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\title{Improved Estimate on the Number of Bound States of
Negatively Charged Bosonic Atoms}
\author{Mary Beth Ruskai\thanks{supported by National Science
Foundation Grants DMS-89-08125 and HRD-91-03315}\\Department of
Mathematics\\ University of Massachusetts $\bullet$ Lowell\\Lowell,
MA 01854 USA \\ {\normalsize bruskai@cs.ulowell.edu}}
\begin{document}
\maketitle
\begin{abstract}
It is shown that the number of bound states of an atom whose
``electrons'' satisfy bosonic symmetry conditions is bounded
above by $C_\beta (\log Z)^\kappa$ where,
$\kappa$ and $C_\beta$ are constants, the nuclear charge $Z > Z_\beta$
for some constant $Z_\beta$ and the number of ``electrons'' $N$
satisifies $N > (1+\beta)Z + 1 $ with
$ \beta > 0$. The constant $\kappa$ is universal, but
$Z_\beta$ depends upon $\beta$ and $C_\beta$ is inversely
proportional to a power of $\beta$.
\end{abstract}
\section {Introduction}
In a recent paper, Bach, Lewis, Lieb, and Siedentop \cite{BLLS}
studied the number of bound states, i.e. discrete eigenvalues,
of the Hamiltonian for an atom consisting of a nucleus with
infinite mass and charge $Z$ and $N$ negatively charged spinless
bosons interacting with a Coulomb potential. This system is
described by the Hamiltonian
\begin{eqnarray}\label{eq:ham}
H = \sum_{i=1}^N \left( -\Delta_i - \frac{Z}{|x_i|} \right) +
\sum_{i < j}\frac{1}{|x_i - x_j|}
\end{eqnarray}
restricted to the symmetric subspace of $\otimes [L_2({\bf R}^3]^N$,
i.e., the eigenfunctions of $H$ are required to be symmetric under
interchange of coordinates of any two particles. If the number of
bound states of such a system is denoted as $\nu_b (N,Z)$, they
showed that when $(1 + \beta )Z + 1 < N < (1 + \beta')Z$, then
$\nu_b (N,Z) < C Z^k$ where the constants $C$ and $k$ may depend
upon $\beta$ and $\beta'$. Although their estimates are neither
optimal nor easily extended to fermions, this result is significant
because it is the first explicit bound on the actual number of
bound states of a multi-particle atomic system. The only previous
results gave conditions on $N$ and $Z$ for which the number
of bound states was zero, finite, or infinite. (See \cite{Rneg}
for references.)
The main result of Section 3 of \cite{BLLS} can be stated as
follows.
\begin{thm}
If, the Hamiltonian
$H(N,Z)$ has no bound states supported outside a ball of radius $R$,
then $\nu_b (N,Z) < C n^k$ where
$n = \int_{|x| R \right\} $, where
${\bf x}$ denotes $(x_1, x_2, ... x_n)$.
If we write $R = f(Z)/Z$ and $f(Z) > 1$, then
$$\int_{|x| Z_\beta$, $H(N,Z)$ has no bound states supported outside
a ball of radius $R = C_\beta \log Z /Z$ which yields
$\nu_b (N,Z) < C_\beta (\log Z)^{3k}$. Moreover, we show that this holds
provided $N > (1+\beta)Z +1$, i.e. we show that the upper limit
on $N$ can be removed, and discuss the dependence of $\nu_b (N,Z)$
on $\beta$. Our final result can be stated as follows.
\begin{thm}
For all $\beta > 0$ there exists $Z_\beta$ such that whenever
$Z > Z_\beta$, and $N > \beta Z$, then
$\nu_b (N,Z) < C (\log Z)^{4\kappa}/\beta^{3\kappa}$
where $C$ and $\kappa$ are constants.
\end{thm}
\section{General Remarks on Localization Error}
We now discuss some heuristics about localization error.
