-x_1)^2+V 
=H-\tau(p_2-x_1)+\tau^2
\end{equation*}
extends from $\tau\in\mathbb R$ to an entire analytic family of type A. Hence
assumptions (i) of Theorem B is fulfilled. Assumption (ii) holds because of
the following lemma.

\begin{lem} 
Let $E\notin 2\mathbb N +1$. Then there is an open interval $\Delta\ni E$
and $\alpha>0$ such that
\begin{equation*}
E_\Delta(H)\i[H,x_2]E_\Delta(H)\ge\alpha E_\Delta(H)\;,
\end{equation*}
provided $\|V\|_\infty$ is small enough. 
\label{lem:hs5}
\end{lem} 

  
\smallskip\noindent 
{\bf Proof.} We shall prove the statement in the case $V=0$ first. Because of 
(\ref{eq:hs23}, \ref{eq:hs30}) it suffices to show that the difference 
$Bx_2-K=p_1$ contributes arbitrarily little to the commutator. In a state
$\psi=E_\Delta(H_0)\psi$ this contribution is estimated as
\begin{eqnarray*}
|(H_0\psi,p_1\psi)-(p_1\psi,H_0\psi)|&=&
|((H_0-E)\psi,p_1\psi)-(p_1\psi,(H_0-E)\psi)|\\
&\le& 2\|(H_0-E)\psi\|\|p_1\psi\|
\le 2|\Delta|(|E|+|\Delta|)^{1/2}\|\psi\|^2\;,
\end{eqnarray*}
where the equality is justified by (\ref{eq:hs17}) and 
$\psi\in\mathcal D(H_0)$. The r.h.s. can be made
arbitrarily small by letting $|\Delta|\to 0$. 
\par
We next extend \cite{5} the result to $H$ and note that
$H'(0)=\i[H,x_2]=\i[H_0,x_2]$, which is relatively bounded w.r.t. $H_0$, is 
unchanged. Let $\alpha>0$ and $\Delta\ni E$ be as given by 
Lemma~\ref{lem:hs4} (we may assume $\Delta$ to be centered on $E$). 
For $\widetilde\Delta\ni E$, we split 
\begin{equation*}
E_{\widetilde\Delta}(H)=E_\Delta(H_0)E_{\widetilde\Delta}(H)+
(1-E_\Delta(H_0))E_{\widetilde\Delta}(H)\;,
\end{equation*}
and obtain
\begin{eqnarray*}
E_{\widetilde\Delta}(H)(H'(0)-\alpha)E_{\widetilde\Delta}(H)&\ge&
E_{\widetilde\Delta}(H)E_\Delta(H_0)(H'(0)-\alpha)E_\Delta(H_0)
E_{\widetilde\Delta}(H)-\\
&&2\|(H'(0)-\alpha)(1-E_\Delta(H_0))E_{\widetilde\Delta}(H)\|\;.
\end{eqnarray*}
The first term on the r.h.s. is non--negative, and the second line can be 
estimated by a constant times
\begin{equation*}
\|H_0(1-E_\Delta(H_0))(H_0-E)^{-1}\|\|(H_0-E)E_{\widetilde\Delta}(H)\|
\le(1+2|E||\Delta|^{-1})(|\widetilde\Delta|+\|V\|_\infty)\;.
\end{equation*}
Hence,
\begin{equation*}
E_{\widetilde\Delta}(H)(H'(0)-\alpha)E_{\widetilde\Delta}(H)\ge
-\alpha/2
\end{equation*}
for $|\widetilde\Delta|+\|V\|_\infty$ small enough, and therefore
$E_{\widetilde\Delta}(H)H'(0)E_{\widetilde\Delta}(H)\ge(\alpha/2)
E_{\widetilde\Delta}(H)$.\hfill$\blacksquare$


  
\smallskip\noindent 
{\bf Proof of Theorem \ref{thm:r1}.} Application of Theorem B.
\hfill$\blacksquare$

\section{General domains}
\label{sect:gd}

%\subsection{Geometrical preliminaries}

The precise geometrical assumption made in Theorem \ref{thm:r2} is as follows.

