\documentclass[reqno]{amsart}
\usepackage{euscript}
\AtBeginDocument{{\noindent\small {\em Electronic Journal of
Differential Equations}, Vol. 2003(2003), No. 48, pp. 1--25.
\newline ISSN: 1072-6691. URL: http://ejde.math.swt.edu
or http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu (login: ftp)}
\thanks{\copyright 2003 Southwest Texas State University.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE--2003/48\hfil Magnetic fields and eigenvalues in gaps ]
{Magnetic barriers of compact support and eigenvalues in
spectral gaps}
\author[Rainer Hempel \& Alexander Besch \hfil EJDE--2003/48\hfilneg]
{Rainer Hempel \& Alexander Besch}
\address{Rainer Hempel \hfill\break
Institut f\"ur Analysis, Technische Universit\"at Braunschweig\\
Pockelsstra{\ss}e 14, 38106 Braunschweig\\
Germany}
\email{r.hempel@tu-bs.de}
\address{Alexander Besch \hfill\break
Institut f\"ur Analysis, Technische Universit\"at Braunschweig\\
Pockelsstra{\ss}e 14, 38106 Braunschweig\\
Germany.
Current address: Volkswagen AG, Wolfsburg, Germany}
\email{alexander.besch@volkswagen.de}
\date{}
\thanks{Submitted May 22, 2001; revised July 3, 2002; February 20, 2003.
Published April 24, 2003.}
\subjclass[2000]{35J10, 81Q10, 35P20}
\keywords{Schr\"odinger operator, magnetic field, eigenvalues, spectral gaps,
\hfill\break\indent strong coupling}
\begin{abstract}
We consider Schr\"odinger operators
$H = -\Delta + V$ in $L_2(\mathbb{R}^2)$
with a spectral gap, perturbed by a strong magnetic field $\mathcal{B}$
of compact support.
We assume here that the support of $\mathcal{B}$ is connected and has
a connected complement; the total magnetic flux may be zero
or non-zero. For a fixed point $E$ in the gap, we show that
(for a sequence of couplings tending to $\infty$) the signed
spectral flow across $E$ for the magnetic perturbation is equal
to the flow of eigenvalues produced by a high potential barrier
on the support of the magnetic field. This allows us to use
various estimates that are available for the high barrier case.
\end{abstract}
\maketitle
\newcommand{\lnorm}[1]{\|#1\|}
\newcommand{\dx}{\,{\rm d}x}
\newcommand{\supp}{\mathop{\rm supp}}
\newcommand{\Cci}[1]{C_c^\infty(#1)}
\newcommand{\commutator}[2]{[#1,#2]}
\newcommand{\scaprod}[2]{\langle #1,#2\rangle}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{assum}[theorem]{Assumption}
\newtheorem{corollary}[theorem]{Corollary}
\section{Introduction}
We study the discrete eigenvalues that may appear in a spectral gap of
a Schr\"o\-dinger operator acting in $L_2(\mathbb{R}^2)$,
\[
H = -\Delta + V, \tag{1.1}
\]
under perturbation by a strong magnetic field $\mathcal{B}$ of compact
support. In the present paper we restrict our attention to the case
where the support of $\mathcal{B}$ as well as its complement are connected sets.
For a bounded and measurable potential $V = V(x)$ and a magnetic
vector potential $\vec{a} \in C^1(\mathbb{R}^2;\mathbb{R}^2)$ with $\mathop{\rm curl}\vec{a} = \mathcal{B}$,
the associated Schr\"odinger operator is defined as
\[
H(\lambda\vec{a}) = (-{\rm i}\nabla - \lambda\vec{a}(x))^2 + V(x), \quad \lambda \ge 0.
\tag{1.2}
\]
Our main interest is then in the (signed) flow of the eigenvalues of $H(\lambda\vec{a})$
across a fixed observation point $E$ in a spectral gap of $H$,
as the coupling $\lambda$ tends to infinity. For periodic $H$, the eigenvalues of
$H(\lambda\vec{a})$ can be interpreted as electronic bound states in a thin wafer of
solid matter (semi-conductor or insulator) which is locally
penetrated by a strong magnetic field.
For a sequence of couplings $\lambda_k \to \infty$ we will establish
a direct link between the magnetic problem and the classical
case where $H$ is perturbed by $\mu \chi_\Omega$, for $\mu \to \infty$;
here $\chi_\Omega$ denotes the characteristic function
of the bounded, open set
\[
\Omega = \{ x \in \mathbb{R}^2 : \mathcal{B}(x) \ne 0\}.
\tag{1.3}
\]
To describe the relevant properties of the high barrier problem, let
\[
M = \mathbb{R}^2 \setminus \Omega
\tag{1.4}
\]
and let $H_M$ denote the operator $-\Delta + V$, acting in
$M$ with Dirichlet boundary conditions on $\partial M$. It is well known that
$H + \mu \chi_\Omega$ converges to $H_M$ in the norm resolvent sense,
as $\mu \to \infty$,
and a finite number of eigenvalues of $H + \mu \chi_\Omega$ may
cross a fixed point $E$ inside a gap of $H$; we denote
this number (counting multiplicities) as ${\EuScript N}(\Omega,E)$,
cf.~(3.2), (3.4).
All eigenvalues of $H + \mu \chi_\Omega$
are increasing functions of the coupling $\mu$.
Estimates for ${\EuScript N}(\Omega,E)$ will be discussed in Section 3.
Compactness implies that $\sigma_{\rm ess}(H_M) = \sigma_{\rm ess}(H)$
so that the spectrum of $H_M$ is discrete in $\mathbb{R} \setminus \sigma_{\rm ess}(H)$.
The eigenvalues of $H + \mu \chi_\Omega$ cross $E$ according to the repulsion
produced by the potential barrier $\mu\chi_\Omega$.
By the Avron-Herbst-Simon-bound \cite{AHS} and Dirichlet decoupling \cite{HH},
a similar
repulsive effect is to be expected in the case of a magnetic
perturbation. The magnetic case is more delicate, however,
because eigenvalues will not depend monotonically on
the coupling and, in general, we have to deal with vector potentials that do not
vanish identically in $M$. Because of the lack of monotonicity, we will rather
count the signed flow of spectral multiplicity instead of the number of eigenvalues
that cross $E$. Defining the {\it flux of $\mathcal{B}$},
\[
\Phi = \int \mathcal{B}(x) \dx, \tag{1.5}
\]
we must distinguish, as usual, between the cases $\Phi = 0$ and $\Phi \ne 0$.
For $\Phi=0$, we may pass to an equivalent vector potential that
vanishes in the interior $G$ of $M$ (assuming some regularity of $\partial G$) and
much simpler proofs are available (cf.~Remark\ref {rmk5.6});
the situation is similar in dimensions 3 and higher.
Hence the most difficult case is the one where we are in $\mathbb{R}^2$ and
$\Phi$ is non-zero; in addition, this is the case that is
closest to the experimental situation in physics. Note that
we may assume $\lambda \ge 0$ and $\Phi \ge 0$ without loss of generality since
the operators $H(\lambda\vec{a})$ and $H(-\lambda\vec{a})$ are anti-unitarily equivalent
under complex conjugation.
We will work with the following assumptions:
\begin{assum} \label{a1.1} \rm
The sets $\overline\Omega = \supp\mathcal{B}$ and
$G = (\supp \mathcal{B})^C$ are both connected, with
$\supp\mathcal{B} \ne \emptyset$.
\end{assum}
Loosely speaking, Assumption \ref{a1.1} means that $\supp\mathcal{B}$ is connected and has no holes.
Assumption \ref{a1.1} implies that the fundamental group of $G$ is isomorphic to $\mathbb{Z}$.
\begin{assum} \label{a1.2} \rm
The Dirichlet Laplacian $-\Delta_M$ of
the closed set $M$ has form core $C_c^\infty(G)$.
\end{assum}
The precise definition of $-\Delta_M$ is given in Section 2.
It is sufficient for Assumption \ref{a1.2} to hold that $\partial\Omega$ has measure zero
and that $G$ satisfies the segment condition; cf.\ Section 2
for details and a more general criterion.
Following Safronov \cite{Sa1, Sa2}, we next define
a function that counts the signed spectral multiplicity of $H(\lambda\vec{a})$ crossing
$E$: we let ${\EuScript M}(\lambda;\mathcal{B},E)$ denote the number of eigenvalues (counting
multiplicities) of $H(\mu\vec{a})$ that cross $E$ in upwards
direction minus the number of eigenvalues crossing downwards, at couplings
$\mu \in (0,\lambda)$; the eigenvalues that change direction at $E$ are not
counted. The precise definition of ${\EuScript M}(\lambda;\mathcal{B},E)$ is given in
Equation (2.7). Our main result reads as follows.
\begin{theorem} \label{thm1.3}
Let $H$ and $H_M$ as above and let $\mathcal{B} : \mathbb{R}^2 \to \mathbb{R}$ a continuous function of compact
support such that Assumptions \ref{a1.1} and \ref{a1.2} are satisfied.
Let $E \in \mathbb{R}$, $E \notin \sigma(H) \cup \sigma(H_M)$. Then there exists
$\Lambda_0 \ge 0$ such that
\[
{\EuScript M}(\lambda; \mathcal{B}, E ) = {\EuScript N} (\Omega,E), \quad \lambda \ge \Lambda_0,
\quad \lambda \Phi \in 2\pi \mathbb{Z}.
\tag{1.6}
\]
\end{theorem}
We therefore see that the repulsion produced by a strong magnetic field of
compact support corresponds to the repulsion from a high potential barrier,
supported on the set where $\mathcal{B}$ is non-zero, for a sequence of couplings
going to infinity.
\begin{remark} \label{rmk1.4}\rm
$(i)$ For flux $\Phi =0$, the only restriction on $\lambda$ is $\lambda \ge \Lambda_0$.
For $\Phi > 0$, we obtain a sequence $(\lambda_k)_{k\in\mathbb{N}} \subset (0,\infty)$,
with $\lambda_k = 2\pi k/\Phi$, such that (1.6)
holds for $\lambda_k \ge \Lambda_0$.
\noindent$(ii)$ For $E < \inf \sigma(H)$, we have ${\EuScript N} (\Omega,E) =0 $ by monotonicity
and ${\EuScript M}(\lambda; \mathcal{B}, E ) =0 $ by the diamagnetic inequality.
\noindent $(iii)$ In between two successive $\lambda_k$'s a certain number of
eigenvalues of $H(\lambda\vec{a})$ may cross and cross back. While the results of
Herbst and Nakamura \cite{HN} imply that the eigenvalues in the gap will
approach periodic functions, as $\lambda \to \infty$, it appears to be
rather difficult to find conditions that would guarantee these perodic
limiting functions to be non-constant. At the same time,
it is rather unlikely that these periodic functions are constant.
Under the assumptions of Theorem \ref{thm1.3} there exists a constant $c(E,\vec{a}) \ge 0$
such that
\[
| {\EuScript M}(\lambda; \mathcal{B}, E ) - {\EuScript N} (\Omega,E) | \le c(E,\vec{a}), \quad
\lambda \ge \Lambda_1,
\tag{1.7}
\]
for some $\Lambda_1 \ge 0$; cf.~Remark \ref{rmk5.5} for a proof.
More precisely, it is shown in Remark \ref{rmk5.5} that
the constant $c(E,\vec{a})$ in (1.7) can be estimated in
terms of a (non-magnetic) eigenvalue problem for which the Birman-Schwinger
principle is applicable.
\noindent$(iv)$ In a subsequent paper
we will discuss magnetic perturbations where
the sets $G = M^{\rm int}$ and $\supp\mathcal{B}$ may have more than one component.
Here a theorem of Dirichlet in
number theory can be used in the construction of a suitable gauge
leading to a ``smallest possible'' vector potential.
\noindent $(v)$ Theorem \ref{thm1.3} assumes $E \notin \sigma(H) \cup \sigma(H_M)$. If $E
\notin \sigma(H)$ belongs to $\sigma(H_M)$, monotonicity with respect to $E$
and Theorem \ref{thm1.3} yield % for the case $\Phi > 0$ and $\lambda_k$ as above
\[ \begin{gathered}
{\EuScript N} (\Omega,E + \epsilon)
\le \liminf_{\lambda \to \infty,\,
\lambda\Phi \in 2\pi\mathbb{Z} } {\EuScript M}(\lambda; \mathcal{B}, E )
\le \limsup_{\lambda \to \infty,\,
\lambda\Phi \in 2\pi\mathbb{Z}} {\EuScript M}(\lambda; \mathcal{B}, E )
\le {\EuScript N} (\Omega,E),
\end{gathered} \tag{1.8}
\]
for any sufficiently small $\epsilon > 0$.
\end{remark}
In the following corollaries, Theorem \ref{thm1.3} is combined with
simple upper and lower estimates
on $ {\EuScript N} (\Omega,E)$ of phase space volume type,
taken from \cite{H2, H3}; cf. Section~3.
The first corollary provides an upper bound:
\begin{corollary} \label{coro1.5}
Suppose that the assumptions of Theorem \ref{thm1.3} are
satisfied.
Then there exist constant $C_1$, $C_2$, independent of $\mathcal{B}$, such that
\[
\lim _{\lambda \to \infty,\, \lambda\Phi \in 2\pi\mathbb{Z}}
{\EuScript M}(\lambda; \mathcal{B}, E ) \le C_1 R^2 + C_2,
\tag{1.9}
\]
provided $ \Omega = \{ x ; \mathcal{B}(x) \ne 0 \}$ is contained in some disk of radius $R$.
\end{corollary}
The corresponding lower bound requires an additional assumption on the
spectrum of $H$ below $E$:
\begin{assum} \label{a1.6} \rm
Let $\dim_E(H_{B_R})$ denote the number of eigenvalues of $H_{B_R}$
below $E$, counting multiplicities, where $H_{B_R} = -\Delta +V$
on $B_R := \{ x \in \mathbb{R}^2 : |x| < R\}$, with Dirichlet boundary conditions.
Let $E \in \mathbb{R} \setminus \sigma(H)$. We assume that there exist constants
$c > 0$ and $R_0 \ge 0$ such that
\[
\dim_E(H_{B_R}) \ge c R^2, \quad R \ge R_0.
\tag{1.10}
\]
\end{assum}
For this assumption to hold it is sufficient that the i.d.s.\ for $H$
exists (cf.\ \cite{RS-IV, KM}) and has a positive value at $E$.
