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\begin{document}
\footnotetext[1]{Partially supported
by the US NSF through
grant DMS-9211624.}
\footnotetext[2]{Part of this research was conducted
during the US-Sweden Workshop on Spectral Methods
sponsored by the NSF under grant INT-9217529.}
\footnotetext[3]{Work related to
Doctoral thesis in
preparation at Georgia Institute of Technology.}
\vspace*{3.7cm}
\noindent{\Large\bf Commutator Bounds for Eigenvalues
\newline of Some Differential Operators}
\vspace{1.cm}
\noindent EVANS M. HARRELL II\footnotemark[1]
\newline \noindent PATRICIA L. MICHEL\footnotemark[2]
\footnotemark[3]
\vspace{.6cm}
\noindent School of Mathematics,
Georgia Institute of
Technology
\noindent School of Mathematics,
Georgia Institute of
Technology
\addvspace{3.7cm}
\section{INTRODUCTION}
\addvspace{.6cm}
\noindent Consider the evolution due to the linear diffusion equation
\[ u_{t} = \kappa\Delta u \]
or to similar equations with popular additional complications, such as
\[ u_{t}(x,t) = {\rm div}\left(P(x)
{\rm grad}\; u(x,t)\right) + V(x) u(x,t)\,,\]
the heat equation with inhomogeneities and endothermic reactions, or
\[i\hbar u_{t}(x,t) = ({\bf p} - {\bf A}(x))^2
u(x,t) + V(x) u(x,t),\]
where
\[ {\bf p} := -i\hbar \nabla,\]
the Schr\"{o}dinger equation with a magnetic field. Here $div$
and $grad$ may be tensorially defined.
Typically the generator of the semigroup $\exp(-tH),$ where $H$ is the linear
operator on the right side, has two important positivity properties, i.e.,
$H$ is semibounded and $\exp(-tH)$ is positivity improving, which is essentially
equivalent to having a positive integral kernel. In addition, often at least
the lower portion of the spectrum is discrete, especially if the equation is
defined on a bounded domain or manifold ${\bf \Omega}.$
The positivity properties of the evolution imply in a standard way that
the lowest eigenvalue is nondegenerate and has a positive
eigenfunction. Well-known arguments establish that estimates of gaps
between eigenvalues correspond to exponential rates of convergence of
the system to equilibrium: If the initial condition is $u(x,0)$, then, for
example for the heat equation with Neumann boundary conditions:
\[u(x,t) - \langle u,u_{0}\rangle u_0 = \sum_{n=1}^{\infty}\exp(-\mu_{n}t)
\langle u,u_{n}\rangle u_{n} = {\bf O}(\exp(-\mu_{1}t)),\]
where $ \langle \cdot , \cdot \rangle$ denotes the inner product on
$L^2({\bf \Omega})$
and the eigenfunctions are denoted $u_{n},$ and in this case
$\lambda_0 = 0,$ with $ u_{0} = 1/\sqrt{|{\bf \Omega}|}.$
Analogous things occur for other problems.
Estimates of this kind have been familiar at least since \cite{kn:Kac68}.
The work discussed here concerns upper bounds on eigenvalue gaps
(and, implicitly, on rates of convergence of evolution equations),
using a theorem we recently proved \cite{kn:Har-MiA}. This in turn
is closely related to earlier work of Hook \cite{kn:Ho90b}, Harrell
\cite{kn:Har93}, and Davies and Harrell (appendix to \cite{kn:Har88}).
The essential reason for the association between eigenvalue gaps and
commutators is elementary. Let $H$ be a self-adjoint operator such that
$H u_{k} = \lambda_{k} u_{k},$ and suppose $G$ is an auxiliary operator.
Then a formal calculation (ignoring domain questions) shows that
\[ \langle u_{k},[H,G]u_{j}\rangle =
(\lambda_{k} - \lambda_{j})\langle u_{k},Gu_{j}\rangle.\]
A great many of the good estimates known for eigenvalue gaps can be
derived from this formula. As with all variational techniques, the test
function -- in this case $G$ -- must be chosen cleverly in
order to get a good
estimate. For further discussion of these points, see \cite{kn:Har88}.
