Update to v0.1: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.

3 files changed, 515 insertions, 464 deletions

diff --git a/Carlen_Jauslin_Lieb_2019.tex b/Carlen_Jauslin_Lieb_2019.tex
index 9e020a5..3676d8a 100644
--- a/Carlen_Jauslin_Lieb_2019.tex
+++ b/Carlen_Jauslin_Lieb_2019.tex
@@ -13,22 +13,22 @@
 \hbox{}
 \hfil{\bf\LARGE Analysis of a simple equation for the
 \par
-\vskip10pt
+\vskip8pt
 \hfil ground state energy of the Bose gas
 }
-\vfill
+\vskip10pt
 
-\hfil{\bf\large Eric Carlen}\par
+\hfil{\bf Eric A. Carlen}\par
 \hfil{\it Department of Mathematics, Rutgers University}\par
 \hfil{\tt\color{blue}\href{mailto:carlen@rutgers.edu}{carlen@rutgers.edu}}\par
 \vskip20pt
 
-\hfil{\bf\large Ian Jauslin}\par
+\hfil{\bf Ian Jauslin}\par
 \hfil{\it Department of Physics, Princeton University}\par
 \hfil{\tt\color{blue}\href{mailto:ijauslin@princeton.edu}{ijauslin@princeton.edu}}\par
 \vskip20pt
 
-\hfil{\bf\large Elliott H. Lieb}\par
+\hfil{\bf Elliott H. Lieb}\par
 \hfil{\it Departments of Mathematics and Physics, Princeton University}\par
 \hfil{\tt\color{blue}\href{mailto:lieb@princeton.edu}{lieb@princeton.edu}}\par
 
@@ -102,15 +102,39 @@ provides a useful and illuminating route to the computation of the properties of
 However, the convolution nonlinearity in\-~(\ref{simpleq}) makes it non-local, and very different from\-~(\ref{simpleq2}).
 \bigskip
 
-\indent As explained in\-~\cite{Li63} the solutions of physical interest are integrable, and satisfy
+\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 $u \geqslant 0$:
+
+\theoname{Theorem}{Positivity}\label{positivity}   Suppose that $\mathcal{V}$ is  non-negative and integrable and that $u$ is an integrable solution of 
+(\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
+\begin{equation}\label{slow}
+\int |x|u(x)d x = \infty \ .
+\end{equation}
+Thus, any physical solutions of (\ref{simpleq})-(\ref{energy}) must necessarily satisfy the {\em pair} of inequalities
 \begin{equation}
- 0 \leqslant u(x) \leqslant 1 \quad{\rm for \ all}\ x \ .
- \label{con1}
+  0  \leqslant u(x) \leqslant 1 \quad{\rm for \ all}\ x \ .
+  \label{con1}
 \end{equation}
-We shall see that any non-negative solution automatically satisfies this upper bound.
+\endtheo
+
 \bigskip
+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 $L^1(\mathbb R^d)$.
+While $u(x)\leqslant 1$ is a physical requirement, $u(x)\geqslant0$ is not, see section\-~\ref{sec:bosegas} for details.
+\bigskip
+
+The converse of Theorem\-~\ref{positivity} also holds, as stated in the following theorem.
 
-\indent Before stating our main theorems, we make a few observations.
+\theo{Theorem}\label{theorem:leq1}
+  Let $\mathcal V\in L^1(\mathbb{R}^d)\cap L^p(\mathbb{R}^d)$, $p>\max\{\frac d2,1\}$, be non-negative.
+  If $u$ is an integrable solution of 
+(\ref{simpleq})-(\ref{energy}) such that  $u(x) \geqslant 0$ for all $x$, then $u(x) \leqslant 1$ for all $x$.
+\endtheo
+
+
+{\bf Remark}: We have thus proved that $u\geqslant0$ if and only if $u\leqslant1$.
+This, in principle, leaves the door open to solutions that are sometimes $>1$ and sometimes $<0$, though we do not believe such solutions exist.
+\bigskip
+
+\indent Before stating our main theorems, we make a few observations. 
 \medskip
 
 \point The system\-~(\ref{simpleq})-(\ref{energy}) is actually equivalent to\-~(\ref{simpleq}) and the constraint
@@ -213,10 +237,10 @@ In particular, this shows that the system\-~(\ref{simpleq})-(\ref{energy}) does
 In fact, as is stated in the following theorem, $\rho$ and $e$ are constrained to be related by a functional equation.
 \bigskip
 
-\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. 
+\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. 
 Then there is a constructively defined continuous function $\rho(e)$ on $(0,\infty)$ such that 
  $\lim_{e\to 0}\rho(e) = 0$ and $\lim_{e\to \infty} \rho(e) = \infty$ and such that for any $e\geqslant 0$ and $\rho = \rho(e)$, 
- 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}).
+ 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}).
 \endtheo
 \bigskip
 
@@ -236,7 +260,7 @@ monotone increasing function. In that case, the functional relation could be inv
 \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.
 \bigskip
 
-\theo{Theorem}[asymptotics of the energy for $d=3$]\label{theorem:asymptotics}
+\theoname{Theorem}{asymptotics of the energy for $d=3$}\label{theorem:asymptotics}
  Consider the case $d=3$.
  Let $\mathcal V$ be non-negative, integrable and square-integrable. Then, for each 
  $\rho>0$ 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 $\rho$).
@@ -263,17 +287,20 @@ monotone increasing function. In that case, the functional relation could be inv
 \end{itemize}
 \bigskip
 
-\theo{Theorem}[decay of $u$ in $d=3$]\label{theorem:decay}
- For $d=3$, if $\mathcal V$ is non-negative, square-integrable, spherically symmetric (that is, $\mathcal V(x)=\mathcal V(|x|)$), and, for $|x|>R$,
- \begin{equation}
- \mathcal V(|x|)\leqslant Ae^{-B|x|}
- \label{expdecay}
- \end{equation}
- for some $A,B>0$ then there exists $\alpha>0$ such that
- \begin{equation}
- u(|x|)\mathop\sim_{|x|\to\infty}\frac\alpha{|x|^4}
- .
- \end{equation}
+\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
+\begin{equation}\label{gendecay}
+\int |x| u(x) dx = \infty \quad{\rm and}\quad \int |x|^r u(x) dx < \infty \quad{\rm for \ all} \ 0 < r < 1\ .
+\end{equation}
+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.
+For $d=3$, if $\mathcal V$ is non-negative, square-integrable, spherically symmetric (that is, $\mathcal V(x)=\mathcal V(|x|)$), and, for $|x|>R$,
+  \begin{equation}
+    \mathcal V(|x|)\leqslant Ae^{-B|x|}
+    \label{expdecay}
+  \end{equation}
+  for some $A,B>0$ then there exists $\alpha>0$ such that
+  \begin{equation}
+    u(x)\mathop\sim_{|x|\to\infty}\frac\alpha{|x|^4}   .
+  \end{equation}
 \endtheo
 \bigskip
 
@@ -294,11 +321,57 @@ The simple equation\-~(\ref{simpleq}) is actually an approximation of a richer e
 \bigskip
 
 \indent The paper is organized as follows.
-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.
+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}.
 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.
 In section\-~\ref{sec:open} we discuss a few open problems and extensions.
 
