From 6a019dc4f91c3db3731193aa0351444462b44a24 Mon Sep 17 00:00:00 2001
From: Ian Jauslin <ian@jauslin.org>
Date: Mon, 1 Mar 2021 16:51:33 -0500
Subject: Update to v0.2:

  Added: Comparison of the energy for two different potentials that
  share the same scattering length and integral (new figure).

  Added: Lee-Huang-Yang prediction in figures.

  Fixed: Wrong exponent in definition of correlation function.

  Added: More precise comparison of the big equation prediction for the
  hard core potential (new figure).

  Changed: Miscellaneous minor fixes and clarifications.
---
 Carlen_Holzmann_Jauslin_Lieb_2020.tex | 127 +++++++++++++++++++++++++++-------
 1 file changed, 101 insertions(+), 26 deletions(-)

(limited to 'Carlen_Holzmann_Jauslin_Lieb_2020.tex')

diff --git a/Carlen_Holzmann_Jauslin_Lieb_2020.tex b/Carlen_Holzmann_Jauslin_Lieb_2020.tex
index 3e2a478..67398df 100644
--- a/Carlen_Holzmann_Jauslin_Lieb_2020.tex
+++ b/Carlen_Holzmann_Jauslin_Lieb_2020.tex
@@ -35,13 +35,13 @@
 \email{lieb@princeton.edu}
 
 \begin{abstract}
-In 1963, a {\it Simple Approach} was developed to study the ground state energy of an interacting Bose gas.
+In 1963, a {\it Simplified Approach} was developed to study the ground state energy of an interacting Bose gas with a purely repulsive potential.
 It consists in the derivation of an Equation, which is not based on perturbation theory, and which gives the exact expansion of the energy at low densities.
 This Equation is expressed directly in the thermodynamic limit, and only involves functions of $3$ variables, rather than $3N$.
-Here, we revisit this approach, and show that the Equation yields accurate predictions for various observables for {\it all} densities.
-Specifically, in addition to the ground state energy, we have shown that the Simple Approach gives predictions for the condensate fraction, two-point correlation function, and momentum distribution.
-We have carried out a variety of tests by comparing the predictions of the Equation with Quantum Monte Carlo calculations, and have found remarkable agreement.
-We thus show that the Simple Approach provides a new theoretical tool to understand the behavior of the many-body Bose gas, not only in the small and large density ranges, which have been studied before, but also in the range of intermediate density, for which little is known.
+Here, we revisit this approach, introduce two more equations and show that these yields accurate predictions for various observables for {\it all} densities for repulsive potentials with positive Fourier transform.
+Specifically, in addition to the ground state energy, we have shown that the Simplified Approach gives predictions for the condensate fraction, two-point correlation function, and momentum distribution.
+We have carried out a variety of tests by comparing the predictions of the Equations with Quantum Monte Carlo calculations for exponential interaction potentials as well as a different, finite range potential of positive type, and have found remarkable agreement.
+We thus show that the Simplified Approach provides a new theoretical tool to understand the behavior of the many-body Bose gas, not only in the small and large density ranges, which have been studied before, but also in the range of intermediate density, for which much less is known.
 \end{abstract}
 
 \maketitle
@@ -51,7 +51,7 @@ We thus show that the Simple Approach provides a new theoretical tool to underst
 Bose gases are one of the foundational objects in the statistical mechanics of quantum systems, and have been the focus of much scrutiny, dating back to the early days of quantum mechanics\-~\cite{Le29}.
 Nevertheless, there are still several important problems to be solved, in the case of interacting Bose gases, in which the correlations between particles make the analysis very difficult.
 In this case, observables may be computed by either performing numerical computations using finite-size approximations and extrapolations, or by devising effective theories which capture some of the correlations between particles, while remaining integrable.
-In this paper, we present an effective theory which goes back to 1963\-~\cite{Li63}, and which we have found gives astonishingly accurate predictions in the thermodynamic limit at {\it all} densities that have been verified numerically by Quantum Monte Carlo (QMC) computations.
+In this paper, we present an effective theory which goes back to 1963\-~\cite{Li63}, and which we have found gives accurate predictions in the thermodynamic limit at {\it all} densities that have been verified numerically by Quantum Monte Carlo (QMC) computations.
 This remarkable agreement leads us to suggest that this may be a new way of understanding and analyzing the quantum many-body problem.
 \bigskip
 