\smallskip\noindent
{\bf Assumption (GA)} {\it Let $\Omega\subset\mathbb R^2$ be an open set.
\begin{itemize}
\item[(i)] $\Omega$ has the uniform ${\rm C}^3$--property in the sense of 
\cite{7};
\item[(ii)] Let $\partial\Omega$ consist of finitely many connected components 
$\gamma$, each parametrized by its
arclength $s$ (with the induced orientation). Let there be a function 
$s\in{\rm C}^2(\overline\Omega)$ extending
arclength from $\partial\Omega$ to $\Omega$, i.e., $s(\gamma(s'))=s'$ for 
$s'\in\mathbb R$, satisfying
\begin{equation}
\|\partial_i s\|_\infty<\infty\;,\qquad
\|\partial_i\partial_j s\|_\infty<\infty\;.
\label{eq:gp3}
\end{equation}
\end{itemize}
}

\medskip\noindent
We note that, by (i), a bounded component of $\partial\Omega$ would be a
closed curve, but by (ii) no such curve is allowed. Hence $\Omega$ is simply 
connected and unbounded. Also, (GA) is not affected by the ambiguity 
$s\mapsto s-s_0$ implicit in the definition of arclength.

We illustrate (GA) with an example. 
Let $\Omega\subset\mathbb R^2$ be simply connected open set with oriented 
boundary
$\partial\Omega$ consisting of a single unbounded smooth curve 
$\gamma\in{\rm C}^3_{\rm unif}(\mathbb R)$
parametrized by arclength $s\in\mathbb R$. For simplicity, we shall assume 
that $\gamma$ is {\it asymptotically straight}, i.e., 
\begin{equation}
\ddot\gamma\in{\rm L}^1(\d s)\;,
\label{eq:gd4}
\end{equation}
with $\dot{}=\d/\d s$. The {\it overall bending} of $\gamma$ is 
\begin{equation*}
\beta=\int_{-\infty}^\infty\ddot\gamma(s)\d s\;,
\end{equation*}
and takes values in $[-\pi,\pi]$. 

\begin{exa} If 
\begin{equation}
\beta\neq\pi\;,
\label{eq:gp5}
\end{equation}
then $\Omega$ as described satisfies (GA). In particular, the domains (a, b)
in Figure \ref{figure:fig1}
are allowed, but not (c).
\end{exa}



\smallskip\noindent 
{\bf Proof.} Only part (ii) of (GA) requires proof. Because of 
$\gamma\in{\rm C}^3_{\rm unif}(\mathbb R)$ and of (\ref{eq:gd4}) we also have
$\lim_{s\to\pm\infty}\ddot\gamma(s)=0$. Elementary geometric 
considerations show that, for large
enough $r>0$, the equation $|\gamma(s)|=r$ has exactly two solutions
$s=s_\pm(r)$, with $s_\pm(r)\to\pm\infty,\,(r\to\infty)$. We define
functions
$\varphi_\pm(r)$ through 
$\gamma(s_\pm(r))=r(\cos\varphi_\pm(r),\sin\varphi_\pm(r))$
in such a way that $\varphi_\pm(r)$ are continuous and that 
$\varphi_-(r)-\varphi_+(r)\in(0,2\pi)$. Their limiting values 
$\varphi_\pm^\infty=\lim_{r\to\infty}\varphi_\pm(r)$ exist and satisfy
$\beta+(\varphi_-^\infty-\varphi_+^\infty)=\pi$. Condition 
(\ref{eq:gp5}) implies
\begin{equation}
\varphi_-^\infty-\varphi_+^\infty>0\;.
\label{eq:gp6}
\end{equation} 
Under our assumptions,
\begin{equation}
\frac{\d s_\pm}{\d r}\to\pm 1\;,\quad
\frac{\d^2 s_\pm}{\d r^2}\to 0\;,\quad
r\frac{\d\varphi_\pm}{\d r}\to 0\;,\quad
r\frac{\d^2\varphi_\pm}{\d r^2}\to 0\;,
\label{eq:gp8}
\end{equation}
as $r\to\infty$. For $R>0$ large enough we define $s(x)$ on 
\begin{equation*}
\overline\Omega\cap\{x\mid |x|>R\}=
\{x=r(\cos\varphi,\sin\varphi)\mid r>R,\, 
\varphi_+(r)\le\varphi\le\varphi_-(r)\}
\end{equation*}
by linear interpolation along arcs $|x|=r$, i.e., by
\begin{equation*}
s(x)=s_-(r)\frac{\varphi-\varphi_+(r)}{\varphi_-(r)-\varphi_+(r)}
+s_+(r)\frac{\varphi_-(r)-\varphi}{\varphi_-(r)-\varphi_+(r)}\;;
\end{equation*}
we then smoothly extend $s$ further to the compact complement 
$\overline\Omega\cap\{x\mid |x|\le R\}$.
Now (\ref{eq:gp3}) follows from (\ref{eq:gp6}, \ref{eq:gp8}).
\hfill$\blacksquare$