-\section{Proof of Theorem\-~\expandonce{\ref{theorem:existence}}}\label{sec:existence}
+\section{Proof of Theorem \expandonce{\ref{positivity}}}\label{sec:pos}
+
+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 $\int |x| u(x) dx = \infty$.   
+
+Our starting point is the form of (\ref{simpleq}) given in (\ref{simpleq3}). For an integrable solution $u$, define
+\begin{equation}\label{pos1}
+f :=  2e\rho Y_{4e}*u\ .
+\end{equation}
+If (\ref{energy}) is satisfied,  then
+\begin{equation}\label{pos2}
+\int f dx = \frac 12\ .
+\end{equation}
+and (\ref{simpleq3}) can be written as
+\begin{equation}\label{pos3}
+u = Y_{4e}*(\mathcal V (1- u)) +  f\ast u\ .
+\end{equation}
+
+\theo{Lemma}
+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 $x$.
+\endtheo
+
+
+
+{\bf Proof} Since $Y_{4e}*(\mathcal V (1- u(x)) \geqslant 0$, it follows that 
+\begin{equation}
+  u_- \leqslant  (f*u)_- =  (f*u_+ - f*u_-)_-  \leqslant f*u_-\ .
+\end{equation}
+Integrating, we find
+${\displaystyle \int u_- {\rm d}x \leqslant \frac12 \int u_-{\rm d}x}$,
+and this implies that $u_- = 0$. \qed
+
+
+{\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 
+\begin{equation}\label{pos5}
+f(x) \geqslant f\ast f(x)
+\end{equation}
+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 
+\begin{equation}
+  \int |x| f(x)\ dx = \infty\ .
+\end{equation}
+However, 
+\begin{equation}
+  \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 \ .
+\end{equation}
+Then since $\lim_{x\to \infty}\left(4e|x|^{-1} Y_{4e} \ast  |x|\right) = 1$, (\ref{slow}) follows. \qed
+
+\section{Proof of Theorems \expandonce{\ref{theorem:leq1}} and \expandonce{\ref{theorem:existence}}}\label{sec:existence}
 
 \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.
 As it turns out, only the latter one
@@ -311,7 +384,7 @@ Starting with $u_0(x) = 0$, define
 \begin{equation}
  u_n(x) = \Phi(u_{n-1})(x)
 \end{equation}
-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}). 
+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}). 
 
 \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
 \begin{equation}
@@ -339,7 +412,7 @@ We proceed by induction. By definition, $u_0 =0$ and $\rho_0 = 2e\left(\int_{\ma
 $u_1 = K_e\mathcal V \geqslant u_0$ and $\rho_1 = 2e\left( \int \mathcal V(1- K_e\mathcal{V})dx \right)^{-1}$.
  As noted in the discussion between\-~(\ref{con4}) and\-~(\ref{con4B}), 
 \begin{equation}
-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.
+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.
 \end{equation}
 Since $t\mapsto t^{-1}$ is monotone decreasing on $(0,\infty)$, this shows that $\rho_1 > \rho_0$, and that\-~(\ref{simple17}) holds for $n=1$. 
 \bigskip
@@ -359,7 +432,7 @@ Integrating both sides of $u_{n+1} = G_e \mathcal {V}(1- u_{n+1}) + 2e \rho_n G_
 \end{equation}
 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 
 \begin{equation}
- 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\ .
+ 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\ .
 \end{equation}
 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}) $.
 This proves\-~(\ref{simple17}) for $n+1$, and shows that 
@@ -381,7 +454,7 @@ and then, as before, $\rho_{n+1}\geqslant \rho_n$.
 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 $n=1$. 
 \bigskip
 
-\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 
+\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 
 $\mathcal{V} + 2e\rho_n u_n\ast u_n \in L^p(\mathbb{R}^d)$. Therefore, 
 $u_{n+1}= K_e(\mathcal{V} + 2e\rho_n u_n\ast u_n) \in W^{2,p}(\mathbb{R}^d)$. It follows as before that $u_{n+1}$ is continuous and vanishing at infinity, and in particular, bounded, and 
 \begin{eqnarray*}
@@ -407,9 +480,9 @@ Then both limits exist, $u\in W^{2,p}(\mathbb{R}^d)$ and $u$ satisfies\-~(\ref{s
 {\bf Proof:} 
 By Lemma~\ref{lem1}, both limits exist, and by\-~(\ref{simple17}), $\rho(e) \leqslant \left(\int_{\mathbb{R}^d} K_e\mathcal{V}dx\right)^{-1}$.
 Also by Lemma~\ref{lem1}, $\int_{\mathbb{R}^d} \leqslant \frac{1}{2e}\int_{\mathbb{R}^d}\mathcal{V}(x)dx$,
-$u$ 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
+$u$ 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
 \begin{equation}
- \|u\ast u - u_n\ast u_n\|_p \leq \|u_n\|_1\|u_n -u\|_p + \leq \|u\|_1\|u_n -u\|_p\ .
+ \|u\ast u - u_n\ast u_n\|_p \leqslant \|u_n\|_1\|u_n -u\|_p + \leqslant \|u\|_1\|u_n -u\|_p\ .
 \end{equation}
 Therefore, $\lim_{n\to\infty}(\mathcal{V} +2e\rho_n(e) u_n\ast u_n) = (\mathcal{V} +2e\rho(e) u\ast u)$
 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
@@ -417,7 +490,7 @@ $L^p(\mathbb{R}^d)$. It now follows that $u = K_e(\mathcal{V} +2e\rho(e) u\ast u
 By remarks made above, this means that $u$ satisfies\-~(\ref{simpleq}) and\-~(\ref{energy}). \qed
 \bigskip
 
