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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,
Georgia Institute of Technology, 1994.}
\vspace*{.25in}
\begin{center}{\bf COMMUTATOR BOUNDS FOR EIGENVALUES, \\
WITH APPLICATIONS TO SPECTRAL GEOMETRY} \\
\vspace{.8cm}
Evans M. Harrell II\footnotemark[1] \\
\vspace{.2cm}
Center for Dynamical Systems and Nonlinear Studies \\
School of Mathematics \\
Georgia Institute of Technology \\
Atlanta, Georgia 30332-0160 \\
harrell@math.gatech.edu \\
\vspace{.5cm}
Patricia L. Michel\footnotemark[2] \footnotemark[3] \\
\vspace{.2cm}
School of Mathematics \\
Georgia Institute of Technology \\
Atlanta, Georgia 30332-0160 \\
michel@math.gatech.edu \\
\vspace{1.5cm}
\noindent{\bf Abstract}
%\vspace{.1cm}
\end{center}
\noindent We prove a purely algebraic version of an eigenvalue inequality
of Hile and Protter, and derive corollaries bounding differences of
eigenvalues of Laplace--Beltrami operators on manifolds. We
significantly improve earlier bounds of Yang and Yau, Li, and Harrell.
\section{Introduction}
In a recent paper \cite{kn:Har92}, one of us produced an algebraic version of a
well--known bound on differences between eigenvalues due to
Payne, P\'{o}lya, and Weinberger \cite{kn:PPW}. They had shown that the
difference of successive eigenvalues of the Dirichlet Laplacian on a
domain in ${\bf R}^\nu $ is bounded by a universal constant times the sum
of all the lower eigenvalues. It is easy to see, however, that if the
analogous problem is considered on a manifold, then the geometry
must enter into the relationship between the differences and the
sum, and \cite{kn:Har92} showed specifically that for certain manifolds
admitting global Fermi coordinate systems, the relationship depends in a
certain way on the sectional curvature of the manifold. The algebraic
bound in that article involves auxiliary operators, which can be
specially chosen to reveal the geometric content of the eigenvalue
differences. The philosophy of this article will be the same, except
that we shall prove a different algebraic bound, which allows sharper
estimates.
In Section 3 we provide new upper bounds on eigenvalue differences and related
quantities in many situations, including the Dirichlet problem on subdomains of
spheres (Corollary 3.1) and hyperbolic space (Corollary 3.2); and the
eigenvalues of the Laplace--Beltrami operator on minimally immersed
submanifolds (Corollary 3.7) and homogeneous manifolds (Corollary 3.8). These
bounds are expressed in terms of a constant $C_{HP}$ which we shall define
below.
A reasonable way to frame the problem is as follows.
Consider a Riemannian manifold ${\cal M},$
with subdomains ${\bf \Omega}$. The Laplace--Beltrami
operator acts on smooth scalar functions on ${\bf \Omega}$ by
$\Delta f = \nabla \cdot \nabla f,$ where $\nabla \cdot $ and
$\nabla $ are the covariantly defined divergence and gradient.
In local coordinates it has the form
\begin{equation}
\Delta f = \frac{1}{\sqrt{g}}\,\sum_{i,j} \partial_i \left( g^{ij}\,
\sqrt{g}
\,\partial_j f \right)
\label{eq:LB}
\end{equation}
and it is defined as a self--adjoint operator on appropriate
Sobolev spaces incorporating the boundary conditions. Here $g$ is the
determinant of the metric tensor $g_{ij},$ and $g^{ij}$ is the
contravariant (inverse) metric tensor. For details
on Laplace--Beltrami eigenvalue problems, we refer the reader to
\cite{kn:Chav}, \cite{kn:Buser}, and \cite{kn:Oss-Wein}.
The Dirichlet eigenvalues of the
Laplace--Beltrami operator on a given domain will be denoted
$\lambda_{\ell},\: \ell = 1,2,\ldots,n \:;$ these
eigenvalues form a sequence of positive numbers
accumulating at infinity.
Let $\langle {f}_{\ell}\rangle_{\ell \leq n} $
denote the average of an expression involving eigenvalues over all $\ell
\leq n.$ Thus, for example $\langle \lambda_\ell \rangle_{\ell \leq n}
= \frac{1}{n} \sum_{\ell \leq
n}\lambda_\ell.$ If, as will sometimes occur below, $\ell = 0, 1, \ldots,
n\:,$ we divide by $n+1.$ We now define a constant which will turn
out to reflect the geometry of ${\cal M}$ by
\[C_{PPW}({\cal M}):= \sup_{n,{\bf \Omega} \subset {\cal M}}
\frac{\lambda_{n+1} - \lambda_{n}}
{\langle\lambda_\ell\rangle_{\ell \leq n}}\,.\]
In the original article of Payne, P\'{o}lya, and Weinberger \cite{kn:PPW}, the
manifold ${\cal M}$ was Euclidean, and
$C_{PPW} = \frac{4}{\nu},$
where $\nu :=$ dimension of ${\cal M}$ was actually fixed at 2. Many
extensions of this have been made;
the most up--to--date survey of
the situation is \cite{kn:Ash93}. For other related work and background
see \cite{kn:Har88}, \cite{kn:Prot}, and \cite{kn:Chav}.