A novel feature of the bounds we discuss in this article
is that they involve the
interplay between first and second commutators. As was shown in
\cite{kn:Har93}, when particularized to $H$ = the Laplace-Beltrami
operator on a Riemannian manifold, similar bounds show how the
spectral gaps are controlled by the curvature.
One application we make below
is to sharpen some of the estimates of that paper. Other applications are
made to bounds for Neumann boundary conditions and for bi-Laplacians.
The operator under study is a self-adjoint operator $H,$ and there
are
two families of symmetric ``test operators,'' which
we call $G_{j}$ and $\Pi_{j}.$ (The
$\Pi_{j}$'s are often analogues of the momentum
operator of quantum mechanics, accounting for our notation. Hook
\cite{kn:Ho90b} earlier had a similar theorem, and a rough
correlation with his notation is that our $G'$s correspond to his $B'$s and
our $\Pi'$s correspond to his $T'$s times $i$.)
\begin{tnum}
\label{tn:main}
Let $H$ be self-adjoint on a Hilbert space $\cal H,$ and suppose
that the lower portion of its spectrum consists of discrete
eigenvalues
$\lambda_{1} \leq \lambda_{2} \leq \ldots
\leq \lambda_{n} < \lambda_{n+1} \leq \ldots$
Let $P$ be the spectral projection for $ \lambda_{1}, \lambda_{2},
\ldots ,
\lambda_{n},$ and let $\{ G_{j}\}$ and $\{ \Pi_{j}\}$ be two families
of
symmetric operators such that all products of the form
$\Pi_{j}G_{j},$
$G_{j}\Pi_{j},$ $G_{j}^{2}H,$ $HG_{j}^{2},$ and $G_{j}HG_{j}$ are well
defined.
Then
\begin{equation}
\sum_{j=1}^{m} Tr\Big((\lambda_{n+1}I - H)^{-1}P\Pi_{j}^{2}\Big) \geq
\frac{\Big|\sum_{j=1}^{m} Tr\Big(P[\Pi_{j},G_{j}]\Big)
\Big|^{2}}{2 \sum_{j=1}^{m}
Tr\Big(P[G_{j},[H,G_{j}]]\Big)},
\label{eq:main}
\end{equation}
assuming that these three traces are finite and nonzero.
\end{tnum}
\vspace{.3cm}
\noindent REMARKS: \ Note that because of the hypothesis
$\lambda_{n+1} > \lambda_n\,,$ the operator $\lambda_{n+1}I
- H $ is uniquely invertible from the range of $P$ to itself.
The natural setting for this theorem is that of $C^\ast$-algebras,
in which the assumption on products of operators is unnecessary.
Here, however, we are interested in unbounded operators, so domain
questions must be considered carefully. While there is a certain
amount of freedom in the choice of the auxiliary operators $\Pi_j$
and $G_j,$ it is important that the $G_j'$s are chosen in such a way that
$HG_j$ is defined
on the given domain, i.e. for $u \in {\cal D}(H)$ we must have $G_ju \in
{\cal D}(H).$ Similarly for $G_j^2u.$
An upper bound for $\lambda_{n+1} - \lambda_{n}$ is obtained if the
left hand side is increased to \linebreak
$(\lambda_{n+1} - \lambda_{n})^{-1}\sum_{j=1}^{m}Tr(P\Pi_{j}^{2})$
and then one solves for the gap $\lambda_{n+1} - \lambda_{n}.$
The resulting bounds are analogous to the classic
Payne-P\'{o}lya-Weinberger bounds \cite{kn:PPW} (see \cite{kn:Ash92}
for a discussion of this bound)
and those derived in \cite{kn:Har93}. In most applications,
$m$ will be taken to be $\nu,$ the dimension of the underlying space, but
other choices are possible and will in some cases give better results.
