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\title{\bf Semiclassical eigenvalue estimates for the Pauli operator
with strong non-homogeneous magnetic fields\\ \bigskip
\large \bf I. Non-asymptotic Lieb-Thirring type estimate}
\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 19, 1996}
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\begin{document}
\maketitle
\begin{abstract}
We give the first Lieb-Thirring type estimate on the sum of the
negative eigenvalues of the Pauli operator that
behaves as the corresponding semiclassical expression
even in the case of strong non-homogeneous magnetic fields.
This enables us, in the companion paper \cite{ES-II},
to obtain the leading order semiclassical eigenvalue asymptotic,
which, in turn, leads to the proof of the validity of the
magnetic Thomas-Fermi theory of \cite{LSY-II}. Our
work generalizes the results of \cite{LSY-II} to non-homogeneous
magnetic fields.
\end{abstract}
\vfill\pagebreak
\tableofcontents
\bigskip
\section{Introduction}
In this paper and its companion \cite{ES-II} we shall study the
semiclassical limit of the Pauli operator
with both electric and magnetic fields. Our main concern
is to allow for non-homogeneous magnetic fields. As we hope to
illustrate, the transition from homogeneous to non-homogeneous
fields is highly non-trivial.
This is so not only because of technical difficulties
but also because the case of non-homogeneous field is qualitatively
different.
We shall be concerned here mainly with dimension three. In the follow up
paper we will consider also 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 $-i\nabla=\bp$.
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 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
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,
the location of the essential spectrum can be very different for the
two operators
(recall that the physically most interesting eigenvalues are those below the
essential spectrum).
Typically, the bottom
of the essential spectrum for the Schr\"odinger operator
depends on the magnetic field.
But it turns out that, under very general circumstances,
the essential spectrum for the Pauli operator
in both two and three dimensions
starts at zero. It was proven in \cite{HNW} for $V=0$, but
the proof works as well for potentials vanishing at infinity.
Therefore we can simply study the negative eigenvalues.
For the Schr\"odinger operator the right question would be
to study the eigenvalues below the bottom of the essential
spectrum, but since the essential spectrum is in general
not known a-priori (for a constant magnetic
field it is known) these eigenvalues are very difficult to locate.
Note that it is a bonus from Nature that the Pauli operator,
which describes the electron correctly, including its spin, behaves
in a more stable way.
The physically as well as mathematically interesting
quantities connected with the eigenvalues are the number
and the sum of 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
(e.g. \cite{Sol}, \cite{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. In fact, the
asymptotics of how the number of eigenvalues
accumulate near the essential spectrum is studied in
\cite{I, Sob-1986}. 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. In fact, \cite{LSY-II,LSY-III} establish
Lieb-Thirring type estimates on the eigenvalue sum which only require
the negative part of the potential to be in appropriate $L^p$ spaces.
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
$\bB$. 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.
Note that it is a great simplification that we, as discussed above,
can concentrate on the eigenvalues in a fixed interval
(the negative eigenvalues)
and not have to consider eigenvalues in a $\bB$ dependent interval,
as it would be the case for the 'magnetic' Schr\"odinger operator.
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 and its continuation 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.
First we would like to motivate the importance of studying these issues.
Historically, magnetic fields have occured as small perturbations in
the Hamiltonian, e.g., as in the standard treatment of the Zeeman
effect. In later years, however, the investigation of strong magnetic
fields has become relevant in important physical systems.
In the early 70's it became clear to astrophysicists that the strong
magnetic fields expected to exist on a neutron star would have a
significant influence on the structure of the atoms existing on the
surface of the star. This was one of the main motivations for the study in
\cite{LSY-II}. (A fairly extensive reference to the physical literature
can be found in \cite{LSY-II}).
In the 80's strong magnetic fields became of importance even in
laboratory experiments. This occured as a result of new experimental
techniques which allowed one to study effectively two dimensional systems,
e.g., the quantum Hall experiments and the heterostructures such
as quantum dots. In these systems the electrons
move in crystal structures where their effective mass and charge
change
to such an extent that laboratory magnetic fields become effectively
very strong and can no longer be treated perturbatively.
In all of these systems it is a good approximation
to consider the magnetic field as homogeneous.
There are several reasons 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 estimates allowing one to control
errors and contributions coming from non-semiclassical regions. These
estimates, which for the sum of the eigenvalues are often
refered to as Lieb-Thirring estimates, show that the sum of the
negative eigenvalues is finite and can be controlled from below by
expressions, which behave like the semiclassical formula one would like
to prove. They need, however, not be asymptotically exact since they are
only used for controlling errors.
The second part of the semiclassical study is to show that,
when all the errors have been controlled, one can indeed
get the asymptotic formulas.
Our approach to the problem, which we treat in the continuation
to this paper, is the coherent state method also
used in \cite{LSY-II,LSY-III}.
The present paper treats the somewhat more
fundamental problem of establishing a Lieb-Thirring estimate.
This part requires many novel ideas and most of them will also be
needed in the subsequent asymptotic study of the second paper.
Having
proved Lieb-Thirring estimates, one can prove leading order
semiclassics with very weak regularity assumptions on the potential
and the magnetic field. In particular, we can treat the case of atomic
potentials with the Coulomb singularity. The explicit treatment of the
leading order energy for large atoms in strong non-homogeneous
magnetic fields can be found in the second paper, where we prove,
as an application of the
semiclassical result, that
the `Magnetic Thomas Fermi (MTF) theory' introduced even for
non-homogeneous fields in
\cite{LSY-II} gives the asymptotically correct ground state energy of
atoms for large nuclear charge.
The first generalizations of the Lieb-Thirring inequality
of \cite{LSY-II} and \cite{LSY-III}
to the case of non-homogeneous magnetic fields
were first given in \cite{E-1995} and later improved in
\cite{Sob-1996(1)} and \cite{Sob-1996(2)}.
These works were mainly concerned with allowing fields growing at
infinity. However, for the purpose of applications in semiclassical
proofs, it is more important that the size of the Lieb-Thirring estimate
be comparable with the semiclassical expression.
In the case of dimension two and in the
case of constant {\it direction} field in dimension three,
the estimates obtained in \cite{E-1995},
\cite{Sob-1996(1)} and \cite{Sob-1996(2)} are adequate
for the purpose of studying the semiclassical limit. For three
dimensions in general, none of the previously established
Lieb-Thirring estimates
(including the one in \cite{LLS}) can be applied
to the semiclassical problem, because none of them scale correctly
for large magnetic fields.
The purpose of the present paper is to establish an estimate which is
adequate in three dimensions.
The issue is somewhat complicated and we shall now try to
illucidate it.
The three dimensional case is complicated by the fact
that there exist fields $\bB$ (nonconstant direction)
such that
the corresponding Pauli-operator with $V=0$, although non-negative,
can have eigenfunctions with eigenvalue zero, we shall refer to
them as zero modes.
Examples of such fields were
given in \cite{LY}. The same is true in two dimensions,
but there these eigenfunctions are expected from the index theorem
or more precisely the Aharonov-Casher Theorem (see
\cite{CFKS}). Even in two dimensions the zero modes make
the problem non-trivial, but at least there one has
a control on the density of zero modes (see \cite{E-1993}).
In three dimensions the issue is much more complicated.
{F}rom (\ref{ESC}) it is natural to conjecture
that the correct Lieb-Thirring estimate would bound the
sum of the negative eigenvalues from below by
$$
-Ch^{-3}\int_{\bR^3}|h\bB(x)|[V(x)]_-^{3/2}\rd x
-Ch^{-3}\int_{\bR^3} [V(x)]_-^{5/2}\rd x.
$$
This is indeed the result for constant magnetic field
proved in \cite{LSY-II}.
The existence of zero modes together with the variational principle,
however, immediately implies that such an estimate is {\it wrong} in
general. Thus there is a fundamental difference here
between
homogeneous and non-homogeneous fields. In fact,
if $u_1,\ldots$ denote the (orthonormalized) zero modes, i.e.,
if $\bsigma\cdot(h\bp + \bA (x))u_i=0$, then the variational
principle implies that the sum of the negative eigenvalues is bounded
above by
$$
\sum_i\langle u_i|H(h,\bA , V)|u_i\rangle=
\int_{\bR^3}\sum_i|u_i(x)|^2V(x)\rd x.
$$
Note that for small $V$ this expression can in general not be
dominated by the two terms above. The Lieb-Thirring inequality
must therefore contain a term of the form
$$
-\int_{\bR^3}n(x) [V(x)]_-\rd x,
$$
where $n(x)$ is an upper bound to, $\sum_i|u_i(x)|^2$,
the density of states of zero modes.
We shall not go into details about the results proved in
\cite{E-1995,Sob-1996(1),Sob-1996(2),LLS}, but just remark that
the function $n(x)$ in all of these behaves qualitatively like
$|\bB(x)|^{3/2}$.
Indeed, $|\bB(x)|^{-1/2}$ has the dimension of a length and therefore
$|\bB(x)|^{3/2}$ behaves dimensionally as a density, i.e.,
like a negative third power of a length.
Of course the function $n(x)$, which appears must have the
the correct dimensionality. [When we say that $|\bB(x)|^{-1/2}$
has the dimension of a length, we of course simply
refer to how it behaves under scaling transformations.]
Now we immediately see that if we replace $n(x)$ by $|\bB(x)|^{3/2}$
then the above term has a completely different behavior than the
semiclassical expression. In fact, if we insert the semiclassical
parameter $h$ (i.e., replace $\bB$ by $h\bB$ and
multiply the eigenvalue sum with $h^{-3}$) we see that the extra term
above with
$n(x)\sim \|\bB\|^{3/2}$ would behave like $h^{-3} h^{3/2} \|B\|^{3/2}
=h^{-3/2}\|B\|^{3/2}$. It is, however, clear that (if, say,
$|\bB|$ is bounded above and below away from zero then)
$E_{scl}\sim h^{-2}\|B\|+h^{-3}$. It is therefore obvious that unless
$\|B\|\leq O(h^{-1})$ the above term will be bigger than
the semiclassical expression. Thus, to be able to derive the
semiclassical formula (\ref{ESC}) for $\|\bB\|\gg h^{-1}$, we
must prove a new Lieb-Thirring estimate. Recall, that
$\|B\|\sim h^{-1}$ is exactly the borderline for when the
semiclassical expression (\ref{ESC}) no longer behaves like the
Weyl term.
We shall throughout the paper
assume the following conditions on
the magnetic field ($\|\cdot\|$ denotes the supremum norm).
\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.
The potential $V$ is assumed to be in $L^{5/2}\cap W^{1,1}$, and
we restrict ourselves to the case when $V\geq 0$
(since the Lieb-Thirring
inequality is a lower bound on the eigenvalue sum one can always
replace $V$ by its negative part $-[V]_-$).
The following theorem, which we state for $h=1$, is the main result
of this paper.
\begin{theorem}\label{LTtheorem} With the notations above, there
exists a universal constant $C_1$
such that the following estimate is valid for the sum of the negative
eigenvalues
$\{ e^{(m)}_{\rm Pauli} \}_{m=1,2,\ldots}$ of the Pauli operator
$[\bsigma\cdot(\bp + \bA)]^2 - V$
\be
\sum_m |e^{(m)}_{\rm Pauli}|
\label{LT}\ee
\[
\leq C_1\left(\int_{{\bf R}^3}V^{5/2}+
\int_{{\bf R}^3}(B + d_0^{-2})V^{3/2} +
\int_{{\bf R}^3}(B+d_0^{-2})d_0^{-1} V +
\int_{{\bf R}^3}(B+d_0^{-2})|\nabla V| \right),
\] where
\be d_0:= \min \{ \|B\|^{-1/4}l^{1/2}, L, l \}.
\label{d0}\ee
\end{theorem}
{\it Remark 1. (Comparison with semiclassics)}
Inserting the semiclassical parameter $h<1$, assuming
$\|B\| \geq \max\{l^{-2},L^{-2}\}$, and bounding $B\leq \| B\|$,
we obtain a bound of order
\be
h^{-3}\int_{{\bf R}^3}V^{5/2} +
\|B\| h^{-2}\int_{{\bf R}^3}V^{3/2} +
\|B\|^{5/4}h^{-5/4}l^{-1/2}\int_{{\bf R}^3}V +
\|B\| h^{-1} \int_{{\bf R}^3}\left( |\nabla V| + V \right) .
\label{SC}\ee
The sum of the first two terms is exactly of the same
order of magnitude as the semiclassical expression.
The point is that the error terms (last two terms)
are smaller than the
semiclassical terms if
$$
\|B\|l(\bB)^{-2}=o(h^{-3})
$$
(assuming that $V$ and its first derivatives are fixed).
Thus this LT inequality will suffice to show the semiclassical
formula when this condition is satisfied.
This is the real application of our theorem and the methods are
developed to satisfy the needs of a semiclassical statement.
In particular, we obtain semiclassics uniformly in
$\|B\|$ for constant direction field ($l=\infty$).
The
reason for the restriction $\|B\|l(\bB)^{-2}=o(h^{-3})$ is the following.
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 quantum 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 $r_L=B^{-1/2}h^{1/2}$
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 $l^{-1}s^2\ll r_L$. This condition is simply that
$Bl^{-2}\ll h^{-3}e^2$.
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.
{\it Remark 2. (The last term)}
The last term above which depends on the gradient of $V$
is a conceptual error coming from our method and we have no reason to
believe that it could not possibly be removed.
It, however, scales in a way that it still allows for proving
the semiclassical asymptotics.
{\it Remark 3. (Zero modes)} In this Lieb-Thirring
inequality the function
corresponding to the density
$n(x)$ discussed above is the function
$(B(x)+d_0^{-2})d_0^{-1}$. In our inequality, however, this is
not the only term which is linear in $V$. The last gradient term
also has this property. We can therefore
not say that $(B(x)+d_0^{-2})d_0^{-1}$ bounds the
density of zero modes.
{\it Remark 4. (Constant field)}
The Lieb-Thirring inequality for a constant magnetic field
(\cite{LSY-II})
with our notations asserts the standard bound
\be
C_1\left(\int V^{5/2} + \int B \, V^{3/2}\right).
\label{constLT}\ee
To get a constant field, we have to let $l$ and $L$ go to infinity.
This reduces our bound to (\ref{constLT}) with the extra error term
of size $ \int B \, |\nabla V|$.
{\it Remark 5. (Large field.)} We know that if the field has a large gradient,
the standard estimate can be violated (see
\cite{E-1995}). This is reflected by the fact that
$d_0^{-1}$ blows up in (\ref{LT}) as $l^{-1}$ or $L^{-1}$ increases.
{\it Remark 6. (Constant direction field.)}
In case of the constant direction field with a uniform positive lower
bound, we expect the standard (\ref{constLT}) to hold. This case is
equivalent to
$l=\infty$ in our setup. This does not suppress the extra terms in our
theorem. We do not focus on this
point since the constant direction LT under the assumption
(\ref{asp1}) (no gradient bounds) has been proven by much simpler
methods in \cite{E-1995}. With more involved methods, the condition
(\ref{asp1}) was weakened first in \cite{E-1995}, then later
\cite{Sob-1996(2)} gave an essentially optimal result.
Before going into the proof of the Lieb-Thirring
estimate (\ref{LT}) we now state the semiclassical
results that are proved in the second paper \cite{ES-II}.
Proving these fundamental theorems is the real motivation for
establishing the Lieb-Thirring estimate of the present paper.