Since $1 + C_{PPW}$ is an upper
bound on the ratio $\lambda_{2}/\lambda_{1}$, and this ratio is
arbitrarily large in the geometric setting, $C_{PPW}$ is generally more
complicated on manifolds. For example, the n-sphere with a small
cap removed will have $\lambda_{1}$ close to 0, but $\lambda_{2}$ is
bounded away from 0.
Several people (e.g. \cite{kn:Li}, \cite{kn:Leung}, \cite{kn:Yang-Yau})
have studied bounds on gaps and ratios of eigenvalues
to see how Riemannian geometry is revealed in the analysis of the
Laplace--Beltrami operator. Most progress has occured in the context of
spaces of high symmetry; one of our goals here is to sharpen several
of the more significant of these bounds. Following \cite{kn:Har92} we also
produce some bounds for fairly general Riemannian manifolds.
One of the extensions of \cite{kn:PPW}, due to Hile and Protter \cite{kn:Hi-Pr}
at first sight may seem to be a mere technical improvement, since it
has a more complicated relationship between the eigenvalue
differences and the sums, which reduces to that of Payne, P\'{o}lya,
and Weinberger in what is usually the most interesting case, that of
the gap between $\lambda_{2}$ and $\lambda_{1}.$ The proof seems to
follow the steps of the earlier result for the most part, except for the
order of the steps and the introduction of free parameters (as it
turns out, unnecessarily). In our abstract version of the Hile--Protter
inequality, the underlying algebra is actually somewhat distinct from
the PPW bound, and in addition it allows more flexibility because
there are two families of auxiliary operators, as opposed to the one
in \cite{kn:Har92}. These differences are not important for the original
problem with a Euclidean Laplacian, but are quite helpful in the
situations we consider below.
Earlier, Hook produced an abstract algebraic bound of the
Hile--Protter type and made several applications, recovering as special
cases several prior bounds on various operators on
domains in ${\bf R}^\nu $ (see \cite{kn:Ho90a} and \cite{kn:Ho90b}).
Hook followed the original argument of Hile and
Protter rather closely, and in particular, his result contains the
same free parameter as in \cite{kn:Hi-Pr}. In order to recover the prior
bounds, he makes special choices of the operators in the abstract
theorem and optimizes over the parameter. Our Theorem \ref{tn:main} is
similar in nature, but avoids the parameter altogether, and recovers
a variety of results directly with special choices of the abstract
operators. Some non-geometric applications of this technique are
dealt with in a separate paper \cite{kn:Ha-Mib}.
The Hile--Protter inequality is usually expressed as a lower bound,
but for comparison with the PPW inequality, we define a comparable
constant:
\[C_{HP}({\cal M}):= \sup_{n,{\bf \Omega} \subset {\cal M}}
\left< \frac{\lambda_\ell}{\lambda_{n+1} - \lambda_\ell}
\right>_{\ell \leq n}^{-1}.\]
Observe that $C_{HP} \geq C_{PPW},$ so for upper bounds on
eigenvalue differences, it is not necessary to distinguish the two
constants.
\section{A theorem for abstract operators on Hilbert space}
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. A rough
correlation with Hook's 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 $P_{\le n}$ be the spectral projection for $ \lambda_{1}, \lambda_{2},
\ldots ,
\lambda_{n},$ and for any positive $m$ let $\{ G_{j}\}_{j=1}^m$ and $\{
\Pi_{j}\}_{j=1}^m$ 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\left(\left(\lambda_{n+1}I - H\right)^{-1}
P_{\le n}\Pi_{j}^{2}\right) \geq
\frac{\left|\sum_{j=1}^{m} Tr\left(P_{\le n}\left[
\Pi_{j},G_{j}\right]\right)
\right|^{2}}{2 \sum_{j=1}^{m}
Tr\left(P_{\le n}\left[G_{j},\left[H,G_{j}\right]\right]\right)}\,,
\label{eq:main}
\end{equation}
assuming that these three traces are finite and nonzero.
\end{tnum}
\vspace{.2cm}
{\em Remarks:} The operator $\lambda_{n+1}I - H $ is uniquely
invertible as a positive operator from the range of $P_{\le n}$ 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
$G_j u \in {\cal D}(H).$ Similarly for $G_j^2u.$
In many applications the operators $\Pi_{j}$ are chosen so
that, in the
quadratic--form sense,
\[\sum_{j=1}^{m} \Pi_{j}^{2} \leq \beta H\]
for some constant $\beta$ (which could be scaled to 1).
Thus, in terms of the constants defined in the
introduction,
\[ \sum_{j=1}^m Tr \left( \left( \lambda_{n+1} - H \right)^{-1}
P_{\le n} \Pi_j^2 \right)
\le \beta \, \sum_{j=1}^m \frac{\lambda_j}{\lambda_{n+1} - \lambda_j}
= \beta \, n \, C_{HP}^{-1} \]
and by (\ref{eq:main}),
\begin{equation}
C_{PPW} \leq C_{HP} \leq \frac{2n \beta
\sum_{j=1}^{m}
Tr\left(P_{\le n}[G_j ,[H,G_j ]]\right)}
{\left|\sum_{j=1}^{m} Tr\left(P_{\le n}[\Pi_j ,G_j]\right)\right|^2}.
\label{eq:upbnd}
\end{equation}
Most frequently, the auxiliary operators will be chosen to
satisfy
\begin{equation}
\Pi_j = i[H,G_j] = i(-\Delta G_j - 2\nabla G_j \cdot
\nabla).