\section{EXAMPLES}
\addvspace{.6cm}
In this section we list a few applications of our technique,
some of which are to a certain extent already known in some form,
but which we feel we obtain more efficiently.
Theorem \ref{tn:main} is an abstract version of the Hile--Protter inequality
\cite{kn:Hi-Pr}. Hook's earlier abstract Hile--Protter
bound in \cite{kn:Ho90b} contains a free parameter,
which he needs to optimize in order to get some of the applications
which we obtain directly below.
As in \cite{kn:Har-MiA} we define two universal constants $C_{PPW}$ and
$C_{HP}$ as follows
\[C_{PPW}({\cal M}):= \sup_{n,{\bf \Omega} \subset {\cal M}}
\frac{\lambda_{n+1} - \lambda_{n}}
{\langle\lambda_\ell\rangle_{\ell \leq n}} \]
and
\[\hspace*{1.3cm} C_{HP}({\cal M}):= \sup_{n,{\bf \Omega} \subset {\cal M}}
\left< \frac{\lambda_\ell}{\lambda_{n+1} - \lambda_\ell}
\right>_{\ell \leq n}^{-1}\]
where $\langle {f}_{\ell}\rangle_{\ell \leq n} $
denotes the average of an expression involving eigenvalues over all $\ell
\leq n.$ Thus, for example $\langle \lambda_\ell \rangle_{\ell \leq n}
= \frac{1}{n} \sum_{\ell \leq
n}\lambda_\ell.$ Or, as will occur later, if the sum starts at $\ell = 0$
then we divide by $n+1.$ According to these definitions we always have
$C_{HP} \ge C_{PPW}.$
\vspace{.8cm}
\pagebreak
\noindent {\bf Example 1: Operators with continuous spectra}
\vspace{.3cm}
\newline
We begin with a remark about operators with continuous
spectra, which immediately follows from the original estimates
of Payne, P\'{o}lya, and Weinberger \cite{kn:PPW}
and of Hile and Protter \cite{kn:Hi-Pr}, but which does not appear to have been
appreciated. Under wide circumstances it is known that
continuous spectra of Schr\"{o}dinger operators, Sturm--Liouville
operators, Jacobi matrices, etc.
can be quantified by using
a measure known as the density of states, defined by cut-off procedures
(for example, see \cite{kn:CFKS}, \cite{kn:Kir-Wer}, and \cite{kn:Ger-Har}).
Let us call the density of states measure $dn(\lambda),$
and assume that it is
well--defined (as a weak-$\ast$ limit and has bounded first moment)
for the operator $H$, which is a local operator on $L^2(\bf R^{\nu})$
or $L^2(\bf Z^{\nu}).$ Suppose that $\Gamma = (a,b)$ is a gap
in the spectrum of $H$ (i.e., an open interval of length $|\Gamma
|$ in the resolvent set, the ends of which are in the spectrum),
and, moreover, that when $H$ is restricted to a bounded set with
Dirichlet boundary conditions (vanishing conditions, in the
discrete case), there is a finite $C_{HP}.$ Then, when we restrict
$H$ to rectangles of side $R$ and let $R \rightarrow \infty$, we get:
$${|\Gamma | \int_{-\infty }^{a}dn\left({\lambda }\right)
\over \int_{-\infty }^{a}\lambda dn\left({\lambda }
\right)}\ \le C_{PPW}\ \le C_{HP}.$$
For example, if $H$ is a Schr\"odinger operator in $\nu$ dimensions
with a smooth, periodic, non-negative potential, then
$$|\Gamma | \le {4\int_{-\infty }^{a}\lambda
dn\left({\lambda }\right) \over \nu \int_{-\infty
}^{a} dn\left({\lambda }\right)}\ \le
{4a \over \nu }.$$
In particular cases, where estimates can be made of the density of states,
much sharper bounds than $4 a / \nu$ are possible.