If (1.10) holds, $H$ must have some essential spectrum below $E$.
\begin{corollary} \label{coro1.7}
In addition to the conditions of Theorem \ref{thm1.3}, suppose that
Assumption \ref{a1.6} is satisfied.
Then there exist constants $c_1 >0$, $c_2 \ge 0$, independent of $\mathcal{B}$, such that
\[ \begin{gathered}
\lim_{\lambda \to \infty,\,
\lambda\Phi \in 2\pi\mathbb{Z}}
{\EuScript M}(\lambda; \mathcal{B}, E ) \ge c_1 R^2 - c_2,
\end{gathered} \tag{1.11}
\]
provided $\supp \mathcal{B}$ contains a disk of radius $R$.
\end{corollary}
The paper is organized as follows. In Section 2, we provide basic
definitions and describe the fundamental approximation procedure
by eigenvalue problems on large disks $B_n$. In particular, we
construct a family of approximating operators $\tilde{H}_n(\lambda\vec{a})$, for
$\lambda \ge 0$, acting in $L_2(B_n)$ with Dirichlet boundary conditions.
In Section 3 we recall basic properties of the discrete eigenvalues
of $H+ \mu \chi_U$, for $0 \le \mu \to \infty$,
where $U \subset \mathbb{R}^m$ is open and bounded.
Section 4 contains the key estimate for the number of eigenvalues
below $E$ for the approximating operators $\tilde{H}_n(\lambda\vec{a})$ on large disks $B_n$.
Here we consider the counting functions
\[
{\EuScript M}_n(\lambda;\mathcal{B},E)
= \dim_E(\tilde{H}_n(0)) - \dim_E(\tilde{H}_n(\lambda\vec{a})),
\tag{1.12}
\]
which correspond to the loss of eigenvalues below $E$ due to the
magnetic perturbation for the approximating problems on $B_n$.
In a first step, we decouple the problem by a natural
Dirichlet boundary condition along $\partial G$, for $\lambda$
large. Then, for $\lambda \Phi \in 2\pi\mathbb{Z}$, it is shown that the operator on
the ``annulus'' $B_n \cap G$,
with Dirichlet boundary conditions, is unitarily equivalent
to an operator without magnetic terms. For the latter
the number of eigenvalues below $E$ is easy to estimate.
We also obtain a preliminary result on the existence of
$\lim_{n\to \infty}{\EuScript M}_n(\lambda; \mathcal{B}, E)$, for $\lambda$ large, $\lambda\Phi \in 2\pi\mathbb{Z}$,
and we show that this limit is equal to ${\EuScript N}(\Omega,E)$.
In Section 5, we finally show that
\[
\lim_{n\to \infty}{\EuScript M}_n(\lambda; \mathcal{B}, E) = {\EuScript M}(\lambda; \mathcal{B}, E),
\tag{1.13}
\]
provided $\lambda$ is such that $ E \notin \sigma(H(\lambda\vec{a}))$.
We then prove Theorem \ref{thm1.3} as well as Corollaries 1.5 and 1.7.
There are three
appendices (Sections 6---8) containing some of the more technical arguments.
We conclude the introduction with a few remarks on related work in
the literature. General information on magnetic Schr\"odinger
operators can be found in Avron, Herbst and Simon \cite{AHS},
Mohamed and Raikov \cite{MR}. Scattering for decreasing
magnetic fields has been studied by Loss and Thaller \cite{LTh}.
The problem studied
in the present paper is somewhat related to the Bohm-Aharonov effect
(cf.~Helffer \cite{Hel}, Weder \cite{W}),
but with a (periodic) electric background potential.
A situation which has attracted some attention is
that of a constant magnetic background, combined with localized
electric perturbations (cf., e.g., Birman and Raikov \cite{BR},
Hempel and Levendorski \cite{HL}, Levendorskii \cite{L}, Raikov \cite{R}).
The existence of eigenvalues in spectral gaps for a perturbation by
a magnetic field of compact support has been studied previously in
Hempel and Laitenberger \cite{HLai} under the assumption of zero flux,
in a rather special case. In \cite{HL} there is a result for the
case where both the magnetic field and the vector potential
decay exponentially at infinity; again, the flux is zero.
The paper \cite{H4} constructs examples of Schr\"odinger operators
$H(\lambda\vec{a})$ having an eigenvalue in the gap
that asymptotically approaches a
periodic or quasi-periodic function of the coupling $\lambda$; here
the magnetic field has compact support consisting of a disk
enclosed by an annulus.
It is interesting to note that the situation
can be strikingly different for the Pauli operator in $\mathbb{R}^2$.
Here some (purely magnetic) cases
can be analyzed in depth by using the property of
supersymmetry (cf. \cite{CFrKS}). It turns out that, under
suitable assumptions, eigenvalues will in fact move downwards
where one might at first expect an upwards movement (cf. \cite{Be1, Be2}).
We take as the ``unperturbed'' operator the Pauli Hamiltonian of
a constant magnetic field $\mathcal{B}_0 > 0$, and add a perturbation by a magnetic
field $\mathcal{B} \le 0$ of compact support. We therefore consider
the pair of operators
\[
H_\pm(\lambda) = (-{\rm i}\nabla -\vec{a}_0 - \lambda \vec{a})^2 \mp \mathcal{B}_0
\mp \lambda \mathcal{B},
\tag{1.14}
\]
where we refer to \cite{CFrKS} for the construction of the Pauli Hamiltonian in
$\mathbb{R}^2$. Looking at $H_+(\lambda)$, one might expect that the repulsive
effect of the potential barrier $-\lambda\mathcal{B} \ge 0$, combined with some
repulsion coming from the magnetic term $\lambda\vec{a}$, should shift a
finite number of eigenvalues from below through each spectral gap
of $H_+$. But this cannot be true for the first gap $(0,2\mathcal{B}_0)$:
by supersymmetry, each non-zero point in
the spectrum of $H_+(\lambda)$ must also appear
in the spectrum of $H_-(\lambda)$,
yet $H_-(0)$ has no spectrum below $2\mathcal{B}_0$.
On the other hand, it can be shown that there exist eigenvalues
that move downwards through the first gap, as $\lambda$ increases.
A more detailed discussion of the related phenomena can be found
in \cite{Be1, Be2}.
\section {Notation and Preliminaries} % Section 2
\subsection*{General Notation}
We write $x=(x_1,x_2) \in \mathbb{R}^2$ and $B_s = \{x \in \mathbb{R}^2 : |x| < s \}$;
$\chi_s$ denotes the characteristic function of the ball $B_s$, for $s > 0$.
The function spaces $C^k(U)$, $L_2(U)$, for $U \subset \mathbb{R}^d$ open,
are defined as usual; the norm of $u \in \mathcal{L}_2(U)$ is given by
$\lnorm{u}^2 = \int_U |u|^2 \dx$. Scalar products in Hilbert space are
written $\langle .,. \rangle $.
The Sobolev space $\mathcal{H}^1(\mathbb{R}^2)$ consists of those
functions $u \in L_2(\mathbb{R}^2)$ that have weak first derivatives
$\partial_j u \in L_2(\mathbb{R}^2)$. $\mathcal{H}^1(\mathbb{R}^2)$ is a Hilbert space
with norm $\lnorm{u}_1^2 = \lnorm{u}^2 + \sum_j \lnorm{\partial_j u}^2$.
For a self-adjoint operator $T$ acting in a Hilbert space $\mathcal{H}$ we
denote the spectrum by $\sigma(T)$, the essential spectrum by
$\sigma_{\rm ess}(T)$, and the spectral projections by $P_{I}(T)$, for any
interval $I$ of the real line. If $T$ has only discrete spectrum
in the interval $I$, then
$\dim \mathop{\rm ran} P_I(T) = \mathop{\rm trace} P_I(T)$ is the number of eigenvalues of
$T$ in $I$, counting multiplicities.
For $\alpha \le \beta \in \mathbb{R}$ and $\eta \in \mathbb{R}$, we will write
\[
\dim_{(\alpha,\beta)}(T)
= \dim \mathop{\rm ran} P_{(\alpha,\beta)}(T), \quad
\dim_\eta(T) = \dim \mathop{\rm ran} P_{(-\infty,\eta)}(T).
\tag{2.1}
\]
\subsection*{The Dirichlet Laplacian of a Closed Set}
We will mostly be concerned with self-adjoint operators
defined via quadratic forms, with form domain given by a
suitable subspace of the Sobolev space $\mathcal{H}^1(\mathbb{R}^2)$.
For any closed set $K \subset \mathbb{R}^2$, a natural subspace of
$\mathcal{H}^1(\mathbb{R}^2)$ \cite{Hed, HZ} is
\[
\tilde{\mathcal{H}}^1_0(K): = \{ u \in \mathcal{H}^1(\mathbb{R}^2); u(x) = 0\
\hbox{ a.e. in } K^C \},
\tag{2.2}
\]
where $K^C$ denotes the complement of $K$.
The self-adjoint operator associated with the quadratic form $\int |\nabla u|^2 \dx$
on $\tilde{\mathcal{H}}^1_0(K)$ via the usual representation theorem
(\cite[Thm.~VI-2.1]{K}) will be called the {\it Dirichlet Laplacian on}
$K$, denoted as $-\Delta_K$. Operators of this type
occur naturally; for instance, if $U = K^C$, then $-\Delta + \mu \chi_U$ converges
to $-\Delta_K$ in the strong resolvent sense, as $\mu \to \infty$.
If $D$ is an open set, there is also the standard Sobolev
space $\mathcal{H}^1_0(D)$, obtained by taking the closure
of $\Cci{D}$ in the $\lnorm{\cdot}_1$-norm.
For $K$ closed, an alternative choice of a form domain
for a Dirichlet Laplacian is $\mathcal{H}^1_0(K^{\rm int}) \subset {\tilde{\mathcal{H}}}^1_0(K)$;
note that $\mathcal{H}^1_0(K^{\rm int})$ may be considerably smaller than
${\tilde{\mathcal{H}}}^1_0(K)$. We say that the
Dirichlet Laplacian associated with $K$ is {\it unique},
if $\mathcal{H}^1_0(K^{\rm int}) = {\tilde{\mathcal{H}}}^1_0(K)$.
Returning to the sets $M = \overline{M}$, $\Omega = M^C$
and $G = M^{\rm int}$ of Section 1,
Assumption \ref{a1.2} is equivalent to the uniqueness of the
Dirichlet Laplacian on $M = \Omega^C$. It follows directly from results in
\cite{HZ} that the Laplacian of $M = \overline{M}$ is unique if
$\partial G$ satisfies the segment condition and if
$M \setminus {\overline G} = {\overline \Omega}^{\rm int}
\setminus \Omega$ has measure zero.
While the above discussion of Dirichlet Laplacians applies
also to $\mathbb{R}^d$, we note as an aside that there is a stronger criterion
\cite{Her} that is specific to $\mathbb{R}^2$:
in fact, the Dirichlet Laplacian of a closed set $M \subset \mathbb{R}^2$ will be unique
if for each point $ x \in \partial M$ there
exists a continuous function $f:[0,1] \to \mathbb{R}^2$ such that
$f(0) = x$ and $f(s) \notin M$, for all $s \in (0,1]$.
This criterion follows from \cite[Thms.~2.1, 2.5]{HZ},
combined with the fact that Brownian paths in
$\mathbb{R}^2$ will immediately ``spiral'' around
their starting point with probability 1 (cf.\ \cite[Ch.~2, Section 7]{PS};
we warn the reader to be careful about the definition of $\tau_M$
in \cite{HZ} and of $\tau_B$ in \cite{PS}).
\subsection*{Magnetic Schr\"odinger operators}
We next turn to magnetic Schr\"odinger operators. As above,
we assume that $ \vec{a} \in C^1(\mathbb{R}^2;\mathbb{R}^2)$ is bounded and
such that $\mathcal{B} = \mathop{\rm curl} \vec{a} = \partial_2 a_1
- \partial_1 a_2$ has compact support.
For an electric potential $V \in L_\infty(\mathbb{R}^2,\mathbb{R})$
the magnetic Schr\"odinger operators
$ H(\vec{a}) = (-{\rm i}\nabla - \vec{a}(x))^2 + V(x)$
can be easily realized as self-adjoint operators
with form domain $\mathcal{H}^1(\mathbb{R}^2)$; cf., e.g., \cite{CFrKS}.
Magnetic operators defined on a closed set
$K \subset \mathbb{R}^2$ with Dirichlet boundary conditions
are obtained as for the Dirichlet Laplacian.
We will also use the following simple commutator identities:
for $\phi \in C_c^\infty$,
we have $ \commutator{-{\rm i}\nabla - \vec{a}}{ \phi} = -{\rm i}\nabla \phi$ and
$\commutator{(-{\rm i}\nabla - \vec{a})^2}{ \phi}
= -2{\rm i} \nabla \phi \cdot (-{\rm i}\nabla - \vec{a}) - \Delta \phi$.
The vector potential $\vec{a}$ is not
uniquely determined by the field $\mathcal{B}$; in fact, if we take any function
$f \in C^2(\mathbb{R}^2)$, then $\vec{a} + \nabla f$ produces the same field
as $\vec{a}$. Passing from $\vec{a}$ to $\vec{a} + \nabla f$
is called a ``gauge transformation''; any two $C^1$ vector potentials \
on $\mathbb{R}^2$ associated with the same field are connected via a gauge
transformation.
It is a well-known and simple fact (cf. \cite{CFrKS}) that $H(\vec{a})$ and $H(\vec{b})$
are unitarily equivalent if $\vec{b} = \vec{a} + \nabla f$; this is
also true for unbounded vector potentials. In particular, the spectrum
and also the usual parts of the spectrum are gauge-independent.
If the field $\mathcal{B}$ has compact support contained in $B_R$, and if the
flux $\Phi$ is zero, then we can easily find a vector potential
$\vec{a}$ such that $\mathop{\rm curl}\vec{a} = \mathcal{B}$ and $\vec{a}(x) = 0$ for $|x|\ge R$.
For $\Phi > 0$, we have the following lemma.
\begin{lemma} \label{lm2.1}
Let $\mathcal{B} : \mathbb{R}^2 \to \mathbb{R}$ be a continuous function of compact
support and suppose $\Phi = \int \mathcal{B} > 0$. Let $R > 0$ be such
that $\supp \mathcal{B} \subset B_R$.
