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

arXiv:0808.4066v1 [math-ph] 29 Aug 2008

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

Jun Yin

Department of Mathematics, Harvard University, Email: JYin@math.harvard.edu

August 29th, 2008

Abstract

The leading term of the ground state energy/particle of a dilute gas of bosons with massm in the thermodynamic limit is 2π~²a̺/m when the density of the gas is̺, the interaction potential is non-negative and the scattering lengthais 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.

1 Introduction and main theorems

In Dyson’s work [9] and Lieb, Yngvason and Seiringer’s work [7, 6], it is rigor- ously proved that the leading term of the ground state energy/particle of a three dimensional dilute bose gas of mass min the thermodynamic limit with density̺ is 2π~²a̺/m, i.e.,

e(̺, m) = 2π~²a̺/m(1 +o(1)) if a³̺≪1 (1.1) where they assumed that the interaction potential is non-negative, the scattering length ais 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.

***

c 2008 by the author. This paper may be reproduced, in its entirety, for non-commercial purposes.

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 Λ =L³ in the following way (in units where ~= 2m= 1):

H_N ≡ XN

i=1

−∆_i+ X

1≤i<j≤N

v^a(x_i−x_j) (1.2)

Here ∆ denotes the Laplacian on Λ with periodic boundary condition and v^a is a scaled interaction potential, i.e.,

v^a(x) =a⁻²·v(x/a), a >0 (1.3) The pair interaction potential v is spherically symmetric and supported on the set {x∈R³:|x| ≤R₀} for someR₀>0.

DEFINITION 1 (Scattering Length). Assume that w is a pair spherically sym- metric interaction potential with compact support. Denote E[φ] as the energy of the complex-valued function φon R³ as follows,

E[φ] = Z

R³|∇φ(x)|²+¹₂w(x)|φ(x)|²dx. (1.4) Define the scattering length SL(w) of potential w as the following minimum energy.

SL(w)≡min

φ

1

4πE[φ] : lim

|x|→∞φ(x) = 1

(1.5) Note: IfSL(w)>−∞, one can easily prove that the Hamiltonian−∆ + ¹₂whas no bound state. In particular, whenw≥0, we haveSL(w)≥0 andwhas 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 andv^a in (1.3), we can assume that

SL(v) = 1, SL(v^a) =a (1.6)

Letf₁(x) be the solution of the zero-energy scattering equation ofv, i.e.,

−∆f₁(x) +¹₂v(x)f₁(x) = 0, (1.7) then we have that f_a(x)≡f₁(x/a) is the solution of the following zero-energy scat- tering equation of v^a.

−∆f_a(x) + ¹₂v^a(x)f_a(x) = 0, (1.8) As in [5], one can prove that if fa is normalized as lim_|x|→∞fa(x) = 1, then

f_a(x) = 1−a/x, for |x|> R₀a (1.9) In this paper, we are interested in the ground energyE(N,Λ) of H_N in the thermo- dynamic limit that Λ → ∞, N → ∞ and N/Λ = ̺. Low density means that the average inter-particle distance ̺^−1/3 is much larger than the scattering lengtha, i.e.

a³̺≪1.

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

THEOREM 1. Fix vwithSL[v] = 1 andv^a satisfying (1.3). Letf₁ be the solution of zero-energy equation of v and normalized as: f₁(∞) = 1. In the thermodynamic limit, limN→∞N/Λ = ̺, we have the following upper bound on E(N,Λ), which is the ground energy of H_N in (1.2),

lim sup

N→∞

E(N,Λ)

4πa̺N ≤1 + const.(a³̺)^1/4, (1.10) for some constant depending on kf₁k∞, provided that ^4π₃ a³̺≤1.

Note: So far, the best proof of the error term on upper bound, when v ≥ 0, is O(a³̺)^1/3, as in [5].

On the other hand, for the lower bound in (1.1), we prove that as long asvhas 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 ifv^ais a continuous function on R³ and H_N has no bound state for any N, v^a satisfies the above two requirements, i.e.,

v^a(0)>0, minv^a(r)>−∞ (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)≥0, v₋(x)≥ −λ₋, λ₋>0 and v₊ has a positive core, i.e. ∃r₁, such that v₊(x) ≥λ₊>0 for |x| ≤r₁. Here v₋ need not be negative.

There existc₁(R₀/r₁)andc₂(R₀/r₁), which are greater than one and only depend on R₀/r₁, such that the following holds.