\medskip\noindent 

The proof will just address the changes required for the generalization from 
Theorem~\ref{thm:r1} to Theorem~\ref{thm:r2}. For notational simplicity we 
first consider the Hamiltonian on $\Omega$. The
required scaling $\Omega\to\lambda\Omega$ (see the end of 
Section \ref{sect:r}) will be done later.  
\smallskip

 We define the components of the velocity as 
$\pi_i=-\i\partial_i-A_i$ with domain ${\rm W}_1(\Omega, A)$; see eq. 
(\ref{eq:r5}).
For $\omega,\rho\in {\rm W}_1(\Omega, A)$ we have the integration by parts
identity 
\begin{equation}
(\pi_i\omega,\rho)-(\omega,\pi_i\rho)=\i\langle\omega,n_i\rho\rangle\;,
%(\psi,\pi_i\rho)-(\pi_i\psi,\rho)=\langle\psi, n_i\rho\rangle\;,
\label{eq:gp25}
\end{equation}
where $n_i$ are the components of the {\it outer unit normal} 
$n=n(s)=-\varepsilon\dot\gamma(s)$. Also,
$H_0\psi=(\pi_1^2+\pi_2^2)\psi$ for $\psi\in\mathcal D(H_0)$.


\smallskip
We next extend the trace theorem to the present setting.

\begin{lem}  The map
\begin{equation}
L:\;{\rm W}^2(\Omega, A)\to
{\rm W}^{3/2}(\partial\Omega, A\cdot\dot\gamma)\times
{\rm W}^{1/2}(\partial\Omega, A\cdot\dot\gamma)\;,\qquad
\psi\mapsto(\psi\restriction\partial\Omega,\pi\cdot n\psi\restriction\partial
\Omega)
\label{eq:gp30}
\end{equation}
is continuous, has $\ker L={\rm W}^2_0(\Omega, A)$, and admits a continuous
right inverse $R$: $LR=1$. The norms of $L$ and $R$ depend only on the 
${\rm C}^3$--regularity data of $\Omega$.
\end{lem}

\smallskip\noindent 
{\bf Proof.} The uniform ${\rm C}^3$--property is associated with a cover
$\{U_j\}$ of $\partial\Omega$ and corresponding diffeomorphisms (of class 
${\rm C}^3$) $\Psi_j:\,B=\{x\in\mathbb R^2\mid |x|<1\}\to U_j$, such that
$\Psi_j(B\cap\{x_1<0\})=\Omega\cap U_j$; see \cite{7}, Section 4.6 for details.
The statement for $A=0$ is \cite{7}, Theorem 7.53 (see also Section 4.29 
allowing for unbounded $\partial\Omega$; or \cite{13}, Theorem 5.9). The
proof makes use of a partition of unity for $\partial\Omega$ subordinate to
$\{U_j\}$, with the effect of reducing the statement to the analogous
restriction property from $B_-\equiv B\cap\{x_1<0\}$ to $B_0\equiv
B\cap\{x_1=0\}$. Similarly, in our case ($A\neq 0$) matters are reduced to
(\ref{eq:gp30}) with the replacements $\Omega\to B_-,\,\partial\Omega\to B_0,\,
A\to\tilde A=\Psi_j^*A$ and $A\cdot\dot\gamma\to\tilde A_2$, where
$\tilde A$ is the pull--back of $A$ under $\Psi_j$, in the sense of
1--forms. The claim so left to prove is gauge covariant, and we 
may choose the special gauge
\begin{equation*}
\tilde A(x_1,x_2)=(0, \int_0^{x_1}\tilde\omega(\xi,x_2)\d\xi)\;,
\end{equation*}
where $\tilde \omega(x_1,x_2)\d x_1\wedge\d x_2=\Psi_j^*(\d x_1\wedge\d x_2)$
is the pull--back of the area 2--form on $U_j$. We note that 
$\tilde \omega,\tilde A \in{\rm C}^2(B_-)$, with
bounded derivatives. Moreover, $\tilde A=0$ on $B_0$. Now the claim again
follows from the case $\tilde A=0$, since the norms of ${\rm W}^2(B_-)$ and of
${\rm W}^2(B_-,\tilde A)$ are equivalent.\hfill$\blacksquare$

\smallskip\noindent 
{\bf Remark.} As in proof of Lemma \ref{lem:hs4}, we shall also
need the boundedness of the map 
${\rm W}^{5/2}(\Omega, A)\to{\rm W}^1(\partial\Omega, A\cdot\dot\gamma),\,
\psi\mapsto \pi\cdot n\psi\restriction\partial\Omega$. The same argument
applies and, in fact, it is only here that a uniform ${\rm C}^2$--property
does not suffice.


