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Eigenvalue asymptotics, Schr\"{o}dinger and Dirac operators, magnetic field
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% 
%     Eigenvalue asymptotics for the 
%     even-dimensional perturbed
%     Dirac and Schr\"{o}dinger operators 
%     with constant magnetic fields
% 
% LaTeX2e VERSION
% 
% March 14, 2002.
% Last modified: March 14, 2002.


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\hyphenation{Toep-litz o-pe-ra-tor po-ly-no-mi-als 
             per-tur-bed clo-sed hy-per-sur-fa-ce}

\begin{document} 

%% title page 

\title{EIGENVALUE ASYMPTOTICS FOR \\ 
       EVEN-DIMENSIONAL PERTURBED \\
       DIRAC AND SCHR\"{O}DINGER  \\
       OPERATORS WITH CONSTANT  \\
       MAGNETIC FIELDS}
\author{M. Melgaard\thanks{The first author is a Marie Curie Post-Doc Fellow, supported by the European Union under grant no. HPMF-CT-2000-00973.} {} and G. Rozenblum \\
        Department of Mathematics \\
        Chalmers University of Technology \\
        and University of Gothenburg \\
        Eklandagatan 86, S-412 96 Gothenburg, Sweden \\
        E-mail: melgaard@math.chalmers.se \\
        \hspace*{1.35cm}and grigori@math.chalmers.se}
%\thanks{} 
%\subjclass{Primary ; Secondary , }

\date{March 14, 2002}

\maketitle 

\begin{abstract}
The even-dimensional Dirac and Schr\"{o}dinger operators 
with a constant magnetic field have purely essential spectrum 
consisting of isolated eigenvalues, so-called Landau levels. 
For a sign-definite electric potential $V$ which tends to zero 
at infinity, {\em not too fast}, it is known for the Schr\"{o}dinger
operator that the number of eigenvalues near each Landau level is 
infinite and their leading (quasi-classical) asymptotics depends on 
the rate of decay for $V$. We show, both for Schr\"{o}dinger and
Dirac operators, that, for {\em any} sign-definite, bounded $V$ 
which tends to zero at infinity, there still are an infinite number 
of eigenvalues near each Landau level. For compactly supported $V$ 
we establish the {\em non-classical} formula, not depending on $V$, 
describing how the eigenvalues converge to the Landau levels 
asymptotically. 
\end{abstract}



%% end of title page

\section{INTRODUCTION}
\label{sect:intro}
In nonrelativistic quantum mechanics a spinless particle confined
to the $xy$-plane, and subject to a constant magnetic field 
$\mbf{B}=(0,0,b)$ aligned along the $z$-axis as well as a 
real-valued electrostatic potential $\pm V(x,y)$, is described 
by the Schr\"{o}dinger operator 
\begin{eqnarray}
H_{0}\pm V & = & (\mbf{P}-\mbf{A})^{2}\pm V  \nonumber \\ 
& = & \left(-i\frac{\partial}{\partial x}-\frac{b}{2}y \right)^{2}+
\left(-i\frac{\partial}{\partial y}+\frac{b}{2}x \right)^{2}\pm V(x,y)
\label{sect:intro-eq1}
\end{eqnarray} 
acting in the Hilbert space $L^{2}(\Rl^{2})$. Here $\mbf{P}=-i \nabla$ 
is the momentum operator and $\mbf{A}=1/2(\mbf{B} \wedge \mbf{r})$, 
$\mbf{r}=(x,y)$, is the (magnetic) vector potential. For simplicity
we have not included any physical parameters (i.e., the particle mass,
the particle charge, the speed of light, or Planck's constant) in 
(\ref{sect:intro-eq1}). Such ``simple'' systems has in recent years
attracted renewed attention in connection with nanotechnology (quantum
dots, boxes etc).

The unperturbed operator $H_{0}$ has rather peculiar spectral properties:
Its spectrum consists of eigenvalues of infinite multiplicity, the so-called 
Landau levels $\Lambda_{q}=b(2q-1)$, $q \in \Nl$; (we shall use the abbreviation
LL for Landau level). When the operator is perturbed by a potential $V$ which 
tends to zero at infinity, new eigenvalues may appear, with only possible limit 
points at LLs, which follows from Weyl's essential spectrum theorem. Therefore the 
natural question arises, whether there are a finite or infinite number of 
eigenvalues, and in the latter case, to find their asymptotical distribution.

We may formulate the same problem if, instead of two, the underlying
Euclidean space has an even dimension $2d$. Again, the unperturbed operator $H_{0}$ 
has purely essential spectrum consisting of LLs 
$\{ \Lambda_{\lmbf{q}} \}$, $\mbf{q} \in \Nl^{d}$, which can be expressed as 
$\Lambda_{\lmbf{q}}=\sum_{j=1}^{d} b_{j}(2q_{j}-1)$, for fixed numbers 
$b_{j} > 0$, $j=1, 2, \ldots, d$, determining the magnetic field. The number 
$\varkappa$ of different sets $\mbf{q}=\{ q_{1}, \ldots, q_{d} \}$ which determine one and the same level is called the multiplicity of $\Lambda_{\lmbf{q}}$.

The analogous problem in three dimensions, where the spectrum of $H_{0}$ fills
a semi-axis, has been studied intensively. Avron {\em et al}~\cite{ahs78}, 
Solnyshkin~\cite{solnysh82} and Sobolev~\cite{sobolev84} has shown that even 
for a compactly supported $V \geq 0$, there are infinitely many eigenvalues 
below the lowest point of the essential spectrum of $H_{0}-V$. 
For fast, but not too fast decaying $V$, Sobolev obtained the leading 
asymptotics of power-like type.

Extending the work by Avron {\em et al}, Solnyshkin, Sobolev, and
also Tamura \cite{tamura88}, Raikov \cite{raikov90} obtained the most
detailed results on the leading eigenvalue asymptotics near
the boundary points of the essential spectrum in dimensions $\geq 2$; e.g.\
in even dimensions, if $V$ decays at infinity not too fast, the eigenvalues
of $H_{0} \pm V$ near each LL have power-like asymptotics. However, 
neither Raikov's work nor later work by Boyarchenko-Levendorski~\cite{levendorski97}, 
Hempel-Levendorski~\cite{hempel98} and Ivrii~\cite{ivrii98} cover very fast 
decaying or even compactly supported $V$. 

The complication arising when considering very fast decaying or compactly supported
potentials is related to the following circumstance. During the last
two decades, the progress in studying spectral asymptotics problems
was achieved by refining microlocal analysis methods, using pseudodifferential 
and Fourier integral operators (see, e.g.~\cite{ivrii98,safarov97}).
Such methods were used in the papers cited above, as well. However, for
very fast decaying potentials the pseudodifferential methods do not work.
The reason for this can be explained by noting which properties of symbol classes
are critical for the pseudodifferential analysis. In fact, usually all classes of 
symbols arising in various problems have the common feature: They improve their 
decay under differentiation (with respect to some, or all, variables). This property 
is observed, say, by symbols with a power-like behaviour. However, if the symbol 
decays rapidly, say, as $\exp(-|\xi|^{2})$, differentiation worsens the behaviour of 
the symbol, instead of improving it. It is exactly this sort of behaviour of the 
symbol that one encounters when reducing the spectral problem for the
magnetic Schr\"{o}dinger operator with fast decaying potentials to ones
for pseudodifferential operators as it was done 
in~\cite{sobolev84,tamura88,raikov90,levendorski97,ivrii98}. This is the 
reason why the methods of the latter works do not apply to such potentials; 
even the question of finiteness of the number of eigenvalues was not resolved. 
It is this kind of potentials we are going to consider in the present paper.

Specifically, we consider potentials satisfying the following assumption.

\begin{assumption}
Let $V$ be a nonnegative, bounded, measurable function on $\Rl^{2d}$
having compact support. In addition, assume that $V >0$ holds on 
a nonempty, open set.
\label{assump0}
\end{assumption}

Fix the LL $\Lambda=\Lambda_{\lmbf{q}}$, $\mbf{q} \in \Nl^{d}$ and let 
$\L^{\pm}$ denote the nearest LL lying to the left or 
right of $\L$, respectively. Choose any $\mu \in (\Lambda^{-}, \Lambda)$ 
(respectively, $\mu  \in (\Lambda, \Lambda^{+})$) and let 
$N_{\L}^{-}(\lambda) = N(\mu_{1}, \Lambda-\lambda \, | \, H_{0}-V)$ 
(respectively, $N_{\L}^{+}(\lambda)=N(\Lambda+\lambda, \mu_{2} \, | \, H_{0}+V)$), 
for $\lambda> 0$ small enough, denote the number (counting multiplicity) of 
eigenvalues of $H_{0} \pm V$ in the gap $(\mu_{1}, \Lambda-\lambda)$ 
(respectively, $(\Lambda+\lambda, \mu_{2})$). 

Under Assumption~\ref{assump0} we establish the following theorem.

\begin{theorem}
Let $V$ satisfy Assumption~\ref{assump0} on $\Rl^{2d}$, $d\in \Nl$. 
Then the number of eigenvalues for $H_{0} \pm V$ near any LL 
$\Lambda$ is infinite. Moreover, if $\mu_{1} \in (\Lambda^{-}, \Lambda)$ 
and $\mu_{2} \in (\Lambda, \Lambda^{+})$ then
\begin{displaymath}
N(\mu_{1}, \Lambda-\lambda \, | \, H_{0}-V) \sim 
N(\Lambda+\lambda, \mu_{2} \, | \, H_{0}+V) \sim 
\varkappa \,  \varXi_{d}(\lambda) \mbox{ as } \lambda \downarrow 0,
\end{displaymath}
where $\varkappa$ denotes the multiplicity of the LL $\Lambda$ and
\begin{displaymath}
\varXi_{d}(\lambda)= \frac{1}{d!} \left( \! \frac{| \ln \lambda |
}{\ln | \ln \lambda |} \! \right)^{d}.
\end{displaymath}
\label{sect:intro-thm1}
\end{theorem}

Hence, under Assumption~\ref{assump0} there still are an infinite
number of eigenvalues in each gap, they converge to the LLs very fast 
and their leading asymptotics do not depend on $V$. The asymptotics 
is not described by the quasi-classical formulas known for the case where 
$V$ admits power-like decay. In this sense it is non-classical. Such 
behaviour is rather exceptional for differential operators in 
mathematical physics. 

As a spin-off we manage to strengthen the first statement in 
Theorem~\ref{sect:intro-thm1} to {\em any} nonnegative bounded $V$ which tends 
to zero at infinity (see Proposition~\ref{sechtwo1:prop1}).

Another operator of mathematical physics with spectral structure similar
to the one for the magnetic Schr\"{o}dinger operator is the Dirac operator
with a constant magnetic field in even dimension $2d$.

The spectrum of the unperturbed Dirac operator $\dop_{d}$ is purely essential 
and it consists of infinitely many isolated eigenvalues, called LLs, 
$\mu_{\lmbf{q}}^{\pm}$, $\mbf{q} \in \Nl^{d}$. Here 
$\mu_{\lmbf{q}}^{+} =-\mu_{\lmbf{q}}^{-}$ except for the ``lowest'' eigenvalue 
$\mu_{\lmbf{0}}$, which is {\em either} $-1$ {\em or} $1$. In order to study the 
discrete spectrum of the operator $\dop_{d}$ perturbed by an electric potential 
$VI$, where $I_{2^{d}}$ is the $2^{d} \times 2^{d}$ identity matrix and the 
potential $V$ is a real-valued function on $\Rl^{2d}$ satisfying 
Assumption~\ref{assump0}, we need a detailed description of 
the eigenspaces corresponding to the LLs of $\dop_{d}$. We were unable to 
find this description in the general situation in the literature. Even the analysis 
in~\cite{thaller92} performed for $d=1$ is not sufficient for our 
purposes since the eigenspaces are described rather implicitly there. Therefore we 
devote Section~\ref{edirac} to the spectral analysis of $\dop_{d}$. Having this
description at our disposal, we establish the following result.  

\begin{theorem}
Let $V$ satisfy Assumption~\ref{assump0} on $\Rl^{2d}$. Then
the number of eigenvalues for the perturbed Dirac operator $\dop_{d} \pm V$ near any 
LL $\mu=\mu_{\lmbf{q}}^{\pm}$ is infinite. Moreover, if 
$\mu^{-}$ and $\mu^{+}$ denote the neighboring LLs lying below and
above $\mu$, respectively, and if $s_{1} \in (\mu^{-}, \mu)$ and 
$s_{2} \in (\mu, \mu^{+})$ then
\begin{displaymath}
N(s_{1}, \mu-\lambda \, | \, \dop_{d}-V) \sim
N(\mu+\lambda, s_{2} \, | \, \dop_{d}+V) \sim 
\varpi \, \varXi_{d}(\lambda) \mbox{ as } \lambda \downarrow 0,
\end{displaymath} 
where $\varpi=\varpi(\mu)$ is a certain integer (given explicitly in 
(\ref{ediracs5:prop2-eq1})). 
\label{edirac:thm1} 
\end{theorem}

Thus, just as for the Schr\"{o}dinger operator, there are an infinite number of 
eigenvalues in each gap and the asymptotic behaviour of the eigenvalues near the 
LLs is expressed by the same formula (up to a multiplicative constant) 
as in Theorem~\ref{sect:intro-thm1}. 

Both for the Schr\"{o}dinger and Dirac operators, the proofs of asymptotical 
eigenvalue formulas are based on the analysis of Toeplitz-like operators having 
the form $T=PVP$, where $P$ is the projection onto the Landau eigenspace of the 
corresponding operator. The operator $T$ was in the center of study in most of 
the preceding papers devoted to the subject. It is here the afore-mentioned 
microlocal methods fail for our case. To handle the problem, we apply the 
old-fashioned but still powerful variational method. Nevertheless, the treatment 
of Toeplitz operators is fairly technical, and our presentation is divided into 
three sections, Sections~\ref{secvk}-\ref{multi} dealing with the Schr\"{o}dinger 
case and Section~\ref{ediracs8} with the Dirac case. The variational approach is 
also used in Section~\ref{sechtwo}, where the perturbational reasoning carries over 
the results for Toeplitz operators to the perturbed Schr\"{o}dinger and Dirac 
operators. In Appendix we establish a number theoretical result 
(Lemma~\ref{gaussprop1}) which is crucial in dimensions $> 2$.

When this paper was near its completion, Prof. G. Raikov informed the authors
about the manuscript~\cite{raikov02}, where the eigenvalue 
asymptotics for the Schr\"{o}dinger operator with compactly supported or 
very fast decaying potentials is considered in two and three dimensions. 
The results in two dimensions cover the ones herein. Higher dimensions, 
however, create considerable additional problems which are resolved in the 
present paper and, even for two dimensions, our reasoning is, probably, more 
transparent. Moreover, the analogous problem for the Dirac operator is not 
considered in~\cite{raikov02} (or elsewhere).


\begin{flushleft} 
{\small {\bf Acknowledgement}. We are grateful to Dr. G.R.H. Greaves 
for providing us with Lemma~\ref{gausslem1}, which enable us to give a 
more streamlined proof of Lemma~\ref{gaussprop1} compared to an initial, 
more {\em ad hoc} proof. Moreover, the authors thank Prof. G. Raikov for 
useful discussions and for sending them the manuscript~\cite{raikov02}}.
\end{flushleft} 

\section{VARIATIONAL METHOD}
\label{sect:prelim}
Let ${\CH}$ be a separable Hilbert space with its norm
and scalar product denoted by $\| \cdot \|$ and $\langle
\cdot, \cdot \rangle$, respectively. For a self-adjoint operator 
$T$ the distribution function $N(l,r \, | \, T)$ counts the total 
number of points of the spectrum $\sigma(T)$ of $T$ (taking into account 
their multiplicity) in the interval $(l,r)$. If a point
of the essential spectrum belongs to $(l,r)$ then 
$N(l,r \, | \, T)=\infty$. We define $N_{-}(s,T)=N(-\infty, s \, | \, T)$ 
and $N_{+}(s,T) = N(s,\infty \, | \, T)$. If $T$ is compact, we set
$n_{\pm}(s,T)=N_{\pm}(s,T)$.

We shall use the min-max description of eigenvalues (following directly
from the Spectral Theorem).

\begin{lemma} 
Let $T$ be a self-adjoint operator. Then, for $l < r$, 
\begin{eqnarray} 
\hspace*{-0.7cm} N(l,r | T) \hspace*{-0.1cm} \! & \!\!  = \hspace*{-0.1cm}\!\! & \!  \max  {\rm dim}  \left\{ {\mathcal L}  \subset  \dom{T} :  \| (T-s)u \|^{2}  <  t^{2} \| u \|^{2},  u \in  {\mathcal L} \backslash \{ 0 \} \right\} \label{sectprelim:lem1-eq1} \\
\hspace*{-0.7cm}  \hspace*{-0.1cm} \! & \!\! = \hspace*{-0.1cm} \!\! & \! \min {\rm codim}\! \left\{ {\mathcal L} \! \subset \! \dom{T} : \| (T\!-\!s)u\|^{2} \! \geq \!t^{2} \| u \|^{2}, u \! \in \! {\mathcal L} \right\}, 
\label{sectprelim:lem1-eq2}
\end{eqnarray} 
where $s=(l+r)/2$ and $t=(r-l)/2$. 
\label{sectprelim:lem1}
\end{lemma} 

For a nonnegative, compact operator $T$ it is convenient to use another form
of (\ref{sectprelim:lem1-eq1})-(\ref{sectprelim:lem1-eq2}):
\begin{eqnarray}
\hspace*{-0.7cm} n_{+}(s, T) \hspace*{-0.1cm} \! & \!  =  \hspace*{-0.1cm} \!\!& \!  \max {\rm dim}\, \! \left\{ {\mathcal L} \subset {\CH} \, : \, \langle Tu,u \rangle  > s \| u \|^{2}, \: u \in {\mathcal L} \backslash \{ 0 \}  \right\} \label{sectprelim:lem1-eq3} \\ 
\hspace*{-0.7cm}  \hspace*{-0.1cm} \! & \! = \hspace*{-0.1cm} \!\! & \! \min {\rm codim}\,\!  \left\{ {\mathcal L} \subset {\CH} \, : \, \langle Tu,u \rangle \leq  s \| u \|^{2}, \!\:u \in {\mathcal L} \, \right\}, s > 0. \label{sectprelim:lem1-eq4}
\end{eqnarray} 

We shall also use the following perturbational result found in, e.g., \cite{birman87}.

\begin{lemma}
Let $T$ be self-adjoint and let $S$ be bounded on ${\CH}$
such that $\sigma(S) \subset [s_{1},s_{2}]$. Then for any
finite interval $(l,r)$ one has that
\begin{displaymath}
N(l+s_{1},r+s_{2} \, | \, T+S) \geq N(l,r \, | \, T).
\end{displaymath}
\label{sectprelim:lem2}
\end{lemma}

In addition to the standard order symbols we use, for two real-valued
functions $f(\lambda)$ and $g(\lambda)$, the notation $f(\lambda) \lesssim
g(\lambda)$ which means that $f(\lambda) \leq g(\lambda)(1+o(1))$ as
$\lambda \rightarrow 0$.

\section{THE UNPERTURBED SCHR\"{O}DINGER OPERATOR}
We summarize some well-known facts about the unperturbed 
Schr\"{o}dinger operator with a constant magnetic field 
(see, e.g., \cite{helffer88}).
 
\subsection{Two-dimensional Case}
\label{sectfham}
In $\Rl^{2}$ we consider a charged, spinless particle in a
homogeneous magnetic field. We assume that the magnetic field 
$\mbf{B}$ has constant strength $b > 0$ and is aligned in 
the $z$ direction, viz. $\mbf{B}=(0,0,b)$. In appropiate units the 
Hamiltonian of the particle is $H_{0} =  (\mbf{P}+\mbf{A})^{2}$, where 
$\mbf{P}=-i\nabla$ is the momentum operator and $\mbf{A}$ is some 
vector potential associated with the field, viz.\ 
$\mbf{B} = \nabla \times \mbf{A}$, and defined up to a gauge 
transformation. We choose the gauge in which 
$\mbf{A} = \frac{1}{2} (\mbf{B} \wedge \mbf{r})$, $\mbf{r}=(x,y)$. The 
resulting Hamiltonian
\begin{equation}
H_{0}=\left(-i\frac{\partial}{\partial x} -\frac{b}{2}y \right)^{2}
+\left( -i\frac{\partial}{\partial y} +\frac{b}{2}x \right)^{2} 
\label{sectfham:eq1}
\end{equation}
is essentially self-adjoint on $C_{0}^{\infty}(\Rl^{2})$~\cite{rs2}. 
It is well-known that $H_{0}$ has a purely essential
spectrum consisting of isolated eigenvalues, viz.
\begin{equation}
\sigma(H_{0})= \{ \, \Lambda_{q}=b(2q-1) \, | \, q=1, 2, 3, \ldots \}.
\label{sectfham2:eq3}
\end{equation}
In literature on physics the eigenvalues $\Lambda_{q}$ are referred to as 
LLs. For a fixed LL $\Lambda_{q}=b(2q-1)$ one
can choose an orthonormal basis in the Landau eigenspace consisting of
functions 
\begin{eqnarray}
\lefteqn{ \hspace*{-1.6cm} f_{m}(\rho,\theta)= f_{m,q}(\rho,\theta;b)= (2\pi)^{-1/2} 
b^{(1+|m|)/2} 2^{-|m|/2} \left( \frac{q!}{(q+|m|)!} \right)^{1/2} } \nonumber \\
&& \hspace*{-1.7cm} \times e^{-i m \theta} \rho^{|m|} e^{-\frac{b}{4} \rho^{2}} 
L_{q}^{(|m|)}\left(\frac{b}{2}\rho^{2}\right), \:\: m=-q+1,-q+2, \ldots,
\label{sectfham1:eq17}
\end{eqnarray}
in polar coordinates $(\rho, \theta)$. Here $L_{q}^{(|m|)}(\xi)$ are the 
generalized Laguerre polynomials
\begin{equation}
L_{q}^{|m|}(\xi) = \sum_{l=0}^{q} d_{l,|m|} (-\xi)^{l}\: ; \:\:d_{l,|m|}=\left( \begin{array}{c} q+|m| \\ q-l \end{array} \right) \frac{1}{l!}.
\label{secfham1:eq119}
\end{equation}

\subsection{Even-dimensional Case}
\label{secteham} 
Introduce the vector potential 
$\mbf{a}=(a_{1}, a_{2}, \ldots, a_{2d}) \in C^{1}(\Rl^{2d};\Rl^{2d})$ 
and the $1$-form ${\mathcal A} =\sum_{j=1}^{2d} a_{j} \, dx_{j}$. Define the 
magnetic field $2$-form
\begin{displaymath}
{\mathcal B} =d{\mathcal A}= \sum_{i=1}^{2d} \sum_{j=1}^{2d}
b_{ij} \, dx_{i} \wedge dx_{j}, \:\:\:\:b_{ij} =\frac{1}{2}
(\partial_{x_{i}}  a_{j}-\partial_{x_{j}} a_{i}).
\end{displaymath} 
We assume that ${\mathcal B} \not \equiv 0$. Moreover, we suppose that all 
the entries of the antisymmetric matrix $\mbf{B}=\{ b_{ij} \}_{i,j}^{2d}$ 
are constant. The unperturbed Hamiltonian 
$H_{0}=  \sum_{j=1}^{2d} (-i\partial_{x_{j}} -a_{j})^{2}$ defined on
$C_{0}^{\infty}(\Rl^{2d})$ is essentially self-adjoint in $L^{2}(\Rl^{2d})$. 
Its unique self-adjoint extension is also denoted by $H_{0}$. The eigenvalues 
of $\mbf{B}$ form a subset of the imaginary axis, symmetric with respect 
to the origin. Let $b_{j} \in \Rl$, $j=1, 2, \ldots, d$ be such numbers that the 
nonzero eigenvalues of $\mbf{B}$ coincide (counting 
multiplicity) with the imaginary numbers $-ib_{j}$ and $ib_{j}$, 
$j=1, \ldots, n$. Thus we have $2n={\rm rank}\, \mbf{B}$, $0 < 2n \leq 2d$. 