In the two dimensional case we have a completely uniform statement.
\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
$\|B \|<\infty$, $\|\, |\nabla B|/|B| \, \| <\infty$.
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}
\bigskip
As discussed above the three dimensional case requires a
condition on the magnetic field strength.
\begin{theorem}{\bf (3D Semiclassics)}\label{thm:3dsc}
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:3dsc}\ee
\end{theorem}
In fact, the results proved in \cite{ES-II} are somewhat stronger,
e.g., in the three dimensional case $V$ is allowed to
depend on the limiting parameter $\tau$
in a certain way. This generalization may seem to be an
unnecessary complication. It turns out, though,
that it is vital to the application in the theory of
atoms in magnetic fields.
In order to prove our Lieb-Thirring inequality we
localize in regions of space where we can approximate
by a constant magnetic field. In these regions we then
use standard Lieb-Thirring estimates. As already mentioned,
the optimal regions are cylinders parallel to the
magnetic field. Approximating the magnetic
field also requires approximating the vector potential.
This involves choosing optimal gauges locally.
One of the main ingredients in the proof is a particularly
good gauge choice in a cylindrical domain. The well known Poincar\'e
formula for choosing a gauge, is well suited for spherical
domains, but is not optimal in elongated domains.
We discuss approximating magnetic fields in
Section~\ref{sec:geometry} and the cylindrical gauge choice
is described in \ref{sec:gauge}.
When localizing we have to control two type of errors:
the localization errors (the energy it costs to localize)
and the approximation errors. Of course controlling
the localization error require choosing large regions, whereas
the approximation errors are small when the regions are small.
It turns out that we must do the localization in two steps.
The first localization uses essentially `isotropic' regions (cubes).
On these cubes we approximate the magnetic field, not by a constant
field, but only by a constant {\it direction} field.
This can in general be done on a larger region than if we were
approximating by a constant field immediately.
Since we now have a distinguished direction on each of these
cubes we can separate the subspaces corresponding to spin up and
spin down relative to this direction. This is
done in Section~\ref{cubsection}.
The spin up subspace is fairly easy to treat since the free
Pauli operator on this subspace can be controlled by the
Schr\"odinger operator, which is then in turn controlled by
the diamagnetic inequality. This is Section \ref{Pupsec}.
The spin down subspace is studied in Section~\ref{Pmin}.
Here we need to localize further into cylinders (which are typically
elongated ones, i.e., if $l\gg L$) and we approximate the magnetic field
by a {\it constant} field on each cylinder.
This is done by introducing Neumann boundary conditions
on the cylinders (due to the narrowness of the cylinders, Dirichlet
localization would cost too much). In each cylinder we then use a standard
supersymmetry argument to separate
the lowest Landau level from the higher levels. The higher levels
are then treated somewhat similarly to the spin up space.
The lowest level is studied using explicit estimates on the Neumann
ground state density. These estimates are derived in
Section~\ref{gsdens}. The main difficulty here is that the
Neumann density diverges near the boundary.
We get around this difficulty by choosing overlapping cylinders
allowing us in each cylinder to consider the potential
to be supported in what we call the cores, which
are away from the boundary.
The reader may wonder why we need two localizations. The main reason
is that localizing in narrow cylinders with
Dirichlet boundary conditions is very expensive.
We can only afford that on the orthogonal complement to
the lowest level. Fortunately, for the lowest
level the supersymmetry gives us additional information, so
the Neumann problem is actually tractable. This
requires, however, that we know the lowest level, and this we do
only if we have a constant direction field.
This delicate sequence of steps is characteristic for the problem,
one cannot afford being sloppy in the estimates. The structure
of the proof is very tight and in each step we have to use the
full strength of the estimate involved.
Finally, we explain how the errors are controlled. If we consider
a cube, where the potential is in some sense large (see the beginning
of Section~\ref{sec:split} for the precise notion)
we absorb the errors into the potential. If the potential
is small on a cube we do not need the full kinetic energy
to control it and we may simply ignore a fraction
of the kinetic energy thereby only keeping a fraction
of the errors. This simple, but crucial idea has been used
several times before in studying magnetic problems (see, e.g.,
\cite{F, FLL, LLS}).
Since the localization errors from one cube often
has to be controlled in the neighboring cubes
(allocation of errors), we must have
that the sizes of the potential in neighboring cubes are comparable.
In Section~\ref{sec:alloc} we show how to replace the original
potential with a potential with this property.
\section{The geometry of the magnetic field}\label{sec:geometry}
\setcounter{equation}{0}
In this section we prove two propositions. Their proofs are
a bit technical, but the statements carry the core of
our main proof. We suggest skipping the proofs in a first
reading.
\subsection{Approximating the magnetic field}\label{apprsec}
The following proposition will be used to approximate a general
magnetic field by a {\it constant direction} field. This
statement will be used twice; the crucial advantage of a
constant direction field is that it allows the separation of the
third direction and that it carries a natural supersymmetric
structure, which is essential to establish a spectral gap.
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
$\Omega$ 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
Before giving the proof of this proposition we first discuss
another approximation result, which will be a simple corollary
of the proposition. 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}
{\it Proof.} We apply the proposition to the large cube $\Omega'$.
Let $\tilde \bB$ denote the approximating constant direction field.
Within $\Omega'$ (in particular within $C_P$)
it satisfies (\ref{const}) with $\lambda$ replaced by
$2w+\sqrt{3}\lambda$. Since $\tilde\bB$ is divergence free, and points
along the third direction in the new coordinate system, we conclude that
$\tilde \bB$ is independent of $\xi_3$.
{F}rom (\ref{derest}) we see therefore that
$|\tilde \bB(x)-\tilde \bB(P)|\leq
\left\{\sup_{|x-Q|<5(\sqrt{3}\lambda+2w)}B(x)\right\}w$
for $x\in C_P$. Thus we
can choose $\tilde {\bf B}_P=\tilde {\bf B}(P)$. $\,\,\Box$
\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
$\Omega$.
\medskip
{\it Proof of Proposition \ref{Bappr}.}
We work in the coordinate system described before Corollary
\ref{apprcor}. Consider the larger cube $\tilde\Omega :=
[-\sqrt{3}\lambda/2, \sqrt{3}\lambda/2]^3$,
obviously $\Omega\subset\tilde\Omega$, but note that their
sides may not be parallel.
We define the constant direction (divergence free) field
\be
\tilde \bB(\xi) := (0,0, B_3(\xi_1, \xi_2, 0)).
\label{Btildedef}\ee
For the proof of the gradient bound (\ref{derest}), we note that
$|\nabla\tilde \bB|\leq |\nabla\bB|$ by construction, and
\be
|\nabla \bB | \leq |\nabla B| + |B| | \nabla (\bB / B)|
\label{derest1}\ee
pointwise.
Next, we show the estimate (\ref{Btilde}) for the coordinate plane
${\cal P}:=\{ \xi\, : \, \xi_3=0\}$.
We have, for $\xi\in{\cal P}$,
\be
| \bB(\xi) - \tilde\bB(\xi)| = \sup_{\Vert\bu\Vert =1, u_3=0}
(\bB(\xi) - \tilde\bB(\xi))\cdot \bu = \sup_{\Vert\bu\Vert =1, u_3=0}
\bB(\xi)\cdot\bu .
\label{supu}\ee
On the other hand, since $\bB(0) \cdot \bu =0$ if $u_3=0$,
\be
|\bB(\xi)\cdot\bu|
\leq B(\xi)\left|\left(\frac{\bB(\xi)}{B(\xi)}-\frac{\bB(0)}{B(0)}\right)
\cdot\bu\right| \leq \left\{\sup_{|\xi|\leq5\lambda}B(\xi)\right\} l(\bB)^{-1}
\lambda\sqrt{\frac{3}{2}},
\label{Bksi}\ee
using $|\xi| \leq \lambda\sqrt{3/2}$ which proves (\ref{Btilde}) for
points in $\xi\in{\cal P}\cap\tilde\Omega$.
Now we consider an arbitrary point $\xi \in \Omega$
with $\xi_3>0$ (the case $\xi_3<0$ is treated similarly),
and the arclength parametrized
field line curve $\gamma_{\xi}(t)$, which passes through
$\xi$ at time $t=0$,
and satisfies $\rd \gamma_{\xi}/\rd t = -(\bB/B) (\gamma_{\xi}(t))$.
Clearly, $|\gamma_{\xi} (t) - \gamma_{\xi} (0)| \leq t$.
We now show that
\be
|\gamma_{\xi}(t) - (\xi_1, \xi_2, \xi_3 - t)|
\leq t(3\lambda/2 +t)\Vert \nabla (\bB/B)\Vert.
\label{curve}\ee
In fact, $\eta (t) := \gamma_{\xi}(t) - (\xi_1, \xi_2, \xi_3 -t)$
satisfies $\eta (0)=0$, and
\[
|\rd\eta/\rd t|
\leq |(\bB/B)(\gamma_{\xi}) (t) - (0,0,1)|
\leq \|\nabla(\bB/B)\| |\gamma_{\xi}(t)|
\]
\be
\leq\Vert\nabla (\bB/B)\Vert (|\gamma_{\xi}(0)|+|\gamma_{\xi} (t) -
\gamma_{\xi} (0)|)
\leq (3\lambda/2 +t)l(\bB)^{-1} ,
\label{eta}\ee
where we used that $\bB/B(0)=(0,0,1)$.
Since we assumed (\ref{const}), there exists a time $t_{\xi} <
\sqrt{3}\lambda$ such that
$P_{\xi}:=\gamma_{\xi}(t_{\xi})\in{\cal P}$, since for short times
$t\leq \sqrt{3}\lambda$,
the points $\gamma_{\xi}(t)$ and $(\xi_1, \xi_2, \xi_3 -t)$
are still $(3+3\sqrt{3}/2)\lambda^2l(\bB)^{-1}
\leq \lambda/2$ close to each other, but at time
$t=\sqrt{3}\lambda$, the latter
point is already at least at a distance $\sqrt{3}\lambda/2> \lambda/2$
on the opposite side of ${\cal P}$ than $\xi$.
Note that for $0\leq t\leq t_\xi$ the curve $|\gamma_{\xi}(t)|\leq 5\lambda$.
Armed with these observations and recalling the notation $\xi_{\perp} =
(\xi_1, \xi_2, 0)$, we have
\be
|\bB (\xi) -\tilde\bB (\xi)|
\leq |\bB (\xi) - \bB(P_{\xi})|
+ |\bB (P_{\xi}) - \bB(\xi_{\perp})|
+ |\bB(\xi_{\perp})-
\tilde \bB (\xi_{\perp})|
\label{Bdecomp}\ee
since $\tilde\bB (\xi_{\perp})= \tilde\bB(\xi)$.
The last term has been treated above, since $\xi_{\perp}\in{\cal P}
\cap\tilde\Omega$.
The second term is bounded by
\[
\left\{\sup_{|\xi'|\leq5\lambda}|\nabla\bB(\xi')|\right\} |P_{\xi}-\xi_{\perp}|
\leq \left\{\sup_{|\xi'|\leq5\lambda}|\nabla\bB(\xi')|\right\}
t_{\xi}(3\lambda/2 +t_{\xi})\Vert \nabla (\bB/B)\Vert
\]
\be
\leq (3+3\sqrt{3}/2)\lambda^2
\left\{\sup_{|\xi'|\leq 5\lambda}B(\xi')\right\}(L(\bB)^{-1}+ l(\bB)^{-1}) l(\bB)^{-1}
\label{II}\ee
using that $|P_{\xi} -\xi_{\perp}|^2 =
|\eta(t_{\xi})|^2 - |\xi_3 -t_{\xi}|^2\leq |\eta (t_{\xi})|^2$.
To estimate the first term in (\ref{Bdecomp}), we use the fact that $\bB$ is
divergence free, in particular
\be
0=\mbox{div} \left( B\frac{\bB}{B}\right) = B \, \mbox{div}\frac{\bB}{B}
+ \frac{\bB}{B}\cdot\nabla B,
\label{div}\ee
which implies the pointwise estimate
\be
\left|\left(\frac{\bB}{B}\cdot\nabla\right)\bB\right|\leq
\left|\frac{\bB}{B}\cdot\nabla B\right|+ B\left|\nabla\left(\frac{\bB}{B}
\right)\right|
\leq 4B |\nabla (\bB/B)|.
\label{divest}\ee
Therefore
\be
|\bB (\xi) - \bB(P_{\xi})|\leq t_{\xi}
\left\{\sup_{|\xi|<5\lambda}
\left|\left(\frac{\bB}{B}(\xi)\cdot \nabla\right) \bB(\xi)
\right|\right\}
\leq
4\sqrt{3}\lambda \left\{\sup_{|\xi|<5\lambda} B(\xi)\right\}
l(\bB)^{-1},
\label{fin1}\ee
which completes the proof of Lemma \ref{Bappr}.
$\,\,\Box$
\subsection{Choice of a good gauge}\label{sec:gauge}
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
(\ref{general}) later. 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 (\ref{general})
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
\medskip
{\it Proof.}
We may assume that $C=\{ \, x\in\bR^3 \, :\, |x_{\perp} | \leq w\}$.
The following formula gives $\alpha$ explicitly:
\be
\alpha_1(y) = -\int_0^1 t\beta_3(ty) y_2 \rd t
\label{Aformula}\ee
\[
\alpha_2(y) = \int_0^1 t\beta_3(ty)y_1 \rd t
+ \int_0^{y_1} \beta_3(u, y_2, y_3)\rd u
- \int_0^1 \rd t \int_0^{y_1} \rd u
[ t^2u\partial_1\beta_3 + t^2y_2\partial_2\beta_3
+ 2t\beta_3](tu, ty_2, ty_3)
\]
\[
\alpha_3(y) = -\int_0^{y_1}\beta_2(u, y_2, y_3)\rd u +
\int_0^{y_2} \beta_1(0, v, y_3)\rd v
\]
\[
+\int_0^1 \rd t \biggl[ -t\beta_2(ty)y_1 + t\beta_1(ty)y_2
+ \int_0^{y_1} \rd u [ t \beta_2 + t^2\partial_2
\beta_2y_2 + t^2\partial_1\beta_2u](tu, ty_2, ty_3)
\]
\[
- \int_0^{y_2}\rd v [t\beta_1 + t^2\partial_2\beta_1v]
(0, tv, ty_3)\biggr].
\]
One can verify directly that this vector potential generates the $\beta$
field (one has to use $\mbox{div}\beta =0$ several times).
It is more instructive to show how one can find such a formula.
We can start with the Poincar\'e gauge choice $\alpha^{(P)}$:
\be
\alpha_1^{(P)}(y) = \int_0^1 t\beta_2(ty)y_3 \rd t - \int_0^1 t\beta_3(ty) y_2
\rd t
\label{general}\ee
\[
\alpha_2^{(P)}(y)
= \int_0^1 t\beta_3(ty)y_1 \rd t - \int_0^1 t\beta_1(ty) y_3 \rd t
\]
\[
\alpha_3^{(P)}(y) = \int_0^1 t\beta_1(ty) y_2 \rd t
-\int_0^1 t\beta_2(ty)y_1 \rd t
\]
The bad terms are those which contain $y_3$, therefore we eliminate them.
The first term in $\alpha^{(P)}_1$ is eliminated by adding $\nabla\varphi$ to
$\alpha^{(P)}$, where
\be
\varphi(y): = -\int_0^1 t y_3 \int_0^{y_1} \beta_2(tu, ty_2, ty_3)\rd u\rd t.
