Update to v0.1:
Added: Theorem on positivity of solutions.
Added: Reference to [CJLL20]
Added: Theorem on decay rate is now more general.
Fixed: Clarified the discussion in point 2-2 of the proof of the theorem
on decay.
Removed: open problem about positivity of solutions.
Fixed: Format of named theorems.
Fixed: Minor formatting fixes.
Fixed: In proof of decay: indenting error.
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@ -13,22 +13,22 @@
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\hbox{}
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\hfil{\bf\LARGE Analysis of a simple equation for the
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\par
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\vskip10pt
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\vskip8pt
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\hfil ground state energy of the Bose gas
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}
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\vfill
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\vskip10pt
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\hfil{\bf\large Eric Carlen}\par
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\hfil{\bf Eric A. Carlen}\par
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\hfil{\it Department of Mathematics, Rutgers University}\par
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\hfil{\tt\color{blue}\href{mailto:carlen@rutgers.edu}{carlen@rutgers.edu}}\par
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\vskip20pt
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\hfil{\bf\large Ian Jauslin}\par
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\hfil{\bf Ian Jauslin}\par
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\hfil{\it Department of Physics, Princeton University}\par
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\hfil{\tt\color{blue}\href{mailto:ijauslin@princeton.edu}{ijauslin@princeton.edu}}\par
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\vskip20pt
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\hfil{\bf\large Elliott H. Lieb}\par
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\hfil{\bf Elliott H. Lieb}\par
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\hfil{\it Departments of Mathematics and Physics, Princeton University}\par
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\hfil{\tt\color{blue}\href{mailto:lieb@princeton.edu}{lieb@princeton.edu}}\par
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@ -102,12 +102,36 @@ provides a useful and illuminating route to the computation of the properties of
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However, the convolution nonlinearity in\-~(\ref{simpleq}) makes it non-local, and very different from\-~(\ref{simpleq2}).
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\bigskip
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\indent As explained in\-~\cite{Li63} the solutions of physical interest are integrable, and satisfy
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\indent As explained in\-~\cite{Li63} the solutions of physical interest are integrable and {\em must} satisfy $u(x) \leqslant 1$ for all $x$. Our first result is that for integrable solutions of the system\-~(\ref{simpleq})-(\ref{energy}), the upper bound $u \leqslant 1$ implies the lower bound $u \geqslant 0$:
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\theoname{Theorem}{Positivity}\label{positivity} Suppose that $\mathcal{V}$ is non-negative and integrable and that $u$ is an integrable solution of
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(\ref{simpleq})-(\ref{energy}) such that $u(x) \leqslant 1$ for all $x$. Then $u(x) \geqslant 0$ for all $x$, and all such solutions have fairly slow decay at infinity in that they satisfy
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\begin{equation}\label{slow}
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\int |x|u(x)d x = \infty \ .
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\end{equation}
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Thus, any physical solutions of (\ref{simpleq})-(\ref{energy}) must necessarily satisfy the {\em pair} of inequalities
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\begin{equation}
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0 \leqslant u(x) \leqslant 1 \quad{\rm for \ all}\ x \ .
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\label{con1}
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\end{equation}
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We shall see that any non-negative solution automatically satisfies this upper bound.
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\endtheo
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\bigskip
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This {\em a-priori} result that we prove before we take up existence and uniqueness, turns on results \cite{CJLL20} obtained in collaboration with Michael Loss on the convolution inequality $f \geqslant f\ast f$ in $L^1(\mathbb R^d)$.
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While $u(x)\leqslant 1$ is a physical requirement, $u(x)\geqslant0$ is not, see section\-~\ref{sec:bosegas} for details.
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\bigskip
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The converse of Theorem\-~\ref{positivity} also holds, as stated in the following theorem.
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\theo{Theorem}\label{theorem:leq1}
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Let $\mathcal V\in L^1(\mathbb{R}^d)\cap L^p(\mathbb{R}^d)$, $p>\max\{\frac d2,1\}$, be non-negative.
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If $u$ is an integrable solution of
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(\ref{simpleq})-(\ref{energy}) such that $u(x) \geqslant 0$ for all $x$, then $u(x) \leqslant 1$ for all $x$.
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\endtheo
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{\bf Remark}: We have thus proved that $u\geqslant0$ if and only if $u\leqslant1$.
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This, in principle, leaves the door open to solutions that are sometimes $>1$ and sometimes $<0$, though we do not believe such solutions exist.
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\bigskip
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\indent Before stating our main theorems, we make a few observations.
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@ -213,10 +237,10 @@ In particular, this shows that the system\-~(\ref{simpleq})-(\ref{energy}) does
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In fact, as is stated in the following theorem, $\rho$ and $e$ are constrained to be related by a functional equation.
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\bigskip
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\theo{Theorem}[existence and uniqueness]\label{theorem:existence} Let $\mathcal V\in L^1(\mathbb{R}^d)\cap L^p(\mathbb{R}^d)$, $p>\max\{\frac d2,1\}$, be non-negative.
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\theoname{Theorem}{existence and uniqueness}\label{theorem:existence} Let $\mathcal V\in L^1(\mathbb{R}^d)\cap L^p(\mathbb{R}^d)$, $p>\max\{\frac d2,1\}$, be non-negative.
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Then there is a constructively defined continuous function $\rho(e)$ on $(0,\infty)$ such that
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$\lim_{e\to 0}\rho(e) = 0$ and $\lim_{e\to \infty} \rho(e) = \infty$ and such that for any $e\geqslant 0$ and $\rho = \rho(e)$,
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the system\-~(\ref{simpleq}) and\-~(\ref{energy}) has a unique integrable solution $u(x)$ satisfying\-~(\ref{con1}). Moreover, if $\rho \neq \rho(e)$, the system\-~(\ref{simpleq}) and\-~(\ref{energy}) has {\em no} integrable solution $u(x)$ satisfying\-~(\ref{con1}).