\label{eq:eqnd}
\end{equation}
The bounds that result with this specialization closely
resemble
those of \cite{kn:Har92}, but improve them a bit. This is because in
this case,
\begin{equation}
\frac{2n \sum_{j=1}^{m}
Tr\left(P_{\le n}[G_j ,[H,G_j ]]\right)}
{\left|\sum_{j=1}^{m} Tr\left(P_{\le n}[\Pi_j ,G_j]\right)\right|^2}
= \frac{2n}{ \sum_{j=1}^{m}
Tr\left(P_{\le n}[G_j ,[H,G_j ]]\right)}\,.
\label{eq:eqna}
\end{equation}
\vspace{.2cm}
We recall two familiar properties of the trace, which will be
used frequently, without comment, in the
proof of the theorem:
\begin{enumerate}
\item the cyclic property of the trace:
$Tr(AB) = Tr(BA),$
\item $Tr (A^\ast B)$
is an inner product on $A$ and $B.$
In particular, the Cauchy--Schwarz inequality holds
in the form: $|Tr(A^\ast B)|^2\leq Tr(A^\ast A)\;Tr(B^\ast B).$
\end{enumerate}
Since we are taking traces of products of operators, each product
contains a finite projection, so there are no convergence difficulties.
We also use the fact that spectral projections commute with $H$ and
with one another.
The proof will be given as a series of simple lemmas.
\begin{lnum}
\label{ln:L1}
Let $P$ be a finite rank orthogonal projection,
$Q := 1-P$,
and let $G$ and $\Pi$ be symmetric. Then,
\[\Im m\left(Tr(P{\mit \Pi}G)\right)
=\Im m\left(Tr(P{\mit \Pi}QG)\right).\]
\end{lnum}
\noindent {\bf Proof:}
This is equivalent to
\[\Im m\left(Tr(P{\mit \Pi}PG)\right)=0,\]
which follows from
\[Tr(P {\mit \Pi} P G) = Tr(G P {\mit \Pi} P) = \overline{Tr(P{\mit
\Pi}PG)}.\;\;\;\Box\]
\begin{lnum} For $P, Q, \Pi, G $ as in lemma \ref{ln:L1},
\label{ln:L2}
\[Tr(P[{\mit \Pi},G])
=2i\:\Im m\left(Tr(P{\mit \Pi}QG)\right).\]
\end{lnum}
\noindent {\bf Proof:}
\[\begin{array}{ll}
Tr(P[{\mit \Pi},G])&=\;\;Tr(P{\mit \Pi}G)-Tr(PG{\mit \Pi})\\
\\
&=\;\;Tr(P{\mit \Pi}G)-\overline{Tr({\mit \Pi}GP)}\\
\\
&=\;\;Tr(P{\mit \Pi}G)-\overline{Tr(P{\mit \Pi}G)}\\
\\
&=\;\;2i\;\Im m\;Tr(P{\mit \Pi}G)\\
\\
&=\;\;2i\;\Im m\;Tr(P{\mit \Pi}QG).\;\;\;\Box\end{array}\]
\begin{lnum}
\label{ln:L3} Let $G$ and $H$ be as in theorem \ref{tn:main}. Let
$P$ be a spectral projection for $H$ and $Q = I - P,$ then
\[Tr(P\left[G,[H,G]\right])\;=\;2\;Tr(PGQ[H,G]).\]
\end{lnum}
\noindent {\bf Proof:}
\[\begin{array}{ll}
Tr(P\left[G,[H,G]\right])&=\;\;
2\;Tr(PGHG)-Tr(PG^2H)-Tr(PHG^2)\\
\\
&=\;\;2\;Tr(PGHG)-2\;Tr(PG^2H)\\
\\
&=\;\;2\;Tr(PG[H,G]).\end{array}\]
Since $\left[G,[H,G]\right]$ is symmetric, the left side of
this is real. This implies that $Tr(PG[H,G])$ is real but $i\, [H,G]$
is symmetric so by Lemma \ref{ln:L1}
\[\begin{array}{ll}
i\,Tr(PG[H,G])& = Im\,Tr(PG(i[H,G])) \\
\\
& = Im\,Tr(PGQ(i[H,G])) \\
\\
& = i\,Re\,Tr(PGQ[H,G]).
\end{array}\]
But $Tr(PGQ[H,G])$ is real since it can be expressed as trace of the difference
of two symmetric operators,
$Tr(AHA^*) - Tr(A^*HA)$ where
$A = PGQ.
\;\;\;\Box $
\begin{lnum} Let $H$ and $G$ be as in Theorem \ref{tn:main} and let
$P = P_{\le n}$ be a spectral projection for $H,$ then
\label{ln:L4}
\begin{equation}
0 \leq Tr((\lambda_{n+1}I-H)PGQG)\;\leq\;Tr(PGQ[H,G]).
\label{eq:L4}
\end{equation}
\end{lnum}
\noindent {\bf Proof:}
The right side of inequality (\ref{eq:L4}) is
\[ Tr(PGQ[H,G]) = Tr(PGQHG) - Tr(PGQGH) = Tr(PGQHG) - Tr(HPGQG), \]
and the first of these terms is bounded below as follows
\[\begin{array}{ll}
Tr(PGQHG)&=\;\;
Tr(PGQHQGP)\\
\\
&\geq\;\;Tr(PGQ\lambda_{n+1}QGP)\\
\\
&=\;\;Tr(\lambda_{n+1}PGQG).