-\theo{Lemma}\label{lem3}
+\theo{Lemma}\label{lem4}
  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 $\rho = \rho(e)$.
  Moreover, there does not exist such any such solution when $\rho \neq \rho(e)$. 
 \endtheo
@@ -434,7 +507,7 @@ By remarks made above, this means that $u$ satisfies\-~(\ref{simpleq}) and\-~(\r
  \begin{equation}
  \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))
  \end{equation}
- Since $u_{n-1} =0$, the positivity of $\tilde u$ implies the positivity of $\tilde u(x) - u_1(x)$.
+ Since $u_0 =0$, the positivity of $\tilde u$ implies the positivity of $\tilde u(x) - u_1(x)$.
  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.
  This proves that both $\tilde\rho\geqslant\rho$ and $\tilde u\geqslant u$. However, integrating both sides of the latter inequality yields 
  \begin{equation}
@@ -445,7 +518,7 @@ By remarks made above, this means that $u$ satisfies\-~(\ref{simpleq}) and\-~(\r
 \qed
 \bigskip 
  
-\theo{Lemma}\label{lem4}
+\theo{Lemma}\label{lem5}
  The function $\rho(e)$ is continuous on $(0,\infty)$, with
  \begin{equation}
  \lim_{e\to 0}\rho(e) = 0
@@ -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)
@@ -842,355 +918,371 @@ 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}):
- \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}
- where $S$ was defined in\-~(\ref{S}):
- \begin{equation}
- S(|k|):=\frac\rho{2e}\int e^{ikx}(1-u(|x|))\mathcal V(|x|)\ dx
- .
- \end{equation}
- We split
- \begin{equation}
- \hat u(|k|)=\widehat{\mathcal U}_1(|k|)+\widehat{\mathcal U}_2(|k|)
- \label{hatusplit}
- \end{equation}
- with
- \begin{equation}
- \widehat{\mathcal U}_1(|k|):=\frac{2eS(|k|)}{\rho(1+k^2)}
- \end{equation}
- so that, taking the large $|k|$ limit in\-~(\ref{hatu}),
- \begin{equation}
- \widehat{\mathcal U}_2(|k|)=O(|k|^{-4}S^2(|k|))
- \end{equation}
- so $\widehat{\mathcal U}_2$ is integrable.
- \bigskip
-
- \point{\bf Decay of $\mathcal U_1$}. We first show that
- \begin{equation}
- \mathcal U_1(|x|):=\frac1{(2\pi)^3}\int e^{-ikx}\widehat{\mathcal U}_1(|k|)\ dk
- \end{equation}
- decays exponentially in $|x|$. We have
- \begin{equation}
- \mathcal U_1(|x|)=(-\Delta+1)^{-1}(1-u(|x|))\mathcal V(|x|)
- =Y_1\ast((1-u)\mathcal V)(|x|)
- \end{equation}
- with
- \begin{equation}
- Y_1(|x|):=\frac{e^{-|x|}}{4\pi|x|}
- .
- \end{equation}
- Therefore, by\-~(\ref{expdecay}),
- \begin{equation}
- \mathcal U_1(|x|)
- \leqslant
- \frac A{4\pi}\int_{|y|>R} \frac{e^{-|x-y|-B|y|}}{|x-y|}\ dy
- +
- \frac 1{4\pi}\int_{|y|<R} \frac{e^{-|x-y|}}{|x-y|}\mathcal V(|y|)\ dy
- \end{equation}
- 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{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{Ae^{-b|x|}}{4b^2}(b|x|+1)
- \end{equation}
- we have
- \begin{equation}
- \mathcal U_1(|x|)\leqslant
- \frac{Ae^{-b|x|}}{4b^2}(b|x|+1)
- +
- \frac{e^{-(|x|-R)}}{4\pi(|x|-R)}\int\mathcal V(|y|)\ dy
- .
- \label{U1}
- \end{equation}
- \bigskip
-
- \point{\bf Analyticity of $\mathcal U_2$}.
- We now turn to
- \begin{equation}
- \mathcal U_2(|x|):=\frac1{(2\pi)^3}\int e^{-ikx}\widehat{\mathcal U}_2(|k|)\ dk
- =\frac1{4i\pi^2|x|}\sum_{\eta=\pm}\eta\int_0^\infty e^{i\eta\kappa|x|}\kappa\widehat{\mathcal U}_2(\kappa)\ d\kappa
- .
- \label{U2}
- \end{equation}
- We start by proving some analytic properties of $\widehat{\mathcal U}_2$, which, we recall from\-~(\ref{hatu}) and\-~(\ref{hatusplit}), is
- \begin{equation}
- \widehat{\mathcal U}_2(|k|)=\frac1\rho\left(\frac{k^2}{4e}+1-\sqrt{\left(\frac{k^2}{4e}+1\right)^2-S(|k|)}-\frac{2eS(|k|)}{1+k^2}\right)
- .
- \end{equation}
- \bigskip
-
- \subpoint First of all, $S$ is analytic in a strip about the real axis:
- \begin{equation}
- S(\kappa)=4\pi\int_0^\infty \mathrm{sinc}(\kappa r)r^2\mathcal V(r)(1-u(r))\ dr
- ,\quad
- \mathrm{sinc}(\xi):=\frac{\sin(\xi)}\xi
- \end{equation}
- so
- \begin{equation}
- \partial^nS(\kappa)=4\pi\int_0^\infty \partial^n\mathrm{sinc}(\kappa r)r^{n+2}\mathcal V(r)(1-u(r))\ dr
- .
- \end{equation}
- We will show that if $\mathcal Im(\kappa)\leqslant \frac B2$ (the factor $\frac12$ can be improved to any factor that is $<1$, but this does not matter here), then there exists $C>0$ which only depends on $A$ and $B$ such that
- \begin{equation}
- |\partial^nS(\kappa)|\leqslant n!C^n
- .
- \label{decaydS}
- \end{equation}
- As a consequence, $S$ is analytic in a strip around the real line of height $\frac B2$.
- In particular, if we define the strip
- \begin{equation}
- H_\tau:=\{z:\ |\mathcal Im(z)|\leqslant r^{-\tau},\ \mathcal Re(z)>0\}
- \end{equation}
- with $0<\tau<1$, and take
- \begin{equation}
- r>\left(\frac B2\right)^{-\frac1\tau}
- \end{equation}
- then $S$ is analytic in $H_\tau$.
- \bigskip
-
- \bigskip
-
- \subsubpoint We now prove\-~(\ref{decaydS}).
- We first treat the case $|\kappa|\leqslant\frac B2$. We have
- \begin{equation}
- \mathrm{sinc}(\xi)=\sum_{p=0}^\infty\frac{(-1)^p\xi^{2p}}{(2p+1)!}
- \end{equation}
- so
- \begin{equation}
- \partial^n\mathrm{sinc}(\xi)=\sum_{p=\lceil\frac n2\rceil}^\infty\frac{(-1)^p\xi^{2p-n}}{(2p+1)(2p-n)!}
- .
- \end{equation}
- Therefore
- \begin{equation}
- |\partial^n\mathrm{sinc}(\xi)|\leqslant
