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\centerline{\bf DIAMAGNETIC BEHAVIOR OF SUMS OF DIRICHLET EIGENVALUES}
\bigskip\bigskip
{\baselineskip=2.5ex
\vfootnote{}{\eightpoint
\noindent\copyright 1999 by the authors.
Reproduction of this article, in its entirety, by any means is permitted
for non-commercial purposes.}}
{\baselineskip = 12pt
\halign{\qquad\qquad\quad#\hfil\qquad\hfil&#\hfil\qquad\hfil&#\hfil\cr
{\bf L\'aszl\'o Erd\H os }
\footnote {$^*$}{Work supported by N.S.F. grant DMS-9970323}&
{\bf Michael Loss}
\footnote {$^{**}$}{Work supported by N.S.F. grant DMS-9500840}&
{\bf Vitali Vougalter{$^{**}$}}
\cr}}
\medskip\bigskip
\centerline{ School of Mathematics,}
\centerline{Georgia Institute of Technology,}
\centerline{Atlanta, GA 30332, USA}
\bigskip
\bigskip
\bigskip
{\bf Abstract.}
The Li-Yau semiclassical
lower bound for the sum of the first $N$ eigenvalues of the
Dirichlet--Laplacian is extended to Dirichlet--
Laplacians with constant magnetic fields . Our method involves a new
diamagnetic inequality for constant magnetic fields.
\bigskip
\bigskip
\bigskip
{\bf 1. Introduction}
\bigskip
When studying Schr\"odinger operators with magnetic fields, the diamagnetic
inequality is the only general tool available for apriori estimates and
comparisons with the free Laplacian.
In its simplest form it says that
$$
%\int_{\Bbb{R}^n}
\Big|\left( -i \nabla + A(x) \right)\psi(x)\Big|^2
%{\rm d} x
\geq
%\int_{\bR^n}
\Big|\nabla |\psi(x)|\Big|^2
% {\rm d} x
\ ,\eqno(1.1)
$$
but it also appears in the form of the following estimates on the
heat kernel and the Green's function
$$
\Big| e^{- t(-i\nabla+A)^2} (x,y)\Big|
\leq e^{\Delta t} (x,y) \qquad t>0\ ,
\eqno(1.2)
$$
and
$$
\Bigg|{1 \over (-i \nabla + A)^2+E}(x,y)\Bigg|
\leq {1 \over -\Delta +E}(x,y)\ , \qquad E\ge 0 .\eqno(1.3)
$$
(see [K 72], [S 77,79], [HSU 77], also [AHS 78] and [CFKS 87]).
Here $A(x)$ is a one-form on $\Bbb{R}^n$ and the magnetic field is
a two-form $B(x)$ given by
$$
B(x) = \rd A(x)\ .\eqno(1.4)
$$
The magnetic field determines the vector potential only up to an exact
one-form $\rd\phi$. In particular, in one dimension
the vector potential can be gauged away, i.e., it can be removed by a suitable
choice of $\phi$.
Thus we restrict our attention to the case $n \geq 2$.
One of the numerous applications of these estimates is the
magnetic Lieb--Thirring bound,
i.e. a bound on
the moments of negative eigenvalues for
the Schr\"odinger operator of the form
$$
(-i \nabla + A(x))^2 + V(x) \eqno(1.5)
$$
acting on $L^2(\Bbb{R}^n) \ , $
where $V$ is an external potential
(see [LT 75], [LT 76], [L 80], [AHS 78]).
These estimates do not depend on $B$, in particular
they do no improve as the magnetic field is increased, despite
that negative eigenvalues typically disappear in strong fields.
Moreover, for most cases, the constants in these bounds
are not sharp. The notable exceptions are the recent bounds
of Laptev and Weidl for higher Riesz means of the
eigenvalues of (1.5) ([LW 99]).
Denote by $\lambda_j=\lambda_j(B, V)$, $j=1, 2, \ldots$
the negative eigenvalues of the operator (1.5), which depend only on
the magnetic field $B$ by gauge invariance.
Laptev and Weidl showed that for $\gamma \geq 3/2$
$$
\sum_j (-\lambda_j)^{\gamma} \leq L_{cl} \int_{\bR^n}
{\rm max}(-V(x),0)^{n/2+\gamma}
{\rm d} x \eqno(1.6)
$$
where
$$
L_{cl}: =
{\Gamma(\gamma + 1) \over 2^n \pi^{n/2} \Gamma(\gamma+{n \over 2} +1)}
\eqno(1.7)
$$
is the classical constant that appears in Weyl's asymptotic formula.
Related to the Lieb--Thirring bounds, the following inequalities of
Li and Yau ([LY 83]) are known
for the sum of eigenvalues of the Dirichlet Laplacian
on a domain $U\subset \bR^n$ with volume $|U|$.
$$
\sum_{j=1}^N \lambda_j \geq C_n {n \over n+2} N^{n+2 \over n} |U|^{-{2 \over n}}
\eqno(1.8)
$$
where $C_n := (2 \pi)^2 |B_n|^{-2/n}$, $B_n$ is the unit ball in
$\Bbb R^n$ and $|B_n|$ is its volume.
Again, it follows from Weyl's asymptotic formula
that the constant $C_n$ is the best possible.
Note that the Li--Yau result does not follow from the Laptev--Weidl result.
In this paper we prove a modest extension of the Li--Yau result
to the magnetic Dirichlet Laplacian with a constant magnetic field.
More specifically, for any domain $U \subset \Bbb R^n$ of finite volume we
consider the operator
$$
H= \Big( -i\nabla +A(x) \Big)^2
$$
on $L^2(U)$ given by the closure of the form
$$
(\psi, H \psi):= \int_U \Big|\left( -i\nabla +A(x)
\right)\psi(x)\Big|^2 {\rm d}x \eqno(1.9)
$$
on $C^{\infty}_c (U)$.
The one-form $A(x)$ satisfies $\rd A=B \ , $ where $B$ is a constant two-form.