Then there exists a vector potential $\vec{a}$ of class $C^1$ such that
$\mathop{\rm curl}\vec{a} = \mathcal{B}$ in $\mathbb{R}^2$, $\mathop{\rm div}\vec{a}(x) = 0$ for $|x| > R$, and
\[
|\vec{a}(x)| \le \frac{\Phi}{2 \pi} \frac{1}{|x|}, \quad |x| \ge R.
\tag{2.3}
\]
\end{lemma}
\begin{proof} Writing $ r = (x_1^2 + x_2^2)^{1/2}$, we let
$g : [0,\infty) \to \mathbb{R}$ denote a $C^1$-function satisfying $g'(0) = 0$
and
\[
g(r) = \frac{\Phi}{2 \pi}r^{-2}, \quad r \ge R.
\tag{2.4}
\]
Define $\vec{a}_{\rm rad}(x) = (-x_2 g(r) , x_1 g(r))$ and let
$\mathcal{B}_{\rm rad} = \mathop{\rm curl}{\vec{a}_{\rm rad}}$. Then $\mathop{\rm div} \vec{a}_{\rm rad}$ and
$\mathcal{B}_{\rm rad}$ vanish in the exterior of $B_R$, while
$
\int {\mathcal{B}_{\rm rad}}(x) \dx = \int \mathcal{B}(x) \dx = \Phi.
$
Since the flux of $\mathcal{B} - \mathcal{B}_{\rm rad}$ is zero, there exists a vector potential $\vec{b}$ of
class $C^1$, vanishing outside of $B_R$, such that
$
\mathop{\rm curl} \vec{b} = \mathcal{B} - {\mathcal{B}_{\rm rad}},
$
and we see that $\vec{a} = {\vec{a}_{\rm rad}} + \vec{b}$ has the required properties.
\end{proof}
A direct consequence of Lemma \ref{lm2.1} is that
\[
\sigma_{\rm ess}(H(\vec{a})) = \sigma_{\rm ess}(H),
\tag{2.5}
\]
provided $\vec{a} \in C^1$ with $\mathop{\rm curl}\vec{a}$ of compact support.
In fact, since gauge transformations are unitary, we may assume
that $\vec{a}$ is as in Lemma \ref{lm2.1}. It is then easy to see that the resolvent difference
$H^{-1} - H(\vec{a})^{-1}$
is compact.
While $\sigma_{\rm ess} (H(\lambda\vec{a})) = \sigma_{\rm ess}(H)$, for $\lambda \in \mathbb{R}$,
the discrete spectrum of $H(\lambda\vec{a})$ will depend on $\lambda$, in general.
For $\vec{a}$ as in Lemma \ref{lm2.1}, the operators $H(\lambda\vec{a})$ form a holomorphic self-adjoint
family of type (A) in the sense of Kato \cite{K}, and it follows that
the discrete eigenvalues of $H(\lambda\vec{a})$
are described by an (at most countable) family of analytic functions of $\lambda$.
This family is locally finite in the sense that each compact subset of
$\mathbb{R} \times (\mathbb{R} \setminus \sigma_{\rm ess}(H))$ is
intersected by only finitely many of these functions.
It follows that, for $E \notin \sigma(H)$, the set
\[
M(\mathcal{B}, E) = \{ \mu \in \mathbb{R} : E \in \sigma(H(\mu\vec{a})) \} ,
\tag{2.6}
\]
is a discrete subset of the real line.
In order to define the counting function ${\EuScript M}(\lambda;\mathcal{B},E)$ for $\lambda > 0$,
write $M(\mathcal{B},E) \cap (0,\lambda) = \{\mu_1, \ldots, \mu_\ell\}$, for a suitable
$\ell = \ell(\lambda) \in \mathbb{N}$, where $\mu_1 < \ldots < \mu_\ell$.
For any $j = 1, \ldots, \ell$, there is an open set
$U_j \subset \mathbb{R} \times ( \varrho(H) \cap \mathbb{R})$,
with $(\mu_j,E) \in U_j$, such that the
spectrum of the family $(H(\mu\vec{a}); \mu \ge 0)$ in $U_j$ is described by the
union of the graphs of a finite set of (pairwise distinct)
analytic functions $\phi_{js}$, $s=1,\ldots,r_j$, where
$\phi_{js}(\mu_j) = E$.
If $\phi_{js}$ carries multiplicity $m_{js} \in \mathbb{N}$, then the signed spectral
multiplicity of $H(\mu\vec{a})$ crossing $E$ as $\mu$ increases from $0$ to $\lambda$
is given by
\[
{\EuScript M}(\lambda;\mathcal{B},E) = \sum_{j=1}^{\ell(\lambda)} \sum_{s=1}^{r_j} \sigma_{js} m_{js},
\tag{2.7}
\]
where the sign-factors $\sigma_{js} \in \{-1, 0, 1\}$ are defined as follows:
if $k_{js} \in \mathbb{N}$ is the order of the first non-zero derivative of $\phi_{js}$
at $\mu_j$ (i.e., $\phi_{js}^{(k_{js})}(\mu_j) \ne 0$ while
$\phi_{js}^{(m)}(\mu_j) = 0$, for $0 b-a$ and define
\[
\tilde{H}_n(\lambda\vec{a}) = H_n(\lambda\vec{a}) + \gamma \psi_n {\EuScript P}(H_n(\lambda\vec{a})) \psi_n.
\tag{2.15}
\]
The family $ ( \tilde{H}_n(\lambda\vec{a}))^{-1} $ is norm-continuous in $\lambda$.
More generally, if $\vec{b}$ is a bounded vector potential, we define
\[
\tilde{H}_n(\vec{b}) = H_n(\vec{b}) + \gamma \psi_n {\EuScript P}(H_n(\vec{b})) \psi_n.
\tag{2.16}
\]
In the sequel, we will tacitly assume that $n$ is so large that $\psi_n$
vanishes in a neighborhood of $\overline \Omega$.
As a (partial) motivation for the above definition of $\tilde{H}_n(\lambda\vec{a})$, we
point out that
$\tilde{H}_n(0)$ has no spectrum in $[a'',b''] \subset
(a',b')$, for $n$ sufficiently large (cf. \cite{H1, ADH, H2}), i.e., there exists
$n' \in \mathbb{N}$ such that
\[
\sigma(\tilde{H}_n(0)) \cap [a'',b''] = \emptyset, \quad n \ge n'.
\tag{2.17}
\]
We certainly cannot expect the same to be true for $\tilde{H}_n(\lambda\vec{a})$.
However, it can be shown that
$\tilde{H}_n(\lambda\psi_k\vec{a})$ has no spectrum in $[a'',b'']$,
for $k$ sufficiently large and all $n \ge n_k$:
\begin{proposition} \label{prop2.2}
Suppose that $(a,b)$ is a spectral gap of $H$, let $[a',b'] \subset (a,b)$ and
$[a'',b''] \subset (a',b')$, and let $\vec{b}$ be a continuous
vector potential tending to $0$ at infinity. For $\mu \ge 0$,
define $\tilde{H}_n(\mu\psi_k\vec{b})$ as in (2.16), with ${\EuScript P}$ as in (2.11).
Then for $M > 0$ fixed, there exists $k_0 \in \mathbb{N}$ such that
\[
\sigma(\tilde{H}_n(\mu\psi_k\vec{b})) \cap [a'',b''] = \emptyset, \quad
k \ge k_0, \quad n \ge n_k, \quad 0 \le \mu \le M,
\tag{2.18}
\]
for some $n_k > k$.
\end{proposition}
A proof of this result is given in Section 6.
In counting the eigenvalues of $\tilde{H}_n(\lambda\vec{a})$, we will need
two related operators which appear in the process of taking
the limit $\lambda \to \infty$ in $\tilde{H}_n(\lambda\vec{a})$.
Here we first introduce the Dirichlet Laplacian $-\Delta_{G_n}$ on $G_n = B_n \setminus {\bar \Omega}$, and the operator
\[
H_{G_n}(\lambda\vec{a}) = (-{\rm i}\nabla - \lambda\vec{a}(x))^2 + V(x),
\tag{2.19}
\]
acting in $L_2(G_n)$, with Dirichlet boundary conditions (i.e.,
with $C_c^\infty(G_n)$ as a form core). We
have two choices to modify $H_{G_n}(\lambda\vec{a})$ near the boundary $\partial B_n$:
following the construction of $\tilde{H}_n(\lambda\vec{a})$, we first define
\[
\tilde{H}_{G_n}(\lambda\vec{a}) = H_{G_n}(\lambda\vec{a}) + \gamma \psi_n {\EuScript P}(H_{G_n}(\lambda\vec{a})) \psi_n.
\tag{2.20}
\]
Alternatively, instead of $ {\EuScript P}(H_{G_n}(\lambda\vec{a})) $ we may use
$ {\EuScript P}(H_n(\lambda\vec{a}))$ to produce
\[
{\hat H}_{G_n}(\lambda\vec{a}) = H_{G_n}(\lambda\vec{a}) + \gamma \psi_n {\EuScript P}(H_n(\lambda\vec{a})) \psi_n;
\tag{2.21}
\]
we will need to use both operators since taking $\lambda \to \infty$ in $\tilde{H}_n(\lambda\vec{a})$
leads us to $ {\hat H}_{G_n}(\lambda\vec{a}) $, while the natural
gauge transformation
operates on $\tilde{H}_{G_n}(\lambda\vec{a})$. Fortunately,
the norm difference between the
two correction terms tends to zero in operator norm, as $n \to \infty$; cf.\
Proposition \ref{propC.1}.
\section{Schr\"odinger Operators with High Barriers} % sec. 3
In this section, we recall some basic facts about the eigenvalues
and eigenvalue counting functions associated with the family of Schr\"odinger
operators
\[
H + \mu \chi_U, \quad \mu \ge 0,
\tag{3.1}
\]
acting in $L_2(\mathbb{R}^2)$, where $U \subset \mathbb{R}^2$ is open and bounded.
Let $M = \mathbb{R}^2 \setminus U$ and $H_M = -\Delta_M + V$; for simplicity of
notation, we again assume that $V \ge 1$. It is well-known that $H + \mu\chi_U$
converges to $H_M$ in norm resolvent sense; this follows from the monotonicity of
the associated quadratic forms and the fact that the difference of resolvents
$H^{-1} - H_M^{-1}$ is compact (cf., e.g., \cite{HH}). Compactness also implies that
$\sigma_{\rm ess}(H_M) = \sigma_{\rm ess}(H)$ so that $H_M$ has only discrete spectrum in
$\mathbb{R} \setminus \sigma_{\rm ess}(H)$.
Let $(a,b) \cap \sigma(H) = \emptyset$ and let $ E \in (a,b)$.
The eigenvalues of $H + \mu \chi_U$
in $(a,b)$ are analytic, monotonically increasing
functions of the coupling $\mu$. Any compact subset of
$ \mathbb{R} \times (a,b)$ is intersected by at most a finite number of these
analytic functions. For $E \in (a,b)$ and $\lambda > 0$, we let
\[
{\EuScript N}(\lambda ; U, E) = \sum_{0 < \mu < \lambda} \dim \ker (H + \mu \chi_U - E)
\tag{3.2}
\]
denote the total spectral multiplicity crossing
$E$ at couplings $\mu \in (0,\lambda)$. Obviously, ${\EuScript N}(\lambda; U, E)$
is monotonically increasing w.r.t.\ $\lambda$ and
monotonically decreasing w.r.t.\ $E$. Monotonicity with respect
to $U$ is more subtle:
\begin{lemma} \label{lm3.1}
Let $U \subset U'$ be open and bounded subsets of $\mathbb{R}^2$ and let
$E \in \mathbb{R} \setminus \sigma(H)$. Then
\[
{\EuScript N}(\lambda ; U, E) \le {\EuScript N}(\lambda ; U', E), \quad \lambda \ge 0.
\]
\end{lemma}
\begin{proof}
By the Birman-Schwinger principle,
${\EuScript N}(\lambda ; U, E)$ is equal to the number of eigenvalues less than
$-1/\lambda$ of the (compact and symmetric) {\it Birman-Schwinger operator}
\[
\mathcal{K}_U = \chi_U (H-E)^{-1} \chi_U.
\tag{3.3}
\]
Now suppose $\mathcal{L}$ is a linear subspace of $L_2(U)$ such that
$\scaprod{\mathcal{K}_U u}{u} < (-1/ \lambda) \lnorm{u}^2$,
for all $0 \ne u \in \mathcal{L}$. Then
$U' \supset U$ implies that we also have
$\scaprod{\mathcal{K}_{U'} u}{u} < (-1/ \lambda) \lnorm{u}^2$,
for all $0 \ne u \in \mathcal{L}$.
Therefore, the min-max principle of Weyl (\cite[Thm.~XIII-2]{RS-IV}) yields
$ \dim_{-1/\lambda}(\mathcal{K}_{U'}) \ge \dim_{-1/\lambda}(\mathcal{K}_{U})$,
and the result follows.
\end{proof}
The eigenvalues of $H + \mu \chi_U$ either cross the gap or they are asymptotic
to some eigenvalue of $H_M$ in the gap, as $\mu\to\infty$; in fact, for each
$\eta \in \sigma_{\text{disc}}(H_M)$ there
exists an eigenvalue branch of $H + \mu \chi_U$ that is asymptotic to $\eta$,
as $\mu \to \infty$. Finally, we let
\[
{\EuScript N}(U,E) = \limsup_{\lambda\to\infty} {\EuScript N}(\lambda; U, E)
\tag{3.4}
\]
denote the total number of eigenvalues (counting multiplicities) of $H + \mu \chi_U$
that cross $E$ as $\mu$ ranges from $0$ to $\infty$.
${\EuScript N}(U,E)$ is finite for bounded $U$ and there exists
$\lambda_0 \ge 0$ such that ${\EuScript N}(U,E) = {\EuScript N}(\lambda ; U,E)$, for all
$\lambda \ge \lambda_0$. Upper and lower bounds for $ {\EuScript N}(U,E)$ are discussed below.
Central to our approach are approximations on large disks $B_n$ and we
need to study the convergence of such approximations to the corresponding
problem on $\mathbb{R}^2$. For the case of potential barriers,
we consider $\tilde{H}_n = \tilde{H}_n(0)$, as defined in Section~2, and we let
\[
{\EuScript N}_n(U,E) = \sum_{0 < \mu < \infty} \dim\ker(\tilde{H}_n(0) + \mu \chi_U - E)
\tag{3.5}
\]
denote the number of eigenvalues (counting multiplicities) of
$\tilde{H}_n(0) + \mu\chi_U$ that cross $E$ at positive couplings.
