Content-Type: multipart/mixed; boundary="-------------0107120805705" This is a multi-part message in MIME format. ---------------0107120805705 Content-Type: text/plain; name="01-262.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="01-262.keywords" decomposition of radial functions, positive definiteness ---------------0107120805705 Content-Type: application/x-tex; name="askey3.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="askey3.tex" \newcommand{\version}{\today} \documentclass[12pt]{article} \usepackage{amsmath,amsgen,amstext,amsbsy,amsopn,amsthm,amssymb} \pagestyle{myheadings} \swapnumbers \setlength{\voffset}{-.75truein} \setlength{\textheight}{9.25truein} \setlength{\textwidth}{6.5truein} \setlength{\hoffset}{-.7truein} \theoremstyle{plain} \newtheorem{thm}{THEOREM} \newtheorem{cor}[thm]{COROLLARY} \newtheorem{lem}[thm]{LEMMA} \newtheorem{proposition}[thm]{PROPOSITION} \theoremstyle{definition} \newtheorem{rem}[thm]{REMARK} \newtheorem{ex}[thm]{EXAMPLE} \newcommand{\infspec}{{\rm inf\ spec\ }} \newcommand{\R}{{\mathbb R}} \newcommand{\C}{{\mathbb C}} \newcommand{\N}{{\mathbb N}} \newcommand{\D}{{\mathcal D}} \newcommand{\Ll}{{\mathcal L}} \newcommand{\Hh}{{\mathcal H}} \newcommand{\Cc}{{\mathcal C}} \newcommand{\eps}{\varepsilon} \newcommand{\A}{{\bf A}} \newcommand{\abf}{{\bf a}} \newcommand{\B}{{\bf b}} \newcommand{\Aa}{{\bf A}} \newcommand{\x}{{\bf x}} \newcommand{\y}{{\bf y}} \newcommand{\X}{{\bf X}} \newcommand{\0}{{\bf 0}} \newcommand{\rr}{{\bf r}} \newcommand{\bfeta}{{\bf z}} \newcommand{\xpp}{\x^\perp} \newcommand{\ypp}{\y^\perp} \newcommand{\yperp}{\y_\perp} \newcommand{\rperp}{{\bf r}_\perp} \newcommand{\Tr}{{\rm Tr}} \newcommand{\half}{\mbox{$\frac{1}{2}$}} \newcommand{\third}{\mbox{$\frac{1}{3}$}} \newcommand{\rg}{\rho_\Gamma} \newcommand{\dx}{\frac\partial{\partial x}} \newcommand{\dy}{\frac\partial{\partial y}} \newcommand{\bsigma}{\mathord{\hbox{\boldmath $\sigma$}}} \newcommand{\rmd}{{d}} \newcommand{\al}{{\alpha}} \newcommand{\dodd}{\delta_{\rm odd}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \date{\small\version} \begin{document} \markboth{\scriptsize{HS \version}}{\scriptsize{HS \version}} \title{\bf{General decomposition of radial functions on $\R^n$ and refined conditions for positive definiteness}} \author{\vspace{5pt} Christian Hainzl$^1$ and Robert Seiringer$^{2}$\\ \vspace{-4pt}\small{$1.$ Mathematisches Institut, LMU M\"unchen}\\ \vspace{-4pt} \small{Theresienstrasse 39, 80333 M\"unchen, Germany}\\ \small{{\texttt{hainzl@rz.mathematik.uni-muenchen.de}}}\\ \vspace{-4pt}\small{$2.$ Institut f\"ur Theoretische Physik, Universit\"at Wien}\\ \vspace{-4pt} \small{Boltzmanngasse 5, A-1090 Vienna, Austria}\\ \small{{\texttt{rseiring@ap.univie.ac.at}}}} \date{\small\version} \maketitle \begin{abstract} We present a generalization of the Fefferman-de la Llave decomposition of the Coulomb potential to quite arbitrary radial functions $V$ on $\R^n$ going to zero at infinity. As a byproduct, we obtain conditions for positive definiteness of $V$, thereby improving results of Askey. \end{abstract} \bigskip \footnotetext[1]{Marie Curie Fellow} For the description of physical systems interacting with Coulomb forces, the Fefferman-de la Llave decomposition \cite{FL86} of the Coulomb potential