Our main result is the following:
\bigskip
{\bf Theorem 1}\ .
{\it Let $H$ be given by (1.9) where $A$ generates a constant
magnetic field.
Then for any $N$ orthonormal functions $\{ \phi_j\}_{j=1}^N$ in the
form domain of $H$ we have the inequality
$$
\sum_{j=1}^N (\phi_j, H \phi_j) \geq {n \over n+2}C_n
N^{n+2 \over n} |U|^{-{2 \over n}} \ ,\eqno(1.10)
$$
with
$ C_n$ as before. Again the constant $C_n$ is the best possible.
}
\bigskip
{\it Remark 1:} That $C_n$ is best possible follows again from the
Weyl asymptotic formula, noting that the magnetic field does not contribute
to the eigenvalue sum to leading order as $N\to\infty$.
{\it Remark 2:}
Let $\lambda_j(B)$ be the eigenvalues of (1.9). Diamagnetism for eigenvalue
sums in the strongest sense would mean that
$$
\sum_{j=1}^N \lambda_j(B) \ge \sum_{j=1}^N \lambda_j(B=0) \ . \eqno(1.11)
$$
The diamagnetic inequality (1.1) shows that (1.11) is valid for $N=1$, i.e.
the lowest eigenvalue
of the magnetic operator ((1.5) or (1.9)) increases as the
magnetic field is turned on. But (1.11) fails in general even for $N=2$.
To see this, one can consider a planar domain where
the second Dirichlet eigenvalue of $-\Delta$
is twofold degenerate. If we turn on a small constant magnetic
field $B>0$, the first eigenvalue increases quadratically with $B$,
while the second one splits into two eigenvalues; one is raised and the other
one is lowered proportionally to $B$. Thus
the sum of the first two eigenvalues actually decreases for small $B$.
A similar phenomenon can occur for the sum of the first $N$ eigenvalues.
Hence the eigenvalue sum $\sum_{j=1}^N \lambda_j(B)$ may decrease
by turning on a nonzero magnetic field $B$; however
our result says that it does not
decrease so much as to violate the semiclassical bound (1.10).
In this sense, Theorem 1 establishes a weak diamagnetic
behavior for the eigenvalue sum.
This remark also applies to the result of Laptev and
Weidl. The moment of negative eigenvalues in (1.6) may increase
as $B$ is turned on, but it never exceeds the classical value.
\bigskip
Let us say a few words about proofs.
The strategy of [LY 83] does not work as smoothly in our problem as in the
case without a magnetic field. The reason is that while the eigenfunctions
of the problem on the whole space are explicitly known,
the computation becomes
fairly difficult in dimensions larger than two.
Instead, we first reduce the problem to estimates
on the integrated density of states (IDS) for the magnetic Hamiltonian
defined on the
whole space. Then we estimate the magnetic IDS (with a constant field) in
terms of the IDS of the Laplacian without magnetic field.
This estimate generalizes the known
diamagnetic inequality in the following way.
\bigskip
{\bf Proposition 1} {\it (Generalized diamagnetic inequality) \ .
Let $B$ be a constant magnetic field in arbitrary
dimension $n\ge 2$ and let $P=\chi_U$ be the characteristic
function of an open set $U\subseteq {\Bbb {R}^n}$ with finite volume.
Then,
$$ \hbox{Tr}\Big[ Pf\big( (-i\nabla+A)^2\big)P\Big]
\leq \hbox{Tr}\big[ Pf(-\Delta)P\big] \ , \eqno(1.12) $$
or, in its pointwise form
$$ f\big((-i\nabla+A)^2\big)(x,x)\leq f(-\Delta)(x,x) \ . \eqno(1.13) $$
Here $f$ is an arbitrary nonnegative convex function
defined on $\Bbb{R}^+$ with $\displaystyle{\lim_{\lambda \to \infty
}{f(\lambda)}=0}$ .}
\bigskip
It is natural to ask whether our result in Theorem 1
holds for a general magnetic
field. We do not know the answer to this question. However, we show
that our new diamagnetic inequality
does not hold generally for an inhomogeneous magnetic field.
In the following section we prove the two dimensional version of
our theorem in two ways. We give then the proof for arbitrary dimension in the
subsequent sections along with the proof of Proposition 1.
We end the paper with a discussion of the results and the techniques.
\bigskip
{\bf 2. A simple proof for the two dimensional case}
\bigskip
{\bf Theorem 2} \ .
{\it
Let $U\subseteq {\Bbb {R}^2}$ be open, with finite volume. Assume that $A$
is such that ${\rm curl} \, A=B$ with $B$ constant,
e.g. $A(x)={B\over 2}(-x_2, x_1) $.
Let $\{ \phi_j \}^N_{j=1}$ be an orthonormal
set of functions in $H^1_{0} (U)$. Then for any $N \geq 1$
$$
\sum^N_{j=1} \| (-i \nabla+A)\phi_j\|^2 \geq {2\pi N^2\over |U|}.
\eqno (2.1)
$$
}
\bigskip
We give two proofs of this theorem. The first one is a
transcription of the proof given by
Li--Yau for the free Laplace operator. The second one uses IDS
of the infinite volume problem and we explain it in the next section.
{\it First proof.} Without loss of generality we can assume that $B >0$.
Let $ \Pi ^B_{k}$ be the projection onto the $k$-th Landau level
of $(-i\nabla+A)^2 $ defined on all of $L^2(\Bbb{R}^2)$,
$$
(-i\nabla+A)^2 \Pi ^B_{k}=(2k+1)B\ \Pi ^B_{k} \ .\eqno(2.2)
$$
The projection $ \Pi ^B_{k}$ has an explicit integral kernel
whose value on the diagonal is given by
$$
\Pi ^B_{k}(x,x) = {B \over 2 \pi} \ .\eqno(2.3)
$$
\bigskip
A simple calculation for functions $ \phi_j \in C^\infty_{c} (U)$
shows that
$$
\sum^N_{j=1}\|(-i\nabla+A)\phi_j\|^2=
\sum^\infty_{k=0}(2k+1)B\sum^N_{j=1}(\phi_j,\Pi^B_{k}\phi_j) \ ,\eqno(2.4)
$$
which extends to $\phi_j \in H^1_0(U)$ by standard approximation.