- \sum_{p=\lceil\frac n2\rceil}^\infty\frac{|\xi|^{2p-n}}{(2p-n)!}
- \leqslant \cosh(|\xi|)
- .
- \end{equation}
- Thus,
- \begin{equation}
- |\partial^n S(\kappa)|\leqslant 4\pi\int_0^\infty \cosh(|\kappa|r)r^{n+2}\mathcal V(r)(1-u(r))\ dr
- \end{equation}
- so, by\-~(\ref{expdecay}),
- \begin{equation}
- |\partial^n S(\kappa)|\leqslant
- 4A\pi\int_R^\infty \cosh(|\kappa|r)r^{n+2}e^{-Br}\ dr
- +
- 4\pi\int_0^R \cosh(|\kappa|r)r^{n+2}\mathcal V(r)\ dr
- \end{equation}
- and
- \begin{equation}
- |\partial^n S(\kappa)|\leqslant
- 8A\pi\int_0^\infty r^{n+2}e^{-(B-|\kappa|)r}\ dr
- +
- 8\pi e^{|\kappa|R}R^{n}\int r^2\mathcal V(r)\ dr
- \end{equation}
- which, if $|\kappa|\leqslant \frac B2$, implies that
- \begin{equation}
- 8A\pi\int_0^\infty r^{n+2}e^{-(B-|\kappa|)r}\ dr
- \leqslant
- 8A\pi\int_0^\infty r^{n+2}e^{-\frac B2r}\ dr
- =
- \frac{2^{n+6}A\pi}{B^{n+3}}(n+2)!
- \end{equation}
- and
- \begin{equation}
- 8\pi e^{|\kappa|R}R^{n+2}\int\mathcal V(r)\ dr
- \leqslant
- 8\pi e^{\frac B2R}R^{n}\int r^2\mathcal V(r)\ dr
- \end{equation}
- which implies\-~(\ref{decaydS}) in this case.
- \bigskip
-
- \subsubpoint We now turn to $|\kappa|\geqslant \frac B2$:
- \begin{equation}
- \partial^n\mathrm{sinc}(\xi)=\sum_{p=0}^n{n\choose p}\partial^p\sin(\xi)\frac{(n-p)!(-1)^{n-p}}{\xi^{n-p+1}}
- \end{equation}
- so
- \begin{equation}
- |\partial^n\mathrm{sinc}(\xi)|\leqslant 2e^{\mathcal Im(\xi)}\sum_{p=0}^n\frac{n!}{p!}|\xi|^{-(n-p+1)}
- .
- \end{equation}
- Therefore,
- \begin{equation}
- |\partial^nS(\kappa)|\leqslant
- 8\pi \sum_{p=0}^n\frac{n!}{p!|\kappa|^{n-p+1}}
- \int_0^\infty e^{\mathcal Im(\kappa)r}r^{p+1}\mathcal V(r)(1-u(r))\ dr
- \end{equation}
- so, by\-~(\ref{expdecay}),
- \begin{equation}
- |\partial^nS(\kappa)|\leqslant
- \sigma_1+\sigma_2
- \end{equation}
- with
- \begin{equation}
- \sigma_1:=8A\pi\sum_{p=0}^n\frac{n!}{p!|\kappa|^{n-p+1}}
- \int_R^\infty r^{p+1}e^{-(B-\mathcal Im(\kappa))r}\ dr
- \end{equation}
- and
- \begin{equation}
- \sigma_2:=8\pi\sum_{p=0}^n\frac{n!}{p!|\kappa|^{n-p+1}}
- \int_0^R r^{p+1}e^{\mathcal Im(\kappa)r}\mathcal V(r)\ dr
- .
- \end{equation}
- Furthermore,
- \begin{equation}
- \sigma_1
- =
- 8A\pi n!
- \sum_{p=0}^n\frac{p+1}{(B-\mathcal Im(\kappa))^{p+2}|\kappa|^{n-p+1}}
- \end{equation}
- so, as long as $|\kappa|\geqslant \frac12B$ and $\mathcal Im(\kappa)\leqslant\frac12B$,
- \begin{equation}
- \sigma_1
- \leqslant
- \frac{2^{n+6}A\pi}{B^{n+3}}n!\sum_{p=0}^n(p+1)
- =
- \frac{2^{n+5}A\pi}{B^{n+3}}(n+2)!
- .
- \end{equation}
- In addition,
- \begin{equation}
- \sigma_2
- \leqslant
- 8\pi\sum_{p=0}^n\frac{n!}{p!|\kappa|^{n-p+1}}R^{p-1}e^{\mathcal Im(\kappa)R}\int_0^R r^2\mathcal V(r)\ dr
- \end{equation}
- so
- \begin{equation}
- \sigma_2
- \leqslant
- 8\pi\sum_{p=0}^n\frac{n!2^{n-p+1}}{p!B^{n-p+1}}R^{p-1}e^{\mathcal Im(\kappa)R}\int_0^R r^2\mathcal V(r)\ dr
- \leqslant
- \frac{2^{n+4}\pi}{RB^{n+1}} n!e^{RB}\int_0^R r^2\mathcal V(r)\ dr
- \end{equation}
- 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 $\sqrt F$.
- For $\kappa\in\mathbb R$,
- \begin{equation}
- |S(\kappa)|\leqslant 1
- \end{equation}
- so, for $\kappa\in\mathbb R$,
- \begin{equation}
- 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 $0$.
- Thus, $\widehat{\mathcal U}_2$ is analytic in $H_\tau$.
- \bigskip
-
- \point{\bf Decay of $\mathcal U_2$}. We deform the integral to the path
- \begin{equation}
- \{i\eta y,\ 0<y<|x|^{-\tau}\}\cup\{i\eta |x|^{-\tau}+y,\ y>0\}
- \end{equation}
- and find
- \begin{equation}
- \int_0^\infty e^{i\eta\kappa|x|}\kappa\widehat{\mathcal U}_2(\kappa)\ d\kappa
- =
- I_1+I_2
- \label{I12}
- \end{equation}
- with
- \begin{equation}
- I_1:=
- -\int_0^{|x|^{-\tau}}e^{-y|x|}y\widehat{\mathcal U}_2(i\eta y)\ dy
- \end{equation}
- and
- \begin{equation}
- I_2:=
- e^{-|x|^{1-\tau}}
- \int_0^\infty e^{i\eta y|x|}(i\eta|x|^{-\tau}+y)\widehat{\mathcal U}_2(i\eta|x|^{-\tau}+y)\ dy
- .
- \end{equation}
- \bigskip
-
- \subpoint We first estimate $I_1$.
- We expand $S$:
- \begin{equation}
- S(\kappa)=1-\beta \kappa^2+O(|\kappa|^4)
- \end{equation}
- with $\beta>0$ (since $S$ is analytic and symmetric, and $|S(|k|)|\leqslant 1$).
- Therefore, $y\mapsto\widehat{\mathcal U}_2(iy)$ is $\mathcal C^2$ for $y\neq0$, and
- \begin{equation}
- \widehat{\mathcal U}_2(i\eta y)
- =
- \frac1\rho-\frac{i\eta y}\rho\sqrt{\frac1{2e}+\beta}
- +O(y^2)
- .
- \end{equation}
- Furthermore,
- \begin{equation}
- -\int_0^{|x|^{-\tau}}e^{-y|x|}y\ dy
- =
- -\frac1{|x|^2}+\frac{1+|x|^{1-\tau}}{|x|^2}e^{-|x|^{1-\tau}}
- \end{equation}
- \begin{equation}
- -\int_0^{|x|^{-\tau}}e^{-y|x|}y^2\ dy
- =
- -\frac2{|x|^3}+\frac{1+|x|^{1-\tau}(2+x^{1-\tau})}{|x|^3}e^{-|x|^{1-\tau}}
- \end{equation}
- and
- \begin{equation}
- -\int_0^{|x|^{-\tau}}e^{-y|x|}y^3\ dy
- =
- O(|x|^{-4})
- \end{equation}
- \begin{equation}
- I_1=
- -\frac1{\rho|x|^2}
- +\frac{2i\eta}{\rho|x|^3}\sqrt{\frac1{2e}+\beta}
- +O(|x|^{-4})
- \end{equation}
- so
- \begin{equation}
- \frac1{4i\pi^2|x|}\sum_{\eta=\pm}\eta I_1=\frac1{\pi^2\rho|x|^4}\sqrt{\frac1{2e}+\beta}
- +O(|x|^{-5})
- .
- \label{ineqI1}
- \end{equation}
- \bigskip
+{\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
 
