# The Ground State Energy of Dilute Bose Gas in Potentials with Positive
Scattering Length

Jun Yin
Department of Mathematics, Harvard University,

Email<EMAIL_ADDRESS>
(August 29th, 2008)

###### Abstract

The leading term of the ground state energy/particle of a dilute gas of bosons
with mass $m$ in the thermodynamic limit is $2\pi\hbar^{2}a\varrho/m$ when the
density of the gas is $\varrho$, the interaction potential is non-negative and
the scattering length $a$ is positive. In this paper, we generalize the upper
bound part of this result to any interaction potential with positive
scattering length, i.e, $a>0$ and the lower bound part to some interaction
potentials with shallow and/or narrow negative parts.

${\,}$${\,}$footnotetext: ***${\,}$${\,}$footnotetext: © 2008 by the author.
This paper may be reproduced, in its entirety, for non-commercial purposes.

##  1 Introduction and main theorems

In Dyson’s work [9] and Lieb, Yngvason and Seiringer’s work [7, 6], it is
rigorously proved that the leading term of the ground state energy/particle of
a three dimensional dilute bose gas of mass $m$ in the thermodynamic limit
with density $\varrho$ is $2\pi\hbar^{2}a\varrho/m$, i.e.,

$e(\varrho,m)=2\pi\hbar^{2}a\varrho/m(1+o(1))\,\,\,{\rm
if}\,\,\,a^{3}\varrho\ll 1$ (1.1)

where they assumed that the interaction potential is non-negative, the
scattering length $a$ is positive. This result is generalized to a two
dimensional dilute bose gas in [8]. In this paper, first, in Theorem 1, we
generalize the upper bound part of (1.1) to general interaction potentials $v$
with positive scattering length. On the other hand, for the lower bound on the
ground energy, it was conjectured in [7] that the lower bound part of (1.1)
should hold if the scattering length is positive and $v$ has no $N$-body bound
states for any $N$. Recently, it is proved in [11] that in some cases with
partly shallow negative potential the lower bound part of (1.1) holds. In
Theorem 2, we introduce a different method for the lower bound on (1.1) when
$v$ can have shallow and/or narrow negative components and provide
better(smaller) error term.

We begin with describing the questions more precisely. We write the
Hamiltonian of a system of $N$ interacting bosons which are restricted to a
cubic box of volume $\Lambda=L^{3}$ in the following way (in units where
$\hbar=2m=1$):

$\displaystyle H_{N}\equiv\sum_{i=1}^{N}-\Delta_{i}+\sum_{1\leq i<j\leq
N}v^{a}(x_{i}-x_{j})$ (1.2)

Here $\Delta$ denotes the Laplacian on $\Lambda$ with periodic boundary
condition and $v^{a}$ is a scaled interaction potential, i.e.,

$v^{a}(x)={a^{-2}}\cdot v(x/a),\,\,\,a>0$ (1.3)

The pair interaction potential $v$ is spherically symmetric and supported on
the set $\\{x\in{\mathbb{R}}^{3}:|x|\leq R_{0}\\}$ for some $R_{0}>0$.

######  DEFINITION 1 (Scattering Length).

Assume that $w$ is a pair spherically symmetric interaction potential with
compact support. Denote $E[\phi\,]$ as the energy of the complex-valued
function $\phi$ on ${\mathbb{R}}^{3}$ as follows,

$E[\phi\,]=\int_{{\mathbb{R}}^{3}}|\nabla\phi(x)|^{2}+\mbox{$\frac{1}{2}$}w(x)|\phi(x)|^{2}dx.$
(1.4)

Define the scattering length $SL(w)$ of potential $w$ as the following minimum
energy.

$SL(w)\equiv\min_{\phi}\left\\{\frac{1}{4\pi}E[\phi\,]:\lim_{|x|\to\infty}\phi(x)=1\right\\}$
(1.5)

Note: If $SL(w)>-\infty$, one can easily prove that the Hamiltonian
$-\Delta+\mbox{$\frac{1}{2}$}w$ has no bound state. In particular, when $w\geq
0$, we have $SL(w)\geq 0$ and $w$ has no bound state. One can see that this
definition is equivalent to the definition of scattering length in [5] when
$w>0$.

With the relation between $v$ and $v^{a}$ in (1.3), we can assume that

$SL(v)=1,\,\,\,\,SL(v^{a})=a$ (1.6)

Let $f_{1}(x)$ be the solution of the zero-energy scattering equation of $v$,
i.e.,

$-\Delta f_{1}(x)+\mbox{$\frac{1}{2}$}v(x)f_{1}(x)=0,$ (1.7)

then we have that $f_{a}(x)\equiv f_{1}(x/a)$ is the solution of the following
zero-energy scattering equation of $v^{a}$.

$-\Delta f_{a}(x)+\mbox{$\frac{1}{2}$}v^{a}(x)f_{a}(x)=0,$ (1.8)

As in [5], one can prove that if $f_{a}$ is normalized as
$\lim_{|x|\to\infty}f_{a}(x)=1$, then

$f_{a}(x)=1-a/x,\,\,\,{\rm for}\,\,\,|x|>R_{0}a$ (1.9)

In this paper, we are interested in the ground energy $E(N,\Lambda)$ of
$H_{N}$ in the thermodynamic limit that $\Lambda\to\infty$, $N\to\infty$ and
$N/\Lambda=\varrho$. Low density means that the average inter-particle
distance $\varrho^{-1/3}$ is much larger than the scattering length $a$, i.e.
$a^{3}\varrho\ll 1$.

First, we state that for any fixed $v$, the upper bound on (1.1) holds for the
dilute bose gas.

######  THEOREM 1.

Fix $v$ with $SL[v]=1$ and $v^{a}$ satisfying (1.3). Let $f_{1}$ be the
solution of zero-energy equation of $v$ and normalized as: $f_{1}(\infty)=1$.
In the thermodynamic limit, $\lim_{N\to\infty}N/\Lambda=\varrho$, we have the
following upper bound on $E(N,\Lambda)$, which is the ground energy of $H_{N}$
in (1.2),

$\displaystyle\limsup_{N\to\infty}\frac{E(N,\Lambda)}{4\pi a\varrho N}\leq
1+{\rm const.\,}(a^{3}\varrho)^{1/4},$ (1.10)

for some constant depending on $\|f_{1}\|_{\infty}$, provided that
$\frac{4\pi}{3}a^{3}\varrho\leq 1$.

Note: So far, the best proof of the error term on upper bound, when $v\geq 0$,
is $O(a^{3}\varrho)^{1/3}$, as in [5].

On the other hand, for the lower bound in (1.1), we prove that as long as $v$
has a positive core and is bounded from below, (1.1) holds when the negative
part is small enough (shallow and/or narrow). In the appendix, we show that if
$v^{a}$ is a continuous function on ${\mathbb{R}}^{3}$ and $H_{N}$ has no
bound state for any $N$, $v^{a}$ satisfies the above two requirements, i.e.,

$v^{a}(0)>0,\,\,\,\,\min v^{a}(r)>-\infty$ (1.11)

The above two inequalities (1.11) also hold when $v^{a}$ is stable [1] (the
stability of potential is assumed in [11]).

######  THEOREM 2.

We assume that $v(x)=v_{+}(x)+v_{-}(x)$, $v_{+}(x)\geq 0$,
$v_{-}(x)\geq-\lambda_{-}$, $\lambda_{-}>0$ and $v_{+}$ has a positive core,
i.e. $\exists\,r_{1}$, such that $v_{+}(x)\geq\lambda_{+}>0$ for $|x|\leq
r_{1}$. Here $v_{-}$ need not be negative.

There exist $c_{1}(R_{0}/r_{1})$ and $c_{2}(R_{0}/r_{1})$, which are greater
than one and only depend on $R_{0}/r_{1}$, such that the following holds.

If there exists some positive number $t$ satisfying

$SL[c_{1}(R_{0}/r_{1})\cdot(v+tv_{-})]\geq 0\,\,\,{\rm
and}\,\,\,\lambda_{+}\geq(1+t^{-1})\,c_{2}(R_{0}/r_{1})\cdot\lambda_{-},$
(1.12)

we have the following lower bound on $E(N,\Lambda)$,

$\displaystyle\liminf_{N\to\infty}\frac{E(N,\Lambda)}{4\pi a\varrho N}\geq
1-{\rm const.\,}(a^{3}\varrho)^{1/17},$ (1.13)

for some constant depending on $v_{+}$ and $v_{-}$, provided that
$\frac{4\pi}{3}a^{3}\varrho$ is smaller than some constant depending on
$v_{+}$, $v_{-}$ and $t$.

Note: So far, the best estimation of the error term of the lower bound, when
$v>0$, is also $O(a^{3}\varrho)^{1/17}$, as in [5].

This theorem implies the following two corollaries.

######  COROLLARY 1.

Assume that

$v(x)=v_{+}(x)+\lambda_{-}v_{-}(x),\,\,\,v_{+}(x)\geq 0,\,\,\,v_{-}(x)\geq-1$
(1.14)

and $v_{+}$ has a positive core, i.e. $\exists r_{1}$ such that
$v_{+}(x)\geq\lambda_{+}$ for $|x|\leq r_{1}$. There exists
$\lambda_{0}(r_{1},R_{0},\lambda_{+},v_{-})$ such that, if
$0\leq\lambda_{-}\leq\lambda_{0}$, i.e., the potential is shallow enough, we
have the following lower bound on $E(N,\Lambda)$,

$\displaystyle\liminf_{N\to\infty}\frac{E(N,\Lambda)}{4\pi a\varrho N}\geq
1-{\rm const.\,}(a^{3}\varrho)^{1/17}$ (1.15)

provided that $\frac{4\pi}{3}a^{3}\varrho$ is smaller than some constant
depending on $v_{+}$ and $v_{-}$.

###### Proof.

For fixed $R_{0}$, $r_{1}$ and $\lambda_{+}$, when $\lambda_{-}$ is small
enough, we have that

$SL[c_{1}(R_{0}/r_{1})(v_{+}+2\lambda_{-}v_{-})]\geq 0\,\,\,{\rm
and}\,\,\,\lambda_{+}\geq 2\,c_{2}(R_{0}/r_{1})\lambda_{-}.$ (1.16)

Using Theorem 2, with the choice $t=1$, we arrive at the desired result. ∎

######  COROLLARY 2.

Assume that

$v(x)=v_{+}(x)+v_{-}(x),\,\,\,\,v_{+}(x)\geq 0\geq v_{-}(x)\geq-\lambda_{-}$
(1.17)

and $v_{+}$ has a positive core, i.e. $\exists r_{1}$ such that
$v_{+}(x)\geq\lambda_{+}$ for $|x|\leq r_{1}$. There exist
$\lambda_{0}(R_{0}/r_{1},\lambda_{-})$ and
$\varepsilon\,(R_{0},r_{1},\lambda_{-})$ such that, if

$\lambda_{+}\geq\lambda_{0}\,\,\,{\rm
and}\,\,\,\int_{x\in{\mathbb{R}}^{3}}|v_{-}(x)|dx\leq\varepsilon\,(R_{0},r_{1},\lambda_{-}),$
(1.18)

we have the following lower bound on $E(N,\Lambda)$,

$\displaystyle\liminf_{N\to\infty}\frac{E(N,\Lambda)}{4\pi a\varrho N}\geq
1-{\rm const.\,}(a^{3}\varrho)^{1/17}$ (1.19)

provided that $\frac{4\pi}{3}a^{3}\varrho$ is smaller than some constant
depending on $v_{+}$ and $v_{-}$.

###### Proof.

We choose $\lambda_{0}=\max\\{3,2\,c_{2}(R_{0}/r_{1})\\}\lambda_{-}$, then we
have that $\lambda_{+}\geq\lambda_{0}\geq 3\lambda_{-}$, which implies that

$[v+v_{-}](x)\geq\lambda_{+}/3\geq 0\,\,\,{\rm for}\,\,\,|x|\leq r_{1}$ (1.20)

Then we claim that for any $n\geq 1$ and $\lambda_{+}\geq 3\lambda_{-}$, there
exists $\xi(n)>0$,

$\int_{{\mathbb{R}}^{3}}|v_{-}(x)|dx\leq\xi(n)\Rightarrow SL[n(v+v_{-})]\geq
0.$ (1.21)

To prove (1.21), we shall prove that there exists $\xi(n)>0$, if
$\int_{{\mathbb{R}}^{3}}|v_{-}(x)|dx\leq\xi(n)$, for any non-negative radial
function $f$,

$\int_{|x|\leq R_{0}}|\nabla f|^{2}(x)+\frac{n}{2}(v+v_{-})f^{2}(x)dx\geq 0.$
(1.22)

We can see, with (1.20),

$\displaystyle\int_{|x|\leq R_{0}}\left[|\nabla
f|^{2}(x)+\frac{n}{2}(v+v_{-})f^{2}(x)\right]dx$ (1.23) $\displaystyle\geq$
$\displaystyle\int_{r_{1}\leq|x|\leq R_{0}}\left[|\nabla
f|^{2}(x)-\|nv_{-}\|_{\infty}f^{2}(x)\right]dx$ $\displaystyle\geq$
$\displaystyle\int_{r_{1}\leq|x|\leq R_{0}}|\nabla
f|^{2}(x)dx-n\lambda_{-}\int_{r_{1}\leq|x|\leq R_{0}}|f|^{2}(x)dx$

Hence, if (1.22) does not hold, the right side of (1.23) is less than 0. With
Sobolev inequality and Schwarz’s Inequality, we obtain that there exists
$\eta(n)$ such that

$\left(\int_{r_{1}\leq|x|\leq
R_{0}}|f(x)|^{4}dx\right)^{1/2}\leq\eta(n)\int_{r_{1}\leq|x|\leq
R_{0}}|f|^{2}(x)dx$ (1.24)