Set
$$
a_k : = { 2\pi \over {B|U|}}\sum^N_{j=1}(\phi_j,\Pi^B_{k}\phi_j)\ .\eqno(2.5)
$$
Assume that
$\phi_1, \cdots, \phi_N$ is an orthonormal set, we extend it
to an orthonormal basis of $L^2(U)$
and using (2.3) we see that
$$
0\leq a_k \leq { 2\pi \over {B|U|}}
{\sum^ \infty _{j=1}(\phi_j,\Pi^B_{k}\phi_j)}
=
{{2\pi} \over {B|U|}} \hbox {Tr}(\chi_{U}\Pi^B_{k})=1 \ ,\eqno(2.6) $$
where $\chi_U$ is the characteristic function of the set $U$.
Since
$$
\sum^{\infty}_{k=1} {\Pi}^B_{k} =\Bbb{I}
$$
we get
$$
\sum^\infty_{k=0}a_k={2\pi N\over {B|U|}} := \alpha \ .\eqno (2.7)
$$
Thus we have that
$$
\sum^N_{j=1}\|(-i\nabla+A)\phi_j\|^2={B|U|\over
2\pi}\sum^\infty_{k=0}B(2k+1)a_k\ ,
\eqno (2.8)\
$$
and we minimize the right side of (2.8) over all $a_k$ satisfying (2.6)
and (2.7) with a given $\alpha$.
Applying the bathtub principle (see, e.g., [LL 97] p.28) we
learn that the minimizer of this problem is given by
$$
a_k=\left \{\matrix {1 \ ,&0 \leq k \leq [\alpha]-1\cr \alpha-[\alpha ] \ ,&k=
[\alpha ]\cr 0 \ ,& k>[\alpha ]\cr }\right .\eqno(2.9)
$$
Here [ ] brackets denote the integer part. An easy computation shows that
the minimum of (2.8) is:
$$ { {2\pi N^2} \over {\alpha ^2 |U|}}
\Big(\alpha -[\alpha ]-(\alpha -[\alpha ])^2+\alpha ^2\Big)\ ,\eqno(2.10)
$$
which is greater or equal than $\displaystyle{{ 2\pi N^2\over |U|} \ .
}$\hfill\lanbox
\bigskip
{\bf 3. Bathtub principle for the integrated density of states}
\bigskip
First we give an abstract version of the bathtub principle used in Section 2
in terms of the IDS.
Let $H$ be a nonnegative selfadjoint operator on a Hilbert space ${\cal H}$ and
let its spectral decomposition be $ H=\int_0^\infty\lambda \rd E_\lambda $,
where $E_\lambda $ is the spectral family associated with $H$.
Recall the following properties of $E_\lambda $:
\medskip
$$
\lambda\to E_\lambda \ \ {\rm is\ continuous\ from\ the\ right}\ \eqno(3.1)
$$
and
$$
E_\lambda \nearrow \Bbb{I}\ \ {\rm as}\ \ \lambda \to \infty \ .\eqno(3.2)
$$
Further, let ${h}$ be a closed
subspace of ${\cal H}$, and denote by $P$ the projection
from ${\cal H}$ onto $ h$. Let $\phi_j$, $j=1,\dots N$ be
an orthonormal set of functions in the intersection of $ h $
and the form domain of $H$.
\bigskip
{\bf Lemma 1} \ .
{\it Let $f(\lambda):= \hbox {Tr}(PE_\lambda) $, then
$$
\sum_{j=1}^N (\phi_j, H \phi_j)\geq \int_0^{\infty} (N - f (\lambda ))_{+}
\rd \lambda ,\eqno(3.3)
$$
where $(N - f (\lambda ))_{+} : = \hbox{max}\{ N- f(\lambda), 0 \}$
is the positive part of
$(N - f (\lambda ))$.
}
\bigskip
{\it Remark.} In applications $P$ will be the projection of ${\cal H}
=L^2(\bR^n)$ onto $h= L^2(U)$ with some $U\subset \bR^n$.
In this case $|U|^{-1} f(\lambda)$ is the integrated density of
states, i.e. the number of states up to energy $\lambda$ per unit volume.
\bigskip
{\it Proof}.
Let
$$
F_N(\lambda):=\sum^N_{j=1} (\phi_j,E_\lambda \phi_j)\ .\eqno(3.4)
$$
By (3.2) $ F_N(\lambda)$ is increasing towards $N$ as ${\lambda\to\infty }$
and for all $\lambda$
$$
0\leq F_N(\lambda)\leq \hbox {Tr}(PE_\lambda)=f(\lambda)\ . \eqno(3.5)
$$
By these properties, the function $F_N(\lambda)$ defines a measure
$\rd F_N(\lambda)$.
Since
$$\
\sum_{j=1}^N (\phi_j,H \phi_j)
=\int_ 0^\infty \lambda \, \rd F_N(\lambda)=
\int_0^{\infty} \Big( F_N(\infty) - F_N(\alpha)\Big) {\rm d}\alpha
=\int_0^\infty(N-F_N(\alpha))\rd\alpha\ ,\eqno(3.6)
$$
the result follows from (3.5) and the fact that $F_N(\lambda)\leq N$ for all
$\lambda\geq 0$. \hfill\lanbox
\bigskip
The following elementary lemma is useful
for computing explicit lower bounds.
\bigskip
{\bf Lemma 2} \ . {\it
Let $f$ and $g$ be two nondecreasing functions on the positive
line satisfying
$\lim_{\lambda\to \infty}f(\lambda)=
\lim_{\lambda\to \infty}g(\lambda)=+\infty $.