\vspace{.8cm}
\noindent {\bf Example 2: Algebraic formulation of quantum mechanics}
\vspace{.3cm}
\newline
In quantum mechanics it is not strictly necessary to
represent the algebra of observables with
differential operators; for many purposes
an abstract algebra is quite sufficient (cf. \cite{kn:Thir}).
In this setting, the fundamental phenomena of quantum mechanics
have their origin in the quantum canonical commutation relations, which
state that if $H$ is a Hamiltonian operator, then canonically conjugate
classical variables satisfy the Heisenberg uncertainty principle.
In the Heisenberg picture, the
operators $X, \Pi$ are a canonically conjugate pair when
$$\dot{X}\ =\ {\Pi \over m}\ =\ {i \over \hbar}\left[{H,X}\right],$$
where $m$ is the mass of the particle and $\hbar$ is Planck's constant
divided by $2 \pi.$ In this case, the commutation relations state that
$\left[{X,\Pi}\right]\ =\ i\hbar$.
If we identify $X$ with $G,$ and assume that ${\Pi}^{2} \le \beta H$,
then the bound on $C_{HP}$ in this case becomes:
\[{C}_{HP}\ \le \ \frac{2\beta n}
{m Tr\left(P\right)}\ = \frac{2 \beta}{m}\ .\]
If there are $M$ canonical momenta, such that
$$\sum\limits_{j = 1}^{M}
{\Pi}_{j}^{2} \le \beta H \ ,$$
then, similarly,
\begin{equation}
{C}_{HP}\ \le {2\beta n \over mM
Tr\left({P}\right)}\ = \frac{2 \beta}{m M}\ .
\label{eq:chpqm}
\end{equation}
With the usual Euclidean momenta, $\beta = 2 m,$ and a
Hile--Protter type bound results. In the language of
quantum mechanics, inequality (\ref{eq:chpqm})
states that the ratio of the gap excitation energy to the
average of the energy below an energy gap is bounded
by $2 \beta / m M$ divided by the trace of the
density matrix for the unexcited states.
\vspace{.8cm}
\noindent {\bf Example 3: Schr\"{o}dinger operators with magnetic fields and
\newline Dirichlet
boundary conditions}
\vspace{.3cm}
\newline Compare with \cite{kn:Ho90b} p. 628. Consider the case when
$H = \left({\bf p} - {\bf A(x)}\right)^2 + V({\bf x})$ on
${\bf L}^2({\bf \Omega})$
for ${\bf \Omega}$ a bounded domain in ${\bf R}^\nu$. And
where ${\bf p} = -i\nabla$ and ${\bf A(x)} = (A_1({\bf x}), A_2({\bf x}),
\ldots, A_\nu({\bf x}))$ is a magnetic vector potential for
the magnetic field ${\bf B(x)}.$
Suppose $Hu_i = \lambda_i u_i.$ The operator $H$ has some discrete eigenvalues
under very wide circumstances (see \cite{kn:CFKS}).