\ee
After some integration by parts (and $\mbox{div}\beta =0$), one obtains,
for the new gauge $\tilde\alpha := \alpha^{(P)} + \nabla\varphi$,
\be
\tilde\alpha_1(y) = -\int_0^t t\beta_3(ty)y_2 \rd t
\ee
\[
\tilde\alpha_2(y) = \int_0^1 t\beta_3(ty) y_1 \rd t
- \int_0^1 t\beta_1(0,ty_2, ty_3)y_3 \rd t
+\int_0^{y_1} \beta_3(u, y_2, y_3) \rd u
\]
\[
- \int_0^1\rd t
\int_0^{y_1} \rd u(t^2 u\partial_1\beta_3 + t^2y_2\partial_2\beta_3
+ 2t\beta_3)(tu, ty_2, ty_3)
\]
\[
\tilde\alpha_3(y)
= \int_0^1 t[-\beta_2(ty)y_1 + \beta_1(ty)y_2] \rd t - \int_0^{y_1}
\beta_2(u, y_2, y_3) \rd u
\]
\[
+ \int_0^1 \rd t \int_0^{y_1}
\rd u [t\beta_2 + t^2u\partial_1\beta_2 +t^2 y_2
\partial_2\beta_2](tu, ty_2, ty_3).
\]
For example, $\tilde\alpha_2$ is obtained in the following way:
\[
\tilde\alpha_2(y)=\alpha_2^{(P)} + \partial_2\varphi =
\int_0^1 t\beta_3(ty) y_1 \rd t
- \int_0^1 t\beta_1(ty)y_3 \rd t
- \int_0^1 t^2y_3
\int_0^{y_1}\partial_2\beta_2(tu, ty_2, ty_3)\rd u\rd t
\]
\be
= \int_0^1 t\beta_3(ty) y_1 \rd t
- \int_0^1 t\beta_1(ty)y_3 \rd t
\label{partint}\ee
\[
+ \int_0^1 t^2y_3
\int_0^{y_1}\partial_1\beta_1(tu, ty_2, ty_3)\rd u\rd t
+ \int_0^1 t^2y_3
\int_0^{y_1}\partial_3\beta_3(tu, ty_2, ty_3)\rd u\rd t
\]
\[
=\int_0^1 t\beta_3(ty) y_1 \rd t - \int_0^1 t\beta_1(ty) y_3
\rd t
\]
\[
+ \int_0^1 ty_3 \int_0^{ty_1} \partial_1\beta_1 (v, ty_2, ty_3 )
\rd v\rd t
+ \int_0^{y_1} \int_0^1 t^2y_3\partial_3\beta_3 (tu, ty_2, ty_3)
\rd t\,\rd u
\]
\[
=\int_0^1 t\beta_3(ty) y_1 \rd t - \int_0^1 t\beta_1(0,ty_2, ty_3) y_3
\rd t + \int_0^{y_1} \int_0^1 t^2 \frac{\rd}{\rd t}[\beta_3
(tu, ty_2, ty_3)] \rd t \,\rd u
\]
\[
- \int_0^{y_1} \int_0^1 t^2(u \partial_1\beta_3 + y_2\partial_2\beta_3)
(tu, ty_2, ty_3)\rd t\, \rd u
\]
\[
= \int_0^1 t\beta_3(ty) y_1 \rd t
- \int_0^1 t\beta_1(0,ty_2, ty_3)y_3 \rd t
+\int_0^{y_1} \beta_3(u, y_2, y_3) \rd u
\]
\[
- \int_0^1\rd t
\int_0^{y_1} \rd u(t^2 u\partial_1\beta_3 + t^2y_2\partial_2\beta_3
+ 2t\beta_3)(tu, ty_2, ty_3).
\]
The calculation for $\tilde\alpha_3$ is similar.
The only remaining `bad' term in $\tilde\alpha$
is the second one in $\tilde\alpha_2$.
The important thing to notice about this term is that it is
independent of $y_1$.
Adding $\nabla\psi$ to $\tilde\alpha$ with
\be
\psi (y):= \int_0^1 \rd t \, t\int_0^{y_2} \rd v \beta_1(0,tv, ty_3)y_3,
\ee
one can eliminate this term as well, i.e.
the final gauge $\alpha:=\tilde\alpha +\nabla\psi$, after some
calculation, yields (\ref{Aformula}). Note that since
$\psi$ is independent of $y_1$, $\tilde\alpha_1=\tilde\alpha_1+
\partial_1\psi$.
The estimate (\ref{Aest}) is straighforward. $\,\,\Box$.
\bigskip
\section{The Neumann problem
with a constant field on a cylinder}\label{gsdens}
\setcounter{equation}{0}
\subsection{Supersymmetry}\label{susysection}
Consider an arbitrary cylinder $C:=\{ \xi \, : |\xi_{\perp} - P|\leq w,
|\xi_3|\leq
\lambda/2 \}$ with center $P \in {\cal P}= \{ \xi \, :\, \xi_3 =0\}$
as in Section \ref{apprsec}.
Let $\hat\bB$ be a constant magnetic field which is parallel
with the $\xi_3$-axis, i.e. in this coordinate system $\hat\bB$
can be written as $\hat\bB = (0,0,\hat B)$, with $\hat B \in {\bf R}_+$.
Choose a gauge $\hat \bA$, $\nabla\times\bA = \hat \bB$,
which is independent of $\xi_3$ and has a special form $\hat \bA =
(\hat A_{\xi_1}, \hat A_{\xi_2}, 0)$ (for example the standard gauge,
$\hat \bA (\xi) = \frac{1}{2}\hat\bB \times (\xi -P)$, would do).
Let us define the operator $T_C:= (p_{\xi_1} + \hat A_{\xi_1}) - i
(p_{\xi_2} + \hat A_{\xi_2})$ on $C$ with Neumann boundary condition
(more precisely, $D(T_C):= H^1(C)$).
An easy calculation shows that its adjoint is
$T^*_C = (p_{\xi_1} + \hat A_{\xi_1}) + i
(p_{\xi_2} + \hat A_{\xi_2})$ with Dirichlet boundary condition
on the mantel of the cylinder $C=C_P$ (i.e. on the set
$ M_P: = \{ \xi \, : \, |\xi_3|< \lambda/2, \, |\xi_{\perp} - P|= w\}$).
Furthermore, the operator $T_C^*T_C$ is equal to the second
order operator $\sum_{j=1}^2(p_{\xi_j}+\hat A_{\xi_j})^2 - \hat B$
with operator domain $D(T^*_CT_C)=\{ \phi\in D(T_C) \, : \, T_C\phi
\in D(T_C^*)\}$.
Similarly, $T_CT^*_C=\sum_{j=1}^2 (p_{\xi_j}+\hat A_{\xi_j})^2 + \hat B$
on $D(T_CT^*_C)=\{ \phi\in D(T_C^*) \, : \, T_C^*\phi \in D(T_C)\}$ and both
operators are nonnegative and self-adjoint. Clearly
\be
T_CT^*_C = T_C^*T_C + 2\hat B
\label{susy1}\ee
on $D(T_CT^*_C)\cap D(T_C^*T_C)$, in particular, $T_CT^*_C \geq 2\hat B$
on $D(T_CT^*_C)$ (use that this relation is
true on $D(T_CT^*_C)\cap D(T_C^*T_C)$, but $H_0^2(C)$, which is a subset
of this intersection, is a core for $T_CT_C^*$ by the Kato-Rellich theorem).
By supersymmetry, the
spectra of $T_CT^*_C$ and $T_C^*T_C$
coincide except at zero,
therefore
$T_C^*T_C$ has a gap in the spectrum, i.e.,
\be
\mbox{spec}(T_C^*T_C)\subset\{0\}\cup[2\hat B,\infty).
\label{eq:susygap}\ee
Let $\Pi_C$ be the spectral
projection onto the zero energy level in the spectrum of the
operator $T^*_CT_C$. Note that all the operators $T_C, T^*_C$,
and therefore $\Pi_C$
commute with $p_{\xi_3}$. In particular, $T_C$, therefore
$\Pi_C$ act trivially
on functions depending only on the third direction, i.e.
$\Pi_C = \Pi_C^{(2)}\otimes \mbox{Id}_{\bf R}$, where $\Pi_C^{(2)}$ is
the zero energy projection of $T_C^*T_C$ viewed as
a two-dimensional operator acting on $L^2(C\cap {\cal P})$
with Neumann boundary conditions.
\subsection{Estimate of the Neumann ground state density}\label{ACbound}
We use the notations from Section \ref{susysection}, and, for
simplicity, let $P=0$.
\begin{lemma}
Consider a cylinder $C$ (see (\ref{Cyl})), with center $P=0$,
then $\Pi_C^{(2)}$ is an integral operator
with a continuous kernel on the open disk $D:= \mbox{int} (C\cap
{\cal P})$ of radius $w$, and its diagonal element satisfies
\be
\Pi_C^{(2)}(\xi_{\perp}, \xi_{\perp}) \leq e^{\hat B w^2/4}
\frac{w^2}{\pi (w^2 - |\xi_{\perp}|^2)^2}.
\label{diagest}\ee
\end{lemma}
{\it Remark.} With a slight abuse of notation,
we set $\xi_{\perp}: = (\xi_1, \xi_2 )$, while previously it was
$(\xi_1, \xi_2, 0)$.
\medskip
{\it Proof.}
Obviously $\mbox{Ran}\, \Pi_C^{(2)} =\mbox{Ker}\, T_C = \mbox{Ker}\, T_C^*
T_C$, where $T_C$ is viewed as a two-dimensional operator with Neumann
boundary conditions on the disk $D$ of radius $w$.
It is well known (and easy to verify) that any function
$g\in L^2(D)$ of two variables,
satisfying $T_Cg=0$, can be presented in the form $g=h\lambda_{\hat B}$,
where
\be
\lambda_{\hat B}(\xi_{\perp}):=\exp (-\hat B \xi_{\perp}^2/4)
\label{lambda}\ee
and $h$ is an analytic function on $D$.
Notice that $\hat B\xi_{\perp}^2 \leq \hat B w^2$ on $D$, therefore
\be
\exp(-\hat B w^2/4)\leq\lambda_{\hat B}\leq 1
\label{expest}\ee
within the disk. This implies that
$h\in {\cal A}^2(D)$, i.e. it belongs to the set
of square-integrable analytic functions on $D$.
Conversely, if $h\in {\cal A}^2(D)$, then $g=h\lambda_{\hat B}$ is in $L^2(D)$
and pointwise satisfies the equation $T_Cg=0$, therefore it is in
$\mbox{Ker}\, T_C$ (recall that the differential equation
$T_Cg=0$ is equivalent to the Cauchy-Riemann equations for $h$).
This establishes a bounded bijection between $\mbox{Ran}\Pi_C^{(2)}$
and ${\cal A}^2(D)$ with bounded inverse.
Let ${\cal B}$ be the projection operator from $L^2(D)$ to ${\cal A}^2(D)$.
It is an integral operator and we have an explicit formula
for its kernel (Bergman kernel):
\be
{\cal B}(\xi_{\perp}, \eta_{\perp})= \frac{w^2}{\pi (w^2-{\xi_{\perp}}
\overline{\eta_{\perp}})^2},
\label{bergman}\ee
which is valid on the open disk (the Bergman kernel,
in connection with the Pauli operator, was also used in \cite{F}).
The overline here means complex
conjugation as we canonically imbed $D$ into ${\bf C}$.
Let $h_m(z):= \alpha_m z^m$ be the standard orthonormal basis for
${\cal A}^2(D)$ (where $\alpha_m$ is the normalization), then
${\cal B}(\xi_{\perp}, \eta_{\perp})= \sum_{m=0}^{\infty}
h_m(\xi_{\perp})\overline{h_m(\eta_{\perp})}$, and
the sum is absolutely convergent for the interior of the disk.
Therefore, by the bijection above,
$\{ h_m\lambda_{\hat B}\}_{m=0}^{\infty}$ is an orthogonal
basis for $\mbox{Ran}\,\Pi_C^{(2)}$ (orthogonality follows from
integration in the angular variable). Moreover,
\be
\Vert h_m\lambda_{\hat B} \Vert_{L^2(D)} \geq e^{-\hat B w^2/4}\Vert h_m
\Vert_{L^2(D)} = e^{-\hat B w^2/4}.
\label{normest}\ee
This uniform lower bound shows that $\Pi_C^{(2)}$ is an
integral operator on the open disk with continuous kernel
\be
\Pi_C^{(2)}(\xi_{\perp}, \eta_{\perp})= \sum_{m=0}^{\infty}
\frac{{\lambda_{\hat B}(\xi_{\perp})h_m(\xi_{\perp})}
\overline{\lambda_{\hat B}(\eta_{\perp})h_m(\eta_{\perp})}}
{\Vert h_m\lambda_{\hat B}\Vert^2},
\label{Pikernel}\ee
since the sum above is absolutely convergent by the boundedness of
the Bergman kernel.
Consequently we have for the diagonal element
\be
\Pi_C^{(2)}(\xi_{\perp}, \xi_{\perp}) \leq e^{\hat B w^2/4}{\cal B}
(\xi_{\perp}, \xi_{\perp}) \leq e^{\hat B w^2/4}
\frac{w^2}{\pi (w^2 - |\xi_{\perp}|^2)^2}. \,\,\Box
\label{diagest1}\ee
\subsection{Energy contribution from the
Neumann ground states}
If ${\cal U}$ is a non-negative function on $\bR^3$ we shall
study the negative spectrum of the operator
$$
H_C^{low}=\Pi_C(p_{\xi_3}^2-\chi_C {\cal U})\Pi_C
$$
on $L^2(C)$. We consider this operator defined as a
quadratic form with Dirichlet boundary
conditions on the top and bottom of the cylinder, i.e.,
defined on functions in $H^1(C)$, that vanish for $\xi_3=\pm\lambda/2$.
\begin{lemma} \label{lm:LTconf} If $0\leq {\cal U}\in L^{3/2}(\bR^3)$
then the sum of the negative eigenvalues of
$H_C^{low}$ is bounded below by
$$
-L_{1,1}\int_C \Pi_C^{(2)}(\xi_\perp,\xi_\perp)
{\cal U}(\xi)^{3/2} \rd \xi,
$$
where $L_{1,1}$ is the constant in the one dimensional
Lieb-Thirring inequality ((\ref{eq:1dlt}) below).
\end{lemma}
{\it Proof:} For the sake of simplifying notations
we shall assume that $C$ is an infinite cylinder ($\lambda=\infty$).
(This will only lower the sum of the negative eigenvalues.)
Let $u_1,u_2,\ldots$ denote the (orthonormalized) eigenfunctions of
$H_C^{low}$ corresponding to negative eigenvalues.
Then since $\Pi_C u_i=u_i$
we have that the sum of the negative eigenvalues of $H_C^{low}$ is
$$
\sum_i\int_C \left(|\partial_{\xi_3}u_i|^2 -{\cal U}(\xi)|u_i(\xi)|^2
\right)\rd\xi.
$$
We shall show below that for almost all $\xi_\perp$ in the
disc $D$ and all $f\in L^2(\bR)$
we have
\be
0\leq\sum_i \left|\int_{\bR}f(\xi_3)
u_i(\xi_\perp,\xi_3)\rd\xi_3\right|^2
\leq \Pi_C^{(2)}(\xi_\perp,\xi_\perp)
\int_{\bR}|f(\xi_3)|^2\rd\xi_3.
