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\title{\bf Semiclassical eigenvalue estimates for the Pauli operator
with strong non-homogeneous magnetic fields\\ \bigskip
\large \bf II. Leading order asymptotic estimates}
\author{L\'aszl\'o Erd\H os \\ Courant Institute, NYU \\
251 Mercer Str, New York, NY-10012, USA\\
E-mail: {\verb -erdos@cims.nyu.edu-}\\
and \\
Jan Philip Solovej \\ Department of Mathematics\\
Aarhus University\\ Ny Munkegade Bgn. 530\\
DK-8000 Aarhus C, Denmark\\
E-mail: {\verb -solovej@mi.aau.dk-}}
\date{Aug 20, 1996}
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\begin{document}
\maketitle
\begin{abstract}
We give the leading order semiclassical
asymptotics for the sum of the negative
eigenvalues of the Pauli operator (in
dimension two and three) with
a strong non-homogeneous magnetic field.
As in \cite{LSY-II} for homogeneous field,
this result can be used to prove that the
magnetic Thomas-Fermi theory gives the
leading order ground state energy of large atoms.
We develop a new localization scheme
well suited to the anisotropic character of the
strong magnetic field. We also use the basic
Lieb-Thirring estimate obtained in our
companion paper \cite{ES-I}.
\end{abstract}
\vfill\pagebreak
\tableofcontents
\section{Introduction}
This work is the continuation of our previous paper \cite{ES-I}
on studying semiclassical limits of the Pauli operator
with both electric and magnetic fields. Our main concern,
compared to most other works in the subject, is
to allow for non-homogeneous magnetic fields. This
transition from homogeneous to non-homogeneous field is
highly non-trivial, partly because of challenging technical difficulties
and partly because the non-homogeneous field can exhibit qualitatively
different behaviour.
We shall be concerned with dimensions two and three. Though, it may seem
that dimension three is the physically most important case, certain
new experimental techniques (see an extensive review in \cite{LSY-III})
allows one to study effectively two dimensional systems, like quantum dots.
For these two dimensional systems laboratory magnetic fields
actually have a stronger influence on the structure than for most
three dimensional systems.
The other reason why we treat the two dimensional case as well is
pedagogical; some of our basic ideas can be presented without less
technicalities in dimension two.
The three dimensional Pauli
operator is the following operator acting on the space
$L^2(\bR^3;\bC^2)$ of spinor valued functions.
\be
H(h,\bA , V):= [ \bsigma\cdot(- ih\nabla + \bA (x))]^2 +
V(x) =(-ih\nabla +\bA(x))^2 + V(x) +h\bsigma\cdot\bB(x)
\label{Pauli}, \ee
where $\bsigma=(\sigma_1,\sigma_2,\sigma_3)$ is the vector of
Pauli spin matrices, i.e.,
$$
\sigma_1=\left(\matrix{0&1\cr1&0}\right),\
\sigma_2=\left(\matrix{0&-i\cr i&0}\right),\
\sigma_3=\left(\matrix{1&0\cr0&-1}\right).
$$
The magnetic field $\bB:\bR^3\to\bR^3$ is a divergence free
field related to the vector potential $\bA:\bR^3\to\bR^3$ by
$\bB=\nabla\times\bA$. The potential $V(x)$ describes the
electric field. As usual $h$ is the semiclassical parameter.
Throughout the paper we shall use the convention of
writing $-i\nabla=\bp$. Let $B(x):= |\bB(x)|$ and $\bn(x):= \bB(x)/B(x)$
be the field strength and direction, respectively.
The two dimensional Pauli operator has essentially
the same form as the three dimensional operator
above. The modifications are rather obvious.
The magnetic field is a function $B:\bR^2\to\bR$,
the vector potential is a vector field $\bA:\bR^2\to\bR^2$
and we shall write as before $B=\nabla\times\bA$ with the
obvious interpretation. We apply the convention that
$\bsigma\cdot v :=\bsigma
\cdot (v, 0)$ for any $v\in \bR^2$, $\bsigma \cdot \bp = \bsigma \cdot
(\bp ,0)$ for the two dimensional momentum operator $\bp$, and
similarly for the vectorproduct of a 3D and a 2D vector (in
particular, we define $\bn : =(0,0,1)$ and let $\bn \times v \in
\bR^2$ be $\bn \times v : = (-v_2, v_1)$ for $v\in \bR^2$).
We may then write
\be
H^{(2)}(h,\bA , V):= [ \bsigma\cdot(h\bp + \bA (x))]^2 +
V(x) =(h\bp +\bA(x))^2 + V(x) +h\sigma_3B(x).
\label{2dPauli} \ee
Since we shall consider $\bB\in L^{\infty}(\bR^3)$
and $B\in L^{\infty}(\bR^2)$, we consider the last
terms in (\ref{Pauli}) and (\ref{2dPauli})
as a multiplication operator on the corresponding $L^2$ space.
The Pauli operator describes the motion of a
non-relativistic electron, where the electron spin is
important because of its interaction with the magnetic field.
For simplicity we have not included any physical parameters
(i.e., the electron mass, the electron charge, the speed of
light, or Planck's constant $\hbar$)
in the expressions for the operators. In place of
Planck's constant we have the semiclassical parameter
$h$, which we let tend to zero.
The last identities in (\ref{Pauli}) and (\ref{2dPauli})
can easily be checked. If we note that $\bsigma\cdot(h\bp + \bA (x))$
is in fact the three dimensional Dirac operator, we recognize
the last identity in (\ref{Pauli}) as the Lichnerowicz formula.
As a consequence of these identities one sees a
significant difference between the Pauli operator and the `magnetic'
Schr\"odinger operator $ (h\bp + \bA (x))^2+V(x)$.
In particular, for reasonable potentials and reasonable magnetic fields
the essential spectrum of the Pauli operator starts at zero.
(See \cite{ES-I} for more details.)
The physically as well as mathematically interesting
quantities connected with the eigenvalues are the number
and the sum of the eigenvalues below the essential spectrum
(in this case, the negative eigenvalues).
Recall that the sum of the negative eigenvalues
represents the energy of the non-interacting fermi gas in the
external potential $V$ and magnetic field $\bB$.
In the case of a constant magnetic field it is known
\cite{Sol, Sob-1986}
that even for a smooth compactly supported potential $V$,
which is negative, there will be {\it infinitely}
many negative eigenvalues.
This holds in both two and
three dimensions. It was, however, proved in \cite{LSY-II} (three
dimensions) and \cite{LSY-III} (two dimensions) that the
{\it sum} of the negative eigenvalues is finite.
The goal in \cite{LSY-II} was to analyze the eigenvalue sum in the
semiclassical limit, i.e., as the semiclassical parameter $h$ tends to
zero. In the case where one fixes the magnetic field $\bB$ and let
$h\to0$ one finds that the leading order contribution to the sum of the
eigenvalues becomes independent of
the magnetic
field\footnote{We believe that this result in its greatest generality
was also first proved in \cite{LSY-II}, or rather follows from the
result in \cite{LSY-II}. In fact, to prove the semiclassics of the
sum of all the negative eigenvalues for
fixed $\bB$ one needs to know that the sum is finite, which was first
established by the Lieb-Thirring (LT) estimate in \cite{LSY-II}.}. It is
therefore equal to the non-magnetic Weyl term, which in three
dimensions is
$-2(15\pi^2)^{-1}h^{-3}\int_{\bR^3}[V]_-^{5/2}$ and in two dimensions
is $-h^{-2}(8\pi)^{-1}\int_{\bR^2}[V]_-^2$ ($[V]_-$ denotes
the negative part of the function $V$).
This type of semiclassical
limit is therefore not very well suited to the study of the effect of
magnetic fields. One could maybe hope that higher order terms in the
expansion would reveal information about the magnetic field. In this
context we should point out, however, that without some assumptions on
the classical Hamiltonian flow one cannot establish non-vanishing
higher order corrections to the above Weyl term.
The observation made in \cite{LSY-II} for homogeneous magnetic fields
is that one can establish a semiclassical expression for the sum
of the negative eigenvalues which is
asymptotically exact uniformly in the magnetic field strength.
In contrast to the above standard semiclassical Weyl term,
the generalized semiclassical expression, indeed,
depends on the magnetic field.
In case of three dimensions this formula is given by
\begin{equation}
E_{scl}(h, \bB, V):= - h^{-3} \int_{\bR^3} P(h |\bB(x)|,
[V(x)]_-)\rd x
\label{ESC}\ee
with
\be
P(B, W):= \frac{B}{3\pi^2}\left( W^{3/2} + 2\sum_{\nu =1}^{\infty}
[2\nu B - W]_-^{3/2}\right) =
\frac{2}{3\pi}\sum_{\nu =0}^{\infty} d_{\nu}B[2\nu B - W]_-^{3/2}
\label{press}\ee
being the pressure of the three dimensional
Landau gas ($B, W \geq 0$). Here $d_0 = (2\pi)^{-1}$ and $d_{\nu}
=\pi^{-1}$ if $\nu \geq 1$. Observe that if $\|\bB\|=o(h^{-1})$
then $E_{scl}$ reduces to leading order to the standard Weyl term
as $h\to0$. If $B(x)h\to\infty$ for all $x$, then
only the lowest Landau band gives the main
contribution, i.e. $E_{scl}$ reduces to leading order
to a similar expression where only the first term ($\nu =0$)
is kept in (\ref{press}).
Here and throughout the paper $\|\cdot\|$ refers
to the supremum norm.
In \cite{LSY-III} the two dimensional problem was studied, but
since this paper was aimed mainly at applications the semiclassical
formula did not appear explicitly. It is
\be E^{(2)}_{scl}(h, B, V):= - h^{-2} \int_{\bR^2} P^{(2)}(hB(x),
[V(x)]_-) \rd x
\label{2dESC}\ee with
\be P^{(2)}(B,W):=\frac{B}{2\pi} \left(W + 2
\sum_{\nu =1}^{\infty} [2\nu B -W]_{-} \right)
=\sum_{\nu=0}^\infty d_\nu B [2\nu B -W]_{-}
\label{2dpress}
\ee
being the pressure of the two dimensional Landau gas ($d_\nu$ is as above).
Again if $\|\bB\|=o(h^{-1})$ then $E^{(2)}_{scl}$ reduces to
the standard Weyl term.
Our goal in this paper is to show that these
semiclassical formulas are exact also for non-homogeneous fields. Of
course, for non-homogeneous fields it is more subtle exactly what one
means by uniformity in the field, since the field is now no longer
determined by just one parameter. We shall return to this question
later.
The physical motivations for studying these issues are explained in
\cite{ES-I}. Here we just recall, as one of the most important applications,
that the problem of the ground state energy
of large atoms in strong magnetic fields
can be reduced to the semiclassical limit of the eigenvalue sum,
using Thomas-Fermi theories. We shall investigate this question
in our context of non-homogeneous magnetic fields.
In typical applications, it is usually a good approximation
to consider the magnetic field as homogeneous.
There are several reasons, however, why one would still like to extend
the analysis to non-homogeneous fields.
First of all it is of course natural to ask whether the
features found for homogeneous fields are really stable.
Furthermore, a detailed mathematical study
often requires one to be able to locally vary the field,
even if one is mainly interested in the constant field case.
Even though we will find that
the semiclassical results known for constant fields
really carry over to non-constant fields, we will also see that,
in fact, not all features of the constant case are stable.
Of course the problem is also of independent
interest and raises, as we shall see, many extremely interesting
mathematical issues. The homogeneous field case is
comparatively simple because the kinetic energy part is an exactly solvable
quantum mechanical model.
The semiclassical analysis of eigenvalues is really twofold.
One must first of all establish non-asympotic, Lieb-Thirring type estimates
on the sum of the negative eigencvalues, allowing
one to control errors and contributions coming from non-semiclassical
regions. In our context this is the subject of \cite{ES-I}.
The second part of the semiclassical study is to show that
when all the errors have been controlled one can indeed
get the asymptotic formula. This is the main subject of the present paper.
There are a multitude of highly developed methods for this
part of the analysis, e.g.,
pseudo-differential operator and Fourier integral operator
methods (see e.g., \cite{Sob-1994, Sob-1995} which generalizes results
in \cite{LSY-II} using these methods, or the book
\cite{R}, or the preprints \cite{I}).
Depending on certain properties of the classical Hamiltonian
flow, these results even give higher order corrections to
the leading term. However, these methods require, in general,
strong regularity assumptions on the data. Also, without some
non-asymptotic Lieb-Thirring type apriori estimates, these
results usually either refer to only a part of the spectrum which is
strictly away form the essential spectrum or compute only
quantities which are localized in space, what is called
local traces or local eigenvalue moments. Within this
context non-Weyl type formulas similar to (\ref{ESC}) and
(\ref{2dESC}) are studied in the preprints
\cite{I}.
Our approach to the problem
is the more elementary coherent state method also
used in \cite{LSY-II,LSY-III}. In addition to the
conceptual simplicity, it reveals some important geometric features
related to the magnetic field.
As always, semiclassical methods
require controlling localization errors. For a homogeneous field
one simply has to localize in regions where the potential
does not vary too much. In the
case of non-homogeneous fields one is, however, forced to also
localize on the often much shorter scale where the
vector potential is nearly constant.
This scale turns out to be so small that the standard IMS localization
procedure would be too expensive.
We have elaborated a new localization scheme, particularly suitable
for magnetic problems. Similarly to the proof of the Lieb-Thirring
estimate in \cite{ES-I}, we shall use a two step localization.
First we have a larger scale isotropic localization. On this
length scale, the field direction is almost constant and also
the field strength does not vary too much.
As usual
in IMS type arguments, the actual localization function
is not essential, only its length scale is determined.
In dimension two the field direction is of course stable, therefore
this step is only necessary
to control
the variation of the
field strength (this strategy allows us to treat fields whose strength
has no uniform positive lower bound).
The second localization is reminiscent of the cylindrical
localization in the LT proof since the localization function is
supported in a typically elongated cylindrical domain, corresponding
to the effective anisotropic character of the magnetic field.
The key point is that, in contrast to
the standard localization approach, the function itself is very specially
chosen. It essentially must be a zero mode of the associated
two dimensional Pauli operator with a locally constant field.
In a strong magnetic field, the zero mode can be chosen as a well
localized Gaussian function. It allows one to localize very strongly
(much beyond the IMS localization), essentially for free
(the price we pay appears as a slight modification in the magnetic field).
To understand this phenomenon, recall that in the typical IMS scheme
one pays the price of the lowest eigenvalue of the Dirichlet
Laplacian for localization. This is of course an expression of
the uncertainity principle.
It should be, however, noted
that while the uncertainity principle between position and momentum
is a universal fact about
quantum systems, large momentum does not necessarily imply
large energy.
This is exactly the case for the two dimensional Pauli operator,
where the magnetic field and spin coupling can, and in the right sector
does, compensate the ``spinless" kinetic energy. This is the reason
behind the existence of the Aharonov-Casher zero modes (see
\cite{AC}, \cite{CFKS}), which carry large (angular) momentum,
but have no energy.
\medskip
We should mention that probably the first
asymptotic formula for eigenvalues of an operator with non-homogeneous
magnetic field appeared in \cite{CdV, T},
and later was extended in \cite{H},
\cite{Mat-1994}. All these works consider the
large eigenvalue asymptotics for the `magnetic' Schr\"odinger
operator with a magnetic field increasing at infinity (magnetic bottles).