- \subpoint We now bound $I_2$.
- Recall that, for $\kappa\in\mathbb R$, $|S(\kappa)|\leqslant 1$.
- Recalling\-~(\ref{decaydS}),
- \begin{equation}
- |S(\kappa+i\eta|x|^{-\tau})|
- \leqslant
- \sum_{n=0}^\infty\frac1{n!}|\partial^nS(\kappa)|^n|x|^{-n\tau}
- \leqslant
- \frac1{1-C|x|^{-\tau}}
- \leqslant 2
- \end{equation}
- provided $|x|^\tau>2C$.
- Therefore, for large $\kappa$,
- \begin{equation}
- |\widehat{\mathcal U}_2(\kappa+i\eta)|=O(\kappa^{-4})
- \end{equation}
- so
- \begin{equation}
- I_2\leqslant C'e^{-|x|^{1-\tau}}
- \label{ineqI2}
- \end{equation}
- for some constant $C'>0$.
- \bigskip
+We now specialize to $d=3$, and impose the additional assumption on the potential.
+\medskip
 
- \subpoint Inserting\-~(\ref{ineqI1}) and\-~(\ref{ineqI2}) into\-~(\ref{I12}) and\-~(\ref{U2}), we find that
- \begin{equation}
- \mathcal U_2(|x|)=\frac1{\pi^2\rho|x|^4}\sqrt{\frac1{2e}+\beta}+O(|x|^{-5})
- \end{equation}
- which, using\-~(\ref{U1}), concludes the proof of the lemma.
- \bigskip
+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}
+  where $S$ was defined in\-~(\ref{S}):
+  \begin{equation}
+    S(|k|):=\frac\rho{2e}\int e^{ikx}(1-u(|x|))\mathcal V(|x|)\ dx
+    .
+  \end{equation}
+  We split
+  \begin{equation}
+    \hat u(|k|)=\widehat{\mathcal U}_1(|k|)+\widehat{\mathcal U}_2(|k|)
+    \label{hatusplit}
+  \end{equation}
+  with
+  \begin{equation}
+    \widehat{\mathcal U}_1(|k|):=\frac{2eS(|k|)}{\rho(1+k^2)}
+  \end{equation}
+  so that, taking the large $|k|$ limit in\-~(\ref{hatu}),
+  \begin{equation}
+    \widehat{\mathcal U}_2(|k|)=O(|k|^{-4}S^2(|k|))
+  \end{equation}
+  so $\widehat{\mathcal U}_2$ is integrable.
+  \bigskip
+
+  \point{\bf Decay of $\mathcal U_1$}. We first show that
+  \begin{equation}
+    \mathcal U_1(|x|):=\frac1{(2\pi)^3}\int e^{-ikx}\widehat{\mathcal U}_1(|k|)\ dk
+  \end{equation}
+  decays exponentially in $|x|$. We have
+  \begin{equation}
+    \mathcal U_1(|x|)=(-\Delta+1)^{-1}(1-u(|x|))\mathcal V(|x|)
+    =Y_1\ast((1-u)\mathcal V)(|x|)
+  \end{equation}
+  with
+  \begin{equation}
+    Y_1(|x|):=\frac{e^{-|x|}}{4\pi|x|}
+    .
+  \end{equation}
+  Therefore, by\-~(\ref{expdecay}),
+  \begin{equation}
+    \mathcal U_1(|x|)
+    \leqslant
+    \frac A{4\pi}\int_{|y|>R} \frac{e^{-|x-y|-B|y|}}{|x-y|}\ dy
+    +
+    \frac 1{4\pi}\int_{|y|<R} \frac{e^{-|x-y|}}{|x-y|}\mathcal V(|y|)\ dy
+  \end{equation}
+  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{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{Ae^{-b|x|}}{4b^2}(b|x|+1)
+  \end{equation}
+  we have
+  \begin{equation}
+    \mathcal U_1(|x|)\leqslant
+    \frac{Ae^{-b|x|}}{4b^2}(b|x|+1)
+    +
+    \frac{e^{-(|x|-R)}}{4\pi(|x|-R)}\int\mathcal V(|y|)\ dy
+    .
+    \label{U1}
+  \end{equation}
+  \bigskip
+
+  \point{\bf Analyticity of $\mathcal U_2$}.
+  We now turn to
+  \begin{equation}
+    \mathcal U_2(|x|):=\frac1{(2\pi)^3}\int e^{-ikx}\widehat{\mathcal U}_2(|k|)\ dk
+    =\frac1{4i\pi^2|x|}\sum_{\eta=\pm}\eta\int_0^\infty e^{i\eta\kappa|x|}\kappa\widehat{\mathcal U}_2(\kappa)\ d\kappa
+    .
+    \label{U2}
+  \end{equation}
+  We start by proving some analytic properties of $\widehat{\mathcal U}_2$, which, we recall from\-~(\ref{hatu}) and\-~(\ref{hatusplit}), is
+  \begin{equation}
+    \widehat{\mathcal U}_2(|k|)=\frac1\rho\left(\frac{k^2}{4e}+1-\sqrt{\left(\frac{k^2}{4e}+1\right)^2-S(|k|)}-\frac{2eS(|k|)}{1+k^2}\right)
+    .
+  \end{equation}
+  \bigskip
+
+  \subpoint First of all, $S$ is analytic in a strip about the real axis:
+  \begin{equation}
+    S(\kappa)=4\pi\int_0^\infty \mathrm{sinc}(\kappa r)r^2\mathcal V(r)(1-u(r))\ dr
+    ,\quad
+    \mathrm{sinc}(\xi):=\frac{\sin(\xi)}\xi
+  \end{equation}
+  so
+  \begin{equation}
+    \partial^nS(\kappa)=4\pi\int_0^\infty \partial^n\mathrm{sinc}(\kappa r)r^{n+2}\mathcal V(r)(1-u(r))\ dr