Assume that $V({\bf x}) \geq -M.$ The special choices to be made for the
auxiliary operators whose indices will run from $1,\ldots,\nu+1$ are
\begin{eqnarray*}
\Pi_j & =& -i\frac{\partial}{\partial x_j}
- A_j({\bf x})\;,\;\;\;\;\;
j = 1,\ldots,\nu \\
& & \\
\Pi_{\nu+1} & =& (V({\bf x}) + M)^{\frac{1}{2}} \\
& & \\
G_j & = & x_j \;,\;\;\;\;\; j = 1,\ldots,\nu\\
& & \\
G_{\nu +1} & =& 1. \end{eqnarray*}
It is easy to see that $\sum_{j=1}^{\nu +1} \Pi_j^2 = H + M.$
So we see that
the left hand side of inequality (\ref{eq:main}) is,
\[\sum_{i=1}^n (\lambda_{n+1} - \lambda_i)^{-1}\langle u_i,(H+M)u_i\rangle
= \sum_{i=1}^n(\lambda_{n+1} - \lambda_i)^{-1} (\lambda_i + M).\]
For the right hand side we must compute the commutators
\begin{eqnarray*}
[\Pi_j,G_j] = [-i\frac{\partial}{\partial x_j},x_j] & = & -i\;,
\;\;\;\;\;j=1,\ldots,\nu \\
& & \\
\left[ \Pi_{\nu +1} ,G_{\nu +1}\right] & = & 0\;. \\
\end{eqnarray*}
And the double commutators,
\begin{eqnarray*}
\Big[G_j,[H,G_j]\Big] & = &
\Big[x_j,[-\Delta + i\nabla \cdot {\bf A} +
i {\bf A} \cdot \nabla , x_j]\Big] \\
& = & \Big[x_j,(-2 \frac{\partial}{\partial x_j} +
2 i A_j)\Big] \\
& = & 2
\;,\;\;\;\;\; j = 1,\ldots,\nu \\
& & \\
\Big[G_{\nu+1},[H,G_{\nu +1}]\Big] & = & 0\,. \\
\end{eqnarray*}
Thus, the right hand side of inequality (\ref{eq:main}) is
\[\frac{\Big|\sum_{j=1}^\nu
Tr (P)\Big|^2}{2\:\sum_{j=1}^\nu Tr\Big(P\!\cdot\!2\Big)}
= \frac{\nu n}{4}\]
and by the theorem,
\[\hspace*{1.4cm}\sum_{i=1}^n \frac{\lambda_i + M}{\lambda_{n+1} - \lambda_i}
\geq \frac{\nu n}{4}.\;\;\;\Box\]
\vspace{.8cm}
\noindent {\bf Example 4: The bi-Laplacian}
\vspace{.3cm}\newline
The operator $H = \Delta^2$ on $L^2({\bf \Omega})$ where
${\bf \Omega} \subset {\bf R}^\nu$ is a bounded domain with smooth
boundary has applications in the theory of elasticity. Compare with
\cite{kn:Che} and
\cite{kn:Ho90b} p. 633. In particular,
the problem
\begin{eqnarray*}
\Delta^2 u_i & = & \mu_i u_i \;\;\; {\rm in} \;\; {\bf \Omega}\\
& & \\
u_i = \frac{\partial u_i}{\partial n} & = & 0 \;\;\; {\rm on}
\;\; \partial{\bf \Omega}\\
\end{eqnarray*}
is related to the modes of vibration of a clamped plate.
This problem was studied by Payne, P\'{o}lya, and Weinberger in \cite{kn:PPW}
where they obtained the bound
\begin{equation}
\mu_{n+1} - \mu_n \leq \frac{8(\nu + 2)}{\nu^2 n} \sum_{i=1}^n \mu_i
\label{eq:ppwbi}
\end{equation}
and later by Hile and Yeh in \cite{kn:Hi-Ye} where the result was improved
to
\begin{equation}
\sum_{i=1}^n \frac{\sqrt{\mu_i}}{\mu_{n+1} - \mu_i}
\geq \frac{\nu^2 n^{\frac{3}{2}}}{8(\nu + 2)}
\left(\sum_{i=1}^n \mu_i\right)^{-\frac{1}{2}}.
\label{eq:hy}
\end{equation}
We now recover the result of \cite{kn:Hi-Ye} by applying Theorem \ref{tn:main}.
Assume that the eigenvalues are enumerated in the usual way. The
choice for the auxiliary operators for this case is
\begin{eqnarray*}
\Pi_j & =& -i\frac{\partial}{\partial x_j} \\
& & \\
G_j & = & x_j \;
\end{eqnarray*}
where $j = 1, \ldots, \nu \;.$
The left hand side of equation (\ref{eq:main}) is then
\[\sum_{i=1}^n(\mu_{n+1} - \mu_i)^{-1} \langle u_i,-\Delta u_i\rangle
\leq \sum_{i=1}^n(\mu_{n+1} - \mu_i)^{-1}
\langle u_i,u_i\rangle^{\frac{1}{2}}
\langle u_i, \Delta^2 u_i\rangle^\frac{1}{2}\]
\[\hspace{2cm} = \sum_{i=1}^n(\mu_{n+1} - \mu_i)^{-1} \mu_i^\frac{1}{2} \]
where the Schwarz inequality and the boundary
conditions were used to obtain the inequality.