This operator has pure point spectrum (and compact resolvent), therefore
the problem is considerably simpler than ours. The analogy
to our result is that
in both problems the asymptotic formula for the non-homogeneous magnetic
field is obtained by simply inserting the strength of the
non-homogeneous field into the formula for homogeneous field.
\bigskip
The organization of the paper is the following. In the next two
subsections we explain the main results in the simplest setup.
In Section \ref{sec:loc} we present our new localization method,
which is one of the key point of our analysis. The other key point
is the analysis of the geometry of the three dimensional magnetic field
(in particular a careful choice of a suitable gauge), which was
already used in \cite{ES-I}. For the reader's convenience we
recall the necessary results in the Appendix.
In Section \ref{sec:2dsc}
we work out the two dimensional case. We start with a short
presentation of the necessary Lieb-Thirring inequality. Then
we apply the localization scheme to estimate the quadratic form
of the kinetic energy with a non-homogeneous field, by a similar
expression with a locally homogeneous field. Finally we
work out separately the lower bound, using Lieb-Thirring inequality,
and the upper bound, using coherent states and variational
principle. In both cases we heavily rely on
the structure of the Pauli operator
with a constant magnetic field (e.g. Landau levels).
Section \ref{sec:3dsc} contains the three dimensional semiclassics.
Approximating the true kinetic energy by a (locally) constant field kinetic
energy requires choosing an
`economical' gauge for the approximating field, we use the
results from the Appendix. Once this approximation is done,
the lower and upper bounds for the eigenvalue sum are obtained
essentially in the same way as in the two dimensional case.
For the lower bound, one has to use the more complicated
three dimensional Lieb-Thirring inequality.
The formulas are somewhat
lengthy, as one has complicated error terms, but their basic structure
resembles the simpler two dimensional setup. Therefore the
reader is advised to start with the two dimensional proof.
The last Section is devoted to the proof of the validity of the
Magnetic Thomas Fermi (MTF) theory introduced and studied,
even for non-homogeneous fields, in \cite{LSY-II}.
The validity of this theory, as an approximation to the ground state
energy of large atoms in strong magnetic fields, was proved for
homogeneous fields in \cite{LSY-II}.
Here we have to cope with two new difficulties,
compared to the constant field case in \cite{LSY-II}.
First is that our Lieb-Thirring inequality does not
provide a kinetic energy inequality via Legendre transform
(because of an extra gradient term, see (\ref{LTest}) later).
The other complication is that one needs more
information on the magnetic Thomas Fermi potential, again
because of the extra terms in our LT inequality, i.e.,
one has to prove that these terms are negligible for the potential
of MTF theory.
\subsection{Main results in semiclassics}
We shall throughout the paper
assume the following conditions on
the magnetic field, $\bB: \bR^3\to\bR^3$, in dimension three
\be
\|B\| <\infty,
\label{asp1'}\ee
\be
l(\bB)^{-1}:=\| \nabla\bn\|=\|\nabla \frac{\bB}{B}\|<\infty,
\label{asp2'}\ee
\be
L(\bB)^{-1}:=\|\frac{|\nabla B|}{B}\|<\infty.
\label{asp3'}\ee
Here $l=l(\bB)$ describes the length scale on which the field line
geometry changes, while $L=L(\bB)$ is the length scale on which
the field strength varies. Note that the conditions imply
that $B(x):=|\bB(x)|$ never vanishes.
We are especially
interested in the case when $l \gg L$.
In particular, for
$l=\infty$, $L< \infty$, we obtain the constant direction
(but non-homogeneous) case.
In dimension two we assume conditions analoguous to (\ref{asp1'}),
(\ref{asp3'})
\be
\| B \| < \infty
\label{2dasp1'}\ee
\be
\| \frac{|\nabla B|}{B} \| < \infty
\label{2dasp3'}\ee
for the two dimensional magnetic field $B :\bR^2 \to \bR$.
Notice, that we do not assume more than a bound\footnote{
We strictly speaking only need the magnetic field to be Lipschitz}
on the first derivative of the magnetic field in contrast to the
more traditional SC approach which uses pseudodifferential
operators and therefore assumes strong regularity conditions on
the data.
The potential $V$ is naturally assumed to be in the corresponding $L^p$ spaces
which make the semiclassical formulas (\ref{ESC}), (\ref{2dESC})
finite, that is we assume $V\in L^{5/2}(\bR^3)\cap L^{3/2}(\bR^3)$
in the three dimensional case, and $V \in L^1(\bR^2)\cap L^2(\bR^2)$
in dimension two. As it is explained in \cite{ES-I}, in
dimension two these natural assumptions are sufficient since we have
a Lieb-Thirring inequality which scales exactly as the semiclassical
expression. In fact, in the simple case of a bounded field, which
is our concern here, the combination of the methods in \cite{E-1995}
and \cite{LSY-III} gives a very simple proof which we present in
Section \ref{sec:2dlt} (for references on unbounded fields, see
\cite{ES-I}).
However, in dimension three, our Lieb-Thirring inequality
(the main theorem in \cite{ES-I}) bounds the
sum of the negative eigenvalues of $H$ in (\ref{Pauli})
by
\be
C_1\Biggl(h^{-3}\int_{{\bf R}^3}[V]_-^{5/2}+
\int_{{\bf R}^3}(h^{-1}B+d^{-2})h^{-1}\, [V]_-^{3/2}
\label{LTest}\ee
\[
+\int_{{\bf R}^3}(h^{-1}B+d^{-2})d^{-1}\, [V]_- +
\int_{{\bf R}^3}(h^{-1}B+d^{-2})\, |\nabla [V]_-| \Biggr),
\]
where
\[
d= d(h, \bB) := \min \{ h^{1/4}\|B\|^{-1/4}l(\bB)^{1/2}, L(\bB), l(\bB) \}.
\]
This estimate
contains two extra terms which, to be finite,
require $[V]_-\in W^{1,1}(\bR^3)$.
These terms are of lower order as far as the large field semiclassics
is concerned, but at least the term involving $\int [V]_-$
cannot be eliminated
(see \cite{ES-I}). The other term, with $\int |\nabla [V]_-|$, is
a conceptual error coming from our method, also has to be considered.
\bigskip
Our theorem in dimension two is the following.
\begin{theorem}{\bf (2D Semiclassics)}\label{thm:intro2d}
Assume that the potential $V$ satisfies $V\in L^1({\bf R}^2)\cap
L^2({\bf R}^2)$ and the magnetic field $B$ satisfies
(\ref{2dasp1'})-(\ref{2dasp3'}).
For $h, b >0$ let $e_1(h,b),e_2(h,b), \ldots$ denote the negative
eigenvalues of the operator $H= H^{(2)}(h, b\bA, V)$
in (\ref{2dPauli}). Then
$$ \lim_{h\to0}\left|\frac{\sum_k
e_k(h,b)}{E^{(2)}_{scl}(h,bB,V)}-1\right|=0
$$
uniformly in $b$.
\end{theorem}
Later we shall in fact prove a slighty stronger
result.
Instead of considering magnetic fields depending only on the
parameter $b$, we shall prove a statement that is uniform
on more general families of magnetic fields.
We shall make it more precise
in Section \ref{sec:2dsc}.
\bigskip
The formulation of the
three dimensional result is more complicated than the
two dimensional result,
since
we do not prove a fully uniform statement.
\begin{theorem}{\bf (3D Semiclassics)}\label{thm:intro3d}
Assume that the potential $V$ satisfies
$V\in L^{5/2}(\bR^3)\cap L^{3/2}(\bR^3)$,
$[V]_-\in W^{1,1}(\bR^3)$ and the magnetic field $\bB$
satisfies (\ref{asp1'})--(\ref{asp3'}).
For $h,b >0$
let $e_1(h,b) ,e_2(h,b), \ldots$ denote the
negative eigenvalues of the operator
$H=H(h, b\bA, V)$
in (\ref{Pauli}). Then
\be
\lim_{h\to0\atop bh^3\to0}\left|
\frac{\sum_k e_k(h,b)}{E_{scl}(h, b\bB ,V)}-1\right|=0.
\label{eq:intro3dsc}\ee
\end{theorem}
We shall again prove a slightly
stronger result.
The reason for this generalization is
that in our main application the magnetic Thomas-Fermi theory,
the potential will depend slightly on the magnetic field and on the effective
Planck constant. The details are explained
in Section \ref{sec:MTF}.
There is a fundamental
technical reason for the condition
$bh^3\to 0$.
We already
explained it in \cite{ES-I}, since a similar condition plays
a role in the proof of our
Lieb-Thirring inequality.
Part of the motivation for
believing that the semiclassical formula for homogeneous
or even constant direction fields should generalize to
fully non-homogeneous fields is that
these fields on the relevant quantum scales should behave
approximately like constant direction fields.
This is, however, not true if the field is too strong.
A charged particle moving in a magnetic field essentially occupies
a region in space of the shape of a cylinder with axis parallel to
the magnetic field.
For particles of fixed energy $e$ the
radius of the cylinder is the Landau radius $r\sim b^{-1/2}h^{1/2}$
and the height is of order $s=he^{-1/2}$ (particles localized in
regions of length $he^{-1/2}$ in one dimension have energies of
order $e$).
The condition that one can approximate the
magnetic field within this region by a constant direction field is that
the field lines remain within this cylinder, i.e.,
that $l(\bB)^{-1}s^2\ll r$. This condition is simply that
$bl(\bB)^{-2}\ll h^{-3}e^2$.
Although the above restriction on the
magnetic field might seem natural, we believe that
it can be removed by an additional geometrical analysis
which is beyond the scope of the present work.
We intend to return to this issue in the future.
\subsection{Application to the magnetic Thomas-Fermi
theory}\label{sec:intromtf}
As an application of our semiclassical analysis we shall
here generalize Theorem~5.1 in \cite{LSY-II} on the energy of large atoms
in strong {\it exterior} magnetic fields. We shall work in dimension three,
but we are convinced that a very similar analysis can be carried
over in dimension two, similarly to \cite{LSY-III}.
We consider again a magnetic
field $\bB=\nabla\times\bA$ which satisfies
(\ref{asp1'}--\ref{asp3'}). Our generalization will be to allow a much
more general class of three dimensional exterior fields than in
\cite{LSY-II}, where only homogeneous fields were treated.
The quantum mechanical Hamiltonian for an atom with nuclear
charge $Z$ and with $N$ electrons in such an exterior
magnetic field is given by
\be
H_{N,\bA,Z} := \sum^N_{i=1} \left\{[
\bsigma_i \cdot (\bp_i + \bA(x_i))]^2 -Z|x_i|^{-1}\right\} + \sum
\limits^N_{i0$ represents the localization scale.
That this function does not have compact support turns out to be just
a minor difficulty.
\begin{lemma}{\bf (Magnetic localization)} \label{lm:leibniz}
Let $B_0$ be any constant
and $k \in C^{\infty}(\bR^2, \bC^2)$.
Then with the explicit choice of $\eta^{(0)}$ we have for all
$0<\delta<1$
\begin{eqnarray}
\lefteqn{|\eta^{(0)}(x)\bsigma\cdot(h\bp +
(1/2)B_0\bn\times x)k(x)|^2
\label{leibnizlow}}\\
&\geq&(1-\delta)\left|\bsigma\cdot\left(
h\bp +(1/2)(B_0
+4hw^{-2})\bn\times x\right)
(\eta^{(0)} k)(x)\right|^2\nonumber\\
&&{}-c\delta^{-1} h^2 w^{-2} |x w^{-1}|^2
|P_+(\eta^{(0)} k)(x)|^2\nonumber
\end{eqnarray}
and
\begin{eqnarray}
\lefteqn{|\bsigma\cdot(h\bp +
(1/2)B_0\bn\times x)(\eta^{(0)} k)(x)|^2
\label{leibnizup}}\\
&\leq&(1+\delta)\left|\eta^{(0)}(x)\bsigma\cdot\left(
h\bp +(1/2)(B_0
-4hw^{-2})\bn\times x\right)
k(x)\right|^2\nonumber\\
&&{}+c\delta^{-1} h^2 w^{-2} |x w^{-1}|^2
|P_+(\eta^{(0)} k)(x)|^2.\nonumber
\end{eqnarray}
\end{lemma}
\bigskip
{\it Proof.}
Using Leibniz formula a simple computation gives
\begin{eqnarray*}
\lefteqn{\bsigma\cdot \left(h\bp +(1/2)B_0\bn\times x
\right) (\eta^{(0)} k)(x)}\\
&=&\eta^{(0)}(x)\bsigma\cdot\left(h\bp +
(1/2)(B_0-4hw^{-2})\bn\times x\right)k(x)
+4i hw^{-2}\eta^{(0)}(x)(\bsigma \cdot x)P_+k(x).
\end{eqnarray*}
A simple application of a Cauchy-Schwarz inequality then gives
(\ref{leibnizup}). Replacing $B_0$ by $B_0+4hw^{-2}$
we similarly get (\ref{leibnizlow}).$\,\,\Box$
We shall use both the magnetic localization and the IMS Lemma.
The magnetic localization shall be used to approximate
the variable magnetic field by a constant field. We
shall now explain why the magnetic localization is superior to
the IMS formula for this purpose.
Imagine that we attempt to approximate the variable field
by a constant field over a region of length $w$.
The approximation error for the vector potential
is then $\|B\|L^{-1}w^2$ using that $|\nabla B|\leq \|B\| L^{-1}$ and
(\ref{Poinest}). [This error will
appear squared in the estimate on the Hamiltonian
(see e.g. (\ref{2dlinestlow}) and (\ref{2dlinestup}))
below, but this is unimportant for the present discussion.] Since we
want to prove estimates uniform in
$\|B\|$ we must choose $w$ proportional to $\|B\|^{-1/2}$.
The IMS formula would then give an error $w^{-2}\sim \|B\|$,
which is not independent of $\|B\|$.
The magnetic localization seems at first sight to give the same
error. In fact, this is the order of the last terms in
(\ref{leibnizlow}) and (\ref{leibnizup}). [Since
$\int \eta^{(0)}(x)^2 x\rd x= cw$ we should think of
$|xw^{-1}|$ being of order one.] The important observation is
that the error terms in the magnetic localization contain
the spin up projection $P_+$. If the magnetic field $B$
is bounded below then the free Pauli operator restricted
to the spin up subspace is not just positive
but, indeed, bounded below by a positive amount proportional to
the lower bound on $B$ [see (\ref{eq:2dgap})].
If the ratio of the supremum of $B$ is bounded relative the infimum
of $B$ then
the {\it relative} localization error (compared to the main term)
in the magnetic localization is independent of $\|B\|$ and this
is the important fact.
We shall not actually assume that the magnetic field
is bounded below. This is just a minor technical problem. In fact,
as should be clear from the above discussion, it is only the
ratio of the maximum of the field to the minimum that
counts. We therefore simply use the standard IMS Lemma
to localize in regions where this ratio is bounded.
\section{Semiclassics in two dimensions}\label{sec:2dsc}
\setcounter{equation}{0}
In the introduction we stated our semiclassical result
in Theorem \ref{thm:intro2d} for fixed potential.
First we formulate our more general result for
potentials which are allowed to depend mildly on $B$ and $h$.