On the other hand, with (1.20), $v+v_{-}\geq 2v_{-}$ and Schwarz’s Inequality,
we have that

$\displaystyle\int_{|x|\leq R_{0}}|\nabla
f|^{2}(x)+\frac{n}{2}(v+v_{-})f^{2}(x)dx$ (1.25) $\displaystyle\geq$
$\displaystyle\int_{|x|\leq R_{0}}|\nabla f|^{2}(x)+\int_{|x|\leq
r_{1}}\frac{n\lambda_{+}}{6}f^{2}(x)-\left|\int_{r_{1}\leq|x|\leq
R_{0}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!nv_{-}(x)|f(x)|^{2}dx\right|$
$\displaystyle\geq$ $\displaystyle\int_{|x|\leq R_{0}}|\nabla
f|^{2}(x)+\int_{|x|\leq r_{1}}\frac{n\lambda_{+}}{6}f^{2}(x)$
$\displaystyle-n\eta(n)\left|\int_{r_{1}\leq|x|\leq
R_{0}}v^{2}_{-}(x)dx\right|^{1/2}\int_{r_{1}\leq|x|\leq R_{0}}|f|^{2}(x)dx$
$\displaystyle\geq$ $\displaystyle\int_{|x|\leq R_{0}}|\nabla
f|^{2}(x)+\int_{|x|\leq
r_{1}}\frac{n\lambda_{+}}{6}f^{2}(x)-n\eta(n)\lambda_{-}\|v_{-}\|_{1}^{1/2}\int_{|x|\leq
R_{0}}|f|^{2}(x)dx$

Thus, for $n\geq 1$, if

$\displaystyle\|v_{-}\|_{1}^{1/2}\leq\left(\xi(n)\right)^{1/2}\equiv\frac{1}{n\eta(n)}\cdot\min_{f}\frac{\int_{|x|\leq
R_{0}}|\nabla f|^{2}(x)+\int_{|x|\leq
r_{1}}\frac{n\lambda_{+}}{6}f^{2}(x)dx}{\int_{|r|\leq R_{0}}|f|^{2}(x)dx},$

the inequality (1.22) holds. We note that it is easy to see that $\xi(n)>0$.
Hence we arrive at the desired result (1.21). At last, choosing

$\varepsilon\,(R_{0},r_{1},\lambda_{-})=\xi\left(c_{1}(R_{0}/r_{1})\right)$
(1.26)

and using the result of Theorem 2 with $t=1$, we arrive at the desired result
(1.19). ∎

Remark: Compared with the result of [11], we improve the error term (It was
$(a^{3}\varrho)^{1/31}$ in [11]) and generalize the shapes of potentials,
i.e., the negative part of potential can be shallow and/or narrow. In
particular, there is no restriction on the depth of the interaction potential
$v$, i.e. for $\forall\lambda_{-}>0$, there $\exists v$ satisfying
$\min_{x\in{\mathbb{R}}^{3}}v(x)<-\lambda_{-}$ and Theorem 2 holds.

##  2 Proofs

###  2.1 Proof of Theorem one

###### Proof.

As usual, to prove the upper bound on the ground state energy, we only need to
construct a sequence of trial states $\Psi_{N,\Lambda}$ satisfying

$\limsup_{N\to\infty}\frac{\langle\Psi|H_{N}|\Psi\rangle}{N\langle\Psi|\Psi\rangle}\leq
4\pi a\varrho(1+{\rm const.\,}Y)$ (2.1)

for some constant that depends only on $\|f_{1}\|_{\infty}$. Here we denote
$Y$ as

$Y\equiv\left(\frac{4\pi}{3}a^{3}\varrho\right)^{1/4}$ (2.2)

Following the ideas in [9, 6], we construct the trial state of the following
form,

$\Psi_{N}=\prod_{p=1}^{N}F_{p}$ (2.3)

In [9], $F_{p}$ depends on the the nearest particle to the $x_{p}$ among all
the $x_{i}$ with $i<p$, i.e.,

$F_{p}=f(t_{p}),\,\,\,t_{p}=\min_{i<p}\left\\{\left|x_{i}-x_{p}\right|\right\\}$
(2.4)

via the function $f$ which is very close to the zero energy scattering
solution and satisfies

$0\leq f\leq 1,\,\,\,\,f\,^{\prime}\geq 0$ (2.5)

Hence in [9], $F_{p}$ has the following property

$F_{p,i}\cdot f(|x_{p}-x_{i}|)\leq F_{p}\leq F_{p,i}$ (2.6)

Here $F_{p,i}$ is defined in [9] as the value that $F_{p}$ would take if the
point $x_{i}$ were omitted from consideration.

But in our case where the potential has a negative part, the zero energy
scattering solution $f_{a}$ of $v^{a}$ may not be an increasing function or
bounded by 1 (if it was, the proof would be much simpler). Hence we do not
have the property (2.6). For this reason, our choice of $F_{p}$ will be more
complicated. Our $F_{p}$ depends on all particles near the $x_{p}$, not just
the nearest.

We remark that the function $F_{p}$ should have following properties.

  1. 1.

$F_{p}$ is a continuous function of $x_{i}$ ($1\leq i\leq N$).

  2. 2.

When $|x_{i}-x_{p}|$ is large enough, the position of $x_{i}$ does not effect
$F_{p}$, i.e., $\nabla_{x_{i}}F_{p}=0$.

  3. 3.

$F_{p}$ has a similar property as (2.6).

First we define $\theta_{r}(x)$ as the characteristic function of the set
$\\{x:|x|\leq r\\}$ and $\bar{\theta}_{r}\equiv 1-\theta_{r}$. Choosing
$b=a/Y$, we have

$a/b=\frac{4\pi}{3}b^{3}N/\Lambda=Y.$ (2.7)

Without loss of generality, we assume that $b>\max\\{2R_{0}a,4a\\}$, as in [9,
5]. We define $f(x)$ as

$f(x)=\left\\{\begin{array}[]{ll}f_{a}(x)/f_{a}(b)&b\geq|x|\geq 0\\\ 1&{\rm
otherwise}\,,\end{array}\right.$ (2.8)

Here $f_{a}$ is the zero energy scattering solution of $v^{a}$, as in (1.8).
With the equation (1.9), we note that

$f(x)=\frac{1-a/|x|}{1-a/b},\,\,{\rm for}\,\,\,b\geq|x|\geq R_{0}a.$ (2.9)

Let ${\widetilde{R}}=\max\\{R_{0}a,2a\\}$, which implies that
$f({\widetilde{R}})>\mbox{$\frac{1}{2}$}$. We define $\Theta^{in}_{p}$,
$\Theta^{out}_{p}$ ($1<p\leq N$) as

$\displaystyle\Theta^{in}_{p}\equiv\prod_{j<p}\theta_{\widetilde{R}}(x_{j}-x_{p}),\,\,\,\Theta^{out}_{p}\equiv\prod_{j<p}\bar{\theta}_{\widetilde{R}}(x_{j}-x_{p})$
(2.10)

We can see that $\Theta^{in}_{p}=1$ when $|x_{j}-x_{p}|\leq{\widetilde{R}}$
for all $j<p$ and $\Theta^{out}_{p}=1$ when $|x_{j}-x_{p}|>{\widetilde{R}}$
for all $j<p$. With $\Theta^{in}_{p}$ and $\Theta^{out}_{p}$, we can define
$r_{p}(x_{1},\cdots,x_{N})$ and $R_{p}(x_{1},\cdots,x_{N})$ as follows,
($x_{i}\in[0,L]^{3},i=1,\cdots N$)

$\displaystyle r_{p}\equiv$ (2.11)
$\displaystyle(1-\Theta^{out}_{p})\cdot\min_{i<p}\bigg{\\{}|x_{i}-x_{p}|:f(x_{i}-x_{p})=\min_{j<p}\big{\\{}f(x_{j}-x_{p}):|x_{j}-x_{p}|\leq{\widetilde{R}}\big{\\}}\bigg{\\}}$
$\displaystyle
R_{p}\equiv{\widetilde{R}}\cdot\Theta^{in}_{p}+(1-\Theta^{in}_{p})\times\min_{i<p}\bigg{\\{}|x_{i}-x_{p}|:|x_{i}-x_{p}|>{\widetilde{R}}\bigg{\\}}$

With the definition of $R_{p}$ and (2.9), we have that

  1. 1.

$f(R_{p})\leq f(x_{j}-x_{p})$ for any $j<p$ satisfying
$|x_{j}-x_{p}|>{\widetilde{R}}$,

  2. 2.

$R_{p}\leq|x_{j}-x_{p}|$ for any $j<p$ satisfying
$|x_{j}-x_{p}|>{\widetilde{R}}$.

  3. 3.

$R_{p}\geq{\widetilde{R}}$

  4. 4.

When $\Theta^{in}_{p}=0$, there exists $j_{p}$ such that
$|x_{j_{p}}-x_{p}|=R_{p}$

Similarly, we have

  1. 1.

$f(r_{p})\leq f(x_{j}-x_{p})$ for any $j<p$ satisfying
$|x_{j}-x_{p}|\leq{\widetilde{R}}$,

  2. 2.

$r_{p}\leq|x_{j}-x_{p}|$ for any $j<p$ satisfying
$|x_{j}-x_{p}|\leq{\widetilde{R}}$ and $f(x_{j}-x_{p})=f(r_{p})$.

  3. 3.

$r_{p}\leq{\widetilde{R}}$

  4. 4.

When $\Theta^{out}_{p}=0$, there exists $i_{p}$ such that
$|x_{i_{p}}-x_{p}|=r_{p}$

Then, we define a continuous function $T$ on ${\mathbb{R}}$ as follows

$T(|x|)=\left\\{\begin{array}[]{ll}1&2{\widetilde{R}}\geq|x|\\\
(|x|^{-1}-b^{-1})(2{\widetilde{R}}^{-1}-b^{-1})^{-1}&b\geq|x|\geq
2{\widetilde{R}}\\\ 0&{|x|\geq b}\,,\end{array}\right.$ (2.12)

At last we define $F_{p}(x_{1},\cdots,x_{N})$ on $[0,L]^{3N}$ as follows
($1<p\leq N$),

$\displaystyle
F_{p}\equiv\left\\{\begin{array}[]{ll}f(r_{p})&\Theta^{in}_{p}=1\\\
f(R_{p})&\Theta^{out}_{p}=1\\\
f(r_{p})+T(R_{p})\left[f(R_{p})-f(r_{p})\right]_{-}&{otherwise}\,,\end{array}\right.$
(2.16)

and $F_{1}=1$. Here $[\cdot]_{-}$ denotes the negative part, i.e., $[x]_{-}=x$
when $x<0$ and $[x]_{-}=0$ when $x\geq 0$. We note that for any $x$

$[x]_{-}\leq 0$ (2.17)

Note: If $v\geq 0$, it is well known that $f$ is an increasing function, which
implies the $F_{p}$ we defined is equal to the $F_{p}$ in [9].

One can prove that $F_{p}$ is a continuous function of $(x_{1},\cdots,x_{N})$
by checking that, for any $j\neq p>1$ and fixed
$x_{1},\cdots,x_{j-1},x_{j+1},\cdots,x_{N}$, $F_{p}$ is a continuous function
of $x_{j}$. First we can see that it is trivial for $j>p$, since $F_{p}$ is
independent of $x_{j}$ when $j>p$. For $j<p$, it only remains to check that
$F_{p}$ is continuous when $x_{j}$ moves from $|x_{j}-x_{p}|={\widetilde{R}}$
to $|x_{j}-x_{p}|={\widetilde{R}}+0^{+}$. One can see that when
$|x_{j}-x_{p}|={\widetilde{R}}$, $f(R_{p})\geq
f({\widetilde{R}})=f(x_{j}-x_{p})\geq f(r_{p})$, so $F_{p}=f(r_{p})$, i.e.

$F_{p}=\min\bigg{\\{}\min_{k:k\neq
j,k<p}\\{f(x_{k}-x_{p}):|x_{k}-x_{p}|\leq{\widetilde{R}}\\},f(x_{j}-x_{p})\bigg{\\}}$
(2.18)

On the other hand, when $|x_{j}-x_{p}|={\widetilde{R}}+0^{+}\leq
2{\widetilde{R}}$, we can see that $R_{p}=|x_{j}-x_{p}|$, $T(R_{p})=1$ and
$f(R_{p})=f({\widetilde{R}})+0^{+}$. Hence,

$F_{p}=\min\bigg{\\{}\min_{k:k\neq
j,k<p}\\{f(x_{k}-x_{p}):|x_{k}-x_{p}|\leq{\widetilde{R}}\\},f(x_{j}-x_{p})\bigg{\\}}$
(2.19)

Hence we arrive at the desired result that $F_{p}$ is continuous function.

We can also see that $F_{p}$ is non-negative and bounded as follows

$M\equiv\|F_{p}\|_{\infty}=\|f\|_{\infty}\leq(1-a/b)^{-1}\|f_{a}\|_{\infty}=(1-a/b)^{-1}\|f_{1}\|_{\infty}\leq
2\|f_{1}\|_{\infty}.$ (2.20)

Here we use the fact $f_{a}(x)=f_{1}(x/a)$.

By the definition of $F_{p}$, one can see that $F_{p}=1$ when
$\prod_{q<p}\bar{\theta}_{b}(x_{p}-x_{q})=1$ and $F_{p}\leq 1$ when
$\prod_{q<p}\bar{\theta}_{\widetilde{R}}(x_{p}-x_{q})=1$, so

$1-\sum_{q<p}\theta_{b}(x_{p}-x_{q})\leq F_{p}\leq
1+\sum_{q<p}(M-1)\theta_{\widetilde{R}}(x_{p}-x_{q})$ (2.21)

We now construct the state functions $\Phi_{k}$ as follows ($1\leq k\leq N$)

$\Phi_{k}=\prod_{p=1}^{k}F_{p}$

Note: all $\Phi$’s are functions on $[0,L]^{3N}$ and $\Phi_{k}$ is independent
of $x_{l}$ for $l>k$. We will choose $\Psi=\Phi_{N}$ for (2.1).