\medskip
In the context 
of Theorem \ref{thm:r2} a formal candidate for an observable increasing in 
time is 
$K=s(Z)$, where $s$ is arclength as given by (GA), and $Z$ is the guiding
center (\ref{eq:r3}). This
definition is unsuitable, because $p=-\i\nabla$ and hence $Z$ are not 
self--adjoint (vector) operators, and because the two components of $Z$ do not
commute. Instead, we formally linearize $s(Z)$ in $Z-x$ and set
\begin{equation}
K=s(x)-\nabla s\cdot\varepsilon\pi=s(x)-\varepsilon\pi\cdot\nabla s\;,
\label{eq:pc5}
\end{equation}
which is a well--defined (non self--adjoint) operator. The second expression
follows from 
$\varepsilon_{ij}[\pi_j,\partial_is]=
-\i\varepsilon_{ij}\partial_j\partial_is=0$. Here, and in the following, the
summation convention is understood.

\begin{lem} Let $s$ satisfy (\ref{eq:gp3}).
For $\varphi,\psi\in\mathcal D(H_0)\cap \mathcal D(s)$ we have 
\begin{equation}
\i[(H_0\varphi,K\psi)-(K\varphi,H_0\psi)]=
(\varphi,\pi_i(m_{ij}+m_{ji})\pi_j\psi)+
\langle\pi_i\varphi,(\dot\gamma_j\partial_js)\pi_i\psi\rangle\;,
\label{eq:pc6}
\end{equation}
where 
$m_{ij}(x)=\varepsilon_{jk}\partial_k\partial_is$. 
\end{lem}
 
\medskip\noindent
The last term in (\ref{eq:pc6}) can also be written as 
$\langle(\pi\cdot n)\varphi,(\dot\gamma\cdot\nabla s)(\pi\cdot n)\psi\rangle$,
since $\pi_i\psi=n_i(\pi\cdot n)\psi$ on $\gamma$ for 
$\psi\in\mathcal D(H_0)$.
If, as in (GA), $s$ is arclength on $\gamma$ we have
$\dot\gamma\cdot\nabla s=1$ by definition. 