To describe the precise result, we
introduce the {\bf 2D magnetic Lieb-Thirring error functional}
\be
\cE^{(2)}_{h, B} (V):= h^{-2}\int |V|^2 + \|B\| h^{-1}\int
|V|
\label{2dmterr}
\ee
(in Section \ref{sec:2dsc} all integrals are on $\bR^2$, unless otherwise
specified).
With this notation, the Lieb-Thirring inequality
Theorem \ref{thm:2dlt} in Section \ref{sec:2dlt}
states that the sum of the negative eigenvalues
$e_1(H),e_2(H),\ldots$ of $H= H^{(2)}(h, \bA, V)$ satisfies the bound
$$
\sum_k |e_k(H)|\leq c\cE^{(2)}_{h,B}([V]_-)\leq c\cE^{(2)}_{h, B}(V)
$$
if the magnetic field satisfies (\ref{2dasp1'}).
Define the following set for $L>0$
\[
\cC_L:= \Bigl\{ 0 < B(x) \in L^{\infty}(\bR^2) \, : \,
\sup_x \frac{|\nabla B(x)|}{B(x)} \leq L^{-1} \, \Bigr\}.
\]
We may now introduce the conditions on the potential.
\be
C_+(V):=\sup_{B\in {\cal C}_L, \, 00.
\label{2dcon1.1}\ee
Note
that (\ref{2dcon1.2}), (\ref{2dcon1.1}), and
$P^{(2)}(B,W)\leq c(BW +W^2)$ imply that
\be c\leq\frac{\cE^{(2)}_{h, B}(V)}{|E^{(2)}_{scl}(h, B,V)|}
\leq \frac{C_+(V)}{C_-(B, V)}
\label{2dcon5}\ee
for all $00$ introduce the set
$$
{\cal C}_{C, L}(V):=\left\{0**0$ we have
$$ \lim_{h\to0}\left(\sup_{B\in {\cal C}_{C, L}(V)}\left|\frac{\sum_k
e_k(H)}{E^{(2)}_{scl}(h,B,V)}-1\right|\right)=0.
$$
\end{theorem}
\medskip
{\it Remark.} What we really need about $V$ and $B$ for the
semiclassical limit is the conditions (\ref{2dcon1.2})--(\ref{2dcon1.1}).
For a fixed potential, these conditions follow simply from
$V\in L^1\cap L^2$ and $B\in \cC_{C,L}(V)$.
Following \cite{LSY-III}, there are two ingredients in the proof:
Lieb-Thirring inequality and localization.
\subsection{Two dimensional Lieb-Thirring inequality}\label{sec:2dlt}
For completeness, we formulate here the necessary Lieb-Thirring
inequality.
\begin{theorem}\label{thm:2dlt}
For any $\gamma \geq 1$
there exists
a universal constant $C_\gamma$ such that the following estimate
is valid for the $\gamma^{th}$ moment
of the negative eigenvalues $\{ e^{(2)}_m \}_{m=1,2
\ldots}$ of the two dimensional
operator $H^{(2)} (1, \bA, V)= [\bsigma\cdot (\bp + \bA)]^2
+ V$
\be
\sum_m |e^{(2)}_m|^{\gamma} \leq C_\gamma
\left( \Vert B \Vert \int [V]_-^\gamma +
\int [V]_-^{\gamma +1}\right).
\label{eq:2dlt}\ee
In particular we get an estimate for the sum of the eigenvalues
(the case $\gamma=1$).
\end{theorem}
{\it Remark.} Magnetic Lieb-Thirring inequalities for nonhomogeneous
magnetic field were first proven in \cite{E-1995}. There only the
three dimensional case was discussed, though the corresponding
two dimensional results follow analogously. In fact, \cite{E-1995}
mainly focuses on the constant direction case, which essentially requires
a two dimensional analysis. The only difficulty stems from the
fact that the exponent $\gamma =1$ is critical in two dimensions,
which has to be treated, using Fan's theorem, analogously to \cite{LSY-III}.
Later Sobolev \cite{Sob-1996(1)}, with a different approach,
proved Theorem~\ref{thm:2dlt} in a more general setting
which allows fairly general
unbounded magnetic fields. We would like to
point out, however, that Theorem~\ref{thm:2dlt},
in the present form, i.e. for
bounded field, has a simple proof which
follows immediately from \cite{E-1995} and \cite{LSY-III}.
Without going into the details, here we just outline the steps.
{\it Steps of the proof.}
Since $B(x)>0$, it is enough to consider the operator on the
spin-down subspace and we can replace $V$ by $-[V]_-$. Therefore
we are left with the operator $H_-:=(\bp +\bA)^2 - B -[V]_-$ acting
on $L^2(\bR^2)$. We introduce the Birman-Schwinger kernel
$K_E:= [V]_-^{1/2}((\bp +\bA)^2 - B+E)^{-1}[V]_-^{1/2}$
(there is no need to add part of $E$ to the potential, see
the proof of Theorem 5.1. in \cite{LSY-III}), and
decompose it into a lower and an upper part, $K_E = K_{E,L}^< + K_{E,L}^>$
with
\[
K_{E,L}^<:= [V]_-^{1/2}\Pi_L((\bp +\bA)^2 - B+E)^{-1}\Pi_L[V]_-^{1/2}
\]
\[
K_{E,L}^>:= [V]_-^{1/2}(I-\Pi_L)
((\bp +\bA)^2 - B+E)^{-1}(I-\Pi_L)[V]_-^{1/2},
\]
where $\Pi_L$ is the spectral projection,
onto $[0,L]$, of the operator $(\bp +\bA)^2-B$.
Here we cannot
separate the lowest Landau level from the rest of the spectrum, as
the field is not constant. Nevertheless we artifically cut the
spectrum at level $L$ (to be chosen $2\Vert B \Vert$ later) by
inserting the spectral projections $\Pi_L$, $I-\Pi_L$
(similarly to (26), (27) in \cite{E-1995},
but we now omit the kinetic energy in the third direction).
Using the method of [LSY-III] one easily gets, as in
(49) of \cite{E-1995}, the following
bound on $N_E$, the number of eigenvalues of $H_-$ below $-E$,
\[
N_E \leq \# \{ \mbox{ev.'s of } \, \Pi_L[V]_-\Pi_L \,\mbox{bigger
than}\, E/4\} + 4 \mbox{Tr}[K_{E,L}^>]
\]
Therefore, analogously to (50) in \cite{E-1995}, we have
the following bound on the $\gamma^{th}$ moment of the negative
eigenvalues of $H_-$
\be
\sum_i |e_i(H_-)|^\gamma \leq 4^\gamma\gamma\mbox{Tr}
\left([V]_-^{\gamma/2}\Pi_L[V]_-^{\gamma/2}\right)
+ 4\gamma\int_0^\infty \mbox{Tr}[K_{E,L}^>] E^{\gamma -1}\rd E.
\label{50}\ee
By the diamagnetic inequality
$$
\Pi_L(x,x) \leq e^{tL}e^{-tH_-}(x,x)\leq e^{tL}
e^{t\|B\|} e^{t\Delta}(x,x)
$$
Making the particular choice
of $L=2\| B\|$ and $t= (3\| B\|)^{-1}$ we therefore see
(as in Proposition 3.1. of \cite{E-1995}) from
the explicit formula for the heat kernel $e^{t\Delta}(x,x)$
that $\Pi_L (x,x) \leq
c\Vert B\Vert$. This yields the first term in the present
Lieb-Thirring inequality (\ref{eq:2dlt}).
The second term in (\ref{50})
is estimated by using
the obvious operator inequality
$$
(I-\Pi_L)\left( (\bp +\bA)^2 -B+E\right)^{-1}(I-\Pi_L) \leq
\left( \frac{1}{2}[(\bp + \bA)^2 - B] + \frac{L}{2} + E \right)^{-1},
$$
and the pointwise inequality
\be
\left( \frac{1}{2}[(\bp + \bA)^2 - B] + \frac{L}{2} + E \right)^{-1}(x,x)
\leq \left( \frac{1}{2}\bp^2 + E\right)^{-1}(x,x),
\label{pointwis}\ee
which is obtained by rewriting the resolvent kernel as the Laplace
transform of the heat kernel then using the diamagnetic inequality
and the monotonicity of the nonmagnetic heat kernel, i.e.
\begin{eqnarray*}
\exp \left[ - t\left( \frac{1}{2}[(\bp + \bA)^2 - B]
+ \frac{L}{2} + E\right)\right](x,x)
&\leq & \exp \left[ -t\left( \frac{1}{2}[\bp^2 - B] + \frac{L}{2} + E
\right)\right] (x,x)\\
&\leq& \exp \left[ - t\left( \frac{1}{2}\bp^2 +E\right)\right](x,x),
\end{eqnarray*}
since $B\leq L/2$. From (\ref{pointwis})
one finishes the proof along the lines
of \cite{LSY-III} (the $\gamma >1$ case requires obvious
modification). $\,\,\Box$
\subsection{Constant field approximation}\label{2dconstappr}
In this section we rewrite
the kinetic energy part of the Pauli operator in terms of a spatial
average of operators with constant magnetic field plus error terms.
Define spin-up and spin-down projections $P_\pm := \frac{1}{2} (1\pm
\sigma_3)$ . For each
$u\in\bR^2$ we define the function $\eta^u:= \eta^{(0),u}\zeta$, where
$\eta^{(0),u}(x):=c_{norm,u}w_u^{-1}e^{-w_u^{-2}x^2}$ (modulo the constant)
is the magnetic localization function introduced
in Lemma~\ref{lm:leibniz} and $\zeta \in
C^{\infty}_0(\bR^2)$, $\zeta \equiv 1$ on $B(0,\lambda(h)/2)$,
$\mbox{supp}\,\zeta \subset B(0,\lambda(h))$ and
$|\nabla\zeta|\leq c\lambda(h)^{-1}$.
The scale $w_u$ of $\eta^{(0),u}$ we choose to be
$$
w_u:=w_u(h,B)=\Lambda(h)B^{\#}(u)^{-1/2}h^{1/2},
$$
where $\Lambda(h)$ is any function satisfying $\Lambda(h)\to\infty$
and $\Lambda(h)h^{1/2}\to0$ as $h\to0$. We have introduced the
notation
$$
B^{\#}(u):=\sup_{|x-u|<2L}\{B(x)\}.
$$
We also require
$\Lambda(h)\geq 4e$.
The constant
$c_{norm,u}$ is chosen such that
$\int(\eta^u)^2=1$. In particular we have an estimate of the moments
\be
\int \eta^u(v)^2 v^{2k}\rd v \leq c\min\{w_u,\lambda(h)\}^{2k}
\label{eq:2dmoments}\ee for $k=0,1,2$.
It is easy to see that there exists a constant
$c$ such that we then also have
$$
c^{-1}<\int_{\mbox{supp}\,\zeta } (\eta^{(0),u})^20$, depending only on $h$ and $L$
such that $\varepsilon(h)\to0$
as $h\to0$; and for each $u,v\in \bR^2$ there
exist a (phase) function $\phi_{u,v}$ in $C^1(\bR^2)$ and
constant magnetic fields $\hat B^\pm_{u,v}$
satisfying
\be
|B(u) - \hat B^{\pm}_{u,v}|\leq \varepsilon(h) B(u),\quad
\mbox{for all $v$ with $|u-v|\leq 2\lambda(h)$}
\label{2dfieldcomp}\ee
such that the following is valid. For any $f\in C^{\infty}(\bR^2, \bC^2)$
and $g\in C_0^\infty(B(u,\lambda(h)),\bR)$ we have
\begin{eqnarray}
\lefteqn{\int |\bsigma\cdot (h\bp + \bA)gf|^2
\label{2dbelowfin1}}\\
&\geq & \int\int \left[(1-\varepsilon(h)) |\bsigma\cdot
(h\bp+\hat\bA_{u,v}^+) (e^{i\phi_{u,v}}\eta^u_v gf)(x)|^2
-\varepsilon(h)|(e^{i\phi_{u,v}}\eta^u_v gf)(x)|^2
\right] \rd x\rd v\nonumber
\end{eqnarray}
and for any fixed $v\in \bR^2$
\begin{eqnarray}
\lefteqn{\int
|\bsigma\cdot(h\bp+\bA)(e^{-i\phi_{u,v}}\eta^u_vgf) (x)|^2
\rd x
\label{2dabovefin1}}\\
&\leq& \int (\eta^u_vg)^2(x)
\left[(1+\varepsilon(h)) |\bsigma\cdot
(h\bp+\hat\bA_{u,v}^-)f(x)|^2
+\varepsilon(h)W_{u,v}(x)|f(x)|^2
\right.\nonumber\\
&&\left.
+\varepsilon(h)W^+_{u,v}(x)| P_+f(x)|^2\right] \rd x
+ch^2\int|\nabla(\zeta_vg)(x)|^2|(\eta^{(0),u}_{v}f)(x)|^2
\rd x \nonumber,
\end{eqnarray}
where
$$
W_{u,v}(x)=w_u^{-4}(x -v)^4
\quad\mbox{and}\quad
W^+_{u,v}(x)= cB^{\#}(u)hw_u^{-2}(x-v)^2.
$$
and the vector potentials
\be
\hat\bA_{u,v}^{\pm}(x):= \frac{1}{2}\hat{B}_{u,v}^\pm\bn\times x,
\label{2dhatadef}\ee
generate the constant magnetic fields $\hat{B}_{u,v}^\pm$
\end{proposition}
\bigskip
{\it Proof.}
Since $u$ is fixed we shall omit the $u$ subscript in the proof.
\bigskip
{\it Step 1. Separation}
\medskip
\noindent
The separation of the spin up and spin down subspaces is trivial
since $P_\pm$ commutes with $\sigma_3 B$:
\be
\int |\bsigma \cdot(h\bp + \bA)gf|^2
=\int |\bsigma \cdot(h\bp + \bA)gP_-f|^2
+ \int |\bsigma \cdot(h\bp+ \bA)gP_+f|^2
\label{2dspinsep}\ee
\bigskip
{\it Step 2. Localization}
\bigskip
We separately consider the kinetic energies of
$P_{\pm}f =: f_{\pm}$.
For the lower bound we write
\be
\int |\bsigma\cdot (h\bp + \bA)gf_{\pm}|^2 = \int\int
\eta_v(x)^2 |\bsigma\cdot (h\bp + \bA )(gf_{\pm})(x)|^2
\rd x \,\rd v .
\ee
Note that the above integrals can be restricted to
$x,v\in B(u,2\lambda(h))$.
For all $x,v\in B(u,2\lambda(h))$,
we can, since $\lambda(h)0$ for the moment, and
let $M(\nu, u)$ be the characteristic function of the set
\[
\{ (\nu, u) \, : \, 2\nu hB(u)<
[V(u)]_-, |u|\leq \varrho\}.
\]
Note that $M(\nu, u)=0$ if $V(u)\geq0$.
Define the operator $\gamma$ on $L^2(\bR^2, \bC^2)$ by
\[
\gamma = \sum_{\nu=0}^\infty
\int M(\nu , u)
\Pi (\nu, u,v) \rd v \rd u,
\]
which satisfies the density matrix condition $0\leq \gamma \leq
1_{L^2(\bR^2, \bC^2)}$ by (\ref{2d316}).