As in [7], for proving the upper bound on the total energy
$\langle\Phi_{N}|H_{N}|\Phi_{N}\rangle\|\Phi_{N}\|^{-2}_{2}$, we shall
estimate the upper bounds on

$\|\Phi_{N}\|^{-2}_{2}\int\sum_{i}|\nabla_{i}\Phi_{N}|^{2}\prod_{j=1}^{N}dx_{j}\,\,\,\,\,\,{\rm
and}\,\,\,\,\,\,\|\Phi_{N}\|^{-2}_{2}\int\sum_{i<j}v^{a}(x_{i}-x_{j})|\Phi_{N}|^{2}\prod_{k=1}^{N}dx_{k}$
(2.22)

Since in our case $v^{a}$ has negative parts, our strategy is more
complicated, i.e., we need to estimate the upper bounds on

$\|\Phi_{N}\|^{-2}_{2}\int\sum_{i}|\nabla_{i}\Phi_{N}|^{2}\prod_{j=1}^{N}dx_{j}\,\,\,\,\,\,{\rm
and}\,\,\,\,\,\,\|\Phi_{N}\|^{-2}_{2}\int\sum_{i<j}[v^{a}]_{+}(x_{i}-x_{j})|\Phi_{N}|^{2}\prod_{k=1}^{N}dx_{k}$
(2.23)

and the lower bound on

$\|\Phi_{N}\|^{-2}_{2}\int\sum_{i<j}\bigg{|}[v^{a}]_{-}(x_{i}-x_{j})\bigg{|}\cdot|\Phi_{N}|^{2}\prod_{k=1}^{N}dx_{k}$
(2.24)

In the remainder of this section we are going to prove the following three
inequalities

  * •

$\|\Phi_{N}\|^{-2}_{2}\int\sum_{i}|\nabla_{i}\Phi_{N}|^{2}\prod_{j=1}^{N}dx_{j}\leq(1+o(1))\frac{N^{2}}{\Lambda}\int_{{\mathbb{R}}^{3}}|\nabla
f(x)|^{2}dx$

  * •

$\|\Phi_{N}\|^{-2}_{2}\int\sum_{i<j}[v^{a}]_{+}(x_{i}-x_{j})|\Phi_{N}|^{2}\prod_{k=1}^{N}dx_{k}\leq(1+o(1))\frac{N^{2}}{\Lambda}\int_{{\mathbb{R}}^{3}}\mbox{$\frac{1}{2}$}[v]_{+}|f(x)|^{2}dx$

  * •

$\|\Phi_{N}\|^{-2}_{2}\int\sum_{i<j}\bigg{|}[v^{a}]_{-}(x_{i}-x_{j})\bigg{|}\cdot|\Phi_{N}|^{2}\prod_{k=1}^{N}dx_{k}\geq(1-o(1))\frac{N^{2}}{\Lambda}\int_{{\mathbb{R}}^{3}}\mbox{$\frac{1}{2}$}\left|[v]_{-}\right|\cdot|f(x)|^{2}dx.$

To prove these inequalities, we begin with proving the following three
inequalities (all $\Phi$’s are functions on $[0,L]^{3N}$):

  1. 1.

For any $m$-variable function $g_{m}(x_{i_{1}}\cdots x_{i_{m}})$, $m<k\leq N$,
$i_{j}\neq i_{k}$ for $j\neq k$, we have

$\displaystyle\|\Phi_{k}F^{-1}_{i_{1}}\cdots
F^{-1}_{i_{m}}g_{m}\|^{2}_{2}\leq(2M)^{2m}\Lambda^{-m}\|\Phi_{k-m}\|^{2}_{2}\|g_{m}\|^{2}_{2}$
(2.25)

  2. 2.

For any two variable function $g_{2}(x_{i},x_{i^{\prime}})$ ($i<i^{\prime}$),
we have

$\|\Phi_{N}F^{-1}_{i^{\prime}}g_{2}\|^{2}_{2}\leq\|g_{2}\|^{2}_{2}\Lambda^{-2}\|\Phi_{N}\|^{2}_{2}(1+{\rm
const.\,}Y)$ (2.26)

  3. 3.

Let $f_{i,i^{\prime}}=f(x_{i}-x_{i^{\prime}})$, for any two variable function
$g_{2}(x_{i},x_{i^{\prime}})$ ($i<i^{\prime}$), we have

$\displaystyle\|\Phi_{N}f^{-1}_{i,i^{\prime}}g_{2}\|^{2}_{2}\geq\|g_{2}\|^{2}_{2}\Lambda^{-2}\|\Phi_{N}\|^{2}_{2}(1-{\rm
const.\,}Y)$ (2.27)

Note: if $f_{i,i^{\prime}}=0$ and $i<i^{\prime}$, then $F_{i^{\prime}}=0$, so
$\Phi_{N}f^{-1}_{i,i^{\prime}}$ is definable.

We will use (2.25) for controlling the error terms. The inequalities (2.26)
and (2.27) will be used in estimating the terms (2.23) and (2.24),
respectively.

We begin with deriving a lower bound on $\|\Phi_{N}\|^{2}_{2}$. For $i\neq p$,
Let $F_{p,i}$ be the value that $F_{p}$ would take if changing the order of
particles as follows,

$F_{p,i}(x_{1}\cdots x_{N})\equiv F_{n(p,i)}(x_{1}\cdots x_{i-1},x_{i+1}\cdots
x_{N},x_{i}\,)$ (2.28)

Here $n(p,i)$ is defined as follows ($i\neq p$)

$n(p,i)=\left\\{\begin{array}[]{ll}p&i>p\\\ p-1&i<p\,,\end{array}\right.$
(2.29)

Similarly, we can define $F_{p,i,j}(x_{1}\cdots x_{N})$ as

$F_{p,i,j}(x_{1}\cdots x_{N})\equiv F_{m(p,i,j)}(x_{1}\cdots
x_{i-1},x_{i+1}\cdots x_{j-1},x_{j+1}\cdots
x_{N},x_{i},x_{j}\,)\,\,\,for\,\,\,i<j$ (2.30)

and $F_{p,i,j}=F_{p,j,i}$ for $j<i$. Here $m(p,i,j)$ is defined as the number
of the elements of the set $\\{1,\cdots,p\\}\setminus\\{i,j\\}.$

Note: As we mentioned $F_{p}$ we defined is equal to the $F_{p}$ in [6] in the
case when $v\geq 0$. Furthermore, one can see that our definitions of
$F_{p,i}$ and $F_{p,i,j}$ are equivalent to those definitions in [6] when
$v\geq 0$.

With the definitions of $F_{p}$ and $F_{p,i}$, we obtain that $F_{p,i}$ is
independent of $x_{i}$ and $F_{p}$ is bounded from below as follows

$F_{p}(x_{1}\cdots x_{N})\geq\left\\{\begin{array}[]{ll}F_{p,i}&i>p\\\
F_{p,i}\bar{\theta}_{b}(x_{p}-x_{i})&i<p\,,\end{array}\right.$ (2.31)

and

$F_{p}\geq\prod_{i<p}\bar{\theta}_{b}(x_{p}-x_{i}).$

Then $\Phi^{2}_{N}$ is bounded from below, for any fixed $i$, by

$\displaystyle\bigg{|}F_{1}\cdot F_{2}\cdots F_{N}\bigg{|}^{2}$
$\displaystyle\geq\bigg{|}F_{1,\,i}\cdots F_{i-1,\,i}F_{i+1,\,i}\cdots
F_{N,\,i}\bigg{|}^{2}\times\prod_{j\neq i}\bar{\theta}_{b}(x_{i}-x_{j})$
$\displaystyle\geq\bigg{|}F_{1,\,i}\cdots F_{i-1,\,i}F_{i+1,\,i}\cdots
F_{N,\,i}\bigg{|}^{2}\times\bigg{(}1-\sum_{j\neq
i}\theta_{b}(x_{i}-x_{j})\bigg{)}$

Integrating both sides with $\int\prod_{j=1}^{N}dx_{j}$, we obtain that

$\displaystyle\|\Phi_{N}\|_{2}^{2}\geq\|\Phi_{N-1}\|_{2}^{2}\left(1-\frac{4\pi
b^{3}}{3}N/\Lambda\right)=\|\Phi_{N-1}\|_{2}^{2}(1-Y)$ (2.33)

Here we used the fact that

$\|\Phi_{N-1}\|^{2}_{2}=\int\bigg{|}F_{1,\,i}\cdots
F_{i-1,\,i}F_{i+1,\,i}\cdots F_{N,\,i}\bigg{|}^{2}\prod_{j}dx_{j}.$ (2.34)

Similarly, one can also prove that for $k\leq N$,

$\displaystyle\|\Phi_{k}\|_{2}^{2}\geq\|\Phi_{k-1}\|_{2}^{2}(1-Y)$ (2.35)

Next we are going to prove (2.25) in the case $m=1,k=N$, i.e.,

$\displaystyle\|\Phi_{N}F_{i}^{-1}g_{1}\|^{2}_{2}\leq
4M^{2}\Lambda^{-1}\|\Phi_{N-1}\|^{2}_{2}\,\|g_{1}\|^{2}_{2}$ (2.36)

One can check that $F_{p}>F_{p,i}$ only when the following conditions are
satisfied:

  1. 1.

$i<p$

  2. 2.

$|x_{i}-x_{p}|\leq{\widetilde{R}}$,

  3. 3.

for any other $j<p$, $|x_{j}-x_{p}|$ is greater than ${\widetilde{R}}$,

  4. 4.

$T(R_{p})<1$, i.e. for any other $j<p$, $|x_{j}-x_{p}|>2{\widetilde{R}}$,

i.e.,

$F_{p}>F_{p,i}\Rightarrow
G_{p,i}\equiv\theta_{\widetilde{R}}(x_{p}-x_{i})\prod_{j<p,j\neq
i}\bar{\theta}_{2{\widetilde{R}}}(x_{j}-x_{p})=1$ (2.37)

On the other hand, using the fact that
$f({\widetilde{R}})>\mbox{$\frac{1}{2}$}$, one obtains that if
$F_{p}>F_{p,i}$,

$F_{p,i}\prod_{j<p,j\neq
i}\bar{\theta}_{2{\widetilde{R}}}(x_{j}-x_{p})>f({\widetilde{R}})\prod_{j<p,j\neq
i}\bar{\theta}_{2{\widetilde{R}}}(x_{j}-x_{p})\geq\frac{1}{2}\prod_{j<p,j\neq
i}\bar{\theta}_{2{\widetilde{R}}}(x_{j}-x_{p})$

Hence when $G_{p,i}=1$, we have $2MF_{p,i}G_{p,i}\geq M\geq F_{p}$, i.e.,

$F_{p}\leq\left\\{\begin{array}[]{ll}F_{p,i}&i>p\\\
F_{p,i}\bigg{(}1+(2M-1)G_{p,i}\bigg{)}&i<p\,.\end{array}\right.$ (2.38)

By the definition of $G$’s, one can see that if $p,q>i$ and $p\neq q$,

$G_{p,i}G_{q,i}=0.$ (2.39)

Hence, we have that

$\displaystyle\prod_{p>i}\bigg{(}1+(2M-1)G_{p,i}\bigg{)}\leq 2M$ (2.40)

Combining (2.38) and (2.40), we have the upper bound on $|\Phi_{N}F_{i}^{-1}|$
as follows,

$\displaystyle\bigg{|}F_{1}\cdot F_{2}\cdots F_{N}F_{i}^{-1}\bigg{|}^{2}\leq
4M^{2}\bigg{|}F_{1,\,i}\cdots F_{i-1,\,i}F_{i+1,\,i}\cdots
F_{N,\,i}\bigg{|}^{2},$ (2.41)

which implies the desired result (2.36) with (2.34). Furthermore, for any
$m$-variable function $g_{m}(x_{i_{1}}\cdots x_{i_{m}})$

$\displaystyle\|\Phi_{N}F^{-1}_{i_{1}}\cdots
F^{-1}_{i_{m}}g_{m}\|^{2}_{2}\leq(2M)^{2m}\Lambda^{-m}\|\Phi_{N-m}\|^{2}_{2}\|g_{m}\|^{2}_{2}$
(2.42)

With the inequality (2.33) and the fact $F_{i}\leq M$ for any $i\leq N$, we
get

$\displaystyle\|\Phi_{N}g_{m}\|^{2}_{2}\leq$
$\displaystyle(2M^{2})^{2m}\|\Phi_{N-m}\|^{2}_{2}\|g_{m}\|^{2}_{2}$ (2.43)
$\displaystyle\leq$
$\displaystyle(1-Y)^{-m}(2M^{2})^{2m}{\Lambda}^{-m}\|\Phi_{N}\|^{2}_{2}\|g_{m}\|^{2}_{2}$

Similarly, we can generalize this result to $m<k\leq N$

$\displaystyle\|\Phi_{k}g_{m}\|^{2}_{2}\leq$
$\displaystyle(2M^{2})^{2m}\|\Phi_{k-m}\|^{2}_{2}\|g_{m}\|^{2}_{2}$ (2.44)
$\displaystyle\leq$
$\displaystyle(1-Y)^{-m}(2M^{2})^{2m}{\Lambda}^{-m}\|\Phi_{k}\|^{2}_{2}\|g_{m}\|^{2}_{2}$

Now we shall prove the upper bound on $\|\Phi_{N}\|^{2}_{2}$ with (2.44).
Choosing $p=N$, with the bounds of $F_{p}$ in (2.21), we get that