Except for Simon's enhancement of the ``Sigal-Simon
localization trick'' the contents of this section are
well-known. However, because computing the localization error
precisely is rather tedious, it is useful to review some
aspects of its general behavior. Since details are
available in the literature (and one can always check a
specific case explicitly by differentiation),
we concentrate on heuristics here
and refer the reader to \cite{CFKS,Ru,SiAP} for further details.
In many problems one wants to break up configuration space
into pieces, i.e. one wants to consider
\begin{eqnarray}\label{eq:brkup}
\langle \Psi, H \Psi \rangle = \int_{{\bf R}^{3N}} \Psi H \Psi d^3{\bf x}
= \sum_k \int_{\Omega_k}\Psi H \Psi d^3{\bf x}.
\end{eqnarray}
However, this leads to technical difficulties unless $\Psi = 0$
on the boundary of each $\Omega_k$. In order to achieve this
one needs a set of localizing functions $F_k$ such that
$0 \leq F_k \leq 1$, $\hbox{supp}(F_k) \subset \Omega_k$
and $\sum_k F_k^2 = 1$. Then instead of (\ref{eq:brkup}),
one has
\begin{eqnarray}\label{eq:loc}
\langle \Psi, H \Psi \rangle =
\sum_k \langle ~(F_k \Psi) H (F_k \Psi)~ \rangle -
\sum_k \sum_i \mu_i \int_{{\bf R}^{3N}} |\nabla_i F_k|^2 |\Psi|^2
\end{eqnarray}
The quantity $LE =
\sum_k \sum_i \mu_i \int_{{\bf R}^{3N}} |\nabla_i F_k|^2 |\Psi|^2$
is called the localization error and there is term containing
$\mu_i$ and $\nabla_i$ for each Laplacian $- \Delta_i$ with
coefficient $\mu_i$ in the Hamiltonian $H$.
Now suppose there are $M$ localizing functions $F_k$ (i.e. $k = 1...M$)
and $N$ gradient terms (i.e. $i = 1 ... N$). If one is dealing with
a problem with permutational symmetry, one expects each of these
terms to be roughly the same size (except possibly for a few
exceptional terms which will not affect the large $M, N$ behavior)
so that one would expect the total localization error (LE) to be
propertional to $MN$. Therefore, it is somewhat surprising that,
at least for atomic problems, it is often possible to choose
$F_k$ so that LE grows only like $(\log M)^2$!
Beforing sketching the construction which yields this behavior, we
review a few other facts. By a simple scaling argument, it is clear
that LE is proportional to $1/(\mbox{distance})^2$.
Since $F_k$ is usually chosen
so that $F_k = 1$ on most of $\Omega_k$, $0$ outside
$\Omega_k$, and decreases smoothly from $1$ to $0$ near the
boundary of $\Omega_k$, it will have a non-zero derivative only
near the boundary. Now one typically chooses the localization
so that the boundaries of $\Omega_k$ are precisely the places where
certain critical distances are equal, or at least comparable in
a well-specified sense. Therefore, one has a natural parameter
$r$ so that $|\nabla_i F_k|^2$ is proportional to $1/r^2$.
It is often easier to first find functions $G_k$ with
$0 \leq G_k \leq 1$, $\hbox{supp}(G_k) \subset \Omega_k$
and $\sum_k G_k^2 \geq 1$. One then defines
$F_k = G_k /\sqrt{ \sum_j G_j^2 }$ and easily
verifies that
\begin{eqnarray}\label{eq:locG}
LE \leq \frac
{\sum_k \sum_i \mu_i \int_{{\bf R}^{3N}} |\nabla_i G_k|^2 |\Psi|^2}
{\sum_k G_k^2}
\end{eqnarray}
Now one typically chooses $G_K$ to have the form
$G_k(x) = \chi(|x_k| / \|{\bf x}\|_p)$, where, {\bf x}
is often replaced by a specified subset of $x_1, x_2, ... x_n$;
$~~\chi(t) : {\bf R}^+ \mapsto [0,1]$, $\chi(t) = 1$ if $t \geq 1$,
and $\chi(t) = 0$ if $t \leq t_o$ for some chosen value of $t_o$.