{F}rom the variational principle, (\ref{2d322}) and (\ref{2d319})
we have (recall that $\lambda(h)^{-2}h^2\leq \varepsilon(h)$)
\begin{eqnarray}
\sum_k e_k(H)&\leq &\mbox{Tr}[H\gamma]\label{2dvarprin}\\
&\leq& \sum_\nu \int
M(\nu , u) d_\nu h^{-1} (1+\varepsilon(h))B(u)\Biggl\{
\Bigl[
(1+c\varepsilon(h)) 2\nu hB(u)
+c\varepsilon(h)\Bigr]\nonumber\\
&&{} + \left(\left\{[V]_+ -
\frac{1-\varepsilon(h)}{1+\varepsilon(h)}[V]_-\right\}
*(\theta_\lambda)^2\right)(u)
\Biggr\} \rd u.\nonumber
\end{eqnarray}
Moreover, for $u$ such that $V(u)<0$ and $|u|\leq \varrho$
we have
\[
\sum_\nu d_\nu h^{-1} B(u)
M(\nu, u)
= h^{-2}\partial_2 P^{(2)}(hB(u), [V(u)]_-)
\]
therefore we can continue the estimate (\ref{2dvarprin})
\begin{eqnarray*}
\sum_k e_k(H)&\leq&
-(1+c\varepsilon(h))^2
h^{-2}\int_{|u|\leq \varrho} P^{(2)}(hB(u), [V(u)]_-)
\rd u + \mbox{Error}(h)\\
&\leq&-h^{-2}\int_{|u|\leq \varrho} P^{(2)}(hB(u), [V(u)]_-)
\rd u + \mbox{Error}(h)
\end{eqnarray*}
with
\begin{eqnarray}
\mbox{Error}(h)&:=&h^{-2}\int_{|u|\leq \varrho\atop V(u)<0}
(1+\varepsilon(h))\partial_2 P^{(2)}\left(
hB(u), [V(u)]_- \right)\label{2dError}\\
&&{}\times\left((1+c\varepsilon(h))[V(u)]_- +
\left(\left\{[V]_+ -
\frac{1-\varepsilon(h)}{1+\varepsilon(h)}
[V]_-\right\}*(\theta_\lambda)^2\right)(u) +c\varepsilon(h)\right)
\rd u.\nonumber
\end{eqnarray}
Therefore, using (\ref{2dpart}),
we have if ($\varepsilon(h)<1$)
\begin{eqnarray}
\mbox{Error}(h) &\leq& h^{-2}\int_{|u|\leq\varrho}c(h \|B\|
+[V(u)]_-)\Big[ \left| [V]_- - [V]_-*(\theta_\lambda)^2
\right|(u)
\label{2der}\\
&&+ \left|[V]_+*(\theta_\lambda)^2 -[V]_+\right|(u)
+ \varepsilon(h)[V(u)]_- + \varepsilon(h)\Big] \rd u
\nonumber\\
&\leq&c\Biggl\{\varepsilon(h)[h^{-1}\|B\|+h^{-2}]\varrho^2 +
\varepsilon(h)\cE^{(2)}_{h, B}([V]_-) \nonumber\\
&&+\sum_{\pm}\left(\cE^{(2)}_{h, B}\left([V]_\pm -
[V]_\pm*(\theta_\lambda)^2\right) +
\cE^{(2)}_{h, B}\left([V]_\pm - [V]_\pm*(\theta_\lambda)^2
\right)^{1/2}
\cE^{(2)}_{h, B}([V]_-) ^{1/2}\right)\Biggr\}\nonumber
\end{eqnarray}
where we used also used (\ref{v2+1}) as in the lower bound.
Considering (\ref{2dcon1.2}--\ref{2dcon1.1}), and (\ref{2dcon2}) we
get, using Jensen's inequality as in the lower bound, that
for fixed $\varrho$
$$
\frac{\mbox{Error}(h)}{|E_{scl}^{(2)}(h, B, V)|}\to0
$$
uniformly for $B$ in ${\cal C}_{C, L}(V)$ as $h\to0$. Hence for all
$\varrho>0$ we have
\be
\liminf_{h\to0}\frac{\sum_k e_k(H)}{E^{(2)}_{scl}(h,B,V)}
\geq \left(
1-\limsup_{h\to0}\frac{h^{-2}\int_{|u|\geq \varrho}
P^{(2)}(hB(u), [V(u)]_-) \rd u}{|
E_{scl}^{(2)}(h, B, V)|}\right).
\label{2dupend}\ee
By (\ref{2dcon3}) and $C_-(B,V) \geq C$,
\[
h^{-2}\int_{|u|\geq \varrho} P^{(2)}(hB(u), [V(u)]_-) \rd u =
h^{-2}\int P^{(2)}(hB(u), [V(1-\chi_{\varrho})(u)]_-) \rd u
\]
\[
\leq \cE^{(2)}_{h, B}
([V-V\chi_\varrho]_-)= \cE^{(2)}_{h, B} ([V\chi_\varrho]_-
- [V]_-) \leq \varepsilon_2(V,\varrho) |E_{scl}^{(2)}
(h, B, V)|/C_-(B,V)
\]
we obtain the
final result (\ref{2dupperend}).
$\,\,\Box$
\section{Semiclassics in three dimensions}\label{sec:3dsc}
\setcounter{equation}{0}
In the introduction we stated our semiclassical result
in Theorem \ref{thm:intro3d} for fixed potential
and a simple one parameter family of magnetic fields but
we shall in fact prove a slightly
stronger result,
which includes more general family of magnetic fields
and which allows the potential to depend
mildly on $\bB$ and $h$.
We always
take $h\rightarrow 0$, and we also would like to
allow $\Vert \bB \Vert \to \infty$ simultaneously (otherwise
the leading term in the
semiclassical limit becomes independent of the magnetic field),
but always with the restrictions
that $hL(\bB)^{-1}\to0$, $hl(\bB)^{-1}\to0$
and $h^3\Vert \bB \Vert l(\bB)^{-2}\to 0$.
The first two conditions are
natural, as they require that the magnetic field should not change
considerably on the usual semiclassical distance scale $h$.
The role of the third condition was explained in
the Introduction after Theorem \ref{thm:intro3d}.
This means that instead of a single magnetic field,
we consider a one-parameter family of magnetic fields, $\bB_\tau$,
parametrized by a real parameter $\tau \in (0,1)$.
We also allow the potential $V=V_\tau$ and
the semiclassical parameter $h=h_\tau$ depend on
$\tau$ in such a way that $h_\tau <1$, $\lim_{\tau\to0} h_\tau=0$, i.e.
we consider a triple of one-parameter family of data $(h_\tau, \bB_\tau,
V_\tau)$. Let
$\mu(h, \bB):= h \max \{ L(\bB)^{-1}, l(\bB)^{-1}\}$,
$\kappa(h, \bB ):= h^3\Vert \bB \Vert l(\bB)^{-2} $ and
let $\mu(\tau):= \mu(h_\tau, \bB_\tau)$, $\kappa(\tau):= \kappa
(h_\tau, \bB_\tau)$ for shortness, then
we require that these functions go to zero as $\tau\to 0$.
To describe how the potential is allowed to depend on $\bB$ and $h$
(via $\tau$),
we introduce the {\bf full 3D magnetic Lieb-Thirring error functional}
\be
\cF_{h, \bB}(V):=
h^{-3}\int|V|^{5/2}
+ (h^{-1}\Vert \bB \Vert + d(h, \bB)^{-2}) h^{-1}\int|V|^{3/2}
\label{mtferr}\ee
\[
+ (h^{-1}\Vert \bB \Vert + d (h, \bB )^{-2})d(h, \bB)^{-1}\int
|V| + (h^{-1}\Vert \bB \Vert +d(h, \bB )^{-2}) \int |\nabla V|,
\]
where
$$
d(h, \bB):= \min \left\{h^{1/4}
\Vert \bB \Vert^{-1/4}l(\bB)^{1/2}, L(\bB), l(\bB) \right\},
$$
and we also introduce the {\bf reduced 3D magnetic Lieb-Thirring error
functional}
\be
\cE_{h, \bB} (V):= h^{-3}\int |V|^{5/2}
+ \Vert \bB \Vert h^{-2}\int |V|^{3/2}
\label{mterr}\ee
(in Section \ref{sec:3dsc} all the integrals are on $\bR^3$ unless
otherwise specified).
Recall that in the case of a large magnetic field (which is our
main concern), namely if
$\Vert \bB \Vert \geq \max \{ L(\bB)^{-2}, l(\bB)^{-2}\}$,
$\cF_{h, \bB}$ is of order
\be
h^{-3}\int |V|^{5/2} +
\|B\| h^{-2}\int |V|^{3/2} +
\|B\|^{5/4}h^{-5/4}l^{-1/2}\int |V| +
\|B\| h^{-1} \int\left( |\nabla V| + |V| \right).
\label{SC}\ee
With this notation, the Lieb-Thirring inequality in
\cite{ES-I}
states that the sum of the negative eigenvalues
$e_1(H), e_2(H), \ldots$
of the operator $H = H(h, \bA , V)$ satisfies the bound
\[
\sum_k |e_k(H)| \leq c\cF_{h, \bB}([V]_-)\leq c\cF_{h, \bB}(V)
\]
for any magnetic field $\bB$ satisfying
(\ref{asp1'}), (\ref{asp2'}), (\ref{asp3'}).
We may now introduce that set of conditions on the triple $(h_\tau,
\bB_\tau, V_\tau)$ which involve
the potential. We require the following
\be
C_-:=\liminf_{\tau\to 0} C_-(\tau):=\liminf_{\tau\to 0}
\frac{|E_{scl}(h_\tau,\bB_\tau,V_\tau)|}{\|\bB_\tau\| h_\tau^{-2}
+ h_\tau^{-3}}>0,
\label{con1.1}\ee
\be
C_+:=\limsup_{\tau\to0} C_+(\tau):=\limsup_{\tau\to0}
\frac{\cE_{h_\tau, \bB_\tau,}([V_\tau]_-)}{\|\bB_\tau\|
h_\tau^{-2} +h_\tau^{-3}}<\infty,
\label{con1.2}\ee
and
\be
\lim_{\tau, r\to 0}\sup_{\sigma\leq \tau}
\varepsilon_\pm (\sigma, r)=0,
\quad \lim_{\varrho\to\infty}\sup_{\tau\in (0,1)}
\eps_2(\tau, \varrho)=0
\label{con1.3}\ee
where
\be
\varepsilon_+(\tau, r):=
\sup_{|y|\leq r}
\frac{\cE_{h_\tau,
\bB_\tau}([V_\tau]_+-[V_\tau(\cdot-y)]_+)}{\|\bB_\tau\|
h_\tau^{-2} +h_\tau^{-3}},
\label{con2}\ee
\be
\varepsilon_- (\tau ,r) :=\sup_{|y|\leq r}
\frac{\cF_{h_\tau, \bB_\tau}
([V_\tau]_-- [V_\tau(\cdot - y)]_-)}{\|\bB_\tau\| h_\tau^{-2}+
h_\tau^{-3}},
\label{con3}\ee
\be
\varepsilon_2(\tau, \varrho):=
\frac{\cF_{h_\tau, \bB_\tau}([V_\tau\chi_\varrho]_-
-[V_\tau]_-)}{\|\bB_\tau\| h_\tau^{-2} + h_\tau^{-3}}
\label{con4}\ee
and here $\chi_\varrho$ denotes the characteristic
function of the ball of radius $\varrho$ centered at the origin.
Note that (\ref{con1.1}), (\ref{con1.2}) and $P(B,W)\leq c(BW^{3/2} +
W^{5/2})$ imply that
\be
c\leq\frac{\cE_{h_\tau, \bB_\tau}(V)}{|E_{scl}(h_\tau, \bB_\tau,
V_\tau)|} \leq \frac{C_+(\tau)}{C_-(\tau)}.
\label{con5}\ee
{\it Remark.}
If $V$ is independent of $\tau$ (i.e. of $\bB_\tau$ and $ h_\tau$),
then the conditions
follow simply from $V\in L^{5/2}\cap L^{3/2}$, $[V]_-\in
W^{1,1}$, $|E_{scl}(h\tau, \bB_\tau, V)|\geq C(\|\bB_\tau\| h_\tau^{-2}
+ h_\tau^{-3})$
and $\mu(\tau), \kappa(\tau)\to 0$ (use that
$d(h, \bB)^{-2}\leq h^{-2}(\mu^2(h, \bB) +\kappa(h, \bB)^{1/2})$).
\begin{theorem}{\bf (3D Semiclassics)}\label{thm:3dsc}
Let $e_1(H_\tau) ,e_2(H_\tau), \ldots$ denote the
negative eigenvalues of the operator $H_\tau=H(h_\tau, \bA_\tau, V_\tau)$
in (\ref{Pauli}). Assume that the triple of the physical data
$(h_\tau, \bB_\tau, V_\tau)$ is such that $\mu(\tau)=
h_\tau\max \{ L(\bB_\tau)^{-1}, l(\bB_\tau)^{-1}\}\to0$, $\kappa(\tau)=
h^3_\tau \Vert \bB_\tau\Vert l(\bB_\tau)^{-2}\to0$ (as $\tau\to0$), and
it satisfies
(\ref{con1.1}--\ref{con1.3}) then
\be
\lim_{\tau\to0}\left|
\frac{\sum_k e_k(H_\tau)}{E_{scl}(h_\tau,\bB_\tau ,V_\tau)}-1\right|=0.
\label{eq:3dsc}\ee
\end{theorem}
{\it Remark 1.} From our proof it will be clear that in order
to get the corresponding upper bound on the eigenvalue sum
in (\ref{eq:3dsc}), we can relax the condition (\ref{con1.3}) involving
$\eps_-$ and $\eps_2$
by replacing $\cF_{h_\tau, \bB_\tau}$ to $\cE_{h_\tau, \bB_\tau}$
in their definitions (\ref{con3})-(\ref{con4}).
{\it Remark 2.} The reader might wonder to what extent our result
is uniform, i.e. if we can say something without restricting
ourselves to a one-parameter family of data.
Our proof, in fact, immediately provides
a uniform statement but its precise formulation is fairly
complicated due to the interrelations between various control
parameters (like $\kappa, \mu, \eps_\pm, \eps_2$), therefore we omit it.
\subsection{Constant field approximation}\label{constappr}
Fix $\mu, \kappa >0$, $h\in (0,1)$ and consider
a magnetic field $\bB$ such that $hL(\bB)^{-1}, hl(\bB)^{-1}\leq \mu$ and
$h^3\|\bB\|l(\bB)^{-2}\leq \kappa$.
The goal of this section is to rewrite the kinetic energy
part of the Pauli operator in terms of a spatial average
of operators with constant magnetic field plus error
terms. The error terms will go to zero as $\mu, \kappa, h \to 0$,
(notice that, in addition to $h$,
$\kappa$ and $\mu$ also play the role of the small parameters).
Later, this representation will be used to obtain
precise lower and upper bounds on the eigenvalue sum.
The final result of this section is
Proposition \ref{constprop} below, but we have
to introduce some notations before stating it.