We limit ourselves to the {\em non-degenerate case} $n=d$. 
In proper coordinates $(\mbf{x},\mbf{y})\in \Rl^{2d}$, 
$\mbf{x},\mbf{y} \in \Rl^{d}$, the $2$-form ${\mathcal B}$ block-diagonalizes: 
${\mathcal B} =\sum_{j=1}^{d} b_{j} \, dy_{j} \wedge dx_{j}$. By changing
variables $x_{j} \mapsto -x_{j}$ one may reduce the problem to the case
when all $b_{j}$ are positive. In these coordinates $H_{0}$ is unitary equivalent 
to (again we use the notation $H_{0}$) the operator
\begin{equation}
H_{0} =  \sum_{j=1}^{d} \tilde{H}_{0,j}=\sum_{j=1}^{d} \left( I \otimes \cdots \otimes H_{0,j} \otimes \cdots I \right)
\label{secteham:eq1}
\end{equation} 
in $L^{2}(\Rl^{2d})=\bigotimes_{j=1}^{d} L^{2}(\Rl_{j}^{2})$, where 
\begin{equation}
H_{0,j} =   \left( -i \frac{\partial}{\partial x_{j}} -
\frac{b_{j}}{2} y_{j} \right)^{2} + \left(
-i\frac{\partial}{\partial y_{j}} +  \frac{b_{j}}{2} x_{j} \right)^{2}
\label{secteham:eq2}
\end{equation} 
and
\begin{equation} 
\sigma(H_{0,j})=\sigma_{ess}(H_{0,j})=\{ b_{j}(2q_{j}-1) \, | \, q_{j}=1,2,3, \ldots \,\}
\label{secteham:eq3}
\end{equation}
with a basis of eigenfunctions of the form (\ref{sectfham1:eq17}), with 
$(\rho,\theta)$ replaced by the polar coordinates $(\rho_{j},\theta_{j})$, 
for the corresponding eigenfunctions $f_{m_{j},q_{j}}^{(j)}(\rho_{j},\theta_{j})=f_{m_{j},q_{j}}(\rho_{j},\theta_{j};b_{j})$. 

This gives us that the operator $H_{0}$ is essentially self-adjoint on
$\dom{H_{0}}=\bigotimes_{j=1}^{d}\dom{H_{0,j}}= \bigotimes_{j=1}^{d}
C_{0}^{\infty}(\Rl_{j}^{2})=C_{0}^{\infty}(\Rl^{2d})$ and, in view of
(\ref{secteham:eq1}) and (\ref{secteham:eq3}), its spectrum equals
\begin{equation} 
\sigma(H_{0})=\sigma_{ess}(H_{0})=\{ \, \Lambda_{\lmbf{q}} \, | \, \mbf{q}=(q_{1}, \ldots, q_{d}) \in \Nl^{d} \, \},
\label{secteham:eq4}
\end{equation} 
where the isolated eigenvalues $\{ \Lambda_{\lmbf{q}} \}$, $\mbf{q} \in \Nl^{d}$, have 
the form 
\begin{equation} 
\L=\Lambda_{\lmbf{q}} = \sum_{j=1}^{d} b_{j}(2q_{j}-1),\:\:q_{j} \in \Nl.
\label{secteham:l-levelsdef}
\end{equation} 
As above, the numbers $\Lambda_{\lmbf{q}}$ are called LLs. The number 
$\varkappa$ of different sets $\mbf{q}=\{ q_{1}, q_{2}, \ldots, q_{d}\}$ which 
determine one and the same level $\Lambda$ according to (\ref{secteham:l-levelsdef}) 
is called the multiplicity of $\Lambda$. 

We discuss multiple LLs in more details. Let, e.g.\ $d=2$, and 
let the matrix $\mbf{B}$ have eigenvalues $b_{1}$, $b_{2}$, $-b_{1}$, 
$-b_{2}$, $b_{j}>0$. Then the Schr\"{o}dinger operator has the form
\begin{displaymath}
H_{0} = H_{1} \otimes 1+1 \otimes H_{2}=\tilde{H}_{1} +\tilde{H}_{2},
\end{displaymath}
where $H_{j}$ is given in (\ref{secteham:eq2}). The LLs of 
$H_{0}$ are sums of LLs for $H_{1}$ and $H_{2}$, i.e.\ for a fixed level 
$\Lambda$ we may write
\begin{equation}
\Lambda=\Lambda_{1}+\Lambda_{2}=b_{1}(2q_{1}-1)+b_{2}(2q_{2}-1).
\label{multi:eq2}
\end{equation}
It may turn out that a given $\Lambda$ can be represented in the form 
(\ref{multi:eq2}) in several different ways, for example
\begin{equation}
\Lambda=b_{1}(2q_{1}^{'}-1)+b_{2}(2q_{2}^{'}-1)=
b_{1}(2q_{1}^{''}-1)+b_{2}(2q_{2}^{''}-1).
\label{multi:eq3}
\end{equation}
The number of such representations is the multiplicity $\varkappa$ of
the LL $\L$. According to (\ref{multi:eq3}), the Landau eigenspace 
of $H_{0}$ for $\Lambda$ is a direct sum of eigenspaces corresponding to 
different representations. For example, in the case (\ref{multi:eq3}),
\begin{equation}
{\CH}_{\Lambda}={\CH}_{\Lambda}^{'}\oplus
{\CH}_{\Lambda}^{''}= \left( {\CH}_{q_{1}^{'}}^{1} \otimes {\CH}_{q_{2}^{'}}^{2}\right)\oplus \left( {\CH}_{q_{1}^{''}}^{1} \otimes {\CH}_{q_{2}^{''}}^{2}\right)  
\label{multi:eq4}
\end{equation}
where, e.g.\ ${\CH}_{q_{j}^{'}}^{j}$, $j=1,2$, is the Landau eigenspace of 
$H_{j}$ corresponding to the LL $b_{j}(2q_{j}^{'}-1)$. Generally, for 
the multiplicity $\varkappa$ in dimension $2d$, we have
\begin{equation}
\Lambda=\sum_{j=1}^{d} \Lambda_{j}^{(\alpha)}=
\sum_{j=1}^{d} b_{j}(2q_{j}^{(\alpha)} -1),
\:\:\alpha=1, \ldots, \varkappa,
\label{multi:eq5}
\end{equation}
with corresponding eigenspace and projection given by
\begin{eqnarray}
{\CH}_{\Lambda} & = & \bigoplus_{\alpha=1}^{\varkappa}
\bigotimes_{j=1}^{d} {\CH}_{q_{j}^{(\alpha)}}^{j} =
\bigoplus {\CH}_{\Lambda}^{(\alpha)}, \nonumber \\
P & = & \sum P^{(\alpha)} \mbox{ with } P^{(\alpha)}=\otimes_{j=1}^{d} 
P_{j,q_{j}^{(\alpha)}},
\label{multi:eq6}
\end{eqnarray}
where $P_{j,q_{j}^{(\alpha)}}$ is the spectral projection onto
${\CH}_{q_{j}^{(\alpha)}}^{j}$. For all LLs to have multiplicity 
$1$, it is sufficient that $b_{1}, \ldots, b_{d}$ are linearly independent 
over odd integers. Thus, in the general situation, in particular for 
$b_{1}=\cdots=b_{d}$, one must expect that multiple LLs arise.

\section{UNPERTURBED DIRAC OPERATOR}
\label{edirac}
In this section we study the even-dimensional (unperturbed) Dirac operator 
$\dop_{d}$ with a constant magnetic field. In analogy with the Schr\"{o}dinger 
operator $H_{0}$ (see (\ref{secteham:eq1})) the spectrum of 
$\dop_{d}$ turns out to be purely essential and it consists of infinitely many 
isolated eigenvalues $\mu_{\lmbf{q}}^{\pm}$, $\mbf{q} \in \Nl^{d}$, called Landau 
levels (LL). Here $\mu_{\lmbf{q}}^{+} =-\mu_{\lmbf{q}}^{-}$ except for the ``lowest'' eigenvalue $\mu_{\lmbf{0}}$, which is {\em either} $-1$ {\em or} $1$. We find, moreover, an explicit representation of the corresponding Landau eigenspaces. 

\subsection{Dirac Matrices and Operators}
\label{ediracs1}
In dimension $2d$, $d \in \Nl$, there are $2d+1$ Dirac matrices,
denoted $\sigma_{1}^{(d)}$, $\sigma_{2}^{(d)}$, $\ldots$, $\sigma_{2d}^{(d)}$ and 
$\sigma_{0}^{(d)}$, each of size $2^{d}$, and they satisfy the relations
\begin{equation}
\hspace*{-0.2cm} (\sigma_{j}^{(d)})^{\ast}\! =\! \sigma_{j}^{(d)}, \:\:\sigma_{j}^{(d)}\sigma_{k}^{(d)}+\sigma_{k}^{(d)}\sigma{j}^{(d)}\!=\! 2\delta_{jk}I_{2^{d}},\:\:j\!=\! 0,1,2,\ldots,2d.
\label{ediracs1:eq1}
\end{equation}
The matrices may be defined inductively: For $d=1$ they are given by
\begin{displaymath}
\sigma_{1}=\left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right), \:\:
\sigma_{2}=\left( \begin{array}{cc} 0 & i \\ -i & 0 \end{array} \right) \mbox{ and }
\sigma_{0}=\left( \begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array} \right).
\end{displaymath}
If $\sigma_{1}^{(d)}, \sigma_{2}^{(d)}, \ldots, \sigma_{2d}^{(d)}$ and 
$\sigma_{0}^{(d)}$ are given for $d$, the matrices for $d+1$ are 
\begin{eqnarray*}
\sigma_{1}^{(d+1)} & = & \left( \begin{array}{cc}  \mbf{0} & \sigma_{1}^{(d)} \\ \sigma_{1}^{(d)} & \mbf{0} \end{array} \right), \:\: \ldots, \: \sigma_{2d-2}^{(d+1)} =  \left( \begin{array}{cc} \mbf{0} & \sigma_{2d-2}^{(d)} \\ \sigma_{2d-2}^{(d)} & \mbf{0} \end{array} \right), \\
\sigma_{2d-1}^{(d+1)} & = & \left( \begin{array}{cc} \mbf{0} & \sigma_{0}^{(d)} \\ \sigma_{0}^{(d)} & \mbf{0} \end{array} \right), \:\:\:\: \sigma_{2d}^{(d+1)} = \left( \begin{array}{cc} \mbf{0} & iI \\ -iI & \mbf{0} \end{array} \right)
\end{eqnarray*}
and
\begin{displaymath}
\sigma_{0}^{(d+1)} = \left( \begin{array}{cc} I & \mbf{0} \\ \mbf{0} & -I \end{array} \right) ; \:\:I=I_{2^{d-1}}.
\end{displaymath}
Vector components corresponding to upper blocks in these block-matrices
will be referred to as the upper half, the rest as the lower half.
 
For $b_{j} \in \Rl$, $j=1, 2, \ldots,d$, set $\beta_{j}=| b_{j}|$. Define 
\begin{displaymath}
P_{2j-1}=-i \frac{\partial}{\partial x_{j}}+\frac{b_{j}}{2} y_{j}, \:\:\: P_{2j}=-i \frac{\partial}{\partial y_{j}}-\frac{b_{j}}{2} x_{j}.
\end{displaymath}
Note that, since we handle first order operators now, we cannot use transformations
$x_{j} \mapsto -x_{j}$ to make all $b_{j}$ positive. Thus, here we have to consider
$b_{j}$ with arbitrary signs. 

The $2d$-dimensional unperturbed Dirac operator with a constant magnetic field, 
acting in $L^{2}(\Rl^{2})^{d}$, is defined by
\begin{displaymath}
\dop_{d} = \aop_{d}+\sigma_{0}^{(d)}, \mbox{ where } \aop_{d}=\sum_{j=1}^{2d} \sigma_{j}^{(d)} P_{j}.
\end{displaymath}
For $j=1, \ldots, d$ put
\begin{equation}
\Pi_{j}=P_{2j-1}+iP_{2j}=-i\frac{\partial}{\partial x_{j}} +\frac{b_{j}}{2} y_{j} + \frac{\partial}{\partial y_{j}}-i\frac{b_{j}}{2} x_{j}
\label{ediracs1:eq1-1}
\end{equation}
and
\begin{equation}
{\mathcal T}_{j} =  \left( \begin{array}{cc} \Pi_{j}^{\ast} & \mbf{0} \\ \mbf{0} & -\Pi_{j} \end{array} \right) = P_{2j}+\sigma_{0}^{(j)} P_{2j-1}.
\label{ediracs1:eq1-2}
\end{equation}
It is convenient to describe $\dop_{d}$ inductively as follows. For $d=1$ we write
\begin{equation}
\dop_{1} = \aop_{1}+\sigma_{0}^{(1)}, \mbox{ where } \aop_{1}=\left( \begin{array}{cc}
0 & \Pi_{1}^{\ast} \\ \Pi_{1} & 0 \end{array} \right)
\label{ediracs1:eq1-3}
\end{equation}
and, generally for an arbitrary $d$,
\begin{equation}
\dop_{d} = \aop_{d}+\sigma_{0}^{(d)}, \mbox{ where } \aop_{d}=\left( \begin{array}{cc}
\mbf{0} & \aop_{d-1}+{\mathcal T}_{d}^{\ast} \\ \aop_{d-1}+{\mathcal T}_{d} & \mbf{0} \end{array} \right).
\label{ediracs1:eq2}
\end{equation}

The afore-defined operators satisfy the following relations:
\begin{eqnarray}
\hspace*{-0.4cm} [ \Pi_{j},\Pi_{k} ] & = & [ \Pi_{j},\Pi_{k}^{\ast} ] =0, \:\:\: j \neq k, \label{ediracs1:eq3a} \\
\hspace*{-0.4cm} \Pi_{j} \Pi_{j}^{\ast} & = & \Pi_{j}^{\ast} \Pi_{j}+2b_{j} =H_{0,j}+b_{j},\: j=1, 2, \ldots, d, \label{ediracs1:eq3b} \\
\hspace*{-0.4cm} {\rm [}\,  \aop_{j}, \Pi_{j} \, {\rm ]} & = & 0, \:\:\:j=1, 2, \ldots, d, \label{ediracs1:eq3c} \\
\hspace*{-0.4cm} \aop_{j-1}{\mathcal T}_{j}+{\mathcal T}_{j}^{\ast} \aop_{j-1} & = & 0, \:\:\:j=1, 2, \ldots, d, \label{ediracs1:eq3d} 
\end{eqnarray}
with $H_{0,j}$ being the two-dimensional Schr\"{o}dinger operator (\ref{secteham:eq2})
associated with $b_{j}$. Moreover, 
\begin{equation}
{\mathcal T}_{j}{\mathcal T}_{j}^{\ast} = {\mathcal T}_{j}^{\ast} {\mathcal T}_{j} -2b_{j} \sigma_{0}^{(j)}=H_{0,j} I_{2^{j}} -b_{j} \sigma_{0}^{(j)}
\label{ediracs1:eq3e}
\end{equation}
and 
\begin{eqnarray*}
\sigma({\mathcal T}_{j}{\mathcal T}_{j}^{\ast})\! & \!= \! & \! \{ \L_{q_{j}} +b_{j} \, | \, q_{j}\in \Nl \, \}= \left\{ \begin{array}{ll}
          \{ \beta_{j}2q_{j} \, | \, q_{j} \in \Nl \, \} & \mbox{ if } b_{j} > 0, \\
          \{ \beta_{j}(2q_{j}-2) \, | \, q_{j} \in \Nl \, \} & \mbox{ if } b_{j} < 0, \end{array} \right. 
\end{eqnarray*} 

\subsection{The Operator $\aop_{d}^{2}$}
\label{ediracs2}
From the relations (\ref{ediracs1:eq3a})-(\ref{ediracs1:eq3e}) we deduce that
\begin{eqnarray}
\hspace*{-0.7cm} \aop_{d}^{2} & = & \left( \sum_{j=1}^{d} H_{0,j}\right) \underbrace{I \otimes \cdots \otimes I}_{\#=d} -b_{1} \underbrace{I \otimes \cdots \otimes I}_{\#=d-1} \otimes \sigma_{0} \nonumber \\
\hspace*{-0.7cm}& + & \!\!\!b_{2} \underbrace{I\otimes \cdots \otimes I}_{\#=d-2} \otimes \sigma_{0} \otimes \sigma_{0} +b_{3} \underbrace{ I \otimes \cdots \otimes I}_{\#=d-3} \otimes \sigma_{0} \otimes I \otimes \sigma_{0} \nonumber \\
\hspace*{-0.7cm} & + & \!\!\! \cdots+b_{d-1} I \otimes \sigma_{0} \otimes \underbrace{I \otimes \cdots \otimes I}_{\#=d-3} \otimes \sigma_{0} + b_{d} \sigma_{0} \otimes \underbrace{ I \otimes \cdots \otimes I}_{\#=d-2} \otimes \sigma_{0}.
\label{ediracs2:eq1}
\end{eqnarray}
Thus one may represent $\aop_{d}^{2}$ as
\begin{equation}
\aop_{d}^{2} = H_{0} I_{2^{d}} -\Delta_{d}(\mbf{b}).
\label{ediracs2:eq2}
\end{equation}
Here $H_{0}=\sum_{j=1}^{d} H_{0,j}$ is the $2d$-dimensional Schr\"{o}dinger 
operator (\ref{secteham:eq1}), $\mbf{b}=(b_{1}, \ldots, b_{d})$ and
$\Delta_{d}(\mbf{b})$ is the diagonal $2^{d} \times 2^{d}$ matrix having
on the diagonal the sums $b_{\lmbf{\e}}=\sum_{j=1}^{d} \e_{j}b_{j}$, where
$\mbf{\e}$ belongs to the set
\begin{displaymath}
E^{d} =\{ (\e_{1}, \ldots, \e_{d}) \, | \, \mbox{ all possible combinations of } \e_{j}=\pm 1 \, \}.
\end{displaymath}
The way in which $b_{\lmbf{\e}}$ enter on the diagonal of $\Delta_{d}(\mbf{b})$ 
determines an order for elements $\mbf{\e} \in E^{d}$. We denote by $E_{+}^{d}$ and
$E_{-}^{d}$ the sets of those $\mbf{\e} \in E^{d}$ for which $b_{\lmbf{\e}}$ enters 
in the upper half or lower half of $\Delta_{d}(\mbf{b})$, respectively. The 
particular order in which $b_{\lmbf{\e}}$ enter on the diagonal is of no importance, 
however, one property of $b_{\lmbf{\e}}$, which follows easily inductively from the 
inductive definition of Dirac matrices and (\ref{ediracs2:eq1}) is the following. Fix 
some subset $\Jl \subset \{ 1, 2, \ldots, d \}$ with $\nu$ elements, 
$1 \leq \nu \leq d$. Then of $2^{d-\nu}$ combinations of $\mbf{\e}$ with $\e_{j}$ 
fixed for $j \in \Jl$, exactly one half, i.e.\ $2^{d-\nu-1}$, belong to $E_{+}^{d}$, 
and $2^{d-\nu-1}$ belong to $E_{-}^{d}$. 