\label{eq:dmestimate}\ee
Let us first show how this estimate implies the lemma.
For fixed $\xi_\perp\in D$ we consider the one dimensional operator
$p_{\xi_3}^2-{\cal U}(\xi_\perp,\cdot)$ acting on $L^2(\bR)$.
It follows from the one dimensional
Lieb-Thirring inequality that the sum of the negative eigenvalues
of this operator is bounded below by
\be
-L_{1,1}\int_{\bR} {\cal U}(\xi)^{3/2}\rd\xi_3,
\label{eq:1dlt}\ee
which by assumption is finite for almost all $\xi_\perp\in D$.
For almost all $\xi_\perp$ in $D$
we therefore have the following inequality for $v\in L^2(\bR)$
$$
\int_{\bR} \left(|p_{\xi_3}v(\xi_3)|^2-{\cal U}(\xi)|v(\xi_3)|^2
\right)\rd\xi_3\geq
\sum_m \mu^{(m)}_{\xi_\perp}
\left|\int_{\bR} f^{(m)}_{\xi_\perp}(\xi_3)v(\xi_3)\rd\xi_3\right|^2,
$$
where $f^{(m)}_{\xi_\perp}$ (the eigenfunctions of the
one dimensional operator) are orthonormal in $L^2(\bR)$
and $\mu^{(m)}_{\xi_\perp}$ (the eigenvalues of the one dimensional
operator) satisfy $\sum_m \mu^{(m)}_{\xi_\perp}\geq
-L_{1,1}\int_{\bR} {\cal U}(\xi)^{3/2}\rd\xi_3$. The lemma follows
easily from this.
We return to proving (\ref{eq:dmestimate}). We first use that
since $u_i=\Pi_C u_i$ we have for fixed $\xi_3$
$$
u_i(\xi_\perp,\xi_3)=\int_D \Pi_C^{(2)}(\xi_\perp,\eta_\perp)
u_i(\eta_\perp,\xi_3)\rd\eta_\perp.
$$
Thus
$$
\int_{\bR} f(\eta_3)u_i(\xi_\perp,\eta_3)\rd\eta_3
=\int_{C} f(\eta_3) \Pi_C^{(2)}(\xi_\perp,\eta_\perp)
u_i(\eta)\rd \eta.
$$
It is easy to see from (\ref{Pikernel}) that
$f(\eta_3)\Pi_C^{(2)}(\xi_\perp,\eta_\perp)$ is in
$L^2(C)$ as a function of $\eta$ for each fixed $\xi_\perp\in D$.
Since $u_1,u_2,\ldots$ are orthonormal it follows from
Plancherel's formula that
\begin{eqnarray*}
\sum_i \left|\int_{\bR}f(\eta_3)
u_i(\xi_\perp,\eta_3)\rd\eta_3\right|^2&=&
\sum_i\left|\int_{C} f(\eta_3)\Pi_C^{(2)}(\xi_\perp,\eta_\perp)
u_i(\xi_\perp,\eta_3)\rd\eta
\right|^2\\&\leq &\int_{C} |f(\eta_3)|^2
|\Pi_C^{(2)}(\xi_\perp,\eta_\perp)|^2 \rd\eta\\
&=&\Pi_C^{(2)}(\xi_\perp,\xi_\perp)\int_{\bR} |f(\eta_3)|^2
\rd\eta_3,
\end{eqnarray*}
where in the last line we used that since $\Pi_C^{(2)}$ is a projection
we have
$$
\int_D |\Pi_C^{(2)}(\xi_\perp,\eta_\perp)|^2\rd\eta_\perp
=\Pi_C^{(2)}(\xi_\perp,\xi_\perp). \,\,\,\Box
$$
\section{Proof of the Lieb-Thirring inequality}\label{sec:proof}
\setcounter{equation}{0}
\subsection{Localization into cubes}\label{cubsection}
Consider a regular cubic tiling of the space by closed cubes
$\{ \Omega_i^0 \, : \, i\in {\bf Z}^3\}$ with edge-length
\be
d:=k_1\min \{\|B\|^{-1/4}l^{1/2}, l, L \},
\label{d}\ee
and centers at the points of the lattice
$(d{\bf Z})^3$. Here $k_1$ is an adjustable
constant which we shall choose later, such that
it satisfies $k_1\leq1$.
Let $\Omega_i$ denote the ten times bigger cubes,
with the same center, we shall refer to $\Omega_i$ as cubes and
to $\Omega^0_i$ as small cubes.
The cubes $\Omega_i$
cover the whole space with an overlap bounded by a universal
constant.
\subsubsection{Allocating the potential}\label{sec:alloc}
We shall allocate the given potential $V$
using the following lemma.
\begin{lemma}\label{alloclemma} Given any non-negative function
$V\in L^{5/2}\cap W^{1,1}$, there exists
$\tilde V \in L^{5/2}\cap W^{1,1}$
with the following properties:
\be
\tilde V \geq V
\label{propV1}\ee
\be
\int_{\Omega_i} |\tilde V|^p \sim \int_{\Omega_j} |\tilde V|^p
\quad \mbox{if} \quad \Omega_i \cap \Omega_j \neq \emptyset
\label{propV2}\ee
\be
\int_{{\bf R}^3} |\tilde V|^p \leq c \int_{{\bf R}^3} | V|^p
\qquad \mbox{and}\qquad
\int_{{\bf R}^3} |\nabla\tilde V| \leq c \int_{{\bf R}^3} |\nabla V|,
\label{propV3}\ee
where $p =1, \, 3/2$ or $5/2$, and the notation $X\sim Y$ means that
there is a universal constant $c>0$ such that $c^{-1}\leq X/Y \leq c$.
\end{lemma}
\bigskip
{\it Proof:} We define $\tilde V := V \ast \eta$, where $\eta$ is the
following distribution
\be
\eta : =\sum_{{\bf n}\in {\bf Z}^3} 2^{-|{\bf n}|}\cdot \delta_{d{\bf n}}
\label{lattice}\ee
Property (\ref{propV1}) is immediate since $\eta \geq \delta_0$;
property (\ref{propV3})
follows from $\Vert \eta \Vert_1 \leq c$ and Young's inequality.
To check the property (\ref{propV2}), first observe that it
is enough to show
\be
\int_{\Omega_i^0} |\tilde V|^p \sim \int_{\Omega_j^0} |\tilde V|^p
\label{propV2'}\ee
for the
smaller (tiling) cubes with $\Omega_i^0 \cap \Omega_j^0 \neq \emptyset$,
i.e. for neighboring small cubes.
Denoting the vector, pointing from the center of $\Omega_i^0$ to that
of $\Omega_j^0$, by $d{\bf v}$, we have
\[
\int_{\Omega_i^0} |\tilde V|^p
= \int_{\Omega_i^0} \left(
\sum_{{\bf n}\in {\bf Z}^3} 2^{-|{\bf n}|} V(x- d{\bf n})\right)^p \rd x
=\int_{\Omega_j^0} \left( \sum_{{\bf n}\in {\bf Z}^3}
2^{-|{\bf n}|} V(x- d({\bf n} + {\bf v}))\right)^p\rd x
\]
\[
=\int_{\Omega_j^0} \left( \sum_{{\bf n}\in {\bf Z}^3} 2^{-|{\bf n}- {\bf v}|}
V(x- d{\bf n})\right)^p \rd x
\leq 2^{p\sqrt{3}} \int_{\Omega_j^0} \left(
\sum_{{\bf n}\in {\bf Z}^3} 2^{-|{\bf n}|} V(x- d{\bf n}))\right)^p\rd x
\]
\be
= 2^{p\sqrt{3}}\int_{\Omega_j^0} |\tilde V|^p,
\label{comp}\ee
using that $|{\bf n}| \leq \sqrt{3} + |{\bf n}- {\bf v}|$. $\,\,\Box$
We may therefore assume that our potential $V$ satisfies
the property (\ref{propV2}), since,
otherwise, in our theorem we can replace $V$ by $\tilde V$
constructed above.
\subsubsection{Splitting into overall spinup and
spindown within a fixed cube}\label{sec:split}
We begin with a definition.
\bigskip
\begin{definition}
We call a cube $\Omega_j$ weak if
\be
k_2\left(\int_{\Omega_j} V^{3/2}\right)^{2/3}< 1,
\label{defweak}\ee
and we define
\be
\delta_j := k_2 \left(\int_{\Omega_j} V^{3/2}\right)^{2/3}.
\label{delta}\ee
Otherwise, we call the cube $\Omega_j$ strong, and we let $\delta_j : =1$.
Here $k_2$ denotes a constant to be chosen later.
\end{definition}
Notice that if $V$ satisfies property (\ref{propV2})
then $\delta_i \sim \delta_j$ for overlapping
$\Omega_i$ and $\Omega_j$ we call this the
{\it regular variation property}.
Choose a partition of unity,
consisting of $C^{\infty}$-cutoff functions $\theta_i$, such that
$\mbox{supp}\, \theta_j \subset \Omega_j$,
$\theta_j \geq c$ on $\Omega_j^0$,
$\sum_j \theta^2_j\equiv 1$, and $|\nabla\theta_j|
\leq cd^{-1}$.
\begin{lemma} \label{lm:fullspl}
Let $P_-^{(j)}$ and $P_+^{(j)}$ denote the spin down and
spin up projections
relative to the magnetic field at the center $Q_j$ of $\Omega_j$
(i.e. on the image of $P_{\pm}^{(j)}$ we have $\bsigma\cdot\bB(Q_j) =
\pm B(Q_j)$). Assume that the potential $V$ satisfies (\ref{propV2}).
For any $f\in H^1(\bR^3;\bC^2)$ we have
\be
\int_{{\bf R}^3} \left( |\bsigma\cdot(\bp + \bA )f|^2 - V|f|^2 \right)
\geq T_+(f) + T_-(f)
\label{fullspl}\ee
with
\be
T_+(f):= \sum_j \delta_j T_+^{\Omega_j} (P_+^{(j)}\theta_jf), \quad
T_-(f):= \sum_j \delta_j T_-^{\Omega_j} (P_-^{(j)}\theta_j f) ,
\label{T}\ee
where
\be
T_+^{\Omega_j}(g):=\int_{\Omega_j}
\left( |\bsigma\cdot(\bp + \bA )g|^2
- (V\delta_j^{-1} + cd^{-2}+ ck_1^4B_j^\#)|g|^2
\right)
\label{T+}\ee
\be
T_-^{\Omega_j}(g):=\int_{\Omega_j}\left(
|\bsigma\cdot(\bp + \bA )g|^2
- (V\delta_j^{-1} + cd^{-2})|g|^2\right)
\label{T-}\ee
are quadratic forms defined on spinors $g\in H^1_0 (\Omega_j, \bC^2)$
and $P_+^{(j)}g=g$, $P_-^{(j)}g=g$, respectively.
Here
\be
B_j^\#=\sup_{|x-Q_j|<200L}B(x).
\label{eq:bsharp}\ee
\end{lemma}
The upper indices in the spin projections indicate that they
depend on $\Omega_j$, as they are determined by
the direction of the magnetic field in the center of the cube.
Sometimes, for shortness, we shall drop the index $j$
and use the notation $T_{\pm}^{\Omega}(g)$.
To get the statement of the main Theorem~\ref{LTtheorem}, we have to
sum up (\ref{fullspl}) for an arbitrary finite set of orthonormal
spinors $\{ f_m \}_{m=1}^N$, and give a uniform lower bound for
the right hand side of
\be
\sum_{m=1}^N
\int_{{\bf R}^3} \left( |\bsigma\cdot(\bp + \bA )f_m|^2 - V|f_m|^2 \right)
\geq \sum_{m=1}^N \left( T_+(f_m)+T_-(f_m)\right).
\label{fin}\ee
The rest of this section contains the proof of Lemma~\ref{lm:fullspl}.
In Section~\ref{Pupsec} we estimate $T_+$ in Section~\ref{Pmin}
we study $T_-$ and finally in Section~\ref{end} we complete
the proof of the main Theorem~\ref{LTtheorem}.
{\it Proof of Lemma~\ref{lm:fullspl}:}
Let $f$ be an arbitrary spinor, then,
by the localization formula and $\delta_j\leq 1$,
\[
\int_{{\bf R}^3} \left( |\bsigma\cdot(\bp + \bA )f|^2 - V|f|^2 \right)
\geq \sum_j \int_{\bR^3} \left(
\delta_j\theta_j^2|\bsigma\cdot(\bp + \bA )f|^2
-V|\theta_jf|^2\right)
\]
\[
=
\sum_j \delta_j\int_{\bR^3} \left( |\bsigma\cdot(\bp + \bA )(\theta_jf)|^2
-|\nabla\theta_j|^2|f|^2 - V\delta^{-1}_j|\theta_jf|^2 \right)
\]
\be
\geq\sum_j \delta_j\int_{\bR^3}\left(
|\bsigma\cdot(\bp + \bA )(\theta_jf)|^2
- (V\delta_j^{-1} + cd^{-2})|\theta_jf|^2\right) ,
\label{ims}\ee
where, in the last step,
we used the finite overlapping property of the cubes and
the regular variation of the $\delta$'s to reallocate the localization
errors.
Recall that the assumption (\ref{propV2}) on the potential
implies the regular variation of the $\delta$'s.
Now we estimate the contribution of each cube individually.
We consider a fixed cube $\Omega:=\Omega_j$, and let $\psi$ (in applications,
$\psi =\theta_j f$)
be a spinor with Dirichlet boundary conditions on $\Omega$. We omit
all references to the index $j$.
We can split the kinetic energy as follows (all integrals are over $\Omega$,
but can be considered over ${\bf R}^3$, since we may extend $\psi$ to be
zero outside $\Omega$):
\[
\int |\bsigma\cdot(\bp + \bA) \psi|^2=
\int |(\bp +\bA)\psi|^2 -\int (\psi,\bsigma\cdot\bB \psi)
\]
\[
=\int |(\bp +\bA)P_-\psi|^2 -\int (P_-\psi ,\bsigma\cdot\bB P_-\psi)
+\int |(\bp +\bA)P_+\psi|^2 -\int (P_+\psi ,\bsigma\cdot\bB P_+\psi)
\]
\[
-\int (P_-\psi ,\bsigma\cdot\bB P_+\psi )
-\int (P_+\psi ,\bsigma\cdot\bB P_-\psi )
\]
\be
=\int |\bsigma\cdot(\bp +\bA) P_-\psi |^2+\int |\bsigma\cdot
(\bp +\bA) P_+\psi |^2
-\int (P_-\psi ,\bsigma\cdot\bB P_+\psi)
-\int (P_+\psi ,\bsigma\cdot\bB P_-\psi).
\label{kinspl}\ee
We here used that $P_{\pm}$ commute with $\bp + \bA $.
The cross terms are estimated by using $|\bB(x)/B(x) - \bB(Q)/B(Q)|
\leq 20l^{-1}d$
which follows from (\ref{asp2}) for any $x\in \Omega$ (as $|x-Q|\leq
10\sqrt{3}d< 20d$) and
that $(P_+\psi , \bsigma\cdot\bB(Q) P_-\psi )=0$:
\[
|(P_+\psi (x),\bB(x)\cdot\bsigma P_-\psi(x))|
= B(x) \left|\left(P_+\psi(x),
\bsigma\cdot\left(\frac{\bB(x)}{B(x)}-\frac{\bB(Q)}{B(Q)}\right)
P_-\psi(x)\right)\right|
\]
\be
\leq 20B^\#l^{-1}d |P_-\psi(x)||P_+\psi(x)|
\leq 10B^\#l^{-1}d(\alpha |P_-\psi(x)|^2 + \alpha^{-1}|P_+\psi(x)|^2)
\label{cross}\ee
by the arithmetic-geometric mean inequality.