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the system\-~(\ref{simpleq}) and\-~(\ref{energy}) has a unique integrable solution $u(x)$ satisfying $u(x)\leqslant 1$. Moreover, if $\rho \neq \rho(e)$, the system\-~(\ref{simpleq}) and\-~(\ref{energy}) has {\em no} integrable solution $u(x)$ satisfying\-~(\ref{con1}).
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\endtheo
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\bigskip
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@ -236,7 +260,7 @@ monotone increasing function. In that case, the functional relation could be inv
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\indent Having proved that the solution to the simple equation is unique, our second main result is an asymptotic expression for $e(\rho)$, both for low and for high density.
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\bigskip
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\theo{Theorem}[asymptotics of the energy for $d=3$]\label{theorem:asymptotics}
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\theoname{Theorem}{asymptotics of the energy for $d=3$}\label{theorem:asymptotics}
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Consider the case $d=3$.
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Let $\mathcal V$ be non-negative, integrable and square-integrable. Then, for each
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$\rho>0$ there is at least one $e>0$ such that $\rho = \rho(e)$. For any such $\rho $ and $e$ we have the following bounds for low and high density (i.e., small and large $\rho$).
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@ -263,7 +287,11 @@ monotone increasing function. In that case, the functional relation could be inv
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\end{itemize}
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\bigskip
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\theo{Theorem}[decay of $u$ in $d=3$]\label{theorem:decay}
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\theoname{Theorem}{decay of $u$ at infinity}\label{theorem:decay} In all dimensions, provided $\mathcal{V}$ is spherically symmetric with $\int |x|^2\mathcal{V} dx <\infty $ in addition to satisfying the hypotheses imposed in Theorem~\ref{theorem:existence}, all integrable solutions of (\ref{simpleq})-(\ref{energy}) with $u(x) \leqslant1$ for all $x$ satisfy
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\begin{equation}\label{gendecay}
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\int |x| u(x) dx = \infty \quad{\rm and}\quad \int |x|^r u(x) dx < \infty \quad{\rm for \ all} \ 0 < r < 1\ .
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\end{equation}
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Thus, if $u(x) \sim |x|^{-m}$ for some $m$, the only possibility is $m = d+1$. Under stronger assumptions on the potential, this is actually the case.
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For $d=3$, if $\mathcal V$ is non-negative, square-integrable, spherically symmetric (that is, $\mathcal V(x)=\mathcal V(|x|)$), and, for $|x|>R$,
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\begin{equation}
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\mathcal V(|x|)\leqslant Ae^{-B|x|}
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@ -271,8 +299,7 @@ monotone increasing function. In that case, the functional relation could be inv
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\end{equation}
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for some $A,B>0$ then there exists $\alpha>0$ such that
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\begin{equation}
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u(|x|)\mathop\sim_{|x|\to\infty}\frac\alpha{|x|^4}
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.
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u(x)\mathop\sim_{|x|\to\infty}\frac\alpha{|x|^4} .
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\end{equation}
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\endtheo
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\bigskip
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@ -294,11 +321,57 @@ The simple equation\-~(\ref{simpleq}) is actually an approximation of a richer e
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\bigskip
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\indent The paper is organized as follows.
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We prove theorems\-~\ref{theorem:existence}, \ref{theorem:asymptotics} and\-~\ref{theorem:decay} in sections\-~\ref{sec:existence}, \ref{sec:asymptotics} and\-~\ref{sec:decay} respectively.
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We prove theorem\-~\ref{positivity} in section\-~\ref{sec:pos}, theorems\-~\ref{theorem:leq1} and\-~\ref{theorem:existence} in section\-~\ref{sec:existence}, theorem\-~\ref{theorem:asymptotics} in section\-~\ref{sec:asymptotics}, and theorem\-~\ref{theorem:decay} in section\-~\ref{sec:decay}.
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In section\-~\ref{sec:bosegas}, we explain how the simple equation is related to the Bose gas, and present some numerical evidence that it is very good at predicting the ground state energy.
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In section\-~\ref{sec:open} we discuss a few open problems and extensions.
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\section{Proof of Theorem\-~\expandonce{\ref{theorem:existence}}}\label{sec:existence}
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\section{Proof of Theorem \expandonce{\ref{positivity}}}\label{sec:pos}
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As explained in the introduction, the solutions of (\ref{simpleq})-(\ref{energy}) that are of physical interest are those that are integrable and satisfy $u(x) \leqslant 1$ for all $x$. In this section we prove, making no assumptions on the potential $\mathcal{V}$ other than its positivity and integrability, that all such solutions are non negative, and have slow decay so that $\int |x| u(x) dx = \infty$.
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Our starting point is the form of (\ref{simpleq}) given in (\ref{simpleq3}). For an integrable solution $u$, define
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\begin{equation}\label{pos1}
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f := 2e\rho Y_{4e}*u\ .
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\end{equation}
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If (\ref{energy}) is satisfied, then
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\begin{equation}\label{pos2}
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\int f dx = \frac 12\ .
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\end{equation}
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and (\ref{simpleq3}) can be written as
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\begin{equation}\label{pos3}
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u = Y_{4e}*(\mathcal V (1- u)) + f\ast u\ .
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\end{equation}
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\theo{Lemma}
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Let $u(x)$ be an integrable solution of the system (\ref{simpleq})-(\ref{energy}) such that $u(x) \leqslant 1$ for all $x$. Let $f$ be defined in terms of $u$, $e$ and $\rho$ by (\ref{pos1}). If $f(c) \geqslant 0$ for all $x$, then $u(x) \geqslant 0$ for all $x$.