It follows by regular perturbation theory and monotonicity that
\[
{\EuScript N}_n(U,E) = \dim_E(\tilde{H}_n) - \dim_E({\hat H}_{G_n}),
\tag{3.6}
\]
with ${\hat H}_{G_n} = {\hat H}_{G_n}(0) $ as in (2.21). For later
use we note that, for $E \notin \sigma(H_M)$ and $n$ large, we have
$\dim_E({\hat H}_{G_n}) = \dim_E({\tilde{H}}_{G_n})$,
as $ \lnorm{\psi_n \left( {\EuScript P}(H_n) - {\EuScript P}(H_{G_n}) \right) \psi_n } \to 0$, $n \to
\infty$, by Proposition \ref{propC.1}. Therefore,
\[
{\EuScript N}_n(U,E) = \dim_E(\tilde{H}_n) - \dim_E({\tilde{H}}_{G_n}),
\tag{3.7}
\]
for $n$ large, provided $E \notin \sigma(H_M)$.
The main result of this section reads as follows.
\begin{proposition} \label{prop3.2}
Let $U \subset \mathbb{R}^2$ be open and bounded,
let $M = \mathbb{R}^2 \setminus U$, and suppose that $ E \in \mathbb{R}$,
$E \notin \sigma(H) \cup \sigma(H_M)$. We then have
\[
{\EuScript N}_n(U, E) \to {\EuScript N}(U, E), \quad n \to \infty.
\tag{3.8}
\]
\end{proposition}
\begin{proof}
(1) We denote the associated Birman-Schwinger operators as
\[
\mathcal{K} = \chi_U(H-E)^{-1} \chi_U,
\quad \mathcal{K}_n = \chi_U (\tilde{H}_n - E)^{-1} \chi_U;
\tag{3.9}
\]
then \cite[Corollary 2.2]{H2} implies
\[
\lnorm{\mathcal{K}_n - \mathcal{K}} \to 0, \quad n \to \infty,
\tag{3.10}
\]
where $K_n$ has been extended by zero outside of $B_n$.
\noindent(2) As ${\EuScript N}(U, E)$ is finite, there exists $\eta_0 > 0$ such
that $\mathcal{K}$ has no spectrum in the interval $(-\eta_0,0)$. There
also exist $\lambda_0 > 0$ and $\delta_0 > 0$ such that
$H + \mu \chi_U$ has no eigenvalues in the interval $(E - \delta_0,
E + \delta_0)$, for $\mu \ge \lambda_0$.
\noindent(3) We now claim that there exist $\eta_1 > 0$ and $n_1 \in \mathbb{N}$
such that the Birman-Schwinger operators $\mathcal{K}_n$ have no
spectrum in the interval $(-\eta_1,0)$, for $n \ge n_1$.
Assuming the claim not to be true, it follows that there exist
sequences $(n_j) \subset \mathbb{N}$, $(\lambda_j) \subset (0,\infty)$ such that
$E \in \sigma(\tilde{H}_{n_j} + \lambda_j \chi_U)$ and $n_j \to \infty$,
$\lambda_j \to \infty$.
Let $u_j \in D(\tilde{H}_{n_j})$ satisfy $\lnorm{u_j} = 1$ and
$ (\tilde{H}_{n_j} + \lambda_j \chi_U ) u_j = E u_j$. We
now consider $v_j = \phi_{n_j/3} u_j$ and
$w_j = u_j - v_j = \psi_{n_j/3} u_j$, where $\phi_k$, $\psi_k$ are as
in (2.12), (2.13).
By the familiar calculations we find (assuming that $n_j$ is so large
that $\chi_U \psi_{n_j/3} = 0$, without restriction)
\[
(\tilde{H}_{n_j} - E )(w_j) =
(\tilde{H}_{n_j} + \lambda_j \chi_U - E )(w_j) =
- 2 \nabla \psi_{n_j/3} \cdot \nabla u_j - (\Delta \psi_{n_j/3}) u_j.
\tag{3.11}
\]
By construction of $\tilde{H}_n$, there exists a constant $\beta > 0$ such that
$ \lnorm{(\tilde{H}_n - E )w} \ge \beta \lnorm{w}$, for all $w \in D(\tilde{H}_n)$, and so
we can use (2.14) to find a constant $C$ such that
\[
\beta \lnorm{w_j} \le C n_j^{-1}\, \left( \lnorm{\nabla u_j} + \lnorm{u_j}\right).
\tag{3.12}
\]
It is easy to see that there is a constant $C' >0$ such that
$\lnorm{\nabla u_j} \le C'$, since $\lambda_j \chi_U \ge 0$.
As a consequence,
\[
\lnorm{w_j} \to 0, \quad \lnorm{v_j} \to 1, \quad j \to \infty.
\tag{3.13}
\]
By a similar calculation we also find that
\[
(\tilde{H}_{n_j} + \lambda_j \chi_U - E )v_j \to 0,
\quad j \to \infty.
\tag{3.14}
\]
Obviously, $v_j \in D(H)$ and $(H + \lambda_j \chi_U - E) v_j
= (\tilde{H}_{n_j} + \lambda_j \chi_U - E) v_j \to 0$.
Since $\lnorm{v_j} \to 1$, it follows that
there exists a sequence $(\epsilon_j)$ of positive numbers, $\epsilon_j
\to 0$, such that $H + \lambda_j \chi_U$ has spectrum in the interval
$( E -\epsilon_j, E + \epsilon_j)$, in contradiction to what was obtained in (1).
\noindent(4) Let $\eta = \min\{\eta_0, \eta_1\}$. Then (2) and (3) imply that
the Birman-Schwinger operators $\mathcal{K}$ and $\mathcal{K}_n$, $n \ge n_1$,
have no eigenvalues in the interval $(-\eta,0)$, and the desired result
follows from (3.10).
\end{proof}
Upper and lower estimates for ${\EuScript N}(U,E)$ are discussed, e.g., in
\cite{H2, H3}. The simplest result reads as follows:
\begin{proposition} \label{prop3.3}
$(i)$ There exist constants $C_1$, $C_2$ such that
\[
{\EuScript N}(U,E) \le C_1 R^2 + C_2, \quad R > 0,
\tag{3.15}
\]
for all open and bounded sets $ U \subset \mathbb{R}^2$ satisfying $U \subset B_R$.
\noindent$(ii)$ Suppose $H$ satisfies Assumption \ref{a1.6}.
Then there exist constants $c_1 > 0$, $c_2 \ge 0$ such that
\[
{\EuScript N}(U,E) \ge c_1 R^2 - c_2, \quad R > 0,
\tag{3.16}
\]
for all open and bounded $U \subset \mathbb{R}^2$ that satisfy $U \supset B_R$.
\end{proposition}
\begin{proof}
By the monotonicity property of Lemma \ref{lm3.1}, we only need estimates for
$U = B_R$. For the statement $(i)$, we can directly refer to
\cite[Theorem 3.6]{H3}, since $\dim_{E}(H_R) \le
\dim_E(-\Delta_{B_R}) \le c R^2$.
Similarly, Assumption \ref{a1.6} and \cite[Theorem~4.4]{H3} yield $(ii)$.
\end{proof}
\begin{remark} \label{rmk3.4} \rm
It is not easy to give sharp upper or lower bounds for
$ {\EuScript N}(U,E)$ since ${\EuScript N}(U,E)$ is more or less unrelated
to the volume of $U$ (cf. \cite{H2, H3}).
On the one hand, the counting function will not be affected in the
limit $\lambda \to \infty$ if $U$ has many tiny holes (``Swiss cheese'').
Hence there may be a large number of eigenvalues crossing $E$ although
$U$ has small volume.
Conversely, if $U$ consists of small and well separated pieces,
the volume of $U$ may be large while $ {\EuScript N}(U,E) = 0$.
\end{remark}
\section{Eigenvalue Counting on Large Disks} % sec. 4
In this section, we create the link that connects the magnetic problem with
the high barrier case: we will show that, for large couplings $\lambda$ that
satisfy $\lambda\Phi \in 2\pi\mathbb{Z}$,
the signed spectral flow across $E$ on $B_n$ is the same
for magnetic perturbations as for potential barriers. We will use the function
\[
{\EuScript M}_n(\lambda; \mathcal{B}, E) = \dim_E(\tilde{H}_n(0)) - \dim_E(\tilde{H}_n(\lambda\vec{a})),
\quad \lambda > 0,
\tag{4.1}
\]
which corresponds to the loss of spectral multiplicity below $E$
for the operators $\tilde{H}_n(\mu\vec{a})$ as the coupling $\mu$ increases from $0$ to $\lambda$.
\begin{proposition} \label{prop4.1}
Let $ E \in \mathbb{R}$, $E \notin \sigma(H) \cup \sigma(H_M)$
and let $\eta > 0$ be so small that $[E -\eta, E + \eta]$ does not intersect
$\sigma(H) \cup \sigma(H_M)$.
Then there exists $\Lambda \ge 0$ such that for all $\lambda \ge \Lambda$
that satisfy $\lambda\Phi \in 2 \pi\mathbb{Z}$ there exists $n_\lambda$ such that
\[
{\EuScript N}_n(\Omega, E + \eta) \le {\EuScript M}_n(\lambda ; \mathcal{B}, E)
\le {\EuScript N}_n(\Omega, E - \eta), \quad n \ge n_\lambda.
\tag{4.2}
\]
\end{proposition}
The proof, given at the end of this section,
relies on the following two basic facts:
\noindent(1) By the results of \cite{HH, HN}, the resolvent of ${\tilde H}_{G_n}(\lambda\vec{a})$
yields a good approximation of the resolvent of $ \tilde{H}_n(\lambda\vec{a})$,
for $\lambda$ large. (Here the resolvent of ${\tilde H}_{G_n}(\lambda\vec{a})$
acts on $L_2(G_n)$, with $G_n = B_n \setminus {\bar \Omega}$, and we
take the direct sum with the zero operator on $L_2({\bar\Omega})$
without making this explicit in the notation.)
\noindent(2) Second, for $\lambda\Phi \in 2 \pi \mathbb{Z}$ we may use a gauge transformation
to eliminate the vector potential in the region $G_n$. \medskip
Our first lemma is an adaptation of results of \cite{HH, HN} on the
emergence of Dirichlet boundary conditions at the boundary of
a strong magnetic field. There are two additional difficulties:
we need estimates that are uniform in $n$, for $n$ large,
and we have to take
care of the non-local terms $\psi_n {\EuScript P}(H_n(\lambda\vec{a})) \psi_n$.
\begin{lemma} \label{lm4.2}
Suppose $\vec{a}$ is a bounded vector potential of class $C^1$
such that $\mathcal{B} = \mathop{\rm curl} \vec{a}$ has compact support and
Assumption \ref{a1.2} is satisfied.
For any $\epsilon > 0$ there exists $\Lambda_\epsilon \ge 0$ such that
\[
\lnorm{ \tilde{H}_n(\lambda\vec{a})^{-1}
- \tilde{H}_{G_n}(\lambda\vec{a})^{-1}
} < \epsilon, \quad \lambda \ge \Lambda_\epsilon,
\tag{4.3}
\]
for all sufficiently large $n$ (i.e., $n \ge N_{\epsilon,\lambda}$).
\end{lemma}
\begin{proof} By Proposition \ref{propB.2}, for $\epsilon > 0$ given
there exists $\Lambda_\epsilon$ such that
\[
\lnorm{H_n(\lambda\vec{a})^{-1} - H_{G_n}(\lambda\vec{a})^{-1} } < \epsilon,
\tag{4.4}
\]
for all $\lambda \ge \Lambda_\epsilon$ and $n \ge N_{\epsilon,\lambda}$.
We now add the term $\gamma\psi_n {\EuScript P}(H_n(\lambda\vec{a})) \psi_n$
to both operators in (4.4) to produce $\tilde{H}_n(\lambda\vec{a})$ and
${\hat H}_{G_n}(\lambda\vec{a})$, as defined in (2.15), (2.21).
It follows from (4.4) and Lemma \ref{lm4.3}, below, that
\[
\lnorm{\tilde{H}_n(\lambda\vec{a})^{-1} - {\hat H}_{G_n}(\lambda\vec{a})^{-1} } < 4 \epsilon,
\quad \lambda \ge \Lambda_\epsilon, \quad n \ge N_{\epsilon,\lambda}.
\tag{4.5}
\]
By (2.15), (2.21), we have
\[
\tilde{H}_{G_n}(\lambda\vec{a}) - {\hat H}_{G_n}(\lambda\vec{a}) =
\gamma \psi_n \left({\EuScript P}(H_{G_n}(\lambda\vec{a})) - {\EuScript P}(H_n(\lambda\vec{a})) \right)
\psi_n,
\tag{4.6}
\]
and Proposition \ref{propC.1} concludes the proof.
\end{proof}
We have singled out the following lemma from the proof of Lemma \ref{lm4.2}.
\begin{lemma} \label{lm4.3}
Let $A$, $B$ denote self-adjoint
operators that satisfy $A \ge c$ and $B \ge c$,
for some $c > 0$, $D(A) = D(B)$, and
$\lnorm{ A^{-1} - B^{-1}} \le \epsilon$,
for some $\epsilon \in (0,1)$. Let $C$ denote a bounded,
symmetric operator. If $\lnorm{C} \le c/2$, then
\[
\lnorm{ (A + C)^{-1} - (B + C)^{-1} } \le 4 \epsilon.
\tag{4.7}
\]
\end{lemma}
\begin{proof} First note that $A+C \ge c/2$ implies that
$A+C$ is invertible with $\lnorm{(A+C)^{-1}} \le 2/c$.
By the second resolvent equation, we have
\[
(A+C)^{-1} - (B+C)^{-1} =
- (A+C)^{-1}A A^{-1} ( A - B ) B^{-1} B (B+C)^{-1},
\tag{4.8}
\]
and the desired result is now immediate from
$\lnorm{A(A+C)^{-1}} \le 2$, $\lnorm{B(B+C)^{-1}} \le 2$, and
$B^{-1} (A-B) A^{-1} = B^{-1} - A^{-1}$.
