\magnification=\magstep1 \baselineskip=20pt \parindent=0pt \font\mfrm=cmr5 scaled \magstep3 \def\lemma#1{\underbar{\bf{Lemma.}}\ {#1}\vskip 3em} \def\proof{{\bf{Proof: }}} \def\reals#1{{\bf R}^{#1}} \def\lnorm#1{\left|\!\left|{#1}\right|\!\right|} \def\parv{\par\vskip 1em} \def\lapl{\Delta} \def\repulsion{{1\over 2}\sum_{i\ne j}{1\over {|x_i-x_j|}}} \def\dzdr{\>{{dz\,dR}\over {R^5}}} \def\dr{{{dR}\over {R^5}}} \def\2int{\int\!\!\!\int} \font\BBigrm=cmr10 scaled \magstep4 \font\Bbigrm=cmr10 scaled \magstep3 \font\Bigrm=cmr10 scaled \magstep2 \def\sss{\scriptscriptstyle} \def\scapro#1,#2{\left<#1,#2\right>} \def\smallc{\scriptstyle c} \def\frac#1,#2{{}^{#1}\!\!{/}\!{}_{#2}} \def\fra#1,#2{\leavevmode\kern.1em\raise.5ex\hbox{\the\scriptfont0 $ \scriptstyle #1$ }\kern-.1em\!\!/\kern-.15em\lower.25ex\hbox{\the\scriptfont0 $ \scriptstyle #2$}} \def\BBracket#1{\Biggl(#1\Biggr)} \def\Bracket#1{\Bigl(#1\Bigr)} \def\bracket#1{\bigl(#1\bigr)} \def\bbracket#1{\biggl(#1\biggr)} \def\bra#1{\left(#1\right)} \def\vs#1,#2{#1_1,\ldots,#1_{#2}} \def\abs#1{\left|#1\right|} \def\muchbigger{>\!\!\!>} \def\muchsmaller{<\!\!\!<} \def\bover#1,#2{{\displaystyle{#1}\over{\displaystyle{#2}}}} \def\today{\ifcase\month\or January\or February\or March\or April\or May\or June\or July\or August\or September\or October\or November\or December\fi \space\number\day, \number\year} \def\Vol#1{{\rm\ Vol\ }\bra{#1}} \def\sign{{\rm sign\ }} \def\for{\qquad{\rm\ for\ }\quad} \def\and{\qquad{\rm\ and\ }\qquad} \def\bigo#1{{\it O}\!\bra{#1}} \def\littleo#1{{\it o}\!\bra{#1}} \def\eqbydef{\;{\buildrel {\rm def} \over =}\;} \def\est{\hbox{\mfrm Estimate}} \def\tfd{\rho_{\scriptscriptstyle TF}} Mathematics \vskip 2em {\BBigrm An Upper Bound for the Number of} \par {\BBigrm Electrons in a Large Ion} \vskip2em Asymptotic neutrality/excess charge/quantum mechanics \vskip2em {\Bigrm C. L. Fefferman and L. A. Seco}\vskip1em Department of Mathematics, Princeton University, Princeton NJ 08544. \vskip1em \vfil \eject {\bf Abstract} Let $E(Z,N)$ be the ground-state energy of $N$ quantized electrons and a single nucleus of charge $Z$. For fixed $Z$, $E(Z,N)$ is independent of $N$ for $N\ge N_{\rm critical}(Z)$. Physically, this means that at most $N_{\rm critical}$ electrons can bind to the nucleus. We prove that $N_{\rm critical}\le Z+CZ^a$ with $a=0.84$. \vfil \eject Consider the Hamiltonian for a nucleus of charge $Z$ and $N$ quantized electrons, $$H_{Z,N} = \sum_{i=1}^N \biggl [(-\lapl _{x_i}) - {Z \over {|x_i|}} \biggr ] + {1 \over 2} \sum_{i \ne j} {1 \over {|x_i - x_j|}} = -\lapl + V_{\rm Coulomb}.