Assume further that
$$
\int _ 0^E f(\lambda){\rm d} \lambda \leq
\int_0^E g(\lambda){\rm d}\lambda \eqno(3.7)
$$
for all $E\geq 0$ .
Then
$$
\int_0^\infty (N-f(\lambda))_+{\rm d}\lambda \geq \int_0^\infty
(N-g(\lambda))_+{\rm d} \lambda \ ,\eqno(3.8)
$$
for all $N\geq 0$.
}
\bigskip
{\it Proof}.
With the definitions $\lambda_0:={\inf}\{\lambda : f(\lambda)\geq N\}$ and
$ \mu_0:={\inf}\{\mu : g(\mu)\geq N\}$ the problem is reduced to
showing
$$
\int_ 0^{\mu_{ 0}} g -\int_ 0^{\lambda_{0 }}f \geq (\mu_0-\lambda_0)N \ .
\eqno(3.9)
$$
In the case $ \mu_0\geq \lambda_0$ we write the left side as
$$
\int _0^{\mu_0}( g-f) +\int _{\lambda_0}^{\mu_0} f \ .\eqno(3.10)
$$
Since the first term in this sum is nonnegative and $ f \geq N $
on $( \lambda_0,
\mu_ 0)$ we obtain (3.9).
In the case $\mu_0 \leq \lambda_ 0$ we write the left side of (3.9) as
$$
\int_ 0^{\mu_0} (g-f) -\int_ {\mu_0}^{\lambda_0} f \ .\eqno(3.11)
$$
Again the first term is nonnegative and on $(\mu_0,\lambda_0)$ we have
$f\leq N \ , $
which yields the result.\hfill\lanbox
\bigskip
{\it Remark.} Armed with Lemma 2, one can use Berezin's trace inequality
[B 72]
to give an alternative proof of Lemma 1 without bathtub principle
(we are grateful to Timo Weidl for pointing this out to us).
Let $\widetilde E_\lambda$ be the spectral resolution of $PHP$.
By the variational principle and the spectral theorem
$$
\sum_{j=1}^N (\phi_j, H \phi_j)\geq \int_0^\infty
\Big( N - \hbox{Tr} \, \,
\widetilde E_\lambda\Big)_+ \rd \lambda \ .
$$
Hence (3.3) would follow from Lemma 2 once we prove that for any $E\ge 0$
$$
\int_0^E \hbox{Tr} \,\, \widetilde E_\lambda \; \rd \lambda
\leq \int_0^E \hbox{Tr} \big( P E_\lambda P\big) \; \rd \lambda \ ,
$$
but this is just Berezin's inequality $\hbox{Tr} \;
\varphi(PHP) \leq \hbox{Tr} \;
P\varphi(H)P$ for the convex function $\varphi(u) := (E-u)_+$.
Here we used that
$$
\int_0^E E_\lambda\rd \lambda = (E-H)_+ \ .\eqno(3.12)
$$
and a similar relation for $\widetilde E_\lambda$.
\medskip
With the help of these Lemmas we give now a second proof of Theorem 2
that can be easily generalized to higher dimensions.
\bigskip
{\bf 4. Second proof of Theorem 2}
\bigskip
Let $ H=(-i\nabla+A)^2$ be the constant
field operator, ${\cal H}={ L^2 (\Bbb{R}^2)}$, $h=L^2(U)$.
Let $P$ be the orthogonal projection from ${\cal H}$ to $h$,
in other words, the
multiplication by the characteristic function $\chi_U$.
Then
$$
{1\over |U|}\hbox{Tr}(PE_\lambda)={1\over |U|}\int_U E^B_\lambda(x,x)\rd x
= E_\lambda^B
\ ,\eqno(4.1)
$$
is the integrated density of states,
where
$$
E^B_\lambda(x,x):=\sum _{(2k+1)B\leq \lambda } \Pi^B_k(x,x) = {B\over
2\pi}\Bigg[{\lambda\over 2B}+{1\over 2}\Bigg]=:E^B_\lambda\ ,\eqno(4.2)
$$
using (2.3). By translation invariance $E_\lambda^B (x,x)$ is
clearly independent of $x$.
\bigskip
Thus, by Lemma 1
$$
\int _0^\infty \Big (N- |U|{B\over 2\pi}\Big[{\lambda \over 2B}+{1\over 2
}\Big]\Big)_+ \rd\lambda \ \eqno(4.3)
$$
is a lower bound to the left side of (2.1).
Finally, the bound
$$
\int _0^\infty \Big (N- |U|{B\over 2\pi}\Big[{\lambda \over 2B}+{1\over 2
}\Big]\Big)_+ \rd\lambda
\geq \int _ 0^\infty \Big(N-{\lambda\over 4\pi} |U|\Big)_+\rd\lambda
={2\pi N^2\over |U|} \ ,\eqno(4.4)
$$
is a consequence of Lemma 2 and the elementary but
important observation that
$$
\int_0^E E_\lambda^B \rd \lambda = \int_ 0^E {B\over 2\pi} \Big[{\lambda
\over 2B}+
{1\over 2} \Big]\rd\lambda\leq \int_0^E {\lambda\over 4\pi} \rd\lambda
= \int_0^E E_\lambda^{0} \rd \lambda \ ,
\eqno(4.5)
$$
for all $E \geq 0$. Here $E_\lambda^{0} : = \lambda/( 4\pi)$
is the IDS of the free Laplacian.
It is useful to view (4.5) as a comparison of the
IDS of two Laplace operators in the whole space:
one with and the other without magnetic field.
A nice way to see inequality (4.5) is to notice that the left side
is the integral (up to $E$) of the Landau staircase function $\lambda
\to E_\lambda^B$, which is a function
that has jumps of height $B/(2\pi)$ at the points $(2k+1)B, k=0,1,\dots $.
By comparing this function with the IDS
of the free Laplacian, which is a straight line of slope $1/(4 \pi)$
going through the middle point of each of the stairs, (4.5) follows easily.
Note that this inequality is saturated exactly at values $E=2kB$, $k=0, 1,
\ldots $.