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\endtheo
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{\bf Proof} Since $Y_{4e}*(\mathcal V (1- u(x)) \geqslant 0$, it follows that
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\begin{equation}
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u_- \leqslant (f*u)_- = (f*u_+ - f*u_-)_- \leqslant f*u_-\ .
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\end{equation}
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Integrating, we find
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${\displaystyle \int u_- {\rm d}x \leqslant \frac12 \int u_-{\rm d}x}$,
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and this implies that $u_- = 0$. \qed
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{\bf Proof of Theorem~\ref{positivity}} Multiply (\ref{pos3}) through by $2e\rho$, and then convolve both sides with $Y_{4e}$. The result is $f = 2e\rho Y_{4e}*(Y_{4e}*(\mathcal V (1- u)) + f\ast f$, and since $Y_{4e}*(Y_{4e}*(\mathcal V (1- u)) \geqslant 0$, $f$ is an integrable solution of
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\begin{equation}\label{pos5}
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f(x) \geqslant f\ast f(x)
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\end{equation}
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for all $x$. It is proved in \cite{CJLL20} that all integrable solutions of (\ref{pos5}) are non-negative and have integral no greater than $\frac12$, and that moreover, (\ref{pos2}) and (\ref{pos3}) together imply that
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\begin{equation}
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\int |x| f(x)\ dx = \infty\ .
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\end{equation}
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However,
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\begin{equation}
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\int |x| f(x)\ dx = 2e\rho\int |x| Y_{4e}\ast u(x)\ dx= 2e\rho\int (Y_{4e} \ast |x|) u(x)\ dx \ .
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\end{equation}
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Then since $\lim_{x\to \infty}\left(4e|x|^{-1} Y_{4e} \ast |x|\right) = 1$, (\ref{slow}) follows. \qed
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\section{Proof of Theorems \expandonce{\ref{theorem:leq1}} and \expandonce{\ref{theorem:existence}}}\label{sec:existence}
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\indent As was shown in\-~(\ref{simpleq3}) and\-~(\ref{simpleq4}), there are at least two ways to write\-~(\ref{simpleq}) as a fixed point equation.
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As it turns out, only the latter one
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@ -311,7 +384,7 @@ Starting with $u_0(x) = 0$, define
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\begin{equation}
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u_n(x) = \Phi(u_{n-1})(x)
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\end{equation}
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for $n\geq1$. It is easy to see that for arbitrary $e,\rho \geqslant 0$, this produces a monotone increasing sequence of non-negative integrable functions. Thus, $u(x) := \lim_{n\to \infty}u_n(x)$ will exist, but it need not be integrable and it need not satisfy\-~(\ref{energy}) or\-~(\ref{con1}).
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for $n\geqslant1$. It is easy to see that for arbitrary $e,\rho \geqslant 0$, this produces a monotone increasing sequence of non-negative integrable functions. Thus, $u(x) := \lim_{n\to \infty}u_n(x)$ will exist, but it need not be integrable and it need not satisfy\-~(\ref{energy}) or\-~(\ref{con1}).
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\indent To bring\-~(\ref{energy}) into the iteration scheme, we take $e$ as the independent parameter, and define a sequence $\{\rho_n\}$ along with the sequence $\{u_n(x)\}$, both depending on $e$, through
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\begin{equation}
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@ -339,7 +412,7 @@ We proceed by induction. By definition, $u_0 =0$ and $\rho_0 = 2e\left(\int_{\ma
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$u_1 = K_e\mathcal V \geqslant u_0$ and $\rho_1 = 2e\left( \int \mathcal V(1- K_e\mathcal{V})dx \right)^{-1}$.
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As noted in the discussion between\-~(\ref{con4}) and\-~(\ref{con4B}),
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\begin{equation}
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2\int_{\mathbb{R}^d} u_1dx = \frac{1}{e}\int_{\mathbb{R}^d}\mathcal{V}(1-u_1)dx \leq \ \frac{1}{e}\int_{\mathbb{R}^d}\mathcal{V}dx.
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2\int_{\mathbb{R}^d} u_1dx = \frac{1}{e}\int_{\mathbb{R}^d}\mathcal{V}(1-u_1)dx \leqslant \ \frac{1}{e}\int_{\mathbb{R}^d}\mathcal{V}dx.
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\end{equation}
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Since $t\mapsto t^{-1}$ is monotone decreasing on $(0,\infty)$, this shows that $\rho_1 > \rho_0$, and that\-~(\ref{simple17}) holds for $n=1$.
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\bigskip
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@ -359,7 +432,7 @@ Integrating both sides of $u_{n+1} = G_e \mathcal {V}(1- u_{n+1}) + 2e \rho_n G_
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\end{equation}
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Then since $\int_{\mathbb{R}^d} u_{n}{\rm d}x < \frac{1}{2e}\int_{\mathbb{R}^d}\mathcal V(1-u_{n}) = \frac{1}{\rho_n}$, (\ref{simple15}) implies
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\begin{equation}
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2\int_{\mathbb{R}^d} u_n{\rm d}x \leq \frac{1}{2e}\int_{\mathbb{R}^d}\mathcal{V}(1-u_n) + \int_{\mathbb{R}^d}u_{n-1}{\rm d}x\ .
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2\int_{\mathbb{R}^d} u_n{\rm d}x \leqslant \frac{1}{2e}\int_{\mathbb{R}^d}\mathcal{V}(1-u_n) + \int_{\mathbb{R}^d}u_{n-1}{\rm d}x\ .