has proved very useful. It states that for $x\in \R^3$ \begin{equation}\label{feff} \frac 1{|x|}=\frac 1\pi \int_0^\infty dr \frac 1{r^5} \chi_r*\chi_r(x), \end{equation} where $\chi_r=\Theta(r-|x|)$ is the characteristic function of a ball of radius $r$ centered at the origin, and $*$ denotes convolution on $\R^3$. Except for the constant $1/\pi$, it is easily checked that (\ref{feff}) holds true, since the right hand side is a radial, homogeneous function of order $-1$. However, for a general function $V$ on $\R^n$, it is not known whether a decomposition of the form (\ref{feff}) with a general weight function $g(r)$ replacing $1/r^5$ exists, except for the case $n=1$ \cite{HS01}. We show that this is indeed the case, provided some decrease and regularity properties of $V$ are satisfied. Our main result, stated in Theorem \ref{thm1} below, gives this decomposition, with a weight $g(r)$ that is related to the $[n/2]+2$'th derivative of $V$. Here $[\cdot]$ denotes the Gau\ss\ bracket, i.e., $[m]=\max\{n\in\N_0, n\leq m\}$. As a byproduct, we see from this decomposition that a function $V$ is positive definite, if the corresponding weight function $g(r)$ is positive. This gives conditions on $V$ which are less restrictive than the ones obtained before by Askey \cite{A73a}. The decomposition (\ref{vg}) may especially be of interest in generalizing results so far only applicable to the Coulomb potential, for instance estimating the \lq\lq indirect part\rq\rq\ of the interaction energy in an $N$-particle quantum system, as done in the Coulomb case in \cite{LO81}, respectively estimating the validity of Hartree-Fock approximations (see \cite{B92} and \cite{GS94}). \medskip Our main result is the following: \begin{thm}\label{thm1} Let $V:\R^n\to \R$ be a radial function that is $[n/2]+2$ times differentiable away from $x=0$. For $m\in\N_0$ denote $V^{(m)}(|x|)=d^m/d|x|^m V(x)$. Assume further that $\lim_{|x|\to\infty}|x|^m V^{(m)}(|x|)=0$ for all $0\leq m\leq [n/2]+1$. Let $\chi_r(x)=\Theta(r-|x|)$, and $n\geq 2$. Then \begin{equation}\label{vg} V(x)=\int_0^\infty dr g(r) \chi_{r/2}*\chi_{r/2}(x), \end{equation} where \begin{eqnarray}\nonumber g(r)=\frac{(-1)^{[n/2]}}{\Gamma(\frac{n-1}2)}\frac 2{(\pi r^2)^{(n-1)/2}} &\Biggl(&\int_r^\infty ds V^{([n/2]+2)}(s) \left(\frac d{ds}\right)^{n-1-[n/2]} s (s^2-r^2)^{\half(n-3)} \\ \label{defg} &&+\, \dodd\, V^{([n/2]+2)}(r) r (2r)^{\half(n-3)} \Gamma(\mbox{$\frac{n-1}2$}) \Biggl), \end{eqnarray} and $\dodd=1$ for $n$ odd, $\dodd=0$ for $n$ even. \end{thm} Note the $r/2$ in (\ref{vg}), which is chosen for convenience. \begin{proof} Elementary considerations show that \begin{equation} \chi_{r/2}*\chi_{r/2}(x)=\frac 1{\Gamma(\frac{n+1}2)}\left(\frac \pi 4\right)^{(n-1)/2} \int_{|x|}^r dy (r^2-y^2)^{\half(n-1)} \end{equation} for $|x|\leq r$, and $0$ otherwise. Therefore we can write \begin{eqnarray}\nonumber &&\left( \frac { (-1)^{[n/2]}}{\Gamma(\frac{n-1}2)\Gamma(\frac{n+1}2)}\frac 1{2^{n-2}} \right)^{-1}\int_0^\infty dr g(r) \chi_{r/2}*\chi_{r/2}(x)=\\ \nonumber && \int_{|x|}^\infty dr \frac 1{r^{n-1}} \int_r^\infty ds V^{([n/2]+2)}(s) \left(\frac d{ds}\right)^{n-1-[n/2]} s(s^2-r^2)^{\half(n-3)} \int_{|x|}^r dy (r^2-y^2)^{\half(n-1)}\\ &&+\,\dodd \int_{|x|}^\infty dr \frac 