Choosing $\alpha := d^{-3}l(B^\#)^{-1}$ gives, for any $x\in \Omega$,
$$
|(P_+\psi (x),\bsigma\cdot\bB(x) P_-\psi(x))|
\leq 10d^{-2}|P_-\psi(x)|^2 +10k_1^4B^\#|P_+\psi(x)|^2,
$$
where we have used that $d^4l^{-2}(B^\#)^2\leq
k_1^4\|B\|^{-1}(B^\#)^2\leq k_1^4B^\#$.
We also split the potential term (using the pointwise
identity $|\psi(x)|^2 = |P_+\psi(x)|^2 + |P_-\psi(x)|^2$):
\be
\int (V\delta^{-1} + cd^{-2})|\psi|^2
= \int (V\delta^{-1} + cd^{-2}) |P_-\psi|^2
+ \int (V\delta^{-1} + cd^{-2})|P_+\psi|^2.
\label{potspl}\ee
Altogether, we have
\[
\int \left( |\bsigma\cdot(\bp + \bA )\psi|^2 - (V\delta^{-1} + cd^{-2})
|\psi|^2
\right)
\]
\[
\geq
\int \left( |\bsigma\cdot(\bp + \bA )P_+\psi|^2 - (V\delta^{-1} +
cd^{-2}+ ck_1^4B^\#)|P_+\psi|^2 \right)
\]
\be +\int \left( |\bsigma\cdot(\bp + \bA )P_-\psi|^2 -
(V\delta^{-1} + cd^{-2})|P_-\psi |^2 \right) .
\label{splomega}\ee
The lemma follows if we insert
the formula above for
each cube $\Omega_j$ and for $\psi=\theta_j f$
into (\ref{ims}). $\,\,\Box$
\subsection{Treating the spin-up part in a cube}\label{Pupsec}
First, we focus on the contribution
$T_+^{\Omega_j}(P_+^{(j)}\theta_j f)$, of a single cube $\Omega_j$,
in $T_+(f)$, so for a while we drop the indices and
$\Omega$ will denote any of the $\Omega_j$'s.
The notations $P_+$ and $\theta$ are used accordingly.
\begin{theorem}\label{T+theorem}
Let us consider a cube $\Omega$ with center $Q$ and
edge length $10d$, where
$d=k_1\min
\{ \|B\|^{-1/4}l^{1/2}, L, l\}$,
and let $P_+$ be the spin-up projection according to the magnetic
field in the center of $\Omega$. Let $\delta:= \min \{ 1,
k_2\left(\int_{\Omega} V^{3/2}\right)^{2/3}\}$
and $B^\#=\left\{\sup_{|x-Q|\leq200L}B(x)\right\}$.
Define a quadratic form on spinors $g\in H^1_0 (\Omega, \bC^2)$,
$P_+g=g$ by
\be
T_+^{\Omega}(g):=
\int_{\Omega} \left( |\bsigma\cdot(\bp + \bA )g|^2
- (V\delta^{-1} + cd^{-2} + ck_1^4B^\#)|g|^2 \right).
\label{T+def}\ee
Then there exists a universal constant $c'$ such that
if $k_1< c'$ and if $2^{3/2}L_{3,1}k_2^{-3/2}< 1$
[where $L_{3,1}$ is the Lieb-Thirring constant (see \cite{LT})]
then
for any orthonormal family of spinors $\{ f_m \}_{m=1}^N$ and
for any $\theta \in C^{\infty}_0(\Omega)$, $0\leq \theta \leq 1$,
we have
\be
\delta\sum_{m=1}^N T_+^{\Omega} (P_+\theta f_m)
\geq -c d^{-2}(k_2^{-1/2} + k_2)
\left( d^{-1}\int_{\Omega}V +
\int_{\Omega}|\nabla V|\right)
\label{2weak}\ee
if $\delta <1$, and
\be
\delta \sum_{m=1}^N T_+^{\Omega} (P_+\theta f_m)=
\sum_{m=1}^N T_+^{\Omega} (P_+\theta f_m)
\geq -c\int_{\Omega} V^{5/2}
-c k_2d^{-2}\left( d^{-1}\int_{\Omega}V +
\int_{\Omega}|\nabla V|\right)
\label{2strong}\ee
if $\delta =1$,
uniformly in $N$.
\end{theorem}
Before the proof we state a corollary, which shows how we are
going to use this lemma to estimate $T_+$ (see (\ref{T})).
\begin{corollary}\label{Ppluscor} There exists $c'c>0$ such that
if $k_1c'^{-1}$ (recall that $T_+$ depends on $k_1$
and $k_2$ through $d$ and $\delta$)
then we have for any orthonormal
family of spinors $\{ f_m \}_{m=1}^N$
\be
\sum_{m=1}^N T_+(f_m)
\geq -c\int_{{\bf R}^3} V^{5/2} - c(k_2^{-1/2} + k_2)d^{-2}
\left( d^{-1}\int_{{\bf R}^3}V + \int_{{\bf R}^3}
|\nabla V|\right).
\label{Pplus}\ee
\end{corollary}
\bigskip
{\it Proof of Corollary \ref{Ppluscor}.} We can apply
Theorem \ref{T+theorem} for each cube
$\Omega =\Omega_j$ and $\theta = \theta_j$.
Putting together (\ref{2weak}) and (\ref{2strong}), using
the universally bounded overlap of the cubes $\{ \Omega_j \}_j$, we get
\be
\sum_{m=1}^N T_+(f_m) =
\sum_{m=1}^N\sum_j \delta_j T_+^{\Omega_j}(P_+^{(j)}\theta_j f_m)
\geq -c\int_{{\bf R}^3} V^{5/2} - c(k_2^{-1/2} + k_2)d^{-2}
\left( d^{-1}\int_{{\bf R}^3}V + \int_{{\bf R}^3}|\nabla V|\right),
\ee
which completes the proof. $\,\,\Box$.
\bigskip
{\it Proof of Theorem \ref{T+theorem}}.
If $k_1<(10(6+3\sqrt{3}))^{-1}$ then
\be
10(6+3\sqrt{3})d l^{-1} \leq 10(6+3\sqrt{3}) k_1 \leq 1,
\label{const1'}\ee
This allows us to use Proposition \ref{Bappr} for the
cube $\Omega$, with edge length $10d$.
Using Lemma \ref{Bappr} for $\Omega$ (i.e. $\lambda = 10d$),
by the Poincar\'e formula
(\ref{general}) and the estimate (\ref{Poinest}), there exists
a vectorpotential $\tilde\bA$ (depending on the fixed
cube $\Omega$), generating a constant direction field
$\tilde\bB$ (see (\ref{Btildedef})), parallel with $\bB (Q)$, such that
\begin{eqnarray*}
|\bA(x) - \tilde\bA(x)|
&\leq& 10\sqrt{3}d\sup_{\Omega}|\bB(x) -\tilde\bB(x)|
\leq cd^2B^\#l^{-1}(1+d(l^{-1}+L^{-1}))\nonumber\\
&\leq &ck_1^2 (B^\#/\|B\|)^{1/2}(B^\#)^{1/2}\leq ck_1^2 (B^\#)^{1/2}
\label{Adiff}
\end{eqnarray*}
for all $x\in\Omega$. In the last step we used (\ref{Btilde}),
and the definition of $d$.
For brevity, we did not keep track of the
explicit constant, but we applied the convention on the constants
$c$ mentioned at the beginning.
Therefore using a Cauchy-Schwarz inequality
\be
\int_{\Omega} |\bsigma\cdot(\bp + \bA )g|^2
\geq \frac{1}{2}
\int_{\Omega} \left( |\bsigma\cdot(\bp + \tilde\bA )g|^2
- ck_1^4B^\# |g|^2 \right) .
\label{ACappr}\ee
We shall now prove that if $k_1$ is sufficiently small then
there exists a universal constant $\tilde c$ such that
$
|\tilde \bB (x)|\geq \tilde c B^\#
$
for any $x\in \Omega$. To see this use that since
$d\leq k_1l$ and $d\leq k_1L$,
we have
\be
||\bB(x)|-|\tilde\bB(x)||\leq |\bB(x)-\tilde\bB(x)|
\leq cd l^{-1} B^\#(1+ d(L^{-1} + l^{-1}))\leq ck_1 B^\#.
\label{Btilpr}\ee
On the other hand for $x\in\Omega$ we have
$$
\log(B^\#/B(x))\leq 210L\|\nabla\log B\|
=cLL^{-1} \leq c,
$$
therefore
\be
B(x)\geq \exp(-c) B^\#.
\label{sharfest}\ee
Thus we may find
$0<\tilde c<1$ and then choose $k_1$ small enough
to ensure that
$\tilde\bB(x)$, which points in the same direction as $\bB (Q)$
for all $x\in\Omega$, satisfies
\be
\tilde c B^\#\leq\tilde B (x) \leq (\tilde c)^{-1} B^\#
\label{Blowpr}\ee
Using the lower bound in (\ref{Blowpr}), we shall now prove the
inequality
\be
(g,[\bsigma\cdot(\bp + \tilde\bA )]^2g)
\geq (g,(\bp + \tilde\bA)^2g) + \tilde c B^\#(g,g)
\label{Paulitrick}\ee
on spinors $g$ in the range of $P_+$ and
with Dirichlet boundary condition on the
cube $\Omega$.
The inequality (\ref{Paulitrick}) follows from
\be
P_+[\bsigma\cdot(\bp + \tilde\bA )]^2P_+
= P_+[(\bp +\tilde\bA)^2 +\bsigma\cdot\tilde\bB]P_+
= P_+[(\bp +\tilde\bA)^2 + |\tilde\bB|]P_+,
\ee
where, in the last step, we used that $\tilde\bB$ points
in the same direction as ${\bf B}(Q)$.
This implies, along with (\ref{ACappr}) that
\[
T_+^{\Omega}(g)=\int_{\Omega} \left( |\bsigma\cdot(\bp + \bA )g|^2
- (V\delta^{-1} + cd^{-2} + c k_1^4 B^\#)|g|^2 \right)
\]
\be
\geq\frac{1}{2}(g, ((\bp + \tilde\bA)^2 - W)g),
\label{diam}\ee
with
\be
W:= 2V\delta^{-1} + cd^{-2}+ (ck_1^4-\tilde c)B^\#.
\label{W}\ee
Thus if $ck_1^4\leq \tilde c$ we obtain
\be
H(W)\geq
(\bp + \tilde\bA)^2 - 2V\delta^{-1} - cd^{-2}
\label{tildeest}\ee
where
\be
H(W):= (\bp +\tilde\bA)^2 -W
\label{HW}\ee
with Dirichlet boundary conditions on $\Omega$.
\bigskip
\noindent
{\it Weak cube} ($\delta <1$):
\medskip
\noindent
Using the assumption
$2^{3/2}L_{3,1}k_2^{-3/2} < 1$, the operator $(\bp +
\tilde\bA)^2 -2V\delta^{-1}$, with Dirichlet boundary conditions on
$\Omega$, is seen to be positive by the CLR-bound
(recall that the CLR bound \cite{L} is true for Schr\"odinger
operators with the magnetic Laplacian, $(\bp +
\tilde\bA)^2$, as well, as it was pointed out in
\cite{AHS}, see also \cite{MR})
since $L_{3,1}\int_{\Omega} (2V\delta^{-1})^{3/2}
\leq 2^{3/2}L_{3,1}k_2^{-3/2} < 1$.
Note that
we could use the CLR-bound on ${\bf R}^3$, since
the Dirichlet boundary condition for $P_+\theta f$ allows us to extend the
functions onto the whole space.
Therefore, we have a lower bound $H(W)=(\bp + \tilde\bA)^2- W
\geq -cd^{-2}$; in particular this is a lower
bound on all eigenvalues.
On the other hand, again by the CLR bound,
the dimension of the spectral projection $\Pi_{\{H(W)<0\}}$
onto the negative spectrum of $H(W)$ (in this case this is just
the number of negative eigenvalues of $H(W)$) is bounded by
\be
\mbox{dim} \, \Pi_{\{ H(W) < 0 \}}
= \mbox{Tr}\, \Pi_{\{ H(W) < 0 \}} \leq
L_{3,1}\int_{\Omega} (2V\delta^{-1} + cd^{-2})^{3/2} \leq
ck_2^{-3/2} + c.
\label{Or}\ee
Let $\{ f_m \}_{m=1}^N$ be an arbitrary orthonormal set of spinors
and $\theta \in C^{\infty}_0 (\Omega)$, $0\leq \theta \leq 1$.
Then, from (\ref{diam}), we have for weak cubes $\Omega$ (i.e.
$\delta \leq 1$)
\[
\delta\sum_{m=1}^N T_+^{\Omega} (P_+\theta f_m) =
\sum_{m=1}^N \delta\int_{\Omega}
\left( |\bsigma\cdot(\bp + \bA)(P_+\theta f_m)|^2
- (V\delta^{-1} + cd^{-2} + ck_1^4B^\#)|P_+\theta f_m|^2\right)
\]
\be
\geq c\delta
\sum_{m=1}^N (P_+\theta f_m, ((\bp + \tilde\bA)^2 - W)P_+\theta f_m)
=c\delta\mbox{Tr}(H(W)P_+\theta\gamma_N\theta P_+)
\label{gammatr}\ee
where $\gamma_N$ is the projection (written in Dirac notation)
\be
\gamma_N:= \sum_{m=1}^N |f_m \rangle\langle f_m|.
\label{gammadef}\ee
Since we have the operator inequality
$0\leq P_+\theta\gamma_N\theta P_+\leq 1$ and $H(W)\geq
-cd^{-2} $ we have
\be
\mbox{Tr}(H(W)P_+\theta\gamma_N\theta P_+)
\geq -cd^{-2}\mbox{Tr}\Pi_{\{ H(W)< 0\}}
\label{trhwg}\ee
and using (\ref{Or}) and the definition of $\delta$ for weak cubes,
we obtain
\be
\delta\sum_{m=1}^N T_+^{\Omega} (f_m)
\geq - c\delta d^{-2}(k_2^{-3/2}+c)
\geq -cd^{-2}(k_2^{-1/2} + k_2)
\left( \int_{\Omega} V^{3/2}\right)^{2/3}
\label{2weak'}\ee
\[
\geq -c (k_2^{-1/2} + k_2)d^{-2}\left( d^{-1}\int_{\Omega}V +
\int_{\Omega}|\nabla V|\right),
\]
which completes the proof of (\ref{2weak}).
Here, in the last step, we used Sobolev's inequality, which
asserts that for any nonnegative function $V$ on an arbitrary
cube $\Omega$, with edge length $\lambda$,
\be
\left( \int_{\Omega} V^{3/2}\right)^{2/3} \leq |\Omega|^{-1/3}\int_{\Omega}
V + \int_{\Omega} |\nabla V|
= \lambda^{-1}\int_{\Omega}V + \int_{\Omega} |\nabla V| .