\hfill\lanbox
\bigskip
{\it Remark.} Formula (3.12) implies
that (4.5) is equivalent to
$$
\hbox{Tr}\Big[P(E-(-i\nabla+A)^2)_+\Big]
\leq \hbox{Tr}\Big[ P(E+\Delta)_+\Big] \ ,\ \ P=\chi_U \ .
\eqno (4.6)
$$
\bigskip
{\bf 5. Higher dimensions}
\bigskip
The following lemma is the main device for passing to higher dimensions.
\medskip
{\bf Lemma 3} \ .
{\it Let ${\cal H}_1$ and ${\cal H}_2$ be Hilbert spaces and let $A_j, B_j$
be nonnegative selfadjoint
operators on ${\cal H}_j$, $j=1, 2$.
By a slight abuse of notation we denote by $A_1+A_2$ the operator
that acts on ${\cal H}=
{\cal H}_1\otimes {\cal H}_2$ and $B_1+B_2$ acts in a similar
way.
Let $P=P_1 \otimes P_2$ , where $P_1$ and $P_2 $ are nonnegative selfadjoint
operators acting
on ${\cal H}_1$ and ${\cal H}_2$, respectively.
Assume further that
$$
{\rm Tr}\Big[ P_1(E-A_1)_+\Big] \leq {\rm Tr}\Big[ P_1(E-B_1)_+\Big]
\eqno(5.1)
$$
and
$$
{\rm Tr}\Big[ P_2(E-A_2)_+\Big] \leq {\rm Tr}\Big[ P_2(E-B_2)_+\Big]
\eqno(5.2)
$$
hold for all $E\geq 0.$ Then
$$
{\rm Tr}\Big[P(E-A_1 -A_2)_+\Big]\leq
{\rm Tr} \Big[ P(E-B_1-B_2)_+\Big] \eqno(5.3)
$$
for all $E \geq 0$.
(The traces are taken on the respective Hilbert spaces where the operators
are defined.)
}
\bigskip
{\it Proof.}
For all real numbers $x$ and $y$ the following identity holds
$$
(E-x-y)_+=\int _0^\infty \theta (E-\beta -y)\big( 1-\theta (x-\beta)
\big)\rd\beta\
,\eqno(5.4)
$$
where
$$
\theta(t)= \cases{$1$ & for $t\geq 0$ \cr $0$ & for $t< 0$ \cr} \ . \eqno(5.5)
$$
Via spectral calculus,
this formula yields
$$
{\rm Tr}\Big[P(E-A_1-A_2)_+\Big]=\int _0^\infty {\rm
Tr}\Big[P_1\theta(E-\beta-A_1)\Big] \,\,
{\rm Tr}\Big[P_2(1-\theta(A_2-\beta))\Big] \rd \beta \ .\eqno(5.6)
$$
The function $ f(\beta)=\hbox {Tr}\Big[P_2(1-\theta(A_2-\beta))\Big] $
is obviously monotonically increasing. By the layer cake
representation (see [LL 97], p. 26)
it can be written as
$$
f(\beta)=\int _0^\infty \chi _{\{\nu:f(\beta)>\nu\}}d\nu\ .\eqno(5.7)
$$
For any $\nu$ fixed,
$\beta \to \chi _{\{\nu:f(\beta)>\nu
\}}$ are characteristic functions of half--lines
starting at
$$\beta_0(\nu):=\inf\{\beta:f(\beta)>\nu\}\ .\eqno(5.8) $$
In the case when $ \{\beta:f(\beta)>\nu\}$ is an empty set we assume
$\beta_0(\nu)= +\infty$.
Hence we obtain
$$
{\rm Tr} \Big[ P(E-A_1-A_2)_+\Big]
= \int _0^\infty \rd\nu \int _0^\infty \rd\beta \
{\rm Tr}\Big[P_1\theta(E-\beta-A_1)\Big] \ \chi_{\{\nu:f(\beta)>\nu\}}=
$$
$$
= \int _0^\infty \rd\nu \ {\rm Tr}\Big[P_1(E-\beta_0(\nu)-A_1)_+\Big] \leq
$$
$$
\leq \ \int_0^{\infty} \rd\nu \ {\rm Tr}\Big[
P_1(E-\beta_0(\nu)-B_1)_+\Big]= {\rm Tr}\Big[ P(E-B_1-A_2)_+\Big] \ .
\eqno(5.9)
$$
Here we used (5.1).
By the same reasoning we have
$$
{\rm Tr}\Big[ P(E-B_1-A_2)_+\Big]\leq {\rm Tr}\Big[ P(E-B_1-B_2)_+\Big]
\ .\eqno(5.10)
$$
\hfill\lanbox
Now we are ready to prove Theorem 1.
\bigskip
{\it Proof of Theorem 1.}
If the dimension $n$ is even,
then the
operator $(-i\nabla +A)^2$ acting on all $\Bbb{R}^n$ is unitarily
equivalent to a sum of ${n \over 2}$ two dimensional
magnetic Schr\"odinger operators that
act on $L^2(\Bbb{R}^2)$.
This follows from the fact that
the magnetic field, being a two form with constant coefficients, can be
transformed into the form
$$
\sum_{k=1}^{n \over 2} B_k \, \, \rd x_{2k-1} \wedge \rd x_{2k}\ , \eqno(5.11)
$$
by an orthogonal change of coordinates.
If the dimension $n$ is odd the magnetic field looks as above ( \ except the
summation is up to ${n-1 \over 2}$ \ ) but the
operator is of the form
$$
\sum_{k=1}^{{n -1 \over 2}} H(B_k) + H_0 \ . \eqno(5.12)
$$
Here $H(B_k)$ is a two dimensional magnetic Schr\"odinger operator
with constant field $B_k$
acting on the $(x_{2k-1}, x_{2k})$ coordinate plane
and $H_0 : = -\partial^2_{x_n}$.