\end{array}\]
\newline
Finally, $Tr((\lambda_{n+1}I - H)PGQG) = Tr(QGP(\lambda_{n+1}I - H)PGQ) \ge 0 $
since \linebreak $(\lambda_{n+1}I - H) \ge 0$ on the range of $P_{\le n}.
\;\;\;\Box$
\vspace{.3cm}
\noindent {\bf Proof of the theorem:} for brevity we drop the subscript
on $P_{\le n}.$
\noindent Lemmas \ref{ln:L3} and \ref{ln:L4} imply
\[Tr\left((\lambda_{n+1}I-H)PG_jQG_j\right)
\leq\;\frac{1}{2}\;Tr(P[G_j,[H,G_j]])\]
and by lemma \ref{ln:L2}
\[\sum^m_{j=1}Tr\left(P[{\mit \Pi}_j,G_j]\right)\;=\;
2i\;\Im m\;\sum^m_{j=1}Tr\left(P{\mit \Pi}_jQG_j\right) .\]
\noindent Then,
\begin{eqnarray*}
&& \hspace{-1.cm} \frac{\left|\sum_j Tr\left(
P\left[\Pi_j,G_j\right]\right)\right|^2}{4} \\
& \leq & \displaystyle \left| \sum_j Tr\left(P
\Pi_j Q G_j\right)\right|^2\\
& = & \displaystyle \left|\sum_j Tr\left(\left(\lambda_{n+1}I-H\right)^{1/2}
\left(\lambda_{n+1}I-H\right)^{-1/2} P \Pi_j Q G_j\right)\right|^2\\
& = & \displaystyle \left| \sum_j Tr\left(
\left(\lambda_{n+1}I-H\right)^{-1/2}
P \Pi_j Q G_j \left(\lambda_{n+1}I-H\right)^{1/2}P\right)\right|^2\\
& \leq & \displaystyle \left|\sum_j Tr\left( \Pi_j P
\left(\lambda_{n+1}I-H\right)^{-1}P \Pi_j\right)\right|
\; \displaystyle \left| \sum_j Tr\left(Q G_j
\left(\lambda_{n+1}I-H\right)P G_j Q\right)\right|\\
& = & \displaystyle \left|\sum_j Tr\left(
\left(\lambda_{n+1}I-H\right)^{-1}P
\Pi_j^2\right)\right|
\;\displaystyle \left|\sum_j Tr\left(
\left(\lambda_{n+1}I-H\right)P G_j Q G_j\right)\right|\\
& \leq &\;\frac{1}{2}\;\displaystyle \left(\sum_j Tr\left(
\left(\lambda_{n+1}I-H\right)^{-1}P
\Pi_j^2\right)\right)\;\displaystyle \left(\sum_j Tr\left(
P\left[G_j,\left[H,G_j\right]\right]\right)\right).
\end{eqnarray*}
\noindent Dividing both sides by $\frac{1}{2} \displaystyle \sum_j
Tr(P[G_j,[H,G_j]])$ yields equation (\ref{eq:main}).$\;\;\;\Box$
As remarked above $(\lambda_{n+1} - H)$ is positive on the range of $P$
so by the square root lemma (see \cite{kn:Re-Si}) the powers of this
operator used above are all well-defined.
\section{Applications}
We begin by improving some of the bounds of \cite{kn:Har92}.
\vspace{.7cm}
\noindent{\bf Spherical domains}
\vspace{.5cm}
Let ${\cal M}=S^{\nu} \setminus B_{\rho}, \;\;\nu \geq 2,$
where $B_{\rho}$ is a geodesic ball of radius $\rho > 0.$
We assume the radius of $S^{\nu}$ is $1$ (other radii are
included by scaling).
The Laplace--Beltrami
operator on a spherical domain is the same as the
angular momentum operator in quantum mechanics.
It was shown in \cite{kn:Har92} that
\[{C}_{PPW}\ \le \ {16\,{\left({1+\displaystyle
{\left({\nu \rm -2}\right)\sin \rho
\over \rm 2\left({1\ -\ \cos \rho }\right)\sqrt {{\lambda
}_{1}}}}\right)}^{2} \over {\left({1\ -\ \cos \rho }\right)}^{2}\nu }\
.\]
Except in dimension 2, this has an unneeded and unpleasant factor,
for we now have:
\begin{cnum}
Let ${\cal M} = S^{\nu} \setminus B_{\rho},$ as above. Then,
\begin{equation}
C_{HP} \leq \frac{16}{(1 - \cos\rho )^2 \nu}.