$\displaystyle\Phi_{N}^{2}\leq$ $\displaystyle F^{2}_{1}\cdot F^{2}_{2}\cdots
F^{2}_{N-1}\left(1+\sum_{j<N}M^{2}\theta_{\widetilde{R}}(x_{j}-x_{N})\right)$
(2.45) $\displaystyle=$
$\displaystyle\Phi_{N-1}^{2}+\Phi_{N-1}^{2}\sum_{j<N}M^{2}\theta_{\widetilde{R}}(x_{j}-x_{N})$

Hence, using the inequalities (2.44)($m=1$) and (2.35), we obtain that

$\displaystyle\|\Phi_{N}\|^{2}_{2}$
$\displaystyle\leq\|\Phi_{N-1}\|^{2}_{2}+(1-Y)^{-1}(2M^{2})^{2}Y\|\Phi_{N-2}\|^{2}_{2}$
$\displaystyle\leq\|\Phi_{N-1}\|^{2}_{2}(1+{\rm const.\,}Y)$

Putting (2.1) and (2.33) together, we obtain the relation between
$\|\Phi_{N}\|$ and $\|\Phi_{N-1}\|$

$\|\Phi_{N}\|^{2}_{2}=\|\Phi_{N-1}\|^{2}_{2}(1+O(Y))$ (2.47)

Similarly, for $k\leq N$

$\|\Phi_{k}\|^{2}_{2}=\|\Phi_{k-1}\|^{2}_{2}(1+O(Y))$ (2.48)

Next, we shall prove (2.26), i.e.,

$\displaystyle\|\Phi_{N}F^{-1}_{i^{\prime}}g_{2}\|^{2}_{2}\leq\|g_{2}\|^{2}_{2}\Lambda^{-2}\|\Phi_{N}\|^{2}_{2}(1+{\rm
const.\,}Y).$ (2.49)

Using the inequalities
$F_{i}\leq\left(1+\sum_{l<i}(M-1)\theta_{\widetilde{R}}(x_{l}-x_{i})\right)$
and (2.38), with the property of the $G$’s in (2.39), we get

$\displaystyle\prod_{k\neq i^{\prime}}F_{k}$
$\displaystyle\leq\left(1+\sum_{l<i}(M-1)\theta_{\widetilde{R}}(x_{l}-x_{i})\right)\prod_{k\neq
i^{\prime},i}F_{k\,,i}\cdot\bigg{(}1+\sum_{j>i}2MG_{j,i}\bigg{)}$ (2.50)

Similarly, replacing $F_{p,i}$’s with $F_{p,i,i^{\prime}}$’s and using the
fact that $G_{k,l}\leq\theta_{\widetilde{R}}(x_{k}-x_{l})$, we get

$\displaystyle\Phi_{N}F^{-1}_{i^{\prime}}\leq$ $\displaystyle\prod_{k\neq
i^{\prime},i}F_{k,i,i^{\prime}}\left(1+\sum_{l<i}M\theta_{\widetilde{R}}(x_{l}-x_{i})\right)$
$\displaystyle\times\bigg{(}1+\sum_{j>i}2M\theta_{\widetilde{R}}(x_{j}-x_{i})\bigg{)}\times\bigg{(}1+\sum_{j^{\prime}>i^{\prime}}2M\theta_{\widetilde{R}}(x_{j}^{\prime}-x_{i}^{\prime})\bigg{)}$

Expanding (2.1), multiplying $g_{2}(x_{i},x_{i^{\prime}})$ to each side and
integrating them with $\prod_{k=1}^{N}dx_{k}$, with the result of (2.44,
2.48), we obtain that

$\|\Phi_{N}F^{-1}_{i^{\prime}}g_{2}\|^{2}_{2}\leq\|g_{2}\|^{2}_{2}\Lambda^{-2}\|\Phi_{N}\|^{2}_{2}(1+{\rm
const.\,}Y)$ (2.52)

So far we proved some upper bounds of the expectation value of $\Phi_{N}$.
Next we are going to prove the following lower bound on
$\|\Phi_{N}f^{-1}_{i,i^{\prime}}g_{2}\|^{2}_{2}$:

$\displaystyle\|\Phi_{N}f^{-1}_{i,i^{\prime}}g_{2}\|^{2}_{2}\geq\|g_{2}\|^{2}_{2}\Lambda^{-2}\|\Phi_{N}\|^{2}_{2}(1-{\rm
const.\,}Y)$ (2.53)

Here we denote $f_{i,i^{\prime}}=f(x_{i}-x_{i^{\prime}})$ ($i<i^{\prime}$).
First, by the definition of $F_{i^{\prime}}$, one can see that
$F_{i^{\prime}}=f(x_{i}-x_{i^{\prime}})$ when $\prod_{k<i^{\prime},k\neq
i}\bar{\theta}_{b}(x_{k}-x_{i}^{\prime})=1$, i.e.,

$F^{2}_{i^{\prime}}\geq
f(x_{i}-x_{i^{\prime}})^{2}\left(1-\sum_{k<i^{\prime},k\neq
i}\theta_{b}(x_{k}-x_{i^{\prime}})\right)$ (2.54)

Using this inequality and (2.42) with $m=3$, $i_{1}=i$, $i_{2}=i^{\prime}$ and
$i_{3}=k$, we obtain that

$\displaystyle\left\|\Phi_{N}f^{-1}_{i,i^{\prime}}g_{2}\right\|^{2}_{2}\geq\left\|\Phi_{N}F^{-1}_{i^{\prime}}g_{2}\right\|^{2}_{2}-{\rm
const.\,}Y\left\|g_{2}\right\|^{2}_{2}\Lambda^{-2}\left\|\Phi_{N}\right\|^{2}_{2}$
(2.55)

Then with the lower bound on $F_{i}^{2}$ in (2.21), i.e., $F^{2}_{i}\geq
1-\sum_{k<i}\theta_{b}(x_{k}-x_{i})$, we obtain that

$\displaystyle\|\Phi_{N}f^{-1}_{i,i^{\prime}}g_{2}\|^{2}_{2}\geq\|\Phi_{N}F^{-1}_{i}F^{-1}_{i^{\prime}}g_{2}\|^{2}_{2}-{\rm
const.\,}Y\|g_{2}\|^{2}_{2}\Lambda^{-2}\|\Phi_{N}\|^{2}_{2}$ (2.56)

Again, using the bound on $F_{p}$ in (2.31), we see that

$\displaystyle\Phi_{N}F^{-1}_{i}F^{-1}_{i^{\prime}}\geq$
$\displaystyle\prod_{k\neq
i,i^{\prime}}F_{k,i,i^{\prime}}\left(1-\sum_{l<i}\theta_{b}(x_{l}-x_{i})\right)\times\left(1-\sum_{l^{\prime}<{i^{\prime}},\,\,l^{\prime}\neq
i}\theta_{b}(x_{l^{\prime}}-x_{i^{\prime}})\right)$

Then using (2.44) and (2.48), we arrive at the desired result (2.53).

So far, we have proved the inequalities we need for calculating the value of
$\langle\Phi_{N}|\sum_{i,j}v^{a}(x_{i}-x_{j})|\Phi_{N}\rangle$. Then we need
to calculate $\nabla_{i}\Phi_{N}$. We denote $i_{p}$ as the particle
satisfying ${i_{p}}<p$ and $|x_{i_{p}}-x_{p}|=r_{p}$ and $n^{r}_{p}$ as the
unit vector in the direction of $x_{p}-x_{i_{p}}$. Similarly, denote $j_{p}$
as the particle satisfying ${j_{p}}<p$ and $|x_{j_{p}}-x_{p}|=R_{p}$ and
$n^{R}_{p}$ as the unit vector in the direction of $x_{p}-x_{j_{p}}$. We
remark that such $i_{p}$ or $j_{p}$ may not exist in some cases, but we do
define them as $0$. We denote $\nabla_{0}F_{p}=0$. Recall the definition of
$F_{p}$ in (2.16). We have

$\displaystyle-\nabla_{p}F_{p}$
$\displaystyle=\nabla_{i_{p}}F_{p}+\nabla_{j_{p}}F_{p}$ (2.57)
$\displaystyle\nabla_{i_{p}}F_{p}$
$\displaystyle=-n^{r}_{p}f\,^{\prime}(r_{p})\bigg{(}\Theta^{in}_{p}+\Theta^{-}_{p}(1-T(R_{p}))+\Theta^{+}_{p}\bigg{)}$
$\displaystyle\nabla_{j_{p}}F_{p}$
$\displaystyle=-n^{R}_{p}\bigg{(}\Theta^{out}_{p}f\,^{\prime}(R_{p})+\Theta^{-}_{p}T(R_{p})f^{\prime}(R_{p})+\Theta^{-}_{p}T\,^{\prime}(R_{p})\big{(}f(R_{p})-f(r_{p})\big{)}\bigg{)}$

Here $\Theta^{+}_{p}$ is the function of $(x_{1}\cdots x_{N})$ which is
defined as

$\Theta^{+}_{p}\equiv\big{[}1-\Theta^{in}-\Theta^{out}\big{]}\cdot
h\big{[}f(R_{p})-f(r_{p})\big{]}$ (2.58)

and $\Theta^{-}_{p}$ is defined as

$\Theta^{-}_{p}\equiv\big{[}1-\Theta^{in}-\Theta^{out}\big{]}\cdot
h\big{[}f(r_{p})-f(R_{p})\big{]}$ (2.59)

Here $h$ is the Heaviside step function. By the definition of $\Phi_{N}$, we
obtain that

$\displaystyle\frac{|\nabla_{p}\Phi_{N}|^{2}}{|\Phi_{N}|^{2}}=\left|-F_{p}^{-1}\nabla_{i_{p}}F_{p}-F_{p}^{-1}\nabla_{j_{p}}F_{p}+\sum_{q,i_{q}=p}F_{q}^{-1}\nabla_{p}F_{q}+\sum_{q,j_{q}=p}F_{q}^{-1}\nabla_{p}F_{q}\right|^{2}$

Then with (2.57), we have that

$\displaystyle\sum_{p}|\nabla_{p}\Phi_{N}|^{2}\leq$ $\displaystyle
2|\Phi_{N}|^{2}\sum_{p}F_{p}^{-2}\bigg{(}|f\,^{\prime}(r_{p})|^{2}\big{(}\Theta^{in}_{p}+\Theta^{-}_{p}|1-T(R_{p})|^{2}+\Theta^{+}_{p}\big{)}$
$\displaystyle+|T(R_{p})f\,^{\prime}(R_{p})|^{2}\Theta^{-}_{p}+|T\,^{\prime}(R_{p})|^{2}|f(R_{p})-f(r_{p})|^{2}\Theta^{-}_{p}$
$\displaystyle+|T(R_{p})|\cdot|1-T(R_{p})|\cdot|f\,^{\prime}(r_{p})|\cdot|f\,^{\prime}(R_{p})|\Theta^{-}_{p}+|f\,^{\prime}(R_{p})|^{2}\Theta^{out}_{p}\bigg{)}$
$\displaystyle+2|\Phi_{N}|^{2}\sum_{k<p<q}F_{p}^{-1}F_{q}^{-1}\bigg{(}|\nabla_{k}F_{p}|\cdot|\nabla_{p}F_{q}|+|\nabla_{k}F_{p}|\cdot|\nabla_{k}F_{q}|\bigg{)}$

Because $0\leq T\leq 1$ and $i_{p}\neq j_{p}$, one can easily prove that for
any fixed $p$,

$\displaystyle\bigg{(}|f\,^{\prime}(r_{p})|^{2}\big{(}\Theta^{in}_{p}+\Theta^{-}_{p}|1-T(R_{p})|^{2}+\Theta^{+}_{p}\big{)}+|f\,^{\prime}(R_{p})|^{2}\Theta^{out}_{p}$
$\displaystyle+$
$\displaystyle|T(R_{p})f\,^{\prime}(R_{p})|^{2}\Theta^{-}_{p}+|T(R_{p})|\cdot|1-T(R_{p})|\cdot|f\,^{\prime}(r_{p})|\cdot|f\,^{\prime}(R_{p})|\Theta^{-}_{p}\bigg{)}$
$\displaystyle\leq$
$\displaystyle\sum_{k:k<p}f\,^{\prime}(|x_{p}-x_{k}|)^{2},$

and

$|T\,^{\prime}(R_{p})|^{2}|f(R_{p})-f(r_{p})|^{2}\Theta^{-}_{p}\leq
M^{2}\sum_{k:k<p}\left(T\,^{\prime}(|x_{p}-x_{k}|)^{2}\sum_{j:j\neq
k,\,p}\theta_{\widetilde{R}}(x_{j}-x_{p})\right)$ (2.60)

Hence, we obtain that

$\displaystyle\langle\Phi_{N}|H_{N}|\Phi_{N}\rangle\leq
2\sum_{i<j}{\int|\Phi_{N}|^{2}F_{j}^{-2}\bigg{(}\mbox{$\frac{1}{2}$}f(x_{i}-x_{j})^{2}[v(x_{i}-x_{j})]_{+}+f\,^{\prime}(x_{i}-x_{j})^{2}\bigg{)}}$
$\displaystyle-2\sum_{i<j}{\int|\Phi_{N}|^{2}f^{-2}(x_{i}-x_{j})\bigg{(}\mbox{$\frac{1}{2}$}f^{2}(x_{i}-x_{j})\bigg{|}[v(x_{i}-x_{j})]_{-}\bigg{|}\bigg{)}}$
$\displaystyle+2\sum_{i<j}\int|\Phi_{N}|^{2}M^{2}\sum_{k<p}\left(T\,^{\prime}(|x_{p}-x_{k}|)^{2}\sum_{j:j\neq
k,\,p}\theta_{\widetilde{R}}(x_{j}-x_{p})\right)$ (2.61)
$\displaystyle+2\sum_{k<p<q}\int|\Phi_{N}|^{2}F_{p}^{-1}F_{q}^{-1}\bigg{(}|\nabla_{k}F_{p}|\cdot|\nabla_{p}F_{q}|+|\nabla_{k}F_{p}|\cdot|\nabla_{k}F_{q}|\bigg{)}$