By Lemma 1 we know that
$$
\sum_{j=1}^N (\phi_j, H \phi_j) \geq \int_0^{\infty} (N-f(\lambda))_+\rd
\lambda
\ ,\eqno(5.13)$$
where
$$
f(\lambda) = {\rm Tr}(PE_\lambda) \eqno(5.14)
$$
is determined by the integrated density of states of the operator $(-i\nabla +
A)^2$ in the
whole space and $P=\chi_U$.
We make an induction argument over dimensions.
>From Theorem 2 we know that
$$
{\rm Tr}\Big[ P\big( E-(-i\nabla+A)^2\big)_+\Big]
\leq {\rm Tr}\Big[ P(E+\Delta)_+\Big]
\ , \eqno(5.15)
$$
is true in $\Bbb{R}^2$ \ , where $P=\chi_U$ \ .
Suppose (5.15) holds in $\Bbb{R}^n$ , where $ n$ is even.
Then it is valid in $\Bbb{R}^{n+1}$ by means of Lemma \ 3.
We choose $A_1$ to be the $n$--dimensional
magnetic Schr\"odinger operator, $B_1$ is the $n$--dimensional minus
Laplacian both acting on the $L^2$ space of the first $n$ coordinates.
Finally $A_2$ and $B_2$ are both equal to $ -\partial^2_{x_{n+1}} $ as
in (5.12).
To prove (5.15)
in $\Bbb{R}^{n+2}$ we choose $A_1$ and $B_1$ as above; and we let $A_2$ and
$B_2$ be two dimensional Laplacians on the $(x_{n+1}, x_{n+2})$ coordinate
plane with and without magnetic field, respectively.
This induction argument works not only for domains of the form $U_1\times U_2
\subseteq \Bbb{R}^n\times \Bbb{R}$ and $U_1\times U_2\subseteq \Bbb{R}^n\times
\Bbb{R}^2$ suggested by Lemma \ 3 but also for any finite volume domain.
The reason is that
all the operators considered are actually on the full Euclidean space, hence
they are translation invariant up to a gauge
and their kernels at the $(x,x)$ diagonal are independent of $x$.
Therefore $\hbox
{Tr}(PE_\lambda)$ equals to the integrated density of states multiplied by
$|U|$. So actually we proved the
pointwise form of (5.15)
$$
\Big( E - (-i\nabla + A)^2\Big)_+(x,x) \leq
\big( E + \Delta \big) (x,x) \ . \eqno(5.16)
$$
By applying (5.13)-(5.15) and Lemma 2
(via the identity (3.12) the inequality (5.15)
plays the role of (3.7) in Lemma 2),
we obtain the lower bound
$$\sum ^N _{j=1}
( \phi_ j,H\phi_ j) \ge
\int _0^\infty \Big (N- {{\lambda^{n\over 2}}\over (2 \pi)^n}|B_{n}||U|
\Big )_+ \rd\lambda \ , \eqno(5.17)
$$
where $\displaystyle{{{\lambda^{n\over 2}}\over {(2\pi)}^n}|B_n|}$ is the
integrated density of states of $-\Delta$ acting on $L^2(\Bbb{R}^n).$
An easy computation shows that the right side of (5.17) is equal to
$${ n\over n+2}C_n N^{n+2\over n} |U|^{-{2\over n}} \ , \eqno(5.18)$$
with
$$ C_n= (2\pi )^2 {|B_n| }^{-{2\over n}} \ , $$
where $B_n$ is the unit ball in $\bR^n$.
This completes the proof of our main Theorem.
\hfill\lanbox
\bigskip
{\it Proof of Proposition 1:}
Using the spectral calculus and the identity
$f(\lambda)=\int _0^\infty(E-\lambda)_+f''(E)\rd E$ the result follows from
(5.15) and (5.16). \hfill\lanbox
\bigskip
{\it Remark:} Due to the close connection between the magnetic
operator with a constant field
and the harmonic oscillator it is natural to ask whether
the analogue of (1.13) is true for the $n$--dimensional
harmonic oscillator $H_\omega =
-\Delta + \omega^2 x^2$. While it is true for the two dimensional harmonic
oscillator and hence for even dimensional ones, it definitely fails in one
dimension. This can be seen by straightforward calculations.
\bigskip
{\bf 6. The generalized diamagnetic inequality for
inhomogeneous large fields}
\bigskip
Now we wish to consider whether our technique of comparing IDS
works for general
magnetic fields. We can show that our generalized diamagnetic inequality
(5.16) [ hence (1.13) ]
remains valid for inhomogeneous magnetic fields
in the large field limit. However, by constructing a counterexample
in the next section,
we also show that it fails in general.
To consider the large field limit case, we
work in $n=2$ dimensions for simplicity.
\bigskip
{\bf Proposition 2.} {\it Let the function
$\psi:\Bbb{R}^2\to \Bbb{R}^+$ be nonnegative, smooth and compactly
supported.
Let $A$ be smooth, generating a magnetic field satisfying $B(x)\geq c>0$
for all $x\in \Bbb{R}^2$. Assume that $|\nabla B(x)|\geq c>0$ on an open set
$U$, where $c$ is any fixed positive constant. Then for any fixed $E>0$
$$ \lim_{\lambda \to \infty} {1 \over \lambda^2} \hbox{Tr}
\Big[
\psi \, \big( \lambda E - (-i\nabla +\lambda A)^2\big)_+\Big]
\leq \lim_{\lambda \to \infty}
{1\over \lambda^2} \hbox{Tr}\Big[ \psi \, ( \lambda E +\Delta )_+\Big]
\ , \eqno(6.1) $$
where $\psi$ acts as a multiplication operator.
The right side of (6.1) is independent of $\lambda$ even before the limit and
equals to
$\displaystyle{{E^2\over 8\pi}\int \psi \ .}$ The inequality is sharp if
$\hbox{supp} \,\, \psi \cap U \ne \emptyset$.
}
\bigskip
The heart of the proof is the following semiclassical statement.