+    .
+  \end{equation}
+  We will show that if $\mathcal Im(\kappa)\leqslant \frac B2$ (the factor $\frac12$ can be improved to any factor that is $<1$, but this does not matter here), then there exists $C>0$ which only depends on $A$ and $B$ such that
+  \begin{equation}
+    |\partial^nS(\kappa)|\leqslant n!C^n
+    .
+    \label{decaydS}
+  \end{equation}
+  As a consequence, $S$ is analytic in a strip around the real line of height $\frac B2$.
+  In particular, if we define the strip
+  \begin{equation}
+    H_\tau:=\{z:\ |\mathcal Im(z)|\leqslant r^{-\tau},\ \mathcal Re(z)>0\}
+  \end{equation}
+  with $0<\tau<1$, and take
+  \begin{equation}
+    r>\left(\frac B2\right)^{-\frac1\tau}
+  \end{equation}
+  then $S$ is analytic in $H_\tau$.
+  \bigskip
+
+  \bigskip
+
+  \subsubpoint We now prove\-~(\ref{decaydS}).
+  We first treat the case $|\kappa|\leqslant\frac B2$. We have
+  \begin{equation}
+    \mathrm{sinc}(\xi)=\sum_{p=0}^\infty\frac{(-1)^p\xi^{2p}}{(2p+1)!}
+  \end{equation}
+  so
+  \begin{equation}
+    \partial^n\mathrm{sinc}(\xi)=\sum_{p=\lceil\frac n2\rceil}^\infty\frac{(-1)^p\xi^{2p-n}}{(2p+1)(2p-n)!}
+    .
+  \end{equation}
+  Therefore
+  \begin{equation}
+    |\partial^n\mathrm{sinc}(\xi)|\leqslant
+    \sum_{p=\lceil\frac n2\rceil}^\infty\frac{|\xi|^{2p-n}}{(2p-n)!}
+    \leqslant \cosh(|\xi|)
+    .
+  \end{equation}
+  Thus,
+  \begin{equation}
+    |\partial^n S(\kappa)|\leqslant 4\pi\int_0^\infty \cosh(|\kappa|r)r^{n+2}\mathcal V(r)(1-u(r))\ dr
+  \end{equation}
+  so, by\-~(\ref{expdecay}),
+  \begin{equation}
+    |\partial^n S(\kappa)|\leqslant
+    4A\pi\int_R^\infty \cosh(|\kappa|r)r^{n+2}e^{-Br}\ dr
+    +
+    4\pi\int_0^R \cosh(|\kappa|r)r^{n+2}\mathcal V(r)\ dr
+  \end{equation}
+  and
+  \begin{equation}
+    |\partial^n S(\kappa)|\leqslant
+    8A\pi\int_0^\infty r^{n+2}e^{-(B-|\kappa|)r}\ dr
+    +
+    8\pi e^{|\kappa|R}R^{n}\int r^2\mathcal V(r)\ dr
+  \end{equation}
+  which, if $|\kappa|\leqslant \frac B2$, implies that
+  \begin{equation}
+    8A\pi\int_0^\infty r^{n+2}e^{-(B-|\kappa|)r}\ dr
+    \leqslant
+    8A\pi\int_0^\infty r^{n+2}e^{-\frac B2r}\ dr
+    =
+    \frac{2^{n+6}A\pi}{B^{n+3}}(n+2)!
+  \end{equation}
+  and
+  \begin{equation}
+    8\pi e^{|\kappa|R}R^{n+2}\int\mathcal V(r)\ dr
+    \leqslant
+    8\pi e^{\frac B2R}R^{n}\int r^2\mathcal V(r)\ dr
+  \end{equation}
+  which implies\-~(\ref{decaydS}) in this case.
+  \bigskip
+
+  \subsubpoint We now turn to $|\kappa|\geqslant \frac B2$:
+  \begin{equation}
+    \partial^n\mathrm{sinc}(\xi)=\sum_{p=0}^n{n\choose p}\partial^p\sin(\xi)\frac{(n-p)!(-1)^{n-p}}{\xi^{n-p+1}}
+  \end{equation}
+  so
+  \begin{equation}
+    |\partial^n\mathrm{sinc}(\xi)|\leqslant 2e^{\mathcal Im(\xi)}\sum_{p=0}^n\frac{n!}{p!}|\xi|^{-(n-p+1)}
+    .
+  \end{equation}
+  Therefore,
+  \begin{equation}
+    |\partial^nS(\kappa)|\leqslant
+    8\pi \sum_{p=0}^n\frac{n!}{p!|\kappa|^{n-p+1}}
+    \int_0^\infty e^{\mathcal Im(\kappa)r}r^{p+1}\mathcal V(r)(1-u(r))\ dr
+  \end{equation}
+  so, by\-~(\ref{expdecay}),
+  \begin{equation}
+    |\partial^nS(\kappa)|\leqslant
+    \sigma_1+\sigma_2
+  \end{equation}
+  with
+  \begin{equation}
+    \sigma_1:=8A\pi\sum_{p=0}^n\frac{n!}{p!|\kappa|^{n-p+1}}
+    \int_R^\infty r^{p+1}e^{-(B-\mathcal Im(\kappa))r}\ dr
+  \end{equation}
+  and
+  \begin{equation}
+    \sigma_2:=8\pi\sum_{p=0}^n\frac{n!}{p!|\kappa|^{n-p+1}}
+    \int_0^R r^{p+1}e^{\mathcal Im(\kappa)r}\mathcal V(r)\ dr
+    .
+  \end{equation}
+  Furthermore,
+  \begin{equation}
+    \sigma_1
+    =
+    8A\pi n!
+    \sum_{p=0}^n\frac{p+1}{(B-\mathcal Im(\kappa))^{p+2}|\kappa|^{n-p+1}}
+  \end{equation}
+  so, as long as $|\kappa|\geqslant \frac12B$ and $\mathcal Im(\kappa)\leqslant\frac12B$,
+  \begin{equation}
+    \sigma_1
+    \leqslant
+    \frac{2^{n+6}A\pi}{B^{n+3}}n!\sum_{p=0}^n(p+1)