$$ The ground state energy is then $$ E(Z) = \inf_N E(Z,N) =\inf_{N}\,\inf\Biggl\{ \Bigl < H_{Z,N}\,\psi\, , \,\psi \Bigr > \quad|\quad \psi\in \wedge_{i=1}^N (L^2(\reals{3})\otimes {\bf C}^q),\quad {\lnorm{\psi}_2 = 1}\Biggr\}. $$ For each $Z$, call $N(Z)$ the smallest number for which $E(Z)=E(Z,N)$. It is an interesting problem to obtain sharp estimates for $N(Z)$. The sharpest known result appears in (1). In particular, $N(Z)/Z \to 1$ as $Z\to\infty$, although there were no estimates for the rate of convergence. Our main result is the following: \smallskip {\bf\Bigrm Theorem:} $$ N(Z) = Z + O(Z^\alpha) \qquad{\rm for\ some\ } \alpha<1$$ For the proof we will be interested only in the case $Z\le N\le 2Z$. Recall that $E(Z)/Z^{\frac 7,3}\to 1$ as $Z\to\infty$. \smallskip {\Bigrm\bf Definitions:} {\parindent=20pt \item{1.} Given a ball of center $0$ and radius $R$, call $N_{\sss R}=N_{\sss R}(x_1,\ldots,x_N)$ the number of $x_i$ that belong to it. \item{2.}We say that $\est(\bar\epsilon,\epsilon,R)$ holds if for a nucleus of charge $Z$ at the origin and $N$ quantized electrons {\sl confined to the ball} $B(0,R)$ we have $$ \Bigl< H_{Z,N}\,\psi, \psi\Bigr> \ge E_0(Z) + {\bar\epsilon Z \over R}\Bigl(N - (1+\epsilon)Z\Bigr) $$ where $$E_0(Z) = \inf_{0\le N\le (1+\epsilon_\sharp)Z}\,E(Z,N) $$ for $\epsilon_\sharp$ to be picked later.\vskip0em By {\sl N quantized electrons confined to the ball $B(0,R)$} we mean that the support of $\psi$ is included in the set $\{{\bf x}| N_R({\bf x}) = N\}$. \item{3.}Fix once and for all an even approximation to the identity, $\phi$, supported in $B(0,Z^{\frac -2,3})$. Given points $x_1,\ldots,x_N$ set $\rho(x)=\sum_i\phi(x-x_i)$. \item{4.}Take a smooth function $\chi_{\sss R}$ that is equal to $1$ if $|x|2R$, and define $N_{\chi,R}(x)=\sum_{i=1}^N\phi(x-x_i)\cdot\chi_{\sss R}(x)$. Obviously, $N_{\sss R/2} \le N_{\chi,R} \le N_{\sss 2R} $. \item{5.}If $\tfd$ is the Thomas-Fermi density, $N_{\chi,\sss R}^{\sss TF} = \int_{B(0,R)} \tfd(x)\,\chi(x)\,dx$. Note that if $R=Z^{{\frac -1,3}+\gamma}$, then $Z\ge N_{\chi,\sss R}^{\sss TF}\ge Z-cZ^{1-3\gamma}$, $\gamma>0$. } \smallskip {\bf Key Result: } $$ \Bigl \ge E_0(Z) + C{|\bigl< N_{\chi}\psi,\psi\bigr> - N_{\chi}^{\sss TF}|^2\over R} - O(Z^{{\frac 7,3}-b}) $$ for any $\psi$; in particular we do not assume that there are any number of electrons confined to any ball. {\Bigrm\bf Proof of Key Result: } It follows from W. Hughes (2) and H. Siedentop-R. Weikard (3) that if we set $$ K(x_1,\ldots,x_N) = \2int{(\rho-\tfd)(x)(\rho-\tfd) (y)\over|x-y|}dx\,dy $$ then $H_{Z,N}\ge E_0(Z) + K(x_1,\ldots,x_N) - O(Z^{{\frac 