If there exists some positive number t satisfying

SL[c₁(R₀/r₁)·(v+tv₋)]≥0 and λ₊≥(1 +t⁻¹)c₂(R₀/r₁)·λ₋, (1.12) we have the following lower bound on E(N,Λ),

lim inf

N→∞

E(N,Λ)

4πa̺N ≥1−const.(a³̺)^1/17, (1.13) for some constant depending onv₊ andv₋, provided that ^4π₃ a³̺is 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 alsoO(a³̺)^1/17, as in [5].

This theorem implies the following two corollaries.

COROLLARY 1. Assume that

v(x) =v₊(x) +λ₋v₋(x), v₊(x)≥0, v₋(x)≥ −1 (1.14)

and v₊ has a positive core, i.e. ∃r₁ such that v₊(x)≥λ₊ for |x| ≤r₁. There exists λ₀(r₁, R₀, λ₊, v₋) such that, if 0≤λ₋≤λ₀, i.e., the potential is shallow enough, we have the following lower bound on E(N,Λ),

lim inf

N→∞

E(N,Λ)

4πa̺N ≥1−const.(a³̺)^1/17 (1.15) provided that ^4π₃ a³̺ is smaller than some constant depending on v+ and v₋.

Proof. For fixed R₀,r₁ and λ₊, when λ₋ is small enough, we have that

SL[c₁(R₀/r₁)(v₊+ 2λ₋v₋)]≥0 and λ₊≥2c₂(R₀/r₁)λ₋. (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)≥0≥v₋(x)≥ −λ₋ (1.17) and v₊ has a positive core, i.e. ∃r₁ such that v₊(x) ≥λ₊ for |x| ≤ r₁. There exist λ₀(R₀/r₁, λ₋) and ε(R₀, r₁, λ₋) such that, if

λ₊≥λ₀ and Z

x∈R³|v₋(x)|dx≤ε(R₀, r₁, λ₋), (1.18) we have the following lower bound on E(N,Λ),

lim inf

N→∞

E(N,Λ)

4πa̺N ≥1−const.(a³̺)^1/17 (1.19) provided that ^4π₃ a³̺ is smaller than some constant depending on v+ and v₋.

Proof. We choose λ₀ = max{3,2c₂(R₀/r₁)}λ₋, then we have that λ₊ ≥λ₀ ≥3λ₋, which implies that

[v+v₋](x)≥λ₊/3≥0 for |x| ≤r₁ (1.20) Then we claim that for any n≥1 andλ+≥3λ₋, there exists ξ(n)>0,

Z

R³|v₋(x)|dx≤ξ(n)⇒SL[n(v+v₋)]≥0. (1.21) To prove (1.21), we shall prove that there exists ξ(n) > 0, if R

R³|v₋(x)|dx ≤ ξ(n), for any non-negative radial function f,

Z

|x|≤R0

|∇f|²(x) +n

2(v+v₋)f²(x)dx≥0. (1.22) We can see, with (1.20),

Z

|x|≤R0

h

|∇f|²(x) +n

2(v+v₋)f²(x)i

dx (1.23)

≥ Z

r1≤|x|≤R0

|∇f|²(x)− knv₋k∞f²(x) dx

≥ Z

r1≤|x|≤R0

|∇f|²(x)dx−nλ₋ Z

r1≤|x|≤R0

|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 η(n) such that

Z

r1≤|x|≤R0

|f(x)|⁴dx

!1/2

≤η(n) Z

r1≤|x|≤R0

|f|²(x)dx (1.24) On the other hand, with (1.20), v+v₋≥2v₋ and Schwarz’s Inequality, we have that

Z

|x|≤R0

|∇f|²(x) +n

2(v+v₋)f²(x)dx (1.25)

≥ Z

|x|≤R0

|∇f|²(x) + Z

|x|≤r1

nλ₊

6 f²(x)− Z

r1≤|x|≤R0

nv₋(x)|f(x)|²dx

≥ Z

|x|≤R0

|∇f|²(x) + Z

|x|≤r1

nλ+

6 f²(x)

−nη(n) Z

r1≤|x|≤R0

v²₋(x)dx

1/2Z

r1≤|x|≤R0

|f|²(x)dx

≥ Z

|x|≤R0

|∇f|²(x) + Z

|x|≤r1

nλ₊

6 f²(x)−nη(n)λ₋kv₋k^1/21

Z

|x|≤R0

|f|²(x)dx

Thus, for n≥1, if

kv₋k^1/21 ≤(ξ(n))^1/2 ≡ 1

nη(n) ·min

f

R

|x|≤R0|∇f|²(x) +R

|x|≤r1

nλ+

6 f²(x)dx R

|r|≤R0|f|²(x)dx , the inequality (1.22) holds. We note that it is easy to see that ξ(n)>0. Hence we arrive at the desired result (1.21). At last, choosing