Since $\aop_{d}^{2}$ is a direct sum of operators of the form $H_{0}-b_{\lmbf{\e}}$, 
the spectrum of $\aop_{d}^{2}$ is determined by the spectrum of $H_{0}$. To define 
the eigenspaces of $\aop_{d}^{2}$, we introduce the following notation. Let 
${\CH}_{\lmbf{\e}} \subset L^{2}(\Rl^{2d})_{\lmbf{\e}}$, 
$\mbf{\e} \in E^{d}$, be closed subspaces. Let $\iota_{\lmbf{\e}}$ be 
the natural embedding of $L^{2}(\Rl^{2d})_{\lmbf{\e}}$ into 
$L^{2}(\Rl^{2d}, \Cl^{2^{d}})$ (to an element $\psi$ in $L^{2}(\Rl^{2d})_{\lmbf{\e}}$ 
we assign the $2^{d}$-dimensional vector in $L^{2}(\Rl^{2d}, \Cl^{2^{d}})$ for which
all entries are zero expect the ``$\mbf{\e}$th'', which equals $\psi$). Then we set 
$\boxplus_{\lmbf{\e}} {\CH}_{\lmbf{\e}} = \oplus_{\lmbf{\e}} \iota_{\lmbf{\e}} {\CH}_{\lmbf{\e}}$. This will be called the exterior direct sum. The same notation will be 
used for elements in ${\CH}_{\lmbf{\e}}$.

\begin{proposition}
Let $\L:=\L_{\lmbf{q}}=\sum_{j=1}^{d} \beta_{j}(2q_{j}-1)$ be a LL,
with multiplicity one, of the $2d$-dimensional Schr\"{o}dinger operator 
$H_{0}$ and let ${\CH}_{\L}$ be the corresponding eigenspace. For any 
$\mbf{\e} \in E^{d}$ define
\begin{displaymath}
\mathfrak{l}_{\lmbf{\e}} =\sum_{j=1}^{d} \beta_{j}[2q_{j}-2-(1-\e_{j})].
\end{displaymath}
Then the largest of these $\mathfrak{l}_{\lmbf{\e}}$, 
$\mathfrak{l}=2 \sum_{j=1}^{d} \beta_{j}(q_{j}-1)$ is an eigenvalue of 
$\aop_{d}^{2}$. To each $\mathfrak{l}_{\lmbf{\e}}$ one associates a subspace
${\CH}_{\L_{\lmbf{\e}}}$ in $L^{2}(\Rl^{2d})_{\lmbf{\e}}$ as follows: Define
\begin{displaymath}
{\mathcal E}(\mbf{q})=\{ \, \mbf{\e} \in E^{d} \, | \, 2q_{j}-1-(1-\e_{j}) > 0, \: \forall \e_{j} \, \}.
\end{displaymath}
If $\mbf{\e} \in {\mathcal E}(\mbf{q})$ then one puts ${\CH}_{\L_{\lmbf{\e}}}$ equal 
to the eigenspace associated with the LL 
$\L_{\lmbf{\e}}:=\sum_{j=1}^{d} \beta_{j}[2q_{j}-1-(1-\e_{j})]$ of $H_{0}$. 
Otherwise, if $\mbf{\e} \not \in {\mathcal E}(\mbf{q})$ then $\L_{\lmbf{\e}}$ is
not a LL for $H_{0}$ and one puts ${\CH}_{\L_{\lmbf{\e}}}=\{ 0 \}$. Then 
$\yrum_{\mathfrak{l}} = \boxplus_{\lmbf{\e} \in {\mathcal E}(\lmbf{q})} 
{\CH}_{\L_{\lmbf{\e}}}$ is the eigenspace associated with the eigenvalue 
$\mathfrak{l}$ of $\aop_{d}^{2}$. Moreover, if $\nu(\mbf{q})$ is the number of those
$q_{j}$ which are larger than one, then 
${\rm Card}\,{\mathcal E}(\mbf{q})=2^{\nu(\lmbf{q})}$ and, if $\nu(\mbf{q}) > 0$,
exactly half of the elements in ${\mathcal E}(\mbf{q})$, i.e.\ $2^{\nu(\lmbf{q})-1}$,
belongs to $E_{+}^{d}$, and the other half belongs to $E_{-}^{d}$.
\label{ediracs5:prop1}
\end{proposition}

\begin{proof}
Assume that $\mathfrak{l}$ is an eigenvalue of $\aop_{d}^{2}$, i.e.\
\begin{equation}
\aop_{d}^{2} \Psi = \mathfrak{l} \Psi
\label{ediracs5:prop1eq3}
\end{equation}
is solved for $\Psi= ( \psi_{\lmbf{\e}} )_{\lmbf{\e}} \in \yrum_{\mathfrak{l}}$, 
where the latter is some eigenspace to be determined. Now, (\ref{ediracs5:prop1eq3}) 
can be written as the system 
\begin{equation}
H_{0} \psi_{\lmbf{\e}} = (\mathfrak{l}+b_{\lmbf{\e}})\psi_{\lmbf{\e}},
\label{ediracs5:prop1eq2}
\end{equation}
where $b_{\lmbf{\e}}=\sum_{j=1}^{d} \e_{j}b_{j}$. Thus, at least one of these
$\mathfrak{l}+b_{\lmbf{\e}}$ must be a LL, which implies that the 
largest of these $\mathfrak{l}+b_{\lmbf{\e}}$, namely $\L=\mathfrak{l}+\sum \beta_{j}$ is a LL of $H_{0}$, i.e.\ $\L=\sum \beta_{j}(2q_{j}-1)$ for some $q_{j} \in \Nl$. 
Indeed, if the largest one is not a LL of $H_{0}$ then
the smaller ones  are not LLs of $H_{0}$ either. Consequently, 
$\mathfrak{l}=2\sum \beta_{j}(q_{j}-1)$. To find the associated eigenspace, we 
consider all other possible LLs for $H_{0}$, corresponding to this 
$\mathfrak{l}$, for all $\mbf{\e}$. According to (\ref{ediracs5:prop1eq2}) they may 
be represented as
\begin{equation}
\L_{\lmbf{\e}} \!=\! \L-\left( \sum_{j=1}^{d} \beta_{j} -
\sum_{j=1}^{d} \e_{j} b_{j} \right) \! = \! \sum_{j=1}^{d} 
\beta_{j}[2q_{j}-1-(1-\e_{j})].
\label{ediracs5:prop1eq2-1}
\end{equation}
It is clear that these smaller $\L_{\lmbf{\e}}$ need not all
be LLs of $H_{0}$. Indeed, if a certain $q_{j}$ equals 
$1$ (i.e.\ attains its smallest possible value) and the corresponding 
$\e_{j}$ equals $-1$ then the corresponding $\beta_{j}$ is not present 
in the expression for $\L_{\lmbf{\e}}$, viz.\ $\L_{\lmbf{\e}}$ is not 
a LL of $H_{0}$. As a consequence, we define
$\mathfrak{l}_{\lmbf{\e}}=\sum \beta_{j}[2q_{j}-2-(1-\e_{j})]$ (of which
the largest is the eigenvalue $\mathfrak{l}$ of $\aop_{d}^{2}$). Then
we assign a subspace ${\CH}_{\L_{\lmbf{\e}}}$ to each of
these $\mathfrak{l}_{\lmbf{\e}}$ as described in the statement of the
Proposition, and it follows immediately that
$\yrum_{\mathfrak{l}}=\boxplus_{\lmbf{\e} \in {\mathcal E}(\lmbf{q})} {\CH}_{\L_{\lmbf{\e}}}$ is the eigenspace associated with the eigenvalue $\mathfrak{l}$ of 
$\aop_{d}^{2}$. This family of $\L_{\lmbf{\e}}$ will be referred to as the
``ladder'' corresponding to the eigenvalue $\mathfrak{l}$.

On the other hand, any LL of $H_{0}$ can act as the largest one for some eigenvalue
of $\aop_{d}^{2}$. In particular, if we have $\L=\sum_{j} \beta_{j}$ (the lowest
LL of $H_{0}$) and, consequently, $\mathfrak{l}=0$, then $\L$ is the only LL of
$H_{0}$ included in the above-mentioned ladder. 

Define $\nu(\mbf{q})$ as in the Proposition, i.e.\ there are $d-\nu(\mbf{q})$
elements in $\mbf{q}$ for which $q_{j}=1$. Let $\Jl$ denote the set of indices 
for the latter $q_{j}$'s. To ensure that ${\CH}_{\L_{\lmbf{\e}}} \neq \{ 0\}$ 
for a particular $\mbf{\e}$, each of the elements $\e_{j}$, $j \in \Jl$, has to
be chosen as $+1$. For the remaining $\nu(\mbf{q})$ entries in each $\mbf{\e}$, 
we may choose $\e_{j}=\pm 1$. Hence, there are $2^{\nu(\lmbf{q})}$ different 
$\mbf{\e}$ for which ${\CH}_{\L_{\lmbf{\e}}} \neq \{ 0\}$, i.e.
${\rm Card}\, {\mathcal E}(\mbf{q})=2^{\nu(\lmbf{q})}$. Provided $\nu(\mbf{q}) >0$,
the property mentioned just above the Proposition, asserts that exactly half of the 
$2^{\nu(\lmbf{q})}$ elements in ${\mathcal E}(\mbf{q})$, i.e. 
$2^{\nu(\lmbf{q})-1}$, belongs to $E_{+}^{d}$. The other half belongs to $E_{-}^{d}$. 
\end{proof}

Define $\yrum_{\mathfrak{l}}^{(\pm)}=\boxplus_{\lmbf{\e} \in {\mathcal E}(\lmbf{q}) \cap E_{\pm}^{d}}{\CH}_{\L_{\lmbf{\e}}}$. According to Proposition~\ref{ediracs5:prop1}, 
the number of summands in $\yrum_{\mathfrak{l}}^{(\pm)}$ equals $2^{\nu(\lmbf{q})-1}$,
provided $\nu(\mbf{q}) >0$.

\vspace*{0.2cm}
\noindent
For multiple LLs of $H_{0}$ we have to perform the above
``ladder'' construction for each representation of the LL $\L$.

\begin{proposition}
Let $\L=\L_{\lmbf{q}^{(\alpha)}} = \sum_{j=1}^{d} \beta_{j} (2q_{j}^{(\alpha)}-1)$, 
$\alpha=1, 2, \ldots,\varkappa$, be a LL, with multiplicity $\varkappa$, of 
the $2d$-dimensional Schr\"{o}dinger operator $H_{0}$. Let ${\CH}_{\L}$ be the corresponding 
eigenspace. For any $\mbf{\e} \in E^{d}$ define
\begin{displaymath}
\mathfrak{l}_{\lmbf{\e}} =\sum_{j=1}^{d} \beta_{j}[2q_{j}^{(\alpha)}-2-(1-\e_{j})].
\end{displaymath}
Then the largest of these $\mathfrak{l}_{\lmbf{\e}}$, 
$\mathfrak{l}=2 \sum_{j=1}^{d} \beta_{j}(q_{j}^{(\alpha)}-1)$ is an eigenvalue of 
$\aop_{d}^{2}$. To each $\mathfrak{l}_{\lmbf{\e}}$ we associate a subspace
${\CH}_{\L_{\lmbf{\e}}}^{(\alpha)}$ in $L^{2}(\Rl^{2d})$ as follows: Define
\begin{equation}
{\mathcal E}(\mbf{q}^{(\alpha)})=\{ \, \mbf{\e} \in E^{d} \, | \, 2q_{j}^{(\alpha)}-1-(1-\e_{j}) > 0, \:\forall \e_{j} \, \}.
\label{ediracs5:prop2-eq0}
\end{equation}
If $\mbf{\e} \in {\mathcal E}(\mbf{q}^{(\alpha)})$ then one puts 
${\CH}_{\L_{\lmbf{\e}}}^{(\alpha)}$ equal to the eigenspace associated with the LL 
$\L_{\lmbf{\e}}=\sum_{j=1}^{d} \beta_{j}[2q_{j}^{(\alpha)}-1-(1-\e_{j})]$ of
$H_{0}$. Otherwise, if $\mbf{\e} \not \in {\mathcal E}(\mbf{q}^{(\alpha)})$ then $\L_{\lmbf{\e}}$ is not a LL for $H_{0}$ and one puts 
${\CH}_{\L_{\lmbf{\e}}}^{(\alpha)}=\{ 0 \}$. If $\nu(\mbf{q}^{(\alpha)})$ 
is the number of those $q_{j}$ which are larger than one, then 
${\rm Card}\,{\mathcal E}(\mbf{q}^{(\alpha)})=2^{\nu(\lmbf{q})}$ and, provided 
$\nu(\mbf{q}^{(\alpha)}) > 0$, exactly half of the elements in 
${\mathcal E}(\mbf{q}^{(\alpha)})$, i.e.\ $2^{\nu(\lmbf{q}^{(\alpha)})-1}$,
belongs to $E_{+}^{d}$, and the other half belongs to $E_{-}^{d}$.

Define $\yrum_{\mathfrak{l}}^{(\alpha)}= \boxplus_{\lmbf{\e} \in {\mathcal E}(\lmbf{q}^{(\alpha)})}{\CH}_{\L_{\lmbf{\e}}}^{(\alpha)}$. Then $\yrum_{\mathfrak{l}} = \oplus_{\alpha} \yrum_{\mathfrak{l}}^{(\alpha)}$ is the eigenspace associated with the eigenvalue $\mathfrak{l}$ of $\aop_{d}^{2}$. 
\label{ediracs5:prop2}
\end{proposition}

In Proposition~\ref{ediracs5:prop2} the eigenvalue $\mathfrak{l}$ of 
$\aop_{d}^{2}$ can be represented in $\varkappa$ different ways, so we say 
that it has multiplicity $\varkappa$.

It follows from Proposition~\ref{ediracs5:prop2} that the number of summands in 
$\yrum_{\mathfrak{l}}^{(\alpha)}$ equals ${\rm Card}\,{\mathcal E}(\mbf{q}^{(\alpha)})=2^{\nu(\lmbf{q})}$ and there are $\sum_{\alpha=1}^{\varkappa} 2^{\nu(\lmbf{q}^{(\alpha)})}$ summands in $\yrum_{\mathfrak{l}}$. Let $\yrum_{\mathfrak{l}}^{(\pm,\alpha)}=\boxplus_{\lmbf{\e} \in {\mathcal E}(\lmbf{q}^{(\alpha)}) \cap E_{\pm}^{d}}{\CH}_{\L_{\lmbf{\e}}}^{(\alpha)}$ and define $\yrum_{\mathfrak{l}}^{(\pm)}=\oplus_{\alpha=1}^{\varkappa} \yrum_{\mathfrak{l}}^{(\pm,\alpha)}$. Clearly the number of summands in $\yrum_{\mathfrak{l}}^{(\pm)}$ equals 
\begin{equation}
\varpi:=\sum_{\alpha=1}^{\varkappa} 2^{\nu(\lmbf{q}^{(\alpha)})-1}.
\label{ediracs5:prop2-eq1}
\end{equation}


\subsection{The Operator $\dop_{d}$}
\label{ediracs6}
Since $\dop_{d}=\aop_{d}+\sigma_{0}$, we have $\dop_{d}^{2}=\aop_{d}^{2}+1$. 
According to Proposition~\ref{ediracs5:prop2} the eigenvalues of $\aop_{d}^{2}$ equal 
$\mathfrak{l}_{\lmbf{q}}=2\sum_{j=1}^{d} \beta_{j}(q_{j}^{(\alpha)}-1)$. Thus, the 
eigenvalues of $\dop_{d}^{2}$ equal $\mathfrak{l}_{\lmbf{q}}+1$ and the associated 
eigenspaces are the spaces $\yrum_{\mathfrak{l}_{\lmbf{q}}}$ also given in 
the proposition. 

Since $\dop_{d}^{2}=\aop_{d}^{2}+1$, the eigenvalues of $\dop_{d}$ are 
$\mu_{\lmbf{q}}^{\pm}=\pm \sqrt{\mathfrak{l}_{\lmbf{q}}+1}$. The eigenvalues $\mu_{\lmbf{q}}^{\pm}$ are
called LLs in analogy with the eigenvalues of $H_{0}$. We are
going now to give an explicit description of the eigenspaces of $\dop_{d}$
(which are some subspaces of $\yrum_{\mathfrak{l}_{\lmbf{q}}}$).

The equation $\dop_{d} \Phi = \mu_{\lmbf{q}}^{\pm} \Phi$, where $\Phi$ has the form
\begin{equation}
\Phi(\mbf{\rho},\mbf{\theta})  = \left( \begin{array}{c} \Phi^{(+)}(\mbf{\rho},\mbf{\theta}) \\ \Phi^{(-)}(\mbf{\rho},\mbf{\theta}) \end{array} \right),
\label{ediracs8:lem1-eq1}
\end{equation}
can be written as
\begin{displaymath}
\left( \begin{array}{cc} 1 & \aop_{d-1}+{\mathcal T}_{d}^{\ast} \\
\aop_{d-1}+{\mathcal T}_{d}  & -1 \end{array} \right) \left( \begin{array}{c}
\Phi^{(+)} \\ \Phi^{(-)} \end{array} \right) = \mu_{\lmbf{q}}^{\pm}
\left( \begin{array}{c} \Phi^{(+)} \\ \Phi^{(-)} \end{array} \right).
\end{displaymath}
Here $\Phi^{(\pm)}$ belong to some subspaces 
$\yrum_{\mathfrak{l}_{\lmbf{q}}}^{(\pm)}$. The latter equation can
be rewritten as 
\begin{eqnarray}
(1-\mu_{\lmbf{q}}^{\pm}) \Phi^{(+)} & = & -(\aop_{d-1}+{\mathcal T}_{d}^{\ast}) \Phi^{(-)}, \label{ediracs7:eq1} \\
(1+\mu_{\lmbf{q}}^{\pm}) \Phi^{(-)} & = & (\aop_{d-1}+{\mathcal T}_{d})\Phi^{(+)}.
\label{ediracs7:eq2}
\end{eqnarray}

Suppose first that $\mu_{\lmbf{q}}^{\pm} \neq \pm 1$. In other
words we do not consider the ``lowest'' LLs of
$\dop_{d}$. In this case (\ref{ediracs7:eq1}) and (\ref{ediracs7:eq2})
are clearly equivalent: If, say, (\ref{ediracs7:eq2}) holds then
(\ref{ediracs7:eq1}) follows from (\ref{ediracs5:prop1eq3}).

Of the spaces $\yrum_{\mathfrak{l}_{\lmbf{q}}}^{(\pm)}$ at least one is non-empty 
(and then automatically the second is). Choose 
$\Phi^{(+)} \in \yrum_{\mathfrak{l}_{\lmbf{q}}}^{(+)}$ arbitrarily and set
\begin{equation}
\Phi^{(-)}=(1+\mu_{\lmbf{q}})^{-1}(\aop_{d-1}+{\mathcal T}_{d})\Phi^{(+)}.
\label{ediracs8:lem1-eq2}
\end{equation}
Then the subspace $\wrum_{\mu_{\lmbf{q}}^{\pm}} \subset \yrum_{\mathfrak{l}_{\lmbf{q}}}$ given by
\begin{displaymath}
\wrum_{\mu_{\lmbf{q}}^{\pm}}=\left\{ \left( \begin{array}{c} \Phi^{(+)} \\
\Phi^{(-)} \end{array} \right) \, | \, \Phi^{(+)} \in
\yrum_{\mathfrak{l}_{\lmbf{q}}}^{(+)}, \Phi^{(-)}=(1+\mu_{\lmbf{q}}^{\pm})^{-1}
(\aop_{d-1}+{\mathcal T}_{d}) \Phi^{(+)} \, \right\}
\end{displaymath}
is the eigenspace corresponding to the eigenvalue $\mu_{\lmbf{q}}^{\pm}$ of 
$\dop_{d}$. Thus, in $\yrum_{\mathfrak{l}_{\lmbf{q}}}$ we find two eigenspaces 
corresponding to the eigenvalues $\mu_{\lmbf{q}}^{\pm}=\pm \sqrt{\mathfrak{l}_{\lmbf{q}}+1}$, respectively. It is easy to see that 
$\yrum_{\mathfrak{l}_{\lmbf{q}}}= \wrum_{\mu_{\lmbf{q}}^{+}} \oplus \wrum_{\mu_{\lmbf{q}}^{-}}$. Furthermore, for $\mu_{\lmbf{q}}^{\pm} \neq \pm 1$ and any 
$\Phi^{(+)} \in \yrum_{\mathfrak{l}_{\lmbf{q}}}^{(+)}$,
\begin{displaymath}
\Phi_{\mu_{\lmbf{q}}^{\pm}}(\mbf{\rho},\mbf{\theta}) = \left(
\frac{\mu_{\lmbf{q}}^{\pm} + 1}{2\mu_{\lmbf{q}}^{\pm}} \right)^{1/2}
\left( \begin{array}{cc} \Phi^{(+)}(\mbf{\rho},\mbf{\theta}) \\
(1 +\mu_{\lmbf{q}}^{\pm})^{-1} [(\aop_{d-1}+T_{d})
\Phi^{(+)}](\mbf{\rho},\mbf{\theta}) \end{array} \right)
\end{displaymath}
is a normalized eigenvector associated to the eigenvalue
$\mu_{\lmbf{q}}^{\pm}$ of $\dop_{d}$. Moreover, the space 
$\yrum^{(+)} \equiv \yrum_{\mathfrak{l}_{\lmbf{q}}}^{(+)}$ splits into a 
direct sum of $\varkappa$ subspaces $\yrum^{(+,\alpha)}$ corresponding to 
different representations in (\ref{secteham:l-levelsdef}). In view of 
Proposition~\ref{ediracs5:prop1}, in its turn, each $\yrum^{(+,\alpha)}$ is 
the (exterior) direct sum of the subspaces ${\CH}_{\L_{\lmbf{\e}}}^{(\alpha)}$ 
in $L^{2}(\Rl^{2d})$, which are nondegenerate Landau eigenspaces of the 
Schr\"{o}dinger operator $H_{0}$, corresponding to the particular 
$\mbf{q}^{(\alpha)}$ and $\mbf{\e}$. Of these subspaces exactly 
$2^{\nu(\lmbf{q}^{(\alpha)})-1}$ are nonempty. According to this description, 
the functions in $\yrum^{(+)}$ have the form
\begin{equation}
\Phi^{(+)}(\mbf{\rho},\mbf{\theta})=\sum_{\alpha=1}^{\varkappa} \Phi^{(+,\alpha)}(\mbf{\rho},\mbf{\theta}), \:\:\:  \Phi^{(+,\alpha)} \in \yrum^{(+,\alpha)},
\label{ediracs8:lem1-eq2a}
\end{equation}
with
\begin{equation}
\Phi^{(+,\alpha)}(\mbf{\rho},\mbf{\theta})=\boxplus_{\lmbf{\e}\in E_{+}^{d}} \Phi_{\lmbf{\e}}^{(+,\alpha)}(\mbf{\rho},\mbf{\theta}), \:\:\: \Phi_{\lmbf{\e}}^{(+,\alpha)} \in {\CH}_{\L_{\lmbf{\e}}}^{(\alpha)},
\label{ediracs8:lem1-eq2b}
\end{equation}
where
\begin{equation}
\Phi_{\lmbf{\e}}^{(+,\alpha)}(\mbf{\rho},\mbf{\theta}) =\sum_{\lmbf{m} \in \Nl_{\lmbf{q}^{(\alpha)}}^{d}}  c_{\lmbf{m},\lmbf{\e}} \prod_{j=1}^{d} f_{m_{j},q_{j}^{(\alpha)}}^{(j)} (\rho_{j},\theta_{j}),
\label{ediracs8:lem1-eq2c}
\end{equation}
$\Nl_{\lmbf{q}^{(\alpha)}}^{d}$ is defined in (\ref{secvkde:eq0}) and $f_{m_{j},q_{j}^{(\alpha)}}^{(j)}=f_{m_{j},q_{j}^{(\alpha)}}(\rho_{j},\theta_{j};b_{j})$ are given in (\ref{sectfham1:eq17}).


\subsection{The ``Lowest'' LL of $\dop_{d}$}
\label{ediracs7}
The symmetry of $\mu_{\lmbf{q}}^{\pm}$ breaks down for the lowest LL 
$\mu_{\lmbf{0}}^{\pm}=\pm \sqrt{\mathfrak{l}_{\lmbf{0}} +1}=\pm 1$ corresponding 
to $\mbf{q}=(1, 1, \ldots, 1)$. Here only one of the numbers 
$\mathfrak{l}_{\lmbf{\e}}$ is an eigenvalue of $\aop_{d}^{2}$, according to 
Proposition~\ref{ediracs5:prop1}, actually, the one corresponding to 
$\mbf{\e}=\mbf{\e}_{\lmbf{0}}=( \e_{j})=(\sgn b_{j})$, so there is only one nonzero 
term in the direct sum $\yrum_{\mathfrak{l}} = \boxplus_{\lmbf{\e}} {\CH}_{\L_{\lmbf{\e}}}=:{\CH}_{\L_{\lmbf{\e}_{\lmbf{0}}}} \boxplus \left( \boxplus_{\lmbf{\e} \neq \lmbf{\e}_{\lmbf{0}}} 0 \right)$. This term may enter in $\wrum_{\mu_{\lmbf{q}}^{+}}$ or $\wrum_{\mu_{\lmbf{q}}^{-}}$, depending on the particular combination of the signs of 
$b_{j}$. Suppose that the former case takes place. Then in (\ref{ediracs7:eq1}), 
$\Phi^{(-)}$ equals zero and therefore $(1-\mu) \Phi^{(+)} =(\aop_{d}+{\mathcal T}_{d})\Phi^{(-)}=0$. Since $\Phi^{(+)} \neq 0$, this means that $\mu=1$; therefore the whole 
$\yrum_{\mathfrak{l}}$ is the eigenspace of $\dop_{d}$ corresponding to the eigenvalue $1$, and $-1$ is not an eigenvalue of $\dop_{d}$. In the opposite case if 
$\mbf{\e}_{\lmbf{0}}$ lies in the lower half, $\Phi^{(+)}=0$ and therefore 
$(1+\mu) \Phi^{(-)} =(\aop_{d}+{\mathcal T}_{d})\Phi^{(+)}=0$ according to 
(\ref{ediracs7:eq2}). Thus, $\mu=-1$ is an eigenvalue, $\mu=1$ is not an eigenvalue, 
and $\yrum_{\mathfrak{l}}$ is the eigenspace of $\dop_{d}$ corresponding to $\mu=-1$.