\end{proof}
We next employ a gauge transformation on $G_n = B_n \setminus {\bar\Omega}$
to eliminate the vector potential $\lambda\vec{a}$, for $\lambda\Phi \in 2 \pi \mathbb{Z}$.
\begin{lemma} \label{lm4.4}
If $\lambda \in \mathbb{R}$ satisfies $\lambda \Phi \in 2\pi \mathbb{Z}$, then the operators
$\tilde{H}_{G_n}(\lambda\vec{a})$ and $\tilde{H}_{G_n}(0)$ are unitarily
equivalent.
\end{lemma}
\begin{proof}
Fixing a base point $x_0 \in G_n$, we define a (multi-valued) function $F$
by a line integral,
\[
F(x) = \int_{\gamma_{x}} \vec{a}(y) \cdot {\rm d} y,
\tag{4.9}
\]
where $\gamma_{x}$ is a smooth path connecting $x_0$ and $x$
within $G_n$.
The values of $F$ at $x \in G_n$ differ by
integer multiples of the flux $\Phi$. Each branch of $F$ is
(locally) $C^2$ and $\nabla F = \vec{a}$.
Thus, for $\lambda \Phi \in 2\pi \mathbb{Z}$,
the function $u_\lambda = {\rm e}^{{\rm i} \lambda F}$ is in $C^2(G_n)$
and has modulus 1.
Hence multiplication by $u_\lambda$ defines a unitary operator
$U_\lambda$ on $L_2(G_n)$.
To check that $U_\lambda$ establishes a unitary equivalence between
$\tilde{H}_{G_n}(\lambda\vec{a})$ and $\tilde{H}_{G_n}(0)$, we
first note that $C^1_c(G_n)$, the space of $C^1$-functions of
compact support in $G_n$, is a core for the quadratic
forms of
both operators and that $C^1_c(G_n)$ is invariant under $U_\lambda$.
Furthermore, it is easy to see that for all
$f$, $g \in C_c^1(G_n)$,
\[
\langle u_\lambda^{-1}(-{\rm i}\nabla - \lambda\vec{a}) u_\lambda f ,
u_\lambda^{-1}(-{\rm i}\nabla - \lambda\vec{a}) u_\lambda g \rangle
= \langle -{\rm i}\nabla f , -{\rm i}\nabla g \rangle.
\tag{4.10}
\]
{}From this it easily follows that $U_\lambda^{-1} H_{G_n}(\lambda\vec{a})U_\lambda = H_{G_n}(0)$.
Therefore, by the Spectral Calculus,
\[
U_\lambda^{-1} {\EuScript P}(H_{G_n}(\lambda\vec{a}))U_\lambda = {\EuScript P}(H_{G_n}(0)).
\tag{4.11}
\]
Finally, $\commutator{U_\lambda}{ \psi_n} =0$ since $U_\lambda$
is a multiplication operator, and we are done.
\end{proof}
\begin{proof}[Proof of Proposition \ref{prop4.1}]
It follows from Lemma \ref{lm4.2} that there exists $\Lambda \ge 0$ such that
\[
\dim_E( \tilde{H}_n(\lambda\vec{a}) ) \le \dim_{E + \eta}( \tilde{H}_{G_n}(\lambda\vec{a}) ),
\quad \lambda \ge \Lambda,
\tag{4.12}
\]
for $n$ large, $n \ge n_\lambda$, say.
By Lemma \ref{lm4.4}, $ \tilde{H}_{G_n}(\lambda\vec{a})$ is unitarily equivalent to
$\tilde{H}_{G_n} = \tilde{H}_{G_n}(0)$, provided $\lambda\Phi \in 2\pi\mathbb{Z}$,
and we conclude that for $\lambda \ge \Lambda$, $\lambda\Phi \in 2\pi\mathbb{Z}$,
and $n \ge n_\lambda$ we have
\begin{align*}
{\EuScript M}_n(\lambda;\mathcal{B},E)
&= \dim_E(\tilde{H}_n) - \dim_E( \tilde{H}_n(\lambda\vec{a}) ) \\
& \ge \dim_E(\tilde{H}_n) - \dim_{ E + \eta}(\tilde{H}_{G_n}) \tag{4.13} \\
& = \dim_{E +\eta}(\tilde{H}_n) - \dim_{ E + \eta}(\tilde{H}_{G_n}),
\end{align*}
as $\dim_t(\tilde{H}_n)$ is constant for $a < t < b$. Recalling (3.7),
this proves the first inequality.
The proof of the second inequality in Proposition \ref{prop4.1} is analogous and omitted.
\end{proof}
Combining the convergence results of Section 3 with Proposition \ref{prop4.1} we obtain
the following preliminary result.
\begin{proposition} \label{prop4.5}
Let $E \in \mathbb{R}$, $E \notin \sigma(H) \cup \sigma(H_M)$.
Then there exists $\Lambda_0 \in \mathbb{N}$ such
that for $\lambda \ge \Lambda_0$ satisfying $\lambda\Phi \in 2\pi\mathbb{Z}$, there exists $n_\lambda$
such that
\[
{\EuScript M}_n(\lambda ; \mathcal{B}, E) = {\EuScript N}(\Omega, E), \quad n \ge n_\lambda .
\tag{4.14}
\]
\end{proposition}
\begin{proof} Let $\eta > 0$ be such that $[E - \eta, E+ \eta]$
doesn't intersect the spectrum of either $H$ or $H_M$; in particular,
$ {\EuScript N}(\Omega, E \pm \eta) = {\EuScript N}(\Omega, E)$.
By Proposition \ref{prop3.2}, ${\EuScript N}_n(\Omega, E\pm\eta) \to {\EuScript N}(\Omega, E \pm \eta)
= {\EuScript N}(\Omega, E)$. Now Proposition \ref{prop4.1} implies that
${\EuScript M}_n(\lambda ;\mathcal{B},E) = {\EuScript N}(\Omega, E)$, for $\lambda \ge \Lambda$,
$\lambda\Phi \in 2\pi\mathbb{Z}$, and for all $n \ge n_{\lambda}$, and we are done.
\end{proof}
\section{The Convergence Step} %sec. 5
In Section 4, we have seen that $\lim_{n\to\infty}{\EuScript M}_n(\lambda ; \mathcal{B}, E)$
exists and is equal to ${\EuScript N}(\Omega, E)$,
provided $\lambda \ge \Lambda$ and $\lambda\Phi \in 2 \pi \mathbb{Z}$.
Nowe, we show that
$\lim_{n\to\infty}{\EuScript M}_n(\lambda ; \mathcal{B}, E)$ is equal to
${\EuScript M}(\lambda ; \mathcal{B}, E)$ for most $\lambda \ge 0$.
The main result of this section reads as follows.
\begin{proposition} \label{prop5.1}
Let $\mathcal{B}$ be a continuous function of compact support, and let
$\vec{a} \in C^1(\mathbb{R}^2;\mathbb{R}^2)$ be such that $\mathop{\rm curl}\vec{a} = \mathcal{B}$,
with $\vec{a}$ tending to zero at infinity and $\mathop{\rm div}\vec{a} (x) = 0$ outside some ball
$B_R$. Let $E \in \mathbb{R} \setminus \sigma(H)$ and
define $ {\EuScript M}_n(\lambda; \mathcal{B}, E)$ and $ {\EuScript M}(\lambda ; \mathcal{B}, E)$ as before.
Let $\lambda \ge 0$ be such that $E \notin \sigma(H(\lambda\vec{a}))$. Then
\[
{\EuScript M}_n(\lambda; \mathcal{B}, E) \to {\EuScript M}(\lambda ; \mathcal{B}, E),
\quad n \to \infty.
\tag{5.1}
\]
\end{proposition}
\begin{proof}
(1) Fix $a < a' < b' < b$ such that $(a,b) \cap \sigma(H) = \emptyset$ and
$a' < E < b'$. Also fix $\lambda > 0$ such that $ E \notin \sigma(H(\lambda\vec{a}))$.
By analyticity and
compactness, the spectrum of $H(\mu\vec{a})$ inside the compact set
$K_\lambda = [0,\lambda + 1] \times [a',b'] \subset [0,\infty) \times (a,b)$
is the union of the graphs $\Gamma(\phi_j)$ of
a finite number of
analytic functions $\phi_j$, defined on suitable intervals of the real line,
with values in $(a,b)$.
Regular perturbation theory yields a constant $C$
such that $|\phi_j'| \le C$, for all $j$.
By compactness and analyticity again, the number of points in $K_\lambda$
where the graphs of two or more of the functions $\phi_j$ intersect is finite.
Similarly, the number of points in $K_\lambda$ where any of the functions
$\phi_j$ has derivative zero, is finite too.
Therefore, we can find levels $E_+ > E$
and $E_- < E$ with the following properties:
\begin{itemize}
\item $E_\pm \in (a',b')$ and ${\EuScript M}(\lambda ; \mathcal{B}, E_\pm)
= {\EuScript M}(\lambda ; \mathcal{B}, E)$;
\item $E_\pm \notin \sigma(H(\lambda\vec{a}))$;
\item For all $j$ we have $\phi_j'(t) \ne 0$ whenever $t \in (0,\lambda+1]$
is such that $\phi_j(t) = E_\pm$;
\item For any $t \in (0,\lambda+1]$, there is at most one $j$ such that
$\phi_j(t) = E_+$ or $\phi_j(t) = E_-$.
\end{itemize}
(2) We now consider $E_+$. A simple compactness argument and
the properties collected in (1) imply that there exists
a finite number of open rectangles
\[
\mathcal{R}_k : = (t_k - \tau_k,t_k + \tau_k) \times
(E_+ - \eta_k, E_+ + \eta_k), \quad k = 1, \ldots, k_\lambda,
\tag{5.2}
\]
with $\tau_k, \eta_k > 0$, such that $0 \le t_k < t_{k+1}$, and
\noindent(a) the union of the intervals
$ (t_k - \tau_k, t_k + \tau_k)$, $k = 1, \ldots, k_\lambda$,
covers $[0,\lambda]$;
\noindent(b) for each $k$, there is {\it at most one} $j = j_k$
such that $\Gamma(\phi_{j_k}) \cap \mathcal{R}_k \ne \emptyset$.
\noindent(c) suppose there exists $j = j_k$ such that $\Gamma(\phi_{j_k})$ intersects $\mathcal{R}_k$;
then we may assume that $\phi_{j_k}(t_k) = E_+$, i.e.,
$\phi_{j_k}$ crosses the level $E_+$ at the center of $\mathcal{R}_k$.
In addition, we may assume that $\phi_{j_k}'(t) \ne 0$, for $t_k -
\tau_k < t < t_k + \tau_k$, and that
$|\phi_{j_k}(t_k \pm \tau_k) - E_+| < \eta_k/2$.
\noindent(d) By decreasing the $\tau_k$'s, if necessary, we may assume without
restriction that $t_{k-1} < t_k - \tau_k < t_k + \tau_k < t_{k+1}$.
\noindent(3) Suppose now that $\mathcal{R}_k \cap \Gamma(\phi_j) = \emptyset$, for all $j$.
Then it is a direct consequence of Lemma \ref{lm5.2}, below, that
$(E_+ - \eta_k/2, E_+ + \eta_k/2)$ doesn't contain any eigenvalues
of $\tilde{H}_n(\mu\vec{a})$, for $|\mu - t_k | < \tau_k$ and for $n \ge n_k$.
Therefore, the counting functions ${\EuScript M}(. ; \mathcal{B}, E) $ and
${\EuScript M}_n(. ; \mathcal{B}, E)$ are constant in the interval
$(t_k - \tau_k, t_k + \tau_k)$, for $n$ large.
\noindent(4) We next consider the case where for some $j = j_k$ we have
$\Gamma(\phi_{j_k}) \cap \mathcal{R}_k \ne \emptyset$. Here we define the band
of width $\delta < \eta_k$ around the graph of $\phi_{j_k}$,
\[
\Gamma_{\delta}(\phi_{j_k}) =
\{ ( t,e) : | t - t_k| < \tau_k, | e - \phi_{j_k}(t) | < \delta \}.
\tag{5.3}
\]
Lemma \ref{lm5.2} implies that there exists a sequence of positive numbers
$\delta_n \downarrow 0$ such that
$\{(t,e) ; |t - t_k |< \tau_k, e \in \sigma(\tilde{H}_n(t\vec{a})) \cap
(E_+ - 3\eta_k/4, E_+ + 3\eta_k/4) \}$ is contained in
$\Gamma_{\delta_n}(\phi_{j_k}) \cap \mathcal{R}_k$, for $n \ge n_k$.
In view of (5.7), below, we also assume that $\delta_n \ge 1/\log n$,
without restriction.
According to the properties gathered in part (c) of (2), $\phi_{j_k}'$ is either
strictly positive or strictly negative in $(t_k - \tau_k, t_k + \tau_k)$.
We now consider the case where $\phi_{j_k}'(t_k) > 0$, so that for $n$ large
\[
\begin{aligned}
m_k & : = {\EuScript M}(t_k + \tau_k ; \mathcal{B}, E)
- {\EuScript M}(t_k - \tau_k ; \mathcal{B}, E) \\
& = \dim_{(E_+ - \delta_n, E_+ + \delta_n)}(H(t_k\vec{a})).
\end{aligned} \tag{5.4}
\]
We also let $m_k^{(n)}$ denote the loss or gain
in spectral multiplicity below $E_+$ for $\tilde{H}_n(t \vec{a})$,
as $t$ ranges between $t_k - \tau_k$ and $t_k + \tau_k$, i.e.,
\[
\begin{aligned}
m_k^{(n)} & =
{\EuScript M}_n(t_k + \tau_k ; \mathcal{B}, E) -
{\EuScript M}_n(t_k - \tau_k ; \mathcal{B}, E) \\
& = \dim_{E_+}(\tilde{H}_n((t_k - \tau_k)\vec{a}))
- \dim_{E_+}(\tilde{H}_n((t_k + \tau_k)\vec{a})). \
\end{aligned} \tag{5.5}
\]
Here we conclude from the above that
\[
m_k^{(n)} = \dim_{(E_+ - \delta_n, E_+ + \delta_n)}(\tilde{H}_n(t_k\vec{a})),
\tag{5.6}
\]
for $n$ large. In fact, the number of eigenvalues of $\tilde{H}_n(t\vec{a})$ inside
the intervals $(\phi_{j_k}(t) - \delta_n, \phi_{j_k}(t) + \delta_n )$ is constant
for $t_k - \tau_k < t < t_k + \tau_k$, while at the same time
$\phi_{j_k}(t_k - \tau_k) + \delta_n < E_+$ and
$\phi_{j_k}(t_k + \tau_k) - \delta_n > E_+$, for $n$ large. Hence all eigenvalue
branches of $\tilde{H}_n(t\vec{a})$ that intersect
$\tilde{\mathcal{R}}_k = (t_k - \tau_k,t_k + \tau_k) \times
(E_+ - 3\eta_k/4, E_+ + 3\eta_k/4) $ enter below the
level $E_+$ (at $t = t_k - \tau_k$) and leave $\tilde{\mathcal{R}}_k$ above the
level $E_+$ (at $t = t_k + \tau_k$).