First we introduce
a spherically symmetric functions $\theta_{\lambda}= \lambda^{-3/2}
\theta (x/\lambda )$ with $\theta \in C_0^{\infty}(\bR^3)$,
which localizes at scale
$\lambda =\lambda(\kappa, \mu, h)$, i.e. $\int_{\bR^3} \theta_{\lambda}^2 =1$,
$\mbox{supp}\,\theta_\lambda\subset B(0, \lambda)$
and $\int (\nabla \theta_\lambda)^2 \leq c\lambda^{-2}$.
Here the function $\lambda =\lambda(\kappa, \mu ,h)$ will be chosen later,
but it is required to go to zero as $\kappa,\mu, h\to0$.
Let $f\in C^{\infty}_0 (\bR^3, \bC^2)$, then
by the IMS localization formula (Lemma \ref{lm:IMS})
\be
\langle f | [\bsigma \cdot (h\bp + \bA )]^2 | f \rangle =
\iint |\bsigma \cdot (h\bp + \bA (x) )\theta_\lambda (x-u)f(x)|^2
\rd x\, \rd u - h^2
\langle f|f \rangle \int(\nabla \theta_\lambda)^2,
\label{ims}\ee
(the gradient operator $\bp$ always acts on
the $x$ variable).
We fix a point $u$ and now study
\be
\int |\bsigma \cdot (h\bp + \bA (x) )
\theta_\lambda(x-u)f(x)|^2 \rd x
= \int |\bsigma\cdot (h\bp + \bA )gf|^2,
\label{tostudy}\ee
where $g(x):=
\theta_\lambda (x-u)$ (for the rest of this section $u$ and $\lambda$ remain
fixed, so we omit them from the notation).
Note that (\ref{tostudy}) is
invariant under orthogonal coordinate transformation,
so we can choose a coordinate system such that $\bB(u) = (0, 0, B(u))$,
i.e. $\bn(u) = (0,0,1)$. Throughout this section we shall work
in this coordinate system. We also define the spin projections as
\be
P^u_{\pm} = P_{\pm} := \frac{1}{2}(1\pm \bsigma\cdot \bn (u)).
\label{spinpr}\ee
Moreover, we define the transversal coordinates, denoted by $x_{\perp}
\in \bR^2$, by
\[
x= (x_{\perp}, x_3):=(\pi_{1,2}(x - x\cdot \bn(u)), x\cdot \bn(u))
\]
($\pi_{1,2}$ is the standard projection on the first two components
from $\bR^3 \to \bR^2$),
and $p_{\perp}$, $\sigma_{\perp}$
are defined analogously. Strictly speaking the notions of
"perpendicular" and "third direction"
depend on $u$, i.e. formally the notation
$(x_\perp, x_3)= (x_{\perp (u)}, x_{3(u)})$ would be meticulous,
but in this Section we omit the $u$ dependence.
We shall need a second localization, which is essentially identical
to the localization given in Section \ref{2dconstappr}.
For the reader's convenience we recall that for any $u$ it is given by
a function $\eta^u \in C_0^{\infty}(\bR^2)$
with $\mbox{supp} \, \eta^u \subset B(0,\lambda(\kappa, \mu ,h))
\subset \bR^2$ which has the form
$\eta^u := \eta^{(0),u}\zeta$.
Here $\zeta \in C^{\infty}_0(\bR^2)$, $\mbox{supp}\,\zeta\subset
B(0, \lambda(\kappa, \mu ,h))$,
$|\nabla\zeta|\leq c\lambda(\kappa,\mu ,h)^{-1}$, $\zeta \equiv 1$
on $B(0,\lambda(\kappa,\mu ,h)/2)$, $0\leq \zeta \leq 1$;
and the function $\eta^{(0)}=\eta^{(0),u}$ is defined as
\[
\eta^{(0)}(v)=\eta^{(0),u}(v):=c_{norm, u}w_u^{-1}e^{-w_u^{-2}v^2}
\]
with a normalization constant chosen such that $\int_{\bR^2}(\eta^u)^2=1$.
It is easy to see that
\[
c^{-1} < \int_{\mbox{supp} \, \zeta}(\eta^{(0),u})^2 < c
\]
with a universal constant.
The scale $w_u=w_u(\kappa, \mu, h)$ of $\eta^{(0),u}$ is chosen as
\[
w_u:=\Lambda (\kappa ,\mu,h)(B^{\#}(u))^{-1/2}h^{1/2},
\]
where $\Lambda(\kappa ,\mu,h)$ is a function to be chosen later,
which satisfies $\Lambda(\kappa, \mu,h)\to\infty$
as $\kappa ,\mu, h \to0$. Here we introduced the
notation
\[
B^{\#}(u):=\sup_{x\, :\, |x-u|<2L(\bB)}\{ |\bB(x)| \},
\]
and since $e^{-2}\leq B(x)/B(y)\leq e^2$ for all $|x-y|\leq 2L(\bB)$,
we note that
\be
e^{-2}B^{\#}(u)\leq B(u)\leq B^{\#}(u).
\label{sharp}\ee
Notice that $w_u$, the scale of $\eta^{(0),u}$,
depends on $u$, but in general we omit
it from the notation in this section.
In particular we have
\be
\int_{\bR^2} \eta^u(v)^2v^{2k}\rd v \leq c\min \{w_u,
\lambda(\kappa,\mu ,h)\}^{2k}
\label{moments}\ee
for $k=0,1,2$.
We define
\[
\eta^u_{v}(x)=\eta_v(x):=\eta(x_{\perp} - v),
\]
and similarly
\[
\zeta^u_{v}(x)=\zeta_v(x):=\zeta(x_{\perp} - v),
\]
\[
\eta^{(0),u}_{v}(x)=\eta^{(0)}_v(x):=\eta^{(0)}(x_{\perp} - v)
\]
(note that $\eta_v$, $\eta^{(0)}_v$ and $\zeta_v$ are functions
on $\bR^3$ and the $u$-dependence is hidden in $\perp = \perp(u)$).
\bigskip
Armed with these notations and definitions,
we can state the main result of this section.
\begin{proposition}\label{constprop}
There exist two universal constants $c_0\leq 1$
and $c_1\geq 1$, and
there exists a function $\eps(\kappa ,\mu, h)
: (0,c_0)\times (0,c_0)\times (0,c_0)\to (0,1/4)$,
$\eps (\kappa, \mu, h)\to0$ as $\kappa, \mu, h\to0$
(in fact the function $\eps(\kappa, \mu, h)= c_1\max\{\kappa^{1/6},
\mu^{1/3}, h^{2/3}\}$ would do) with
following property:
for any $00$
let $M_{\tau, \varrho}(\nu, u, p)$ be the characteristic function of the set
\[
\{ (\nu, u, p) \, : \, h_\tau^2(p^2 +2\nu h_\tau^{-1}
B_\tau(u))
< [V_\tau(u)]_-, |u|\leq \varrho\}.
\]
Note that $M_{\tau, \varrho}(\nu, u, p) =0$ if $V_\tau(u)\geq 0$.
Define the operator $\gamma_{\tau, \varrho}$ on $L^2(\bR^3, \bC^2)$ by
\[
\gamma_{\tau, \varrho} = (2\pi)^{-1} \sum_{\nu=0}^\infty
\iint_{\bR} M_{\tau, \varrho}(\nu , u, p)\int_{\bR^2}
\Pi^-_\tau (\nu, u, v, p) \rd v\rd p \rd u.
\]
\begin{theorem}\label{thm:upper3dsc}
With the notations above, for any $\varrho>0$, the operator $\gamma_{\tau,
\varrho}$
satisfies the density matrix condition $0\leq \gamma_{\tau, \varrho} \leq
1_{L^2(\bR^3, \bC^2)}$, its density function,
$\rho_{\gamma_{\tau, \varrho}}(x):= \mbox{Tr}_{\bC^2}(\gamma_{\tau,
\varrho} (x,x))$
satisfies the estimate
\be
(1-\tilde\eps(\tau))\leq \frac{\rho_{\gamma_{\tau, \varrho}}
(x)}{h_\tau^{-3}
\int_{|u|\leq \varrho} \partial_2 P(h_\tau, B_\tau(u), [V_\tau(u)]_-)
\theta_\lambda(x-u)^2 \rd u} \leq (1+\tilde\eps(\tau)),
\label{trialdensest}\ee
(it should be understood
such that we allow both the nominator and the denominator be simultaneously
zero),
and, as a trial density matrix, $\gamma_{\tau, \varrho}$
gives the exact semiclassical upper bound
to the energy asymptotics, i.e. for any
$\varrho >0$
\be
\liminf_{\tau\to0}
\frac{\mbox{Tr}[H_\tau\gamma_{\tau, \varrho}]}{E_{scl}
(h_\tau, \bB_\tau, V_\tau)}
\geq 1- \limsup_{\tau\to0}\frac{\eps_2(\tau, \varrho)}{C_-(\tau)} .
\label{eq:upper3dsc}\ee
\end{theorem}
Letting $\varrho\to\infty$,
(\ref{eq:upper3dsc}), (\ref{con1.1}) and (\ref{con1.3}) obviously
imply (\ref{upperend}) by using the variational principle:
$\sum_k e_k (H_\tau) \leq \mbox{Tr}[H_\tau\gamma_{\tau, \varrho}]$.
\bigskip
{\it Proof:}
The following relations are immediate to check
\begin{eqnarray}
(2\pi)^{-1}\sum_{\nu =0}^{\infty} \iint_{\bR}\int_{\bR^2}
\Pi^-_\tau (\nu, u, v, p) \rd v \rd p \rd u &=& 1_{L^2(\bR^3, \bC^2)},
\label{316}\\
\mbox{Tr}_{\bC^2}[\Pi^-_\tau (\nu, u, v, p)(x,x)] &=& d_\nu h^{-1}
\hat B^-_{u,v} (\theta_\lambda
(x-u)\eta^u(x_{\perp(u)}-v))^2.
\label{316.5}\end{eqnarray}
In particular (\ref{316}) implies that $\gamma_{\tau, \varrho}$ satisfies
the density matrix condition.
The relation
\be
(2\pi)^{-1}\sum_\nu d_\nu h_\tau^{-1}B_\tau(u)
\int_\bR M_{\tau, \varrho}(\nu, u, p) \rd p
= h_\tau^{-3}\partial_2 P(h_\tau B_\tau (u), [V_\tau(u)]_-),
\label{partder}\ee
for $|u|\leq \varrho$ and (\ref{316.5})
immediately give (\ref{trialdensest}).
For $\mbox{Tr}[H_\tau \gamma_{\tau, \varrho}]$ we shall need
\be
\mbox{Tr} \Pi^-_\tau(\nu, u, v, p):=\mbox{Tr}_{L^2(\bR^3, \bC^2)}
\Pi^-_\tau (\nu, u, v, p)= d_\nu h^{-1} \hat B_{u,v}^- \Phi (u,v)
\label{317}\ee
with
\[
\Phi (u,v):=\int (\theta_{\lambda} (x-u)^2\eta^u
(x_{\perp (u)}-v))^2 \rd x,
\]
and
\be
\int_{\bR^2} \mbox{Tr}[ V_\tau\Pi^-_\tau(\nu, u, v, p)]\rd v
\leq d_\nu h_\tau^{-1} B_\tau(u)\left(\left\{(1+\tilde\varepsilon
(\tau))[V_\tau]_+ -
(1-\tilde\varepsilon(\tau))[V_\tau]_-\right\}*
(\theta_\lambda)^2\right)(u)
\label{322}\ee
obtained from (\ref{fieldcomp}).
In order to calculate the kinetic energy we note, similarly to
the two dimensional case, that
$\Pi_\tau^-(\nu, u, v, p) = \Xi_\tau (\nu, u, v, p)\Xi_\tau (\nu, u, v, p)^*$,
where $\Xi_\tau (\nu, u, v, p)$ has an integral kernel
\[
\Xi_\tau (\nu, u, v, p)(x,y)=
\theta_\lambda (x-u)\eta^u (x_{\perp (u)}-v)
e^{-i\phi_{u,v} (x)}\Pi_{u,v}^{\nu, -}(x_{\perp (u)},
y_{\perp (u)}) e^{ip (x_{3(u)} -y_{(3)})}.
\]
Let, furthermore, $\Gamma_{u,v,p}^{\nu}$ be the positive operator
with kernel
\[
\Gamma_{u,v,p}^{\nu}(x,y):= \Pi_{u,v}^{\nu, -}(x_{\perp (u)},
y_{\perp (u)}) e^{ip (x_{3(u)} -y_{(3)})}.
\]
We therefore have
\begin{eqnarray}
\lefteqn{
\int_{\bR^2}\mbox{Tr}\left( [\bsigma\cdot (h_\tau\bp + \bA_\tau)]^2
\Pi^-_\tau (\nu, u, v, p)\right) \rd v}&&
\nonumber\\
&=&
\int_{\bR^2}\mbox{Tr}\left[\left(\bsigma\cdot (h_\tau\bp + \bA_\tau)
\Xi_\tau (\nu, u, v, p)\right)^*
\left(\bsigma\cdot (h_\tau\bp + \bA_\tau)
\Xi_\tau (\nu, u, v, p)\right)\right] \rd v
\nonumber\\
&=& \int_{\bR^2}\int\!\int |[\bsigma\cdot (h_\tau\bp + \bA_\tau)
\Xi_\tau (\nu, u, v, p)](x,y)|^2 \rd x \rd y \rd v.
\label{eq:upfin1}\end{eqnarray}
To estimate this, we
use (\ref{abovefin1}) for each fixed $y$, with
for $g:=\theta_{u,\lambda}$
and $f^v(x)= \Gamma_{u,v,p}^\nu (x,y)$.
We begin by examining the error terms. Exactly as in the
two dimensional case, from the spectral density
expression (\ref{eq:density}), the moment estimate (\ref{moments}),
the field comparison (\ref{fieldcomp}) and the normalization of
$\theta_{\lambda,u}$ we find
\begin{eqnarray}
\lefteqn{\int (\eta_v^u\theta_{\lambda,u})^2(x)
W_{u,v}(x_\perp)|\Pi_{u,v}^{\nu, -}(x_\perp ,y_\perp)|^2
\rd y\rd x\rd v}&&\\
&=&\int \left( w_u^{-4}(x-v)^4
+w_u^{-2}(x-v)^2 + 1\right)(\eta_v^u(x))^2\theta_{\lambda,u}^2(x)
|\Pi_{u,v}^{\nu, -}(x_\perp ,x_\perp )|\rd x\rd v\nonumber\\
&\leq& c d_\nu h_\tau^{-1} (1+\tilde\varepsilon(\tau)) B_\tau(u),
\label{eq:uperr1}
\end{eqnarray}
where the first identity follows since $\Pi_{u,v}^{\nu, -}$ is a projection.
Likewise, for the second error term we get
\begin{eqnarray}
\int (\eta_v^u\theta_{\lambda,u})^2(x)
W_{u,v}^+(x_\perp)|P_+\Pi_{u,v}^{\nu, -}(x_\perp ,y_\perp)|^2
\rd y\rd x\rd v
&\leq& \nu h_\tau B_\tau^{\#}(u)d_\nu h_\tau^{-1} (1+\tilde
\varepsilon(\tau)) B_\tau(u)
\nonumber\\
&\leq& c\nu (h_\tau B_\tau(u))d_\nu h_\tau^{-1}
(1+\tilde\varepsilon(\tau)) B_\tau(u).