7,3}-b})$. Since $$K=\int_{\reals{3}}|\xi|^{-2}|\hat\rho(\xi)-\hat\tfd(\xi)|^2d\xi $$ and $$N_{\chi}-N_{\chi}^{\sss TF} = \int(\rho-\tfd)\chi\,dx= \int(\hat\rho-\hat\tfd)\hat\chi\,d\xi$$ Cauchy-Schwarz yields $$ |N_{\chi}-N_{\chi}^{\sss TF}|^2 \le \int|\chi(\xi)|^2\cdot|\xi|^2\,d\xi \int|(\hat\rho-\hat\tfd)|^2\cdot|\xi|^{-2}\,d\xi \le CR\cdot K$$ And Cauchy-Schwarz again gives the result. \medskip {\bf Corollary 1:} For any $\beta_1$, $\beta_2$, satisfying $$2\beta_1+\beta_2 < Z-cZ^{1-\gamma_1}$, and $$3\gamma_2>\gamma_1\qquad 7\gamma_22Z^{{\frac -1,3}+\gamma}$, with $$\gamma>\gamma_2\qquad \gamma>\beta_2 \eqno(\rm{ineq}\,3)$$ Let's rewrite the Hamiltonian $H_{Z,N+N'}$ as $$H_{Z,N+N'}= -\lapl_{x_i \ldots x_N}+V-\lapl_{\rm extra}+V_{\rm extra} $$ with $$\eqalign{\lapl_{\rm extra} &= \lapl_{x'_1,\ldots,x'_{N'}}\cr V &= -\sum_{i=1,\ldots,N}{Z\over |x_i|}+\repulsion\cr V_{\rm extra}&= -\sum_{i=1,\ldots,N'}{Z\over |x'_i|} + \sum_{{i=1,\ldots,N'}\atop{j=1,\ldots,N}}{1\over |x'_i-x_j|} + {1\over 2}\sum_{{{i=1,\ldots,N'}\atop{j=1,\ldots,N'}}\atop{i\ne j}}{1\over |x'_i-x'_j|} \cr}$$ Corollary 2 now implies that $${\scapro V_{\rm extra}\psi,\psi;}\ge {c\epsilon Z\over R}N' \eqno({\rm A})$$ or else $$\bigl \ge E_0(Z)+cZ^{{\frac 7,3}-7\gamma_2}$$ since the fact that $\bigl$ is at least $Z-Z^{1-\gamma_1}$ makes the interaction of the nucleus and of the electrons in $B(0,R/Z^{\gamma-\gamma_2})$ with a fixed electron $x'_j$ in $B(0,2R)-B(0,R/2)$ contribute approximately with $(-Z+N_{R/Z^{\gamma-\gamma_2}})/ |x'_j| \ge -Z^{1-\gamma_1}/|x'_j|$ to the potential energy. We omit the details and we simply point out that you need enough electrons to almost cancel the effect of the nucleus. Precisely, $$ Z^{-\gamma_2}>\epsilon \eqno(\rm{ineq}\,4)$$ \bigskip \indent We now use this estimate to go from $\est(\bar\epsilon,\epsilon,R)$ to $\est(\bar\epsilon,\epsilon',2R)$. To see this, simply take a wave function $\psi$ living in $B(0,2R)$ and a partition of unity $\theta_0$, $\theta_1$, adapted to $B(0,R)$ and $B(0,2R)-B(0,R/2)$, and set $$\psi_{i_1,\ldots,i_N}(x_1,\ldots,x_N) =\theta_{i_1}(x_1)\cdots\theta_{i_N}(x_N)\cdot\psi (x_1,\ldots,x_N)$$ For each $\psi_{i_1,\ldots,i_N}$ we have $N_1$ electrons living in $B(0,R)$ and $N_2$ electrons living in $B(0,2R)-B(0,R/2)$, with $N_1+N_2=N$; now, assume $\est(\bar\epsilon,\epsilon,R)$ holds, apply Corollary 1 and 2 and estimate (A), and sum over all possible choices of $i_1,\ldots,i_N$ to obtain $$\eqalign{\Bigl&\ge E_0(Z) + {\bar\epsilon Z\over R}\Bigl(N_1- (1+\epsilon) Z\Bigr) + {\scapro V_{\rm extra}\psi,\psi;}-{CN\over R^2}\cr &\qquad\ge E_0(Z) + {\bar\epsilon Z\over R}\Bigl(N_1+N_2- (1+\epsilon) Z\Bigr) -{CN\over R^2}\cr &\qquad= E_0(Z) + {\bar\epsilon Z\over R}\Bigl(N- (1+\epsilon) Z\Bigr) -{CN\over R^2}\qquad(\bar\epsilon\le c\epsilon)\cr}$$ The terms $CN/R^2$ come from the Laplacian hitting $\theta_0$ and $\theta_1$. Again we omit the details. Now observe that $$\max\biggl(0,{\bar\epsilon Z\over R}\bigl(N-(1+\epsilon)Z \bigr)-{CN\over R^2} \biggr) \ge {\bar\epsilon Z\over 2R}\bigl(N-(1+\epsilon')Z\bigr) $$ for $\epsilon'-\epsilon={4C\over \bar\epsilon RZ}$; we also need that $R\ge {2C\over \bar\epsilon Z}$, which certainly holds in this case. This proves that $\est(\bar\epsilon,\epsilon',2R)$ holds. \bigskip \indent If $$\beta_2>\gamma_2 \eqno({\rm ineq}\,5)$$ we can use corollary~1 and conclude that $\est(\bar\epsilon_0,\epsilon_n,2^n R_0)$ holds for $$\epsilon_{n+1}=\epsilon_n+{4C\over 2^n\bar\epsilon_0 R_0 Z}.$$ Note that $ \epsilon_n \le C\epsilon_0=\epsilon_\sharp$, for $C$ independent of $n$. Therefore, $\est(\bar\epsilon_0,C\epsilon_0,R)$ holds for all $R\ge R_0$ and so $$ N(Z) - Z \le C\epsilon_0 \le C Z^{1-\beta_1}$$ The value for $\alpha=1-\beta_1$ depends on $b$ and on the parameters $\beta_1$, $\beta_2$, $\gamma_1$ and $\gamma_2$ subject to the constraints (ineq$1$)---(ineq$5$). For each $b$, the optimum value in the closure of this set of parameters is attained for $\beta_1={\frac 3b,7}$. The value for $\alpha$ is then $1-{\frac 3b,7}$. >From Hughes and Siedentop-Weikard it follows that we can take $b={\frac 21,{56}}$. This allows us to take any $\alpha>{\frac 47,{56}}\approx 0.84$. Further improvement of $b$ will lead to a better estimate for $\alpha$. What Hughes and Siedentop-Weikard did is something in fact much harder that just estimate $K$, and it is clear that better estimates can be obtained; maybe we can take $b$ to be ${\frac 2,3}$; this would give the result for any $\alpha > {\frac 5,7}$. \vskip1em {\bf Acknowledgements.}The first author was partially supported by a National Science Foundation Grant. The second author was supported by a Sloan Foundation Graduate Dissertation Fellowship. \vskip1em \parindent13pt \item{1.} Lieb, E. H, Sigal, I., Simon, B. \& Thirring, W. (1988) ``Approximate Neutrality of Large-Z Ions'' {\it Communications in Mathematical Physics} {\bf 116(4)} 635-644.\par \item{2.} Hughes, W. To appear in {\it Advances in Mathematics}.\par \item{3.} Siedentop, H., Weikard, R. (1987) ``On the Leading Energy Correction for the Statistical Model of the Atom: Interacting Case'' {\it Communications in Mathematical Physics} {\bf 112} 471-490\par \end