We now show how to remove the dependence of LE on $N$. Since
$\nabla_i \|{\bf x}\|_p = x_i^{p-1}/\|{\bf x}\|_p^{p-1}$,
it follows that if $G_k$ has the above form, then for $i \neq k$
\begin{eqnarray}
|\nabla_i G_k|^2 \leq
\frac{C}{r^2} \frac{x_i^{2p-2}} {\|{\bf x}\|_p^{2p-2}} \leq
\frac{C}{r^2} \frac{x_i^p} {\|{\bf x}\|_p^p}
\end{eqnarray}
if $p \geq 2$. Since $ \nabla_i G_k \neq 0 \Rightarrow
\|{\bf x}\|_p \geq x_k \geq 2\|{\bf x}\|_p$, one can choose
either $r = x_k$ or $r = \|{\bf x}\|_p$ as appropriate. In either case,
\begin{eqnarray}
\sum_{i=1}^N |\nabla_i G_k|^2 \leq \frac{C}{r^2}
\frac{ 1}{\|{\bf x}\|_p^p} \sum_{i \neq k} x_i^p
= \frac{C}{r^2}
\end{eqnarray}
rather than $NC/r^2$. The reason for the apparent
disappearance of the expected growth
of $LE$ with $N$ is that, especially when $p$ is large, only
one derivative of the form $\nabla_i \|x\|_p (i = 1 ...N)$
dominates in any region of ${\bf R}^{3N}$. Localizations with
p-norms (and $p \rightarrow \infty$) seems to have first been
used in \cite{Ru} (Zhislin \cite{Zh} and others
having previously used localizations with $p=2$);
Simon \cite{CFKS} subsequently advocated sacrificing some smoothness
and using sup norms, for which it is somewhat clearer that, for a
fixed $k$, the regions on which $\nabla_i G_k ~~(i=1...N)$
is non-zero do not overlap.
We now describe the``Sigal-Simon Localization Trick" for minimizing
the dependence of $LE$ on the number of localizing
functions $M$. Let
\begin{eqnarray}\label{eq:simloc}
\chi(t) = \left\{ \begin{array}{ccc}
0 &~& 0 \leq t < \half \\
\half \psi(t)^a = \half (4t-2)^a &~& \half \leq t <
{\textstyle \frac{3}{4}}\\
1 - \half \psi(t)^a = 1 - \half [4(1-t)]^a &~&
{\textstyle \frac{3}{4}} \leq t < 1 \\
1 &~& 1 \leq t
\end{array} \right.
\end{eqnarray}
where $a > 1$ and $\psi(t) = 1 - |4t-3|$. [The precise form of
$\psi$ and the cut-off at $\half$ are chosen for simplicity and
are not essential. Moving the cutoff from $\half$ to $t_o$ closer
to $1$ may be desirable for some purposes but will increase
the localization error by a factor of $1/(1-t_o)^2$.]
With $\chi$ given by (\ref{eq:simloc})
\begin{eqnarray}\label{eq:deriv}
|\chi'(t)| = \left\{ \begin{array}{ccc}
2a \psi(t)^{a-1} &~& \half \leq t \leq 1 \\
0 &~& \hbox{otherwise}
\end{array} \right.
\end{eqnarray}
Now consider (as a function of $\eta > 0$) the expression
$$ \frac{|\chi'|^2}
{{\textstyle \frac{1}{4}} \psi^{2a}\eta^{-1} + \eta^{a-1} }
= 16 a^2 \psi^{-2} \frac{\eta}{1 + 4 \eta^a \psi^{-2a}} $$
which has its maximum when $\eta^a =\psi^{2a}/4(a-1)$
so that
\begin{eqnarray}\label{eq:etamax}
|\chi'|^2 \leq 16 a^2 (\chi^2 \eta^{-1} + \eta^{a-1})
~~\forall ~~a > 1 ~~\hbox{and}~~ \eta > 0.