\label{eq:newchp}
\end{equation}
\end{cnum}
This constant diverges, as it must, when $\rho
\rightarrow 0,$ although not in the optimal way, and reduces to
the Euclidean $C_{HP} = 4/\nu$ as ${\cal M}$ becomes small
$(\rho \rightarrow \pi).$
\newline
\noindent {\bf Proof:}
The special choices are as follows. Embed $S^{\nu}$ in ${\bf R}^{\nu
+1},$
and let $G_j$ be the $j^{th}$ stereographic coordinate, except
for $G_0,$ which is a dummy operator.
\begin{eqnarray*}
G_0 & := & 0, \\
G_j & := & \frac{x^j}{1 - x^0}\;\;\;{\rm for}\; 1 \leq j \leq
\nu
\end{eqnarray*}
\noindent where the Euclidean coordinates are denoted $(x^0, \ldots
,x^{\nu}),$
with $x^0$ oriented towards the center of $B_\rho.$
Define \[\Pi_j := -i\:{\cal R}\: \partial_j\: {\cal E},\]
where ${\cal E}$ is the extension of a function on
$S^{\nu}$ to ${\bf R}^{\nu + 1}\setminus \{0\}$ by writing a function $f$
on $S^{\nu}$ in Euclidean coordinates with the restriction
$\sum_{j=0}^{\nu} (x^j)^2 = 1,$ and then letting
\[{\cal E}f(x^0, \ldots ,x^{\nu}) :=
f\left(\frac{x^0}{r}, \ldots ,\frac{x^{\nu}}{r}\right),
\;\;{\rm where}\;\;
r := \left(\sum_{j=0}^{\nu}\left(x^j\right)^2\right)^{\frac{1}{2}}\]
\noindent and ${\cal R}$ is the restriction of a function in
${\bf R}^{\nu +1}$
to $S^{\nu}.$
Because of the
embedding, we can calculate with the usual Euclidean Laplacian acting
on functions on
${\bf R}^{\nu+1}$ independent of $r.$
It is clear that
$\sum_{j=0}^{\nu} \Pi_j^2 = -\Delta.$
\newline
The choice of the $G$'s is equivalent to that in \cite{kn:Har92},
where it was found that
\[\sum_{j=1}^{\nu} [G_j,[H,G_j]] = \sum_{j=1}^{\nu} 2 |\nabla G_j|^2 =
\frac{2\nu}{(1 - x^0)^2}\,.\]
The numerator in (\ref{eq:upbnd}) is therefore
bounded above by
\[\frac{4\nu n^2}{(1 - \cos\rho )^2}\,.\]
Meanwhile, since $\Pi_j$ satisfies Leibniz's rule for
derivatives, we find that
\[\sum_{j=1}^{\nu} [\Pi_j,G_j] = -i\frac{\nu - 1 - x^0}{1 - x^0}
= -i\left(\frac{\nu - 2}{1 - x^0} + 1\right)\,,\]
so \[|\sum_j Tr(P_{\le n}[\Pi_j,G_j])|^2
\geq \left(\frac{\nu - 2}{2} + 1\right)^2 (TrP_{\le n})^2 =
\left( \frac{\nu n}{2}\right) ^2 \]
yielding the inequality (\ref{eq:newchp}).$\;\;\;\Box$
\vspace{1.0cm}
\noindent{\bf Hyperbolic domains}
\vspace{.5cm}
In \cite{kn:Har92} one of us derived an upper bound on $C_{PPW}$ for
subdomains
of the two--dimensional hyperbolic space ${\cal H}^2$, but
the bound diverges as the size of the domain becomes
infinite.
Unlike the case of spherical domains, this should not
happen, since
\begin{equation}
- \Delta \geq {1 \over 4}
\label{eq:neweqnc}
\end{equation}
in the sense of quadratic forms, which prevents
$\lambda_1$
from approaching 0.
This drawback can be evaded by choosing the auxiliary
operators
from a semigeodesic (Fermi) coordinate system. Recall
that there is a semigeodesic coordinate
system $(t,r)$ for ${\cal H}^2$ with the metric
$ds^2 = dt^2 + cosh^2 t\; dr^2$
(cf. \cite{kn:Chav}, p. 263, where, however, there are some
misprints).
We shall choose $G_1 := t,$ $G_2 := r,$ and
$\Pi_j := i[H,G_j] = i(-\Delta G_j - 2\nabla G_j \cdot \nabla)$ for
$j = 1,2.$
Calculations with equation (\ref{eq:LB}) readily show that
\begin{eqnarray}
\Delta t & = & \tanh{(t)}\,,\nonumber\\
\Delta r & = & 0\,, \nonumber\\
2 \nabla t \cdot \nabla \zeta & = & 2 \; \frac{\partial
\zeta}{\partial t}\,,\nonumber\\
2 \nabla r \cdot \nabla \zeta & = & \frac{2}
{\cosh^2{(t)}}\,\frac{\partial \zeta}{\partial r}\,.