\bigskip
{\bf Proposition 3.} {\it Under the stated conditions,
$$
\lim_{\lambda\to\infty} {1\over \lambda^2}
\hbox{Tr}\Big[ \psi \, \big( \lambda E - (-i\nabla+\lambda A)^2
\big)_+\Big]
= \int_{{\Bbb R}^2} \psi(x) {B(x)\over 2\pi}
\sum_{k=0}^\infty \Big( E- (2k+1)B(x)\Big)_+ \rd x \eqno(6.2)
$$
}
>From this result Proposition 2 follows easily since
$$
{B(x)\over 2\pi}
\sum_{k=0}^\infty\Big( E- (2k+1)B(x) \Big)_+
\leq {E^2\over 8\pi} \eqno(6.3)
$$
for each individual $x$, by the staircase argument. Integrating
(6.3) against $\psi(x)$ we obtain (6.1).
Since $B(x)$ is continuous but not constant on the
support of $\psi$, we see that the staircase inequality (6.3)
is strict on an
open set of $x$ inside the support of $\psi$.
\bigskip
The proof of Proposition 3 is a microlocal result
(see Theorem 6.4.13, statement (6.4.57) in [Iv 98]).
We rescale the problem
so that the strong field limit becomes the standard semiclassical
strong field limit. Clearly
$$
\lim_{\lambda\to\infty} {1\over \lambda^2}
\hbox{Tr}\Big[ \psi \, \big( \lambda E - (-i\nabla+\lambda A)^2\big)_+
\Big]
= \lim_{h\to 0} h^2
\hbox{Tr} \Big[ \psi \big( E - (-ih\nabla+ \mu A)^2)_+\big)\Big]
\eqno(6.4)
$$
with $\mu:=h^{-1}$ and $h:=\lambda^{-1/2}$. Moreover,
$\hbox{Tr}\Big[ \psi \, \big( E - (-ih\nabla+ \mu A)^2\big)_+\Big]$
is exactly the expression
$$
\int \rd x\int (E-\tau)_+ \psi(x) \rd_{\tau}e(x,x, -\infty, \tau')
\eqno (6.5)
$$
in (6.4.50) [Iv 98] (changing $\tau' $ to $\tau$). Here
$$
e(x,x, -\infty, \tau) : = \Pi_{(-\infty, \tau)} (H) (x, x) \eqno(6.6)
$$
is the kernel of the spectral projection on the diagonal with
$$
H = H(h, \mu) = (-ih\nabla+ \mu A)^2 \ . \eqno(6.7)
$$
The other term in (6.4.50) [Iv 98] is
$$
h^{-2}\int \rd x \int (E-\tau)_+ \psi(x)
{B(x)\over 2\pi} \rd_{\tau}\Big(
\sum_{k=0}^\infty \theta( \tau - (2k+1)B(x)) \Big)=
$$
$$
=h^{-2} \int \psi(x) {B(x)\over 2\pi}
\sum_{k=0}^\infty\Big( E - (2k+1)B(x) \Big)_+ \rd x \ , \eqno(6.8)
$$
where $\theta (t)$ is given in (5.5).
Hence (6.4.57) [Iv 98] says that with some constant $C$, depending
on the smoothness of $B$
$$
\Bigg| h^2 \hbox{Tr}\Big[ \psi \big( E - (-ih\nabla+ \mu A)^2\big)_+
\Big]
- \int \psi(x) {B(x)\over 2\pi}
\sum_{k=0}^\infty\Big( E - (2k+1)B(x) \Big)_+ \rd x
\Bigg| \leq Ch^2 \ , \eqno(6.9)
$$
which clearly goes to zero as $h\to 0$.
It is easy to check that the required
conditions in Theorem 6.4.13 of [Iv 98] are satisfied
in our situation. \hfill\lanbox
\bigskip
{\bf 7. Counterexample to the generalized diamagnetic inequality}
\bigskip
The following counterexample shows that (5.16) is not true
for arbitrary magnetic field.
\bigskip
Let us consider the magnetic field in $\Bbb {R}^3$ which is given
by a
vector--potential $\lambda A(|x|)$ that
satisfies the following conditions:
1) $A$ is smooth, compactly supported, $\hbox{supp}A=\Big \{x\in \Bbb{R}^3 :
\ r_1\leq |x|\leq r_2 \Big \}$ ;
\medskip
2) $x \cdot A(|x|)=0$ for all $x \in \hbox{supp}A$ ;
\medskip
3) $ \hbox{div} A=0$ ,
\medskip\noindent
and $\lambda$ is a small coupling constant.
By the spectral resolution
$$ \Big (E-(-i\nabla+\lambda A)^2\Big)_+(x,x)
=\int _{\Bbb{R}^3}(E-k^2)_+|\varphi
_{k}(x)|^2\rd k \ , \eqno(7.1) $$
where $\varphi_{k}(x)$ satisfies the Lippmann--Schwinger equation ([RS 79]):
$$ \varphi_{k}(x)=
{e^{ik\cdot x}\over (2\pi)^{3\over 2}}-{1\over 4\pi}\int_{\Bbb{R}^3}
{e^{i|k||x-y|}\over |x-y|}(V_\lambda\varphi_{k})(y)\rd y \eqno(7.2) $$
with
$$
V_\lambda=-2i\lambda A\cdot\nabla +\lambda^2 A^2 \ . \eqno(7.3)
$$
We solve (7.2)
for $\varphi_{k}(x)$ perturbatively and use them in (7.1) keeping only
the terms of the order up to $\lambda ^2$.