\label{Sobolev}\ee
The reason for using Sobolev's inequality is that finally we will have
to add up the contributions from each cube, and $(\int_{\Omega}
V^{3/2})^{2/3}$ is not additive.
Note that this step introduces the
extra terms, $\int V$ and $\int |\nabla V|$, which
do not appear in Lieb-Thirring type estimates for constant
direction field \cite{E-1995}, \cite{Sob-1996(2)}.
\bigskip
\noindent
{\it Strong cube ($\delta =1$):}
\medskip
\noindent
{F}rom (\ref{diam}) and
(\ref{tildeest})
we have, for strong cubes $\Omega$,
\[
\sum_{m=1}^N T_+^{\Omega}(P_+\theta f_m)=
\sum_{m=1}^N \int_{\Omega}
\left( |\bsigma\cdot(\bp + \bA)(P_+\theta f_m)|^2
- (V\delta^{-1} + cd^{-2} + ck_1^4 B^\#)|P_+\theta f_m|^2\right)
\]
\be
\geq c \sum_{m=1}^N (P_+\theta f_m, ((\bp +\tilde\bA)^2
- 2V - cd^{-2})P_+\theta f_m)=\mbox{Tr}(P_+\theta\gamma_N\theta P_+
H(2V+cd^{-2} )),
\label{gammatr2}\ee
where $H(2V+cd^{-2})= (\bp+\tilde\bA)^2 - 2V -cd^{-2}$ with
Dirichlet boundary
conditions on $\Omega$ and $\gamma_N$ was given (\ref{gammadef}).
Again using the operator inequality $0\leq
P_+\theta \gamma_N\theta P_+\leq 1$ we can estimate
$\mbox{Tr}(P_+\theta \gamma_N\theta P_+H(2V+cd^{-2}))$
below, using the variational principle, by the sum of the negativ
eigenvalues of $H(2V+cd^{-2})$. By the
standard Lieb-Thirring inequality (which is also valid for the
magnetic Laplacian by the diamagnetic inequality)
this eigenvalue sum is bounded below by
$-2^{5/2}L_{3,1}\int_{\Omega} (V + cd^{-2})^{5/2}$.
Therefore
\be
\sum_{m=1}^N T_+^{\Omega}(P_+\theta f_m)\geq
-c\int_{\Omega}(V+cd^{-2} )^{5/2}
\geq - c\int_{\Omega}V^{5/2} - cd^{-2}
\label{2strong'}\ee
\[
\geq - c\int_{\Omega}V^{5/2}
- ck_2 d^{_2}\left( \int_{\Omega} V^{3/2}\right)^{2/3}
\geq -c \int_{\Omega}V^{5/2}
-ck_2d^{-2}\left( d^{-1}\int_{\Omega} V
+ \int_{\Omega}|\nabla V| \right),
\]
again using Sobolev's inequality (\ref{Sobolev}) in the last step.
$\,\,\Box$
\subsection{Treating the spin-down part in a fixed cube}\label{Pmin}
Similarly to Section \ref{Pupsec}, first we focus on the contribution
$T_-^{\Omega_j}(f)$, of a single cube, in $T_-(f)$ (see (\ref{T})), so we
drop the indices (similarly for $P_-$ and
$\theta$).
\begin{theorem}\label{T-theorem}
We again consider a cube $\Omega$ with center $Q$ and edge length $10d$,
where $d=k_1\min
\{ \|B\|^{-1/4}l^{1/2}, L, l\}$, and let $P_-$ be the spin-down
projection according to the magnetic field in the center of $\Omega$.
Again, let
$\delta:= \min \{ 1, k_2\left(\int_{\Omega} V^{3/2}\right)^{2/3}\}$.
Define a quadratic form on spinors $g\in H^1_0(\Omega, \bC^2)$, with
$P_-g=g$,
\be
T_-^{\Omega}(g):=
\int_{\Omega} \left( |\bsigma\cdot(\bp + \bA )g|^2
- (V\delta^{-1} + cd^{-2})|g|^2 \right).
\label{T-def}\ee
Define
\be
w:= \min \{ k_3(B^\#)^{-1/2}, 10d \},
\label{w}
\ee
where $B^\#=\sup_{|x-Q|\leq200L}B(x)$.
There exists a universal constant $c'$ such that
if $k_1c'^{-1}$
then for any orthonormal family of spinors $\{ f_m \}_{m=1}^N$ and
for any $\theta \in C_0^{\infty}(\Omega)$, $0\leq \theta \leq 1$,
we have
\be
\delta \sum_{m=1}^N T_-^{\Omega}(P_-\theta f_m) \geq
-C_1(k_1,k_2,k_3)w^{-2}\left(
d^{-1}\int_\Omega V + \int_\Omega |\nabla V|\right)
\label{-weak}\ee
if $\delta < 1$ (weak cube), and
\be
\delta \sum_{m=1}^N T_-^{\Omega}(P_-\theta f_m)
=\sum_{m=1}^N T_-^{\Omega}(P_-\theta f_m) \geq
-c\int_\Omega V^{5/2} -C_2(k_1,k_2,k_3)w^{-2}
\int_\Omega V^{3/2}
\label{-strong}\ee
if $\delta = 1$ (strong cube). Here $C_{1,2}(k_1,k_2,k_3)$ are
positive constants depending on $k_1$, $k_2$, and $k_3$.
Note that the estimates are uniform in $N$.
\end{theorem}
Before the proof we state an important corollary.
\begin{corollary}\label{Pminuscor}
We may choose $k_1$ and $k_2$ (recall that
$T_-$ depends on $k_1$ and $k_2$ through $d$ and $\delta$)
such that
for any orthonormal family of spinors
$\{ f_m \}_{m=1}^N$ we have, uniformly in $N$,
\be
\sum_{m=1}^N T_-(f_m)
\label{Pminus}\ee
\[
\geq -c \int_{\bR^3}V^{5/2}
-c\left( \int_{{\bf R}^3}(B + d^{-2})V^{3/2}
+ \int_{{\bf R}^3} (B+d^{-2})d^{-1}\, V
+ \int_{{\bf R}^3}(B+d^{-2})|\nabla V| \right)
\]
\end{corollary}
\bigskip
{\it Proof of Corollary \ref{Pminuscor}.}
\bigskip
We can apply Theorem
\ref{T-theorem} for each cube $\Omega_j$ separately. Since
\[
\sum_{m=1}^N T_-(f_m)
= \sum_{m=1}^N\sum_j \delta_j T_-^{\Omega_j}(P_-^{(j)}\theta_j f_m)
\]
by (\ref{T}), using (\ref{-weak}), (\ref{-strong}) and
the universally bounded overlap of the
cubes, we obtain (\ref{Pminus}). In the calculation we used
\be
w^{-2} \leq c(k_3^{-2}B^\# + d^{-2}) \leq c(B (x)+d^{-2})
\ee
for any $x\in\Omega$ by (\ref{w}) and (\ref{sharfest}). $\,\,\,\Box$
\bigskip
The rest of this Section contains the proof of Theorem (\ref{T-theorem}).
This will involve several steps. First we need a further decomposition of
the cube
$\Omega$ into {\it cylinders} (Section~\ref{cylsect}). Then in
Section~\ref{splsect} the kinetic energy is considered within each
cylinder, and we treat the contributions from the lowest and the higher
Landau levels separately. The main result of that section is summarized
in Proposition~\ref{summ}. In Section \ref{puttog} we include the
potential. Finally, in Sections \ref{finlow} and \ref{finup} we treat the
lowest and the higher level contributions separately.
\subsubsection{Localizing into cylinders}\label{cylsect}
Throughout Sections \ref{splsect} and \ref{puttog},
$g$ will denote a
spinor with Dirichlet boundary conditions on $\Omega$ and with
$P_-g=g$. Recall that $P_-$ is the spin space projection
corresponding to the direction of the magnetic field in the center
of $\Omega$. In applications, $g$ will be $P_-\theta f$.
Corresponding to the cube $\Omega$ of side length
$\lambda=10d$ and the magnetic field $\bB$
we can, as in Section~\ref{apprsec},
construct cylinders $C_P$ of base radius $w$ given by (\ref{w}).
These cylinders were defined
by
\be
C_P= \{ \xi \, : \, |\xi_3|\leq 10d,
|\xi_{\perp} - P|\leq w\};
\label{cyldef}\ee
in terms of
the coordinate system $\{ \xi_i \}^3_{i=1}$
of Section~\ref{apprsec} and of the points
$P\in{\cal P}= \{ \xi \, |\, \xi_3 =0 \}$.
Choose a regular square lattice $\Lambda$ with spacing $w/2$ in the plane
${\cal P}$. We consider only the cylinders
$C_P$ corresponding to $P\in\Lambda$. We define also the {\it core} of
the cylinder
$C_P$ as
\be C_P^{(0)}: =\{ \xi \, : \, |\xi_3|\leq 10d, |\xi_{\perp} - P|\leq
w/2\}.
\label{cordef}\ee Recall that all cylinders (and cores) are perfectly
aligned.
Consider the finite subset of $\Lambda' \subset\Lambda$,
consisting of points $P$,
such that the corresponding core $C^{(0)}_P$ intersects the cube
$\Omega$. {F}rom now on we shall consider only cylinders and cores
corresponding to $P\in\Lambda'$.
It is straightforward from the construction above that the cores
$\{ C_P^{(0)} \}_{P\in \Lambda'}$ cover the whole cube $\Omega$.
It also follows from the geometry that the number of cylinders
that overlap at a fixed point is bounded by a universal constant.
If we assume that $k_1$ is small enough we can apply
Corollary~\ref{apprcor} and conclude that we can find a
constant magnetic field
$\tilde \bB_P$ such that
the magnetic field $ \bB_C^*(\xi): = \bB (\xi) - \tilde
\bB_P$ within
the cylinder $C=C_P$ satisfies the bound
$$
\Vert \bB^*_C \Vert_{L^{\infty}(C)} \leq
cl^{-1}dB^\#(1+d(l^{-1}+L^{-1}))
+wB^\#(l^{-1}+L^{-1}).
$$
If we use $w\leq 10d$ we obtain
\be
\Vert \bB^*_C \Vert_{L^{\infty}(C)} \leq ck_1 B^\#
\label{eq:Bstarsimple}\ee
and if we use $w\leq k_3 (B^\#)^{1/2}$ we find
\begin{eqnarray}
\Vert \bB^*_C \Vert_{L^{\infty}(C)}
&\leq&c(B^\#)^{1/2}d^{-1}\left(d^2(B^\#)^{1/2}l^{-1}(1+k_1)
+k_3k_1\right)\nonumber\\
&\leq& ck_1(k_1(1+k_1)+k_3)(B^\#)^{1/2}d^{-1}\label{Bstarsup}.
\end{eqnarray}
By (\ref{derest}) we get for the gradient
\be
\Vert \nabla \bB^*_C \Vert_{L^{\infty}(C)} =
\Vert \nabla \bB\Vert_{L^{\infty}(C)} \leq B^\#(l^{-1} + L^{-1})
\leq 2k_1B^\#d^{-1}.
\label{Bstargrad}\ee
Using Proposition~\ref{Aprop} and the
estimate (\ref{Aest}) we find that there exists a vectorpotential
$\bA_C^*$ (depending on $C$),
with $\nabla\times\bA^*_C = \bB_C^*$, such that
\be
\Vert \bA^*_C \Vert_{L^{\infty}(C)} \leq
c\left[k_1(k_1(1+k_1)+k_3)(B^\#)^{1/2}d^{-1}w
+ k_1B^\#d^{-1}w^2\right]\leq Kd^{-1}
\label{Astarest}\ee
using (\ref{w}),
where
\be
K:= ck_1k_3[k_1(k_1+1) + k_3].
\label{K}\ee
Define $\bA^{(0)}_C(\xi):=\frac{1}{2}\tilde\bB_P \times (\xi -P)$
the standard gauge depending on $C$, which
generates the constant field $\tilde\bB_P$
in the $\{\xi_i\}$ coordinate system (determined by $\Omega$).
Note that $\bA^{(0)}_C = (A_{C,\xi_1}^{(0)}, A_{C,\xi_2}^{(0)}, 0)$, and
$\bA^{(0)}_C$ depends only on $\xi_1$ and $\xi_2$.
The direction of $\tilde\bB_P$ is independent of $P$
since it has the same direction as $\bB (Q)$.
The notion of spin-up and spin-down is therefore independent of the cylinder.
Following Section \ref{susysection},
let us define the operator $T_C:= (p_{\xi_1} + A_{C,\xi_1}^{(0)}) - i
(p_{\xi_2} + A_{C,\xi_2}^{(0)})$ on $C$ with Neumann boundary condition
(more precisely, $D(T_C):= H^1(C)$).
Let $\Pi_C$ be the zero level spectral projection of $T_C^*T_C$.
{F}rom (\ref{eq:susygap}) we get
\be
(I-\Pi_C)T_C^*T_C(I-\Pi_C) \geq 2\tilde B_P(I-\Pi_C)
\label{eq:susygap1}\ee
and from (\ref{susy1})
\be
T^*_CT_C = \sum_{j=1}^2(p_{\xi_j}+A^{(0)}_{C,\xi_j})^2 -\tilde B_P,
\label{eq:Lichne}\ee
on $D(T_C^*T_C)$ defined in Section~\ref{susysection}.
Note that as in (\ref{Blowpr}) we get from (\ref{eq:Bstarsimple})
that
\be
\tilde cB^\#\leq \tilde B_P\leq (\tilde c)^{-1}B^\#.
\label{Blowpr1}
\ee
We need the following lemma to estimate the kinetic energy
within the cylinders.
We remark, that the operators $T_C$ and $\Pi_C$ are apriori
defined on a subspace of $L^2(C)$ but they naturally
act on the corresponding subspace
of the spinor space $L^2(C, {\bf C}^2)$ as well,
acting separately on both components. We shall not indicate
this fact, i.e., in the notations, we do not distinguish
between $T_C$ and $T_C \otimes \mbox{Id}_{{\bf C}^2}$.
\begin{lemma}\label{cyllemma}
For each cylinder $C=C_P$ with $P\in\Lambda'$
there exists a gauge function $\varphi_C:C\to\bR$ such that
the following lower bound holds
for any $g$ with $P_-g=g$
\be
\int_C |\bsigma\cdot (\bp +\bA)g|^2
\geq
c\int_C \left( |T_C e^{i\varphi_C}g|^2
+ |p_{\xi_3}e^{i\varphi_C}g|^2\right) - cK^2d^{-2}\int_C |g|^2.
\label{Pminusf}\ee
\end{lemma}
{\it Proof.}
The vectorpotential $\bA^{(0)}_C + \bA^*_C$ generates the
original field $\bB$ within $C$. Therefore there exists
a real function $\varphi_C$ (depending on $C$) on
the cylinder $C$ such that $\bA^{(0)}_C + \bA^*_C +\nabla\varphi_C =
\bA$, and
\be
\int_C |\bsigma\cdot (\bp + \bA) g|^2 =
\int_C |\bsigma\cdot (\bp + \bA^{(0)}_C + \bA^*_C )e^{i\varphi_C}g|^2.
\label{gaugetr}\ee
Now we can use a Cauchy-Schwarz inequality to estimate
\be
\int_C |\bsigma\cdot (\bp +\bA)g|^2
\geq \frac{1}{2} \int_C |\bsigma\cdot (\bp +\bA^{(0)}_C)e^{i\varphi_C}g|^2
- 2\,(\sup_C |\bA^*_C|)^2 \int_C |g|^2
\label{Astarsep}\ee
\[
\geq\frac{1}{2} \int_C |\bsigma\cdot (\bp +\bA_C^{(0)})e^{i\varphi_C}g|^2
- cK^2d^{-2}\int_C |g|^2.