Here $[\cdot]_{+}$ and $[\cdot]_{-}$ denote the positive and negative part,
respectively and we used the fact that $F_{j}\leq f(x_{i}-x_{j})$ when $i<j$
and $|x_{i}-x_{j}|\leq{\widetilde{R}}$, which implies that

$[v(x_{i}-x_{j})]_{+}\leq F_{j}^{-2}f(x_{i}-x_{j})^{2}[v(x_{i}-x_{j})]_{+}$
(2.62)

With the results in (2.52) and (2.53), we can obtain the upper bound on the
main part of $\langle\Phi_{N}|H_{N}|\Phi_{N}\rangle$, i.e.,

$\displaystyle 2\sum_{i<j}\int|\Phi_{N}|^{2}\times$
$\displaystyle\frac{1}{2}\bigg{(}F_{j}^{-2}\left[f^{2}(x_{i}-x_{j})[v(x_{i}-x_{j})]_{+}+f\,^{\prime}(x_{i}-x_{j})^{2}\right]-\frac{f^{2}(x_{i}-x_{j})}{f^{2}(x_{i}-x_{j})}\bigg{|}[v(x_{i}-x_{j})]_{-}\bigg{|}\bigg{)}$
$\displaystyle\leq 4\pi aN^{2}/\Lambda(1+{\rm const.\,}Y)\|\Phi_{N}\|^{2}_{2}$
(2.63)

With the definition of $T$ in (2.12) and (2.43), we obtain that the third line
of (2.61) is bounded as ${\rm const.\,}aN^{2}Y\|\Phi_{N}\|^{2}_{2}/\Lambda$.
For the other terms, we have

$|\nabla_{i_{p}}F_{p}|\leq|f\,^{\prime}(|x_{i_{p}}-x_{p}|)|,\,\,\,\,|\nabla_{j_{p}}F_{p}|\leq|f\,^{\prime}(|x_{j_{p}}-x_{p}|)|+MT\,^{\prime}(|x_{j_{p}}-x_{p}|)$
(2.64)

Hence, with the inequality (2.42), we can prove that the last line in (2.61)
are bounded as

${\rm const.\,}N^{3}\Lambda^{-2}(K+L)^{2}\|\Phi_{N}\|^{2}_{2}$ (2.65)

Here $K$ and $L$ are defined as follows

$\displaystyle
K\equiv\int_{{\mathbb{R}}^{3}}|f\,^{\prime}\left(|x-y|\right)|dy\,\,\,\,\,\,\,L\equiv\int_{{\mathbb{R}}^{3}}T\,^{\prime}\left(|x-y|\right)dy.$
(2.66)

Note that $K$ and $L$ are independent of $x$. By the definitions of $f$ in
(2.8) and $T$ in (2.12), we get that

$K=O(ab),\,\,\,\ L=O({\widetilde{R}}b)=O(ab)$ (2.67)

Hence we obtain that the last line in (2.61) are bounded by ${\rm
const.\,}aN^{2}Y^{2}$. Combining this result with (2.63), we get the following
result,

$\frac{\langle\Phi_{N}|H_{N}|\Phi_{N}\rangle}{\|\Phi_{N}\|^{2}_{2}}\leq 4\pi
aN^{2}/\Lambda(1+{\rm const.\,}Y)$ (2.68)

At last, by choosing $\Psi=\Phi_{N}$, we arrive at the desired result (2.1),
which implies Theorem 1. ∎

###  2.2 Proof of Theorem Two

###### Proof.

Following the ideas in [7], we need to replace the hard potential by a soft
potential at the expense of local kinetic energy. This method has been used in
many papers on dilute bose or fermi gases [7, 8, 6, 2, 3]. But in this method
the kinetic energy of particle $i$ only can be used for the hard-soft
potential replacement between the particle $i$ and one other $j$ (the nearest
particle [7]). In our case that $v^{a}$ is partly negative, we can not ignore
the potential between $i$ and other $k$’s for the lower bound on the energy.
To solve this problem, we begin with separating the whole Hamiltonian into two
parts, (1) The Hamiltonian of the energy when two particles are close to each
other and they are far away from the others. (2) The Hamiltonian of the
remaining energy. In the remainder of this section, we prove that the first
part is greater than $4\pi aN^{2}\Lambda^{-1}(1-O(a^{3}\varrho)^{1/17})$ and
the second part is non-negative.

Another important property Lieb and Yngvason used in [7] is the
superadditivity of the ground energy $E(n,\ell)$ of $n$ particles in
$[0,\ell]^{3}$ with Neumann boundary condition, i.e.,

$E(n+n^{\prime},\ell)\geq E(n,\ell)+E(n^{\prime},\ell)$ (2.69)

This property is trivial in the case $v^{a}\geq 0$. In our proof, we are not
going to prove any similar property, actually we only need the property
(2.141) that for fixed $\ell$, when $n$ is larger than $4\varrho\ell^{3}$, the
energy/particle is greater than $8\pi a\varrho$, as in (2.62) of [5], i.e.,

$E(n,\ell)/n\geq 8\pi a\varrho(1-{\rm const.\,}(a^{3}\varrho)^{1/17})$ (2.70)

which will be proved in Lemma 1.

Choosing

$R=a(a^{3}\varrho)^{-5/17}\geq 2R_{0}a,$ (2.71)

we define $F_{i,j}$ for $i\neq j$ as follows:

$F_{i,j}=\theta_{R}(x_{i}-x_{j})\prod_{k\neq
i,j}\bar{\theta}_{2R}(x_{i}-x_{k})$ (2.72)

Here $\theta_{R}$ is the characteristic function of the open set $|x|<R$, and
$\bar{\theta}_{R}=1-\theta_{R}$. We note that $F_{i,j}\neq F_{j,i}$ and
$F_{i,j}$ is equal to $1$ only when $x_{j}$ is close to $x_{i}$, but the other
$x_{k}$’s are not. It is easy to check that $\sum_{i:i\neq j}F_{i,j}\leq 1$,
so

$-\nabla_{j}\sum_{i:i\neq j}F_{i,j}\nabla_{j}\leq-\Delta_{j}$ (2.73)

for any fixed $x_{1},\cdots x_{j-1},x_{j+1},\cdots,x_{N}$.

Then we denote $v_{+}^{a}$ and $v_{-}^{a}$ as scaled potentials as follows,

$v_{+}^{a}(r)=a^{-2}v_{+}(r/a),\,\,\,v_{-}^{a}(r)=a^{-2}v_{-}(r/a)$ (2.74)

Choosing

$Y=(a^{3}\varrho)^{1/17}$ (2.75)

and $\varepsilon$ satisfying

$3\cdot\bigg{(}\min\big{\\{}1,SL[v_{+}]\big{\\}}\bigg{)}^{-1}\cdot
Y=\varepsilon<\frac{t}{2(1+t)},$ (2.76)

with the definition

$v_{\varepsilon}^{a}\equiv v^{a}-\varepsilon v^{a}_{+},$ (2.77)

we separate the Hamiltonian $H_{N}$ as follows

$\displaystyle H_{N}=$ (2.78)
$\displaystyle(1-\varepsilon)\sum_{j}-\nabla_{j}\sum_{i}F_{i,j}\nabla_{j}+\sum_{i\neq
j}F_{ij}\frac{v_{\varepsilon}^{a}}{2}(x_{i}-x_{j})-\varepsilon\sum_{j}\Delta_{j}+\varepsilon\sum_{i\neq
j}\frac{v^{a}_{+}}{2}(x_{i}-x_{j})$
$\displaystyle+(1-\varepsilon)\sum_{j}-\nabla_{j}(1-\sum_{i}F_{i,j})\nabla_{j}+\sum_{i\neq
j}(1-F_{ij})\frac{v_{\varepsilon}^{a}}{2}(x_{i}-x_{j})$

First, we claim the following Lemma 1, which will be proved in next section.

######  Lemma 1.

Define $Y$, $F_{i,j}$, $\varepsilon$, $v^{a}_{\varepsilon}$ and $R$ as in
(2.75), (2.72), (2.76), (2.77) and (2.71) respectively. There exists $C$
depending only on $v$ such that

$\displaystyle H^{\prime\prime}\equiv$ (2.79)
$\displaystyle(1-\varepsilon)\sum_{j}-\nabla_{j}\sum_{i}F_{i,j}\nabla_{j}+\sum_{i\neq
j}F_{ij}\frac{v_{\varepsilon}^{a}}{2}(x_{i}-x_{j})-\varepsilon\sum_{j}\Delta_{j}+\varepsilon\sum_{i\neq
j}\frac{v^{a}_{+}}{2}(x_{i}-x_{j})$ $\displaystyle\geq 4\pi
aN^{2}/\Lambda(1-CY)$

Hence, to obtain Theorem 2, it only remains to prove that the last line of
(2.78), as an operator, is bounded from below by zero, i.e.,

$(1-\varepsilon)\sum_{j}-\nabla_{j}(1-\sum_{i}F_{i,j})\nabla_{j}+\sum_{i\neq
j}(1-F_{ij})\mbox{$\frac{1}{2}$}v^{a}_{\varepsilon}(x_{i}-x_{j})\geq 0$

By the assumptions $\varepsilon<\,t\,(2+2t)^{-1}$, we have

$v_{\varepsilon}^{a}\geq\frac{2+t}{2+2\,t}v^{a}_{+}+v^{a}_{-}.$

Hence, it remains to prove that

$0\leq
H_{N}^{\prime}\equiv\frac{2+t}{2+2\,t}\sum_{j}-\nabla_{j}(1-\sum_{i}F_{i,j})\nabla_{j}+\frac{1}{2}\sum_{i\neq
j}(1-F_{ij})\left(\frac{2+t}{2+2\,t}v^{a}_{+}+v^{a}_{-}\right)(x_{i}-x_{j})$

Because $\lim_{N\to\infty}E(N,\Lambda)/N$ exists, for proving Theorem 2, we
can assume that $N$ is even, i.e., $N=2N_{1}$. Consider any partition
$P=(\pi_{1},\pi_{2})$ of ${1,...,N}$ into two disjoint sets with $N_{1}$
integers in $\pi_{1}$ and $\pi_{2}$ respectively. For each $P$, we define that

$\displaystyle H_{P}=H_{(\pi_{1},\pi_{2})}\equiv$
$\displaystyle\frac{2+t}{1+t}\sum_{j\in\pi_{1}}-\nabla_{j}(1-\sum_{i\neq
j}F_{i,j})\nabla_{j}+\sum_{i,j\in\pi_{1}}(1-F_{i,j})\mbox{$\frac{1}{2}$}v_{1,1}^{a}(x_{i}-x_{j})$
$\displaystyle+$
$\displaystyle\sum_{i\in\pi_{2},j\in\pi_{1}}(1-F_{i,j})\mbox{$\frac{1}{2}$}v^{a}_{2,1}(x_{i}-x_{j})+\sum_{i,j\in\pi_{2}}(1-F_{i,j})\mbox{$\frac{1}{2}$}v^{a}_{2,2}(x_{i}-x_{j})$

Here we denote $v^{a}_{\alpha,\beta}$ as the interaction potential between
particles in $\pi_{\alpha}$ and $\pi_{\beta}$, which are chosen as

$v_{1,1}^{a}=v^{a}_{2,2}=\frac{t}{1+t}v^{a}_{+}\geq
0,\,\,\,\,v^{a}_{2,1}=\frac{4}{1+t}v^{a}_{+}+4v^{a}_{-},$ (2.80)

so

$\frac{1}{4}\big{(}v^{a}_{1,1,}+v^{a}_{2,1}+v^{a}_{2,2}\big{)}=\frac{2+t}{2+2\,t}v^{a}_{+}+v^{a}_{-}\leq
v^{a}_{\varepsilon}.$

It is easily to check that

$H^{\prime}_{N}=\sum_{P}H_{P}/\sum_{P}1$ (2.81)

Hence, to obtain $H^{\prime}_{N}\geq 0$, it remains to prove that for $\forall
P$, $H_{P}\geq 0$. Because there is no kinetic energy of particles in
$\pi_{2}$, we can fix the configuration of $x_{i}$’s with $i\in\pi_{2}$. Since
permutation of the labels in $\pi_{1}$ and $\pi_{2}$ is irrelevant, we assume
that $\pi_{1}=\\{1,\cdots,N_{1}\\}$, $\pi_{2}=\\{N_{1}+1,\cdots,N\\}$.

As we can see $v^{a}_{2,1}$ is the only partly negative component in $H_{P}$.
For fixed $\pi_{2}$ particles, we can write $v^{a}_{2,1}(x_{j}-x_{i})$ as

$v^{a}_{2,1}(x_{j}-x_{i})=v^{a}_{2,1}(x_{j}-x_{i})(1-\chi_{A}(x_{i}))+v^{a}_{2,1}(x_{j}-x_{i})\chi_{A}(x_{i})$
(2.82)

Here $\chi_{A}$ is the characteristic function of $A$, which is a subset of
$[0,L]^{3}$ (2.99). We shall show $A$ is the area where the density of
$\pi_{2}$ particles is less than some fixed number. To obtain $H_{P}\geq 0$,
our strategy is to prove that

  1. 1.

The total energy of the interaction potential $v_{1,1}^{a}$ and $v^{a}_{2,2}$
cancels out the negative part of $v^{a}_{2,1}(1-\chi_{A})$.

  2. 2.

The total kinetic energy and the positive part of $v^{a}_{2,1}$ cancels out
the negative part of $v^{a}_{2,1}\chi_{A}$.