1{r^{n-1}} V^{([n/2]+2)}(r) r (2r)^{\half(n-3)} \Gamma(\mbox{$\frac{n-1}2$})\int_{|x|}^r dy (r^2-y^2)^{\half(n-1)}. \end{eqnarray} Using the fact that \begin{eqnarray}\nonumber &&\left(\frac d{ds}\right)^{n-1-[n/2]} \int_y^s dr \frac 1{r^{n-1}} (r^2-y^2)^{\half(n-1)} s (s^2-r^2)^{\half (n-3)}=\\ \nonumber && \int_y^s dr \frac 1{r^{n-1}} (r^2-y^2)^{\half(n-1)} \left(\frac d{ds}\right)^{n-1-[n/2]} s (s^2-r^2)^{\half (n-3)}\\ &&+\, \dodd\, \frac 1{s^{n-1}}(s^2-y^2)^{\half(n-1)}s (2s)^{\half(n-3)}\Gamma(\mbox{$\frac{n-1}2$}), \end{eqnarray} we can change the order of integration to get \begin{eqnarray}\nonumber &&\left( \frac { (-1)^{[n/2]}}{\Gamma(\frac{n-1}2)\Gamma(\frac{n+1}2)}\frac 1{2^{n-2}} \right)^{-1}\int_0^\infty dr g(r) \chi_{r/2}*\chi_{r/2}(x)=\\ && \int_{|x|}^\infty ds V^{([n/2]+2)}(s) \int_{|x|}^s dy \left(\frac d{ds}\right)^{n-1-[n/2]} \int_y^s dr \frac 1{r^{n-1}} s(s^2-r^2)^{\half(n-3)} (r^2-y^2)^{\half(n-1)}. \end{eqnarray} Now \begin{equation} \int_y^s dr \frac 1{r^{n-1}} (s^2-r^2)^{\half(n-3)} (r^2-y^2)^{\half(n-1)}=\frac{\Gamma(\mbox{$\frac{n-1}2$}) \Gamma(\mbox{$\frac{n+1}2$})2^{n-2}(s-y)^{n-1}}{s\Gamma(n)}, \end{equation} and therefore \begin{equation} \int_0^\infty dr g(r) \chi_{r/2}*\chi_{r/2}(x)=\frac {(-1)^{[n/2]}}{\Gamma([n/2]+2)}\int_{|x|}^\infty ds V^{([n/2]+2)}(s) (s-|x|)^{[n/2]+1} = V(x), \end{equation} where we integrated by parts, using the demanded decrease properties of $|x|^m V^{(m)}(|x|)$. \end{proof} \begin{rem} For $n=1$, (\ref{vg}) holds with $g(r)=V^{''}(r)$ \cite{HS01}. \end{rem} \begin{rem} If $V(x)$ is $n+1$ times differentiable away from $x=0$, we can use partial integration to write $g(r)$ in the more compact form \begin{equation}\label{comp} g(r)=\frac{(-1)^{n+1}}{\Gamma(\frac{n-1}2)}\frac 2{(\pi r^2)^{(n-1)/2}} \int_r^\infty ds V^{(n+1)}(s) s (s^2-r^2)^{(n-3)/2} . \end{equation} \end{rem} For simplicity, we have restricted ourselves to formulating Theorem \ref{thm1} for differentiable functions only, but as is obvious from the proof, (\ref{defg}) and (\ref{comp}) hold for a more general class of potentials, if the derivatives are interpreted in the sense of distributions. \begin{ex} For the Coulomb potential $V(x)=1/|x|$, we have \begin{equation} g(r)=\frac 12 \frac{\Gamma(n+2)}{\Gamma(n/2+1)} \frac{\pi^{1-n/2}}{r^{n+2}}. \end{equation} \end{ex} The decomposition (\ref{vg}) provides conditions for positive definiteness of $V$. Since $\chi_r*\chi_r(x)$ is obviously positive definite (as a function of $x$), any $V$ with a positive weight function $g(r)$ is positive definite. This is the content of the following corollary: \begin{cor} If $g(r)\geq 0$, then $V(x)$ is positive definite. \end{cor} It is easy to see that \begin{equation}\label{pos} \left(\frac d{ds}\right)^{n-1-[n/2]} s (s^2-r^2)^{\half(n-3)}\geq 0 \quad \mbox{ for }s\geq r, \end{equation} and therefore positivity of $(-1)^{[n/2]} V^{([n/2]+2)}$ implies that $V(x)$ is positive definite. Moreover, one needs only slightly less regularity assumptions on $V$, and can use Theorem \ref{thm1} to prove the following theorem, which was also proved before by Askey \cite{A73a}, who deduces it from certain positivity properties of Bessel functions proven in \cite{A73b} and \cite{FI75}. \begin{thm} If $V(x)$ is $[n/2]$ times differentiable away