\label{eq:uperr2}
\end{eqnarray}
In the last line we used (\ref{sharp}) to estimate $B_\tau^{\#}(u)$
in terms of $B_\tau(u)$. Note that we inserted a $\nu$ in the estimate.
This is clearly allowed for $\nu\geq1$. For $\nu=0$ it follows simply because
$P_+\Pi_{u,v}^{0, -}=0$ (i.e., the lowest Landau level contains only spinors
with spin down).
For the last error term in (\ref{abovefin1}) we just have to recall that
$\int_{\mbox{supp}\,\zeta } (\eta^{(0),u})^20$. Hence for all $\varrho >0$ we have
\be
\liminf_{\tau\to0}\frac{\mbox{Tr}[H_\tau \gamma_\tau])}{E_{scl}(h_\tau,
\bB_\tau, V_\tau)}
\geq \left(
1-\limsup_{\tau\to0}\frac{h_\tau^{-3}\int_{|u|\geq \varrho}
P(h_\tau B_\tau(u), [V_\tau(u)]_-) \rd u}{|
E_{scl}(h_\tau, \bB_\tau, V_\tau)|}\right) .
\label{upend}\ee
By the definition of $C_-(\tau)$ (see (\ref{con1.1})) and (\ref{con4})
\[
h_\tau^{-3}\int_{|u|\geq \varrho} P(h_\tau B_\tau(u), [V_\tau(u)]_-)
\rd u =
h_\tau^{-3}\int P(h_\tau B_\tau(u), [V_\tau
(1-\chi_{\varrho})(u)]_-) \rd u
\]
\[
\leq \cE_{h_\tau, \bB_\tau} ([V_\tau -V_\tau\chi_\varrho]_-)=
\cE_{h_\tau, \bB_\tau}
([V_\tau\chi_\varrho]_-
- [V_\tau]_-) \leq \eps_2(\tau,\varrho)|E_{scl}(h_\tau, \bB_\tau,
V_\tau)|/C_-(\tau),
\]
hence we obtain (\ref{eq:upper3dsc}). $\,\,\Box$
\section{Magnetic Thomas-Fermi theory}\label{sec:MTF}
\setcounter{equation}{0}
In Section \ref{sec:intromtf} of the
introduction we stated our simplest theorem on
the asymptotic validity of the magnetic Thomas-Fermi (MTF) theory
as the limit of quantum mechanics and we also introduced the
basic notations. Here we recall some further results
on the MTF theory, obtained in \cite{LSY-II}.
The first important result (Proposition~4.3 of \cite{LSY-II})
is that the MTF energy $E^{\rm MTF}(N, B, Z)$
(defined in (\ref{mtf:energy})) is always finite,
$E^{\rm MTF} (N,B,Z)>-\infty$, as long as $B$ is a
locally bounded function. Furthermore, it was proved in
Theorems~4.5--4.7 that there is a unique minimizer $\rho^{\rm MTF}$,
which satisfies the Thomas-Fermi equation (see \cite{LSY-II} equation
(4.27))
\be
\rho^{\rm MTF}(x)=\partial_2P(B(x), [V^{\rm MTF}(x)]_-),
\label{mtf:equation1}
\ee
where
\be
V^{\rm MTF}(x)=-Z|x|^{-1}+\rho^{\rm MTF}(x)*|x|^{-1}+\mu
\label{mtf:equation2}\ee
with $\mu:=\mu(N,B,Z):=-\partial E^{\rm MTF}(N,B,Z)/\partial N \geq0$
being the chemical potential (see \cite{LSY-II}
Theorem~4.8). Conversely, if the pair $(\rho,\mu)$
satisfies (\ref{mtf:equation1}) and (\ref{mtf:equation2})
(with $\rho$ instead of $\rho^{\rm MTF}$) then there exists $N$
such that $\rho$ is the minimizer of $\cE$ with
$\int\rho\leq N$
and $\mu=\mu(N,B,Z)$.
Note that according to \cite{LSY-II} Proposition~4.2
the minimizer $\rho^{\rm MTF}$
is in $L^{5/3}_{\rm loc}(\bR^3)\cap L^1(\bR)$. The
convolution integral $\rho^{\rm MTF}*|x|^{-1}$
therefore makes sense
and for $x\ne0$ we have
\be
-(4\pi)^{-1}\Delta V^{\rm MTF}(x)
=\rho^{\rm MTF}(x).
\label{mtf:diffeq}\ee
{F}rom Theorem~4.8 in \cite{LSY-II} we see that
$$
\int \rho^{\rm MTF}0$ depending on $\bB$ such that
if we define a rescaled field by
$\bB_{Z,b}(x):=b\bB(x/[s(b,Z)K])$ then the
following result holds.
Assume that $Z,N
\rightarrow \infty$ with $N/Z$ fixed and $b/Z^2\rightarrow 0$,
then
$$
E(N,b\bB_{Z,b},Z)/E^{\rm MTF} (N,bB_{Z,b},Z) \rightarrow 1.
$$
\end{theorem}
We see therefore that the scale on which
we allow the magnetic field
to vary is greater than the size of the atom if $B(0)\gg Z^{2}$.
It is an open question to allow the magnetic field to vary on
the scale of the atom if $B(0)\gg Z^{2}$.
Both Theorem~\ref{thm:mtfweak} and Theorem~\ref{thm:mtfweak2}
are simple consequences of the following stronger result.
\begin{theorem} \label{thm:mtfstrong}
Consider sequences $N_n$ of positive integers and $Z_n$ of
positive real numbers with $N_n,Z_n\to\infty$ as $n\to\infty$
and $N_n/Z_n$ bounded above and below away from zero.
If $k>0$ is a constant then there exists a constant $K>0$ such that
if, $\bB_n:=\nabla\times\bA_n :\bR^3\to\bR^3 $
is a sequence of magnetic fields satisfying $B_n(0)\geq k \|B_n\|$ and
\be
L(\bB_n)\geq Ks(\|B_n\|,Z_n)
\label{mtf:Lcond}\ee
for all $n$,
\be
l(\bB_n)^{-1}s(\|B_n\|,Z_n)
\max\left\{s(\|B_n\|,Z_n)^{-1/2}Z_n^{-1/2},
\|B_n\|^{1/2}Z_n^{-1}\right\}
\to0\quad\hbox{as $n\to\infty$},
\label{mtf:lcond}\ee
and
\be
\|B_n\|Z_n^{-3}\to0
\label{mtf:BZ3}\ee
as $n\to\infty$, then
\be
\lim_{n\to\infty}
E(N_n,\bB_n,Z_n)/E^{\rm MTF} (N_n,B_n,Z_n)
\rightarrow 1
\label{mtf:main}\ee
as $n\to\infty$.
\end{theorem}
The roles of the constants $K$ and $k$ may seem mysterious and the corresponding
conditions could possibly
be weakened. The constant $k$ ensures that we are not
considering a magnetic field
which is much weaker in
the center than its maximum. If this were the case,
$s$, as defined here would not be the correct scale of the atom,
since it presumably
should involve also the typical field strength around the nucleus.
The constant $K$ ensures that the field does not change too fast
on the scale of the atom. If this happened the atom could
actually have two different
relevant scales, one where $B$ is large, another where $B$ is small.
In the following all positive constants, denoted by capital
$C$ or $C_1$ etc.~, may
depend on $k$. Constants that are universal will be denoted by the
common symbol $c$. It is of no importance to the proof whether a constant is
universal or depends on $k$.
We devote the rest of this chapter to the proof of
Theorem~\ref{thm:mtfstrong}. For simplicity we omit the subscript
$n$.
\subsection{Rescaling}\label{sec:resc}
We rescale the Hamiltonian (\ref{mtf:hamiltonian}) using the unitary
$$
(U_s\psi)(x_1,\ldots,x_N)=s^{-3N/2}
\psi(s^{-1}x_1,\ldots,s^{-1}x_N),
$$
where $s=s(\|B\|, Z)$ is given in (\ref{mtf:sdef}). We obtain that
$Z^{-1}sE(N,\bB,Z)$ is the bottom of the spectrum of
the operator
\be
H_{\rm eff} := \sum^N_{i=1} \left\{[
\bsigma_i \cdot (h\bp_i + \bA_{\rm eff}(x_i))]^2
-|x_i|^{-1}\right\} +Z^{-1} \sum
\limits^N_{i0$,
$\varphi_a(x)=a^{-3}\varphi(x/a)$ we then have for all
$\tilde\rho:\bR^3\to\bR$
\begin{eqnarray*}
\lefteqn{\sum_{1\leq iR$).
We then have from (\ref{trialdensest}) that
$$
(1-\tilde{\varepsilon}(h,\bB_{\rm eff}))\leq
\frac{\rho_\gamma(x)}{\int h^{-3}\partial_2P(hB_{\rm eff}(u),
[(\Theta_RV_{\rm eff})(u)]_-)\theta_\lambda(x-u)^2\rd u}
\leq (1+\tilde{\varepsilon}(h,\bB_{\rm eff}))
$$
where according to (\ref{mtf:scasp}) we may assume that
$\tilde{\varepsilon}(h,\bB_{\rm eff})\to0$ in the limit we consider.
If we now use the rescaled form of (\ref{mtf:equation1}) we see that
\be
(1-\tilde{\varepsilon}(h,\bB_{\rm eff}))(\chi_R\rho_{\rm eff})
*\theta_\lambda^2(x)\leq\rho_\gamma(x)\leq
(1+\tilde{\varepsilon}(h,\bB_{\rm eff}))(\chi_{2R}\rho_{\rm eff})
*\theta_\lambda^2(x)
\label{mtf:rhogamma}\ee
where $\chi_{R}$ is the characteristic function of the ball of radius $R$.
Since,
$$
\iint\theta_\lambda(x-z)^2|z-w|^{-1}\theta_\lambda(y-w)^2\rd z\rd w
\leq |x-y|^{-1}
$$
we have that
$$
{\textstyle\frac{1}{2}}\iint
\rho_{\gamma}(x)
\vert x-y \vert^{-1}\rho_\gamma(y)\rd x\rd y
\leq {\textstyle\frac{1}{2}}(1+
\tilde{\varepsilon}(h,\bB_{\rm eff}))^2
\iint
\rho_{\rm eff}(x)
\vert x-y \vert^{-1}\rho_{\rm eff}(y)\rd x\rd y.
$$
We therefore get from (\ref{mtf:muN}) and (\ref{mtf:liebvar}) that
\begin{eqnarray}
\lefteqn{Z^{-1}sE(N,\bB,Z)\leq
\mbox{Tr}\left[\gamma \left(
[\bsigma\cdot (h\bp + \bA_{\rm eff})]^2+
V_{\rm eff}\Theta_R\right)\right]-s\mu N Z^{-1}}\nonumber&&\\
&&{}-{\textstyle\frac{1}{2}}Z^{-1}\iint\rho_{\rm eff}(x)
\vert x-y \vert^{-1}\rho_{\rm eff}(y)\rd x\rd
y+\Delta^+E_1+\Delta^+E_2,
\label{mtf:upfin}
\end{eqnarray}
where according to (\ref{mtf:rhogamma}) and assuming $R>2\lambda$
\be
\Delta^+E_1:=\Delta^+E_1(R):=
\mu s Z^{-1}\int(\rho_{\rm eff}-\Theta_R\rho_\gamma)
\leq \mu s Z^{-1}\int\left(\rho_{\rm eff}-
(1-\tilde{\varepsilon}(h,\bB_{\rm eff}))
(\chi_{R/2}\rho_{\rm eff})\right),
\label{mtf:delta+1}\ee
and (again assuming $R>2\lambda$)
\begin{eqnarray}
\Delta^+E_2:=\Delta^+E_1(R)&:=&
Z^{-1}(1+\tilde{\varepsilon}(h,\bB_{\rm eff}))^2
\iint\rho_{\rm eff}(x)
\vert x-y \vert^{-1}\rho_{\rm eff}(y)\rd x\rd y\nonumber
\\
&&-Z^{-1}\iint\rho_{\rm eff}(x)
\vert x-y \vert^{-1}(\Theta_R\rho_\gamma)(y)\rd x\rd y\nonumber
\\
&\leq&Z^{-1}\iint\rho_{\rm eff}(x)
\vert x-y \vert^{-1}\left(\rho_{\rm eff}(y)-
(\chi_{R/2}\rho_{\rm eff})*\theta_\lambda^2(y)\right)\rd x\rd y
\label{mtf:delta+2}\\
&&+c\left(\tilde{\varepsilon}(h,\bB_{\rm eff})+
\tilde{\varepsilon}(h,\bB_{\rm eff})^2\right)
Z^{-1}\iint\rho_{\rm eff}(x)
\vert x-y \vert^{-1}\rho_{\rm eff}(y)\rd x\rd y\nonumber
\end{eqnarray}
\subsection{Properties of the potentials}
In proving our main result we shall apply Theorem~\ref{thm:3dsc}
with $V$ replaced by $V_{a,R}^-$ for the lower bound
and with $\Theta_RV_{\rm eff}$ for the upper bound.
In order to do this we must show that these potentials
satisfy the necessary conditions of Theorem~\ref{thm:3dsc}.
\begin{lemma}\label{lm:vmurho} If $B(0)\geq k \|B\|$ then
there exists constans
$C_0>0$ and $K>0$ (depending on $k$) such that if $\bB$ satisfies
(\ref{mtf:Lcond}) (with this constant $K$) we have
\be
[V_{\rm eff}(x)]_-
\leq C_0\min\{|x|^{-1},|x|^{-4}\},
\label{mtf:vbound}\ee
\be
Z^{-1}\rho_{\rm eff}(x)\leq C_0\min\{|x|^{-3/2},|x|^{-2}\}
\label{mtf:rhobound}\ee
\be
Z^{-1}s\mu\leq C_0(Z/N),
\label{mtf:mubound}\ee
where $s=s(\|B\|, Z)$.
\end{lemma}
{\it Proof.} It is clear from (\ref{mtf:veff}) that
$[V_{\rm eff}(x)]_-\leq |x|^{-1}$.
We consider $|x|\leq r$ for some $r>0$.
Using (\ref{mtf:Lcond}) we obtain that on this set
$$
B_0:=k\|B\|e^{-K^{-1}s^{-1}r}\leq
B(0)e^{-L^{-1}r}\leq B(x).
$$
Consider now the magnetic function $\tilde B_r:\bR^3\to\bR^3$ which is
equal to $B(x)$ for $|x|\leq r$ and which is constantly equal to $B_0$
if $|x|>r$. We may now study the MTF theory of atoms in this `magnetic'
field. (The reader may worry that we have not defined a magnetic field,
but only a scalar function. The observation is that, although this was
not explicit in \cite{LSY-II}, MTF theory makes sense for any locally
bounded scalar function $B(x)$.) It now follows from Theorem~4.11 in
\cite{LSY-II} that the support of this new MTF atom is bounded above by
(see (4.32) in \cite{LSY-II})
$$
r_{\max}\leq c\max\left\{ZB_0^{-1}, Z^{1/5}B_0^{-2/5}\right\},
$$
where $c>0$ is a universal constant.
This means that the new density and the negative part of the
new effective potential vanish
outside this radius (recall the MTF equation (\ref{mtf:equation1}) relating
the density and the effective potential).