\section{INTEGRALS CONTAINING LAGUERRE POLYNOMIALS}
\label{auxlaguer}
For further reference we establish (asymptotic) estimates for certain 
integrals containing Laguerre polynomials $L_{q}^{(m)}$ (see (\ref{secfham1:eq119})).

\begin{lemma}
Assume $V(\rho)=a \chi_{\{ \rho < A \}}(\rho)$ for some $a, A > 0$. Let 
\begin{displaymath}
g_{m}(\rho)=\rho^{m} e^{-\frac{b}{4} \rho^{2}} L_{q}^{(m)}\left(\frac{b}{2} \rho^{2}\right), \:\:\: m \in \Nl,
\end{displaymath}
and let $X_{m}$ be the first order differential operator
\begin{displaymath}
X_{m} =-i\frac{\partial}{\partial \rho} -\frac{i m}{\rho}-\frac{i b}{2} \rho 
\end{displaymath}
Set
\begin{displaymath}
R_{m} =  \int_{0}^{\infty} V(\rho) |g_{m}(\rho)|^{2} \, \rho \, d\rho, \:\:\:\: S_{m} = \int_{0}^{\infty} V(\rho) |(X_{m}g_{m})(\rho)|^{2} \, \rho \, d\rho.
\end{displaymath}
Then, for some $c > 0$, 
\begin{equation}
| \ln R_{m} | \leq c \, m, \:\:\:\: \ln S_{m} \leq c \, m.
\label{auxlaguer:lem1-eq1}
\end{equation}
\label{auxlaguer:lem1}
\end{lemma}

\begin{proof} 
The statement for $R_{m}$ involves estimates from above and from below, the one
for $S_{m}$ only from above. The Laguerre polynomial $L_{q}^{(m)}(\xi)$ 
satisfies
\begin{equation}
|L_{q}^{(m)}(\xi)| \! = \! \left| \sum_{l=0}^{q} d_{l,m} (-\xi)^{l} \right| \! \leq \! \sum_{l=0}^{q} (m+q)^{q-l}\frac{\xi^{l}}{l !} \! \leq \! (m+q)^{q} e^{\xi/(m+q)}. 
\label{auxlaguer:lem1-eq4}
\end{equation}
Consequently,
\begin{eqnarray*}
R_{m} & = & a \int_{0}^{A} \rho^{2m+1} e^{-\frac{b}{2} \rho^{2}} \left( L_{q}^{(m)}\left(\frac{b}{2} \rho^{2}\right) \right)^{2} \, d\rho  \nonumber \\
& \leq & a (m+q)^{2q} e^{b A/(m+q)} \frac{A^{2(m+1)}}{2(m+1)},
\end{eqnarray*}
which, after taking the logarithm on both sides, gives the required estimate from 
above for $\ln R_{m}$. The estimate for $S_{m}$ goes in the same way, since the 
differential operator $X_{m}$ produces only extra factors of order $m$ in the 
coefficients of the polynomial, so they become negligible after taking the logarithm.

To get the estimate for $R_{m}$ from below, note, that for $|\xi| < A$, $q$ fixed, the 
Laguerre polynomial obeys
\begin{equation}
L_{q}^{m}(\xi) > d_{0,m} /2
\label{auxlaguer:lem1-eq5}
\end{equation}
provided $m$ is large enough. Indeed, choose $m$ large enough
to ensure that $d_{l,m}/d_{l+1,m} = (m+l+1)(l+1)/(q-l) > 4A$. Then 
the latter clearly implies that
\begin{displaymath}
L_{q}^{m}(\xi) = \sum_{r=0}^{q/2} \left\{ d_{2r,m} \xi^{2r} -d_{2r+1,m} 
\xi^{2r+1} \right\} \geq  d_{0,m} \left( 1-\frac{d_{1,m}}{d_{0,m}} \xi \right) > \frac{d_{0,m}}{2}.
\end{displaymath}
From (\ref{auxlaguer:lem1-eq5}) it follows that for some $\epsilon >0$, and for $m$ 
large enough, 
\begin{displaymath}
R_{m}  \geq  a \left( \frac{d_{0,m}}{2} \right)^{2} \int_{\epsilon}^{A} \rho^{2m +1} 
e^{-\frac{b}{2}\rho^{2}}  \, d\rho,
\end{displaymath}
which, after taking the logarithm, gives the estimate from below.
\end{proof}

\begin{remark}
It follows from Lemma~\ref{auxlaguer:lem1} that
\begin{equation}
\limsup_{m \rightarrow \infty} \frac{\ln S_{m}}{ \ln R_{m}} \leq 1.
\label{auxlaguer:rem1-eq1}
\end{equation}
This estimate plays a key role when we establish the
upper estimate in the proof of Proposition~\ref{ediracs8:lem1}.
\label{auxlaguer:rem1}
\end{remark} 

We introduce the short-hand notation
\begin{displaymath}
\mbf{\rho} \, \mbf{d} \mbf{\rho}= \left(\prod_{j=1}^{d} \rho_{j} d\rho_{j} \right).
\end{displaymath}
The following technical lemma plays a crucial role in Section~\ref{multi}.

\begin{lemma}
Denote by $f_{q}^{m}(\rho)$ the polynomial
\begin{equation}
f_{q}^{m}(\rho)=\rho^{m} L_{q}^{(m)} (\rho^{2}) q!,
\label{multi:eq8}
\end{equation}
where $L_{q}^{(m)}$ are the generalized Laguerre
polynomials in (\ref{secfham1:eq119}). Let $q_{j}^{(\alpha)}$, 
$\alpha=1, \ldots, \varkappa$, $j=1, \ldots, d$ be a collection of 
positive integers such that all vectors 
$\mbf{q}^{(\alpha)}=(q_{1}^{(\alpha)},\ldots, q_{d}^{(\alpha)})$ 
are different, and let $M$ be one more positive integer. Then there exists 
a function $\gamma(\mbf{m})$, $\mbf{m}=(m_{1}, \ldots, m_{d})$, 
of polynomial growth such that, for any set of coefficients 
$c^{(\alpha)}$, $\alpha=1, \ldots, \varkappa$, 
\begin{eqnarray}
\lefteqn{ \hspace*{-0.4cm} \int_{[0,M^{-1}]^{d}} \left| \sum_{\alpha=1}^{\varkappa} c^{(\alpha)} \prod_{j=1}^{d} f_{q_{j}^{(\alpha)}}^{m_{j}} (\rho_{j}) \right|^{2} \,\mbf{\rho} \, \mbf{d} \mbf{\rho}  } \nonumber \\
&& \hspace*{-1cm} \geq  \gamma(\mbf{m})^{-1} \int_{[0,M^{-1}]^{d}}  \sum_{\alpha=1}^{\varkappa}  |c^{(\alpha)}|^{2} \left( \prod_{j=1}^{d} f_{q^{(\alpha)}}^{m_{j}} (\rho_{j}) \right)^{2} \mbf{\rho} \, \mbf{d} \mbf{\rho}.
\label{multi:eq10}
\end{eqnarray}
\label{multi:lem1}
\end{lemma} 

\begin{remark} Estimate (\ref{multi:eq10}) measures how far are the functions
\begin{equation}
F^{(\alpha)}(\rho_{1}, \ldots,\rho_{d})  = \prod_{j=1}^{d} f_{q_{j}^{(\alpha)}}^{m_{j}}(\rho_{j}), \:\: \alpha=1, \ldots, \varkappa.
\label{multi:eq11}
\end{equation}
from being linearly dependent.
\end{remark}

\begin{proof}
We reduce the inequality in (\ref{multi:eq10}) to an estimate for the 
eigenvalues of some matrix. For fixed 
$\alpha, \beta, q_{j}^{(\alpha)}, q_{j}^{(\beta)}$ we study the 
dependence of integrals on $\mbf{m}$. For this, consider the expression
\begin{equation}
\hspace*{-0.3cm} G(\mbf{m})^{(\alpha,\beta)} = \int_{[0,M^{-1}]^{d}}  \! F^{(\alpha)}(\mbf{\rho}) F^{(\beta)}(\mbf{\rho}) \, \mbf{\rho} \, \mbf{d} \mbf{\rho}, 
\label{multi:lem1eq1}
\end{equation}
where $F^{(\alpha)}$ are defined in (\ref{multi:eq11}).

From (\ref{secfham1:eq119}) and (\ref{multi:eq8}) it can be observed
 that for $m_{j} \geq 0$, $f_{q}^{m_{j}}(\rho)$ is $\rho^{m_{j}}$ times 
a polynomial in $\rho$ variable, of degree $2q$. The coefficients of 
this polynomial are integer, moreover they are polynomials in $m_{j}$ 
of degree no higher than $q$, again with integer coefficients:
\begin{displaymath}
\left( \begin{array}{c} q+m_{j} \\ q-l \end{array} \right) 
\frac{1}{l!} q! = \left( \begin{array}{c}  q \\ l \end{array} \right)
(m_{j}+q)(m_{j}+q-1) \cdots (q+m_{j}-l+1).
\end{displaymath}
Multiplication of several functions $f_{q_{j}^{(\alpha)}}^{m_{j}}$ 
preserves this structure. Thus, 
$F^{(\alpha)}(\mbf{\rho})$ $F^{(\beta)}(\mbf{\rho})$, as a function of 
$\rho_{j}$, equals $\rho_{j}^{2m_{j}}$ times a polynomial of degree 
$2(q_{j}^{(\alpha)} +q_{j}^{(\beta)})$ with integer coefficients, 
being polynomials of degree not higher than 
$(q_{j}^{(\alpha)}+q_{j}^{(\beta)})$ in $m_{j}$. When we integrate
$F^{(\alpha)}(\mbf{\rho}) F^{(\beta)}(\mbf{\rho}) \prod_{j=1}^{d} \rho_{j}$
over $[0,M^{-1}]^{d}$, then for each $j$ we get an expression of the form
\begin{displaymath}
\int_{0}^{M^{-1}} f_{q_{j}^{(\alpha)}}^{m_{j}}
f_{q_{j}^{(\beta)}}^{m_{j}} \, \rho_{j} \, d\rho_{j}
=\sum_{l=0}^{2(q_{j}^{(\alpha)}+q_{j}^{(\beta)})} 
\frac{M^{-2m_{j}-2-l}}{-2m_{j} -2-l} Z_{j}^{(\alpha,\beta)}(m_{j}),
%\label{multi:lem1eq2}
\end{displaymath}
where $Z_{j}^{(\alpha,\beta)}$ is a polynomial of degree
not higher than $q_{j}^{(\alpha)}+q_{j}^{(\beta)}$ with
integer coefficients. Therefore 
\begin{displaymath}
\int_{[0,M^{-1}]^{d}} F^{(\alpha)}(\mbf{\rho}) F^{(\beta)}(\mbf{\rho}) \left( \prod_{j=1}^{d} \rho_{j} \, d\rho_{j} \right) = M^{p(\lmbf{m})} \prod_{j=1}^{d} \frac{P_{j}^{(\alpha,\beta)}(m_{j})}{Q_{j}^{(\alpha,\beta)}(m_{j})},
\end{displaymath}
where $p(\mbf{m})=-2 \sum_{j=1}^{d} m_{j}-2\sum_{j=1}^{d}(q_{j}^{(\alpha)}+q_{j}^{(\beta)}) -2d$, $P_{j}^{(\alpha,\beta)}(m_{j})$ is a polynomial in $m_{j}$, with integer coefficients, having degree at most 
$3(q_{j}^{(\alpha)}+q_{j}^{(\beta)})-1$, and $Q_{j}^{(\alpha,\beta)}(m_{j})=\prod_{l=0}^{2(q_{j}^{(\alpha)}+q_{j}^{(\beta)})} (-2m_{j}-2-l)$, whence
$Q_{j}^{(\alpha,\beta)}(m_{j})$ has no positive integer zeros. It is 
convenient to define $P^{(\alpha,\beta)}(\mbf{m})=\prod_{j=1}^{d} P_{j}^{(\alpha,\beta)}(m_{j})$ and $Q^{(\alpha,\beta)}(\mbf{m})=\prod_{j=1}^{d} Q_{j}^{(\alpha,\beta)}(m_{j})$ such that $P^{(\alpha,\beta)}(\mbf{m})/Q^{(\alpha,\beta)}(\mbf{m})$ is a rational function. Note also that $G^{(\alpha,\alpha)}(\mbf{m})$ is always positive.

Now introduce the quantities
\begin{displaymath}
\tau^{(\alpha)} = c^{(\alpha)} (G^{(\alpha,\alpha)}(\mbf{m}))^{1/2},
\end{displaymath}
\begin{displaymath}
\mbf{M}_{\alpha,\beta}= G^{(\alpha,\beta)}(\mbf{m}) 
(G^{(\alpha,\alpha)}(\mbf{m}) G^{(\beta,\beta)}(\mbf{m}))^{-1/2}.
\end{displaymath}
(Note that factors, exponential in $\mbf{m}$, cancel in $\mbf{M}_{\alpha,\beta}$).
Then the expression on the left-hand side of (\ref{multi:lem1eq1}) can 
be written as $\langle \mbf{M} \mbf{\tau}, \mbf{\tau} \rangle$, where 
$\mbf{\tau}$ is the vector composed of $\tau^{(\alpha)}$ and 
$\mbf{M}$ is the matrix composed of $\mbf{M}_{\alpha,\beta}$. The integral 
on the right-hand side of (\ref{multi:eq10}) then becomes 
$\langle \mbf{\tau}, \mbf{\tau} \rangle$. Therefore, to establish 
(\ref{multi:eq10}), it is sufficient to estimate the lowest eigenvalue of 
the matrix $\mbf{M}$. 

First of all, note that due to the Schwarz inequality
applied to $G^{(\alpha,\beta)}$, all elements of the matrix
$\mbf{M}$ are not greater than $1$ by absolute value.
Therefore, all eigenvalues $\lambda_{\alpha}(\mbf{M})$
are not greater than $\varkappa$.

Let us evaluate the determinant of the matrix $\mbf{M}$. Although the entries in 
$\mbf{M}$ contain square roots of rational functions, in the process of calculation
we have to consider products of elements taken one in each row and column. Thus, each 
$(G^{(\alpha)}(\mbf{m}))^{-1/2}$ enters exactly twice in every product. Therefore
${\rm det}\, \mbf{M}$ is a rational function of $\mbf{m}$, viz.
${\rm det} \, \mbf{M}= P(\mbf{m})/Q(\mbf{m})$, where, again $P$ and $Q$ are 
polynomials in $\mbf{m}$, with degree depending only on $q_{j}^{(\alpha)}$, 
and with integer coefficients. The matrix $\mbf{M}$ is nondegenerate; 
otherwise, we would have $\langle \mbf{M} \mbf{\tau},\mbf{\tau} \rangle =0$ for 
some $\mbf{\tau}$, so this would mean that the functions $F^{(\alpha)}$ 
are linearly dependent which they are not. But since $P$ is a polynomial 
with integer coefficients, the smallest positive value for $|P|$ is $1$. 
Therefore, ${\rm det} \, \mbf{M} \geq 1/Q(\mbf{m})$. Finally, since 
${\rm det}\, \mbf{M}=\prod \lambda_{\alpha}(\mbf{M})$, and all 
$\lambda_{\alpha}(\mbf{M})$ are not greater than $\varkappa$, we get the 
estimate from below:
\begin{displaymath}
\lambda_{min}(\mbf{M}) \geq {\rm det}\, \mbf{M} \varkappa^{-(\varkappa-1)} 
\geq \varkappa^{-(\varkappa-1)} Q(\mbf{m})^{-1}.
\end{displaymath}
Since $Q(\mbf{m})$ is a polynomial, this gives us
the required estimate (\ref{multi:eq10}) with
$\gamma(\mbf{m}) = \varkappa^{\varkappa-1} Q(\mbf{m})$.
\end{proof}

\section{TOEPLITZ OPERATORS RELATED TO
THE \mbox{SCHR\"{O}DINGER} OPERATOR, I}
\label{secvk}
The study of eigenvalues of the Schr\"{o}dinger and Dirac operators, 
$H_{0}\pm V$ and $\dop_{d} \pm V$, $V\geq 0$, near a fixed LL 
will be based on results on the eigenvalue asymptotics of the Toeplitz 
operator of the form $T=PVP$, where $P=P_{\Lambda}$ is the projection onto the 
eigenspace of the unperturbed operator corresponding to the chosen LL. In 
Section~\ref{ediracs8} the results for the Dirac operator will be 
deduced from the ones for the Schr\"{o}dinger operator.  

According to (\ref{sectprelim:lem1-eq3})-(\ref{sectprelim:lem1-eq4}), the 
distribution function $n_{+}(\lambda, T)$, $\lambda > 0$, associated with $T$ 
satisfies the relations: 
\begin{eqnarray}
\hspace*{-0.6cm} n_{+}(\lambda, T) \! & \! \!= \!\! & \! \max {\rm dim}\, 
\left\{ {\mathcal L} \subset {\CH}_{\Lambda} \,\! : \!\, \langle Vu,u \rangle  
> \lambda \| u \|^{2}, \:\! u \! \in \! {\mathcal L} \backslash \{ 0 \} \, \right\} \label{secvk:rel1} \\
\hspace*{-0.6cm} \! & \!\! = \!\! & \!  \min {\rm codim}\,  \left\{ {\mathcal M} \subset {\CH}_{\Lambda} \,\! : \,\!  \langle Vu,u \rangle \leq \lambda \| u \|^{2}, \:\! u \! \in \! {\mathcal M} \, \right\}. \label{secvk:rel2}
\end{eqnarray} 
The formulas (\ref{secvk:rel1})-(\ref{secvk:rel2}) will be systematically
used in both directions.

\subsection{Two-dimensional Case}
\label{secvkd2}
In polar coordinates $(\rho,\theta)$ we set first
\begin{equation}
V(\rho) =a \chi_{ \{ \rho < M \} } (\rho)
\label{secvkd2:assum1-eq1}
\end{equation}
for some $a, M > 0$, and consider the Toeplitz-type operator 
$T_{q}=P_{q}VP_{q}$ in the Hilbert space 
${\CH}_{\Lambda_{q}}= {\rm Ran} \, P_{q}$, where $P_{q}$ is the 
projection onto the eigenspace associated with the $q$th LL 
$\Lambda_{q}$ of $H_{0}$. Henceforth we suppress the index $q$ in 
$T_{q}$, $P_{q}$, ${\CH}_{\Lambda_{q}}$ etc. Clearly, $T$ is a 
nonnegative, compact operator on ${\CH}$. The following holds for 
its distribution function $n_{+}(\lambda, T)$, $\lambda > 0$.

\begin{lemma}
Let $V$ satisfy (\ref{secvkd2:assum1-eq1}). Then $n_{+}(\lambda, T) \sim 
\varXi_{1}(\lambda)$ as $\lambda \downarrow 0$. 
\label{secvkd2:lemma1}
\end{lemma}

\begin{proof}
We establish a lower and an upper estimate, separately.
\newline
\noindent
{\em Lower estimate}. In view of (\ref{secvk:rel1}) it suffices to find a
subspace ${\mathcal L}(\lambda)$ of proper dimension for which 
\begin{equation}
\langle V u, u \rangle > \lambda \| u \|^{2}, \:\: \forall u \in {\mathcal L}(\lambda) \backslash \{ 0 \}. 
\label{secvkd21:eq1}
\end{equation}
We take as our candidate for ${\mathcal L}(\lambda)$,
\begin{equation}
{\mathcal L}(\lambda) = {\rm span}\, \{ f_{m}(\rho,\theta) \, |
\, m=-q+1, -q+2, \ldots, N \, \}.
\label{secvkd21:eq2}
\end{equation}
where $f_{m}$ denote the eigenfunctions in (\ref{sectfham1:eq17}) and $N$ 
will be chosen later depending on $\lambda$. Any 
$u \in {\mathcal L}(\lambda)$, $\|u \|=1$, can be written as
$u = \sum_{m=-q+1}^{N} c_{m} f_{m}$ with $\sum |c_{m}|^{2}=1$. Thus, 
(\ref{secvkd21:eq1}) takes the form
\begin{equation}
2 \pi \,  \int_{0}^{\infty}  \sum_{m=-q+1}^{N} |c_{m}|^{2} |f_{m}(\rho,\theta)|^{2} V(\rho)  \, \rho \, d\rho  > \lambda,
\label{secvkd21:eq3}
\end{equation}
since $V$ is radial and the functions $f_{m}$ are orthogonal on circles $\rho=c$. The 
left-hand side of (\ref{secvkd21:eq6}) is larger than
\begin{equation}
2\pi \, \min_{m \in \{ -q+1, -q+2, \ldots, N \} } \int_{0}^{\infty} |f_{m}(\rho,\theta)|^{2} V(\rho)  \, \rho \, d\rho.
\label{secvkd21:eq4}
\end{equation}
Thus, for $\l$ small, we have to quarantee that
\begin{displaymath}
-\ln \left( \int_{0}^{\infty} |f_{m}(\rho,\theta)|^{2} V(\rho)  \, \rho \, d\rho \right) < | \ln \l |
\end{displaymath}
or, using the notation in Lemma~\ref{auxlaguer:lem1},
\begin{equation}
-\ln \left( c R_{m}[(m+q)!]^{-1} \right) < | \ln \l |.
\label{secvkd21:eq5}
\end{equation}
From Lemma~\ref{auxlaguer:lem1} and Stirling's formula we see that the 
left-hand side of (\ref{secvkd21:eq5}) can be estimated from above by
\begin{equation}
-m \ln m -C_{1} m +C_{2} \ln m
\label{secvkd21:eq6}
\end{equation}
for positive constants $C_{1}$ and $C_{2}$. Therefore, for $\l$ small, since the 
second and third terms in (\ref{secvkd21:eq6}) are negligible compared to the 
leading one, (\ref{secvkd21:eq5}) holds as soon as
\begin{displaymath}
(1+\epsilon' ) m \ln m < | \ln \l|.
\end{displaymath}
Thus, for an arbitrary $\epsilon' > 0$, we can take $N=(1-\epsilon') | \ln \l | /(\ln | \ln \l|)=(1-\epsilon') \varXi_{1}(\lambda)$ in (\ref{secvkd21:eq2}) 
and, consequently, as $\lambda \downarrow 0$,
\begin{displaymath}
n_{+}(\lambda, T) \geq {\rm dim} \, {\mathcal L}_{q}(\lambda) \sim 
\varXi_{1}(\lambda)(1+o(1)).
\end{displaymath}
\noindent
{\em Upper estimate}. Here we use (\ref{secvk:rel2}). As our candidate for the 
subspace ${\mathcal M}(\lambda)$, we take  
\begin{equation}
{\mathcal M}(\lambda) = {\rm span}\, \{ f_{m}(\rho,\theta) \, |
\, m=N_{0}+1, N_{0}+2, \ldots \, \}.
\label{secvkd22:eq1}
\end{equation}
where, as usual, $f_{m}$ denote the eigenfunctions (\ref{sectfham1:eq17})
and $N_{0}=N_{0}(\lambda)$ will be chosen below. For 
$u \in {\mathcal M}(\lambda)$, $\| u \| =1$, $u = \sum_{m=N_{0}+1}^{\infty} c_{m} f_{m}$, the inequality (\ref{secvkd21:eq1}) reduces to 
\begin{equation}
2\pi \, \int_{0}^{\infty} \sum_{m=N_{0}+1}^{\infty} |c_{m}|^{2}
|f_{m}(\rho,\theta)|^{2} V(\rho)  \, \rho \, d\rho  \leq \lambda.
\label{secvkd22:eq2}
\end{equation}
Using Lemma~\ref{auxlaguer:lem1}, the left-hand side of 
(\ref{secvkd22:eq2}) is majorized by 
\begin{equation}
2\pi \, \max_{m > N_{0}} \int_{0}^{\infty} |f_{m}(\rho,\theta)|^{2} V(\rho)  
\, \rho \, d\rho = \max_{m>N_{0}} \left( c R_{m} [(m+q)!]^{-1} \right)
\label{secvkd22:eq3}
\end{equation}
Thus, to quarantee (\ref{secvkd22:eq2}), it is sufficient to have
\begin{equation}
\ln (m+q)! -\ln R_{m} -C > | \ln \l |
\label{secvkd22:eq4}
\end{equation}
Due to Lemma~\ref{auxlaguer:lem1}, again, like in the estimate from below, 
the second and third terms in (\ref{secvkd22:eq4}) are negligible compared with 
the leading one. After applying Stirling's formula, we come to
\begin{displaymath}
m \ln m- C_{1}m - C_{2} \ln m > | \ln \l |, 
\end{displaymath}
which can be obtained by taking $N_{0}=(1+\epsilon^{''}) | \ln \l |/(\ln | \ln \l |)$ in (\ref{secvkd22:eq1}). Consequently, as $\lambda \downarrow 0$, we obtain the upper 
estimate
\begin{equation}
n_{+}(\lambda, T) \leq {\rm codim} \, {\mathcal M}_{q}(\lambda) \sim \varXi_{1}(\lambda)\left( 1+o(1) \right), \:\:\: \lambda \downarrow 0.
\label{secvkd22:eq10}
\end{equation}
\end{proof}