@@ -97,9 +97,9 @@ Note that, whereas Hartree theory is accurate at asymptotically large densities,
 
 \indent
 Therefore, the Bose gas is described by Bogolubov theory at low density, and Hartree theory or the MSA at high density.
-In this paper, we will discuss another effective theory for the ground state of the repulsive Bose gas with a positive type potential, which is highly accurate at all densities.
+In this paper, we will discuss another effective theory for the ground state of the repulsive Bose gas with a positive type potential, which is highly accurate at all densities, which is {\it exact} at low and high densities, and highly accurate at all intermediate densities.
 In other words, it is a physically descriptive interpolation between Bogolubov and Hartree theory.
-To justify our claim that it is in good {\it quantitative} agreement  the physics all all densities, we rely on with QMC simulations of the Bose gas for intermediate densities.
+To justify our claim that it is in good {\it quantitative} agreement with the physics at all densities, we rely on with QMC simulations of the Bose gas for intermediate densities.
 This equation was originally introduced in 1963\-~\cite{Li63}, and studied for the high density Jellium\-~\cite{LS64}, and in one dimension\-~\cite{LL64}.
 There has been no research progress since then.
 The merit of this equation is twofold.
@@ -209,7 +209,7 @@ In the present paper, we discuss some more quantitative results, with more of a
 We will consider potentials that are of positive type, with a special focus on exponential potentials of the form $\alpha e^{-|\mathbf x|}$.
 We have found that the prediction for the energy is very accurate for {\it all} densities, see Figure\-~\ref{fig:energy}.
 In the case $\alpha=1$, the relative error compared to the QMC simulation is as small as $0.1\%$, and is comparable to the error made by a Bijl-Dingle-Jastrow function Ansatz \cite{Bi40,Di49,Ja55}, see Figure\-~\ref{fig:cmp_energy}, even though the solution of the Big Equation is much easier to compute numerically than the Bijl-Dingle-Jastrow optimizer.
-The prediction for the condensate fraction is less accurate, though still remarkably good for small values of $\alpha$, see Figure\-~\ref{fig:condensate0.5}.
+The prediction for the condensate fraction is less accurate in the intermediate density regime, though still remarkably good for small values of $\alpha$, see Figure\-~\ref{fig:condensate0.5}.
 For larger $\alpha$, the Big Equation is off the mark, see Figure\-~\ref{fig:condensate16}, although the qualitative features of the condensate fraction are still well reproduced.
 We have also carried out similar computations for the hard core potential, for which we also find good agreement, see Figure\-~\ref{fig:hardcore}.
 
@@ -238,10 +238,16 @@ For the Big and Simple Equations discussed in this paper, we have found that thi
   \sqrt{\rho a_0}\ll|\mathbf k|\ll1
 \end{equation}
 which is another confirmation of the accuracy of the effective equation at small densities.
-However, if $\sqrt\rho\gtrsim1$, then the Tan regime does not exist, and the picture in terms of strongly coupled few-particle configurations inherent to the analysis of unitary Bose gases\-~\cite{CW11,SBe14} breaks down, as the Bose gas transitions to a strongly correlated liquid.
+However, if $\sqrt\rho\gtrsim1$, then the universal Tan regime does not exist, and the picture in terms of strongly coupled few-particle configurations inherent to the analysis of unitary Bose gases\-~\cite{CW11,SBe14} breaks down, as the Bose gas transitions to a strongly correlated liquid.
 This is confirmed for the prediction of the Big Equation, see Figure\-~\ref{fig:tan}.
 \bigskip
 