Now Lemma \ref{lm5.3} and Equations (5.4), (5.6)
imply that $\limsup_{n\to\infty} m_k^{(n)} \le m_k$.
In the other direction, we apply simple cut-offs to
a basis of $\ker (H(t_k\vec{a}) - E_+)$ to show that, for $n$ large,
\[
\dim_{(E_+ - \delta_n, E_+ + \delta_n)}( \tilde{H}_n(t_k\vec{a}) ) \ge
m_k,
\tag{5.7}
\]
and so $m_k^{(n)} \ge m_k$, for $n$ large. In conclusion, we have
shown that $m_k^{(n)} = m_k$, for $n$ large.
\noindent(5) From the above we conclude that
${\EuScript M}(\lambda;\mathcal{B},E_+) = {\EuScript M}_n(\lambda;\mathcal{B},E_+)$, for $n$ large.
By monotonicity, we have ${\EuScript M}_n(\lambda; \mathcal{B}, E_+) \le
{\EuScript M}_n(\lambda; \mathcal{B}, E)$, while $E_\pm$ have been chosen so that
${\EuScript M}(\lambda ; \mathcal{B}, E_\pm) = {\EuScript M}(\lambda ; \mathcal{B}, E)$. As a consequence,
$
\liminf {\EuScript M}_n(\lambda; \mathcal{B}, E) \ge \liminf {\EuScript M}_n(\lambda; \mathcal{B}, E_+)
= \lim {\EuScript M}_n(\lambda; \mathcal{B}, E_+) = {\EuScript M}(\lambda ; \mathcal{B}, E_+)
= {\EuScript M}(\lambda ; \mathcal{B}, E)$.
\noindent
(6) Similarly, using $E_-$ in place of $E_+$, we obtain
$ \limsup {\EuScript M}_n(\lambda; \mathcal{B}, E) \le {\EuScript M}(\lambda ; \mathcal{B}, E)$,
and we are done. \end{proof}
In the proof of Proposition \ref{prop5.1}, we have been using two lemmas
where the finer parts of our construction come into play. The
first lemma does not take care of multiplicities.
\begin{lemma} \label{lm5.2}
Under the assumptions of Proposition \ref{prop5.1},
let $(a,b) \cap \sigma(H) = \emptyset$, let
$\lambda > 0$ and suppose we are given sequences
$(\mu_n) \subset (0,\lambda]$ and $(E_n) \subset (a,b)$ such that
$E_n$ is an eigenvalue of $\tilde{H}_n(\mu_n\vec{a})$, for all $n \in \mathbb{N}$.
If $\mu_n \to \mu_0 \in [0,\lambda]$ and $E_n \to E_0 \in (a,b)$, as $n \to \infty$,
then $E_0$ is an eigenvalue of $H({\mu_0\,\vec{a}})$.
\end{lemma}
\begin{proof} Let $[a',b'] \subset (a,b)$ and $[a'',b''] \subset (a',b')$
such that $E_0 \in (a'',b'')$.
By assumption, there exist $u_n \in D(\tilde{H}_n) = D(H_n)$, $\lnorm{u_n} = 1$,
such that
\[
\tilde{H}_n(\mu_n\vec{a}) u_n = E_n u_n, \quad n \in \mathbb{N}.
\tag{5.8}
\]
With $\phi_k$ and $\psi_k$ as in (2.12), (2.13), we let
\[
v_n = \phi_{n/3} u_n, \quad w_n = \psi_{n/3} u_n = 1 - v_n.
\tag{5.9}
\]
Clearly, $v_n \in D(H)$. Below, we will prove that
\[
\lnorm{v_n} \to 1, \quad
(H(\mu_0\vec{a}) - E_0) v_n \to 0, \quad n \to \infty.
\tag{5.10}
\]
It is then immediate from (5.10) that $E_0$ belongs to the
spectrum of $H(\mu_0\vec{a})$.
For the proof of (5.10) we first observe that
$\commutator{\psi_n {\EuScript P}(H_n(\mu_n\vec{a})) \psi_n}{\psi_{n/3}} = 0$ because
$(1 - \psi_{n/3}) \psi_n = 0$.
We next compute
\[
\begin{aligned}
(\tilde{H}_n(\mu_n \vec{a}) - E_n) (\psi_{n/3} u_n)
&= \commutator{\tilde{H}_n(\mu_n \vec{a})}{ \psi_{n/3}} u_n \\
&= \commutator{H_n(\mu_n \vec{a})}{ \psi_{n/3}} u_n \\
&= -2{\rm i}\nabla \psi_{n/3} \cdot (-{\rm i}\nabla - \mu_n\vec{a}) u_n -
\Delta \psi_{n/3} u_n.
\end{aligned} \tag{5.11}
\]
Since $\lnorm{(-{\rm i}\nabla - \mu_n\vec{a}) u_n}^2$ is clearly bounded while the
$\psi_{n/3}$ satisfy the estimates given in (2.14), we see that
\[
(\tilde{H}_n(\mu_n\vec{a}) - E_n) (\psi_{n/3} u_n) \to 0, \quad n \to \infty.
\tag{5.12}
\]
In (5.12), we are now going to replace $\vec{a}$ with $\psi_k \vec{a}$,
for suitable $k$. This is the most subtle point of our construction
and it is here that the specific definition of the operators $\tilde{H}_n(\vec{b})$
bears fruit.
First, Proposition \ref{prop2.2} yields a $k \in \mathbb{N}$ such that
the interval $[a'',b'']$ is free of spectrum of $\tilde{H}_n(\mu \psi_k \vec{a})$,
for $n \ge n_k$, and for all $0 \le \mu \le \lambda$.
Second, for $n \ge 9k$, we have $\psi_k \psi_{n/3} = \psi_{n/3}$ so that
$H_n(\mu_n\vec{a})(\psi_{n/3} u_n) = H_n(\mu_n\psi_k\vec{a})(\psi_{n/3} u_n)$.
Third, Proposition \ref{propC.1} implies that
\[
\lnorm{\psi_n \left[ {\EuScript P} (H_n(\mu_n\vec{a})) -
{\EuScript P} (H_n(\mu_n \psi_k \vec{a})) \right] \psi_n } \to 0,
\quad n \to \infty,
\tag{5.13}
\]
and we conclude from (5.12), (5.13) that
\[
(\tilde{H}_n(\mu_n\psi_k\vec{a}) - E_n) (w_n) \to 0, \quad n \to \infty.
\tag{5.14}
\]
Let $\eta > 0$ denote the smaller of the numbers $b''-E_0$ and $E_0 - a''$.
By the Spectral Theorem, we have
\[
\eta \lnorm{v} \le \lnorm{(\tilde{H}_n(\mu \psi_k \vec{a}) - E_0)v},
\quad v \in D(H_n), \quad n \ge n_k,
\tag{5.15}
\]
and it now follows from (5.14) that $w_n \to 0$, proving the first part of (5.10).
As for the second part of (5.10), we first compute
\[
\begin{aligned}
(H(\mu_n \vec{a}) - E_n) ((1 - \psi_{n/3}) u_n) & =
(\tilde{H}_n(\mu_n \vec{a}) - E_n) ((1 - \psi_{n/3}) u_n) \\
& =
- \commutator{\tilde{H}_n(\mu_n \vec{a})}{ \psi_{n/3}} u_n.
\end{aligned} \tag{5.16}
\]
It was shown above that $ \commutator{\tilde{H}_n(\mu_n \vec{a})}{ \psi_{n/3}} u_n \to 0$, as
$n \to \infty$, and it follows that $ (H(\mu_n \vec{a}) - E_n)v_n \to 0$, as $n \to \infty$.
We finally observe that
\[
\begin{aligned}
\lnorm{\left(H(\mu_n\vec{a}) - H(\mu_0\vec{a}) \right)v_n}
\le & 2 |\mu_n - \mu_0|^2 \lnorm{|\vec{a}|^2\, v_n} \\
& + |\mu_n - \mu_0|\, \left( 2 \lnorm{\vec{a} \cdot \nabla v_n}
+ \lnorm{(\mathop{\rm div}\vec{a}) v_n}\right).
\end{aligned} \tag{5.17}
\]
Since $\vec{a}$ and $\mathop{\rm div}\vec{a}$ are bounded, the RHS of (5.17) tends to zero,
as $n \to \infty$, and it follows from (5.16)
that $( H(\mu_0\vec{a}) - E_0) v_n \to 0$, as claimed.
\end{proof}
The next lemma is a variant of Lemma \ref{lm5.2},
where we keep track of multiplicities. The
proof is similar to the proof of Lemma \ref{lm5.2} and it is omitted.
\begin{lemma} \label{lm5.3}
Let ${\mu_0} > 0$, $E_0 \in (a,b)$ and assume that
\[
\dim_{(E_0-\delta_n, E_0+\delta_n)}(\tilde{H}_n({\mu_0}\vec{a})) = m , \quad n \ge n_0,
\tag{5.18}
\]
for some $m, n_0 \in \mathbb{N}$ and for a sequence of positive numbers $\delta_n$
such that $\delta_n \to 0$, as $n \to \infty$. Then
\[
\dim\ker(H({\mu_0}\vec{a}) - E_0) \ge m.
\tag{5.19}
\]
\end{lemma}
Before proceeding to the proof of our main result, Theorem \ref{thm1.3}, we need
yet another lemma.
\begin{lemma} \label{lm5.4}
Let $E \notin \sigma(H) \cup \sigma(H_M)$. Then there exists
$\Lambda' \ge 0$ such that
$E \notin \sigma(H(\lambda\vec{a}))$, for $\lambda \ge \Lambda'$ satisfying
$\lambda\Phi \in 2\pi \mathbb{Z}$.
\end{lemma}
\begin{proof} Let $\gamma > 0$ denote the distance from $E$ to $\sigma(H_M)$.
For $\lambda$ large, $\lambda\Phi \in 2 \pi \mathbb{Z}$,
the spectrum of $H(\lambda\vec{a})$ inside the gap $(a,b)$
is contained in a $\gamma/2$-neighborhood of $\sigma(H_M)$:
in fact, Proposition \ref{propB.1} implies that
\[
\lnorm{(H(\lambda\vec{a}))^{-1} - (H_M(\lambda\vec{a}))^{-1}} \to 0, \quad \lambda \to \infty,
\tag{5.20}
\]
while, by a simple variant of Lemma \ref{lm4.4}, $H_M(\lambda_k\vec{a})$ is unitarily equivalent to
$H_M$, provided $\lambda\Phi \in 2 \pi \mathbb{Z}$,
and so $\sigma(H_M(\lambda\vec{a})) = \sigma(H_M)$. This implies the desired result.
\end{proof}
We are now ready for the proof of our main results as
stated in Section~1.
\begin{proof}[Proof of Theorem \ref{thm1.3}]
By Proposition \ref{prop4.5} we have for all $\lambda \ge \Lambda$ that satisfy
$\lambda \Phi \in 2 \pi \mathbb{Z}$
\[
{\EuScript M}_n(\lambda ; \mathcal{B}, E ) = {\EuScript N}(\Omega, E),
\quad n \ge n_\lambda.
\tag{5.21}
\]
By Lemma \ref{lm5.4} we see that $E \notin \sigma(H(\lambda\vec{a}))$, for $\lambda$ large,
$\lambda\Phi \in 2\pi\mathbb{Z}$, and we may apply Proposition \ref{prop5.1} to the effect that
${\EuScript M}_n(\lambda ; \mathcal{B}, E) \to
{\EuScript M}(\lambda; \mathcal{B}, E)$, as $n \to \infty$.
\end{proof}
Corollaries \ref{coro1.5} and \ref{coro1.7}
follow directly from Theorem \ref{thm1.3} and Proposition \ref{prop3.3}.
\begin{remark} \label{rmk5.5} \rm
In this remark, we give some indications on the proof of (1.7). Let
again $\Phi > 0$ and $E \notin \sigma(H) \cup \sigma(H_M)$.
Also recall that
$ {\EuScript M}(\lambda_k ; \mathcal{B}, E ) = {\EuScript N}(\Omega, E)$, according to
Theorem \ref{thm1.3},
for $\lambda_k := 2 k\pi / \Phi$ large enough.
Writing $d:= \hbox{\rm dist}(E, \sigma(H) \cup \sigma(H_M))$, we let
$E' = E + d/3$ and $E'' = E + 2d/3$.
For any $\epsilon > 0$, Lemma \ref{lm4.2} yields an estimate
\[
\lnorm{ \tilde{H}_n(\lambda\vec{a})^{-1} - \tilde{H}_{G_n}(\lambda\vec{a})^{-1}}< \epsilon,
\tag{5.22}
\]
for $ \lambda \ge \Lambda_\epsilon$, $ n \ge N(\epsilon,\lambda)$, and it follows that,
for $\lambda \ge \Lambda'$ and $n \ge n'(\lambda)$,
\[
\dim_E( \tilde{H}_n(\lambda\vec{a}) ) \le \dim_{E'}( \tilde{H}_{G_n}(\lambda\vec{a}) )
= \dim_{E'}( \tilde{H}_{G_n}((\lambda - \lambda_k)\vec{a}) ),
\tag{5.23}
\]
for $k\in \mathbb{N}$,
by a simple variant of Lemma \ref{lm4.4}. For any $\lambda \ge 0$ we can pick $k$ such that
$| \lambda - \lambda_k | \le \pi/\Phi =: \mu_0$.