\end{eqnarray}
If one then uses this $\chi$ to construct a set of localizing
functions $F_k$ via $G_k$ as above, one finds
\begin{eqnarray}\label{eq:locerr}
\sum_k |\nabla_i F_k|^2 &\leq &16 a^2 \frac
{\sum_k \eta^{-1} G_k^2 + \eta^{a-1}}{\sum_k G_k^2} \nonumber \\
& \leq& 16 a^2 (\eta^{-1} + M\eta^{a-1})
\end{eqnarray}
If one now chooses $\eta = M^{-1/a}$ and combines this estimate
with those above, one finds a net localization error satisfying
$LE \leq C a^2 M^{1/a} ~~\forall~ a > 1$. Writing $M = b^a$, or
equivalently, choosing $a = (\log M)_b$ one gets the final
estimate $LE \leq C (\log M)^2$ for some constant $C$.
This trick was first used by Sigal \cite{Si1,SiAP} in the special case
$a = 2$, which yields a localization error proportional to $\sqrt{M}$
(and for which the bound in (\ref{eq:etamax}) can be obtained by
a simple squaring argument).
The idea of reducing this to $M^\sigma$ with $\sigma > 0$ arbitrarily
small is due to Simon; it is sketched in \cite{SiAP} (see Sect. 5)
and alluded to in \cite{CFKS} (see the remark on p. 47) and in
\cite{LSST,Rneg}; however, the details have not previously appeared
in the literature and the reduction to $\log M$ seems new.
Finally it is worth remarking that this analysis is essentially
unchanged if $G_k$ is a product of a {\em fixed} number of
$\chi$. If however, one needs a product of $N$ functions,
as in \cite{FS,Rmol}, this approach does not work.
\bigskip
\section{Shrinking the Size of the Inner Ball}
We now show how to apply these ideas to improve the estimates
in \cite{BLLS}. Although this involves only
the localization for the second partition in Section 2 of
\cite{BLLS}, which we will subsequently show (see Section 4 below) can
actually be eliminated, the argument is simple and
illustrates the power of the Sigal-Simon
localization trick. Moreover, the final estimates obtained here will
be slightly better than those in Section 4. Let
$J_{kl} =
\chi(|x_k|/\|\hat{{\bf x}}_k\|_\infty)
\chi(|x_\ell|/\|\hat{{\bf x}}_{k\ell}\|_\infty) A J_{n+1}$
where $\hat{{\bf x}}_k$ denotes ${\bf x}$ with $x_k$ removed, etc.,
$\chi$ is as in (\ref{eq:simloc})
and the localizing functions $A$ and $J_{n+1}$ are defined in
\cite{BLLS}. We need here only the facts that $A$ and $J_{n+1}$ make
a contribution of $C/\delta^2 R^2$ to LE, and that $A J_{n+1}$
is supported on a region where at least two ``electrons'' are
outside a ball of radius $R \delta$ so that
\begin{eqnarray}
\mbox{supp} J_{k\ell} \subset \left\{ {\bf x} : 4|x_k| > 2|x_\ell| > |x_i|
~\forall~i \neq k,\ell ~\mbox{and}~ |x_\ell| > R\delta \right\}
\end{eqnarray}
where $\delta < \beta$ and $R$ will be chosen later.
Now the number of $J_{k\ell}$ is $M = N(N-1)/2 \approx N^2/2$.
As discussed in Section 2 above, one easily checks that
the localization error from this
partition satisifes $LE \leq C(\log N)^2/\delta^2 R^2$. Thus,
eq. (10) of \cite{BLLS} now can be replaced by the following estimate
which is valid on supp$(J_{k\ell})$
\begin{eqnarray}\label{eq:blls10}
H(N,Z) - E_o(N-1,Z) ~~~\geq
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \\
-\Delta_1 -\Delta_2 - \frac{Z}{|x_k|} - \frac{Z}{|x_\ell|}
+ I_o(N-1,Z) - \frac{C(\log N)^2}{\delta^2 R^2} \nonumber
\end{eqnarray}
where $I_o(N-1,Z) = E_o(N-2,Z) - E_o(N-1,Z)$ denotes the ionization
potential of the $(N-1)-st$ ``electron''.