Since $B$ and $\tilde B_r$ agree for $|x|\leq r$ we
conclude by uniqueness of the minimizer to the MTF
equations (\ref{mtf:equation1}) and (\ref{mtf:equation2})
that the original atom has radius $r_{\max}$ if
\be
r_{\max}\leq r.
\label{mtf:req0}\ee
We shall now show that we may choose
$r$ such that this condition is satisfied.
We shall attempt to make a choice consistent with
\be
e^{K^{-1}s^{-1}r}\leq 2.
\label{mtf:req}\ee
Then $k\|B\|/2\leq B_0\leq \|B\|$ and
$$
r_{\max} \leq C_1s(\|B\|,Z)
$$
if $\|B\|>CZ^{4/3}$ for some $C,C_1>0$ depending
on $k$. If we choose $r=C_1s$ we
have satisfied (\ref{mtf:req0}) and it is clear that if $K$ is large
enough then (\ref{mtf:req}) is also satisified.
We have thus proved that if $\|B\|>CZ^{4/3}$ then
$$
[V^{\rm MTF}(x)]_-=0
$$
if $|x|\geq C_1s$. Recalling the definition (\ref{mtf:veff})
of $V_{\rm eff}$ this identity implies (\ref{mtf:vbound})
if $\|B\|>CZ^{4/3}$.
We now turn to the case $\|B\|\leq CZ^{4/3}$. Since $\partial_2
P(B,W)\geq cW^{3/2}$, we see from (\ref{mtf:diffeq}) and
(\ref{mtf:equation1}) that for $x\ne0$,
$V^{\rm MTF}(x)$ satisfies
$$
-(4\pi)^{-1}\Delta V^{\rm MTF}(x)=\rho^{\rm MTF}(x)
=\partial_2P(B(x),[V^{\rm MTF}(x)]_-)\geq c [V^{\rm MTF}(x)]_-^{3/2}.
$$
Since $\Delta |x|^{-4} =c (|x|^{-4})^{3/2}$ it follows from a simple
comparison argument, using that $V^{\rm MTF}(x)\geq -c |x|^{-4}$
for small enough $|x|$ and that $V^{\rm MTF}(x)\geq
-Z|x|^{-1}\to0$ as $|x|\to\infty$,
that
$
V^{\rm MTF}(x)\geq -c|x|^{-4}
$
for all $x\ne 0$. This is true for all $B$. If we now use that
$\|B\|\leq CZ^{4/3}$ and hence
$s\geq CZ^{-1/3}$ it then also follows that $V_{\rm eff}(x)\geq
-C|x|^{-4}$. We have thus proved (\ref{mtf:vbound}).
The Thomas-Fermi equation (\ref{mtf:equation1}) implies that
$$
\rho^{\rm MTF}(x)=\partial_2P(B(x),[V^{\rm MTF}(x)]_-)
\leq c\|B\|[V^{\rm MTF}(x)]_-^{1/2}+
c[V^{\rm MTF}(x)]_-^{3/2}.
$$
If we insert the bound $[V^{\rm MTF}(x)]_-\leq c|x|^{-4}$
we obtain
\be
\rho_{\rm eff}(x)=s^3\rho^{\rm MTF}(sx)
\leq cs \|B\| |x|^{-2}+cs^{-3}|x|^{-6},
\label{mtf:rho1}\ee
while the bound $[V^{\rm MTF}(x)]_-
\leq Z|x|^{-1}$ gives
\be
\rho_{\rm eff}(x)\leq cs^{5/2}\|B\| Z^{1/2}|x|^{-1/2} +cs^{3/2}Z^{3/2}
|x|^{-3/2}.
\label{mtf:rho2}\ee
If $\|B\|\leq CZ^{4/3}$ we arrive at (\ref{mtf:rhobound}) using
the bound (\ref{mtf:rho1})
for large $|x|$ and (\ref{mtf:rho2}) for small $|x|$.
If $\|B\|\geq CZ^{4/3}$ we prove (\ref{mtf:rhobound}) using
(\ref{mtf:rho2}) and that, as proved above, $\rho_{\rm eff}(x)=0$ if
$|x|\geq C$.
In order to prove the bound on $\mu$ we observe from
(\ref{mtf:equation1}) and (\ref{mtf:equation2})
that $\rho^{\rm MTF}(x)=0$ if $Z|x|^{-1}\leq \mu$. Thus from
(\ref{mtf:rhobound}) we find that if $\mu\ne0$ then
$$
N=\int\rho^{\rm eff}\leq c Z\int_{|x|\leq \mu^{-1}Zs^{-1}}|x|^{-2}\rd x
=c\mu^{-1}Z^2 s^{-1},
$$
which implies (\ref{mtf:mubound}).
$\,\,\Box$
We note that the bound in Lemma~\ref{lm:vmurho} on $\rho_{\rm eff}$ is
not integrable. It follows, however, from Theorem~4.9 in
\cite{LSY-II} that
$Z^{-1}\int
\rho_{\rm eff}(x)
\rd x\leq 1$. In fact, it follows from the proof of that theorem that
\be
Z^{-1}\rho_{\rm eff}*|x|^{-1}\leq |x|^{-1}.
\label{mtf:lsy49}\ee
We shall now prove a stronger bound than (\ref{mtf:lsy49})
for small $|x|$.
\begin{lemma} With the same assumptions as in
Lemma~\ref{lm:vmurho} and if $a>0$, we obtain the estimates
\be
Z^{-1}\rho_{\rm eff}*|x|^{-1}\leq C\min\{1, |x|^{-1}\}
\label{mtf:rho*bound}\ee
and
\be
\int\left|\nabla(Z^{-1}\Theta_R(x)\rho_{\rm eff}*\varphi_a*|x|^{-1})
\right|\rd x\leq CR^2\ \ \hbox{and}\
\int\left|\nabla(Z^{-1}\Theta_R(x)\rho_{\rm eff}*|x|^{-1})
\right|\rd x\leq CR^2.
\label{mtf:nablarhobound}\ee
\end{lemma}
{\it Proof:} Considering (\ref{mtf:lsy49}) it is enough, in
order to prove (\ref{mtf:rho*bound}), to
show that $Z^{-1}\rho_{\rm eff}*|x|^{-1}\leq C$ for $|x|\leq 1$.
Using (\ref{mtf:rhobound}) and $\int \rho_{\rm eff}\leq N$ we find
for $|x|\leq 1$
$$
Z^{-1}\rho_{\rm eff}*|x|^{-1}\leq C\int_{|y|\leq 2}
|y|^{-3/2}|x-y|^{-1}\rd y + N/Z\leq C.
$$
To prove (\ref{mtf:nablarhobound}) we write
$$
\int\left|\nabla(Z^{-1}\Theta_R(x)\rho_{\rm eff}*|x|^{-1})
\right|\rd x\leq R^{-1}Z^{-1}\int_{|x|<2R}\rho_{\rm eff}*|x|^{-1}\rd x
+Z^{-1}\int_{|x|<2R}\rho_{\rm eff}*|x|^{-2}\rd x.
$$
Inserting the bounds
(\ref{mtf:rhobound}) and (\ref{mtf:rho*bound}) we obtain
(\ref{mtf:nablarhobound}). Note that bounds
similar to (\ref{mtf:rhobound}) and
(\ref{mtf:rho*bound}), possibly with different constants, hold also if $\rho_{\rm eff}$
is replaced by $\rho_{\rm eff}*\varphi_a$. In fact, to prove (\ref{mtf:rho*bound})
for $\rho_{\rm eff}*\varphi_a$ (with the same constant)
simply note that $\rho_{\rm eff}*|x|^{-1}$
is superharmonic. To prove (\ref{mtf:rhobound}) for $\rho_{\rm eff}*\varphi_a$
one simply computes the convolution on both sides of (\ref{mtf:rhobound}).
$\,\,\Box$
We are now ready to control the quantities in
(\ref{con1.1}--\ref{con2}).
\begin{lemma}\label{lm:C-cond} There exists a
constant $C>0$ (depending on only $k$) such that if $R>1$ and $a<1$ we
have the estimates
\begin{eqnarray}
|E_{scl}(h,B_{\rm eff}, V_{a,R}^-)|&\geq& C (\|B_{\rm
eff}\|h^{-2}+h^{-3})\label{mtf:C-low}
\\
|E_{scl}(h,B_{\rm eff}, \Theta_RV_{\rm eff})|&\geq& C (\|B_{\rm
eff}\|h^{-2}+h^{-3})\label{mtf:C-up}
\\
\cE_{h,\bB_{\rm eff}}([V_{a,R}^-]_-)&\leq &C (\|B_{\rm
eff}\|h^{-2}+h^{-3})\label{mtf:C+low}
\\
\noalign{\hbox{and }}\nonumber\\
\cE_{h,\bB_{\rm eff}}([\Theta_RV_{\rm eff}]_-)
\leq \cE_{h,\bB_{\rm eff}}([V_{\rm eff}]_-)&\leq& C
(\|B_{\rm eff}\|h^{-2}+h^{-3}).\label{mtf:C+up}
\end{eqnarray}
\end{lemma}
{\it Proof:} Since $\rho_{\rm eff}*|x|^{-1}$ is superharmonic
we have $\rho_{\rm eff}*\varphi_a*|x|^{-1}\leq
\rho_{\rm eff}*|x|^{-1}$. Hence
$
[V_{a,R}^-]_-\geq [V_{\rm eff}]_-\Theta_R.
$
Using (\ref{mtf:rho*bound}) and (\ref{mtf:mubound}) and
recalling that $N/Z$ is bounded away from zero
we see that for $|x|c a$.
Since $0<|x|^{-1}-\varphi_a*|x|^{-1}\leq |x|^{-1}$
the estimate (\ref{mtf:C+low}) follows immediately.
$\,\,\Box$
\begin{lemma}\label{lm:epsilon-cond} If $R>1$ and $a<1$ then
both for $V=V_{a,R}^-$ and $V=V_{\rm eff}\Theta_R$ we have
\begin{eqnarray}
\frac{\cE_{h,\bB_{\rm eff}}([V]_\pm-[V(\cdot-y)]_\pm)}
{\|B_{\rm eff}\|h^{-2}+h^{-3}}&\leq& C\left(|y|^{1/2}
+|y|^{3/2}\left(1+|\ln(|y|/R)|\right)\right).\label{mtf:cEy}
\\ \nonumber\\
\noalign{\noindent Likewise,}\nonumber\\
\frac{\cF_{h,\bB_{\rm eff}}([V]_\pm-[V(\cdot-y)]_\pm)}
{\|B_{\rm eff}\|h^{-2}+h^{-3}}&\leq&
C\Bigl(|y|^{1/2}
+|y|^{3/2}\left(1+|\ln(|y|/R)|\right)
+|y|R+h R^2\Bigr).
\label{mtf:cFy}
\end{eqnarray}
\end{lemma}
{\it Proof:}
Note that for all $V$, $\left|[V(x)]_\pm-[V(x-y)]_\pm\right|
\leq \left|V(x)-V(x-y)\right|$.
Using the simple case, $\|u*v\|_p\leq\|u\|_1\|v\|_p$,
of Young's inequality for $p=1$, $p=3/2$ or
$p=5/2$ we find for both cases $V=V_{a,R}^-$ and
$V=\Theta_RV_{\rm eff}$ that
\begin{eqnarray}
\|V(\cdot)-V(\cdot-y)\|_p^p
&\leq&(1+(N/Z))^p
\left(\left\|\Theta_R(\cdot)|\cdot|^{-1}-\Theta_R(\cdot-y)
|\cdot-y|^{-1}\right\|_p^p\right)\nonumber\\
&\leq&C\cases{|y|R,& if $p=1$\cr
|y|^{1/2},& if $p=5/2$\cr
|y|^{3/2}\left(1+|\ln(|y|/R)|\right),& if
$p=3/2$}\quad.\label{mtf:pnorm}
\end{eqnarray}
This gives (\ref{mtf:cEy}).
We next turn to the estimates on $\cF_{h,\bB_{\rm eff}}$.
First we note that the requirements (\ref{mtf:scasp})
on $\bB_{\rm eff}$, $l(\bB_{\rm eff})$ and $L(\bB_{\rm eff})$
imply that
$
d(h,\bB_{\rm eff})^{-1}\leq Ch^{-1}.
$
Thus for all $W$
$$
\cF_{h,\bB_{\rm eff}}(W)\leq C\left(\cE_{h,\bB_{\rm eff}}(W)
+(\|B_{\rm eff}\|h^{-2}+h^{-3})\int|W|
+(\|B_{\rm eff}\|h^{-2}+h^{-3}) h \int|\nabla W|
\right).
$$
In order to prove (\ref{mtf:cFy}) it therefore remains
to control $\int|V(x)-V(x-y)|\rd x$ and
$\int|\nabla V(x)-\nabla V(x-y)|\rd x$ for the
two cases $V=V_{a,R}^-$ and $V=\Theta_RV_{\rm eff}$.
The first integral was controlled in (\ref{mtf:pnorm}).
For the gradient we use (\ref{mtf:nablarhobound})
and the trivial estimate $\int_{|x|\leq 2R}|\nabla |x|^{-1}|\rd x
\leq CR^2$ to arrive at
$$
\int\left|\nabla V(x)-\nabla V(x-y)\right|\rd x\leq
\int\left(|\nabla V(x)|+|\nabla V(x-y)|\right)\rd x\leq CR^2,
$$
in both cases.
$\,\,\Box$
\begin{corollary}\label{cl:mtfbound}
There exist constants $C_\pm>0$ (depending only on $k$) such that
the MTF energy satisfies
\be
-C_-\left(\|B_{\rm eff}\|h^{-2}+h^{-3}\right)\geq Z^{-1}s
E^{\rm MTF}(N,B,Z)\geq -C_+
\left(\|B_{\rm eff}\|h^{-2}+h^{-3}\right).
\label{mtf:energybound}\ee
Note, in particular, that $E^{\rm MTF}(N,B,Z)$ is negative.
\end{corollary}
{\it Proof:} Recall that according to (\ref{mtf:leadingorder}),
$\|B_{\rm eff}\|h^{-2}+h^{-3}\sim Z$.
We shall use the expression (\ref{mtf:escale}) for $E^{\rm MTF}$.
{F}rom (\ref{mtf:vbound}) and (\ref{mtf:C-up}) we find that
$$
0Zs^{-1}\mu^{-1}$. Thus if $R/2\geq Zs^{-1}\mu^{-1}$
we get from (\ref{mtf:mubound})
$$
\Delta^+_1E\leq \tilde{\varepsilon}(h,\bB_{\rm eff})Z^{-1}s\mu
\int\rho_{\rm eff}
\leq \tilde{\varepsilon}(h,\bB_{\rm eff}) C_0(Z/N) \int\rho_{\rm eff}
\leq C \tilde{\varepsilon}(h,\bB_{\rm eff}) Z,
$$
where in the last inequality we used that $\int\rho_{\rm eff}\leq Z$
and the
assumption that
$N/Z$ is bounded below.
On the other hand if $R/2\leq Zs^{-1}\mu^{-1}$, i.e.,
if $\mu s Z^{-1}\leq 2R^{-1}$ then (assuming
$\tilde\varepsilon(h,\bB_{\rm eff})\leq 1$)
$\Delta^+_1E\leq 2 R^{-1} Z$.