For the right hand side we again have $[\Pi_j,G_j] = -i.$ And now the
double commutators are
\[\Big[G_j,[H,G_j]\Big] = \Big[x_j,[\Delta^2,x_j]\Big]
= \Big[x_j, 4\partial_j \Delta \Big]
= -4(\Delta + 2\partial_j^2)\]
so that
\[\sum_{j=1}^\nu \Big[G_j,[H,G_j]\Big] = 4(\nu +2)(-\Delta).\]
Then, again applying the Schwarz inequality and the boundary conditions
we have
\[\sum_{i=1}^n \langle u_i, -\Delta u_i\rangle
\leq \left(\sum_{i=1}^n\langle u_i, u_i \rangle \right)^\frac{1}{2}
\left(\sum_{i=1}^n\langle u_i, \Delta^2 u_i \rangle \right)^\frac{1}{2}.\]
So for the right hand side of equation (\ref{eq:main}) we have
\[\frac{(\nu n)^2}{8(\nu +2)n^\frac{1}{2}}
\left(\sum_{i=1}^n \mu_i\right)^{-\frac{1}{2}}\]
and so by the theorem we have
\[\sum_{i=1}^n \frac{\mu_i^\frac{1}{2}}{\mu_{n+1} - \mu_i}
\geq \frac{\nu^2 n^\frac{3}{2}}{8(\nu +2)}
\left(\sum_{i=1}^n \mu_i\right)^{-\frac{1}{2}}\]
as in \cite{kn:Hi-Ye}.$\;\;\;\Box$
\section{NEUMANN BOUNDS}
\addvspace{.6cm}
As a final use of our techniques, we obtain an entirely new bound
for the spectral geometry of the Laplacian with Neumann
boundary conditions. We shall show that
gaps in the spectrum of the Laplacian are controlled, inversely,
by the inradius, which by definition is the radius of the
largest inscribed ball. The inradius is known to control
the ground state of the Dirichlet Laplacian, and thus to
play an important role in the spectral geometry for
that operator (cf. \cite{kn:Chav} and \cite{kn:Davies}).
On the other hand, we believe that the constant we obtain
is far from optimal.
Since the Neumann Laplacian has a zero eigenvalue, and in
our technique we need it to dominate another expression, we let
$H_M := - \Delta + M,$ where $M$ is a positive constant to be
specified below. The addition of the constant $M$
obviously does not affect any commutators with $H.$
Let $\widetilde{u}$ be the first nonconstant
radial Neumann $L^2$ -- normalized eigenfunction for the
unit ball and let $\widetilde{\lambda }$ be the
corresponding eigenvalue.