If one now uses Bach's estimate \cite{Bach} that
$I_o(N,Z) \geq \mu Z^2$ for some constant $\mu$,
it is then evident that one can find constants $C_{\beta}$ and $Z_o$
such that the right side of (\ref{eq:blls10}) is positive
if $R = C_{\beta} \log Z / Z$ and $Z \geq Z_o$. Using this in
Theorem 2 gives the final bound of $\nu_b(N,Z) < C_{\beta} (\log Z)^{3k}$
as explained in the introduction. Note that the choice
$R = C_{\beta} \log Z / Z$ implies that
$-2Z/|x_k| > -2Z/\delta R > - (\mbox{constant}) Z^2/\log Z$.
Since the LE terms are also of the form $-Z^2/\log Z$,
it will be necessary to
choose either $C_\beta$ or $Z_o$ extremely large to ensure that they
are dominated by $\mu Z^2$.
Thus far we, like BLLS \cite{BLLS}, have ignored the question of how $R$ and
$\nu_(N,Z)$ depend upon the lower limit $\beta$. Using the fact
that one must have $\delta < \beta$ and choosing $\delta = \beta/2$,
one finds that the estimates above require $R > C \log Z/\beta Z$.
However, the positivity of eq. (8) of \cite{BLLS} requires the
stronger condition
$R > C \log Z/\beta^3 Z$ so that $f(Z) = c \log Z/\beta^3$
which yields a net bound of
\begin{eqnarray}
\nu_b(N,Z) < C \frac{(\log Z)^{3k}}{\beta^{9k}}
\end{eqnarray}
Since one must have $\beta < \beta_c \approx 0.21$ this estimate
is large even near $\beta_c$ (since $\beta_c^{-9} > 10^6$!)
and becomes enormous as $\beta \rightarrow 0$.
\section {Range of N --- Removing the Upper Bound}
The upper limit of $N < (1+\beta')Z$ in \cite{BLLS} arises because
of the need for the bound on the ionization potential
$E_o(N-1,Z) - E_o(N-2,Z) > \mu Z^2$, which requires that $N$ be in the
range where $H(N,Z)$ still has bound states. However, the
number of bound states is expected to decrease as $N$ approaches
its upper limit of $\beta_c Z \approx 1.21Z$
so that this upper limit on $N$
should not be necessary. We now show how to remove it.
In \cite{BLLS}, their analysis of the outer region begins with
a localization on which the effective potential of the k-th
electron, $V_k^{\hbox{eff}} = -Z/r_k + \sum_{i \neq k} 1/r_{ik}$
is bounded below (see eq. (8) of \cite{BLLS}) as
\begin{eqnarray}\label{eq:elec1}
V_k^{\hbox{eff}} \geq
\frac{-Z + (N-1)/(1+\delta)}{r_k} \geq
\frac{(\beta - \delta)Z}{(1+\delta)r_k}
\end{eqnarray}
which will be positive
only if $ \beta > \delta$. Since this is valid on a region
on which $ r_k \delta > r_i ~\forall~~ i\neq k$, and since $\beta < 0.21$,
such regions will not cover the outer region (which would requires
$\delta \geq 1$) and the resulting localization will not give
a partition of unity. In order to treat the remaining
region they make an additional partition into regions on
which $r_k > r_\ell > r_i ~~\forall i \neq k,\ell$, as discussed
in Section 3 above. In this region they need Bach's estimate
of $\mu Z^2$ on th ionization potential which requires
$N - Z < \beta'Z < \beta_c Z$
In order to eliminate the second partition, we replace the
estimate (\ref{eq:elec1}) by a more refined electrostatic
estimate used
by Lieb, Sigal, Simon and Thirring \cite{LSST} to give the
first proof of asymptotic neutrality. They show that for any
$\epsilon > 0$ there exist $N_\epsilon$
and regions $\Omega_k$ which cover
${\bf R}^{3N}$ when $N > N_\epsilon$ and on which
\begin{eqnarray}\label{eq:elec2}
\sum_{i \neq k} \frac{1}{r_{ik}} \geq \frac {(1 - \epsilon)N}{r_k}
\end{eqnarray}
They also showed that one can find a localization
corresponding to these regions with
$LE \leq C\sqrt{N} (\log N)^2/\epsilon^2 \|{\bf x} \|_\infty^2$.