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\end{equation}
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Then because $\int_{\mathbb{R}^d} u_{n}{\rm d}x < \int_{\mathbb{R}^d} u_{n+1} {\rm d}x $, we have that $\int_{\mathbb{R}^d} u_{n+1}{\rm d}x < \frac{1}{2e}\int_{v}\mathcal{V}(1-u_{n+1}) $.
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This proves\-~(\ref{simple17}) for $n+1$, and shows that
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@ -381,7 +454,7 @@ and then, as before, $\rho_{n+1}\geqslant \rho_n$.
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Since $K_e$ maps $L^p(\mathbb R^d)$ into $W^{2,p}(\mathbb R^d)$, $u_1$ is continuous and vanishes at infinity. Let $A := \{x\ :\ u_1(x) > 1\}$. Then $A$ is open. If $A$ is non-empty, then $u_1$ is subharmonic on $A$, and hence takes on its maximum on the boundary of $A$. Since $u_1$ would equal $1$ on the boundary, this is impossible, and $A$ is empty. This proves the assertion for $n=1$.
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\bigskip
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\indent Now make the inductive hypothesis that $0 \leqslant u_n(x) \leq 1$ for all $x$. Then $\|u_n\|_p^p \leq \|u_n\|_1 \leq\frac{1}{2e}\int_{\mathbb{R}^d}\mathcal{V}dx$. By Young's inequality, $\|u_n\ast u_n\|_p \leq \|u_n\|_p\|u_1\|_1$, and hence
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\indent Now make the inductive hypothesis that $0 \leqslant u_n(x) \leqslant 1$ for all $x$. Then $\|u_n\|_p^p \leqslant \|u_n\|_1 \leqslant\frac{1}{2e}\int_{\mathbb{R}^d}\mathcal{V}dx$. By Young's inequality, $\|u_n\ast u_n\|_p \leqslant \|u_n\|_p\|u_1\|_1$, and hence
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$\mathcal{V} + 2e\rho_n u_n\ast u_n \in L^p(\mathbb{R}^d)$. Therefore,
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$u_{n+1}= K_e(\mathcal{V} + 2e\rho_n u_n\ast u_n) \in W^{2,p}(\mathbb{R}^d)$. It follows as before that $u_{n+1}$ is continuous and vanishing at infinity, and in particular, bounded, and
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\begin{eqnarray*}
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@ -407,9 +480,9 @@ Then both limits exist, $u\in W^{2,p}(\mathbb{R}^d)$ and $u$ satisfies\-~(\ref{s
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{\bf Proof:}
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By Lemma~\ref{lem1}, both limits exist, and by\-~(\ref{simple17}), $\rho(e) \leqslant \left(\int_{\mathbb{R}^d} K_e\mathcal{V}dx\right)^{-1}$.
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Also by Lemma~\ref{lem1}, $\int_{\mathbb{R}^d} \leqslant \frac{1}{2e}\int_{\mathbb{R}^d}\mathcal{V}(x)dx$,
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$u$ is integrable and $\lim_{n\to\infty}\|u_n - u\|_1 = 0$. Moreover, by Lemma~\ref{lem2}, $0 \leq u \leq 1$, and then $\|u\|_p^p \leq \|u\|_1$ and $\|u_n- u\|_p^p \leq (p+1)\|u_n-u\|_1$, and then by Young's Inequality
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$u$ is integrable and $\lim_{n\to\infty}\|u_n - u\|_1 = 0$. Moreover, by Lemma~\ref{lem2}, $0 \leqslant u \leqslant 1$, and then $\|u\|_p^p \leqslant \|u\|_1$ and $\|u_n- u\|_p^p \leqslant (p+1)\|u_n-u\|_1$, and then by Young's Inequality
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\begin{equation}
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\|u\ast u - u_n\ast u_n\|_p \leq \|u_n\|_1\|u_n -u\|_p + \leq \|u\|_1\|u_n -u\|_p\ .
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\|u\ast u - u_n\ast u_n\|_p \leqslant \|u_n\|_1\|u_n -u\|_p + \leqslant \|u\|_1\|u_n -u\|_p\ .
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\end{equation}
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Therefore, $\lim_{n\to\infty}(\mathcal{V} +2e\rho_n(e) u_n\ast u_n) = (\mathcal{V} +2e\rho(e) u\ast u)$
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with convergence in $L^p(\mathbb{R}^d)$. Then $\lim_{n\to\infty}K_e (\mathcal{V} +2e\rho_n(e) u_n\ast u_n) = K_e(\mathcal{V} +2e\rho(e) u\ast u)$ with convergence in $W^{2,p}(\mathbb{R}^d)$, and in particular, in
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@ -417,7 +490,7 @@ $L^p(\mathbb{R}^d)$. It now follows that $u = K_e(\mathcal{V} +2e\rho(e) u\ast u
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By remarks made above, this means that $u$ satisfies\-~(\ref{simpleq}) and\-~(\ref{energy}). \qed
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\bigskip
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\theo{Lemma}\label{lem3}
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\theo{Lemma}\label{lem4}
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For all $e\in (0,\infty)$, the solution $u$ of the system\-~(\ref{simpleq}) and\-~(\ref{energy}) that we have constructed by iteration on Lemma~\ref{lem3} is the unique non-negative integrable solution for $\rho = \rho(e)$.
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Moreover, there does not exist such any such solution when $\rho \neq \rho(e)$.
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\endtheo
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@ -434,7 +507,7 @@ By remarks made above, this means that $u$ satisfies\-~(\ref{simpleq}) and\-~(\r
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\begin{equation}
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\tilde u(x)-u_n(x)=2eK_e(\tilde\rho\tilde u\ast\tilde u(x)-\rho_{n-1}u_{n-1}\ast u_{n-1}(x))
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\end{equation}
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Since $u_{n-1} =0$, the positivity of $\tilde u$ implies the positivity of $\tilde u(x) - u_1(x)$.