+\indent
+As further evidence of the breakdown of universality in the intermediate density regime, we have also compared the ground state energy for two very different potentials, which have the same scattering length and the same integral.
+We have found that the energy for these two potentials is significantly different in the intermediate density regime, see Figure\-~\ref{fig:compare_pots}.
+For these two potentials, we have also found that the Quantum Monte Carlo data fits very well with the prediction of the Big Equation.
+\bigskip
+
 \indent
 The rest of the paper is structured as follows.
 In section\-~\ref{sec:approx}, we detail the approximation needed to get from the many-body Bose gas to the Full Equation, and then discuss the approximations leading to the Big, Medium and Simple Equations.
@@ -261,6 +267,9 @@ We start from the many-body Hamiltonian: denoting the number of particles by $N$
 We confine the $N$ particles in a cubic box $\Lambda$  of volume $V$, and impose periodic boundary conditions.
 Later on, we will take the thermodynamic limit $N,V\to\infty$, $\frac NV=\rho$ fixed.
 
+\indent
+In the derivation presented here, we will rely on the translation invariance of the Hamiltonian, which does not allow us to study a system with a trapping potential at this time.
+
 \indent Let  $E_N$ denote the ground state energy and let $\psi_N(\mathbf x_1,\cdots,\mathbf x_N)$  denote the ground state wave function so that
 \begin{equation}
   H\psi_0(\mathbf x_1,\cdots,\mathbf x_N)=E_N\psi_N(\mathbf x_1,\cdots,\mathbf x_N)
@@ -366,7 +375,7 @@ At high densities, since the system approaches a mean-field regime, one might al
 The Full Equation we have derived is quite difficult to study, even numerically.
 As was discussed in Section\-~\ref{sec:intro}, we will introduce further approximations to simplify the equation.
 The first approximation is to neglect the $\frac12u(\mathbf z)u(\mathbf y-\mathbf x)$ term in\-~(\ref{L}), which is the most difficult term, from a computational point of view. 
-We expect that, at low densities, this term is expected to be of order $\rho^{3/2}$ uniformly $\mathbf x$, whereas the leading order term in $L$ should be of order $\rho$.
+We expect that, at low densities, this term is expected to be of order $\rho^{3/2}$ uniformly in $\mathbf x$, whereas the leading order term in $L$ should be of order $\rho$.
 This leads us to the Big Equation defined in\-~(\ref{bigeq}).
 This equation is easier to solve numerically than the Full Equation, because in Fourier space, it involves only two convolution operators, whereas the Full Equation contains three, which makes it computationally heavier.
 Nevertheless, this equation is still difficult to study analytically, so we make further approximations
@@ -385,7 +394,7 @@ On account of (\ref{EN}), the function $S(\mathbf x)$ defined in (\ref{K}) satis
   \int d\mathbf x\ S(\mathbf x) = \frac{2\tilde e}{\rho}
 \end{equation}
 which is just another way of stating (\ref{erel}).
-There are two different length scales in the problem: the first is the scattering length of the potential $a_0$ and the interparticle distance $\rho^{-1/3}$.
+There are two different length scales in the problem: the first is the scattering length of the potential $a_0$ and the second is the interparticle distance $\rho^{-1/3}$.
 At sufficiently low densities we will have 
 \begin{equation}
   a_0 \ll  \rho^{-1/3}
@@ -429,7 +438,7 @@ mixed-estimator bias occurring for observables different from the ground state e
 In principle, the mixed-estimator bias can be controlled either by systematic improvement of
 the trial wave function \cite{RMH18} or by different projection Monte Carlo methods, e.g. Reptation 
 Monte Carlo \cite{BM99}. For the system under consideration, the mixed estimator bias
-of the pair-product wave function was found to be sufficiently accurate, 
+of the pair-product wave function was found to be sufficiently small, 