To make the strategy more clear, we shall define $A$ where the density of
$\pi_{2}$ particles is less than some fixed number. First we divide the cubic
box $[0,L]^{3}$ into small cubes $B_{n}$ ($n\in{\mathbb{N}}$) of side length
$\ell$, with

$\ell=\mbox{$\frac{1}{2}$}r_{1}a.$

Then, with fixed $x_{i}$’s, $i\in\pi_{2}$, for any $x\in[0,L]^{3}$, we define
the $G(x)$ as the set of $i$’s which satisfy $i\in\pi_{2}$ and $|x_{i}-x|\leq
R_{0}a$, i.e.,

$\displaystyle G(x)\equiv\\{i\in\pi_{2}:|x_{i}-x|\leq R_{0}a\\}$ (2.83)

We denote $|G(x)|$ as the number of the elements of $G(x)$.

We denote $d(x,B_{n})$ as the distance between the cube
$B_{n}\subset{\mathbb{R}}^{3}$ and $x\in{\mathbb{R}}^{3}$. Since $|G(y)|$ is
uniformly bounded ($|G(y)|\leq N_{1}$), there must exist a point
$X(B_{n})\in{\mathbb{R}}^{3}$ satisfying $d(X(B_{n}),B_{n})\leq 2R_{0}a$ and

$\displaystyle|G(X(B_{n}))|=\max\\{|G(y)|:d(y,B_{n})\leq 2R_{0}a\\}$ (2.84)

We define $G(B_{n})\equiv G(X(B_{n}))$. We are going to prove that there
exists $n_{1}\in{\mathbb{N}}$ depending on $R_{0}/r_{1}$ such that

  1. 1.

The total energy of the interaction potential $v_{1,1}^{a}$ and $v^{a}_{2,2}$
cancels out the negative parts of $v^{a}_{2,1}(x_{j},x_{i})$’s when $x_{i}$ is
in a cube $B_{n}$ such that $|G(B_{n})|>n_{1}$.

  2. 2.

The total kinetic energy and the positive part of $v^{a}_{2,1}$ cancel out the
negative part of the remaining $v^{a}_{2,1}$’s.

First, we derive the lower bound on the total energy of $v^{a}_{2,2}$, i.e.
(2.89, 2.91). With the definition of $G(B_{n})=G(X(B_{n}))$, we know that the
set $\\{x_{k}:k\in G(B_{n})\\}$ can be covered by a sphere of radius $R_{0}a$.
So the number of the cubes which one need to cover this set is less than ${\rm
const.\,}(R_{0}/r_{1})^{3}$. We denote these cubes as $B_{n_{1}}\cdots
B_{n_{m}}$ $(m\leq{\rm const.\,}(R_{0}/r_{1})^{3})$ and assume the number of
$i$’s satisfying $i\in G(B_{n})$ and $x_{i}\in B_{n_{k}}$ is $a_{n_{k}}$.
Because the side length of $B_{n_{k}}$ is equal to $r_{1}a/2$, the distance
between the two particles in the same cube is no more than
$\frac{\sqrt{3}}{2}r_{1}a<r_{1}a$. Hence we have

$\displaystyle\sum_{i,j\in G(B_{n})}\theta_{r_{1}a}(x_{i}-x_{j})\geq$
$\displaystyle\sum_{k=1}^{m}\left[(a_{n_{k}})^{2}-(a_{n_{k}})\right]$ (2.85)
$\displaystyle\geq$
$\displaystyle\frac{(\sum_{k=1}^{m}a_{n_{k}})^{2}}{m}-(\sum_{k=1}^{m}a_{n_{k}})$
$\displaystyle\geq$ $\displaystyle{\rm
const.\,}(R_{0}/r_{1})^{-3}|G(B_{n})|^{2}-|G(B_{n})|$

Hence, we obtain that there exist $n_{1}\geq 3$ and $n_{1},n_{2}={\rm
const.\,}(R_{0}/r_{1})^{3}$ such that when $|G(B_{n})|\geq n_{1}$,

$\sum_{i,j\in
G(B_{n})}\theta_{r_{1}a}(x_{i}-x_{j})\geq\frac{1}{n_{2}}|G(B_{n})|^{2},$
(2.86)

which implies

$\sum_{i,j\in
G(B_{n})}v^{a}_{2,2}(x_{i}-x_{j})\geq\frac{t\lambda_{+}a^{-2}}{(1+t)n_{2}}|G(B_{n})|^{2}$
(2.87)

Here, we used (2.80) and (2.74), i.e.,

$v^{a}_{2,2}(r)=\frac{t}{(1+t)}v_{+}^{a}(r)=\frac{t}{(1+t)}a^{-2}v_{+}(r/a)$
(2.88)

Again, with the fact that the set $\\{x_{k}:k\in G(B_{n})\\}$ can be covered
with a sphere of diameter $2R_{0}a\leq R$, one can see that if $i\in G(B_{n})$
and $|G(B_{n})|\geq 3$, we have $F_{i,j}=0$ for any $j\neq i$. Hence we obtain
that, for any fixed $B_{n}$ satisfying $|G(B_{n})|\geq n_{1}$,

$\sum_{i,j\in G(B_{n})}(1-F_{i,j})v^{a}_{2,2}(x_{i}-x_{j})=\sum_{i,j\in
G(B_{n})}v^{a}_{2,2}(x_{i}-x_{j})\geq\frac{t\lambda_{+}a^{-2}}{(1+t)n_{2}}|G(B_{n})|^{2}$
(2.89)

Then, we are going to sum up all the cubes satisfying $|G(B_{n})|\geq n_{1}$.
It is easy to see that

$d(x_{i},B_{n})\leq 3R_{0}a,\,\,\,\,{\rm{for}}\,\,\,\,i\in G(B_{n}),$ (2.90)

which implies that for any fixed $i\in{\pi_{2}}$, the number of cubes
$B_{n}$’s satisfying $i\in G(B_{n})$ is less than some constant $n_{3}$, which
is less than ${\rm const.\,}(R_{0}/r_{1})^{3}$. Hence, summing up all the
blocks satisfying $|G(B_{n})|\geq n_{1}$, with the inequality (2.89), we get
that

$\displaystyle\sum_{i,j\in\pi_{2}}(1-F_{i,j})v^{a}_{2,2}(x_{i}-x_{j})$
$\displaystyle\geq\sum_{n:|G(B_{n})|\geq n_{1}}\sum_{i,j\in
G(B_{n})}(1-F_{i,j})v^{a}_{2,2}(x_{i}-x_{j})$ (2.91)
$\displaystyle\geq\sum_{n:|G(B_{n})|\geq
n_{1}}\frac{t\lambda_{+}a^{-2}}{(1+t)n_{2}n_{3}}|G(B_{n})|^{2}$

Second, we derive the lower bound on the interaction potential between
particles in $\pi_{1}$. Because the distance between any two points in the
same cube is less than $r_{1}a$, we have $v_{1,1}^{a}(x_{i}-x_{j})\geq
a^{-2}\lambda_{+}t(1+t)^{-1}$ when $i,j\in\pi_{1}$ and $x_{i},x_{j}\in B_{n}$,
i.e.,

$\sum_{i,j\in\Pi_{1}(B_{n})}v_{1,1}^{a}(x_{i}-x_{j})\geq\frac{a^{-2}t\lambda_{+}}{1+t}\bigg{(}|\Pi_{1}(B_{n})|^{2}-|\Pi_{1}(B_{n})|\bigg{)}$
(2.92)

Here $\Pi_{1}(B_{n})$ is defined as the set of $i$’s such that $i\in\pi_{1}$
and $x_{i}\in B_{n}$ and $|\Pi_{1}(B_{n})|$ is the number of the elements of
$\Pi_{1}(B_{n})$. Furthermore, if $x_{i}\in B_{n}$ and $|G(B_{n})|\geq 1$,
there must be a $k\in\pi_{2}$ satisfying $|x_{i}-x_{k}|\leq 4R_{0}a\leq 2R$,
hence $F_{i,j}=0$ for any other $j\in\pi_{1}$. Using this result, for any
$B_{n}$ satisfying $|G(B_{n})|\geq 1$, we have that

$\sum_{i,j\in\Pi_{1}(B_{n})}(1-F_{i,j})v_{1,1}^{a}(x_{i}-x_{j})\geq\frac{t\lambda_{+}a^{-2}}{1+t}\bigg{(}|\Pi_{1}(B_{n})|^{2}-|\Pi_{1}(B_{n})|\bigg{)}$
(2.93)

At last, we derive the lower bound on $v^{a}_{2,1}$. By the definitions of
$|G(B_{n})|$ and $v^{a}_{2,1}$, we have that $\forall x\in B_{n}$,

$\sum_{i\in\pi_{2}}[v^{a}_{2,1}]_{-}(x-x_{i})\geq-4\lambda_{-}a^{-2}|G(B_{n})|.$

Here we denote $[v^{a}_{2,1}]_{-}$ as the negative part of $v^{a}_{2,1}$ which
is equal to $4[v^{a}]_{-}$. With the facts $0\geq
4[v^{a}]_{-}\geq-4\lambda_{-}a^{-2}$ and $0\leq F_{i,j}\leq 1$, we have the
following inequality

$\sum_{j\in\Pi_{1}(B_{n}),\,\,i\in\pi_{2}}(1-F_{i,j})[v^{a}_{2,1}]_{-}(x_{i}-x_{j})\geq-4\lambda_{-}a^{-2}\cdot|\Pi_{1}(B_{n})|\cdot|G(B_{n})|$
(2.94)

One can check that if $|G(B_{n})|\geq n_{1}$ and

$\lambda_{+}\geq(1+t^{-1})\lambda_{-}\cdot\max\\{2\sqrt{n_{2}n_{3}},\frac{n_{2}n_{3}}{4n_{1}}\\}\sim{\rm
const.\,}(1+t^{-1})\lambda_{-}(R_{0}/r_{1})^{3},$ (2.95)

the sum of the right sides of (2.93) and (2.94) is bounded from below as
follows,

$\displaystyle-4\lambda_{-}\cdot|\Pi_{1}(B_{n})|\cdot|G(B_{n})|+\frac{t\lambda_{+}}{1+t}\bigg{(}|\Pi_{1}(B_{n})|^{2}-|\Pi_{1}(B_{n})|\bigg{)}$
(2.96) $\displaystyle\geq$
$\displaystyle-\frac{t\lambda_{+}}{(1+t)n_{2}n_{3}}|G(B_{n})|^{2}$

Hence, with (2.93) and (2.94), we obtain that if (2.95) holds and
$|G(B_{n})|\geq n_{1}$,

$\displaystyle 0\leq$
$\displaystyle\frac{t}{1+t}\frac{\lambda_{+}a^{-2}}{n_{2}n_{3}}|G(B_{n})|^{2}+\sum_{i,j\in\Pi_{1}(B_{n})}(1-F_{i,j})v_{1,1}^{a}(x_{i}-x_{j})$
(2.97) $\displaystyle+$
$\displaystyle\sum_{j\in\Pi_{1}(B_{n}),\,\,i\in\pi_{2}}(1-F_{i,j})[v^{a}_{2,1}]_{-}(x_{i}-x_{j}),$

Then summing up all the $B_{n}$’s satisfying $|G(B_{n})|>n_{1}$, with (2.91)
and $v_{11}\geq 0$, we obtain that as long as (2.95) holds,

$\displaystyle 0\leq$
$\displaystyle\sum_{i,j\in\pi_{2}}(1-F_{i,j})\mbox{$\frac{1}{2}$}v^{a}_{2,2}(x_{i}-x_{j})+\sum_{i,j\in\pi_{1}}(1-F_{i,j})\mbox{$\frac{1}{2}$}v_{1,1}^{a}(x_{i}-x_{j})$
$\displaystyle+\sum_{j\in\pi_{1},i\in\pi_{2}}(1-F_{i,j})\mbox{$\frac{1}{2}$}[v^{a}_{2,1}]_{-}(x_{i}-x_{j})\big{(}1-\chi_{A}(x_{j})\big{)}$

Here $A$ is defined as the set $\cup_{|G(B_{n})|\leq n_{1}}B_{n}$.

$A=\cup_{|G(B_{n})|\leq n_{1}}B_{n}$ (2.99)

So far, we proved the interaction potential between particles of the same
groups cancels out the negative part of the $v^{a}_{2,1}(1-\chi_{A})$ term in
(2.82). We shall show that the kinetic energy and the positive part of
$v^{a}_{2,1}$ cancel out the remaining negative part of $v^{a}_{2,1}$.

For the other terms in the Hamiltonian $H_{P}$, we claim that as long as

$SL[4\,n_{1}(v+tv_{-})]\geq 0$ (2.100)

we have

$\displaystyle 0\leq$
$\displaystyle\frac{1}{2}\sum_{j\in\pi_{1},i\in\pi_{2}}(1-F_{i,j})\bigg{(}[v^{a}_{2,1}]_{+}(x_{i}-x_{j})+[v^{a}_{2,1}]_{-}(x_{i}-x_{j})\chi_{A}(x_{j})\bigg{)}$
$\displaystyle+\frac{2+t}{1+t}\sum_{j\in\pi_{1}}-\nabla_{j}\left(1-\sum_{i}F_{i,j}\right)\nabla_{j}$

As we can see that (2.2) and (2.2) implies that $H_{P}\geq 0$ when
$SL[4n_{1}(v+tv_{-})]\geq 0$ and $(\ref{lambda+-})$ holds, i.e.,
$\lambda_{+}\geq{\rm const.\,}(1+t^{-1})\lambda_{-}(R_{0}/r_{1})^{3}$, which
completes the proof of Theorem 2.