Lemma~\ref{secvkd2:lemma1} enables us to establish the following proposition.

\begin{proposition}
Let $V$ satisfy Assumption~\ref{assump0}. Then
\begin{equation}
n_{+}(\lambda, T) \sim \varXi_{1}(\lambda) \mbox{ as } \lambda \downarrow 0. 
\label{secvkd2:prop1-eq1}
\end{equation}
\label{secvkd2:prop1}
\end{proposition}

\begin{proof}
From Lemma~\ref{secvkd2:lemma1} we know that the statement holds for 
the particular $V$ in (\ref{secvkd2:assum1-eq1}). To verify the 
general statement, we first make the usual shift of variables so that $V$ 
becomes positive in a neighborhood of origin. Next, we can clearly find 
two functions $V_{\pm} \geq 0$ having the form in 
(\ref{secvkd2:assum1-eq1}), with compact support, such that 
$V_{-} \leq V \leq V_{+}$. Then the variational principle implies that 
$n_{+}(\lambda, T(V_{-})) \leq n_{+}(\lambda, T(V) ) \leq n_{+}(\lambda, T(V_{+}))$ and since $n_{+}(\lambda, T(V_{\pm}))$ obeys the asymptotics 
(\ref{secvkd2:prop1-eq1}) in view of Lemma~\ref{secvkd2:lemma1}, the same is true for $n_{+}(\lambda, T(V) )$.
\end{proof}


\subsection{Even-dimensional Case for a Simple LL}
\label{secvkde} 
A point in $\Rl^{2d}$, $d\in \Nl$, is represented by 
$(\mbf{\rho},\mbf{\theta})=(\rho_{1}, \ldots, \rho_{d},\theta_{1},\ldots, \theta_{d})$. 
Initially we choose the potential
\begin{equation}
V(\mbf{\rho},\mbf{\theta})=V(\mbf{\rho})=a \prod_{j=1}^{d} V_{j}(\rho_{j})= a \prod_{j=1}^{d} \chi_{ \{ \rho_{j} < M \} }(\rho_{j})
\label{secvkde:potential}
\end{equation}
for some $a, M > 0$, and consider the Toeplitz-type operator 
$T_{\lmbf{q}}=P_{\lmbf{q}}VP_{\lmbf{q}}$ in the Hilbert space 
${\CH}_{\Lambda_{\lmbf{q}}}= {\rm Ran} \, P_{\lmbf{q}}$, where $P_{\lmbf{q}}$ is 
the projection onto the eigenspace associated with the LL 
$\Lambda_{\lmbf{q}}$ of $H_{0}$. 

Throughout this section we assume that the multiplicity of 
$\Lambda_{\lmbf{q}}$ is {\em one}. 

In the sequel we suppress the index $\mbf{q}$ in $T_{\lmbf{q}}$, $P_{\lmbf{q}}$, 
${\CH}_{\Lambda_{\lmbf{q}}}$ etc., and consider the distribution function 
$n_{+}(\lambda,T)$, $\lambda > 0$. For this purpose we introduce the set
\begin{equation}
\Nl_{\lmbf{q}}^{d}=\{ (m_{1}, \ldots, m_{d}) \, | \, m_{j} > -q_{j}+1 \, \}.
\label{secvkde:eq0}
\end{equation} 
When $V$ is chosen as in (\ref{secvkde:potential}) and  $P_{j}$ denotes the operator 
in $L^{2}(\Rl^{2d})$, which acts as $P$ in the variables $(\rho_{j},\theta_{j})$, 
then it is clear that the eigenvalues of $T$ are products of the eigenvalues 
$\nu_{m_{j}}^{(j)}:=\nu_{m_{j}}(T_{j})$ of $T_{j}=P_{j}V_{j}P_{j}$, and the 
eigenfunctions of $T$ are (tensor) products of the eigenfunctions of $T_{j}$. Hence, 
\begin{equation}
n_{+}(\lambda, T) = \# \left\{ \mbf{m} \in \Nl_{\lmbf{q}}^{d} \, | \, \prod_{j=1}^{d} 
\nu_{m_{j}}^{(j)} > \lambda \, \right\}.
\label{secvkde:eq1}
\end{equation}
We are going to estimate (\ref{secvkde:eq1}) from below and above, respectively. A
key ingredient is the following number theoretical result.  

\begin{lemma}
Let $\mbf{m} \in \Nl^{d}$ and $\lambda > 0$. Then
\begin{displaymath}
\# \left\{ \mbf{m} \in {\Nl}^{d} \, |  \,
\prod_{i=1}^{d} {m_{j}}^{-m_{j}} > \lambda \, \right\} \sim 
\varXi_{d}(\lambda) \:\:\: \mbox{ as } \lambda \downarrow 0.
\end{displaymath}
\label{gaussprop1} 
\end{lemma} 
 
The proof of Lemma~\ref{gaussprop1} is given in the Appendix. Having the latter 
lemma at our disposal we are ready to establish the following result.

\begin{proposition}
Let $V$ be chosen as in (\ref{secvkde:potential}). Then
\begin{equation}
n_{+}(\lambda, T) \sim \varXi_{d}(\lambda) \mbox{ as } \lambda \downarrow 0.
\label{secvkde:lem1-eq1}
\end{equation}
\label{secvkde:lem1}
\end{proposition}

\begin{proof} 
Lemma~\ref{secvkd2:lemma1} gives us the result for $d=1$. We will use
induction in $d$. Thus, we assume that the statement holds for all 
$d^{'} < d$ and verify it for $d$. From the induction hypothesis follows
that one may painlessly replace $\Nl_{\lmbf{q}}^{d}$ by $\Nl^{d}$ in
(\ref{secvkde:eq1}) with change of order only $O(\varXi_{d-1}(\l))$.
For $d=1$ Lemma~\ref{secvkd2:lemma1} implies that, given $\epsilon > 0$, there 
exists  $N_{0}=N_{0}(\epsilon)$ such that
\begin{equation}
\left| |\ln \nu_{m}(T^{(1)})| -m \ln m \right| < \epsilon \, m \ln m, \:\: \forall m > N_{0}.
\label{secvkde:eq2b}
\end{equation}
where $\nu_{m}(T^{(1)})$ are eigenvalues of the operator $T=T^{(1)}$ in
this lemma. In the sequel subsets of $\Il=\{1, 2, \ldots, d \}$ are denoted by $\Jl$.
\hspace*{5cm}
\newline
\noindent
{\em Lower estimate}. Using (\ref{secvkde:eq2b}) we have for the right-hand side of 
(\ref{secvkde:eq1}):
\begin{eqnarray*}
\lefteqn{ \hspace*{-1.7cm} \# \! \left\{ \! \mbf{m} \! \in \! \Nl^{d} \,  | \,  \prod_{j=1}^{d}\! \nu_{m_{j}}^{(j)} \! > \! \lambda \,\! \right\} \! \geq \!  \# \! \left\{ \! \mbf{m} \! \in \! \Nl^{d} | m_{j} \! > \! N_{0}, j \! \in \! \Jl, \! \sum_{j=1}^{d}\! m_{j} \! \ln m_{j} \! > \! \frac{\ln \lambda}{1\!-\! \epsilon} \!\, \right\} } \nonumber \\
&& \hspace*{-1cm} \geq \# \left\{ \mbf{m} \in \Nl^{d} \, | \, \sum_{j=1}^{d} m_{j} \ln m_{j} > \frac{\ln \lambda}{1-\epsilon} \, \right\} \nonumber \\
&& \hspace*{-1cm} -\sum_{j^{'}}  \# \left\{ \mbf{m} \in \Nl^{d} \, | \, m_{j^{'}} \leq N_{0}, \sum_{j=1}^{d} m_{j} \ln m_{j} > \frac{\ln \lambda}{1-\epsilon} \, \right\} \nonumber \\
&& \hspace*{-1cm} =: F_{1}(\lambda) - \sum_{j^{'}} F_{2,j^{'}}(\lambda).
\end{eqnarray*}
Consider $F_{2,j^{'}}(\lambda)$ for, say, $j^{'}=1$. We have, due to 
$m_{1} \ln m_{1} \leq N_{0} \ln N_{0}$ and the induction hypothesis that, as
$\lambda \downarrow 0$, 
\begin{eqnarray*}
\lefteqn{ \hspace*{0.8cm} F_{2,1}(\lambda) \! \leq \! N_{0} \# \left\{  \mbf{m} \in \Nl^{d-1} \, | \, \sum_{j=2}^{d} m_{j} \ln m_{j} > \frac{\ln \lambda}{1-\epsilon} -N_{0} \ln N_{0} \, \right\} } \nonumber \\
&& \hspace*{-0.5cm} \! \leq \! N_{0} \# \! \left\{ \mbf{m} \in \Nl^{d-1} | \! \sum_{j=2}^{d} m_{j} \ln m_{j} \! > \! \frac{\ln \lambda}{1\! -\! \epsilon} \, \right\} \lesssim N_{0}\, \varXi_{d-1}(\lambda)=o\left( \varXi_{d}(\lambda) \right).
\end{eqnarray*}
The asymptotics for $F_{1}(\lambda)$ is described by 
Lemma~\ref{gaussprop1}, viz.\ $F_{1}(\lambda) \sim \varXi_{d}(\lambda)$ as
$\lambda \downarrow 0$, and this shows the lower estimate.
\newline
\noindent
{\em Upper estimate}. We have
\begin{displaymath}
n_{+}(\lambda,T) = \# \left\{ \mbf{m} \in \Nl^{d} \, | \, \prod_{j=1}^{d} \nu_{m_{j}}^{(j)} > \lambda \, \right\} = \sum_{\Jl \subset \Il} F_{\Jl}(\lambda),
\end{displaymath}
where
\begin{displaymath}
F_{\Jl}(\lambda)\!=\! \# \left\{ \mbf{m} \! \in \! \Nl^{d} |  \prod_{j=1}^{d} \nu_{m_{j}}^{(j)} \! > \! \lambda,  m_{j} \! \leq \! N_{0} \mbox{ for } j \! \in \! \Jl, m_{j} \! > \! N_{0} \mbox{ for } j \not \in \Jl \, \right\}.
\end{displaymath}
For every $\Jl \neq \emptyset$ (i.e.\ $1 \leq | \Jl |=d^{'} \leq d$) we
estimate $F_{\Jl}(\lambda)$ from above:
\begin{eqnarray*}
\lefteqn{ \hspace*{-2cm} F_{\Jl}(\lambda) \! = \! \# \left\{ \! \mbf{m} \! \in \! \Nl^{d} \, | \, \sum_{j=1}^{d} \ln \nu_{m_{j}}^{(j)} \! > \! \ln \lambda, m_{j} \! \leq \! N_{0}, \! j \in \Jl, \! m_{j} \! > \! N_{0}, \! j \! \not \in \! \Jl \, \right\} } \nonumber \\
&& \hspace*{-1.6cm} \leq N_{0}^{d^{'}} \, \# \left\{ \mbf{m} \in \Nl^{d-d^{'}} \, | \, \sum_{j \not \in \Jl} \ln \nu_{m_{j}}^{(j)} > \ln \lambda \, \right\}.
\end{eqnarray*}
For the last quantity the $d-d^{'}$ dimensional version of our 
induction hypothesis applies:
\begin{displaymath}
\# \left\{ \mbf{m} \in \Nl^{d-d^{'}} \, | \, \sum_{j \not \in \Jl} \ln \nu_{m_{j}}^{(j)} > \ln \lambda, \, \right\} \sim \varXi_{d-d^{'}}(\lambda)=o\left(\varXi_{d}(\lambda) \right) \mbox{ as } \lambda \downarrow 0.
\end{displaymath}
Finally, we consider $\Jl =\emptyset$. Due to (\ref{secvkde:eq2b}) we have that 
\begin{eqnarray}
\lefteqn{ \hspace*{-1cm} F_{\Jl}(\lambda) = \# \left\{ \mbf{m} \in \Nl^{d} \, | \, m_{j} > N_{0},  \prod_{j=1}^{d} \nu_{m_{j}}^{(j)} > \lambda  \, \right\} } \nonumber \\
&& \hspace*{-0.5cm} = \# \left\{ \mbf{m} \in \Nl^{d} \, | \, m_{j} > N_{0}, \sum_{j=1}^{d} \ln \nu_{m_{j}}^{(j)} > \ln \lambda  \, \right\} \nonumber \\
&& \hspace*{-0.5cm} \leq \left\{ \mbf{m} \in \Nl^{d} \, | \, \sum_{j=1}^{d} m_{j} \ln m_{j} > \frac{\ln \lambda}{1+\epsilon} \, \right\} \sim \varXi_{d}(\lambda), \mbox{ as } \lambda \downarrow 0,
\label{secvkde:eq22}
\end{eqnarray}
where the latter asymptotics follows from Lemma~\ref{gaussprop1}. 
\end{proof}

\begin{proposition}
Let $V$ satisfy Assumption~\ref{assump0}. Let $T=P_{\lmbf{q}}VP_{\lmbf{q}}$ be the 
Toeplitz operator associated with the LL $\Lambda_{\lmbf{q}}$ with 
multiplicity one. Then
\begin{equation}
n_{+}(\lambda, T) \sim \varXi_{d}(\lambda) \mbox{ as } \lambda \downarrow 0.
\label{secvkde:prop1:eq1}
\end{equation}
\label{secvkde:prop1}
\end{proposition}

\begin{proof}
The proof is similar to the proof of Proposition~\ref{secvkd2:prop1}.
\end{proof}

To treat the Dirac operator in Section~\ref{ediracs8} we will need a slightly
stronger version of the upper estimate in Proposition~\ref{secvkde:lem1}.

Fix some index $j$, say $j=i$. Suppose that $\tau_{m}$ is a sequence
such that
\begin{equation}
-\ln \tau_{m} \gtrsim -\ln \nu_{m}^{(i)} \mbox{ as } m \rightarrow \infty.
\label{secvkde:eq23}
\end{equation}

\begin{proposition}
Under the assumptions of Proposition~\ref{secvkde:lem1},
\begin{equation}
\# \left\{ \mbf{m} \in \Nl^{d} \, | \, \tau_{m_{i}} \prod_{j \neq i}^{d} \nu_{m_{j}}^{(j)} > \lambda  \, \right\} \lesssim \varXi_{d}(\l). 
\label{secvkde:eq24}
\end{equation}
\label{secvkde:prop2}
\end{proposition}

\begin{proof}
The proof follows the lines of the reasoning leading to the upper estimate in 
the proof of Proposition~\ref{secvkde:lem1}, since the only thing we needed there 
for $\nu_{m_{j}}^{(j)}$ was the estimate 
$-\ln \nu_{m_{j}}^{(j)} \geq (1+\epsilon) m_{j} \ln m_{j}$, but from 
(\ref{secvkde:eq23}), the same holds for $\tau_{m_{i}}$. 
\end{proof}


\section{TOEPLITZ OPERATORS RELATED TO
THE \mbox{SCHR\"{O}DINGER} OPERATOR, II}
\label{multi}
In the case of dimension $2d$, $d >1$, multiple LLs can arise for
the Schr\"{o}dinger operator $H_{0}$. We use notations introduced in 
Section~\ref{secteham}.

\subsection{Even-dimensional Case for Multiple LLs}
Let $V$ satisfy Assumption~\ref{assump0} and let $\Lambda$ be a 
LL of $H_{0}$ with multiplicity $\varkappa$. The main result of this 
section is the following Proposition.

\begin{proposition}
Let $V$ satisfy Assumption~\ref{assump0}. Let $\Lambda$ be a LL 
of $H_{0}$ with multiplicity $\varkappa$ and let $P=P_{\L}$ be the projection
onto the eigenspace associated with $\Lambda$. Then the eigenvalues of the
Toeplitz operator $T=PVP$ obey the asymptotics
\begin{displaymath}
n_{+}(\lambda,T) \sim \varkappa \, \varXi_{d}(\lambda) \mbox{ as } \lambda 
\downarrow 0.
\end{displaymath}
\label{multi:prop1}
\end{proposition}