\]
Using $P_-g = g$, the kinetic energy term splits:
\be
\int_C |\bsigma\cdot (\bp +\bA^{(0)}_C)e^{i\varphi_C}g|^2
\label{thirdsep}\ee
\[
=\int_C \left(
|[\sigma_{\xi_1}(p_{\xi_1} + A^{(0)}_{C,\xi_1}) + \sigma_{\xi_2}
(p_{\xi_2}+A^{(0)}_{C,\xi_2})]e^{i\varphi_C}g|^2 +
|p_{\xi_3}e^{i\varphi_C}g|^2\right) ,
\]
with $p_{\xi_i} = \sum_j e_{\xi_i}^jp_j$, and similarly for
$\sigma_{\xi_i} = \sum_j e_{\xi_i}^j\sigma_j$;
where $e_{\xi_i} = (e_{\xi_i}^1, e_{\xi_i}^2, e_{\xi_i}^3)$
is the $i^{th}$
unit coordinate vector of the $\{\xi_i \}$ system in the standard
basis. The proof of (\ref{thirdsep}) is obvious by
\be
\int_C |\bsigma\cdot (\bp +\bA^{(0)}_C)e^{i\varphi_C}g|^2 =
\int_C |\sum_{j=1}^3
\sigma_{\xi_j}(p_{\xi_j} + A^{(0)}_{C,\xi_j})e^{i\varphi_C}g|^2
\label{baschange}\ee
and by $P_- = (1-\sigma_{\xi_3})/2$, as an easy calculation
shows that the cross terms,
involving $p_{\xi_3}$, are all zero. For example, if $v$ and $w$ are
vectors such that $P_-v=v$ and $P_-w=w$, i.e. $\sigma_{\xi_3}v=-v$
and $\sigma_{\xi_3}w=-w$, then
\be
\overline{\sigma_{\xi_2}v}\cdot\sigma_{\xi_3}w = -\overline{\sigma_{\xi_2}v}
\cdot w = \overline{\sigma_{\xi_2}\sigma_{\xi_3}v}\cdot w =
- \overline{\sigma_{\xi_3}\sigma_{\xi_2}v}\cdot w =
-\overline{\sigma_{\xi_2}v}\cdot\sigma_{\xi_3}w
\label{triv}\ee
shows that the cross terms vanish, where $v$ or $w$ can be any vector
of the form $(p_{\xi_j} + A^{(0)}_{C,\xi_j})e^{i\varphi_C}g$.
By $P_-g =g$, one sees
\[
\int_C |[\sigma_{\xi_1}(p_{\xi_1} + A^{(0)}_{C,\xi_1}) + \sigma_{\xi_2}
(p_{\xi_2}+A^{(0)}_{C,\xi_2})]e^{i\varphi_C}g|^2
\]
\be
=\int_C |[(p_{\xi_1} + A_{C,\xi_1}^{(0)})
-i(p_{\xi_2} + A_{C,\xi_2}^{(0)})]e^{i\varphi_C}g|^2 =\int_C |Te^{i\varphi_C}
g|^2,
\label{spinrot}\ee
since for any two vectors $v$ and $w$ we have, by an easy calculation, that
\be
|\sigma_{\xi_1}P_-v + \sigma_{\xi_2}P_-w|^2 = |P_-(v-iw)|^2.
\label{triv2}\ee
Putting together (\ref{Astarsep}),
(\ref{thirdsep}) and (\ref{spinrot}) we obtain (\ref{Pminusf}). $\,\,\Box$
\subsubsection{Splitting into lower and upper Landau levels}\label{splsect}
In this section we split the kinetic energy within the cube
$\Omega$ into contributions from lower and higher `local' Landau
levels. This means that first we
have to localize the kinetic energy in each cylinder $C:=C_P$
and consider the constant magnetic field $\tilde \bB_P$
within the cylinder .
\begin{proposition}\label{Pminussepprop}
There exist gauge functions $\varphi_C :C\to \bR$ for each cylinder
$C$ intersecting $\Omega$
such that for any spinor $g$ satisfying Dirichlet boundary conditions on
$\Omega$ ($g\in H^1_0(\Omega, \bC^2)$), and $P_- g=g$ we have with
$K$ as defined in (\ref{K})
\be
\int_{\Omega}|\bsigma\cdot (\bp +\bA)g|^2 \geq
c \sum_{C}\int_C |\bsigma\cdot (\bp +\bA)g|^2
\geq c\sum_{C} T_{up}^{\Omega}(g, C)
\label{Pminussep}\ee
\[
+c\sum_{C}\int_C
\left( |p_{\xi_3}\Pi_C e^{i\varphi_C}\chi_Cg|^2
-cK^2d^{-2} \chi_{\Omega}
\chi_{C^{(0)}}|\Pi_C e^{i\varphi_C}\chi_Cg|^2\right)
\]
with
\be
T_{up}^{\Omega}(g, C):=
\label{Tup}\ee
\[
\int_C \left( |T_C(I-\Pi_C)e^{i\varphi_C}\chi_Cg|^2
+ |p_{\xi_3}(I-\Pi_C)e^{i\varphi_C}\chi_Cg|^2
-cK^2d^{-2} \chi_{\Omega}
\chi_{C^{(0)}}|(I-\Pi_C)e^{i\varphi_C}\chi_Cg|^2\right).
\]
Here $\chi_{\Omega}$ and $\chi_{C^{(0)}}$ denote the
characteristic functions of the cube and the core of the cylinder
respectively. The sum $\sum_C$ represents a sum over the family
$\{C=C_P\}_{P\in\Lambda'}$.
\end{proposition}
{\it Remark.}
The importance of introducing the core of the cylinders is
due to the singularity in the spectral density of $\Pi_C$ near
the boundary of $C$ (see (\ref{diagest}).
{\it Proof of Proposition \ref{Pminussepprop}}.
Since the cylinders $C$
cover the cube $\Omega$, with a universally bounded overlap, we have
\be
\int_{\Omega} |\bsigma\cdot (\bp + \bA)g|^2 \geq
c\sum_{C} \int_C |\bsigma\cdot (\bp + \bA)g|^2 .
\label{split}\ee
We estimate the right side of (\ref{split}) using
Lemma~\ref{cyllemma}.
Since the {\it cores} also cover the all of $\Omega$, we
can reallocate the error in (\ref{Pminusf}) to get
\be
\sum_{C}\int_C |\bsigma\cdot (\bp +\bA)g|^2
\label{Pminusfinal}\ee
\[
\geq \sum_C\left[c\int_C \left( |[(p_{\xi_1} + A^{(0)}_{C,\xi_1}) -i
(p_{\xi_2}+A^{(0)}_{C,\xi_2})] e^{i\varphi_C}g|^2
+ |p_{\xi_3}e^{i\varphi_C}g|^2\right) - cK^2d^{-2}\int_C
\chi_{\Omega}\chi_{C^{(0)}}|g|^2\right],
\]
where we have inserted $\chi_{\Omega}$, the characteristic
function of the cube $\Omega$, since $g$ is
supported in $\Omega$. Note that as a consequence
of the reallocation it is the
characteristic function $\chi_{C^{(0)}}$ which appears in
the error term.
Let $\chi_C$ be the
characteristic function of the cylinder $C$; instead of $g$ we shall
use $\chi_C g$ in order to be in the domain of $T_C$ and $\Pi_C$.
{F}rom (\ref{Pminusfinal}) we obtain (recalling that $p_{\xi_3}$ and
$\Pi_C$ commute)
\be
\sum_{C}\int_C |\bsigma\cdot (\bp +\bA)g|^2
\label{Pminussep1}\ee
\[
\geq
c\sum_{C}\int_C \left( |T_C\Pi_C e^{i\varphi_C}\chi_Cg|^2
+ |p_{\xi_3}\Pi_C e^{i\varphi_C}\chi_Cg|^2
-cK^2d^{-2} \chi_{\Omega}\chi_{C^{(0)}}
|\Pi_C e^{i\varphi_C}\chi_Cg|^2\right)
\]
\[
+c\sum_{C}\int_C
\left( |T_C(I-\Pi_C)e^{i\varphi_C}\chi_Cg|^2
+ |p_{\xi_3}(I-\Pi_C)e^{i\varphi_C}\chi_Cg|^2
-cK^2d^{-2} \chi_{\Omega}\chi_{C^{(0)}}
|(I-\Pi_C)e^{i\varphi_C}\chi_Cg|^2\right),
\]
using the pointwise inequality on $C$
\be
|g|^2 = |e^{i\varphi_C}\chi_Cg|^2
\label{Pisplit}\ee
\[
\leq 2|\Pi_Ce^{i\varphi_C}\chi_Cg|^2
+2|(I-\Pi_C)e^{i\varphi_C}\chi_Cg|^2.
\]
Notice that the first term on the right hand side of (\ref{Pminussep1}),
$|T_C\Pi_C e^{i\varphi_C}\chi_Cg|^2$,
is simply zero as $\Pi_C$ projects onto the zero level of $T^*_CT_C$.
This completes the proof of (\ref{Pminussep}). $\,\,\Box$
\bigskip
We next rewrite
the upper spectral part,
$T_{up}^{\Omega}(g, C)$ in order to introduce Dirichlet boundary condition.
This rewriting will
give further contributions to the lower spectral part
(i.e., to the error term containing $\Pi_C$ in (\ref{Pminussep})).
We use a further localization function.
For each cylinder $C=C_P$ we choose a function
$\theta_C\in C^\infty(\bR^3)$,
$0\leq \theta_C\leq 1$, with the properties
\be
\theta_C \,\,\mbox{depends only on}\,\, \xi_{\perp},
\,\, \theta_C \equiv 1 \,\,\mbox{on} \,\, C_P^{(0)},
\,\, |\nabla \theta_C|\leq cw^{-1}
\label{chiperp}\ee
\be
\theta_C\equiv 0 \,\,\, \mbox{for} \,\,\, |\xi_{\perp}-P| \geq w.
\label{supp}\ee
Then, the following lower bound is valid.
\begin{lemma}\label{Tuplemma} There exists $c'>0$ such that if
$k_1< c'$
and $k_3>c'^{-1}$ then for any $g\in H_0^1
(\Omega, \bC^2)$, with $P_-g=g$
\be T_{up}^{\Omega}(g, C) \geq c \int_C \left( |(\bp + \bA^{(0)}_C)
\theta_C(I-\Pi_C)e^{i\varphi_C}\chi_Cg|^2
-c(K^2+1)d^{-2}\chi_{\Omega}|g|^2\right),
\label{Pminup}\ee
where $K$ is given in (\ref{K}).
\end{lemma}
The kinetic energy term is just the magnetic spinless
Laplacian, since $\theta_C$
ensures Dirichlet boundary condition (recall that $g$ is zero outside
of $\Omega$, in particular it automatically satisfies Dirichlet
boundary condition at the ends of the cylinder, since the
cylinders are longer than the cube). The error term will later
(in the next Section~\ref{puttog}) be
reallocated into the cores of the cylinders, where the localization
functions are identically 1.
\bigskip
{\it Proof of Lemma \ref{Tuplemma}}.
\bigskip
Using (\ref{eq:susygap1}) and (\ref{eq:Lichne}) one easily obtains
the operator inequality
\be
(I-\Pi_C)T_C^*T_C(I-\Pi_C) \geq \frac{1}{3}
(1-\Pi_C)(\sum_{j=1}^2(p_{\xi_j}+A^{(0)}_{C,\xi_j})^2 +
3\tilde B_P)(I-\Pi_C)
\label{Ptrickop}\ee
on $D(T^*_CT_C)$.
Using the lower bound in (\ref{Blowpr1}) we find
\be
\int_C |T_C(I-\Pi_C)\phi|^2 \geq \frac{1}{3}\int_C \sum_{j=1}^2
|(p_{\xi_j}+A^{(0)}_{C,\xi_j})(I-\Pi_C)\phi|^2 + \tilde cB^\#
\int_C |(I-\Pi_C)\phi|^2
\label{Ptrickform}\ee
for any $\phi \in D(T_C)$ (note that $I-\Pi_C$ maps $D(T_C)$ into itself,
since $\Pi_C$ maps the whole $L^2(C)$ into $D(T_C)$ as $T_C\Pi_C\phi=0$
for any $\phi\in L^2(C)$).
Now we insert $\theta_C$ (which
commutes with $p_{\xi_3}$) in (\ref{Ptrickform}), and obtain
\begin{eqnarray}
\lefteqn{\int_C |T_C(I-\Pi_C)\phi|^2 \geq \frac{1}{3}\int_C \sum_{j=1}^2\theta_C^2
|(p_{\xi_j}+A^{(0)}_{C,\xi_j})(I-\Pi_C)\phi|^2 + \tilde cB^\#
\int_C |(I-\Pi_C)\phi|^2}&&\nonumber\\
&\geq & \frac{1}{6}\int_C \sum_{j=1}^2
|(p_{\xi_j}+A^{(0)}_{C,\xi_j})\theta_C(I-\Pi_C)\phi|^2
-\frac{1}{3} \int_C |\nabla\theta_C|^2 |(I-\Pi_C)\phi|^2
+ \tilde cB^\#\int_C |(I-\Pi_C)\phi|^2\nonumber\\
&\geq & \frac{1}{6}\int_C \sum_{j=1}^2
|(p_{\xi_j}+A^{(0)}_{C,\xi_j})\theta_C(I-\Pi_C)\phi|^2
+\left((\tilde c-ck_3^{-2})B^\# -cd^{-2}\right) \int_C |(I-\Pi_C)\phi|^2\nonumber\\
&\geq & \frac{1}{6}\int_C \sum_{j=1}^2
|(p_{\xi_j}+A^{(0)}_{C,\xi_j})\theta_C(I-\Pi_C)\phi|^2
-cd^{-2} \int_C |(I-\Pi_C)\phi|^2,\label{Ptrickform1}
\end{eqnarray}
using (\ref{chiperp}), (\ref{w}),
and assuming, in the last step, that the constant $k_3$ is sufficiently large.
We shall substitute (\ref{Ptrickform1}) into the
definition of $T_{up}^{\Omega}(g, C)$ (see (\ref{Tup})).
First, using that $\theta_C$ commutes with $p_{\xi_3}$,
we get (choose $\phi$ to be the components of the
spinor $ e^{i\varphi_C}\chi_Cg$ in (\ref{Ptrickform1})
and use that the operators $\Pi_C$, $T_C$ etc. are diagonal in the spin
space, i.e. they act separately on both components of a spinor)
\[
T_{up}^{\Omega}(g, C)
\geq
c\int_C \left( \sum_{j=1}^2|(p_{\xi_j}+A^{(0)}_{C,\xi_j})
\theta_C(I-\Pi_C)e^{i\varphi_C}\chi_Cg|^2
+ |p_{\xi_3}\theta_C(I-\Pi_C)e^{i\varphi_C}\chi_Cg|^2
\right.
\]
\[
\left.
- c(K^2+1)d^{-2} |(I-\Pi_C)e^{i\varphi_C}\chi_Cg|^2 \right)
\]
\be
\geq c \int_C \left(
|(\bp + \bA^{(0)}_C) \theta_C(I-\Pi_C)e^{i\varphi_C}\chi_Cg|^2
-c(K^2+1)d^{-2}|(I-\Pi_C)e^{i\varphi_C}\chi_Cg|^2\right).