+    =
+    \frac{2^{n+5}A\pi}{B^{n+3}}(n+2)!
+    .
+  \end{equation}
+  In addition,
+  \begin{equation}
+    \sigma_2
+    \leqslant
+    8\pi\sum_{p=0}^n\frac{n!}{p!|\kappa|^{n-p+1}}R^{p-1}e^{\mathcal Im(\kappa)R}\int_0^R r^2\mathcal V(r)\ dr
+  \end{equation}
+  so
+  \begin{equation}
+    \sigma_2
+    \leqslant
+    8\pi\sum_{p=0}^n\frac{n!2^{n-p+1}}{p!B^{n-p+1}}R^{p-1}e^{\mathcal Im(\kappa)R}\int_0^R r^2\mathcal V(r)\ dr
+    \leqslant
+    \frac{2^{n+4}\pi}{RB^{n+1}} n!e^{RB}\int_0^R r^2\mathcal V(r)\ dr
+  \end{equation}
+  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 $\sqrt{F(|k|)}$ with $F(|k|):=(\frac{k^2}{4e}+1)^2-S(|k|)$.
+  For $\kappa\in\mathbb R$,
+  \begin{equation}
+    |S(\kappa)|\leqslant 1
+  \end{equation}
+  so, for $\kappa\in\mathbb R$,
+  \begin{equation}
+    F(\kappa)\geqslant \frac{\kappa^2}{2e}
+    .
+  \end{equation}
+  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
+
+  \point{\bf Decay of $\mathcal U_2$}. We deform the integral to the path
+  \begin{equation}
+    \{i\eta y,\ 0<y<|x|^{-\tau}\}\cup\{i\eta |x|^{-\tau}+y,\ y>0\}
+  \end{equation}
+  and find
+  \begin{equation}
+    \int_0^\infty e^{i\eta\kappa|x|}\kappa\widehat{\mathcal U}_2(\kappa)\ d\kappa
+    =
+    I_1+I_2
+    \label{I12}
+  \end{equation}
+  with
+  \begin{equation}
+    I_1:=
+    -\int_0^{|x|^{-\tau}}e^{-y|x|}y\widehat{\mathcal U}_2(i\eta y)\ dy
+  \end{equation}
+  and
+  \begin{equation}
+    I_2:=
+    e^{-|x|^{1-\tau}}
+    \int_0^\infty e^{i\eta y|x|}(i\eta|x|^{-\tau}+y)\widehat{\mathcal U}_2(i\eta|x|^{-\tau}+y)\ dy
+    .
+  \end{equation}
+  \bigskip
+
+  \subpoint We first estimate $I_1$.
+  We expand $S$:
+  \begin{equation}
+    S(\kappa)=1-\beta \kappa^2+O(|\kappa|^4)
+  \end{equation}
+  with $\beta>0$ (since $S$ is analytic and symmetric, and $|S(|k|)|\leqslant 1$).
+  Therefore, $y\mapsto\widehat{\mathcal U}_2(iy)$ is $\mathcal C^2$ for $y\neq0$, and
+  \begin{equation}
+    \widehat{\mathcal U}_2(i\eta y)
+    =
+    \frac1\rho-\frac{i\eta y}\rho\sqrt{\frac1{2e}+\beta}
+    +O(y^2)
+    .
+  \end{equation}
+  Furthermore,
+  \begin{equation}
+    -\int_0^{|x|^{-\tau}}e^{-y|x|}y\ dy
+    =
+    -\frac1{|x|^2}+\frac{1+|x|^{1-\tau}}{|x|^2}e^{-|x|^{1-\tau}}
+  \end{equation}
+  \begin{equation}
+    -\int_0^{|x|^{-\tau}}e^{-y|x|}y^2\ dy
+    =
+    -\frac2{|x|^3}+\frac{1+|x|^{1-\tau}(2+x^{1-\tau})}{|x|^3}e^{-|x|^{1-\tau}}
+  \end{equation}
+  and
+  \begin{equation}
+    -\int_0^{|x|^{-\tau}}e^{-y|x|}y^3\ dy
+    =
+    O(|x|^{-4})
+  \end{equation}
+  \begin{equation}
+    I_1=
+    -\frac1{\rho|x|^2}
+    +\frac{2i\eta}{\rho|x|^3}\sqrt{\frac1{2e}+\beta}
+    +O(|x|^{-4})
+  \end{equation}
+  so
+  \begin{equation}
+    \frac1{4i\pi^2|x|}\sum_{\eta=\pm}\eta I_1=\frac1{\pi^2\rho|x|^4}\sqrt{\frac1{2e}+\beta}
+    +O(|x|^{-5})
+    .
+    \label{ineqI1}
+  \end{equation}
+  \bigskip
+
+  \subpoint We now bound $I_2$.
+  Recall that, for $\kappa\in\mathbb R$, $|S(\kappa)|\leqslant 1$.
+  Recalling\-~(\ref{decaydS}),
+  \begin{equation}
+    |S(\kappa+i\eta|x|^{-\tau})|
+    \leqslant
+    \sum_{n=0}^\infty\frac1{n!}|\partial^nS(\kappa)|^n|x|^{-n\tau}
+    \leqslant
+    \frac1{1-C|x|^{-\tau}}
+    \leqslant 2
+  \end{equation}
+  provided $|x|^\tau>2C$.
+  Therefore, for large $\kappa$,
+  \begin{equation}
+    |\widehat{\mathcal U}_2(\kappa+i\eta)|=O(\kappa^{-4})
+  \end{equation}
+  so
+  \begin{equation}
+    I_2\leqslant C'e^{-|x|^{1-\tau}}
+    \label{ineqI2}
+  \end{equation}
+  for some constant $C'>0$.
+  \bigskip
+
+  \subpoint Inserting\-~(\ref{ineqI1}) and\-~(\ref{ineqI2}) into\-~(\ref{I12}) and\-~(\ref{U2}), we find that
+  \begin{equation}
+    \mathcal U_2(|x|)=\frac1{\pi^2\rho|x|^4}\sqrt{\frac1{2e}+\beta}+O(|x|^{-5})
+  \end{equation}
+  which, using\-~(\ref{U1}), concludes the proof of the lemma.
+  \bigskip
 \qed
 