from $x=0$, $\lim_{|x|\to\infty}V(x)=0$ and $(-1)^{[n/2]}V^{([n/2])}(r)$ is convex for $r>0$, then $V(x)$ is positive definite. \end{thm} \begin{proof} Convexity of $V^{([n/2])}$ together with the decease of $V$ at infinity implies the decrease properties of $|x|^m V^{(m)}(|x|)$ demanded in Theorem \ref{thm1}. Moreover, $V^{([n/2]+2)}$ defines a positive measure, if the last two derivatives are interpreted in the sense of distributions, and Theorem \ref{thm1} holds. Since $g(r)\geq 0$, $V$ is positive definite. \end{proof} As one sees from (\ref{defg}) and (\ref{pos}), the condition $g(r)\geq 0$, which is sufficient for positive definiteness of $V$, is a weaker assumption than $(-1)^{[n/2]} V^{([n/2]+2)}\geq 0$. Consider for instance the case $n=3$. Here \begin{equation} g(r)=\frac 2{\pi r^2}\left(V^{''}(r)-r V^{'''}(r)\right)=-\frac 2\pi \left(\frac {V^{''}(r)}{r}\right)^{'}, \end{equation} so a monotone decrease of $V^{''}/r$ (instead of $V^{''}$) is sufficient for positive definiteness of $V$. Analogous considerations apply to all dimensions $n\geq 2$. \medskip Positivity of $g$ might be a useful way of checking positive definiteness of a function $V$. However, this condition is of course not necessary, as the following example shows. \begin{ex} For $n=3$, $V(x)=1/(|x|^2+1)$ is positive definite, but the corresponding $g(r)$ is not a positive function. \end{ex} \bigskip \noindent {\it Acknowledgments:} We thank Volker Bach for encouraging the present study, and Michael Loss for providing the references \cite{A73a, A73b, FI75}. C. Hainzl has been supported by a Marie Curie Fellowship of the European Community programme \lq\lq Improving Human Research Potential and the Socio-economic Knowledge Base\rq\rq\ under contract number HPMFCT-2000-00660. R. Seiringer acknowledges warm hospitality at the Mathematical Institute, LMU M\"unchen, where part of this work has been done. \begin{thebibliography}{9} \bibitem{FL86} C.L. Fefferman and R. de la Llave, {\it Relativistic stability of matter I}, Revista Matematica Iberoamericana {\bf 2}, 119--161 (1986) \bibitem{HS01} C. Hainzl and R. Seiringer, {\it Bounds on One-Dimensional Exchange Energies with Application to Lowest Landau Band Quantum Mechanics}, Lett. Math. Phys. {\bf 55}, 133--142 (2001) \bibitem{A73a} R. Askey, {\it Refinements of Abel Summability for Jacobi Series}, in: Harmonic Analysis on Homogeneous Spaces, Proc. Symp. in Pure Math. {\bf 26}, 335--338 (1973) \bibitem{LO81} E.H. Lieb and S. Oxford, {\it Improved Lower Bound on the Indirect Coulomb Energy}, Int. J. Quant. Chem. {\bf 19}, 427--439 (1981) \bibitem{B92} V. Bach, {\it Error bound for the Hartree-Fock energy of atoms and molecules}, Commun. Math. Phys. {\bf 147}, 527--548 (1992) \bibitem{GS94} G.M. Graf and J.P. Solevej, {\it A correlation estimate with applications to quantum systems with coulomb interaction}, Rev. Math. Phys. {\bf 6}, 977--997 (1994) \bibitem{A73b} R. Askey, {\it Summability of Jacobi Series}, Trans. Amer. Math. Soc. {\bf 179}, 71--84 (1973) \bibitem{FI75} J.L. Fields and M.E. Ismail, {\it On the Positivity of some $_1 F_2$'s}, SIAM J. Math. Anal. {\bf 6}, 551--559 (1975) \end{thebibliography} \end{document} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ---------------0107120805705--