 the overall precision being limited rather by
 the finite system size of the QMC calculations.
 \bigskip
@@ -473,6 +482,7 @@ For $\alpha=16$ this is even clearer, and one sees that the Medium Equation is m
   \caption{
     The energy as a function of density for the potential $e^{-|\mathbf x|}$ (top) and $16e^{-|\mathbf x|}$ (bottom).
     We compare the predictions of the Big, Medium and Simple Equations to a QMC simulation.
+    For comparison, we also plot the Lee-Huang-Yang (LHY) energy\-~(\ref{lhy}).
   }
   \label{fig:energy}
 \end{figure}
@@ -506,10 +516,11 @@ The approximations leading to the Big, Simple and Medium Equations reduce the nu
 In doing so, we lose some information, and, in particular, we do not obtain a prediction for the many-body wavefunction $\psi_0$.
 Therefore, computing observables other than the ground state energy is not entirely straightforward.
 To compute the condensate fraction, we first express it in terms of the energy of an auxiliary system, from which we derive an approximation following the prescriptions in section\-~\ref{sec:approx}.
-Specifically, the condensate fraction of the many-body ground state $\psi_0$ is in terms of the projector $P_i\psi_0:=\int\frac{d\mathbf x_i}V\psi_0$ onto the condensate wavefunction (which is the constant function):
+Specifically, the {\it non}-condensed fraction of the many-body ground state $\psi_0$
 \begin{equation}
   \eta_0:=1-\frac1N\sum_{i=1}^N\left<\psi_0\right|P_i\left|\psi_0\right>
 \end{equation}
+is expressed in terms of the projector $P_i\psi_0:=\int\frac{d\mathbf x_i}V\psi_0$ onto the condensate wavefunction (which is the constant function):
 which we re-express in terms of the modified Hamiltonian
 \begin{equation}
   H_\mu=-\frac12\sum_{i=1}^N\Delta_i+\sum_{1\leqslant i<j\leqslant N}v(\mathbf x_i-\mathbf x_j)-\mu\frac1N\sum_{i=1}^NP_i
@@ -571,6 +582,7 @@ The location of the maximum of the non-condensed fraction (or the minimum of the
   \caption{
     The non-condensed fraction as a function of the density for the potential $\frac12e^{-|\mathbf x|}$.
     We compare the predictions of the Big, Medium and Simple Equations to a QMC simulation.
+    The prediction of Bogolubov theory\-~(\ref{eta0}) is also plotted for comparison (Bog).
   }
   \label{fig:condensate0.5}
 \end{figure}
@@ -590,12 +602,12 @@ We first note that this can be rewritten in a way that makes the translation inv
 \end{equation}
 which we can rewrite as a functional derivative of the ground state energy per-particle $e_0$:
 \begin{equation}
-  C_2(\mathbf x)=2\rho\frac{\delta e_0}{\delta v(\mathbf x)}
+  C_2(\mathbf x)=2\rho^2\frac{\delta e_0}{\delta v(\mathbf x)}
   .
 \end{equation}
 The prediction $\tilde C_2$ of the Big and Medium Equations for the two-point correlation function are therefore defined by differentiating $\tilde e$ in\-~(\ref{erel}) with respect to $v$:
 \begin{equation}
-  \tilde C_2(\mathbf x):=2\rho\frac{\delta\tilde e}{\delta v(\mathbf x)}
+  \tilde C_2(\mathbf x):=2\rho^2\frac{\delta\tilde e}{\delta v(\mathbf x)}
   .
   \label{C2}
 \end{equation}
@@ -632,6 +644,7 @@ The prediction of the Big Equation remains quite accurate, when compared to the
 The presence of this local maximum in $g_2$ shows that, in the probability distribution $\psi_0$, there is a larger probability of finding pairs of particles that are separated by a certain fixed distance.
 This indicates the appearance of a new physical length scale at intermediate densities, and indicates that the system exhibits a non-trivial physical behavior in this regime.
 Note that this behavior was observed for the stronger potential $16e^{-|\mathbf x|}$, but seems to be absent for $e^{-|\mathbf x|}$.
+Note, also, that, as will be discussed next, this maximum is also present in the two-point correlation $C_2$, and is, therefore, the manifestation of a physical phenomenon.
 \bigskip
 