ε(R₀, r₁, λ₋) =ξ(c₁(R₀/r₁)) (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 (a³̺)^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 ∀λ₋ > 0, there ∃v satisfying min_x∈R³v(x)<−λ₋ 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 Ψ_N,Λ satisfying

lim sup

N→∞

hΨ|H_N|Ψi

NhΨ|Ψi ≤4πa̺(1 + const. Y) (2.1)

for some constant that depends only on kf₁k∞. Here we denoteY as Y ≡

4π 3 a³̺

1/4

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

Ψ_N = YN

p=1

F_p (2.3)

In [9], F_p depends on the the nearest particle to thex_p among all the x_i with i < p, i.e.,

F_p =f(t_p), t_p= min

i<p {|x_i−x_p|} (2.4)

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

0≤f ≤1, f^′ ≥0 (2.5)

Hence in [9],F_p has the following property

F_p,i·f(|x_p−x_i|)≤F_p ≤F_p,i (2.6) HereF_p,iis defined in [9] as the value thatF_p would take if the pointx_iwere 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 functionF_p should have following properties.

1. F_p is a continuous function ofx_i (1≤i≤N).

2. When |x_i −x_p| is large enough, the position of x_i does not effect F_p, i.e.,

∇^xiFp = 0.

3. F_p has a similar property as (2.6).

First we define θ_r(x) as the characteristic function of the set {x :|x| ≤ r} and θ¯_r ≡1−θ_r. Choosing b=a/Y, we have

a/b= 4π

3 b³N/Λ =Y. (2.7)

Without loss of generality, we assume thatb >max{2R0a,4a}, as in [9, 5]. We define f(x) as

f(x) =

f_a(x)/f_a(b) b≥ |x| ≥0

1 otherwise, (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) = 1−a/|x|

1−a/b , for b≥ |x| ≥R₀a. (2.9) Let Re= max{R₀a,2a}, which implies that f(R)e > ¹₂. We define Θⁱⁿ_p , Θ^out_p (1< p≤ N) as

Θⁱⁿ_p ≡Y

j<p

θ_Re(x_j−x_p), Θ^out_p ≡Y

j<p

θ¯_Re(x_j−x_p) (2.10)

We can see that Θⁱⁿ_p = 1 when |xj −xp| ≤ Re for all j < p and Θ^out_p = 1 when

|x_j −x_p| >Re for all j < p. With Θⁱⁿ_p and Θ^out_p , we can define r_p(x₁,· · ·, x_N) and R_p(x₁,· · · , x_N) as follows, (x_i ∈[0, L]³, i= 1,· · ·N)

r_p ≡ (2.11)

(1−Θ^out_p )·min

i<p

|x_i−x_p|:f(x_i−x_p) = min

j<p

f(x_j−x_p) :|x_j−x_p| ≤Re

R_p ≡Re·Θⁱⁿ_p + (1−Θⁱⁿ_p )×min

i<p

|x_i−x_p|:|x_i−x_p|>Re

With the definition of R_p and (2.9), we have that

1. f(Rp)≤f(xj−xp) for any j < psatisfying |xj−xp|>R,e 2. R_p ≤ |x_j−x_p|for any j < psatisfying |x_j−x_p|>R.e 3. R_p ≥Re

4. When Θⁱⁿ_p = 0, there exists j_p such that|x_jp−x_p|=R_p Similarly, we have

1. f(r_p)≤f(x_j−x_p) for any j < psatisfying |x_j−x_p| ≤R,e

2. r_p ≤ |x_j−x_p|for any j < psatisfying |x_j−x_p| ≤Re and f(x_j −x_p) =f(r_p).

3. rp ≤Re

4. When Θ^out_p = 0, there exists ip such that|xip−xp|=rp

Then, we define a continuous functionT on Ras follows

T(|x|) =





1 2Re≥ |x|

(|x|⁻¹−b⁻¹)(2Re⁻¹−b⁻¹)⁻¹ b≥ |x| ≥2Re

0 |x| ≥b ,

(2.12)

At last we defineF_p(x₁,· · · , x_N) on [0, L]^3N as follows (1< p≤N), F_p ≡





f(r_p) Θⁱⁿ_p = 1

f(R_p) Θ^out_p = 1

f(rp) +T(Rp) [f(Rp)−f(rp)]₋ otherwise ,

(2.13) and F₁ = 1. Here [·]₋ denotes the negative part, i.e., [x]₋ = x when x < 0 and [x]₋= 0 when x≥0. We note that for any x

[x]₋≤0 (2.14)

Note: If v ≥0, it is well known that f is an increasing function, which implies the F_p we defined is equal to theF_p in [9].