Thus we have proved that
$$
\Delta^+_1E\leq CZ\min\left\{\tilde{\varepsilon}(h,\bB_{\rm eff}),
R^{-1}\right\}.
$$
Recalling (\ref{mtf:leadingorder}) and
$\tilde{\varepsilon}(h,\bB_{\rm eff})\to0$ as $n\to\infty$
we conclude (\ref{mtf:upgoal}) for
$\Delta^+_1E$.
It remains to consider $\Delta^+_2E$.
Assuming that $\tilde\varepsilon(h,\bB_{\rm eff})<1$ and $\lambda \leq R/2$
\begin{eqnarray*}
\Delta^+_2E&\leq &Z^{-1}\iint_{|y|\geq R/2}
\rho_{\rm eff}(x)|x-y|^{-1}\rho_{\rm eff}(y)\rd x\rd y
\\
&&+Z^{-1}\left|\iint\rho_{\rm eff}(x)|x-y|^{-1}\left((\chi_{R/2}
\rho_{\rm eff})(y)
-(\chi_{R/2}\rho_{\rm eff})*\theta_\lambda^2(y)\right)\rd x\rd y\right|
+C\tilde\varepsilon(h,\bB_{\rm eff})Z,
\end{eqnarray*}
where we estimated the last term in $\Delta^+_2E$
using (\ref{mtf:rho*bound}) and $\int\rho_{\rm eff}\leq Z$.
{F}rom (\ref{mtf:rho*bound}) and $\int\rho_{\rm eff}\leq Z$
we also see that
$$
Z^{-1}\iint_{|y|\geq R/2}
\rho_{\rm eff}(x)|x-y|^{-1}\rho_{\rm eff}(y)\rd x\rd y
\leq CZR^{-1}.
$$
Finally, using (\ref{mtf:rhobound}) we see that
$\|Z^{-1}\rho_{\rm eff}\|_p2\lambda$ and
is bounded by $|x|^{-1}$ for $|x|\leq 2\lambda$.
Putting these estimates together gives
$$
\Delta^+_2E\leq C Z\left( R^{-1} +\lambda^2
+\tilde\varepsilon(h,\bB_{\rm eff})\right).
$$
Since $\tilde\varepsilon(h,\bB_{\rm eff})\to0$
and $\lambda\to0$ as $n\to0$
we see from (\ref{mtf:leadingorder}) that (\ref{mtf:upgoal}) holds
also for
$\Delta^+_2E$.
$\,\,\Box$
\appendix
\section{The geometry of the three dimensional magnetic field}
\setcounter{equation}{0}
In this Appendix we recall two results from
\cite{ES-I} related to the geometry of a non-homogeneous
three dimensional magnetic field. Here we just give the
statements and the necessary notations for the reader's
convenience, the proofs are found in \cite{ES-I}.
The following proposition will be used to approximate a general
magnetic field by a {\it constant direction} field.
We recall the definitions $l(\bB)^{-1}=\|\nabla (\bB/B)\|$
and $L(\bB)^{-1}=\||\nabla B|/B\|$.
\begin{proposition}\label{Bappr} Consider an arbitrary cube
$\Omega\subset \bR^3$ with center $Q$ and edge length $\lambda$ and
a nonvanishing $C^1$ magnetic (divergence free) field
$\bB:{\bf R}^3\to\bR^3$.
Assume that
\be (6+3\sqrt{3})\lambda l(\bB)^{-1} \leq 1.
\label{const}\ee Then there exists a magnetic (divergence free)
field $\tilde\bB$, with constant direction parallel to the field at
the center $Q$ of $\Omega$, such that for all $x\in\Omega$
\be |\bB(x)-\tilde\bB (x)|
\leq \lambda l(\bB)^{-1} \left\{\sup_{|x-Q|\leq
5\lambda}B(x)\right\}\left(
\frac{\sqrt{3} + 4\sqrt{6}}{\sqrt{2}} +
\frac{6+3\sqrt{3}}{2}\lambda\left(l(\bB)^{-1}+L(\bB)^{-1}\right)\right)
\label{Btilde}\ee and
\be |\nabla \tilde \bB(x) |\leq |\nabla \bB(x) |\leq
\left\{\sup_{|x-Q|\leq 5\lambda}B(x)\right\}(L(\bB)^{-1}+
l(\bB)^{-1}).
\label{derest}\ee
\end{proposition}
\medskip
{\it Remarks.} (i) The assumption (\ref{const}) is a geometric
condition, which states that the field lines of the field $\bB$
should not vary too fast over the scale of the cube.
(ii)
In our application,
where typically $l\gg \lambda$, the approximation in (\ref{Btilde})
will be better than the straightforward choice $\tilde\bB(x) :=
\bB(Q)$ (constant field),
since that would yield only $|\bB(x) - \bB(Q)|\leq
(\sqrt{3}/2)\lambda\sup|\nabla\bB|$, which is of order
$\sup|B|\lambda (l^{-1}+L^{-1})$. This is worse by a factor of
$l\lambda^{-1} \gg 1$ than the similar term in (\ref{Btilde}).
\medskip
We shall, indeed, also need approximations of the
magnetic field by a constant field and not just a constant
direction field. In order to keep the same accuracy in the
approximation we must restrict to a smaller region. It turns
out that we can cover the cube by parallel cylinders such that within
each of these
we, without losing in the approximation, can approximate the
magnetic field by a constant field along the cylinder axis.
To formulate this more precisely we choose an orthonormal
coordinate system $\{ \xi_i \}_{i=1}^3$ in $\bR^3$, such that
the center $Q$ of the cube $\Omega$ is the origin and that
$\bB(Q)=\bB(0)$ points in the positive third direction. Note
that the sides of $\Omega$ need not be parallel with the
coordinate planes in this new coordinate system. We shall
refer to the plane
${\cal P}:= \{ \, \xi \, : \, \xi_3 =0 \}$ as the base plane
of the cube. We consider cylinders,
$C_P$, given in this new coordinate system by
\be C_P = \{ \, \xi \, :\, |\xi_{\perp} - P| \leq w\,\
|\xi_3|\leq\sqrt{3}\lambda/2 \},
\label{Cyl}\ee where
$P\in{\cal P}$ and $w>0$ (here $\xi_{\perp}:= (\xi_1, \xi_2,
0)$). The point $P$ is called the center of the cylinder.
Note that the cylinders are aligned along
$\bB(Q)$, the magnetic field at the center of the cube $\Omega$ and
that the union of all these cylinders covers $\Omega$.
Moreover, all the cylinders $C_P$ such that
$C_P\cap\Omega\ne\emptyset$
are subsets of the larger cube $\Omega'$, that, in the
new coordinate system, is defined by $[-w-\sqrt{3}\lambda/2,
w+\sqrt{3}\lambda/2]^3$.
\medskip
\begin{corollary}\label{apprcor}
Let $\Omega$, $\Omega'$ and $C_P$, for $P\in{\cal P}$ be as
defined above.
Assume that the magnetic field $\bB$ satisfies
\be (6+3\sqrt{3})(2w+\sqrt{3}\lambda) l(\bB)^{-1}
\leq 1.
\label{constnew}\ee
Then within each $C_P$ such that $C_P\cap\Omega\ne\emptyset$,
one can approximate the magnetic field $\bB$
by a {\it constant} field, $\tilde \bB_P$ pointing along the
axis of the cylinder,
with the following precision
\be
{|\bB (x) - \tilde\bB_P|\leq}\label{coreq}\ee
$$
(2w+\sqrt{3}\lambda) l(\bB)^{-1}\left\{\sup_{|x-Q|<5(\sqrt{3}\lambda+2w)}
B(x)\right\}
\Biggl(
\frac{\sqrt{3} + 4\sqrt{6}}{\sqrt{2}}+
\frac{6+3\sqrt{3}}{2}(2w+\sqrt{3}\lambda)
\left(l(\bB)^{-1}+L(\bB)^{-1}\right) \Biggr)
$$
$$
+ w \left\{\sup_{|x-Q|<5(\sqrt{3}\lambda+2w)}B(x)\right\}\left(l(\bB)^{-1}
+L(\bB)^{-1}\right),
$$
for $x\in C_P$.
\end{corollary}
\medskip
{\it Remark.}
Note that it is only the radius of the cylinder
that appears in the
last term in (\ref{coreq}). This is important since in our applications,
typically $w \ll \lambda$, i.e., the cylinder is very thin
compared to the cube.
The size $\lambda$ of the cube appears only together with
$l(\bB)^{-1}$ which,
in our setup, will typically be small. It is in this way that
we will achieve that the constant field approximation
within $C_P$ is as good as the constant direction field approximation within
$\Omega$.
The corollary of the
previous proposition will provide us with a good approximating
constant field within a cylinder (see (\ref{Cyl})).
Here we show that the difference field
(which is supposed to be small) can be generated by a
small vector potential within this cylinder. In general, if one is
given a magnetic field within a domain, then there exists a
vector potential bounded by the supremum of the field times
the largest linear size of the domain (see (\ref{Poinest}) below).
For instance, one
can choose the gauge given by the Poincar\'e formula (see below).
This gives a very
crude bound for domains which are elongated cylinders.
The crucial fact is that, assuming some bound on the first derivative
of the field in addition to its supremum bound, one can
choose a gauge independent of the longest linear size of the domain.
In particular, we can choose a gauge within our cylinder which
is bounded by a constant independent of the length of the
cylinder.
\begin{proposition}\label{Aprop}
Given a $C^1$ magnetic field $\beta: \bR^3\to\bR^3$
and consider a cylinder $C$ with radius $w$.
Then there exists a vector potential
$\alpha: \bR^3\to\bR^3$,
such that $\nabla\times\alpha =\beta$ and
\be
\sup_C \Vert \alpha \Vert \leq 4(w\sup_C \Vert \beta \Vert
+ w^2 \sup_C \Vert \nabla \beta \Vert ).
\label{Aest}\ee
The bound is uniform in the length of the cylinder.
\end{proposition}
\medskip
{\it Remark.} For comparison, the Poincar\'e formula
\be
\alpha (y) = \int_0^1 t (\beta (ty)\times y) \rd t
\ee
obviously
yields a gauge $\alpha$, for any domain $D$, satisfying the bound
\be
\sup_D \Vert \alpha \Vert \leq \sqrt{3}\sup_D \Vert \beta \Vert \cdot
\mbox{diam}(D).
\label{Poinest}\ee
\bigskip\noindent
{\it Acknowledgements.} L. E. gratefully acknowledges financial
support from the Forschungsinstitut f\"ur Mathematik, ETH, Z\"urich,
where this work was started. He is
also grateful for the hospitality and support of Aarhus University
during his visits.
\begin{thebibliography}{hhhhhhhhhh}
\bibitem[AC]{AC} Y. Aharonov and A. Casher, {\em Ground state
of spin-1/2 charged particle in a two-dimensional magnetic field. \/}
Phys. Rev. {\bf A19} (1979), 2461-2462.
\bibitem[CdV]{CdV} Y. Colin de Verdi\'ere, {\em L'asymptotique de Weyl pour
les bouteilles magn\'etiques. \/} Commun. Math. Phys. {\bf 105} (1986),
327-335.
\bibitem[CFKS]{CFKS} H. L. Cycon, R. G. Froese, W. Kirsch and B. Simon,
{\em Schr\"odinger Operators with Application to Quantum Mechanics
and Global Geometry. \/} Springer-Verlag, 1987.
\bibitem[E-1995]{E-1995} L. Erd\H os, {\em Magnetic Lieb-Thirring
inequalities. \/} Commun. Math. Phys. {\bf 170} (1995), 629-668.
\bibitem[ES-I]{ES-I} L. Erd\H os and J. P, Solovej, {\em Semiclassical
eigenvalue estimates for the Pauli operator
with strong non-homogeneous magnetic field.
I. Non-asymptotic Lieb-Thirring type estimate. \/} Preprint, 1996.
\bibitem[H]{H} B. Helffer, {\em Semiclassical analysis for
the Schr\"odinger operator and applications. \/} Lecture Notes in Math.,
{\bf 1336}, Springer, Berlin, 1988.
\bibitem[I]{I} V. Ivrii, {\em Semiclassical microlocal
analysis and precise spectral asymptotics. \/} Preprints
of Centre de Math\'ematiques, Ecole Polytechnique.
\bibitem[L-1981]{L-1981} E. H. Lieb, {\em A variational principle for
many-fermion systems. \/} Phys. Rev. Lett. {\bf 46} (1981), 457-459.
Erratum {\bf 47} (1981), 69.
\bibitem[LO]{LO} E. H. Lieb and S. Oxford, {\em Improved lower
bound on the indirect Coulomb energy. \/} Int. J. Quant. Chem.
{\bf 19} (1981), 427-439.
\bibitem[LSY-II]{LSY-II} E. H. Lieb, J. P. Solovej and J. Yngvason,
{\em Asymptotics of heavy atoms in high magnetic fields: II. Semiclassical
regions. \/} Commun. Math. Phys. {\bf 161} (1994), 77-124.
\bibitem[LSY-III]{LSY-III} E. H. Lieb, J. P. Solovej and J. Yngvason,
{\em Ground states of large quantum dots in magnetic fields. \/}
Phys. Rev. B {\bf 51} (1995), 10646-10665.
\bibitem[Mat-1994]{Mat-1994} H. Matsumoto, {\em Semiclassical
asymptotics of eigenvalue
distributions for Schr\"o\-dinger operators with magnetic fields. \/}
Commun. in PDE. {\bf 19 (5/6)} (1994), 719-759.
\bibitem[R]{R} D. Robert, {\em Autour de l'Approximation
Semiclassique. \/} Progr. Math. {\bf 68}, Birkh\"auser, Boston, 1987.
\bibitem[Sob-1986]{Sob-1986} A. Sobolev, {\em Asymptotic
behavior of the energy levels of a quantum
particle in a homogeneous magnetic field, perturbed by a decreasing
electric field. \/} J. Sov. Math. {\bf 35} (1986), 2201--2212.
\bibitem[Sob-1994]{Sob-1994} A. Sobolev, {\em The quasi-classical
asymptotics of local Riesz means for the Schr\"odinger operator
in a strong homogeneous magnetic field. \/} Duke J. Math. {\bf 74} (1994),
319-428.
\bibitem[Sob-1995]{Sob-1995} A. Sobolev, {\em Quasi-classical asymptotics
of local Riesz means for the Schr\"odinger operator in a moderate
magnetic field. \/} Ann. Inst. H. Poincar\'e Phys. Th\'eor.
{\bf 62} (1995) no.4, 325-360.
\bibitem[Sob-1996(1)]{Sob-1996(1)}
A. Sobolev, {\em On the Lieb-Thirring estimates
for the Pauli operator. \/} To appear in Duke J. Math. (1996)
\bibitem[Sol]{Sol} S. N. Solnyshkin,
{\em The asymptotic behavior of the energy
of bound states of the Schr\"odinger operator in the presence of electric
and magnetic fields. \/} Probl. Mat. Fiz. {\bf 10} (1982), 266--278.
\bibitem[T]{T} H. Tamura, {\em Asymptotic distribution of
eigenvalues for Schr\"odinger operators with magnetic fields. \/}
Nagoya Math. J. {\bf 105} no. 10 (1987), 49-69.
\end{thebibliography}
\end{document}
**