 \begin{figure}
@@ -650,6 +663,7 @@ At low densities, the prediction of the Big Equation agrees rather well with the
 The Simple and Medium Equations are not as accurate.
 At larger densities, the Simple and Medium Equations are quite far from the QMC computation, and the Big Equation is not as accurate as in the case of $\tilde g_2$, but it does reproduce some of the qualitative behavior of the QMC computation.
 In particular, there is a local maximum in the two-point correlation function, which occurs at a length scale that is close to that observed for $\tilde g_2$.
+This suggests the emergence of a non-trivial phase, which resembles a liquid.
 At small $\mathbf x$, $\tilde C_2$ is negative, which is clearly not physical, and those values should be discarded.
 
 \begin{figure}
@@ -739,6 +753,49 @@ This is particularly true in the region in which $|\mathbf k|^{-4}$ transitions
   \label{fig:tan}
 \end{figure}
 
+
+\subsection{Non-universal behavior at intermediate densities}
+\indent
+The low density asymptotics of the energy, given by the Lee-Huang-Yang formula\-~(\ref{lhy}), only depend on the potential through the scattering length.
+At high density\-~(\ref{ehigh}), they only depend on the potential through $\int d\mathbf x\ v(\mathbf x)$.
+In this sense, the low and high density behavior of the Bose gas is {\it universal}.
+In this section, we show that, at intermediate densities, the energy does not only depend on the scattering length and the integral of the potential, thus suggesting that the behavior of the Bose gas at intermediate densities is {\it not universal}.
+
+\indent
+To that end, we have compared the predictions of the Big Equation for the energy for two potentials that have the same scattering length, and the same integral.
+The first potential, $v_{32}^{(0)}$, is defined in the next section, see\-~(\ref{vn}), and the second is an exponential potential
+\begin{equation}
+  \Phi_{\alpha,\beta}(\mathbf x):=\alpha e^{-\beta|\mathbf x|}
+\end{equation}
+where $\alpha$ and $\beta$ are chosen in such a way that the scattering length and integral of $\Phi$ are equal to those of $v_{32}^{(0)}$.
+The scattering length of $v_{32}^{(0)}$ was computed numerically and found to be $\approx0.5878$, and its integral is $\frac{64\pi^2}9$.
+The scattering length of $\Phi_{\alpha,\beta}$ is
+\begin{equation}
+  \frac1\beta\left(\log\frac\alpha{\beta^2}+2\gamma+2\frac{K_0(2\sqrt{\frac\alpha{\beta^2}})}{I_0(2\sqrt{\frac\alpha{\beta^2}})}\right)
+\end{equation}
+where $\gamma$ is the Euler constant and $K_0$ and $I_0$ are modified Bessel functions.
+The integral of $\Phi_{\alpha,\beta}$ is $\frac{8\pi\alpha}{\beta^3}$.
+We thus find that, in order to make the scattering length and integral of $v_{32}^{(0)}$ and $\Phi_{\alpha,\beta}$ coincide, we must choose $\alpha\approx907.2$ and $\beta\approx6.874$.
+
+\indent
+The prediction of the energy for these two potentials is plotted in Figure\-~\ref{fig:compare_pots}.
+We find that, as expected, the energies coincide at low and high density, but they differ significantly in the intermediate density regime.
+We have confirmed this fact by QMC computations, and found good agreement of the QMC data with our prediction for both potentials.
+
+\begin{figure}
+  \hfil\includegraphics[width=8cm]{compare_pots.pdf}
+  \caption{
+    The prediction of the energy by the Big Equation for the potentials $v_{32}^{(0)}$ and $\Phi(\mathbf x)\equiv\alpha e^{-\beta|\mathbf x|}$ with $\alpha\approx 907.2$ and 
+$\beta \approx 6.873$.
+    The potentials are chosen to have the same scattering length, $a_0\approx 0.5878$, as well
+as the same value for their integrals, so they coincide at low and at high densities.
+    They differ signinficantly at intermediate densities.
+    We compare each curve to a few QMC points, which fit well.
+    We also plot the Lee-Huang-Yang (LHY) energy\-~(\ref{lhy}).
+  }
+  \label{fig:compare_pots}
+\end{figure}
+
 \section{Hard-core potential}\label{sec:hardcore}
 
 \indent
@@ -747,18 +804,21 @@ We have investigated two directions to get around this restriction.
 