\begin{proof} 
For the representation (\ref{multi:eq6}), we set
$T^{(\alpha)}=P^{(\alpha)} T P^{(\alpha)}$, where
$P^{(\alpha)}$ is a projection onto ${\CH}_{\Lambda}^{(\alpha)}$.
\newline
\noindent
{\em Estimate from above}. For given $\lambda > 0$, 
Proposition~\ref{secvkde:prop1} asserts that we can find
subspaces ${\mathcal L}^{(\alpha)} \subset {\CH}_{\Lambda}^{(\alpha)}$,
as in (\ref{multi:eq6}), such that
\begin{displaymath}
\int_{\Rl^{2d}} V |u|^{2} \, dx_{1} \cdots dx_{d} \equiv \int_{\Rl^{2d}} V |u|^{2} \, \mbf{d}\mbf{x} \leq \lambda \int_{\Rl^{2d}} |u|^{2} \, \mbf{d}\mbf{x}
\end{displaymath}
on ${\mathcal L}^{(\alpha)}$ and ${\rm codim}\, {\mathcal L}^{(\alpha)}=n_{+}(\lambda, T^{(\alpha)})\sim \varXi_{1}(\lambda)$, $\lambda \downarrow 0$. Then, on ${\mathcal L}= \bigoplus_{\alpha=1}^{\varkappa} {\mathcal L}^{(\alpha)}$ one has
\begin{eqnarray}
\lefteqn{ \hspace*{-1.5cm} \int_{\Rl^{2d}} V |u|^{2} \, \mbf{d}\mbf{x} = \int_{\Rl^{2d}} V \left| \sum_{\alpha=1}^{\varkappa} u^{(\alpha)} \right|^{2} \, \mbf{d}\mbf{x}, \:\: u^{(\alpha)} \in {\mathcal L}^{(\alpha)} \, } \nonumber \\
&& \leq \varkappa \int_{\Rl^{2d}} V \sum_{\alpha=1}^{\varkappa} |u^{(\alpha)}|^{2} \, \mbf{d}\mbf{x} = \varkappa \, \lambda \| u \|^{2}.
\label{multi:eq12}
\end{eqnarray}
The inequality (\ref{multi:eq12}) shows that we have found
the subspace ${\mathcal L}$ in ${\CH}_{\Lambda}$ of
codimension $\sum_{\alpha=1}^{\varkappa} n_{+}(\lambda, T^{(\alpha)})$, so 
that $\langle T_{V}u,u\rangle \leq \varkappa \lambda \| u \|^{2}$ on
${\mathcal L}$. Therefore, $n_{+}(\varkappa \lambda, T) \leq 
\sum_{\alpha} n_{+}(\lambda, T^{(\alpha)})$ and the estimate
from above follows from the asymptotics for $n_{+}(\lambda,T^{(\alpha)})$ 
established in Proposition~\ref{secvkde:prop1}.
\newline
\noindent
{\em Estimate from below}. The proof is based on Lemma~\ref{multi:lem1}. Equipped 
with this lemma we establish the lower estimate first for $V$ having a special form 
and then, using monotonicity as in Section~\ref{secvkd2}, we pass to a general $V$. 
Fix an integer $M$ and set
\begin{equation}
V(\mbf{\rho}) = a \prod_{j=1}^{d} V_{j}(\rho_{j}),
\label{multi:eq12b}
\end{equation}
where $V_{j}$ is the characteristic function of an interval 
$[0,\sqrt{2/b_{j}}M^{-1}]$. According to Proposition~\ref{secvkde:lem1}, 
for such $V$ we can find subspaces 
${\mathcal L}^{(\alpha)} \subset {\CH}^{(\alpha)}$ such that
\begin{equation}
\int_{\Rl^{2d}} V |u|^{2} \, \mbf{d}\mbf{x} > \lambda \int_{\Rl^{2d}} |u|^{2} \,
\mbf{d}\mbf{x}, \:\: u \in {\mathcal L}^{(\alpha)},
\label{multi:eq12c}
\end{equation}
and ${\rm dim}\, {\mathcal L}^{(\alpha)}=n_{+}(\lambda,T^{(\alpha)}) \sim
\varXi_{d}(\lambda)$ as $\lambda \downarrow 0$. Moreover 
${\mathcal L}^{(\alpha)}$ consists of functions which are linear 
combinations of eigenfunctions in ${\CH}^{(\alpha)}$, $u^{(\alpha)} = \sum_{\lmbf{m}} c_{\lmbf{m}}^{(\alpha)} \phi_{\lmbf{m}}^{(\alpha)}$,
%\label{multi:eq13} 
where $\mbf{m}=(m_{1}, \ldots, m_{d})$, and 
\begin{displaymath}
\phi_{\lmbf{m}}^{(\alpha)}(\mbf{\rho},\mbf{\theta}) = \prod_{j=1}^{d} 
h_{m_{j}}^{(\alpha)} (\rho_{j},\theta_{j}),
\end{displaymath}
where $h_{m_{j}}^{(\alpha)}(\rho_{j},\theta_{j})$ are the eigenfunctions
from (\ref{sectfham1:eq17}), the constant factors incorporated into the 
afore-mentioned constants $c_{\lmbf{m}}^{(\alpha)}$, viz.
\begin{equation}
h_{m_{j}}^{(\alpha)}(\rho_{j},\theta_{j})=e^{-im_{j}\theta_{j}}
L_{q_{j}^{(\alpha)}}^{(m_{j})} \left( \frac{b_{j}}{2} \rho_{j}^{2}\right) 
\rho_{j}^{m_{j}} \exp\left( -\frac{b_{j}}{4}\rho_{j}^{2} \right).
\label{multi:eq14}
\end{equation}
We take a linear combination $u=\sum u^{(\alpha)}$ of the functions 
$u^{(\alpha)}$ and estimate $\int_{\Rl^{2d}} V |u|^{2} \, \mbf{d}\mbf{x}$
from below. There are quite a lot of terms in this product, having the 
form
\begin{equation}
\int_{\Rl^{2d}} V u^{(\alpha)} u^{(\beta)} \, \mbf{d}\mbf{x} = 
\sum_{\lmbf{m},\lmbf{m}^{'}} c_{ _{\lmbf{m}}}^{(\alpha)} 
\overline{c_{\lmbf{m}^{'}}^{(\beta)}} \int_{\Rl^{2d}} V 
\phi_{ _{\lmbf{m}}}^{(\alpha)} \overline{\phi_{\lmbf{m}^{'}}^{(\beta)}}
\, \mbf{d}\mbf{x}.
\label{multi:eq15}
\end{equation}
However, most of these terms vanish. In fact, if the
integer multi-indices $\mbf{m}$ and $\mbf{m}^{'}$ differ just at
some position $j$, the integral (\ref{multi:eq15}) equals zero because,
in polar coordinates, the integral in the $\theta_{j}$ variable vanishes. 
Hence, only such terms in (\ref{multi:eq15}) survive for which
$\mbf{m}=\mbf{m}^{'}$. Now, after integration in $\theta$
variables (and incorporating the resulting $2\pi$ into
constants $c_{\lmbf{m}}^{(\alpha)}$) and the change of variables
$\rho_{j} \mapsto \rho_{j} \sqrt{b_{j}/2}$ (again incorporating the 
resulting constants into $c_{\lmbf{m}}^{(\alpha)}$), we obtain that
\begin{displaymath}
\int_{\Rl^{2d}} V | u|^{2} \, \mbf{d}\mbf{x} = \sum_{\lmbf{m}} \int_{[0,M^{-1}]^{d}} V \left| \sum_{\alpha=1}^{\varkappa} c_{\lmbf{m}}^{(\alpha)} \prod_{j=1}^{d} L_{q_{j}^{(\alpha)}}^{(m_{j})}(\rho_{j}) \rho_{j}^{m_{j}} \right|^{2} e^{-\frac{1}{2} \sum \rho_{j}^{2}} \mbf{\rho} \, \mbf{d} \mbf{\rho}.
\end{displaymath}
and, after replacing $V$ by the function $W=c_{1} \prod_{j=1}^{d} \exp(\frac{1}{2} \rho_{j}^{2}) \chi_{\{ \rho_{j} < M^{-1} \}}(\rho_{j})$, with constant $c_{1}$ chosen in such a way that $c_{1}\exp(\frac{1}{2} M^{2})=a$, the latter yields 
\begin{equation}
\int_{\Rl^{2d}}  W  |u|^{2} \, \mbf{d}\mbf{x} = \sum_{\lmbf{m}} \int_{[0,M^{-1}]^{d}} \left| \sum_{\alpha} c_{\lmbf{m}}^{(\alpha)} \prod_{j=1}^{d} L_{q_{j}^{(\alpha)}}^{(m_{j})} (\rho_{j}^{2}) \rho_{j}^{m_{j}}\right|^{2} \mbf{\rho} \, \mbf{d} \mbf{\rho} 
\label{multi:eq16}
\end{equation}
and therefore
\begin{equation} 
\frac{ \int_{\Rl^{2d}} W | u|^{2} \, \mbf{d}\mbf{x}}{\int_{\Rl^{2d}} V |u|^{2} \, \mbf{d}\mbf{x}} \in \left[ \exp(-(1/2) M^{2}),\exp((1/2) M^{-2}) \right]
\label{multi:eq16ratio}
\end{equation}
for any $u$. Using (\ref{multi:eq16}), Lemma~\ref{multi:lem1}, 
and (\ref{multi:eq16ratio}), in this order, we find that
\begin{eqnarray}
\lefteqn{ \hspace*{-1.2cm} \int W |u|^{2} \, \mbf{d}\mbf{x} \geq c_{1}^{-1} e^{-\frac{1}{2}M^{2}}  \sum_{\alpha=1}^{\varkappa}\sum_{\lmbf{m}} \gamma(\mbf{m})^{-1}} \nonumber \\
&& \int V \left| c_{\lmbf{m}}^{(\alpha)} \prod_{j=1}^{d} L_{q_{j}^{(\alpha)}}^{(m_{j})}(\rho_{j}^{2}) \rho_{j}^{m_{j}} e^{-\frac{1}{4} \rho_{j}^{2}} \right|^{2} \mbf{\rho} \, \mbf{d} \mbf{\rho}.
\label{multi:eq170}
\end{eqnarray}
In (\ref{multi:eq170}) $\mbf{m}$ runs over the set of multi-indices
$\mbf{m}=(m_{1}, \ldots, m_{d})$ such that the corresponding functions 
$\phi_{\lmbf{m}}^{(\alpha)}$ belong to our spaces ${\mathcal L}^{(\alpha)}$. 
From the estimate from above in the case of a simple LL, it 
follows that
\begin{displaymath}
m_{j} = O \left( \left( \frac{| \ln \lambda|}{\ln |\ln \lambda|}\right)^{d-1} \right),
\end{displaymath}
and therefore $\gamma(\mbf{m})=O(|\ln \lambda|^{s})$ for some $s> 0$. The
latter, in conjunction with (\ref{multi:eq170}) and 
\begin{displaymath}
\int V \left| \sum_{\lmbf{m}} c_{\lmbf{m}}^{(\alpha)} \phi_{\lmbf{m}}^{(\alpha)} \right|^{2} \, \mbf{d}\mbf{x} = \int V \sum_{\lmbf{m}} |c_{\lmbf{m}}^{(\alpha)}|^{2} | \phi_{\lmbf{m}}^{(\alpha)} |^{2} \, \mbf{d}\mbf{x},
\end{displaymath}
yields that
\begin{displaymath}
\int W |u|^{2} \, \mbf{d}\mbf{x}  \geq  c_{1}^{-1} e^{-\frac{1}{2}M^{2}} |\ln \lambda|^{-s}  \sum_{\alpha=1}^{\varkappa} \int V |u^{(\alpha)}|^{2} \, \mbf{d}\mbf{x}, 
\end{displaymath}
or, using (again) (\ref{multi:eq16ratio}) and afterwards (\ref{multi:eq12c}), we have on ${\mathcal L}=\oplus {\mathcal L}^{(\alpha)}$ 
that, for some $c>0$,
\begin{eqnarray}
\hspace*{-1cm} \int V |u|^{2} \, \mbf{d}\mbf{x} & \geq & c |\ln\lambda|^{-s} 
\sum_{\alpha=1}^{\varkappa} \int V |u^{(\alpha)}|^{2}\, \mbf{d}\mbf{x}  \nonumber \\
\hspace*{-1cm} & \geq & c \lambda |\ln \lambda|^{-s} \sum_{\alpha=1}^{\varkappa}  \int |u^{(\alpha)}|^{2}\, \mbf{d}\mbf{x} = c \lambda |\ln \lambda|^{-s} \| u \|^{2}.
\label{multi:eq17b}
\end{eqnarray}
This means that we have found a subspace of dimension
$\sum_{\alpha=1}^{\varkappa}  n_{+}(\lambda, T^{(\alpha)})$,
where $\langle Tu,u \rangle \geq c \lambda |\ln \lambda|^{-s} \| u \|^{2}$, which implies that $n_{+}(c\lambda |\ln \lambda|^{-s}, T) \geq \sum_{\alpha=1}^{\varkappa} n_{+}(\lambda, T^{(\alpha)})$. Finally, let $\mu=c\lambda |\ln \lambda|^{-s}$ such that $\lambda \sim c^{'} \mu | \ln \mu|^{s}$, and apply the asymptotics for $n_{+}(\lambda, T^{(\alpha)})$ given in Proposition~\ref{secvkde:lem1}. This yields the lower estimate for the $V$ in (\ref{multi:eq12b}). Passage to a general $V$ uses monotonicity.
\end{proof}


\section{TOEPLITZ OPERATORS RELATED TO THE DIRAC OPERATOR}
\label{ediracs8}
For the fixed LL $\mu_{\lmbf{q}}^{\pm}=\pm \sqrt{\mathfrak{l}_{\lmbf{q}}+1}$ 
of the Dirac operator $\dop_{d}$ and the potential $V \equiv VI_{2^{d}}$ we associate
the Toeplitz-type operator $T=T_{\lmbf{q}}^{\pm}=PVP$, where $P=P_{\lmbf{q}}^{\pm}$ is
the spectral projection onto the corresponding eigenspace of $\dop_{d}$. The spectral
asymptotics of $T$ determines the asymptotic behaviour of eigenvalues of 
$\dop_{d} \pm V$. Due to Proposition~\ref{ediracs5:prop2}, there is a one-to-one 
correspondence between $\mathfrak{l}$ and a LL $\L$ of the $2d$-dimensional 
Schr\"{o}dinger operator $H_{0}$ with multiplicity $\varkappa$, so that 
$\L=\L_{\lmbf{q}^{(\alpha)}}$, $\alpha=1, \ldots, \varkappa$. Keeping in mind
this correspondence, we {\em say} that $\mu_{\lmbf{q}}^{\pm}$ has multiplicity $\varkappa$.

In the sequel we adopt the latter terminology and, moreover, we often suppress the 
indices in $T_{\lmbf{q}}^{\pm}$ etc.

\begin{proposition} 
Let $V$ satisfy Assumption~\ref{assump0} and let $\mu \neq \pm 1$ be a LL
of $\dop_{d}$ with multiplicity $\varkappa$. 
%Set 
%\begin{equation}
%\varpi=\varpi(\mu)=\sum_{\alpha=1}^{\varkappa} 2^{\nu(\lmbf{q}^{(\alpha)})-1}, 
%\label{ediracs8:lem1-eqn0}
%\end{equation} 
%where $\nu(\lmbf{q}^{(\alpha)})$ is defined in Proposition~\ref{ediracs5:prop1}. 
Then the eigenvalues of the Toeplitz operator $T=PVP$ obey the asymptotics
\begin{equation}
n_{+}(\lambda, T) \sim   \varpi \, \varXi_{d}(\lambda) \mbox{ as } \lambda 
\downarrow 0,
\label{ediracs8:lem1-eqn1}
\end{equation}
where $\varpi=\varpi(\mu)$ is given in (\ref{ediracs5:prop2-eq1}). The same 
formula with $\varpi=\varpi(\pm 1)=1$ holds for $\mu=1$ or $\mu=-1$, 
depending on which of them is the eigenvalue of $\dop_{d}$.
\label{ediracs8:lem1}
\end{proposition} 


\begin{proof}
We consider $\mu \neq \pm 1$ first. Recall the description of the eigenspaces of
$\dop_{d}$ given in Section~\ref{ediracs6}, in particular (\ref{ediracs8:lem1-eq1})-(\ref{ediracs8:lem1-eq2c}). Due to (\ref{ediracs8:lem1-eq1}), we have for 
$\Phi \in \wrum$,
\begin{equation}
\langle V \Phi, \Phi \rangle = \langle V \Phi^{(+)},\Phi^{(+)} \rangle + 
\langle V \Phi^{(-)},\Phi^{(-)} \rangle.
\label{ediracs8:lem1-eq3}
\end{equation}
We establish a lower and an upper estimate, separately. Again, as before,
we first suppose that $V$ has the form (\ref{secvkde:potential}). 
\newline 
\noindent 
{\em Lower estimate}. In view of (\ref{secvk:rel1}) we are going to find a subspace ${\mathcal L}={\mathcal L}(\l) \subset \wrum$ such that
\begin{equation} 
\langle V \Phi, \Phi \rangle > \lambda \| \Phi \|^{2}, \:\:\:\:\:\: \forall \Phi
 \in {\mathcal L}(\lambda) \backslash \{ 0 \}
\label{ediracs8:lem1-eq4}
\end{equation} 
and 
\begin{equation} 
{\rm dim}\, {\mathcal L}(\lambda) \gtrsim \varpi \, \varXi_{d}(\lambda).
\label{ediracs8:lem1-eq4a}
\end{equation} 
We start by finding a subspace ${\mathcal L}^{(+)}(\lambda)$ in $\yrum^{(+)}$ of
proper dimension, satisfying
\begin{equation}
\langle V \Phi^{(+)}, \Phi^{(+)} \rangle > \lambda \| \Phi^{(+)} \|^{2},  \:\:
\:\:\:\: \forall \Phi^{(+)} \in {\mathcal L}^{(+)}(\lambda) \backslash \{ 0 \}.
\label{ediracs8:lem1-eq5}
\end{equation}
The existence of the latter follows from our considerations for the Schr\"{o}dinger
operator. For each $\mbf{\e} \in \cup_{\alpha=1}^{\varkappa} {\mathcal E}(\mbf{q}^{(\alpha)})$, where ${\mathcal E}(\mbf{q}^{(\alpha)})$ is defined in 
(\ref{ediracs5:prop2-eq0}), Proposition~\ref{multi:prop1} quarantees 
that we can find a subspace ${\mathcal L}_{\lmbf{\e}}^{(+)}(\lambda)$ such that
\begin{equation}
\langle V \Phi_{\lmbf{\e}}^{(+)}, \Phi_{\lmbf{\e}}^{(+)} \rangle > \lambda \| \Phi_{\lmbf{\e}}^{(+)} \|^{2},  \:\:
\:\:\:\: \forall \Phi_{\lmbf{\e}}^{(+)} \in {\mathcal L}_{\lmbf{\e}}^{(+)}(\lambda) 
\backslash \{ 0 \}
\label{ediracs8:lem1-eq5a}
\end{equation}
and
\begin{equation} 
{\rm dim}\,  {\mathcal L}_{\lmbf{\e}}^{(+)}(\lambda) \gtrsim \varkappa_{\lmbf{\e}} \, 
\varXi_{d}(\lambda),
\label{ediracs8:lem1-eq5b}
\end{equation}
where $\varkappa_{\lmbf{\e}}$ is the multiplicity of the LL $\L_{\lmbf{\e}}$, defined in (\ref{ediracs5:prop1eq2-1}), for the Schr\"{o}dinger operator $H_{0}$ 
(in particular, $\varkappa_{\lmbf{\e}}=0$ if $\L_{\lmbf{\e}}$ is {\em not} a 
LL for $H_{0}$).  Taking the exterior direct sum of these 
${\mathcal L}_{\lmbf{\e}}^{(+)}(\lambda)$ over $\mbf{\e} \in (\cup_{\alpha=1}^{\varkappa} {\mathcal E}(\mbf{q}^{(\alpha)})) \cap E_{+}^{d}$, we get the required subspace 
${\mathcal L}^{(+)}(\lambda)$, with dimension 
$\gtrsim \sum_{\lmbf{\e}} \varkappa_{\lmbf{\e}} \, \varXi_{d}(\l)$. It remains to 
notice that $\sum_{\lmbf{\e}} \varkappa_{\lmbf{\e}} = \sum_{\alpha} 2^{\nu(\lmbf{q}^{(\alpha)})-1}=:\varpi$.
 
Having ${\mathcal L}^{(+)}(\lambda)$ at our disposal, we set
\begin{displaymath}
{\mathcal L}(\lambda) \! = \! \left\{ \, (\Phi^{(+)}, (1+\mu)^{-1}(\aop_{d-1}+{\mathcal T}_{d})\Phi^{(+)}) \, | \, \Phi^{(+)} \in {\mathcal L}^{(+)}(\lambda) \, \right\} \! \subset \! \wrum.             
\end{displaymath}
The dimension of ${\mathcal L}(\lambda)$, obviously, equals the dimension of
${\mathcal L}^{(+)}(\lambda)$. At the same time, on ${\mathcal L}(\lambda)$,
\begin{equation}
\langle V \Phi, \Phi \rangle \geq \langle V \Phi^{(+)},\Phi^{(+)} \rangle.
\label{ediracs8:lem1-eq5c}
\end{equation}
On the other hand, (\ref{ediracs1:eq3d})-(\ref{ediracs1:eq3e}) and 
(\ref{ediracs8:lem1-eq2}) imply that
\begin{eqnarray}
\| \Phi \|^{2} & = & \| \Phi^{(+)} \|^{2} + \| (\aop_{d-1}+{\mathcal T}_{d})\Phi^{(+)} \|^{2}  \nonumber \\
& = & \| \Phi^{(+)} \|^{2} +\langle \Phi^{(+)}, (\aop_{d-1}+{\mathcal T}_{d})^{\ast} (\aop_{d-1}+{\mathcal T}_{d}) \Phi^{(+)} \rangle \nonumber \\
& \leq & \| \Phi^{(+)} \|^{2} + \L \| \Phi^{(+)} \|^{2} \leq C \| \Phi^{(+)} \|^{2}.
\label{ediracs8:lem1-eq5d}
\end{eqnarray}
Therefore, (\ref{ediracs8:lem1-eq5c})-(\ref{ediracs8:lem1-eq5d}) yield that
\begin{displaymath}
\langle V \Phi, \Phi \rangle \geq  C \l \| \Phi \|^{2}, \:\:\:\forall \Phi \in {\mathcal L}(\l)\backslash \{ 0 \},
\end{displaymath}
which gives the required lower estimate for the asymptotical eigenvalue distribution 
of $T$. 
\newline
\noindent
{\em Upper estimate.} We begin by constructing some special subspaces in the 
Schr\"{o}dinger Landau eigenspaces. Recall the structure of the operator
$\aop_{d-1}+{\mathcal T}_{d}$, discussed in Section~\ref{ediracs1}. It is a 
$2^{d-1} \times 2^{d-1}$ operator-valued matrix, each entry 
$X_{\lmbf{\e}^{'},\lmbf{\e}}$, $\mbf{\e}^{'} \in E_{+}^{d}$, $\mbf{\e} \in E_{-}^{d}$, 
being either a first order differential operator in {\em one} pair of variables 
$(\rho_{j},\theta_{j})$ or the zero operator.

We fix some $\mbf{\e}^{'},\mbf{\e}$, as well as some $\alpha$, and consider
the subspace ${\CH}_{\L_{\lmbf{\e}}}^{(\alpha)} \subset L^{2}(\Rl^{2d})$.
If this subspace is nonempty, i.e.\ $\mbf{\e} \in {\mathcal E}(\mbf{q}^{(\alpha)})$, 
the eigenfunctions $\Phi_{\lmbf{\e}}^{(+,\alpha)}(\mbf{\rho},\mbf{\theta})$ have the 
form (\ref{ediracs8:lem1-eq2c}). We use the short-hand notation 
$X=X_{\lmbf{\e}^{'},\lmbf{\e}}$, and consider the expression
\begin{equation}
\int V | \Phi_{\lmbf{\e}}^{(+,\alpha)}|^{2} \, \mbf{d} \mbf{x} + \int V |X \Phi_{\lmbf{\e}}^{(+,\alpha)}|^{2} \, \mbf{d} \mbf{x}.
\label{ediracs8:lem1-eq6}
\end{equation}
We write out the the second term:
\begin{displaymath}
\int V |X \Phi_{\lmbf{\e}}^{(+,\alpha)}|^{2} \, \mbf{d} \mbf{x} = \int V |\sum_{\lmbf{m}} X c_{\lmbf{m}} f_{\lmbf{m}}|^{2} \, \mbf{d} \mbf{x}
\end{displaymath}
(we suppress all nonessential sub- and superscripts). Now it follows from
(\ref{ediracs1:eq1-1})-(\ref{ediracs1:eq2}) that the differential operator 
$X$ acting in variables $(\rho_{j_{0}},\theta_{j_{0}})$ on the function $f_{\lmbf{m}}=\prod_{j=1}^{d} f_{m_{j}}^{(j)}(\rho_{j},\theta_{j})= \prod_{j} \exp(i m_{j} \theta_{j}) \tilde{f}_{m_{j}}^{(j)}(\rho_{j})$ replaces all circular factors 
$\exp(i m_{j_{0}} \theta_{j_{0}})$ by $\exp(i (m_{j_{0}}+1) \theta_{j_{0}})$ or
all of them by $\exp(i (m_{j_{0}}-1) \theta_{j_{0}})$. Each pair of functions 
$f_{\lmbf{m}}$, $f_{\lmbf{m}^{'}}$, $\mbf{m} \neq \mbf{m}^{'}$,
are orthogonal with respect to $\mbf{\theta}$-integration. The above property
of $X$ then implies that $X f_{\lmbf{m}}$ and $X f_{\lmbf{m}^{'}}$ are
$\mbf{\theta}$-orthogonal as well. This enables us to transform the integrals
in (\ref{ediracs8:lem1-eq6}) (for $V$ on the form (\ref{secvkde:potential})),
\begin{eqnarray}
\lefteqn{ \hspace*{-0.4cm} \int V | \Phi_{\lmbf{\e}}^{(+,\alpha)}|^{2} \, \mbf{d} \mbf{x} + \int V |X \Phi_{\lmbf{\e}}^{(+,\alpha)}|^{2} \, \mbf{d}  \mbf{x}  } \nonumber \\
&& \hspace*{-0.8cm} \! = \! \sum_{\lmbf{m}} |c_{\lmbf{m}}|^{2} \int V | f_{\lmbf{m}} |^{2} \, \mbf{d} \mbf{x} + \sum_{\lmbf{m}} |c_{\lmbf{m}}|^{2} \int V |X f_{\lmbf{m}}|^{2} \, \mbf{d} \mbf{x} 
\nonumber \\
&& \hspace*{-0.8cm} \! = \! \sum_{\lmbf{m}} |c_{\lmbf{m}}|^{2} \! \int \! V(\rho_{j_{0}}) \left( |f_{m_{j_{0}}}^{(j_{0})}(\rho_{j_{0}},\theta_{j_{0}}) |^{2} \! + \! |X f_{m_{j_{0}}}^{(j_{0})}(\rho_{j_{0}},\theta_{j_{0}}) |^{2}\right) \rho_{j_{0}} \, d\rho_{j_{0}} d\theta_{j_{0}} \nonumber \\
&& \times \prod_{j \neq j_{0}} \int V(\rho_{j}) |f_{m_{j}}^{(j)}(\rho_{j},\theta_{j}) |^{2} \, \rho_{j} \, d\rho_{j} d\theta_{j}.
\label{ediracs8:lem1-eq7}
\end{eqnarray}
We denote the first integral on the right-hand side of (\ref{ediracs8:lem1-eq7})
by $\tau_{m_{j_{0}}}$, and the terms in the product by $\nu_{m_{j}}^{(j)}$. From
Lemma~\ref{auxlaguer:lem1} (see also Remark~\ref{auxlaguer:rem1}), 
$-\ln \tau_{m_{j_{0}}} \gtrsim - \ln \nu_{m_{j_{0}}}^{(j_{0})}$. Now, from 
Proposition~\ref{secvkde:prop2}, it follows that one can find such subset 
$\Ml(\l)$ of integer vectors $\mbf{m} \in \Nl_{\lmbf{q}}^{d}$ so that
\begin{equation}
\tau_{m_{j_{0}}} \prod_{j \neq j_{0}} \nu_{m_{j}}^{(j)} < \l, \:\:\: \mbf{m} \not \in \Ml(\l)
\label{ediracs8:lem1-eq8}
\end{equation}
and ${\rm Card} \, \Ml(\l) \lesssim \varXi_{d}(\l)$. We consider the subspace 
${\mathcal M}_{\lmbf{\e}}^{(+,\alpha)}(\l)$ in ${\CH}_{\L_{\lmbf\e}}^{(\alpha)}$ 
spanned by $f_{\lmbf{m}}$, $\mbf{m} \not \in \Ml(\l)$. This subspace has codimension
not larger than $\varXi_{d}(\l)(1+o(1))$, and, in view of (\ref{ediracs8:lem1-eq8}), 
for any $\Phi_{\lmbf{\e}}^{(+,\alpha)} \in {\mathcal M}_{\lmbf{\e}}^{(+,\alpha)}(\l)$,
 \begin{equation}
\int V | \Phi_{\lmbf{\e}}^{(+,\alpha)}|^{2} \, \mbf{d} \mbf{x} + \int V | X \Phi_{\lmbf{\e}}^{(+,\alpha)}|^{2} \, \mbf{d} \mbf{x} \leq \l \| \Phi_{\lmbf{\e}}^{(+,\alpha)} \|^{2}.
\label{ediracs8:lem1-eq9}
\end{equation}
 