The electric potential $ q(x) := |\vec{a}(x)|^2$ is continuous and satisfies
$0 \le q(x) \le C(1+|x|)^{-2}$, by Lemma \ref{lm2.1}. By (6.1), we have
\[
\tilde{H}_{G_n}(\kappa\vec{a}) \ge (1-\epsilon) \tilde{H}_{G_n}(0) - ( 1 + 1/(2\epsilon)) \mu_0^2\,q ,
\quad 0 \le \kappa \le \mu_0,
\tag{5.24}
\]
in the sense of quadratic forms.
We now fix some $0< \epsilon_0 \le d$ with the property that
$
\dim_{E}(\tilde{H}_{G_n}(0)) = \dim_{E''} (\tilde{H}_{G_n}(0)) =
\dim_{E'}\left((1 - \epsilon_0) \tilde{H}_{G_n}(0)\right) ,
$
for $n$ large; such an $\epsilon_0$ exists
since the spectrum of $\tilde{H}_{G_n}$ approaches the spectrum of $H_M$, as $n \to \infty$.
Now (5.24) implies (writing $\eta_0 := ( 1 + 1/(2\epsilon_0)) \mu_0^2$)
\[
\begin{aligned}
\dim_{E'}( \tilde{H}_{G_n}(\kappa\vec{a}))
& \le \dim_{E'} \left((1-\epsilon_0)\tilde{H}_{G_n}(0) -\eta_0 q\right) \\
& \le \dim_{E'} ((1 - \epsilon_0) \tilde{H}_{G_n}(0)) + C(\eta_0,q)\\
& = \dim_{E} ( \tilde{H}_{G_n}(0)) + C(\eta_0,q),
\end{aligned}\tag{5.25}
\]
with $C(\eta_0,q)$ denoting the number of eigenvalues of the
family $(1-\epsilon_0)\tilde{H}_{G_n}(0) - \xi q$ that cross $E'$ at couplings
$\xi \in (0,\eta_0)$. There are several possibilities to obtain
bounds on $C(\eta_0,q)$ by using the Birman-Schwinger principle, e.g.
We conclude that
\[
\begin{aligned}
{\EuScript M}_n(\lambda ; \mathcal{B}, E )
& = \dim_E(\tilde{H}_n(0)) - \dim_E(\tilde{H}_n(\lambda\vec{a})) \\
& \ge \dim_E(\tilde{H}_n(0)) - \dim_{E}(\tilde{H}_{G_n}(0)) - C(\eta_0,q) \\
& = {\EuScript N}_n(\Omega, E) - C(\eta_0,q), \\
\end{aligned} \tag{5.26}
\]
for $\lambda$ large, where we have also used (3.7).
The upper estimate is obtained in the same way. Now $n\to\infty$
and Propositions 5.1 and 3.2 yield (1.7).
\end{remark}
We conclude this section with a comment on the case where the flux
$\Phi$ is zero.
\begin{remark} \label{rmk5.6} \rm
Our main result (Theorem \ref{thm1.3}) includes the case $\Phi =0$.
If the flux $\Phi$ is zero, substantial
simplifications of the proof are possible
under additional smootheness assumptions on $\partial G$ that
allow to pass to an equivalent vector potential $\vec{b}$
such that $\mathop{\rm curl}\vec{b} = \mathcal{B}$ and $\vec{b}(x) = 0$ for a.e. $x \in M$.
A suitable gauge transformation
can be constructed by starting from a line integral of $\vec{a}$ in
the open set $G$, as in the proof of Lemma \ref{lm4.4}, and
then extending first to the closed set $\overline{G}$ and
then to $\mathbb{R}^2$; both extension steps are non-trivial and
require additional (but mild) assumptions (\cite{St, HH}).
Assuming for the moment that a vector potential $\vec{b}$ with
the above properties exists, we may work
with the $\lambda$-independent ``correction term''
$\psi_n {\EuScript P}(H_n(0)) \psi_n$ throughout. From \cite{HH} and a simple variant
of Proposition \ref{propB.2}, we then infer that
\[
\lnorm{ {\hat H}_n(\lambda\vec{b})^{-1} - {\hat H}^{-1}_{G_n}(0)} < \epsilon,
\quad \lambda \ge \Lambda_\epsilon, \quad n \ge n_{\epsilon,\lambda},
\tag{5.27}
\]
and (3.6) establishes the desired link with the case of a
high barrier on $\Omega$.
\end{remark}
\section{Appendix A: Proof of Proposition \ref{prop2.2}}
We let $\mathbf{h}$ and $\mathbf{h}_{\vec{b}}$ denote the quadratic forms
of the operators $H = -\Delta + V$ and $H(\vec{b}) = (-{\rm i}\nabla -\vec{b})^2 + V$,
respectively, where $\vec{b} = (b_1(x), b_2(x))$ denotes a bounded vector potential.
Recall that we always assume $V \ge 1$. Introducing the electric potential
$q = q_{\vec{b}}(x) := b_1^2(x) + b_2^2(x)$, it is easy to see that
\[
|\mathbf{h}[u] - \mathbf{h}_{\vec{b}}[u] |
\le
\epsilon \mathbf{h}[u] + ( 1 + 1/(2\epsilon))\,\scaprod{q u}{u} ,
\quad u \in \mathcal{H}^1, \quad \epsilon > 0.
\tag{6.1}
\]
In fact, $(6.1)$ follows from the elementary
estimate % $|\scaprod{\vec{b} u}{\vec{b} u}| \le \scaprod{q u}{u}$,
\[
\big|\scaprod{\vec{b} u}{\nabla u}\big|
\le \sum_{j=1}^2 \int | b_j u\, \partial_j u|\, {\rm d} x
\le \sum_{j=1}^2 \big( \epsilon \lnorm{ \partial_j u}^2
+ \frac1{ 4 \epsilon } \scaprod{b_j u}{b_j u} \big),
\quad \epsilon > 0.
\tag{6.2}
\]
The following notation will be useful:
Given a spectral gap $(a,b)$ of $H$, we choose a sequence of (non-empty)
subintervals $(a_j,b_j)$ satisfying
$[a_j, b_j] \subset (a_{j-1},b_{j-1})$, with
$a_0 =a$ and $b_0 = b$, $j \in \mathbb{N}$.
(6.1) and \cite[Thm.~VI-3.9]{K} yield the following lemma.
\begin{lemma} \label{lmA.1}
Suppose $(a,b) \cap \sigma(H) = \emptyset$, and let $(a_1,b_1)$ as above.
Then there exists a constant $\eta > 0$ such that
$\sigma(H(\vec{b})) \cap [a_1,b_1] = \emptyset$,
for all $\vec{b}$ satisfying $\|{\vec{b}}\|_\infty \le \eta$.
\end{lemma}
\begin{proof}[Proof of Proposition \ref{prop2.2}]
By the construction of $\tilde{H}_n(0)$ we have (cf. \cite{DH, ADH, H1, H2})
\[
\sigma(\tilde{H}_n(0)) \cap [a_1,b_1] = \emptyset, \quad n \ge n_1.
\tag{6.3}
\]
We will pass from $\tilde{H}_n(0)$ to $\tilde{H}_n(\psi_k\vec{b})$ by using as
an intermediate the operators
\[
K_{k,n} =
(-{\rm i}\nabla - \psi_k\vec{b})^2 + V(x) + \gamma \psi_n {\EuScript P} (H_n(0)) \psi_n, \quad k, n \in \mathbb{N}.
\tag{6.4}
\]
As $\vec{b}$ tends to zero at infinity, we find as before that the magnetic
terms obtained from $\psi_k \vec{b}$ satisfy the following
estimate: for $\epsilon > 0$, there exists $k_\epsilon$ such that
\[
| \langle \psi_k \vec{b} u, \nabla u \rangle | + \|\psi_k \vec{b} u \|^2
\le \epsilon \langle H_n(0) u, u \rangle
\le \epsilon \langle \tilde{H}_n(0) u, u \rangle, \quad
u \in \mathcal{H}_0^1(B_n),
\tag{6.5}
\]
for $k \ge k_\epsilon$ and for all $n \in \mathbb{N}$; here terms like
$\langle H_n(0)u,u\rangle$ should be read in the sense of quadratic forms.
As before, (6.5) and \cite[Thm.~VI-3.9]{K} imply that the operator
$K_{k,n}$ has no spectrum in the interval $[a_2,b_2]$, for $k \ge k_1$,
and all $n \ge n_1(k)$.
We next provide an estimate for the difference of
$\tilde{H}_n(\psi_k\vec{b})$ and $K_{k,n}$ which we denote as
\[
D_{k,n} = \tilde{H}_n(\psi_k\vec{b}) - K_{k,n}
= \gamma \psi_n \left({\EuScript P}(H_n(\psi_k \vec{b})) - {\EuScript P}(H_n(0))
\right) \psi_n.
\tag{6.6}
\]
As $\|\psi_k \vec{b}\|_\infty \to 0$, as $k \to \infty$, it follows
again from \cite[Thm.~VI-3.9]{K} that, for any $\epsilon > 0$ there exists
$k'_\epsilon$ such that for all $n$
\[
\lnorm{H_n(\psi_k \vec{b})^{-1} - H_n(0)^{-1}} < \epsilon, \quad
k \ge k'_\epsilon.
\tag{6.7}
\]
As a consequence, we find that, for $\epsilon > 0$ given,
there exists $k''_\epsilon \in \mathbb{N}$ such that
\[
\lnorm{{\EuScript P}(H_n(\psi_k \vec{b})) - {\EuScript P}(H_n(0))} < \epsilon, \quad k \ge k''_\epsilon,
\tag{6.8}
\]
and for all $n$. Going from (6.7) to (6.8) is not entirely
trivial. Proceeding as in the proof of \cite[Thm.~VIII.18]{RS-I} we
can infer from (6.7) that the resolvents of $A = H_n(\psi_k \vec{b})^{-1}$
and $B = H_n(0)^{-1}$ satisfy a uniform estimate
\[
\lnorm{(A-\zeta)^{-1} - (B - \zeta)^{-1}} < \epsilon,
\quad \zeta \in \mathbb{C}, \quad |\zeta | = 2,
\tag{6.9}
\]
for $k$ large. We may then express any continuous function of $A$ or
$B$ as a contour integral of the resolvents along $|\zeta | = 2$.
We finally conclude that
$\tilde{H}_n(\psi_k \vec{b}) = K_{k,n} + D_{k,n}$
has no spectrum inside the interval $[a_3,b_3]$, for
$k$ sufficiently large and $n \ge n_1$.
\end{proof}
\section{Appendix B: Strong magnetic fields and Dirichlet boundaries}
Our starting point in this appendix is a result of
\cite{HH, HN} on the emergence of a Dirichlet boundary condition
along the boundary of the support of a strong magnetic field.
We let $H_M(\lambda\vec{a})$ denote the
operator $(-{\rm i}\nabla -\lambda\vec{a})^2 + V$, acting in $L_2(M)$ with
Dirichlet boundary condition. Here $V$ is a bounded, measurable
potential; for simplicity, we again assume $V \ge 1$. We use the
notation of Section 2 for $G$, $G_n$, $H_M$, $H_{G_n}$ etc. The
following proposition is derived in \cite{HN} in the case $V = 0$ only; it
easy to see that one might include a potential $V$ with positive part
locally integrable and negative part in the Kato class.
\begin{proposition}[\cite{HN}] \label{propB.1}
Suppose $\vec{a}$ is a (bounded) vector potential
of class $C^1$ such that $\mathcal{B} = \mathop{\rm curl}\vec{a}$ has compact support and
let $M = \{ x : \mathcal{B}(x) = 0\}$. We then have
\[
\lnorm{(H(\lambda\vec{a}))^{-1} - (H_M(\lambda\vec{a}))^{-1}} \to 0, \quad \lambda \to \infty.
\tag{7.1}
\]
\end{proposition}
In view of Lemma \ref{lm4.2}, we need
an analogous result for operators acting on $B_n$ instead of
$\mathbb{R}^2$, with estimates that are uniform in $n$ large, for $\lambda$ fixed (and
sufficiently large). The precise result is as follows:
\begin{proposition} \label{propB.2}
Let $\vec{a}$ and $\mathcal{B}$ as above and let $\epsilon > 0$.
Then there exists $\Lambda_\epsilon$
such that for all $\lambda \ge \Lambda_\epsilon$ there exists $n_{\epsilon, \lambda} \in \mathbb{N}$
such that
\[
\lnorm{H_n(\lambda\vec{a})^{-1} - H_{G_n}(\lambda\vec{a})^{-1}} < \epsilon,
\quad n \ge n_{\epsilon, \lambda}.
\tag{7.2}
\]
\end{proposition}
For the proof, we will show that the difference of the terms
on the left-hand side of (7.1) and (7.2)
tends to zero, as $n \to \infty$. Here
we first deal with the case $\vec{a}=0$ and $V = 1$ and
then use the Feynman-Kac-It\^o formula to reduce the magnetic case
to the situation of Lemma \ref{lmB.3}.
\begin{lemma} \label{lmB.3}
Let $-\Delta_G$, $-\Delta_{G_n}$ etc.\ as
in Section 2. Then, as $n \to \infty$,
\[
\lnorm{ \left((-\Delta + 1)^{-1} - (-\Delta_G+1)^{-1} \right)
- \left( (-\Delta_{B_n} + 1)^{-1} - (-\Delta_{G_n}+1)^{-1} \right) }
\to 0.
\tag{7.3}
\]
\end{lemma}
\begin{proof} We multiply with $1 = \phi_n + \psi_n$, with $\phi_n$ as in
(2.12), (2.13), and rearrange
the 8 terms in form of the following 4 differences,
\[
\phi_n (-\Delta + 1)^{-1} - \phi_n (-\Delta_{B_n} + 1)^{-1} ,
\quad
\phi_n (-\Delta_G + 1)^{-1} - \phi_n (-\Delta_{G_n} + 1)^{-1} ,
\]
\[
\psi_n (-\Delta + 1)^{-1} - \psi_n (-\Delta_G + 1)^{-1} ,
\quad
\psi_n (-\Delta_{B_n} + 1)^{-1} - \psi_n (-\Delta_{G_n} + 1)^{-1} ;
\]
again, resolvents are extended by zero whenever necessary.
As an example, we
treat the third of the four terms; the argument in the other cases
is similar and omitted. Let
\[
v = (-\Delta + 1)^{-1} f - (-\Delta_G + 1)^{-1} f, \quad f \in L_2,
\tag{7.4}
\]
and find by the usual calculation
$ (-\Delta + 1) \psi_n v = \commutator{\, -\Delta}{ \psi_n} v$,
whence
\[
\psi_n v = (-\Delta + 1)^{-1} \commutator{\kern.1ex -\Delta}{\psi_n}v.