\label{eq:neweqnb}
\end{eqnarray}
\begin{cnum}
Suppose that ${\bf \Omega}$ is a domain in ${\cal H}^2$
such that the distance from any point of ${\bf \Omega}$ to
$\{t=0\}$
is at most $T.$ Then
\[C_{HP} \le \frac{4\, e^{2T}}{1 + \cosh^2{(T)}}.\]
\end{cnum}
{\em Remark:} Because of the freedom to choose the orientation
of
the Fermi coordinate system, $T$ is intuitively an upper
bound on
the thinner dimension of ${\bf \Omega}.$
\vspace{.3cm}
\noindent {\bf Proof:}
Because of
(\ref{eq:neweqnb}),
\[\|\Pi_{1} u\|^2
+ \|\Pi_{2} u\|^2
= \int_\Omega \left(\left(\tanh{(t)} u
+ 2\frac{\partial u}{\partial t}\right)^2
+ \frac{4}{\cosh^4{(t)} }
\left(\frac{\partial u}{\partial r}\right)^2\right)dV.\]
Since
\[\|\nabla u\|^2 =
\int_\Omega \left(\left(\frac{\partial u}{\partial t}\right)^2
+ \frac{1}{\cosh^2{(t)} } \left(\frac{\partial u}{\partial r}\right)^2
\right)dV,\]
we get
\[\|\Pi_1 u\|^2 +
\|\Pi_2 u\|^2 \le
\int_\Omega \left(\tanh^2{(T)} u^2 + 4 \tanh{(T)}u
\left|\nabla u\right| + 4\left|\nabla u \right|^2 \right)dV.\]
Because of (\ref{eq:neweqnc})
and the Cauchy-Schwarz inequality, this is bounded above
by
\[ \tanh^2{T} \|u\|^2 +
4 \tanh{T} \int_\Omega
\left|u\right| \left|\nabla u \right| dV
+ 4 \|\nabla u\|^2 \le
4 \left(1 + \tanh{T}\right)^2 \|\nabla u \|^2.\]
In other words, we can take $\beta = 4\left(1 + \tanh{T}\right)^2.$
According to (\ref{eq:eqna}) and (\ref{eq:neweqnb}),
\[C_{HP} \le \frac{8n\ \left(1 + \tanh{T}\right)^2}
{2\, Tr\!\left(P_{\le n} \left(1 + \frac{1}{\cosh^2{t}} \right) \right)} \le
\frac{4 \left(\cosh{T} + \sinh{T}\right)^2}{ 1 +
\cosh^2{T}}.\]
This simplifies to the statement of the corollary.$\;\;\;\Box$
As $T \rightarrow 0,$ this reduces to the Euclidean bound 2 but as
$T \rightarrow \infty$ then $C_{HP} \le 16.$
\vspace{.7cm}
\noindent{\bf More general manifolds}
\vspace{.5cm}
In \cite{kn:Har92} one of us produced upper bounds on $C_{PPW}$ for
manifolds admitting a global semigeodesic (Fermi) coordinate system.
These bounds reflected the curvature of ${\cal M}$ (specifically, the
Gauss curvature in two dimensions and the sectional curvature in
higher dimensions). Here we remark briefly on the extensions of
those bounds using the main theorem of this article, with
auxiliary operators satisfying (\ref{eq:eqnd}). As remarked earlier,
this leads to a simplification, which we formalize as follows:
\begin{cnum}
Let $G_j$ be real $C^2$ functions, and define the corresponding $\Pi_j$
by ${\rm (\ref{eq:eqnd}).}$
Suppose that $\gamma$ and $\beta$ are constants such that
\[ -\sum_{j=1}^m [H,G_j]^2 \leq \beta H\,,\]
then
\[C_{HP} \le \frac{2n\beta}{
\left|\sum_{j=1}^{m}
Tr\left(P_{\le n}
\left[G_j,\left[H,G_j\right]\right]\right)\right|}.\]
\end{cnum}
In particular, the bounds on $C_{PPW}$ from section 4 of \cite{kn:Har92} all
apply to $C_{HP}.$ As a representative, we cite a two-dimensional
corollary slightly extending a result of that paper:
\begin{cnum}
Let the dimension $\nu = 2$ and suppose that ${\bf \Omega}$ has a semigeodesic
coordinate system with geodesic coordinate $x^1,$ chosen so that
${\cal P} :=\{x^1 = 0\}$ intersects ${\bf \Omega}.$ Let
$D := \sup \left({\rm dist}\left({\cal P,}\partial {\bf \Omega}\right)\right)$
and suppose that the curvature $h_1$ of ${\cal P}$ and the Gauss curvature
$\kappa$ of ${\bf \Omega}$ are bounded above and below by constants:
\begin{eqnarray*}
h_- \; \le & h_1(0,x_2,\ldots) \; \le & h_{+}\, , \\
\kappa_{-} \; \le & \kappa(x_1,x_2,\ldots) \; \le & \kappa_{+}\, .
\end{eqnarray*}
Then
\[C_{HP} \le \left( 2 + \frac{ \sup \left( \left|
r(x^1,a,k)\right| : |x_1| \le D, a = h_{\pm}, k = \kappa_{\pm}
\right)}{\sqrt{\lambda_1}}\right)^2 \, . \]
where $r$ is the function written explicitly in {\rm \cite{kn:Har92}} as
\[r(s,a,k) = \left\{ \begin{array}{ll}
\displaystyle\frac{a - \sqrt{k}\tan{(\sqrt{k}\,s)}}{1 + \frac{a}{\sqrt{k}}\tan
{(\sqrt{k}\,s)}} \,, & k > 0 \\
& \\
\displaystyle\frac{a + \sqrt{|k|} \tanh{(\sqrt{|k|}\,s)}}{1 + \frac{a}{\sqrt{|k|}}
\tanh{(\sqrt{|k|}\,s)}} \,, & k < 0 \\
& \\
\displaystyle\frac{a}{1 + as} \, , & k = 0\,.