To prove (2.2), we only need to prove the following operator inequality, for
any fixed $x_{2},\cdots,x_{N}$,

$\displaystyle 0\leq$
$\displaystyle-\frac{2+t}{1+t}\nabla_{1}(1-\sum_{i=2}^{N}F_{i,1})\nabla_{1}$
$\displaystyle+\mbox{$\frac{1}{2}$}\sum_{j\in\pi_{2}}(1-F_{j,1})\bigg{(}[v^{a}_{2,1}]_{+}(x_{1}-x_{j})+[v^{a}_{2,1}]_{-}(x_{1}-x_{j})\chi_{A}(x_{1})\bigg{)}$

First, if $[v^{a}_{2,1}]_{-}(x_{1}-x_{j})\chi_{A}(x_{1})\neq 0$, then
$d(B^{x_{1}}_{n},x_{j})\leq R_{0}a$, here the $B^{x_{1}}_{n}$ is the cube
where $x_{1}$ is. We obtain that $j\in\pi_{2}^{\prime}\subset\pi_{2}$, here
$\pi^{\prime}_{2}$ is defined as

$\pi^{\prime}_{2}\equiv\\{j\,^{\prime}\in\pi_{2}:\exists
B_{n},D(x_{j\,^{\prime}},B_{n})\leq R_{0}a,|G(B_{n})|\leq n_{1}\\}$ (2.103)

Hence, it only remains to prove that

$\displaystyle 0\leq$
$\displaystyle-\frac{2+t}{1+t}\nabla_{1}(1-\sum_{i=2}^{N}F_{i,1})\nabla_{1}+\mbox{$\frac{1}{2}$}\sum_{j\in\pi^{\prime}_{2}}(1-F_{j,1})v^{a}_{2,1}(x_{1}-x_{j})$
(2.104)

Second, we claim the following inequality which will be proved later.

$n_{1}\left(1-\sum_{i=2}^{N}F_{i,1}\right)\geq\sum_{j\in\pi^{\prime}_{2}}(1-F_{j,1})\theta_{(R_{0}a)}(x_{1}-x_{j})$
(2.105)

which implies that

$-\nabla_{1}\left(1-\sum_{i=2}^{N}F_{i,1}\right)\nabla_{1}\geq-\frac{1}{n_{1}}\nabla_{1}\sum_{j\in\pi^{\prime}_{2}}(1-F_{j,1})\theta_{(R_{0}a)}(x_{1}-x_{j})\nabla_{1}$
(2.106)

With (2.106), we obtain that the right side of (2.104) is not less than

$\displaystyle\sum_{j\in\pi^{\prime}_{2}}\bigg{(}-\frac{2+t}{n_{1}(1+t)}\nabla_{1}(1-F_{j,1})\theta_{(R_{0}a)}(x_{1}-x_{j})\nabla_{1}+\frac{1}{2}(1-F_{j,1})v^{a}_{2,1}(x_{j}-x_{1})\bigg{)}$
$\displaystyle\geq$
$\displaystyle\sum_{j\in\pi^{\prime}_{2}}\bigg{(}1-\prod_{k\neq
1\,or\,j}\bar{\theta}_{2R}(x_{k}-x_{j})\bigg{)}\times\frac{2}{n_{1}(1+t)}$
(2.107)
$\displaystyle\times\bigg{(}-\nabla_{1}\theta_{(R_{0}a)}(x_{1}-x_{j})\nabla_{1}+2n_{1}(v^{a}+tv^{a}_{-})(x_{j}-x_{1})\bigg{)}$

Here we used the definition of $F_{j,1}$ and (2.80), i.e.,
$v^{a}_{2,1}=\frac{4}{1+t}[v^{a}+tv^{a}_{-}]$. With the assumption
$SL[4\,n_{1}(v+v_{-})]\geq 0$, we obtain that (2.107)$\geq 0$, which implies
inequality (2.104).

Hence, it only remains to prove (2.105). For $x_{2},\cdots,x_{N}$ fixed, we
define $\pi_{3}$ as following,

$\pi_{3}=\left\\{2\leq j\leq N:\prod_{2\leq k\leq N,k\neq
j}\bar{\theta}_{2R}(x_{j}-x_{k})=1\right\\}$ (2.108)

With the definition of $\pi_{3}$, we obtain that

$F_{j,1}=\left\\{\begin{array}[]{ll}\theta_{R}(x_{j}-x_{1})&j\in\pi_{3}\\\
0&j\notin\pi_{3}\,,\end{array}\right.$ (2.109)

Hence, it only remains to prove that

$n_{1}\left(1-\sum_{i\in\pi_{3}}\theta_{R}(x_{1}-x_{i})\right)\geq\sum_{j\in\pi^{\prime}_{2},j\notin\pi_{3}}\theta_{(R_{0}a)}(x_{1}-x_{j})$
(2.110)

or

$\max_{x\in{\mathbb{R}}^{3}}\left(n_{1}\sum_{i\in\pi_{3}}\theta_{R}(x-x_{i})+\sum_{j\in\pi^{\prime}_{2},j\notin\pi_{3}}\theta_{(R_{0}a)}(x-x_{j})\right)\leq
n_{1}$ (2.111)

Because the distance between $x_{i}$ ($i\in\pi_{3}$) and $x_{j}$ ($2\leq j\leq
N,j\neq i$) are not less than $2R$, we have that if $i\in\pi_{3}$

$\theta_{R}(x-x_{i})=1\Rightarrow\sum_{j\neq 1,j\neq i}\theta_{R}(x-x_{j})=0$
(2.112)

So, it only remains to prove that

$\max_{x}\left(\sum_{j\in\pi^{\prime}_{2},j\notin\pi_{3}}\theta_{(R_{0}a)}(x-x_{j})\right)\leq
n_{1}$ (2.113)

By the definition of $\pi^{\prime}_{2}$ in (2.103), if $j\in\pi^{\prime}_{2}$
and $\theta_{(R_{0}a)}(x-x_{j})=1$, there exist $B_{n}$ satisfying
$|G(B_{n})|\leq n_{1}$ and $d(x,B_{n})\leq 2R_{0}a$. Hence by the definition
of $G(B_{n})$ in (2.84) and (2.83), we obtain that, for $\forall
x\in{\mathbb{R}}^{3}$

$\sum_{i\in\pi^{\prime}_{2}}\theta_{(R_{0}a)}(x-x_{i})=1\Rightarrow\sum_{i\in\pi_{2}}\theta_{(R_{0}a)}(x-x_{i})\leq
n_{1}$ (2.114)

With the fact that $\pi^{\prime}_{2}\subset\pi_{2}$, we arrive at the desired
result (2.113) and complete the proof of Theorem 2. ∎

###  2.3 Proof of Lemma 1

###### Proof.

Let $\delta\Omega$ be any infinitismal solid angle. With the definition of
scattering length, we have that if $\phi$ is a complex-valued function such
that

$\phi(x)=1\,\,\,{\rm for}\,\,\,x\in\delta\Omega\otimes{\mathbb{R}}\,\,\,{\rm
and}\,\,\,|x|=R^{\prime}\geq R_{0}a$ (2.115)

then

$\delta\Omega\cdot
a\leq\int_{\delta\Omega\otimes[0,R\,^{\prime}]}|\nabla\phi(x)|^{2}+\mbox{$\frac{1}{2}$}v^{a}(x)|\phi(x)|^{2}dx$
(2.116)

Hence we obtain that

$a\int_{{\mathbb{R}}^{3}}\delta(|x|-R^{\prime})|\phi(x)|^{2}dx\leq\int_{|x|\leq
R\,^{\prime}}|\nabla\phi(x)|^{2}+\mbox{$\frac{1}{2}$}v^{a}(x)|\phi(x)|^{2}dx$
(2.117)

which says that for any non-negative radial function $U_{0}(x)$, supported in
the annulus $R_{0}a\leq|x|\leq R$, with
$\int_{{\mathbb{R}}^{3}}U_{0}(x)\,dx=4\pi$, we have

$\displaystyle-\nabla\theta_{R}(x)\nabla+\mbox{$\frac{1}{2}$}v^{a}\geq aU_{0}$
(2.118)

Note: The result of lemma 2.5 of [5] shows the $\theta_{R}(x)$ in above
inequality can be replaced with the characteristic function of any star-shaped
set when $v^{a}\geq 0$.

Furthermore, one can easily prove that for fixed $v$ ($SL[v]=1$), $v_{+}$ and
small enough $\varepsilon$,

$SL[v-\varepsilon v_{+}]>0,\,\,\,\,\,\,|f_{\varepsilon}|_{\infty}\leq{\rm
const.\,}$ (2.119)

Here we denote $f_{\varepsilon}$ as the normalized solution
($\lim_{|x|\to\infty}f_{\varepsilon}(x)=1$) of the zero-energy scattering
equation of $v-\varepsilon v_{+}$. Hence, by the definition of scattering
length, using $f_{\varepsilon}$ as the trial function for $v$, we obtain that

$SL[v]\leq SL[v-\varepsilon
v_{+}]+\varepsilon\|v_{+}\|_{1}\cdot|f_{\varepsilon}|_{\infty}\leq
SL[v-\varepsilon v_{+}]+{\rm const.\,}\varepsilon$ (2.120)

Combining this result with (2.118) and the definition of
$v^{a}_{\varepsilon}$, we have,

$-(1-\varepsilon)\nabla\theta_{R}(x)\nabla+\mbox{$\frac{1}{2}$}v^{a}_{\varepsilon}\geq(1-{\rm
const.\,}\varepsilon)aU_{0}$ (2.121)

and

$-(1-\varepsilon)\nabla_{j}F_{i,j}\nabla_{j}+\mbox{$\frac{1}{2}$}F_{i,j}v^{a}_{\varepsilon}(x_{i}-x_{j})\geq(1-{\rm
const.\,}\varepsilon)aF_{i,j}U_{0}(x_{i}-x_{j})$ (2.122)

Hence, we obtain the following lower bound on $H^{\prime\prime}$, which is
defined in (2.79)

$\displaystyle
H^{\prime\prime}\geq\varepsilon\sum_{j}-\Delta_{j}+\varepsilon\sum_{i\neq
j}\frac{v^{a}_{+}}{2}(x_{i}-x_{j})+(1-{\rm const.\,}\varepsilon)a\sum_{i\neq
j}W_{i,j}$ (2.123)

Here $W_{i,j}$ is defined as

$W_{i,j}=F_{ij}U_{0}(x_{i}-x_{j})\geq 0$ (2.124)

As in [4], we choose

$\ell=aY^{-6}$ (2.125)

and divide $\Lambda$ into small cubes with side length $\ell$. Then we have

$H^{\prime\prime}\geq
H^{(3)}\equiv\varepsilon\sum_{j}-\Delta_{j}+\varepsilon\sum_{i\neq
j}\frac{v^{a}_{+}}{2}(x_{i}-x_{j})+(1-{\rm const.\,}\varepsilon)a\sum_{i\neq
j}W^{\prime}_{i,j}$ (2.126)

Here $W^{\prime}_{i,j}$ is defined as

$W^{\prime}_{i,j}=G_{ij}U_{0}(x_{i}-x_{j}),\,\,\,\,G_{ij}\equiv
F_{ij}\chi(x_{i})\chi(x_{j})\geq 0,$ (2.127)

and $\chi(x)$ is equal to 1 when the distance between $x$ and the edges of the
small cubes is greater than $2R$; otherwise it is equal to 0. As we can see
the particles in different cubes don’t affect each other in $H^{(3)}$.

We are going to estimate the ground energy $E^{(3)}(n,\ell)$ of $H^{(3)}$ for
$n$ particles in $[0,\ell]^{3}$ with Neumann boundary condition.

First, in the case that $n\leq\frac{8}{3}\varrho\ell^{3}Y^{-1}$, with the
definition of $\varepsilon$ in (2.76), we have that

$\displaystyle H^{(3)}$ $\displaystyle\geq 3Y\sum_{j}-\Delta_{j}+(1-{\rm
const.\,}Y)a\sum_{i\neq j}W^{\prime}_{i,j}$ (2.128)

Then with the Temple inequality in [10], as in [5] (Ineq. 2.60, 2.66), we have
that

$\displaystyle\frac{E^{\left(3\right)}\left(n,\ell\right)}{n}\geq$
$\displaystyle 4\pi\frac{an}{\ell^{3}}\left(1-\frac{1}{n}\right)\left(1-{\rm
const.\,}Y\right)\left(1-\frac{2R}{\ell}\right)^{3}\left(1+\frac{4\pi
n}{3}\left(\frac{2R}{\ell}\right)^{3}\right)^{-1}$ (2.129)
$\displaystyle\left(1-\frac{3}{\pi}\frac{an}{\left(R^{3}-\left(aR_{0}\right)^{3}\right)\left(3\pi
Y\ell^{-2}-4a\ell^{-3}n^{2}\right)}\right)$ $\displaystyle\geq$ $\displaystyle
4\pi\frac{an}{\ell^{3}}(1-\frac{1}{n})(1-{\rm
const.\,}Y)(1-\frac{n}{6\ell^{3}\varrho}Y)$

Second, when $n\geq\frac{8}{3}\varrho\ell^{3}Y^{-1}$, using the fact
$W^{\prime}\geq 0$, we obtain that

$H^{(3)}\geq\varepsilon
H^{(4)}\equiv\varepsilon\bigg{(}\sum_{j}-\Delta_{j}+\sum_{i\neq
j}\frac{v^{a}_{+}}{2}(x_{i}-x_{j})\bigg{)}$ (2.130)

Using superadditivity of the ground state energy of $H^{(4)}$, we obtain that
the ground energy $E^{(4)}(n,\ell)$ of $H^{(4)}$ is bounded from below as
follows, ($n\geq p$)

$\displaystyle
E^{(4)}(n,\ell)\geq\left[\frac{n}{p}\right]E^{(4)}(p,\ell)\geq\frac{n}{2p}E^{(4)}(p,\ell)$
(2.131)