In the terminology of Section 2 above, the contribution
of $1 /\epsilon^2$ corresponds to $1/(1-t_o)^2$;
the factor $\sqrt{N}$ corresponds to $M^{1/a}$ when $a=2$
and can be replaced by $(\log N)^2$; and the $(\log N)^2$
already present is an additional factor that arises because of
a cut-off parameter needed, as explained in \cite{LSST}).
Thus one can improve this bound (as already remarked in \cite{LSST})
to $LE \leq C (\log N)^4/\epsilon^2 \|{\bf x} \|_\infty^2$.
Thus, on $\mbox{supp}(\Omega_k)$, the effective potential and
LE can be estimated as
\begin{eqnarray}\label{eq:effpot}
V_k^{\hbox{eff}} - LE & \geq & \frac{(1- \epsilon )N - Z}{r_k} -
\frac {C (\log N)^4}{\epsilon^2 r_k^2} \\
& \geq & \frac{( \beta - \epsilon )Z}{r_k} -
\frac{C'(\log Z)^4}{\epsilon^2 r_k R}
\end{eqnarray}
where we have used the fact \cite{Lb} that $N < 2Z+1$ (but we
really only need that $N < Z^m$ for some $m$ to replace $\log N$
by $\log Z$).
The first term will be positive if $\beta > \epsilon$. Choosing
$\epsilon = \beta/2$, one can conclude that
$V_k^{\hbox{eff}} - LE > 0$ if $R > C(\log Z)^4/\beta^3 Z$.
Using this in Theorem 2 with $f(Z) = C(\log Z)^4/\beta^3$ gives
a final bound of
\begin{eqnarray}
\nu_b(N,Z) \leq C \frac{(\log Z)^{12k}}{\beta^{9k}}
\end{eqnarray}
valid whenever $Z+1 > N - Z > \beta Z$ and $Z > Z_\beta$.
The price one pays for removing the restriction
$N - Z < \beta'Z < \beta_cZ \approx 0.21 Z$
is a higher power of $(\log Z)$.
A more serious price is that we do not have any information about
how $Z_\beta$ depends upon $\beta$. In addition, we have not
been able to improve the dependence of $\nu_b(N,Z)$ on
$1/\beta$, even for $\beta$ near $\beta_c$.
In \cite{BLLS} the restriction that the particles are bosons
was used in the outer region only to estimate the ionization
potential as $\mu Z^2$. Since we have eliminated the need for
this, the argument above should also work for fermions, and
it does. Unfortunately, it only works for $N > \beta Z$ and
we already know \cite{FS,LSST,SSS} that, because fermionic atoms are
asymptotically neutral, $\nu_b(N,Z) = 0$ in this region, i.e. for
fermions the only region of interest is the very delicate region
$Z+1 < N < Z + c Z^\sigma$ with $\sigma < 1$. (By \cite{SSS}
$\sigma < 5/7$).
Indeed, the estimates sketched in this section combined with the
easily proved fact
that no bound states have support such that {\em all} electrons
lie within a ball of radius $O(N^{-1/3})$ suffice to prove
asymptotic neutrality and this is essentially the argument in
\cite{LSST}.
\bigskip
\bigskip
{\bf Acknowledgment} This work was done while the author was
visiting the Mittag-Leffler Institute, University of Trondheim,
Ceremade (Universit\'{e} Parix IX), and the University of Leiden.
The author
is grateful to these institutions for the hospitality and
support. It is a pleasure to also thank Professors H. Siedentop
for suggesting that the author try to improve the localization
error estimates in \cite{BLLS}, R. Brummelhuis for stimulating
discussions about Section 4, and B. Simon for explaining
his localization trick to the author some years ago.
\pagebreak
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\end{document}