\begin{pnum} Let ${\bf \Omega}$ be a domain in $R^{\nu}$ with piecewise
smooth boundary and inradius $r,$
and let $\lambda_k$ be the eigenvalues of the
Neumann Laplacian for ${\bf \Omega.}$ Then
$${\left\langle{{{\lambda }_{\ell} +
M \over {\lambda }_{n+1}-{\lambda }_{\ell}}}
\right\rangle}_{\ell\le n } \ge \frac{K}{n + 1}\, ,$$
where:
$$M := {{\widetilde{\lambda }}^{2}{\|\widetilde{u}\|
}_{\infty }^{2} \over 4{r}^{2}{\|\nabla
\widetilde{u}\|
}_{\infty }^{2}}$$
and
$$\hspace{.6cm}K := \frac{r^\nu \widetilde{\lambda}}{8 \left|{\bf \Omega}
\right|
\left\|\nabla \widetilde{u}\right\|_{\infty}^2} \, .$$
\end{pnum}
\vspace{.3cm}
REMARK: \ This complicated expression
implies a somewhat simpler bound of the form
$${\lambda }_{n+1}-{\lambda }_{n} \le (n + 1)
\left({A\left\langle{{\lambda }_{\ell}}\right\rangle_{\ell \le n}
+B}\right),$$ where $A$ and $B$ depend only on the inradius $r$ and
${\bf \Omega.}$ \vspace{.4cm}
\noindent PROOF:
The special choices are
$G := \widetilde{u},$ centered at the center of the
inscribed ball and with variable $|x|/r$ in the ball
and extended as a
constant outside the ball, and $\Pi := i\left[H,G\right]\
= i\left( -\Delta G -
2\nabla G\cdot \nabla \right).$
According to our usual calculation,
\begin{equation}
\left\langle \frac{\lambda_\ell + M}
{\lambda_{n+1}-\lambda_\ell}
\right\rangle_{\ell\le n} \ge
\frac{\left|Tr\left(
P[G,[H,G]]\right)\right|^2}
{ 2(n+1)\beta Tr\left(P[G,[H,G]]\right)}
= \frac{Tr\left(P\left|\nabla
G\right|^{2}\right)}{(n + 1)\beta }\, ,
\label{eq:NeumA}
\end{equation}
where $\beta$ is any constant such that
$${\|\left({\Delta G}\right)\zeta + 2\nabla
G\cdot \nabla \zeta \|
}^{2}\ \le \beta \left({\|\nabla \zeta \|
}^{\rm 2} + M \| \zeta \|^2 \right)$$
for all $\zeta$ in the quadratic form domain of the Neumann Laplacian.
Since the left side of this expression is
$${\left\|{\widetilde{\lambda } \over
{r}^{2}}\widetilde{u}\left({{x \over r}}\right)\zeta
+ {2 \over r}\nabla \widetilde{u}\left({{x \over r}}\right)\cdot
\nabla \zeta \right\|
}^{2} \le {2{\widetilde{\lambda
}}^{2}{\|\widetilde{u}\|
}_{\infty }^{2} \over {r}^{4}}{\|\zeta \|
}^{2}\ +\ {8 \over {r}^{2}}{\|\nabla
\widetilde{u}\|
}_{\infty }^{2}{\|\nabla \zeta \|
}^{2}$$
(the cross term has been estimated by $2ab \leq a^2 + b^2),$
we can take
$$\beta := {8 \over {r}^{2}}{\|\nabla
\widetilde{u}\|
}_{\infty }^{2}\;\; {\rm and}\;\; M :=
{{\widetilde{\lambda }}^{2}
{\|\widetilde{u}\|
}_{\infty }^{2} \over 4{r}^{2}{\|\nabla
\widetilde{u}\|
}_{\infty }^{2}}\,. $$
Since the ground state is $1 / \sqrt
{|{\bf \Omega} |},$ we estimate the numerator of
equation (\ref{eq:NeumA})
by
\[Tr(P \left|\nabla G\right|^2) \geq
\int{\left|{\nabla G}\right|}^{2}{1 \over
\left|{\bf \Omega}\right|},\]
which
\[ = {r^\nu \widetilde{\lambda } \over
{r}^{2}\left|{\bf \Omega}\right|},\]
and the claim follows.$\;\;\;\Box$
\vspace{.6cm}
\noindent {\bf ACKNOWLEDGEMENTS}
\vspace{.5cm}
\noindent The first author is grateful to the Georgia Tech Foundation
for support and to the Centre de Physique Th\'{e}orique, Luminy,
France, and the Universit\'{e} de Toulon, France, for hospitality
while part of this work was performed.
The second author would like to thank the Mittag-Leffler Institute, Djursholm,
Sweden, for their hospitality while some of this work was carried out.
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\end{document}