Here $[n/p\,]$ is the largest integer not greater than $n/p$. Actually,
$H^{(4)}$ is just the Hamiltonian for the pure non-negative interaction
potential, as in [7]. Denote $a_{+}$ as follows:

$a_{+}=\min\\{SL(v^{a}),SL(v^{a}_{+})\\}\leq a$ (2.132)

Replacing $v_{+}^{a}$ with soft potential, we obtain that,

$H^{(4)}\geq 3Y\sum_{j}-\Delta_{j}+(1-{\rm const.\,}Y)a_{+}\sum_{i\neq
j}W^{\prime}_{i,j}$ (2.133)

As (2.129), we can prove that when $p=\frac{8}{3}\varrho\ell^{3}Y^{-1}$,

$E^{(4)}(p,\ell)/p\geq\frac{32}{3}\pi a_{+}\varrho
Y^{-1}\left(1-\frac{1}{2}\right)(1-{\rm const.\,}Y)\geq\frac{16}{3}\pi
a_{+}\varrho Y^{-1}$ (2.134)

Hence when $n\geq\frac{8}{3}\varrho\ell^{3}Y^{-1}$, we have the following
lower bound on the ground energy $E^{(3)}(n,\ell)$ of $H^{(3)}$

$E^{(3)}(n,\ell)/n\geq\varepsilon
E^{(4)}(n,\ell)/n\geq\frac{8}{3}\pi\varepsilon a_{+}\varrho Y^{-1}\geq 8\pi
a\varrho$ (2.135)

For the last inequality, we used the definition of $\varepsilon$ in (2.76). So
far we proved that

$\displaystyle\frac{E^{(3)}(n,\ell)}{n}\geq\left\\{\begin{array}[]{ll}4\pi\frac{an}{\ell^{3}}(1-\frac{1}{n})(1-{\rm
const.\,}Y)(1-\frac{n}{6\ell^{3}\varrho}Y)&1\leq
n\leq\frac{8}{3}\ell^{3}\varrho Y^{-1}\\\ 8\pi
a\varrho&n\geq\frac{8}{3}\ell^{3}\varrho Y^{-1}\,,\end{array}\right.$ (2.138)

which implies that when $Y$ is small enough,

$\displaystyle\frac{E^{(3)}(n,\ell)}{n(1-{\rm
const.\,}Y)}\geq\left\\{\begin{array}[]{ll}4\pi\frac{an}{\ell^{3}}(1-\frac{1}{n})&1\leq
n\leq 4\ell^{3}\varrho\\\ 8\pi a\varrho&n\geq
4\ell^{3}\varrho\,,\end{array}\right.$ (2.141)

Recall the following two facts,

  1. 1.

the interaction potential $W^{\prime}_{i,j}$ only depends on the particles in
the same cubes as $i$ and $j$,

  2. 2.

the particles in different cubes have no interaction.

Using the inequality above (2.141), with the method in [7] (Ineq. 2.55, 2.56),
one can prove that the ground state $E^{(3)}(N,\Lambda)$ of $H^{(3)}$ of $N$
particles in big cubic $\Lambda$ is greater than

$E^{(3)}(N,\Lambda)/N\geq 4\pi a\varrho(1-{\rm const.\,}Y).$ (2.142)

Here $Y$ is defined in (2.75), which implies the desired result (2.79). ∎

##  3 Appendix

In this appendix, we show that if $v^{a}$ is a continuous function and $H_{N}$
has no bound state for any $N$, $v^{a}$ has a positive core and bounded from
below, i.e.,

$v^{a}(0)>0,\,\,\,\,\min v^{a}(r)\neq-\infty$ (3.1)

And these inequalities also hold when $v^{a}$ is stable [1] in the sense of
(3.2). One can see that $\min v^{a}(r)\neq-\infty$ is trivial when $v^{a}$ is
continuous. So it only remains to prove that $v^{a}(0)>0$.

First, we prove the statement in the case when $v^{a}$ is stable, which is
defined as follows: there exists constant $C$, for any $N$,
$x_{1},\cdots,x_{N}$,

$\sum_{1\leq i\neq j\leq N}v^{a}(x_{i}-x_{j})\geq-CN,$ (3.2)

Inserting

$x_{1}=x_{2}=\cdots=x_{[N/2]}=0,\,\,\,\,x_{[N/2]+1}=x_{[N/2]+2}=\cdots=x_{N}=x_{0}$
(3.3)

into the left side of (3.2), for some $x_{0}\in{\mathbb{R}}^{3}$ satisfying
$v^{a}(x_{0})<0$, we obtain that

${\rm const.\,}v^{a}(0)N^{2}-{\rm const.\,}v^{a}(x_{0})N^{2}\geq-CN,$ (3.4)

which implies the desired result that $v^{a}(0)>0$.

Next, we prove the statement in the case that $H_{N}$ has no bounded state for
any $N$. Because $v^{a}$ is not pure non-negative, there exist
$x_{0}\in{\mathbb{R}}^{3},r_{1},C\in R$ satisfying that

$v^{a}(x)<-C,\,\,\,{\rm for}\,\,\,x\in B(x_{0},r_{1})\subset{\mathbb{R}}^{3}$
(3.5)

Here $B(x_{0},r_{1})$ is the sphere of radius $r_{0}$ centered at $x_{0}$. If
$v^{a}(0)\leq 0$, there exists $r_{2}<r_{1}/2$ satisfying that

$v^{a}(x)<C/2,\,\,\,{\rm for}\,\,\,x\in B(0,r_{2})$ (3.6)

We construct the trial state such that $x_{1},x_{2},\cdots,x_{[N/2]}$ are
localized in $B(0,r_{2})$ with the Dirichelet boundary condition and
$x_{[N/2]+1},x_{[N/2]+2},\cdots,x_{N}$ are localized in $B(x_{0},r_{2})$ with
the same boundary condition. The energy of this state is less than

$-\frac{C}{8}N^{2}+\frac{{\rm const.\,}}{r_{2}^{2}}N$ (3.7)

Here the first term is potential energy and the second term is kinetic energy.
When $N$ goes to infinity, the energy of this trial state is negative and
hence there are bound states, which is a contradiction with our assumptions.
So we arrive at the desired result that $v^{a}(0)>0$.

## References

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# RCSEDv2: the largest database of galaxy properties from a homogeneously
processed multi-wavelength dataset

Igor Chilingarian1,2, Sviatoslav Borisov3,2, Vladimir Goradzhanov2,4, Kirill
Grishin5,2, Anastasia Kasparova2, Ivan Katkov6,2, Vladislav Klochkov2,4,
Evgenii Rubtsov2,4, and Victoria Toptun2,4

###### Abstract

The Reference Catalog of Spectral Energy Distributions of 800,000 galaxies
(RCSED) includes the results of uniform re-processing of 800,000 SDSS DR7
galaxies at redshifts $0.007<z<0.6$ complemented with ultraviolet-to-infrared
photometric data from GALEX, SDSS, and UKIDSS. The key difference between
RCSED and existing databases of galaxy properties (NED, HyperLeda, part of
SIMBAD) is that rather than providing a compilation of literature data, we
perform homogeneous data analysis of spectral and photometric data using our
own tools and publish derived physical properties of galaxies along with re-
calibrated spectra and photometry and their best-fitting models. Here we
present the 2nd release of our catalog, RCSEDv2 where we substantially
expanded the spectral dataset to 4 million objects by including spectral data
analysis for 10 large spectroscopic surveys (SDSS, SDSS/eBOSS, LAMOST,
Hectospec, CfA redshift surveys, 2dFGRS, 6dFGS, DEEP2/3, WiggleZ). The
photometric part has also been expanded by including DESI Legacy Survey, DES,
UHS, ESO Public Surveys, and WISE in addition to GALEX, SDSS, and UKIDSS used
in the original RCSED. This makes RCSEDv2 the largest database of galaxy
properties and homogeneously processed spectral and photometric data up-to-
date and creates a foundation for the analysis of future large-scale spectral
surveys DESI and 4MOST.

1Center for Astrophysics – Harvard and Smithsonian, Cambridge, MA, USA;
<EMAIL_ADDRESS>
2Sternberg Astronomical Institute, Moscow State University, Moscow, Russia

3Department of Astronomy, University of Geneva, Versoix, Switzerland

4Department of Physics, Moscow State University, Moscow, Russia

5Université de Paris, CNRS, Astroparticule et Cosmologie, F-75013 Paris,
France

6New York University Abu Dhabi, Abu Dhabi, UAE

##  1 Motivation and Scope

Statistical studies of galaxy formation and evolution require knowledge of
galaxy properties going beyond simple redshift measurements. Millions of
optical spectra became available over the past 2 decades as a result of wide-
field spectroscopic surveys carried out at different observatories all over
the world. In 2010–2017 we undertook a major effort to re-process 800,000
galaxy spectra from the legacy spectroscopic sample of SDSS. We complemented
spectra with aperture and integrated photometry in the ultraviolet wavelengths
from Galaxy Evolution eXplorer (Martin et al. 2005), optical domain from SDSS,
and 4 near-infrared bands from UKIDSS (Lawrence et al. 2007) converted into
rest-frame using $k$-corrections (Chilingarian et al. 2010) to allow one to
compare photometric measurements for galaxies at different redshifts. This
Reference Catalog of Spectral Energy Distributions (RCSED) of galaxies
(Chilingarian et al. 2017) has already been used for a number of successful
research projects. We decided to substantially expand the catalog by including
virtually all publicly available spectral galaxy surveys and data from new
deep wide-field imaging surveys.

##  2 RCSEDv2: Data and Analysis

Spectroscopic datasets included in the RCSEDv2 comprise: SDSS DR16 galaxy and
QSO specta (Ahumada et al. 2020), 2dF Galaxy Redshift Survey (Colless et al.
2001), 6dF Galaxy Survey (Jones et al. 2004), DEEP2/3 surveys (Newman et al.
2013), WiggleZ (Parkinson et al. 2012), LAMOST DR7, public spectra from the
Hectospec spectrograph, and several CfA redshift surveys from the 60-inch
telescope at Fred Lawrence Whipple Observatory, the most notable being a
recently released public archive of FAST (Mink et al. 2021). The two latter
datasets are collections of uniformly reduced pointed observations from the
corresponding data archives, e.g. these are not survey datasets even though
some subsets of them are indeed galaxy surveys with specific selection
functions. All the remaining datasets are galaxy surveys that is datasets
where the target selection functions are well defined. Currently, we have a
total of 4.6 million galaxy spectra included in RCSEDv2. For every spectrum,
we provide aperture-matched UV-to-IR fluxes from several wide-field imaging
surveys (GALEX, SDSS, Legacy Surveys, DES, UKIDSS, UHS, ESO Public Surveys,
WISE). We also publish extinction- and $k$-corrected integrated fluxes for
most galaxies in the RCSEDv2 sample.

We use the NBursts full spectrum fitting technique (Chilingarian et al. 2007a,
b) with several grids of stellar population models and, for a subset of
spectra where we had available UV and NIR photometric data, we fitted
broadband spectral energy distributions simultaneously with spectra using the
NBursts+phot code (Chilingarian & Katkov 2012). For every data source we
created a pre-processing routine, which converts the original data into the
same format, applies flux calibration and correction for telluric absorption
(if needed), prepares a map of wavelength-dependent spectral resolution for
each spectrum and finally allows us to run our spectral fitting code. The
spectrum fitting returns stellar (absorption-line) and gas (emission-line)
kinematics and a star formation history in a parametric form, typically age,
metallicity, and $\alpha$-enhancement of stellar population. We use several
grids of stellar population models and include the corresponding parameters of
the star formation history in RCSEDv2 for every model grid.

Figure 1.: Redshift distributions of RCSEDv2 galaxies originating from
different spectroscopic datasets. The $y$ axis scale is logarithmic. Figure
2.: A screenshot of an RCSEDv2 web-page for NGC 60 with a summary of its
properties and the first available spectrum with the parameters derived from
it. Red markers over a galaxy image display the positions of available
spectra.

##  3 Technical Solutions and Data Access

We use PostgreSQL database engine with pgSphere (Chilingarian et al. 2004) as
a back-end solution. The intermediate layer for Virtual Observatory access
protocols is provided by GaVO DaCHS (Demleitner et al. 2014). The
intermediate-layer uses Django for database interaction. The front-end is
powered by several modern javascript frameworks and also uses aladin-lite Sky
Atlas (Boch & Fernique 2014).

##  4 Research Highlights and Perspectives

Using RCSED (i) we discovered the universal optical-UV color-color-magnitude
relation of galaxies (Chilingarian & Zolotukhin 2012); (ii) we identified over
70% of known compact elliptical galaxies (Chilingarian & Zolotukhin 2015);
(iii) we found a population of intermediate-mass black holes which power
active galactic nuclei (Chilingarian et al. 2018); (iv) we identified
progenitors of present-day ultra-diffuse galaxies as low-mass spatially
extended post-starburst galaxies in clusters (Grishin et al. 2021). With
RCSEDv2, a new version of our dataset, we have already found young analogs of
compact elliptical galaxies, substantially expanded a sample of intermediate-
mass black holes, and found many active galactic nuclei in rare giant low-
surface brightness galaxies (Saburova et al. 2021).

Numerous interesting research results in the field of galaxy formation and
evolution and in the studies of the large-scale structure of the Universe are
expected to be spawned by the analysis of the largest homogeneously processed
extragalactic dataset provided by RCSEDv2. We expect to use the infrastructure
and data processing and analysis approaches to the data which will be
delivered by the next-generation large spectroscopic surveys DESI and 4MOST.

### Acknowledgments

This project is supported by the Russian Science Foundation Grant 19-12-00281
the Interdisciplinary Scientific and Educational School of Moscow University
“Fundamental and Applied Space Research”.

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