Next, still for $\mbf{\e}^{'}, \mbf{\e}$ fixed, consider all $\alpha$ involved in
the representation of the LL $\L_{\lmbf{\e}}$, and construct, according
to the above reasoning, a subspace ${\mathcal M}_{\lmbf{\e}}^{(+,\alpha)}(\l)$. Then
${\mathcal M}_{\lmbf{\e}}^{(+)}(\l)=\oplus_{\alpha=1}^{\varkappa} {\mathcal M}_{\lmbf{\e}}^{(+,\alpha)}(\l)$ is a subspace in $\yrum_{\lmbf{\e}}^{(+)}$, having there 
codimension not larger than the sum of codimensions of summands 
in $\yrum_{\lmbf{\e}}^{(+,\alpha)}$, i.e.\ $\varkappa_{\lmbf{\e}} \, \varXi_{d}(\l)(1+o(1))$. For any element $\Phi_{\lmbf{\e}}^{(+)}=\sum_{\alpha=1}^{\varkappa} \Phi_{\lmbf{\e}}^{(+,\alpha)}$ on ${\mathcal M}_{\lmbf{\e}}^{(+)}(\l)$, we have that
\begin{eqnarray}
\lefteqn{ \hspace*{-1cm}\int V | \Phi_{\lmbf{\e}}^{(+)} |^{2} \, \mbf{d} \mbf{x} + \int V | X \Phi_{\lmbf{\e}}^{(+)} |^{2} \, \mbf{d} \mbf{x} } \nonumber \\
&& \hspace*{-1.2cm} \! = \! \int V | \sum_{\alpha} \Phi_{\lmbf{\e}}^{(+,\alpha)} |^{2} \, \mbf{d} \mbf{x} + \int V | \sum_{\alpha} X \Phi_{\lmbf{\e}}^{(+,\alpha)} |^{2} \, \mbf{d} \mbf{x}.
\label{ediracs8:lem1-eq10}
\end{eqnarray}
From the latter equality, in combination with the simple inequality $| \sum_{\alpha=1}^{\varkappa} a^{(\alpha)}|^{2}$ $\leq \varkappa \sum | a^{(\alpha)}|^{2}$ and 
(\ref{ediracs8:lem1-eq9}), we get that 
\begin{equation}
\int V | \Phi_{\lmbf{\e}}^{(+)}|^{2} \, \mbf{d} \mbf{x} + \int V | X \Phi_{\lmbf{\e}}^{(+)}|^{2} \, \mbf{d} \mbf{x} \leq \varkappa \, \l \| \Phi_{\lmbf{\e}}^{(+)} \|^{2}.
\label{ediracs8:lem1-eq11}
\end{equation}
Now construct the subspace ${\mathcal M}_{\lmbf{\e}}^{(+)}(\l)$ for each
$\mbf{\e} \in (\cup_{\alpha=1}^{\varkappa} {\mathcal E}(\mbf{q}^{(\alpha)})) \cap E_{+}^{d}$ and set ${\mathcal M}^{(+)}(\l) = \boxplus_{\lmbf{\e} \in (\cup_{\alpha=1}^{\varkappa} {\mathcal E}(\lmbf{q}^{(\alpha)})) \cap E_{+}^{d}} {\mathcal M}_{\lmbf{\e}}^{(+)}(\l)$ with codimension in 
$\yrum^{(+)}=\boxplus \yrum_{\lmbf{\e}}^{(+)}$ not larger than the sum of 
codimensions of ${\mathcal M}_{\lmbf{\e}}^{(+)}(\l)$ in $\yrum_{\lmbf{\e}}^{(+)}$, i.e.
\begin{eqnarray*}
\lefteqn{ \hspace*{-1cm} {\rm codim}\, {\mathcal M}^{(+)}(\l) =\sum_{\lmbf{\e}} {\rm codim}\, {\mathcal M}_{\lmbf{\e}}^{(+)}(\l) } \nonumber \\
&& \lesssim \sum_{\lmbf{\e}} \varkappa_{\lmbf{\e}} \, \varXi_{d}(\l)(1+o(1)) = \varpi \, \varXi_{d}(\l)(1+o(1)).
\end{eqnarray*}
At the same time, due to (\ref{ediracs8:lem1-eq11}) and
the estimate 
\begin{displaymath}
|\sum_{\lmbf{\e}} X_{\lmbf{\e}^{'},\lmbf{\e}} \Phi_{\lmbf{\e}}^{(+)}|^{2} \leq 2^{d} \sum_{\lmbf{\e}} | X_{\lmbf{\e}^{'},\lmbf{\e}} \Phi_{\lmbf{\e}}^{(+)}|^{2},
\end{displaymath}
any $\Phi^{(+)} =\boxplus_{\lmbf{\e} \in (\cup_{\alpha=1}^{\varkappa} {\mathcal E}(\lmbf{q}^{(\alpha)})) \cap E_{+}^{d}} \Phi_{\lmbf{\e}}^{(+)} \in {\mathcal M}^{(+)}(\l)$ satisfies 
\begin{eqnarray}
\lefteqn{ \langle V \Phi^{(+)}, \Phi^{(+)} \rangle +\langle V (\aop_{d-1}+{\mathcal T}_{d}) \Phi^{(+)}, (\aop_{d-1}+{\mathcal T}_{d}) \Phi^{(+)} \rangle } \nonumber \\
&& \leq 2^{d} \sum_{\lmbf{\e}'} \left( \sum_{\lmbf{\e}} \langle V \Phi_{\lmbf{\e}}^{(+)},\Phi_{\lmbf{\e}}^{(+)} \rangle +\sum_{\lmbf{\e}} \langle V X_{\lmbf{\e},\lmbf{\e}^{'}} \Phi_{\lmbf{\e}}^{(+)}, X_{\lmbf{\e},\lmbf{\e}^{'}} \Phi_{\lmbf{\e}}^{(+)} \rangle \right) \nonumber \\
&& \leq 2^{d} \sum_{\lmbf{\e}'} \left( \sum_{\lmbf{\e}} \varkappa  \, \l \| \Phi_{\lmbf{\e}}^{(+)} \|^{2}\right)  = 2^{d} \sum_{\lmbf{\e}'} \varkappa \l \| \Phi^{(+)} \|^{2} \nonumber \\
&& \leq 2^{d}2^{d} \varkappa \, \l \| \Phi^{(+)} \|^{2}=C \l \| \Phi^{(+)} \|^{2}.
\label{ediracs8:lem1-eq12}
\end{eqnarray}
Finally, set 
\begin{displaymath}
{\mathcal M}(\l) = \left\{ \Phi^{(+)}, (1+\mu)^{-1}(\aop_{d-1}+{\mathcal T}_{d}) \Phi^{(+)} \, | \,\Phi^{(+)} \in {\mathcal M}^{(+)}(\l) \right\}.
\end{displaymath}
Due to (\ref{ediracs8:lem1-eq1}), (\ref{ediracs8:lem1-eq2}), the codimension of 
${\mathcal M}^{(+)}(\l)$ in $\yrum^{(+)}$ equals the codimension of 
${\mathcal M}(\l)$ in $\wrum$. On ${\mathcal M}(\l)$, moreover, it follows from
(\ref{ediracs8:lem1-eq12}) that 
\begin{eqnarray*}
\langle V \Phi, \Phi \rangle \! & \! = \! & \! \langle V \Phi^{(+)},\Phi^{(+)} \rangle + (1+\mu)^{-2} \langle V (\aop_{d-1}+{\mathcal T}_{d})\Phi^{(+)},(\aop_{d-1}+{\mathcal T}_{d})\Phi^{(+)} \rangle  \nonumber \\
\! & \! \leq \!  & \! C \l \| \Phi^{(+)} \|^{2} \leq C \l  \| \Phi \|^{2}. 
\end{eqnarray*}
This means that we have constructed the subspace required in (\ref{secvk:rel2}). 
 
For the LL $\mu=1$ or $\mu=-1$, a much more simple reasoning goes through. 
Let, say $\mu=-1$ be the eigenvalue of $\dop_{d}$. Then, according to 
Section~\ref{ediracs7}, $\aop_{d-1}+{\mathcal T}_{d}=0$ on $\yrum_{0}^{(+)}$ and 
the eigenspace of $\dop_{d}$ coincides with the one for the Schr\"{o}dinger operator. 
Consequently, the Toeplitz operators coincide as well, and therefore the results of 
Section~\ref{secvkde} apply (since the lowest LL is, obviously, simple).
\end{proof}

 
\section{THE PERTURBED OPERATORS}
\label{sechtwo}
The proof of asymptotical formulas for eigenvalues goes in the same way for 
Schr\"{o}dinger and Dirac operators. Therefore, we let $\Tl$ stand for
the unperturbed Schr\"{o}dinger operator $H_{0}$ or the unperturbed
Dirac operator $\dop_{d}$, and we let $\mathfrak{l}$ be a LL of $\Tl$
i.e.\ $\mathfrak{l}$ represents either $\L_{\lmbf{q}}$ or $\mu_{\lmbf{q}}^{\pm}$.

We will prove the asymptotics for $\Tl-V$, $V \geq 0$; the proof for
$\Tl+V$ is exactly the same, up to a few changes of signs.

In Propositions~\ref{sechtwo1:prop1} and~\ref{sechtwo2:Propupper} 
we give a lower and an upper estimate, respectively, of the asymptotic behaviour of 
the eigenvalues of $\Tl-V$ accumulating near each LL $\mathfrak{l}$ of 
the operator $\Tl$. Together the propositions establish the main result for the 
perturbed operators, namely Theorems~\ref{sect:intro-thm1}-\ref{edirac:thm1}. 

 
\begin{proposition}
Let $V$ satisfy Assumption~\ref{assump0} and let $\mathfrak{l}$ be a LL of 
$\Tl$ with multiplicity $\varkappa$. Let $\mathfrak{l}^{\pm}$ be the
neighboring LLs of $\Tl$ lying below and above $\mathfrak{l}$, respectively (or any number smaller than $\mathfrak{l}$ if the latter is the lowest LL
of $H_{0}$). Then, for $\lambda > 0$, $\mathfrak{s}_{1} \in (\mathfrak{l}^{-},\mathfrak{l})$ and $\mathfrak{s}_{2} \in (\mathfrak{l},\mathfrak{l}^{+})$, the number of 
eigenvalues for $\Tl \pm V$ in $(\mathfrak{s}_{1},\mathfrak{l}-\lambda)$ or
$(\mathfrak{l}+\lambda,\mathfrak{s}_{2})$, respectively, satisfies the following 
inequality as $\lambda \downarrow 0$:
\begin{equation} 
N(\mathfrak{s}_{1},\mathfrak{l}-\lambda \, | \, \Tl - V) \sim 
N(\mathfrak{l}+\lambda,\mathfrak{s}_{2} \, | \, \Tl + V) \gtrsim 
\varrho \, \varXi_{d}(\lambda),
\label{sechtwo1:prop1-eq1}
\end{equation} 
where the constant $\varrho$ either equals the multiplicity $\varkappa$ of 
$\mathfrak{l}$ if $\Tl$ stands for $H_{0}$, or the constant $\varpi$ in 
(\ref{ediracs5:prop2-eq1}) if $\Tl$ stands for $\dop_{d}$. Moreover, 
(\ref{sechtwo1:prop1-eq1}) holds for any nonnegative, bounded $V$ on $\Rl^{2d}$ 
which tends to zero at infinity. 
\label{sechtwo1:prop1}
\end{proposition}


\begin{proof} We use a simple perturbation argument based on 
Lemma~\ref{sectprelim:lem2}. In~\cite{raikov02} an alternative
approach is used, based on Weyl inequalities and an analysis
of the Birman-Schwinger operator. 

Let $\mathfrak{s}=\mathfrak{s}_{1}$ and let $\mathfrak{g}=\mathfrak{l}-\mathfrak{s}$ be the length of the interval $(\mathfrak{s}, \mathfrak{l}-\lambda)$. 
\newline
\noindent
{\em Step 1: Small $V$.} In this step we show the assertion for a 
function $V$ satisfying $0 \leq V < a$ for some $a> 0$, small enough to 
ensure that $(1/2)- (a/\mathfrak{g}) > 0$. Introduce the parameters
\begin{displaymath}
\mu = (\mathfrak{l}-\lambda+\mathfrak{s})/2, \:\:\:\: 
\tau=(\mathfrak{l}-\lambda-\mathfrak{s})/2.
\end{displaymath}
In view of Lemma~\ref{sectprelim:lem1} it suffices to find some subspace 
${\mathcal L} \equiv {\mathcal L}_{\lmbf{q}}$ for which the condition on the 
right-hand side of (\ref{sectprelim:lem1-eq1}) is fulfilled for all elements in 
${\mathcal L}$.  As the candidate for the subspace ${\mathcal L}$ we take
the span of some finite set of the Landau eigenfunctions of $\Tl$ corresponding
to the LL $\mathfrak{l}$, 
\begin{displaymath}
{\mathcal L} = {\rm span}\, \{ \Phi_{n}(\mbf{\rho},\mbf{\theta}) \, | \, n=1, 2, \ldots, N \, \},
\end{displaymath} 
where $N=N(\lambda)$ will be chosen later. Any $u \in {\mathcal L}$, $\| u \|=1$, 
can be written as $u = \sum_{n=1}^{N} c_{n} \Phi_{n}$. Since each 
$\Phi_{n}$ satisfies $\Tl \Phi_{n}=\mathfrak{l} \Phi_{n}$,
\begin{eqnarray*} 
\| (\Tl-V-\mu) u \|^{2} & = & \langle (\Tl-V-\mu)u,(\Tl-V-\mu)u \rangle  \nonumber \\
 & = & (\mathfrak{l}-\mu)^{2} \| u \|^{2} + \| Vu \|^{2} -2(\mathfrak{l}-\mu) \langle u, Vu \rangle.
\end{eqnarray*} 
Since $\mathfrak{l}-\mu=\tau+\lambda$ and
$(\mathfrak{l}-\mu)^{2} -\tau^{2} =\mathfrak{g} \lambda$, the inequality 
(\ref{sectprelim:lem1-eq1}) becomes
$(2/\mathfrak{g})(\tau+\lambda) \langle u,Vu \rangle-(1/\mathfrak{g}) \| Vu \|^{2} > \lambda$.
The latter is fulfilled if
\begin{equation}
\int \left( \frac{V}{2} -\frac{V^{2}}{\mathfrak{g}} \right) |u|^{2}\, \left( \prod_{j=1}^{d} \rho_{j} \, d\rho_{j} d\theta_{j} \right) > \lambda, \:\: \| u \|=1,
\label{sechtwo1:eq1}
\end{equation} 
is satisfied. Due to $(1/2)-(a/\mathfrak{g}) > 0$, we have 
$(V/2)-(V^{2}/ \mathfrak{g}) \geq 0$ and we arrive at the problem of finding a 
subspace of $u= \sum_{n=1}^{N} c_{n} \Phi_{n}$ such that
\begin{displaymath} 
\left\langle W \left( \sum_{n=1}^{N} c_{n} \Phi_{n} \right), \left(\sum_{n=1}^{N} c_{n} \Phi_{n} \right)  \right\rangle > \lambda, \:\:\: W =\frac{V}{2} -\frac{V^{2}}{\mathfrak{g}} \geq 0,  
\end{displaymath} 
holds for all $N > N_{0}=N_{0}(\l)$. This problem is already solved in 
Section~\ref{secvk}-\ref{multi} or Section~\ref{ediracs8}, respectively, 
depending on whether $\Tl$ plays the role of $H_{0}$ or $\dop_{d}$, with the 
required estimate for the number of elements in ${\mathcal L}$. In this way we obtain
the estimate (\ref{sechtwo1:prop1-eq1}) for small $V$. 
\newline
\noindent
{\em Step 2: General $V$}. We assume that $V$ is a nonnegative, bounded
function on $\Rl^{2d}$, which tends to zero at infinity. In particular, $V$ is 
$\Tl$-compact and, therefore, $\sigma_{ess}(\Tl-V)=\sigma_{ess}(\Tl)$.

Choose $0 \leq V_{0} < \mathfrak{g} /2$ having compact support such that 
$V-V_{0} \geq 0$. The assumptions on $V$ allow us to choose $J$ large enough to ensure
that $V_{j}:=(V-V_{0})/J$ satisfy $\| V_{j} \| < \mathfrak{g}/2$, $j=1, \ldots, J$. 
In addition, $V=\sum_{j=0}^{J} V_{j}$ and $V_{j} \geq 0$. We apply 
Lemma~\ref{sectprelim:lem2} as follows. Put 
\begin{displaymath}
\mathbb{T}_{1} = \Tl-V_{0}-\sum_{j=2}^{J}V_{j}, \:\:\:\: S_{1}=V_{0}+\sum_{j=2}^{J}V_{j}-V=-V_{1}
\end{displaymath} 
such that $S_{1} \leq 0$ and $\gamma_{1}:=\| S_{1} \| < \mathfrak{g}/2$. Moreover, let
\begin{displaymath}
s_{1} =-\gamma_{1}, \:\:\:\: s_{2}=0, \:\:\:\: l_{1} =\mathfrak{s}+\gamma_{1}, \:\:\:\: r_{1}  = \mathfrak{l}-\lambda. 
\end{displaymath}
Then Lemma~\ref{sectprelim:lem2} yields that
\begin{displaymath}
N(\mathfrak{s},\mathfrak{l}-\lambda \,  | \, \Tl-V) \geq N(l_{1}, \mathfrak{l}-\lambda \, | \, \Tl-V_{0}-\sum_{j=2}^{J}V_{j})
\end{displaymath}
Since 
\begin{eqnarray*}
\lefteqn{ \hspace*{-2.5cm} N(l_{1}, \mathfrak{l}-\lambda \, | \, \Tl-V_{0}-\sum_{j=2}^{J}V_{j})= N(\mathfrak{s},\mathfrak{l}-\lambda \,  | \, \Tl-V_{0}-\sum_{j=2}^{J}V_{j}) } \nonumber \\
&& \hspace*{1.5cm} -N(\mathfrak{s},l_{1}\, | \, \Tl-V_{0}-\sum_{j=2}^{J}V_{j})
\end{eqnarray*}
where the last term on the right-hand side is finite (from above
the only possible limits points of the eigenvalues for $\Tl-V$ are 
the LLs, and here the interval $(\mathfrak{s}, l_{1})$ is away from 
the latter), it follows that
\begin{displaymath}
N(\mathfrak{s},\mathfrak{l}-\lambda \,  | \, \Tl-V) \geq N(\mathfrak{s},\mathfrak{l}-\lambda \,  | \, \Tl-V_{0}-\sum_{j=2}^{J}V_{j})-C_{1}
\end{displaymath}
for some constant $C_{1}> 0$. Then we repeat this argument $J-1$ times 
(introducing positive constants $C_{2}, C_{3}, \ldots, C_{J-1}$), using 
Lemma~\ref{sectprelim:lem2} on each step in a way similar to the above, 
and obtain that
\begin{displaymath}
N(\mathfrak{s},\mathfrak{l}-\lambda \,  | \, \Tl-V) \geq N(\mathfrak{s},\mathfrak{l}-\lambda \,  | \, \Tl-V_{0}-V_{J}) -\sum_{j=1}^{J-1} C_{j}
\end{displaymath}
for some constants $C_{j}$. Taking the next step in this procedure
(applying Lemma~\ref{sectprelim:lem2} with $\mathbb{T}_{J}=\Tl-V_{0}$ and 
$S_{J}=-V_{J}$) we obtain that
\begin{equation}
N(\mathfrak{s},\mathfrak{l}-\lambda \,  | \, \Tl-V) \geq N(\mathfrak{s},\mathfrak{l}-\lambda \,  | \, \Tl-V_{0}) -\sum_{j=1}^{J} C_{j}
\label{sechtwo1:prop2-eq1}
\end{equation}
for some constant $C_{J}> 0$. For the operator $\Tl-V_{0}$ the required estimate
from below is obtained in Step 1, which proves the proposition. 
\end{proof}

Next we establish the upper estimate.