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Since $u_0 =0$, the positivity of $\tilde u$ implies the positivity of $\tilde u(x) - u_1(x)$.
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If $\tilde u\geqslant u_{n-1}$, then, by\-~(\ref{rhon}), $\tilde\rho\geqslant\rho_{n-1}$, from which $\tilde u\geqslant u_n$ follows easily.
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This proves that both $\tilde\rho\geqslant\rho$ and $\tilde u\geqslant u$. However, integrating both sides of the latter inequality yields
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\begin{equation}
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@ -445,7 +518,7 @@ By remarks made above, this means that $u$ satisfies\-~(\ref{simpleq}) and\-~(\r
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\qed
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\bigskip
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\theo{Lemma}\label{lem4}
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\theo{Lemma}\label{lem5}
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The function $\rho(e)$ is continuous on $(0,\infty)$, with
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\begin{equation}
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\lim_{e\to 0}\rho(e) = 0
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@ -498,11 +571,14 @@ Now an easy induction shows that $a_n(e)$ is continuous for each $n$. By\-~(\ref
|
||||
(\ref{con4B}). \qed
|
||||
\bigskip
|
||||
|
||||
{\bf Remark}: Note that $\|u - u_n\|_1 = \frac{1}{\rho} - a_n$, and hence by By\-~(\ref{rate}), $\|u - u_n\|_1 \leq Cn^{-1/2}$.
|
||||
{\bf Remark}: Note that $\|u - u_n\|_1 = \frac{1}{\rho} - a_n$, and hence by By\-~(\ref{rate}), $\|u - u_n\|_1 \leqslant Cn^{-1/2}$.
|
||||
In fact, numerically, we find that the rate is significantly faster than this. For example, with $\mathcal V(x)=e^{-|x|}$ and $e=10^{-4}$, $\|u - u_n\|_1$ decays at least as fast as $n^{-3.5}$.
|
||||
\bigskip
|
||||
|
||||
{\bf Proof of Theorem\-~\ref{theorem:existence}} Every statement in the theorem has been established in Lemma~\ref{lem1} through Lemma~\ref{lem4}. \qed
|
||||
{\bf Proof of Theorem\-~\ref{theorem:leq1}} This theorem follows from Lemmas\-~\ref{lem2}, \ref{lem3} and\-~\ref{lem4}. \qed
|
||||
\bigskip
|
||||
|
||||
{\bf Proof of Theorem\-~\ref{theorem:existence}} Every statement in the theorem has been established in Lemma~\ref{lem1} through Lemma~\ref{lem5}. \qed
|
||||
|
||||
\bigskip
|
||||
|
||||
@ -522,7 +598,7 @@ Throughout this section, let $u_\rho$ denote the solution provided by Theorem\-~
|
||||
|
||||
\subsection{High density $\rho$}
|
||||
|
||||
\theo{Lemma}[high density asymptotics]\label{lemma:large}
|
||||
\theoname{Lemma}{high density asymptotics}\label{lemma:large}
|
||||
If $\mathcal V$ is integrable, then as $\rho\to\infty$,
|
||||
\begin{equation}
|
||||
e=\frac\rho2\left(\int \mathcal V(x)\ dx\right)(1+o(1))
|
||||
@ -595,7 +671,7 @@ Note that\-~(\ref{scattering}) can be written as $(-\Delta + \mathcal V)\varphi
|
||||
\begin{equation}\label{scattering2}
|
||||
\varphi(x) = \lim_{e\downarrow 0} K_e \mathcal{V}(x) = \lim_{e\downarrow 0}u_1(x,e)\ ,
|
||||
\end{equation}
|
||||
where $u_1$ is the first term of the iteration introduced in the previous section. It follows from Lemma~\ref{lem2} that $0 \leq \varphi(x) \leq1$ for all $x$.
|
||||
where $u_1$ is the first term of the iteration introduced in the previous section. It follows from Lemma~\ref{lem2} that $0 \leqslant \varphi(x) \leqslant1$ for all $x$.
|
||||
|
||||
\indent We now impose a mild localization hypothesis on $\mathcal{V}$: For
|
||||
$R>0$ define $\mathcal{V}_R(x) = \mathcal{V}(x)$ for $|x| > R$ and otherwise $\mathcal{V}_R(x) =0$. We require that for some $q>1$ and all sufficiently large $R$,
|
||||
@ -624,7 +700,7 @@ where $G(x) = \frac{1}{4\pi|x|}$. Since $p> 3/2$. $p' < 3$, and it is easy to de
|
||||
$G = G_1+G_2$ where $G_1 \in L^{p'}(\mathbb{R}^d)$ and $G_2 \in L^{4}(\mathbb{R}^d)$.
|
||||
Then for all $R$ sufficiently large,
|
||||
\begin{equation}
|
||||
0 \leq G\ast (\mathcal{V_R}(1- \varphi))(x) \leq (\|G_1\|_{p'} + \|G_2\|_4) R^{-q}\ .
|
||||
0 \leqslant G\ast (\mathcal{V_R}(1- \varphi))(x) \leqslant (\|G_1\|_{p'} + \|G_2\|_4) R^{-q}\ .
|
||||
\end{equation}
|
||||
For $0 < r < 1$, then for $|y| < r|x|$,
|
||||
${\displaystyle \frac{1}{1+r} \leqslant \frac{|x|}{|x-y|} \leqslant \frac{1}{1-r}}$.