\smallskip\noindent 
{\bf Proof.} We first claim that for 
%$\varphi\in\mathcal D(H_0)\cap\mathcal D(s),\,
$\rho\in{\rm W}_1(\Omega, A)\cap\mathcal D(s)$ we have
\begin{equation}
\i[(\pi_i\varphi,K\rho)-(K\varphi,\pi_i\rho)]=
(\pi_j\varphi, m_{ij}\rho)\;.
\label{eq:pc7}
\end{equation}
Indeed, the contribution from $s$ in (\ref{eq:pc5}) is
\begin{equation}
\i[(\pi_i\varphi,s\rho)-(s\varphi,\pi_i\rho)]=
\i(\varphi,[\pi_i,s]\rho)=(\varphi,\partial_is\rho)\;,
\label{eq:pc8}
\end{equation}
by using (\ref{eq:gp25}) and $\varphi\in\mathcal D(H_0)$, which makes the 
boundary term vanish. To compute the other contribution we note that
\begin{equation*}
\i[(\pi_i\varphi,\pi_j\rho)-(\pi_j\varphi,\pi_i\rho)]
=\i([\pi_i,\pi_j]\varphi,\rho)+
\langle\pi_i\varphi, n_j\rho\rangle-\langle\pi_j\varphi, n_i\rho\rangle
=-\varepsilon_{ij}(\varphi,\rho)\;,
\end{equation*}
since $\pi_i\varphi=n_i(\pi\cdot n)\varphi$ on $\partial\Omega$, 
and hence, for $f_j=\varepsilon_{kj}\partial_ks$,
\begin{eqnarray}
\i[(\pi_i\varphi,f_j\pi_j\rho)\!-\!(f_j\pi_j\varphi,\pi_i\rho)]
&\!=\!&\i[(\pi_i\varphi,\pi_jf_j\rho)-(\pi_j\varphi,\pi_if_j\rho)]
-(\pi_i\varphi,\partial_jf_j\rho)+(\pi_j\varphi,\partial_if_j\rho)
\nonumber\\
&\!=\!&(\varphi,\partial_is\rho)-(\pi_j\varphi,m_{ij}\rho)\;,
\label{eq:pc10}
\end{eqnarray}
because of $\varepsilon_{ij}f_j=-\partial_is,\,
\partial_jf_j=0$ and of $\partial_if_j=-m_{ij}$. Subtracting (\ref{eq:pc10})
from (\ref{eq:pc8}) yields (\ref{eq:pc7}), which can also be written as
\begin{equation}
\i[(\pi_i\omega,K\psi)-(K\omega,\pi_i\psi)]=
(\omega, m_{ij}\pi_j\psi)\;,
\label{eq:pc12}
\end{equation}
for $\omega\in{\rm W}_1(\Omega, A)\cap\mathcal D(s)$.
Moreover, (\ref{eq:gp25}) yields
\begin{equation}
\i[(K\omega,\rho)-(\omega,K\rho)]=
\langle\omega,\varepsilon_{ji}\partial_jsn_i\rho\rangle
=\langle\omega,\dot\gamma_i\partial_is\rho\rangle\;.
\label{eq:pc14}
\end{equation}
Writing the l.h.s. of (\ref{eq:pc6}) as 
\begin{equation*}
\i[(\pi_i^2\varphi,K\psi)-(K\pi_i\varphi,\pi_i\psi)\;+\;
(K\pi_i\varphi,\pi_i\psi)-(\pi_i\varphi,K\pi_i\psi)\;+\;
(\pi_i\varphi,K\pi_i\psi)-(K\varphi,\pi_i^2\psi)]
\end{equation*}
and using (\ref{eq:pc12}, \ref{eq:pc14}, \ref{eq:pc7}) with 
$\omega=\pi_i\varphi$
and $\rho=\pi_i\psi$ concludes the proof if $\omega,\rho\in\mathcal D(s)$. By
density this suffices.\hfill$\blacksquare$

\smallskip\noindent 
{\bf Proof of Theorem \ref{thm:r2}.} Statement and proof of Lemma
\ref{lem:hs4} hold true with the replacement $p_1\to\pi\cdot n$, with
$\Delta,\,\alpha$ depending only on the regularity data for $\Omega$. These
data are inherited by $\lambda\Omega$ with $\lambda\ge1$. On the other hand 
$\lambda\Omega$ satisfies part (ii) of (GA) with 
$s_\lambda(x)=\lambda s(x/\lambda)$, and correspondingly 
$m_\lambda(x)=\lambda^{-1}m(x/\lambda)$ in eq. (\ref{eq:pc6}). There, the 
first term on the r.h.s. is then estimated in absolute value for 
$\varphi=\psi=E_\Delta(H_\lambda)\psi$ by a constant times
\begin{equation*}
\lambda^{-1}\|\pi E_\Delta(H_\lambda)\psi\|^2\le 
\lambda^{-1}(|E|+|\Delta|)^{1/2}\|\psi\|^2\;.
\end{equation*}
As a result, the positivity of the commutator (\ref{eq:pc6}) on 
$E_\Delta(H_\lambda)$ is preserved for large $\lambda>0$. The conjugate
operator $C=s(x)$ gives raise to the analytic family 
\begin{equation}
H(\tau)=\e^{\i\tau s}H\e^{-\i\tau s}=
H-\tau(\nabla s\cdot\pi+\pi\cdot\nabla s)+\tau^2(\nabla s)^2\;.
\label{eq:pc20}
\end{equation}
Statement and proof of Lemma \ref{lem:hs5} are changed accordingly, with
$s(x)-K=\varepsilon\pi\cdot\nabla s$ replacing $p_1$. In (\ref{eq:pc20}) we 
suppressed the subscript $\lambda$. One should however notice that the
relative bound of $H'(0)=\nabla s\cdot\pi+\pi\cdot\nabla s$
w.r.t. $H$ is independent of $\lambda$.
\hfill$\blacksquare$