\tag{7.5}
\]
It is easy to see that $\lnorm{ (-\Delta + 1)^{-1}
\commutator{\, -\Delta}{ \psi_n}}
\le c/n$, and the result follows.
\end{proof}
With some more effort one could obtain an exponentially small bound in
Lemma \ref{lmB.3}. We are now ready for the proof of Proposition \ref{propB.2}.
\begin{proof}[Proof of Proposition \ref{propB.2}]
The proof relies on arguments used in the proof of \cite[Lemma 1.3]{HH}.
In the following, it will be enough to consider $ 0 \le f \in
C_c^\infty(\mathbb{R}^d)$. We start
from the Feynman-Kac-It\^o-representation for the difference of the
semi-groups associated with $H(\lambda\vec{a})$ and $H_M(\lambda\vec{a})$,
\[
\begin{aligned}
D_\lambda f(x;t) :&=
\big({\rm e}^{-tH(\lambda\vec{a})}f - {\rm e}^{-tH_M(\lambda\vec{a})}f \big)(x) \\
&= \mathbb{E}_x\left\{{\rm e}^{{\rm i}\Phi(\lambda\vec{a},\omega)} {\rm e}^{-\int_0^t V(\omega(s))ds}
\chi_W(\omega) f(\omega(t))
\right\} , \quad t > 0,
\end{aligned} \tag{7.6}
\]
where
$
W = \{\omega \,;\, \omega(0) = x, \medspace \exists s \in [0,t): \omega(s)
\in \Omega \}
$
denotes the set of those Brownian paths $\omega$ (starting at $x$ at time
$t=0$) that enter $\Omega$ before time $t$;
$\Phi(\lambda\vec{a}, \omega)$ is a real phase, given by (15.2) in \cite{S}.
We may write down a completely analogous
formula for the difference of the semi-groups associated with the
operators $H_n(\lambda\vec{a})$ and $H_{G_n}(\lambda\vec{a})$; instead of the set $W$ we
will now have
$
W_n = \{\omega\,;\, \omega(0) =x, \medspace
\forall s \in [0,t):\omega(s) \in {B}_n, \
\exists s \in [0,t) : \omega(s)
\in \Omega \},
$
which consists of those paths $\omega$ starting at $x$ that do not leave
$B_n$ and that enter $\Omega$ before time $t$.
The corresponding difference of semi-groups will be denoted as $D_{\lambda;n}$.
For the difference $D_\lambda - D_{\lambda;n}$, the paths $\omega$ that remain in
the play are given by the set
\[
W \setminus W_n = \{ \omega \,;\, \exists s_1, s_2 \in [0,t):
\omega(s_1) \in \Omega,\, \omega(s_2) \notin {B}_n \}.
\tag{7.7}
\]
Taking the Laplace-transform, we find for a.e.\ $x$
\[
\begin{aligned}
| H(\lambda\vec{a})^{-1}f(x) & - H_{M}(\lambda\vec{a})^{-1}f(x) -
\left( H_n(\lambda\vec{a})^{-1}f(x) - H_{G_n}(\lambda\vec{a})^{-1} f(x) \right) | \\
& \le \int_0^\infty \left| \mathbb{E}_x
\{{\rm e}^{{\rm i}\Phi(\lambda\vec{a},\omega) - \int_0^t V(\omega(s))ds}
\chi_{W \setminus W_n} (\omega) f(\omega(t)) \} \right| dt \\
& \le \int_0^\infty
\mathbb{E}_x \left\{ {\rm e}^{-t}\chi_{W \setminus W_n}(\omega)
f(\omega(t))\right\}
{\rm d} t,
\end{aligned} \tag{7.8}
\]
as $|{\rm e}^{{\rm i}\Phi(\vec{a},\omega)}| = 1$ and
$ {\rm e}^{-\int_0^t V(\omega(s))ds} \le {\rm e}^{-t}$ (recall that $V \ge 1$).
By the Feynman-Kac formula, we can rewrite the last line of (7.8):
\[
\begin{aligned}
& \int_0^\infty \mathbb{E}_x \left\{ {\rm e}^{-t}
\chi_{W \setminus W_n}(\omega)f(\omega(t))\right\} {\rm d} t \\
&=
(-\Delta +1)^{-1}f(x) - (-\Delta_{M} + 1)^{-1}f(x) \\
& \quad - \left(
(-\Delta_n+1)^{-1}f - (-\Delta_{G_n} + 1)^{-1}f \right)(x).
\end{aligned} \tag{7.9}
\]
The result now follows from Lemma \ref{lmB.3} and Proposition \ref{propB.1}.
\end{proof}
\section{Appendix C} % C.
In this appendix we show that the difference of the
operators $\tilde{H}_{G_n}(\lambda\vec{a})$ and ${\hat H}_{G_n}(\lambda\vec{a})$
is small for $n$ large.
We prove here a result that is slightly more general than what
we need in the preceding sections: we compare two magnetic Schr\"odinger
operators on open sets $G_n$, $G_n' \subset B_n$ such that
$G_n, G_n' \supset B_n \setminus B_R$, for
some $0 < R < n$. These operators have
bounded magnetic and electric potentials
$\vec{a}$, $V$ and $\vec{a}'$, $V'$, respectively, which are defined on
all of $\mathbb{R}^d$ and which satisfy
$ \vec{a}(x) = \vec{a}'(x)$, $V(x) = V'(x)$, for $ x \in B_R^C$.
We write
$h_n = (-{\rm i}\nabla - \vec{a})^2 + V$, acting in $L_2(G_n)$, with
Dirichlet boundary conditions
(cf.\ the definition of $H_{G_n}(\lambda\vec{a})$ in (2.19));
$h_n'$ is defined accordingly. Note that we need not assume
that $\vec{a}$, $\vec{a}'$ tend to zero
at infinity nor that the associated magnetic fields have
support inside $B_R$. Furthermore, the result of Proposition \ref{propC.1}
does not depend on the presence of a spectral gap of $-\Delta + V$.
For simplicity of notation, we
again assume $V, V' \ge 1$. We then have:
\begin{proposition} \label{propC.1}
Let $h_n$, $h_n'$ be as above and let $p \in C_0([1, \infty))$, the
space of continuous functions on $[0,\infty)$ tending to $0$ at infinity.
Then
\[
\lnorm{\psi_n \left( p(h_n) - p(h_n')
\right) \psi_n} \to 0, \quad n \to \infty.
\tag{8.1}
\]
Furthermore, for any $\kappa > 0$ fixed,
$(8.1)$ holds uniformly for $\mu\vec{a}$, $\mu\vec{a}'$ in place of
$\vec{a}$, $\vec{a}'$, provided $0 \le \mu \le \kappa$.
\end{proposition}
As in Section 7, we first use the Feynman-Kac-It\^o formula
to eliminate the magnetic vector potentials. In the following
lemma, we write $-\Delta_n = -\Delta_{B_n}$ and
$-\Delta_{R,n} = -\Delta_{B_n \setminus B_R}$.
\begin{lemma} \label{lmC.2}
With the notation and assumptions of
Proposition \ref{propC.1}, we have for $ t > 0$
\[
\big|{\rm e}^{-th_n} f(x) - {\rm e}^{-th_n'} f(x) \big|
\le 2 \big({\rm e}^{-t(-\Delta_n + 1)} |f|(x)
- {\rm e}^{-t(-\Delta_{R,n} + 1)}
|f|(x) \big),
\tag{8.2}
\]
for all $f \in C_c^\infty$ and a.e.\ $x\in \mathbb{R}^d$.
\end{lemma}
\begin{proof} It is enough to consider $f \ge 0$.
Writing down the Feynman-Kac-It\^o formula
for $ {\rm e}^{-t h_n} f(x)$ and for $ {\rm e}^{-t h_n'} f(x)$
(cf.\ (15.2) in \cite{S}, e.g.) and taking
the difference, we see that all
paths drop out from the integration that do not enter
the set $B_R$, for $s \in [0,t)$. We write for fixed
$x \in B_n$
\[
W = \{\omega ; \omega(0) = x,
\exists s \in [0,t) \,:\, \omega(s) \in B_R,\
\forall \tau \in [0,t): \omega(\tau) \in B_n \}
\tag{8.3}
\]
and obtain for $f \ge 0$
\[
\begin{aligned}
& \big| {\rm e}^{-t h_n} f(x) - {\rm e}^{-t h_n'} f(x)\big| \\
& \le \mathbb{E}_x \left\{
( \big| {\rm e}^{{\rm i}\Phi(\vec{a},\omega)
- \int_0^t V(\omega(s))ds} \big|
+\big|{\rm e}^{{\rm i}\Phi(\vec{a}',\omega) -\int_0^t V'(\omega(s))ds}\big|)
\chi_{W}(\omega) f(\omega(t))\right\}.
\end{aligned} \tag{8.4}
\]
As in the proof of Proposition \ref{propB.2}, we find
\[
\begin{aligned}
\big| {\rm e}^{-t h_n} f(x) - {\rm e}^{-t h_n'} f(x) \big|
& \le 2 \mathbb{E}_x \left\{ {\rm e}^{-t} \chi_{W}(\omega)
f(\omega(t)) \right\} \\
& = 2 \big( {\rm e}^{-t(-\Delta_n + 1)}
- {\rm e}^{-t(-\Delta_{R,n} + 1)}\big) f(x),
\end{aligned} \tag{8.5}
\]
by the Feynman-Kac-formula for $ {\rm e}^{-t(-\Delta_n + 1)}$ and
$ {\rm e}^{-t(-\Delta_{R,n} + 1)} $.
\end{proof}
As before, we employ the Laplace-transform to pass from the semi-group
estimates of Lemma \ref{lmC.2} to resolvent estimates:
\begin{lemma} \label{lmC.3}
Under the above assumptions, we have for $k \in \mathbb{N}$
\[
\lnorm{ \psi_n \left(h_n^{-k} - (h_n')^{-k} \right) \psi_n }
\le 2\,\lnorm{ \psi_n \left( (-\Delta_n + 1)^{-k}
- (-\Delta_{R,n} + 1)^{-k} \right) \psi_n }.
\tag{8.6}
\]
\end{lemma}
\begin{lemma} \label{lmC.4}
With the above notation and assumptions, there
exist constants $\alpha > 0$, $c \ge 0$ such that
\[
\lnorm{ \psi_n \left( (-\Delta_n + 1)^{-k}
- (-\Delta_{R,n} + 1)^{-k} \right) \psi_n }
\le c\,{\rm e}^{-\alpha(n - R)}.
\tag{8.7}
\]
\end{lemma}
\begin{proof} We first consider $k=1$. For $u \in C_c^\infty(B_n)$, write
\[
v = \left((-\Delta_n + 1)^{-1} - (-\Delta_{R,n} + 1)^{-1} \right) \psi_n u.
\tag{8.8}
\]
Fix $k_0 \in \mathbb{N}$ such that $\phi_{k_0} = 1$ on $B_R$ and consider
$n$ large enough to have $ \psi_n \psi_{k_0} = \psi_n$. We then have
$\psi_{k_0} v \in D(-\Delta_n)$ and
$
(-\Delta_n + 1)(\psi_{k_0} v) = -2 \nabla\psi_{k_0} \cdot \nabla v
- \Delta\psi_{k_0} v,
$
or
\[
\psi_n v = \psi_n \psi_{k_0} v = \psi_n (-\Delta_n + 1)^{-1}
\left( -2 \nabla\psi_{k_0} \cdot \nabla v - \Delta\psi_{k_0} v \right).
\tag{8.9}
\]
By construction, the set $\{x; \nabla \psi_{k_0} \ne 0 \}$ has distance
$c (n-R)$ from
the support of $\psi_n$. By the maximum principle, the Green's function
of $-\Delta_n + 1$ is pointwise bounded by the Green's function of
$-\Delta + 1$, which decays exponentially off the diagonal. This
implies the desired estimate.
The proof for $k > 1$ is similar and omitted.
\end{proof}
\begin{proof}[Proof of Proposition \ref{propC.1}]
%The proof of Proposition \ref{propC.1} now easily follows from the above lemmas.
If $p$ is a polynomial in $1/x$, then (8.1) holds by
Lemmas \ref{lmC.3}, \ref{lmC.4}. The general result follows via
the Stone-Weierstra\ss-theorem.
\end{proof}
In the following proposition we combine an estimate from Section 6
with Proposition \ref{propC.1}.
\begin{proposition} \label{propC.5}
Under the assumptions of Proposition \ref{prop2.2} (in particular, $\vec{b} \to 0$ at
$\infty$), we have for ${\EuScript P}$ as in (2.11)
and $H_n(\vec{b})$ as in (2.10),
\[
\lnorm{\psi_n \big({\EuScript P}(H_n(0)) - {\EuScript P}(H_n(\vec{b})) \big) \psi_n }
\to 0, \quad n \to \infty.
\tag{8.10}
\]
\end{proposition}
Since $\vec{b}$ decays at infinity, it is natural to expect that, for $n$ large,
there shouldn't be much difference between the various correction terms.
\begin{proof} Given $\epsilon > 0$, it follows from (6.7) that
we can find $k \in \mathbb{N}$ and $n_0 = n_0(\epsilon,k)$ such that
\[
\lnorm{{\EuScript P}(H_n(\psi_k \vec{b})) - {\EuScript P}(H_n(0))} < \epsilon, \quad n \ge n_0.
\tag{8.11}
\]
On the other hand,
Proposition \ref{propC.1} yields that
\[
\lnorm{\psi_n \big( {\EuScript P}(H_n(\vec{b})) - {\EuScript P}(H_n(\psi_k\vec{b}))
\big) \psi_n } \to 0, \quad n \to \infty,
\tag{8.12}
\]
and the result follows.
\end{proof}
\subsection*{Acknowledgements}
The authors gratefully acknowledge partial
support by the Deut\-sche Forschungsgemeinschaft. The first author
(R.~H.) would also like to thank Ira Herbst (Charlottesville) for
several fruitful conversations and, in particular,
for sharing his knowledge of Dirichlet Laplacians on Borel sets.
We also thank the unknown referee for remarks that led us to
reconsider the choice of our fundamental counting functions.
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\end{document}