|
||||
@ -640,7 +716,7 @@ Taking $|x|\to \infty$, and then $r \to 0$ proves\-~(\ref{local}).\qed
|
||||
For this reason, we do not impose the additional condition (\ref{local}) in the statement of Theorem\-~\ref{theorem:asymptotics}: Lemma\-~\ref{sctlem} reconciles the stated definition with the formula (\ref{ephi}).
|
||||
\bigskip
|
||||
|
||||
\theo{Lemma}[low density asymptotics]\label{lemma:small}
|
||||
\theoname{Lemma}{low density asymptotics}\label{lemma:small}
|
||||
If $\mathcal V$ is non-negative and integrable and $d=3$, then
|
||||
\begin{equation}
|
||||
e=2\pi\rho a\left(1+\frac{128}{15\sqrt\pi}\sqrt{\rho a^3}+o(\sqrt\rho)\right)
|
||||
@ -842,8 +918,23 @@ Algebraic decay for $u$ seems natural: by\-~(\ref{simpleq}), $u\ast u$ must deca
|
||||
This is the case if $u$ decays algebraically, but would not be so if, say, it decayed exponentially.
|
||||
\bigskip
|
||||
|
||||
\indent\underline{Proof of Theorem\-~\ref{theorem:decay}}:
|
||||
We recall that the Fourier transform of $u$ (\ref{fourieru}) satisfies\-~(\ref{hatu}):
|
||||
{\bf Proof of theorem\-~\ref{theorem:decay}}:
|
||||
We begin by proving (\ref{gendecay}) in arbitrary dimension. Recall that the first part has already been proved in Theorem~\ref{positivity} without the additional assumption on the potential. For the second part, recall that by the first remark after Theorem~\ref{theorem:existence}, $u$ is also radial, and hence $\mathcal{V}(1-u)$ is non-negative and radial. It then follows from the hypotheses on $\mathcal{V}$ that $g := Y_{4e}\ast Y_{4e}*[\mathcal{V}(1-u)]$
|
||||
satisfies
|
||||
\begin{equation}
|
||||
\int |x|^2 g(x) dx < \infty \quad{\rm and}\quad \int x g(x) d x = 0\ .
|
||||
\end{equation}
|
||||
Then, as explained in Section~\ref{sec:pos}, if $f := 2e\rho Y_{4e}\ast u$, $f - f\ast f = g\geqslant 0$, and then by \cite[Theorem 4]{CJLL20}, the second part of (\ref{gendecay}) follows. Note that if
|
||||
\begin{equation}
|
||||
u(|x|)\mathop\sim_{|x|\to\infty}\frac\alpha{|x|^m}
|
||||
\end{equation}
|
||||
for some $\alpha>0$, then the only choice of $m$ that is consistent with (\ref{gendecay}) is $m = d+1$.
|
||||
\medskip
|
||||
|
||||
We now specialize to $d=3$, and impose the additional assumption on the potential.
|
||||
\medskip
|
||||
|
||||
Recall that the Fourier transform of $u$ (\ref{fourieru}) satisfies\-~(\ref{hatu}):
|
||||
\begin{equation}
|
||||
\hat u(|k|)=\frac1\rho\left(\frac{k^2}{4e}+1-\sqrt{\left(\frac{k^2}{4e}+1\right)^2-S(|k|)}\right)
|
||||
\end{equation}
|
||||
@ -893,13 +984,13 @@ This is the case if $u$ decays algebraically, but would not be so if, say, it de
|
||||
so, denoting $b:=\min(B,1)$,
|
||||
\begin{equation}
|
||||
\mathcal U_1(|x|)\leqslant
|
||||
\frac A{4\pi}\int \frac{e^{-b(|x-y|-|y|)}}{|x-y|}\ dy
|
||||
\frac A{4\pi}\int \frac{e^{-b(|x-y|+|y|)}}{|x-y|}\ dy
|
||||
+
|
||||
\frac{e^{-(|x|-R)}}{4\pi(|x|-R)}\int\mathcal V(|y|)\ dy
|
||||
\end{equation}
|
||||
and since
|
||||
\begin{equation}
|
||||
\frac A{4\pi}\int \frac{e^{-b(|x-y|-|y|)}}{|x-y|}\ dy
|
||||
\frac A{4\pi}\int \frac{e^{-b(|x-y|+|y|)}}{|x-y|}\ dy
|
||||
=
|
||||
\frac{Ae^{-b|x|}}{4b^2}(b|x|+1)
|
||||
\end{equation}
|
||||
@ -1076,7 +1167,7 @@ This is the case if $u$ decays algebraically, but would not be so if, say, it de
|
||||
which implies\-~(\ref{decaydS}) in this case.
|
||||
\bigskip
|
||||
|
||||
\subpoint We have thus proved that $S$ is analytic in $H_\tau$, which implies that the singularities of $\widehat{\mathcal U}_2$ in $H_\tau$ all come from the branch points of $\sqrt F$.
|
||||
\subpoint We have thus proved that $S$ is analytic in $H_\tau$, which implies that the singularities of $\widehat{\mathcal U}_2$ in $H_\tau$ all come from the branch points of $\sqrt{F(|k|)}$ with $F(|k|):=(\frac{k^2}{4e}+1)^2-S(|k|)$.
|
||||
For $\kappa\in\mathbb R$,
|
||||
\begin{equation}
|
||||
|S(\kappa)|\leqslant 1
|
||||
@ -1086,7 +1177,8 @@ This is the case if $u$ decays algebraically, but would not be so if, say, it de
|
||||
F(\kappa)\geqslant \frac{\kappa^2}{2e}
|
||||
.
|
||||
\end{equation}
|
||||
Therefore, since $F$ is analytic in a finite strip around the real axis, $F$ cannot have any roots in the vicinity of the real axis, except at $0$, so the only branch point of $\sqrt F$ near the real axis is $0$.