\label{perpins}\ee
In formula (\ref{perpins}) we have neglected $\chi_{C^{(0)}}$ in the
error term, as we shall have to reallocate that error term again anyway.
The last term in (\ref{perpins}) is estimated trivially,
as
\be
\int_C |(I-\Pi_C)e^{i\varphi_C}\chi_Cg|^2 \leq \int_C |g|^2,
\label{backproj}\ee
using that $I-\Pi_C$ is a
projection on $L^2(C)$; and we get (\ref{Pminup}) from (\ref{perpins})
since we can
insert $\chi_{\Omega}$ for free, as $g\chi_{\Omega}=g$. $\,\,\Box$
\bigskip
The result of this section is summarized in the following
kinetic energy bound.
Its proof is straightforward from
(\ref{Pminussep}), (\ref{Tup}) and (\ref{Pminup}).
\begin{proposition}\label{summ}
There exist a universal constant $c'>0$ and a constant $K_1$
depending on $k_1$ and $k_3$ such that if $k_1c'^{-1}$ then
for any spinor $g \in H^1_0(\Omega, {\bf C}^2)$,
supported in $\Omega$ and satisfying $P_-g=g$, we have
\be
\int_{\Omega} |\bsigma\cdot (\bp + \bA) g|^2 \geq
\sum_{C } \int_C c|\bsigma\cdot (\bp + \bA) g|^2
\ee
\[
\geq \sum_{C} \int_C
\left(c |p_{\xi_3}\Pi_C e^{i\varphi_C}\chi_Cg|^2
-K_1d^{-2} \chi_{\Omega}\chi_{C^{(0)}}|\Pi_C e^{i\varphi_C}\chi_Cg|^2\right)
\]
\[
+ \sum_{C}
\int_C \left( c|(\bp + \bA^{(0)}_C) \theta_C(I-\Pi_C)e^{i\varphi_C}\chi_C g|^2
- K_1d^{-2}\chi_{\Omega} |g|^2\right). \,\,\Box
\]
\end{proposition}
\subsubsection{Including the potential}\label{puttog}
Now we are ready to estimate $T_-^{\Omega}(g)$ (see (\ref{T-def}))
completely
in terms of the contributions from the lowest and higher Landau levels.
The result is
\be
T_-^{\Omega}(g) \geq
\sum_{C} \int_C
\left\{ c|p_{\xi_3}\Pi_C e^{i\varphi_C}\chi_Cg|^2
-K_1d^{-2} \chi_{\Omega}
\chi_{C^{(0)}}|\Pi_C e^{i\varphi_C}\chi_Cg|^2
\right\}
\label{lastline1}\ee
\[
+ \sum_{C}
\int_C \left\{ c |(\bp + \bA^{(0)}_C)
\theta_C(I-\Pi_C)e^{i\varphi_C}\chi_Cg|^2
- K_1d^{-2}\chi_{\Omega}|g|^2 \right.
\]
\[
\left.
-(V\delta^{-1} + cd^{-2})\chi_{\Omega}\chi_{C^{(0)}}
|g|^2 \right\}
\]
(in the last line we used that the cores of the
cylinders, belonging to a cube, cover it,
therefore the potential term in $T_-^{\Omega}(g)$ can
be allocated into the cores, and we again inserted $\chi_{\Omega}$ for free).
After reallocating the $-K_1 d^{-2}\chi_{\Omega}| g|^2$ term into
the cores and including it into the other error term, we have
\be
T_-^{\Omega}(g)\geq
\sum_{C} \int_C
\left\{ c|p_{\xi_3}\Pi_C e^{i\varphi_C}\chi_Cg|^2
-K_1d^{-2} \chi_{\Omega}
\chi_{C^{(0)}}|\Pi_C e^{i\varphi_C}\chi_Cg|^2
\right\}
\label{lastline3}\ee
\[
+ \sum_{C}
\int_C \left\{ c
|(\bp +\bA^{(0)}_C)\theta_C(I-\Pi_C)e^{i\varphi_C}\chi_Cg|^2
-(V\delta^{-1} + (K_1+c)d^{-2})\chi_{\Omega}
\chi_{C^{(0)}}|g|^2\right\},
\]
Now using the pointwise inequality
\be
|\psi|^2 = |e^{i\varphi_C}\chi_C \psi|^2 \leq 2 |\Pi_C
e^{i\varphi_C}\chi_C \psi|^2
+2|(I-\Pi_C)e^{i\varphi_C}\chi_C \psi|^2
\label{pointw}\ee
for any $\psi\in H^1(C, {\bf C}^2)$, we get
\begin{eqnarray}
T_-^{\Omega}(g)&\geq&
\sum_{C} \int_C
\left\{ c|p_{\xi_3}\Pi_C e^{i\varphi_C}\chi_Cg|^2
- c\sum_{C}
\int_C \left( V\delta^{-1} + (K_1+c)d^{-2}\right)
\chi_{\Omega}\chi_{C^{(0)}} |\Pi_Ce^{i\varphi_C}\chi_Cg|^2\right\}
\nonumber\\
&&+ \sum_{C}
\int_C \biggl\{ c |(\bp + \bA^{(0)}_C)
\theta_C(I-\Pi_C)e^{i\varphi_C}\chi_Cg|^2
\nonumber\\
&&-c(V\delta^{-1} + (K_1+c)d^{-2})
\chi_{\Omega}\chi_{C^{(0)}}|(I-\Pi_C)e^{i\varphi_C}\chi_Cg|^2
\biggr\}.\label{lastline4}
\end{eqnarray}
Finally, in the last line we can estimate $\chi_{C^{(0)}}$ by
$\theta_C^2$, since $\theta_C$ is 1 on the core.
We can summarize our result in the following Proposition.
\begin{proposition}\label{togprop}
There exists a universal constant $c'>0$ such that if
$k_1c'^{-1}$ we obtain
for any spinor $g\in H^1_0 (\Omega, \bC^2)$, with $P_-g=g$
\be
T_-^{\Omega}(g)\geq
T_-^{\Omega, \, low}(g) + T_-^{\Omega, \, up}(g)
\label{lastline5}\ee
with
\be
T_-^{\Omega, \, low}(g) :=
\sum_{C} \int_C \left\{
c|p_{\xi_3}\Pi_C e^{i\varphi_C}\chi_Cg|^2
- U \chi_{C^{(0)}} |\Pi_Ce^{i\varphi_C}\chi_Cg|^2
\right\},
\label{T-low}\ee
\be
T_-^{\Omega, \, up}(g):= \sum_{C}
\int_C \left\{ c|(\bp +\bA_C^{(0)})
\theta_C(I-\Pi_C)e^{i\varphi_C}\chi_Cg|^2
-U |\theta_C(I-\Pi_C)e^{i\varphi_C}\chi_C g|^2
\right\} ,
\label{T-up}\ee
where, for shortness, we denoted
\be
U:=c(V\delta^{-1} + K_2d^{-2})\chi_{\Omega},
\label{U}\ee
with $K_2$ depending on $k_1$ and $k_3$.
Note that $U$ depends only on the cube (via $\delta$ and $\chi_{\Omega}$)
but not on the cylinders. $\,\,\Box$
\end{proposition}
\subsubsection{Finishing the lowest Landau level}\label{finlow}
The contribution from $T_-^{\Omega ,\, low}(g)$
is estimated as follows.
\begin{lemma}\label{lowerlemma}
There exist a universal constant $c'>0$ and a constant $K_3$
depending on $k_1$, $k_2$ and $k_3$ such that if $k_10$ and a constant $K_4$
depending on $k_1$, $k_2$ and $k_3$ such that if
$k_1c'^{-1}$ then for any orthonormal family of
spinors $\{ f_m \}_{m=1}^{\infty}$ and
$\theta\in C^{\infty}_0 (\Omega)$, $0\leq \theta \leq 1$ we obtain
\be
\sum_{m=1}^N T_-^{\Omega, \, up}(P_-\theta f_m)
\geq -c \int_{\Omega}V^{5/2} -
K_4d^{-2}
\int_{\Omega}V^{3/2},
\label{upperstr}\ee
if $\Omega$ is a {\it strong} cube
and
\be
\delta\sum_{m=1}^N T_-^{\Omega, \, up}(P_-\theta f_m)
\geq -K_4d^{-2}
\left( d^{-1}\int_{\Omega} V
+ \int_{\Omega}|\nabla V| \right).
\label{upperweak}\ee
if $\Omega$ is a {\it weak} cube.
\end{lemma}
\bigskip
\noindent
{\it Proof of Lemma \ref{upperlemma}}.
\bigskip
As in the proof of Lemma \ref{lowerlemma}, we can write
\be
\sum_{m=1}^N T_-^{\Omega, \, up}(P_-\theta f_m)
\label{upper}\ee
\[
=\sum_{m=1}^N\sum_{C} \int_C
\left\{c|(\bp +\bA_C^{(0)})\theta_C(I-\Pi_C)e^{i\varphi_C}\chi_CP_-
\theta f_m|^2
- U|\theta_C(I-\Pi_C)e^{i\varphi_C}\chi_C P_-\theta f_m|^2\right\}
\]
\[
=
\sum_{C} \mbox{Tr} (H_C^{up}\gamma_{C,N}^{up}),
\]
where
\be
H_C^{up}:= (\bp + \bA^{(0)}_C)^2 -cU\chi_C
\label{hcup}\ee
with Dirichlet boundary conditions on $C$, and
\be
\gamma_{C,N}^{up} := \theta_C(I-\Pi_C)
e^{i\varphi_C}\chi_CP_-\theta
\left(\sum_{m=1}^N |f_m\rangle\langle f_m|\right)
\theta P_-\chi_Ce^{-i\varphi_C}(I-\Pi_C)\theta_C
\label{gammaup}\ee
is an operator (density matrix) between 0 and 1.
We have to consider the cases that $\Omega$ is strong or weak separately.
If $\Omega$ is a {\it strong cube} (i.e. $\delta =1$),
using the variational principle,
the standard magnetic Lieb-Thirring inequality, the
fact that the cylinders have universally bounded overlap,
and finally the definition of $U$, we get
\be
\sum_{m=1}^N T_-^{\Omega, \, up}(P_-\theta f_m)
=\sum_{C} \mbox{Tr}(H_C^{up}
\gamma^{up}_{C,N})\geq
-c \sum_{C} \int_C U^{5/2}
\geq -c\int_{\Omega}U^{5/2}
\label{upperstr'}\ee
\[
\geq -c \int_{\Omega}\left( V^{5/2}+K_2^{5/2}d^{-5}\right)
=-c \int_{\Omega}V^{5/2}
-cK_2^{5/2}d^{-2}
\]
\[
\geq -c \int_{\Omega}V^{5/2} - cd^{-2}k_2^{3/2}K_2^{5/2}
\int_{\Omega}V^{3/2},
\]
where, in the last step, we used that for strong cubes
$k_2^{3/2}\int_{\Omega}V^{3/2}\geq 1$. This proves (\ref{upperstr}).
\bigskip
For $\Omega$ being a {\it weak cube}
(i.e. $\delta < 1$), the operator $H_C^{up}=(\bp +\bA^{(0)}_C)^2 -
cU\chi_C$ with Dirichlet b.c. on $C$ is bounded from below by
$-cK_2d^{-2}$ since $(\bp +\bA^{(0)}_C)^2 -
cV\delta^{-1}\chi_{\Omega}\chi_C$,
defined on $C$ with Dirichlet boundary conditions, is nonnegative
if $k_2$ is large enough. For, by the CLR bound (recall that the
CLR bound is true for spinless magnetic operator as well),
\be L_{3,1}\int_C(cV\delta^{-1}\chi_{\Omega})^{3/2} \leq
cL_{3,1}\delta^{-3/2}
\int_{\Omega}V^{3/2} = cL_{3,1}k_2^{-3/2} < 1
\label{CLR}\ee
if $k_2$ satisfies
$ cL_{3,1}k_2^{-3/2} <1$.
The dimension of the spectral projection $\Pi_{\{ H_C^{up} < 0\}}$ onto
the negative spectrum of $H_C^{up}$ is estimated, by the CLR-bound, as
\be
\mbox{Tr}\,\Pi_{\{ H_C^{up} < 0\}}\leq
cL_{3,1}\int_C \chi_{\Omega}(V\delta^{-1} + K_2d^{-2})^{3/2}
\leq cK_2^{3/2}w^2d^{-2} +c\delta^{-3/2}
\int_C \chi_{\Omega}V^{3/2},
\label{number}\ee
where we used that the volume of the cylinder is $|C|\leq
cdw^2$.
We have a lower bound for each of these eigenvalues, therefore, for $\Omega$
weak,
\begin{eqnarray}
\sum_{m=1}^N T_-^{\Omega, \, up}(P_-\theta f_m)
&= &\sum_{C}\mbox{Tr}
\left( H_C^{up} \gamma^{up}_{C,N}\right) \geq
-cK_2d^{-2}\sum_{C}
\mbox{Tr} \,\Pi_{\{ H_C^{up} <0 \} }
\label{upperweak'}\\
&\geq &-c \sum_{C}
K_2d^{-2}
\left( cK_2^{3/2}w^2d^{-2} +c\delta^{-3/2}
\int_C \chi_{\Omega}V^{3/2}\right)
\nonumber\\
&\geq& -cK_3^{5/2}w^2d^{-4}\left(\frac{cd^2}{w^2}\right)
-c\delta^{-3/2} K_2d^{-2}\int_\Omega \chi_{\Omega}V^{3/2}
\nonumber\\
&\geq&-cK_4d^{-2}\delta^{-1}
\left(\int_{\Omega}V^{3/2}\right)^{2/3}
\nonumber\\
&\geq&-cK_4d^{-2}\delta^{-1}
\left(d^{-1}\int_{\Omega}V + \int_\Omega |\nabla V|\right),
\end{eqnarray}
where, from the second to the third line, we used that the number
of cylinders belonging to a given cube is bounded by
$cd^2w^{-2}$ by elementary geometry. We used, moreover, that
the cylinders have a finite overlap. We also
used the definition of $\delta$ twice, and, in the last line, we
applied Sobolev's inequality (\ref{Sobolev}). $\,\,\Box$
\subsubsection{Completing the treatment of the spin-down part}
\label{sec:compl}
To obtain Theorem~\ref{T-theorem}, we use
Proposition~\ref{togprop} and Lemmas~\ref{lowerlemma} and
\ref{upperlemma}.
Using the estimate
(\ref{lastline5}) one easily
obtains (\ref{-weak}) from (\ref{lowerweak}) and
(\ref{upperweak}) by a simple
calculation and using $w\leq 10d$.
Similarly, (\ref{lowerstr})
and (\ref{upperstr}) imply (\ref{-strong}). This completes
the proof of Theorem~\ref{T-theorem}. $\,\,\Box$
\subsection{Completing the proof of the Lieb-Thirring theorem }
\label{end}
Using Lemma~\ref{alloclemma} we may assume that $V$ satisfies
(\ref{propV2}).
Obviously Lemma~\ref{lm:fullspl} and
Corollaries~\ref{Ppluscor} and \ref{Pminuscor} then give our
Lieb-Thirring inequality (\ref{LT}).
$\,\,\Box$.
\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.
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\end{document}