 \indent
 The first, and most straightforward, is to consider the hard-core potential as a limit of soft core potentials.
+Obviously, this approach will not be accurate at densities approaching close-packing, but as we will see, is rather accurate at smaller densities.
 As was mentioned in section\-~\ref{sec:intro}, it is preferable to only use potentials of positive type (that is, non-negative potentials with a non-negative Fourier transform).
 With this in mind, we consider the sequence of potentials
-\begin{equation}
-  v^{(0)}_n(|\mathbf x|):=\alpha_n\mathds 1_{|\mathbf x|<\frac12}\ast\mathds 1_{|\mathbf x|<\frac12}
-\end{equation}
-that is
 \begin{equation}
   v^{(0)}_n(|\mathbf x|)
-  =\mathds 1_{|\mathbf x|<1}
+  :=\Theta(1-|\mathbf x|)
   \alpha_n\frac{2\pi}{3}(|\mathbf x|-1)^2(|\mathbf x|+2)
 \end{equation}
-where $\mathds 1_{|\mathbf x|<\frac12}$ is the indicator function that $|\mathbf x|<\frac12$, and $\alpha_n\to\infty$ ($v^{(0)}_n$ has positive type because it is the convolution of a function with itself).
+where $\Theta(x)$ is the Heaviside function, which is equal to $1$ for $x>0$ and $0$ otherwise, and $\alpha_n\to\infty$.
+This potential can also be written as
+\begin{equation}
+  v^{(0)}_n(|\mathbf x|)=
+  \alpha_n\int d\mathbf y\ \Theta({\textstyle\frac12-|\mathbf y|})\Theta({\textstyle\frac12-|\mathbf x-\mathbf y|})
+\end{equation}
+which shows that it is of positive type because it is the convolution of the function $\Theta(\frac12-|\mathbf x|)$ with itself.
 In addition, we fix the scattering length of the potential to 1, by rescaling space: denoting the scattering length of $v^{(0)}_n$ by $a_n$, we take the potential to be
 \begin{equation}
   v_n(\mathbf x):=v^{(0)}_n\left({\textstyle\frac{|\mathbf x|}{a_n}}\right)
@@ -786,12 +846,15 @@ For smaller densities, for the Simple Equation, we see that the predictions made
 
 \begin{figure}
   \hfil\includegraphics[width=8cm]{hardcore_energy.pdf}
+  \hfil\includegraphics[width=8cm]{hardcore_compare.pdf}
   \hfil\includegraphics[width=8cm]{hardcore_condensate.pdf}
   \caption{
-    The energy (top) and non-condensed fraction (bottom) as a function of the density for the hard core potential.
+    The energy (top), relative error in the energy $\frac{\tilde e-e_{\mathrm{QMC}}}{e_{\mathrm{QMC}}}$ (middle), and non-condensed fraction (bottom) as a function of the density for the hard core potential.
     The circles were computed by solving the hard core Simple Equation for $|\mathbf x|>1$ (simple hc).
     The lines were computed by approximating the hard core potential by the potential $v_{512}(\mathbf x)$, see\-~(\ref{vn}).
     We compare the predictions of the Big, Medium and Simple Equations to QMC results reported\-~\cite{GBC99}.
+    The prediction of Bogolubov theory\-~(\ref{eta0}) is also plotted for comparison (Bog).
+    The right edge of the plots correspond to the close-packing density $\rho_{\mathrm{cp}}=\sqrt2$\-~\cite{Ha05}.
   }
   \label{fig:hardcore}
 \end{figure}
@@ -811,7 +874,7 @@ It is quite easy to find a counter-example if $v$ is not of positive type.
 For instance, if $v(\mathbf x)=0$ for all $|\mathbf x|<1$, then, consider a wavefunction $\psi$ that is smooth and supported on $|\mathbf x_1|,\cdots,|\mathbf x_N|<\frac 12$.
 Since all particles are at a distance that is $<1$, the potential energy of such a wavefunction is 0, and its kinetic energy is $O(N)$.
 Thus, the energy per particle is of order 1, which, for large $\rho$, is $\ll\frac\rho2\int d\mathbf x\ v(\mathbf x)$.
-(Note that a non-trivial, non-negative potential with $v(\mathbf x)$ cannot be of positive type if $v(0)=0$, since the maximumof a positive type function is attained at $0$.)
+(Note that a non-trivial, non-negative potential with $v(\mathbf x)$ cannot be of positive type if $v(0)=0$, since the maximum of a positive type function is attained at $0$.)
 \bigskip
 