\smallskip\noindent 
{\bf Proof of Theorem \ref{thm:r3}.} We set $s(x)=x_2$, i.e., $K=x_2-\pi_1$
(as in (\ref{eq:hs21}), except for the gauge). Then, in (\ref{eq:pc6}), 
$m_{ij}=0$
and $\dot\gamma\cdot\nabla s=(1+f'(x_2)^2)^{-1/2}\ge\delta$ for some
$\delta>0$ and all $x_2\in\mathbb R$. Upon scaling, 
$f'_\lambda(x_2)=f'(x_2/\lambda)$ has ${\rm C}^2_{\rm unif}(\mathbb R)$--norm
which can be bounded independently of $\lambda\ge\lambda_0$. In particular,
bounds on the norms associated with (\ref{eq:gp30}), as well as $\delta$, are 
independent of $\lambda\ge\lambda_0$.\hfill$\blacksquare$


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\end{document}


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 3060 1215 l 3036 1233 l 2994 1248 l 2937 1265 l 2916 1269 l
 2880 1269 l 2880 1530 l 3510 1530 l 3510 270 l cp gs 0.90 setgray ef gr 
% Polyline
7.500 slw
n 4410 270 m 4410 900 l gs col0 s gr 
% Polyline
n 3240 900 m 3240 270 l gs col0 s gr 
% Polyline
gs  clippath
2490 390 m 2520 270 l 2550 390 l 2550 255 l 2490 255 l cp
clip
n 2520 270 m 2520 900 l gs col0 s gr gr

% arrowhead
n 2490 390 m 2520 270 l 2550 390 l  col0 s
% Polyline
gs  clippath
5100 390 m 5130 270 l 5160 390 l 5160 255 l 5100 255 l cp
clip
n 5130 900 m 5130 270 l gs col0 s gr gr

% arrowhead
n 5100 390 m 5130 270 l 5160 390 l  col0 s
% Polyline
gs  clippath
1576 735 m 1700 740 l 1589 794 l 1721 766 l 1709 708 l cp
clip
n 270 270 m 270 271 l 272 274 l 276 281 l 282 294 l 290 310 l
 300 329 l 309 348 l 318 367 l 327 384 l 335 399 l
 342 413 l 348 426 l 354 438 l 360 450 l 366 462 l
 372 474 l 378 486 l 385 499 l 392 513 l 399 526 l
 406 540 l 413 554 l 421 567 l 428 581 l 436 594 l
 443 606 l 450 618 l 458 630 l 465 642 l 473 654 l
 481 666 l 490 679 l 499 692 l 509 706 l 520 719 l
 530 732 l 541 745 l 551 758 l 562 770 l 572 781 l
 582 792 l 593 803 l 603 813 l 614 823 l 625 833 l
 637 843 l 649 854 l 662 864 l 675 873 l 688 883 l
 701 891 l 713 899 l 725 906 l 736 912 l 747 918 l
 758 923 l 769 927 l 781 931 l 793 934 l 806 937 l
 818 938 l 829 939 l 841 938 l 851 937 l 861 934 l
 870 931 l 878 927 l 885 923 l 892 917 l 898 910 l
 904 903 l 909 894 l 913 885 l 917 874 l 920 864 l
 923 853 l 924 842 l 924 831 l 924 820 l 923 810 l
 920 798 l 916 785 l 910 772 l 904 759 l 896 746 l
 887 733 l 877 722 l 868 713 l 858 704 l 848 698 l
 837 692 l 826 687 l 814 683 l 801 680 l 789 677 l
 776 675 l 764 673 l 753 671 l 744 669 l 735 668 l
 727 665 l 720 662 l 713 659 l 707 655 l 702 650 l
 697 645 l 694 640 l 692 634 l 690 629 l 690 623 l
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 709 577 l 716 569 l 724 561 l 733 554 l 743 548 l
 753 541 l 766 536 l 780 531 l 794 526 l 810 523 l
 826 520 l 840 519 l 854 520 l 867 522 l 878 525 l
 887 530 l 896 537 l 905 545 l 913 555 l 920 567 l
 928 579 l 934 592 l 940 605 l 946 618 l 953 630 l
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 1129 818 l 1141 823 l 1154 826 l 1166 829 l 1177 831 l
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 1671 746 l 1700 740 l gs 0.90 setgray ef gr gs col0 s gr gr

% arrowhead
n 1576 735 m 1700 740 l 1589 794 l  col0 s
$F2psEnd
rs

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