 
@@ -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}
@@ -1388,18 +1459,21 @@ We hope to study this equation numerically in a later publication.
 \begin{thebibliography}{WWW99}
 \small
  
-\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[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[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
+arxiv:{\tt\color{blue}\href{http://arxiv.org/abs/1904.06164}{1904.06164}}.\par\medskip
  
 \bibitem[Ga99]{Ga99}G. Gallavotti - {\it Statistical mechanics, a short treatise}, Springer, 1999.\par\medskip
  
+\bibitem[HJL]{HJL}E.A.\-~Carlen, M.\-~Holzmann, I.\-~Jauslin, E.H.\-~Lieb, in preparation.\par\medskip
+ 
 \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
 doi:{\tt\color{blue}\href{http://dx.doi.org/10.1103/PhysRev.106.1135}{10.1103/PhysRev.106.1135}}.\par\medskip
  
@@ -1430,8 +1504,8 @@ doi:{\tt\color{blue}\href{http://dx.doi.org/10.1007/s00220-012-1576-y}{10.1007/s
 \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\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}
diff --git a/Changelog b/Changelog
new file mode 100644
index 0000000..59e69e0
--- /dev/null
+++ b/Changelog
@@ -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.
+
diff --git a/bibliography/bibliography.tex b/bibliography/bibliography.tex
deleted file mode 100644
index 7236c3e..0000000
--- a/bibliography/bibliography.tex
+++ /dev/null
@@ -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
- 
-\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
- 
-\bibitem[Ga99]{Ga99}G. Gallavotti - {\it Statistical mechanics, a short treatise}, Springer, 1999.\par\medskip
- 
-\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
-doi:{\tt\href{http://dx.doi.org/10.1103/PhysRev.106.1135}{10.1103/PhysRev.106.1135}}.\par\medskip
- 
-\bibitem[Le29]{Le29}W. Lenz - {\it Die Wellenfunktion und Geschwindigkeitsverteilung des entarteten Gases}, Zeitschrift f\"ur Physik, volume\-~56, issue\-~11-12, pages\-~778-789, 1929,\par\penalty10000
-doi:{\tt\href{http://dx.doi.org/10.1007/BF01340138}{10.1007/BF01340138}}.\par\medskip
- 
-\bibitem[Li63]{Li63}E.H. Lieb - {\it Simplified Approach to the Ground-State Energy of an Imperfect Bose Gas}, Physical Review, volume\-~130, issue\-~6, pages\-~2518-2528, 1963,\par\penalty10000
-doi:{\tt\href{http://dx.doi.org/10.1103/PhysRev.130.2518}{10.1103/PhysRev.130.2518}}.\par\medskip
- 
-\bibitem[LL64]{LL64}E.H. Lieb, W. Liniger - {\it Simplified Approach to the Ground-State Energy of an Imperfect Bose Gas. III. Application to the One-Dimensional Model}, Physical Review, volume\-~134, issue\-~2A, pages A312-A315, 1964,\par\penalty10000
-doi:{\tt\href{http://dx.doi.org/10.1103/PhysRev.134.A312}{10.1103/PhysRev.134.A312}}.\par\medskip
- 
-\bibitem[LL01]{LL01}E.H. Lieb, M. Loss - {\it Analysis}, Second edition, Graduate studies in mathematics, Americal Mathematical Society, 2001.\par\medskip
- 
-\bibitem[LY98]{LY98}E.H. Lieb, J. Yngvason - {\it Ground State Energy of the Low Density Bose Gas}, Physical Review Letters, volume\-~80, issue\-~12, pages\-~2504-2507, 1998,\par\penalty10000
-doi:{\tt\href{http://dx.doi.org/10.1103/PhysRevLett.80.2504}{10.1103/PhysRevLett.80.2504}}, arxiv:{\tt\href{http://arxiv.org/abs/cond-mat/9712138}{cond-mat/9712138}}.\par\medskip
- 
-\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
-doi:{\tt\href{http://dx.doi.org/10.1023/A:1010337215241}{10.1023/A:1010337215241}}, arxiv:{\tt\href{http://arxiv.org/abs/math-ph/0002014}{math-ph/0002014}}.\par\medskip
- 
-\bibitem[PT12]{PT12}E. Pulvirenti, D. Tsagkarogiannis - {\it Cluster Expansion in the Canonical Ensemble}, Communications in Mathematical Physics, volume\-~316, issue\-~2, pages\-~289-306, 2012,\par\penalty10000
-doi:{\tt\href{http://dx.doi.org/10.1007/s00220-012-1576-y}{10.1007/s00220-012-1576-y}}, arxiv:{\tt\href{http://arxiv.org/abs/1105.1022}{1105.1022}}.\par\medskip
- 
-\bibitem[RS75]{RS75b}M. Reed, B. Simon - {\it Methods of Modern Mathematical Physics II: Fourier Analysis, Self-Adjointness}, second edition, Academic Press, New York, 1975.\par\medskip
- 
-\bibitem[Ru99]{Ru99}D. Ruelle - {\it Statistical mechanics: rigorous results}, Imperial College Press, World Scientific, (first edition: Benjamin, 1969), 1999.\par\medskip
- 
-\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
-