\begin{proposition}
Let $V$ satisfy Assumption~\ref{assump0} and let $\mathfrak{l}$, $\mathfrak{l}^{\pm}$ 
be the same as in Proposition~\ref{sechtwo1:prop1}. Then, for $\lambda > 0$, 
$\mathfrak{s}_{1} \in (\mathfrak{l}^{-},\mathfrak{l})$ and $\mathfrak{s}_{2} \in (\mathfrak{l},\mathfrak{l}^{+})$, the number of eigenvalues for $\Tl \pm V$ in 
$(\mathfrak{s}_{1},\mathfrak{l}-\lambda)$ or
$(\mathfrak{l}+\lambda,\mathfrak{s}_{2})$, respectively, satisfies the following 
inequality as $\lambda \downarrow 0$:
\begin{equation}
N(\mathfrak{s}_{1},\mathfrak{l}-\lambda \, | \, \Tl - V) \sim 
N(\mathfrak{l}+\lambda,\mathfrak{s}_{2} \, | \, \Tl + V) \lesssim 
\varrho \, \varXi_{d}(\lambda),
\label{sechtwo2:Propupper:eq1}
\end{equation}
where the constant $\varrho$ either equals the multiplicity $\varkappa$ of 
$\mathfrak{l}$ if $\Tl$ stands for $H_{0}$, or the constant $\varpi$ in 
(\ref{ediracs5:prop2-eq1}) if $\Tl$ stands for $\dop_{d}$.
\label{sechtwo2:Propupper}
\end{proposition}

\begin{proof}
Let $\mathfrak{s}=\mathfrak{s}_{1}$ and set
\begin{displaymath}
\mu=(\mathfrak{s}+\mathfrak{l}-\lambda)/2, \:\:\:\tau=(\mathfrak{l}-\lambda-\mathfrak{s})/2.
\end{displaymath}
We are going to construct such a subspace ${\mathcal M}_{\mathfrak{l}}$ that
\begin{equation}
\|  (\Tl-V-\mu)u \|^{2} \geq \tau^{2} \| u \|^{2}
\label{sechtwo2-eq1}
\end{equation}
holds for every $u \in {\mathcal M}_{\mathfrak{l}}$. According to
Lemma~\ref{sectprelim:lem1}, this would mean 
\begin{displaymath}
N(\mathfrak{s},\mathfrak{l}-\lambda \, | \, \Tl-V) \leq {\rm codim}\, {\mathcal M}_{\mathfrak{l}}.
\end{displaymath}
Let $\CH_{\mathfrak{l}}$ be the eigenspace associated with the
eigenvalue $\mathfrak{l}$ of $\Tl$. Introduce its complement
${\CH}_{\mathfrak{l}}^{'}=\dom{\Tl} \ominus {\CH}_{\mathfrak{l}}$. Any $u \in \dom{\Tl}$ has the form $u=u_{1}+u_{2}$,
$u_{1} \in {\CH}_{\mathfrak{l}}$, $u_{2} \in {\CH}_{\mathfrak{l}}^{'}$. Then
(\ref{sechtwo2-eq1}) gives
\begin{equation}
\| (\Tl-V-\mu)(u_{1}+u_{2}) \|^{2} = -\tau^{2} \| u_{1} \|^{2} -\tau^{2} \| u_{2} \|^{2} \geq 0.
\label{sechtwo2-eq2}
\end{equation}
Since $(\Tl-\mu)u_{1}=(\mathfrak{l}-\mu) u_{1}$ and
$\langle u_{1},u_{2} \rangle =0$, the left-hand side of 
(\ref{sechtwo2-eq2}) can be written as follows:
\begin{eqnarray}
\lefteqn{ \hspace*{-1.1cm} [(\mathfrak{l}-\mu)^{2}-\tau^{2}] 
\| u_{1} \|^{2} \! +\!  \| (\Tl-\mu) u_{2} \|^{2} \! -\! \tau^{2} \| u_{2} \|^{2} -2 {\rm Re}\, \langle (\mathfrak{l}-\mu)u_{1}, V(u_{1}+u_{2}) \rangle } \nonumber \\ 
&& \hspace*{-1.2cm} -2 {\rm Re}\, \langle (\Tl-\mu)u_{2}, V u_{2} \rangle -2 {\rm Re}\, \langle (\Tl-\mu)u_{2}, V u_{1} \rangle + \| V(u_{1}+u_{2}) \|^{2}.
\label{sechtwo2-eq3}
\end{eqnarray}

Take some $\epsilon > 0$ (to be chosen later). From Section~\ref{secvk} or
Section~\ref{ediracs8}, depending on whether $\Tl$ stands for $H_{0}$ or $\dop_{d}$,
we know that there exists a subspace ${\mathcal M}_{1,\mathfrak{l}} \subset {\CH}_{\mathfrak{l}}$
such that
\begin{equation}
| \langle u_{1},Vu_{1} \rangle | \leq \lambda \epsilon \| u_{1} \|^{2}
\label{sechtwo2-eq4}
\end{equation}
holds for every $u_{1} \in {\mathcal M}_{1,\mathfrak{l}}$ and, moreover,
\begin{equation}
{\rm codim} \, {\mathcal M}_{1,\mathfrak{l}} \sim \varrho \, \varXi_{d}(\lambda)
\mbox{ as } \lambda \downarrow 0.
\label{sechtwo2-eq5}
\end{equation}

From hereon we suppose that $u_{1} \in {\mathcal M}_{1,\mathfrak{l}}$. Next we estimate 
each of the terms on the right-hand side of (\ref{sechtwo2-eq3}) separately.
\newline
\noindent
{\em Term 1}. Clearly,
\begin{equation}
[(\mathfrak{l}-\mu)^{2}-\tau^{2}] \| u_{1} \|^{2} =
|\mathfrak{l} -\mathfrak{s}| \lambda \| u_{1} \|^{2}.
\label{sechtwo2-eq6}
\end{equation}
\noindent
{\em Term 2}. Since $u_{2} \in \dom{\Tl} \ominus {\CH}_{\mathfrak{l}}$ we have that
\begin{equation}
\| (\Tl-\mu)u_{2} \|^{2} \geq  [{\rm dist}\,(\mu,\mathfrak{s}(\Tl)
\backslash \{ \mathfrak{l} \})]^{2}  \| u_{2} \|^{2} \geq  [(\mathfrak{s}-\mathfrak{l}^{-})^{2}+\tau^{2} ] \| u_{2} \|^{2}
\label{sechtwo2:eq6extra}
\end{equation}
and, therefore, 
\begin{eqnarray}
\lefteqn{ \hspace*{-2.5cm} \| (\Tl-\mu)u_{2} \|^{2}-\tau^{2} \| u_{2} \|^{2} \geq c_{1} \| (\Tl-\mu)u_{2} \|^{2} } \label{sechtwo2-eq7} \\
&& \mbox{ where } \:\: c_{1}=\frac{(\mathfrak{s}-\mathfrak{l}^{-})^{2}}{2(\mathfrak{s}-\mathfrak{l}^{+})^{2}}. \nonumber
\end{eqnarray}
\noindent
{\em Term 3}. For any $u_{1} \in {\mathcal M}_{1,\mathfrak{l}}$ we
have that
\begin{equation}
| 2 {\rm Re} \, \langle (\mathfrak{l}-\mu)u_{1},Vu_{1} \rangle  | \leq 2 (\mathfrak{l}-\mu) |\langle u_{1}, Vu_{1} \rangle | \leq  2 (\mathfrak{l}-\mu) \lambda \epsilon \| u_{1} \|^{2},
\label{sechtwo2-eq8}
\end{equation}
due to the Schwarz inequality and (\ref{sechtwo2-eq4}).
\newline
\noindent
{\em Term 4}. For any $u_{1} \in {\mathcal M}_{1,\mathfrak{l}}$ and $K>0$ we
have, due to (\ref{sechtwo2-eq4}) and (\ref{sechtwo2:eq6extra}), that
\begin{eqnarray}
\lefteqn{ \hspace*{-0.7cm} | 2 {\rm Re} \, \langle
(\mathfrak{l}-\mu)u_{1}, Vu_{2} \rangle | \leq 2 |\mathfrak{l}-\mu| |
\langle Vu_{1}, u_{2} \rangle | \leq  2 |\mathfrak{l}-\mu| 
\| Vu_{1} \| \, \| u_{1} \| }  \nonumber \\ 
&& \leq  2 |\mathfrak{l}-\mu| (\lambda \epsilon \|
u_{1} \|^{2})^{1/2} \left( (\mathfrak{s}-\mathfrak{l}^{-})^{-2}\|
(\Tl-\mu) u_{2} \|^{2} \right)^{1/2} \nonumber \\
&& = \left( 64 K \frac{|\mathfrak{l}-\mu|^{2}}{(\mathfrak{s}-\mathfrak{l}^{-})^{2}}
\lambda \epsilon \| u_{1} \|^{2} \right)^{1/2} \left(
\frac{1}{16 K} \| (\Tl-\mu) u_{2} \|^{2} \right)^{1/2} \nonumber \\
&& \leq 128 K \frac{|\mathfrak{l}-\mu|^{2}}{(\mathfrak{s}-\mathfrak{l}^{-})^{2}}
\lambda \epsilon \| u_{1} \|^{2} +\frac{1}{8 K}  \|
(\Tl-\mu) u_{2} \|^{2} \nonumber \\
&& = c_{2}(K) \lambda \epsilon  \| u_{1} \|^{2}
+\frac{1}{8 K}  \| (\Tl-\mu) u_{2} \|^{2}.
\label{sechtwo2-eq9}
\end{eqnarray}
\noindent
{\em Term 5}. The Schwarz inequality yields
\begin{eqnarray*}
| 2 {\rm Re} \, \langle (\Tl-\mu)u_{2},Vu_{2} \rangle | & \leq & 2 \| (\Tl-\mu) u_{2} \| \, \| V u_{2} \|  \nonumber  \\ 
& \leq & \frac{1}{8K} \| (\Tl-\mu)u_{2} \|^{2} + 8K \| V u_{2} \|^{2}.
\end{eqnarray*}
In the Hilbert space ${\CH}_{\mathfrak{l}}^{-}$ with norm 
$| u_{2} |^{2} = \| (\Tl-\mu)u_{2} \|^{2}$, we
consider the operator, denoted $T$, generated by the
quadratic form $\| V u_{2} \|^{2}$. This operator is, obviously, compact:
\begin{eqnarray*}
n_{+}(t,T) & = & \max {\rm dim}\, \{ {\mathcal L} \subset {\CH}_{\mathfrak{l}}^{-} \, : \, \| V u_{2} \|^{2} > t |u_{2}|^{2} \, \}   \\
& \leq & \max {\rm dim}\, \left\{ {\mathcal
M} \subset \dom{\Tl} \, : \,  \| V u \|^{2} > t
\| (\Tl-\mu)u \|^{2} \, \right\} < \infty,
\end{eqnarray*}
Taking $t = 1/ 64K^{2}$ then we may find a subspace ${\mathcal M}_{2,\mathfrak{l}}
\subset {\CH}_{\mathfrak{l}}^{-}$ with finite codimension, say
\begin{equation}
N: ={\rm codim}\, {\mathcal M}_{2,\mathfrak{l}} < \infty
\label{sechtwo2-eq9a}
\end{equation}
such that $\| V u_{2} \|^{2} \leq \frac{1}{64 K^{2}} \| (\Tl-\mu) u_{2} \|^{2}$ 
for every $u_{2} \in {\mathcal M}_{2,\mathfrak{l}}$.

Therefore, on ${\mathcal M}_{2,\mathfrak{l}} \subset {\CH}_{\mathfrak{l}}^{-}$ we have
\begin{eqnarray}
\lefteqn{ \hspace*{-0.9cm} | 2 {\rm Re} \, \langle (\Tl-\mu)u_{2},Vu_{2} \rangle | 
\leq \frac{1}{8 K} \| (\Tl-\mu)u_{2} \|^{2} + 8 K \| V u_{2} \|^{2} } \nonumber \\ 
&& \hspace*{-1cm}\leq \frac{1}{8K} \| (\Tl-\mu)u_{2} \|^{2} 
+\frac{8K}{64 K^{2}} \| (\Tl-\mu)u_{2} \|^{2} = \frac{1}{4 K} \| (\Tl-\mu)u_{2} \|^{2}
\label{sechtwo2-eq10}
\end{eqnarray}
for every $u_{2} \in {\mathcal M}_{2,\mathfrak{l}}$ and, moreover,
${\rm codim}\, {\mathcal M}_{2,\mathfrak{l}} = N < \infty$.
\newline
\noindent
{\em Term 6}. For every $u_{1} \in {\mathcal M}_{1,\mathfrak{l}}$ and $K >0$ we
have that
\begin{eqnarray}
\lefteqn{ \hspace*{-0.9cm} | 2 {\rm Re} \, \langle (\Tl-\mu)u_{2},Vu_{1} \rangle |   \leq  2 \| (\Tl-\mu) u_{2} \| \, \| V u_{1} \|  } \nonumber \\
&& \hspace*{-1.6cm} \leq  \frac{1}{8 K} \| (\Tl-\mu) u_{2} \|^{2} \! + \! 8 K \| V u_{1} \|^{2} \leq  \frac{1}{8 K} \| (\Tl-\mu) u_{2}
\|^{2} \! +\! 8 K \lambda \epsilon \| u_{1} \|^{2},
\label{sechtwo2-eq12}
\end{eqnarray}
where, again, we used (\ref{sechtwo2-eq4}). 

We substitute the estimates (\ref{sechtwo2-eq6})-(\ref{sechtwo2-eq8}), (\ref{sechtwo2-eq10}) and (\ref{sechtwo2-eq12}) into (\ref{sechtwo2-eq3}):
\begin{eqnarray*}
\lefteqn{ \hspace*{-2.2cm} \| (\Tl-V-\mu)(u_{1}+u_{2}) \|^{2} -\tau^{2} \| u_{1} \|^{2} -\tau^{2} \| u_{2} \|^{2} \geq (c_{1}-\frac{1}{2K}) \| (\Tl-\mu) u_{2} \|^{2} } \nonumber \\
&& + \lambda \left( (\mathfrak{l}-\mathfrak{s}) -2 \epsilon (\mathfrak{l}-\mu)
-c_{2}(K)\epsilon-8K \epsilon \right) \| u_{1} \|^{2}.
\end{eqnarray*}
Take $K$, $c_{1}-1/2 K > 0$ and, afterwards choose $\epsilon$ such that
\begin{displaymath}
\lambda \left( (\mathfrak{l}-\mathfrak{s}) -2
\epsilon(\mathfrak{l}-\mu)-c_{2}(K)\epsilon-8K \epsilon \right) > 0.
\end{displaymath}
This gives us (\ref{sechtwo2-eq2}) for 
$u \in {\mathcal M}_{1,\mathfrak{l}} \oplus {\mathcal M}_{2,\mathfrak{l}}$. 
In conclusion, if we take ${\mathcal M}_{\mathfrak{l}}= {\mathcal M}_{1,\mathfrak{l}} \oplus {\mathcal M}_{2,\mathfrak{l}}$ then (\ref{sechtwo2-eq2}) holds for every 
$u \in {\mathcal M}_{\mathfrak{l}}$ and, in view of (\ref{sechtwo2-eq5}) and 
(\ref{sechtwo2-eq9a}), we clearly have that
\begin{eqnarray*}
\lefteqn{ \hspace*{-0.4cm} {\rm codim} \, {\mathcal M}_{\mathfrak{l}}
\leq  {\rm codim}\, {\mathcal M}_{1,\mathfrak{l}} + {\rm codim} \,
{\mathcal M}_{2,\mathfrak{l}} } \nonumber \\
&& \hspace*{0.9cm} \sim \varrho \, \varXi_{d}(\lambda) +N \lesssim \varrho \, \varXi_{d}(\lambda) \, \mbox{ as } \, \lambda \downarrow 0.
\end{eqnarray*}
\end{proof}


\section*{APPENDIX: LATTICE POINTS INSIDE A CLOSED HYPERSURFACE}
\label{sect:gauss}
We establish Lemma~\ref{gaussprop1}. Thus, we are interested in the number 
$N(\mu)$ of solutions $\mbf{n} \in \Nl^{d}$ to
\begin{equation} 
\sum_{i=1}^{d} n_{i} \ln n_{i} < \mu= \ln\lambda^{-1}
\label{gaussno0} 
\end{equation} 
as $\mu \rightarrow \infty$ (i.e.\ as $\lambda \downarrow 0$). In other words, we 
seek the number of lattice points (with positive integer coordinates) in the region 
$\Gamma$ between the $x_{i}$-axes, $i=1, \ldots, d-1$ and the hypersurface
\begin{equation} 
C: \quad \sum_{i=1}^{d} x_{i} \ln x_{i} = \mu, \quad
x_{i} \geq 1 \mbox{ for } 1 \leq i \leq d.
\label{gaussno1} 
\end{equation} 

The hypersurface $C$ is ``flat'' in the following sense.

\begin{lemma} 
There exist positive numbers $b(\mu)$ and $B(\mu)$such that
\begin{equation} 
b(\mu) \leq \sum_{i=1}^{d} x_{i} \leq B(\mu)
\quad\hbox{ for all points }  \mbf{x} \in C,
\label{gausslem1:eq1}
\end{equation} 
where 
\begin{equation} 
b(\mu) \sim \frac{\mu}{\ln \mu}, \quad B(\mu) \sim \frac{\mu}{\ln \mu}
\quad \mbox{ as } \mu \rightarrow \infty.
\label{gausslem1:eq2}
\end{equation} 
\label{gausslem1} 
\end{lemma} 

\begin{proof} 
Without loss of generality suppose $x_{1} \geq x_{2} \geq
\cdots \geq x_{d}$.

Note that on~$C$ we have $\mu \geq x_{1}$, so that $\ln x_{i}
\leq \ln x_{1} \leq \ln \mu$. Then we observe that
$\mu = \sum_{i=1}^{d} x_{i} \ln x_{i} \leq \sum_{i=1}^{d}  x_{i} \ln \mu$.
This proves the first inequality in the lemma, with $b(\mu) = \mu/\!\ln \mu$.

To establish the second inequality in the lemma, consider the
case $d=2$ first. We will show that on C, which is now
a  curve given by (\ref{gaussno1}),  the maximum value of
$x_{1}+x_{2}$ is attained at the point  where $x_{1}=x_{2}$.
This follows by implicit differentiation of (\ref{gaussno1}), giving
$(1+\ln x_{1})dx_{1}+ (1+\ln x_{2}) dx_{2} = 0$, so
that $dx_{2}/dx_{1} = -(1+\ln x_{1})/(1+\ln x_{2})$. As we move away from the 
point on $C$ with $x_{1}=x_{2}$ into the region $x_{1} > x_{2}$, the value of 
$|dx_{2}/dx_{1}|$ increases. Hence  $x_{1}+x_{2}$ attains its maximum
when $x_{1}=x_{2}$. The dual version of this fact is that if
$x_{1}+x_{2}=B$ is specified then $x_{1} \ln x_{1} + x_{2} \ln x_{2}$ is 
minimized by taking $x_{1}=x_{2}$. 

To see that an analogous result holds in the
$d$-dimensional  case, consider the dual problem of
minimizing $\mu = \sum_{i=1}^{d} x_{i} \ln x_{i}$
subject to the constraint $\sum_{i=1}^{d} x_{i}= B$. If
any two of the numbers $x_{i}$  are unequal, $x_{1} > x_{2}$,
say, then we can increase $x_{2}$ and decrease $x_{1}$ by
the same amount $\delta$,  leaving the constraint fulfilled
but reducing $x_{1} \ln x_{1} + x_{2} \ln x_{2}$, and
therefore reducing $\sum_{i=1}^{d} x_{i} \ln x_{i}$.
Thus the minimum of this  sum occurs when all the numbers
$x_{i}$ are the same. Dually,  the maximum of
$\sum_{i=1}^{d} x_{i}$ subject to
$\sum_{i=1}^{d} x_{i} \ln x_{i}=\mu$  occurs where
$x_{i} \ln x_{i}=\mu/ d$ for each $i$. Then
$x_{i} \sim \mu/d \ln \mu$ as $\mu \rightarrow \infty$, so
that $\sum_{i=1}^{d} x_{i} \sim \mu/ \ln \mu$. This
completes the  proof of the lemma.
\end{proof} 
 

Equipped with Lemma~\ref{gausslem1} we are able to establish
Lemma~\ref{gaussprop1}.

 

\begin{proof}[Proof of Lemma~\ref{gaussprop1}]
First, we use Lemma~\ref{gausslem1} to estimate the
hypervolume $V$ of the region $\Gamma$. We find that
\begin{equation} 
\frac{b(\mu)^{d}}{d!} \leq V \leq
\frac{B(\mu)^{d}}{d!}, \quad  \mbox{ so that }
\quad V \sim \frac{1}{d!}\left(\frac{\mu}{\ln \mu}\right)^{d}.
\label{gaussno3} 
\end{equation} 
Indeed, the second inequality in (\ref{gausslem1:eq1}), e.g., is
an expression for the hypervolume of a polyhedron for which all
the angles at a vertex $O$ are right angles. The value of this
hypervolume equals $\frac{B(\mu)^{d}}{d!}$. Likewise, we determine 
the hypervolume of the polyhedron associated with the first inequality 
in (\ref{gausslem1:eq1}). In this way we obtain (\ref{gaussno3}).
 
To estimate the number $N(\mu)$ of lattice points
satisfying (\ref{gaussno0}), we attach to each positive
integer point $\mbf{n}$ satisfying (\ref{gaussno0}) the
hypercube given by 
\begin{equation} 
n_{i}-1 < x_{i} < n_{i} \quad \mbox{ when } 1 \leq i \leq d. 
\label{gaussno4} 
\end{equation} 
Next we observe that these hypercubes all lie in the
region $\Gamma$, so that $N(\mu) \leq V$, which establishes the upper bound.

To obtain a lower bound, we proceed as follows.
Lemma~\ref{gausslem1} implies that the $N(\mu)$ hypercubes
fill the  region 
\begin{equation} 
\sum_{i=1}^{d} x_{i} < b(\mu) -d.
\label{gaussno5} 
\end{equation} 
Indeed, if a hypercube (\ref{gaussno4}) is not counted
by $N(\mu)$ then the point $\mbf{n}$ lies outside $\Gamma$,
so that, in view of Lemma~\ref{gausslem1}, 
$\sum_{i=1}^{d} n_{i} \geq b(\mu)$. Consequently all the points 
$\mbf{x} \in \Gamma$ not in one of the $N(\mu)$ hypercubes satisfy
$\sum_{i=1}^{d} x_{i} \geq b(\mu) -d$, so that all $\mbf{x}$ in the 
hypercubes satisfy (\ref{gaussno5}) and $x_i > 0$ when 
$1 \leq i \leq \varkappa$.

The region (\ref{gaussno5}) has volume
${(b(\mu)-d)}^{d}/ d!$ (again we compute the
hypervolume of a polyhedron for which all the angles at a
vertex $O$ are right angles), so  that $N(\mu) \geq
(b(\mu)-d)^{d}/ d!$. Using
Lemma~\ref{gausslem1} one more time completes the
verification of the lower bound which, in conjunction with
the upper bound, proves the assertion.
\end{proof} 
 
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