|
||||
Therefore, since $F$ is analytic in a strip around the real axis, there exists an open set containing the real axis in which $F$ has one and only one root, at $0$.
|
||||
Thus the only branch point of $\sqrt F$ on the real axis is $0$.
|
||||
Thus, $\widehat{\mathcal U}_2$ is analytic in $H_\tau$.
|
||||
\bigskip
|
||||
|
||||
@ -1253,6 +1345,14 @@ There are some subtleties to taking this limit, which are explained in\-~\cite{L
|
||||
Defining $u:=1-g_\infty^{(2)}$, the equation for $u$ is\-~\cite[(3.29)]{Li63}.
|
||||
After a few extra reasonable approximations, this equation reduces to\-~(\ref{simpleq}).
|
||||
The equation for the energy\-~(\ref{energy}) is simply the $N\to\infty$ limit of\-~(\ref{energyg}).
|
||||
\bigskip
|
||||
|
||||
\indent In particular, $u$ is related to the correlation function $g^{(2)}$ of the Bose gas.
|
||||
The condition\-~(\ref{con1}) that $u(x)\leqslant1$ is necessary to ensure that $g^{(2)}(x)\geqslant0$.
|
||||
However, $u(x)\geqslant0$ is not a physical requirement, as $g^{(2)}(x)$ could, in principle, be $>1$
|
||||
for some $x$.
|
||||
|
||||
|
||||
|
||||
\subsection{Numerical comparison}\label{subsec:numerics}
|
||||
\indent One of the motivations for studying the simple equation is that it provides a simple tool to approximate the ground state energy of the Bose gas.
|
||||
@ -1319,35 +1419,6 @@ Numerically, it seems quite clear that $\rho e(\rho)$ is convex, see figure\-~\r
|
||||
\label{fig:convexity}
|
||||
\end{figure}
|
||||
|
||||
\point{\bf Solutions with negative values}.
|
||||
In this paper, we solved the simple equation for functions $u$ that satisfy\-~(\ref{con1}).
|
||||
The condition that $u(x)\leqslant 1$ comes from physical considerations, and we are gratified that our simple equation has this property automatically, see\-~(\ref{con1}).
|
||||
The correlation function $g_N^{(2)}$ defined in\-~(\ref{g}) is non-negative, which means that $u(x)\leqslant 1$.
|
||||
However, one may wonder whether the condition $u(x)\geqslant0$ must be imposed, or whether it may follow from the simple equation.
|
||||
In principle, Theorem\-~\ref{theorem:existence} does not exclude the existence of other solutions of\-~(\ref{simpleq})-(\ref{energy}) in which $u(x)<0$ for some $x\in\mathbb R^d$.
|
||||
Proving that there are no such solutions is another interesting open problem.
|
||||
It seems rather unlikely that such solutions exist: defining
|
||||
\begin{equation}
|
||||
\omega(x):=G_eu(x)
|
||||
\end{equation}
|
||||
(\ref{simpleq}) becomes
|
||||
\begin{equation}
|
||||
\omega(x)=
|
||||
G_e^2(1-u)\mathcal V(x)+2e\rho\omega\ast\omega(x)
|
||||
\end{equation}
|
||||
but, $u(x)\leqslant 1$, so $G_e^2(1-u)\mathcal V(x)\geqslant 0$ and
|
||||
\begin{equation}
|
||||
\omega(x)\geqslant 2e\rho\omega\ast\omega(x)
|
||||
.
|
||||
\end{equation}
|
||||
In particular,
|
||||
\begin{equation}
|
||||
\{x:\ \omega(x)<0\}\subset
|
||||
\{x:\ \omega\ast\omega(x)<0\}
|
||||
\end{equation}
|
||||
which does not seem to be possible, although a proof that it is not so has eluded us.
|
||||
\bigskip
|
||||
|
||||
\point {\bf Solution of the full equation}.
|
||||
The simple equation\-~(\ref{simpleq}) is actually a simplified version of an equation that should approximate the Bose gas more accurately\-~\cite{Li63}:
|
||||
\begin{equation}
|
||||
@ -1390,16 +1461,19 @@ We hope to study this equation numerically in a later publication.