Integration by parts and the properties of $A$
imply that at $x=0$ only the $\lambda^2 A^2$
term contributes with respect to the nonmagnetic problem, namely
$$ |\varphi_{k}(0)|^2=
{1\over (2\pi)^3}\Big(1-{\lambda^2\over |k|}\int _{r_1}^{r_2}
A^2 (r)\sin(2|k|r)\rd r\Big) +o(\lambda^2) \eqno(7.4) $$
and, after some calculation
$$ \Big( E-(-i\nabla+\lambda A)^2\Big)_+(0,0)-(E+\Delta)_+(0,0) $$
$$= -{\lambda^2\over (4\pi)^2} \int_{r_1}^{r_2}{A^2 (r)\over r^4}\Big( -\xi^2
\sin\xi -3\xi \cos\xi +3\sin\xi \Big)\rd r +o(\lambda^2) \ , \eqno(7.5) $$
where $\xi :=2r \sqrt E .$
Clearly $r_1 , r_2$ and $E$ can be chosen such that
$$ -\xi^2 \sin\xi -3\xi \cos\xi +3\sin\xi < 0 \eqno(7.6) $$
for all relevant values of $\xi$.
Hence, for sufficiently small values of $\lambda$ the right side of (7.5)
can be
made positive which contradicts to the comparison (5.16).
Since $|x|^{-1}\in L^{3-\epsilon}(\Bbb
{R}^3)+L^{3+\epsilon}(\Bbb {R}^3) \ ,$ $|x|^{-2}\in
L^{{3\over 2}-\epsilon}(\Bbb {R}^3)+L^{{3\over 2}+\epsilon}(\Bbb {R}^3) \ , $
and
$|A| \ ,\ A^2 \in L^p(\Bbb{R}^3)$ for all $1\leq p\leq \infty$, we have that
$$ \int _{\Bbb{R}^3}{|A(|y|)|\over |x-y|}\rd y \ , \int _{\Bbb{R}^3}
{A^2(|y|)\over |x-y|}\rd y \ \ \hbox{and} \ \ \int _{\Bbb{R}^3}
{|A(|y|)|\over |x-y|^2}\rd y
$$
are finite by Young's inequality. This provides the finiteness of
the $L^\infty \to L^\infty$ norm of the integral operator
$$
\big( Q_\lambda \varphi \big)(x) : = -{1\over 4\pi}\int_{\Bbb{R}^3}
{e^{i|k||x-y|}\over |x-y|}(V_\lambda \varphi)(y)\rd y
$$
in the Lippmann--Schwinger equation (7.2).
Moreover, for $\lambda$ sufficiently small
$Q_\lambda$ clearly
becomes a contraction which yields the convergence of the Neumann series
for $\varphi_k$. This
justifies the applicability of the perturbation argument.
\hfill\lanbox
\medskip
This counterexample shows that the approach used
to prove Theorem 1 cannot be
directly generalized to the case of a general magnetic field.
However, the generalization of Theorem 1 to arbitrary magnetic fields
remains open. Another open question:
Which is the most general class of functions
$f$ for which the diamagnetic inequality (1.13) is true?
\bigskip
\noindent
{\bf References}
\bigskip
\item{[AHS 78]} Avron, J.E., Herbst, I. and Simon, B.: {\it
Schr\"odinger operators
with magnetic fields, I. General interactions}, Duke Math. J. {\bf 45} (1978),
847-883.
\medskip
\item{[B 72]} Berezin, F.A.: Convex operator functions, Math. USSR. Sb.
{\bf 17} (1972), 269-277.
\medskip
\item{[CFKS 87]} Cycon, H.L., Froese, R.G., Kirsch, W. and Simon, B.:
Schr\"odinger
operators (with application to Quantum Mechanics and Global Geometry),
Springer 1987.
\medskip
\item{[HSU 77]} Hess, H., Schrader, R. and Uhlenbrock, D.A.:
{\it Domination of semigroups and
generalizations of Kato's inequality},
Duke Math.J. {\bf 44} (1977), 893--904.
\medskip
\item{[Iv 98]} Ivrii, V.:
Microlocal Analysis and Precise Spectral Asymptotics,
Springer (1998).
\medskip
\item{[K 72]} Kato, T.: {\it Schr\"odinger operators with
singular potentials,}
Isr. J. Math. {\bf 13} (1972), 135-148.
\medskip
\item{[L 80]} Lieb, E.H.:
{\it The number of bound states of one--body Schr\"odinger
operators and the
Weyl problem}, Proc. Sym. Pure Math. {\bf 36} (1980), 241--252.
\medskip
\item{[LL 97]} Lieb, E.H., Loss, M.:
Analysis, Graduate Studies in Mathematics,
Volume {\bf 14}, American Mathematical Society (1997).
\medskip
\item{[LT 75]}
Lieb, E.H., Thirring, W.: {\it Bound for the kinetic energy of fermions
which proves
the stability of matter}, Phys. Rev. Lett. {\bf 35} (1975), 687.
\medskip
\item{[LT 76]} Lieb, E.H., Thirring, W.:
{\it Inequalities for the moments of the
eigenvalues of the
Schr\"odinger Hamiltonian and their relation to Sobolev inequalities}.
Studies in Math. Phys., Essays in Honor of Valentine Bargmann.,
Princeton (1976).
\medskip
\item{[LW 99]} Laptev, A., Weidl, T.:
{\it Sharp Lieb-Thirring inequalities in high
dimensions}, to appear in Acta Math.
\medskip
\item{[LY 83]} Li, P., Yau, S.--T.:
{\it On the Schr\"odinger equation and the
eigenvalue problem},
Comm. Math. Phys.,{\bf 88} (1983), 309--318.
\medskip
\item{[RS 79]} Reed, M., Simon, B.:
Methods of Modern Mathematical Physics, III.
Scattering Theory, Academic Press (1979).
\medskip
\item{[S 79]} Simon, B.: Functional integration and Quantum Physics, Academic
Press, 1979.
\medskip
\item{[S 77,79]} Simon, B.:
{\it An abstract Kato inequality for generators of
positivity preserving semigroups},
Ind. Math. J. {\bf 26} (1977), 1067-1073.
{\it Kato's inequality and the comparison of semigroups},
J. Funct. Anal. {\bf 32}
(1979), 97-101.
\bigskip\bigskip
\line{Email addresses: \it lerdos@math.gatech.edu, loss@math.gatech.edu,
vitali@math.gatech.edu \hfill}
\end