 \indent
@@ -825,6 +888,7 @@ This is further confirmed by the computations for the hard core potential, in wh
   \caption{
     The non-condensed fraction as a function of the density for the potential $16e^{-|\mathbf x|}$.
     We compare the predictions of the Big, Medium and Simple Equations to a QMC simulation.
+    The prediction of Bogolubov theory\-~(\ref{eta0}) is also plotted for comparison (Bog).
   }
   \label{fig:condensate16}
 \end{figure}
@@ -832,10 +896,19 @@ This is further confirmed by the computations for the hard core potential, in wh
 \section{Conclusions}\label{sec:conclusions}
 
 \indent
-In this paper we show the astonishing agreement in the predictions of the ground state energy, condensate fraction and correlation function of the interacting Bose gas given by the {\it simplified approach} developed in 1963\-~\cite{Li63} with the values obtained by Quantum Monte-Carlo calculations.
+In this paper we show the good agreement in the predictions of the ground state energy, condensate fraction and correlation function of the repulsive Bose gas given by the {\it simplified approach} developed in 1963\-~\cite{Li63} with the values obtained by Quantum Monte-Carlo calculations, for the potentials $e^{-|\mathbf x|}$ and $16e^{-|\mathbf x|}$.
 The simplified approach was thought to be accurate  only at low densities, in complete agreement with other analyses of the time.
 Here, we show that it is accurate at {\it all} densities.
-This establishes a new paradigm for many body bosonic physics.
+This establishes a new approach to many body bosonic physics.
+Combining this analysis with the exact results in\-~\cite{CJL20,CJL20b} leads us to conjecture that the simplified approach is accurate for any repulsive potential of positive type with a scattering length and an integral that is not too large.
+\bigskip
+
+\indent
+We have discussed three different approximations, the Big, Medium and Simple Equations.
+The Big Equation is the most accurate, but also the most difficult to solve.
+The Medium Equation is obtained by neglecting terms of higher order in $u$, which makes it much more easy to compute with, while remaining rather close to the Big Equation.
+The Simple Equation is then obtained by approximating $g_2(x)v(x)$ by a Dirac-delta function.
+This drastically simplifies the equation, but is also less accurate at intermediate densities (while the low and high densities are still asymptotically exact).
 \bigskip
 
 \indent
@@ -844,6 +917,8 @@ The method provides a promising avenue to approach singular potentials, such as
 In addition, this allows us to approach various physical questions, such as Bose-Einstein condensation, even in the intermediate density regime, away from the dilute and dense limits.
 
 \begin{acknowledgements}
+  We thank two anonymous referees for many helpful comments.
+  E.H.L. thanks the Institute for Advanced study for its  hospitality.
   U.S.~National Science Foundation grants DMS-1764254 (E.A.C.),  DMS-1802170 (I.J.) are gratefully acknowledged.
 \end{acknowledgements}
 
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cgit v1.3