|
||||
|
||||
\bibitem[Bo47]{Bo47}N. Bogolubov - {\it On the theory of superfluidity}, Journal of Physics (USSR), volume\-~11, number\-~1, pages\-~23-32 (translated from the Russian Izv.Akad.Nauk Ser.Fiz, volume\-~11, pages\-~77-90), 1947.\par\medskip
|
||||
|
||||
\bibitem[CHe]{CHe}E.\-~Carlen, M.\-~Holzmann, I.\-~Jauslin, E.H.\-~Lieb, in preparation.\par\medskip
|
||||
\bibitem[CJLL20]{CJLL20}E.A.\-~Carlen, I.\-~Jauslin, E.H.\-~Lieb, M.\-~Loss - {\it On the convolution inequality $f>f*f$}, 2020,\par\penalty10000
|
||||
arxiv:{\tt\color{blue}\href{http://arxiv.org/abs/2002.04184}{2002.04184}}.\par\medskip
|
||||
|
||||
\bibitem[Dy57]{Dy57}F.J. Dyson - {\it Ground-State Energy of a Hard-Sphere Gas}, Physical Review, volume\-~106, issue\-~1, pages\-~20-26, 1957,\par\penalty10000
|
||||
doi:{\tt\color{blue}\href{http://dx.doi.org/10.1103/PhysRev.106.20}{10.1103/PhysRev.106.20}}.\par\medskip
|
||||
|
||||
\bibitem[FS19]{FS19}S.\-~Fournais, J.P.\-~Solovej - {\it The energy of dilute Bose gases}, 2019,\par\penalty10000
|
||||
arxiv:{\tt\href{http://arxiv.org/abs/1904.06164}{1904.06164}}.\par\medskip
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||||
arxiv:{\tt\color{blue}\href{http://arxiv.org/abs/1904.06164}{1904.06164}}.\par\medskip
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||||
|
||||
\bibitem[Ga99]{Ga99}G. Gallavotti - {\it Statistical mechanics, a short treatise}, Springer, 1999.\par\medskip
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||||
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||||
\bibitem[HJL]{HJL}E.A.\-~Carlen, M.\-~Holzmann, I.\-~Jauslin, E.H.\-~Lieb, in preparation.\par\medskip
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||||
|
||||
\bibitem[LHY57]{LHY57}T.D. Lee, K. Huang, C.N. Yang - {\it Eigenvalues and Eigenfunctions of a Bose System of Hard Spheres and Its Low-Temperature Properties}, Physical Review, volume\-~106, issue\-~6, pages\-~1135-1145, 1957,\par\penalty10000
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doi:{\tt\color{blue}\href{http://dx.doi.org/10.1103/PhysRev.106.1135}{10.1103/PhysRev.106.1135}}.\par\medskip
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||||
@ -1430,8 +1504,8 @@ doi:{\tt\color{blue}\href{http://dx.doi.org/10.1007/s00220-012-1576-y}{10.1007/s
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||||
\bibitem[YY09]{YY09}H. Yau, J. Yin - {\it The Second Order Upper Bound for the Ground Energy of a Bose Gas}, Journal of Statistical Physics, volume\-~136, issue\-~3, pages\-~453-503, 2009,\par\penalty10000
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doi:{\tt\color{blue}\href{http://dx.doi.org/10.1007/s10955-009-9792-3}{10.1007/s10955-009-9792-3}}, arxiv:{\tt\color{blue}\href{http://arxiv.org/abs/0903.5347}{0903.5347}}.\par\medskip
|
||||
|
||||
|
||||
\end{thebibliography}
|
||||
|
||||
|
||||
|
||||
\end{document}
|
||||
|
||||
19
Changelog
Normal file
19
Changelog
Normal file
@ -0,0 +1,19 @@
|
||||
v0.1:
|
||||
|
||||
* Added: Theorem on positivity of solutions.
|
||||
|
||||
* Added: Reference to [CJLL20]
|
||||
|
||||
* Added: Theorem on decay rate is now more general.
|
||||
|
||||
* Fixed: Clarified the discussion in point 2-2 of the proof of the theorem
|
||||
on decay.
|
||||
|
||||
* Removed: open problem about positivity of solutions.
|
||||
|
||||
* Fixed: Format of named theorems.
|
||||
|
||||
* Fixed: Minor formatting fixes.
|
||||
|
||||
* Fixed: In proof of decay: indenting error.
|
||||
|
||||
@ -1,42 +0,0 @@
|
||||
\bibitem[Bo47]{Bo47}N. Bogolubov - {\it On the theory of superfluidity}, Journal of Physics (USSR), volume\-~11, number\-~1 , pages\-~23-32 (translated from the Russian Izv.Akad.Nauk Ser.Fiz, volume\-~11, pages\-~77-90), 1947.\par\medskip
|
||||
|
||||
\bibitem[CHe]{CHe}E.\-~Carlen, M.\-~Holzmann, I.\-~Jauslin, E.H.\-~Lieb, in preparation.\par\medskip
|
||||
|
||||
\bibitem[Dy57]{Dy57}F.J. Dyson - {\it Ground-State Energy of a Hard-Sphere Gas}, Physical Review, volume\-~106, issue\-~1, pages\-~20-26, 1957,\par\penalty10000
|
||||
doi:{\tt\href{http://dx.doi.org/10.1103/PhysRev.106.20}{10.1103/PhysRev.106.20}}.\par\medskip
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||||
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||||
\bibitem[FS19]{FS19}S.\-~Fournais, J.P.\-~Solovej - {\it The energy of dilute Bose gases}, 2019,\par\penalty10000
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||||
arxiv:{\tt\href{http://arxiv.org/abs/1904.06164}{1904.06164}}.\par\medskip
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||||
|
||||
\bibitem[Ga99]{Ga99}G. Gallavotti - {\it Statistical mechanics, a short treatise}, Springer, 1999.\par\medskip
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||||
|
||||
\bibitem[LHY57]{LHY57}T.D. Lee, K. Huang, C.N. Yang - {\it Eigenvalues and Eigenfunctions of a Bose System of Hard Spheres and Its Low-Temperature Properties}, Physical Review, volume\-~106, issue\-~6, pages\-~1135-1145, 1957,\par\penalty10000
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||||
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||||
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|
||||
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||||
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||||
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|
||||
|
||||
\bibitem[LY01]{LY01}E.H. Lieb, J. Yngvason - {\it The Ground State Energy of a Dilute Two-Dimensional Bose Gas}, Journal of Statistical Physics, volume\-~103, issue\-~3-4, pages\-~509-526, 2001,\par\penalty10000
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|
||||
|
||||
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||||
|
||||
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||||
|
||||
\bibitem[YY09]{YY09}H. Yau, J. Yin - {\it The Second Order Upper Bound for the Ground Energy of a Bose Gas}, Journal of Statistical Physics, volume\-~136, issue\-~3, pages\-~453-503, 2009,\par\penalty10000
|
||||
doi:{\tt\href{http://dx.doi.org/10.1007/s10955-009-9792-3}{10.1007/s10955-009-9792-3}}, arxiv:{\tt\href{http://arxiv.org/abs/0903.5347}{0903.5347}}.\par\medskip
|
||||
|
||||
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