\end{array}
\right.\]
\end{cnum}
\vspace{1.0cm}
\noindent{\bf Homogeneous spaces and minimally immersed
submanifolds}
\vspace{.5cm}
In this section we study eigenvalue differences for some special
manifolds without boundaries. For background material see \cite{kn:Kob},
\cite{kn:Chav}, and \cite{kn:Aub}. A complicating feature here is that
the lowest eigenvalue $\lambda_0 = 0$ automatically. We number
the eigenvalues so that $0 < \lambda_1$ is the first non-trivial
eigenvalue. Because of this, the appropriate gap bounds will be
of the form \[\lambda_{n+1} - \lambda_n \leq \lambda_1 +
C\langle\lambda_\ell\rangle_{\ell \leq n}\,.\]
\noindent And we define a modified universal constant
\[C_{PPW}' := \sup_{n,{\bf \Omega} \subset {\cal M}}
\frac{\lambda_{n+1} - \lambda_n - \lambda_1}
{\langle\lambda_\ell\rangle_{\ell \leq n}}\,.\]
The analogous Hile--Protter quantity will also gain a term with
$\lambda_1:$
\[C_{HP}' := \sup_{n, {\bf \Omega} \subset {\cal M}}
\frac{1 - \lambda_1 \langle\frac{1}{\lambda_{n+1}
- \lambda_\ell}\rangle_{\ell \leq n}}
{\langle\frac{\lambda_\ell}{\lambda_{n+1} - \lambda_\ell}
\rangle_{\ell \leq n}}\;.\]
As before $C_{PPW}' \leq C_{HP}'.$
\newline
We are interested in studying the eigenvalues of the problem
\[ -\Delta \phi = \lambda \phi \;\;\; {\rm on} \;\; {\cal M}\]
where ${\cal M}$ is a Riemannian manifold, without boundary,
of finite volume $V.$
Two of the more important results in this direction are the
bounds of Yang and Yau \cite{kn:Yang-Yau} for minimally immersed
submanifolds of $S^N(1),$ and Li \cite{kn:Li} for compact
homogeneous spaces.
For details about these spaces, see \cite{kn:Chav}, \cite{kn:Oss}, and
\cite{kn:Helga}.
In our algebraic approach, the key point is that symmetry forces certain
special properties on the eigenfunctions and eigenvalues, which we exploit
in our choices of the operators $G$ and $\Pi.$
In \cite{kn:Yang-Yau}, Yang and Yau use the fact that the coordinate
functions on a compact minimally immersed submanifold ${\cal M}^m
\subset S^N(1) $ are eigenfunctions of $-\Delta$ with degenerate eigenvalue
$m = $ dimension of ${\cal M}^m,$ to obtain a bound on
the gap between successive higher
eigenvalues. In particular they show, as corrected by Leung in \cite{kn:Leung}
\begin{equation}
\lambda_{n+1} - \lambda_n \leq m + \frac{2}{m(n+1)}
\left(\sqrt{\Lambda^2 + m^2\Lambda(n+1)} + \Lambda\right)
\label{eq:YaYa}
\end{equation}
where $\Lambda = \sum_{\ell=0}^n \lambda_\ell.$
Leung takes up the same problem in \cite{kn:Leung} and uses the method
of Hile and Protter (see \cite{kn:Hi-Pr}) to obtain an improved but
very complicated looking version of (\ref{eq:YaYa}).
\newline
In \cite{kn:Li}, Li offers a theorem for more general
homogeneous manifolds:
\begin{lnum} ${\rm (Li,\ Proposition\ 1\ and\ proof\ of\ Proposition\ 1\ of\
\cite{kn:Li})}$
\label{ln:liresults}
Let ${\cal M}$ be a compact homogeneous manifold of finite volume $V$
and take $\{\phi_{1,\alpha}\}_{\alpha = 1}^k $ to be an orthonormal
basis for the $k$-dimensional eigenspace of $\lambda_1$ (the first non-zero
eigenvalue of $-\Delta$). Then,
\begin{equation}
\sum_{\alpha = 1}^k \phi_{1,\alpha}^2 = \frac{k}{V}
\;\;\;{\rm and} \;\;\; \sum_{\alpha =1}^k |\nabla \phi_{1,\alpha}|^2
\leq \frac{\lambda_1 k}{V}.
\label{eq:liresults}
\end{equation}
\end{lnum}
Using these results, Li is able to prove that the eigenvalues of
$-\Delta$ satisfy
\begin{equation}
\lambda_{n+1} - \lambda_n \leq \frac{2}{n+1}
\left(\sqrt{\Lambda^2 + (n+1)\Lambda \lambda_1} + \Lambda \right) + \lambda_1
\label{eq:libound}
\end{equation}
where $\lambda_{n+1} > \lambda_n > \lambda_k.$
It is interesting to note
that this bound does not depend explicitly on the dimension of the
space.
\newline
We now obtain a corollary to Theorem \ref{tn:main}
which we will use to improve the bounds of Yang and Yau (\ref{eq:YaYa})
and Li
(\ref{eq:libound}).
\begin{cnum}
Suppose $\{\phi_{1,\alpha}\}_{\alpha = 1}^k$ are $k$ functions from
the eigenspace of
$\lambda_1 > 0$ of $-\Delta$ on a manifold ${\cal M}$.
