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decomposition of radial functions, positive definiteness
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\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}





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