If, in addition, there exist constants
$a, b > 0$ such that
\[\sum_{\alpha =1}^k \phi_{1,\alpha}^2 = a \;\;\; {\rm and}
\;\;\;\sum_{\alpha=1}^k |\nabla \phi_{1,\alpha} \cdot \nabla \psi|^2
\leq b\,|\nabla \psi |^2 \,,\]
then the eigenvalues of $-\Delta$ satisfy
\[\left<\frac{\lambda_1^2 a + 4 b \lambda_\ell}{\lambda_{n+1} - \lambda_\ell}
\right>_{\ell \leq n} \geq \lambda_1 a\,.\]
\label{cn:maincor}
\end{cnum}
\noindent {\bf Proof:}
The special choices to be made in Theorem \ref{tn:main} are
\[G_\alpha = \phi_{1,\alpha} \;\;\;{\rm for}\;\;\; \alpha = 1,\ldots,k\]
and
\[\Pi_{\alpha} = i[H,G_\alpha] = i(-\Delta G_\alpha - 2\nabla G_\alpha
\cdot \nabla).\]
\newline
First observe that if $G$ is a function in $L^2({\cal M})$ and $H = -\Delta$,
\begin{equation}
[G,[H,G]] = 2 \nabla G \cdot \nabla G = \Delta(G^2) - 2G(\Delta G).
\end{equation}
With the above choices for $G_\alpha$ and $\Pi_\alpha$ the right side
of inequality (\ref{eq:main}) is
\begin{eqnarray*}
\frac{\left|\sum_{\alpha=1}^k Tr\left(P_{\le n}[[H,G_\alpha],G_\alpha]\right)\right|^2}
{2\:\sum_{\alpha=1}^k Tr\left(P_{\le n}[G_\alpha,[H,G_\alpha]]\right)}
& =& \frac{1}{2} \sum_{\alpha=1}^k Tr \left(P_{\le n}\left(\Delta
\left(G_{\alpha}^2\right) -
2G_\alpha\left(\Delta G_\alpha\right)\right)\right)\\
& & \\
& = & \frac{1}{2}
Tr \left(P_{\le n}\left(\Delta\left(\sum_{\alpha=1}^k G_{\alpha}^2\right) +
2 \sum_{\alpha=1}^k G_{\alpha}^2 \lambda_1\right)\right)\\
& & \\
& =& Tr\left(a\lambda_1 P_{\le n}\right)
= a\lambda_1(n+1).
\end{eqnarray*}
We now obtain an upper bound for the left side of (\ref{eq:main}).
We have for each $\ell$
\begin{eqnarray*}
\sum_{\alpha=1}^k
\|[H,G_\alpha]\phi_\ell\|^2
& = &
\int
\sum_{\alpha=1}^k
\left(\lambda_1 G_\alpha \phi_\ell -
2\nabla G_\alpha \cdot \nabla \phi_\ell\right)^2\\
& = &
\lambda_1^2 a + 4\int \sum_{\alpha=1}^k
|\nabla G_\alpha \cdot \nabla \phi_\ell|^2\\
& \le &
\lambda_1^2 a + 4 b \lambda_\ell \,.
\end{eqnarray*}
In the second equality the cross term drops out because it contains the
gradient of $\sum_{\alpha = 1}^k G_\alpha^2 $ which is $0$ because of the
hypothesis. Thus
\begin{eqnarray*}
\sum_{\alpha = 1}^k Tr \left( \left( \lambda_{n+1} - H \right)^{-1}
P_{\le n} \Pi_{\alpha}^2 \right)
& = & \sum_{\ell=0}^n \sum_{\alpha=1}^k
\left(\lambda_{n+1} - \lambda_\ell\right)^{-1}
\|[H,G_\alpha]\phi_\ell\|^2 \\
& \le & \sum_{\ell=0}^n \left(\lambda_{n+1} -
\lambda_\ell\right)^{-1} \left(\lambda_1^2 a + 4 b \lambda_\ell\right).
\end{eqnarray*}
This proves the corollary.$\;\;\;\Box$
\vspace{1cm}
This leads us to
\begin{cnum}Let ${\cal M}^m$ be a minimally immersed submanifold of $S^N(1),$
then
\[C_{PPW}' \leq C_{HP}' \leq \frac{4}{m}\,.\]
\end{cnum}
\noindent {\bf Proof:}
Choose $\{\phi_{1,\alpha}\}_{\alpha = 1}^{k}$ to be the
coordinate functions of ${\bf R}^{N+1}.$
Then $\lambda_1 = m$, $k = N+1$, $ a=1,$ and $b=1$ (see \cite{kn:Chav},
\cite{kn:Leung} and \cite{kn:Yang-Yau}) in Corollary
\ref{cn:maincor}. This yields
\[\sum_{\ell=0}^n \frac{m^2 + 4 \lambda_\ell}{\lambda_{n+1} - \lambda_\ell}
\geq m(n+1)\,.\]
>From which the corollary follows.$\;\;\;\Box$
\begin{cnum}
If ${\cal M}$ is a compact homogeneous manifold, then
\[C_{PPW}' \leq C_{HP}' \leq 4\,.\]
\end{cnum}
\noindent {\bf Proof:}
Choose $\{\phi_{1,\alpha}\}_{\alpha = 1}^{k}$ as in Lemma \ref{ln:liresults}.
Then, by the result of the lemma, $a = \frac{k}{V}$ and $b=\lambda_1 a.$
This implies
\[\sum_{\ell=0}^n \frac{\lambda_1 + 4 \lambda_\ell}
{\lambda_{n+1} - \lambda_\ell}
\geq n + 1\,,\]
which yields the result.$\;\;\;\Box$
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\end{document}