One can prove thatF_p is a continuous function of (x₁,· · · , x_N) by checking that, for any j 6=p > 1 and fixedx₁,· · ·, x_j−1, x_j+1,· · · , x_N,F_p is a continuous function of x_j. First we can see that it is trivial forj > p, sinceF_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| = Re to |x_j −x_p| = Re+ 0⁺. One can see that when |x_j −x_p|= R,e f(R_p)≥f(R) =e f(x_j−x_p)≥f(r_p), soF_p =f(r_p), i.e.

F_p = min

k:kmin6=j,k<p{f(x_k−x_p) :|x_k−x_p| ≤Re}, f(x_j−x_p)

(2.15) On the other hand, when|x_j −x_p|=Re+ 0⁺≤2R, we can see thate R_p =|x_j−x_p|, T(R_p) = 1 and f(R_p) =f(R) + 0e ⁺. Hence,

Fp = min

k:kmin6=j,k<p{f(x_k−xp) :|x_k−xp| ≤Re}, f(xj−xp)

(2.16) Hence we arrive at the desired result thatF_p is continuous function.

We can also see that F_p is non-negative and bounded as follows

M ≡ kF_pk∞=kfk∞≤(1−a/b)⁻¹kf_ak∞= (1−a/b)⁻¹kf₁k∞≤2kf₁k∞. (2.17) Here we use the fact f_a(x) =f₁(x/a).

By the definition ofFp, one can see thatFp= 1 when Q

q<pθ¯_b(xp−xq) = 1 and Fp ≤1 whenQ

q<pθ¯_Re(xp−xq) = 1, so 1−X

q<p

θ_b(xp−xq)≤Fp ≤1 +X

q<p

(M−1)θ_Re(xp−xq) (2.18) We now construct the state functions Φ_k as follows (1≤k≤N)

Φ_k= Yk

p=1

F_p

Note: all Φ’s are functions on [0, L]^3N and Φ_k is independent ofx_l forl > k. We will choose Ψ = Φ_N for (2.1).

As in [7], for proving the upper bound on the total energyhΦ_N|H_N|Φ_NikΦ_Nk⁻2², we shall estimate the upper bounds on

kΦ_Nk⁻2²

Z X

i

|∇iΦ_N|² YN

j=1

dx_j and kΦ_Nk⁻2²

Z X

i<j

v^a(x_i−x_j)|Φ_N|² YN

k=1

dx_k

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

kΦ_Nk⁻2²

Z X

i

|∇iΦ_N|² YN

j=1

dx_j and kΦ_Nk⁻2²

Z X

i<j

[v^a]₊(x_i−x_j)|Φ_N|² YN

k=1

dx_k

(2.20) and the lower bound on

kΦ_Nk⁻2²

Z X

i<j

[v^a]₋(x_i−x_j)

· |Φ_N|² YN

k=1

dx_k (2.21)

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

• kΦNk⁻2²

R P

i|∇ⁱΦN|²QN

j=1dxj ≤(1 +o(1))^N_Λ² R

R³|∇f(x)|²dx

• kΦ_Nk⁻2²

R P

i<j[v^a]₊(x_i−x_j)|Φ_N|²Q_N

k=1dx_k≤(1+o(1))^N_Λ² R

R³ 1

2[v]₊|f(x)|²dx

• kΦ_Nk⁻2²

R P

i<j

[v^a]₋(x_i−x_j)

· |Φ_N|²QN

k=1dx_k ≥ (1−o(1))^N_Λ² R

R³ 1

2|[v]₋| ·

|f(x)|²dx.

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

1. For any m-variable function gm(xi1· · ·xim),m < k ≤N,ij 6=i_k forj 6=k, we have

kΦ_kF_i⁻₁¹· · ·F_i⁻_m¹g_mk²2 ≤(2M)^2mΛ^−mkΦ_k−mk²2kg_mk²2 (2.22) 2. For any two variable function g₂(x_i, x_i′) (i < i^′), we have

kΦ_NF_i⁻_′¹g₂k²2 ≤ kg₂k²2Λ⁻²kΦ_Nk²2(1 + const. Y) (2.23) 3. Letf_i,i′ =f(x_i−x_i′), for any two variable functiong₂(x_i, x_i′) (i < i^′), we have kΦ_Nf_i,i⁻¹′g₂k²2≥ kg₂k²2Λ⁻²kΦ_Nk²2(1−const. Y) (2.24) Note: if f_i,i′ = 0 and i < i^′, thenF_i′ = 0, so Φ_Nf_i,i⁻¹′ is definable.

We will use (2.22) for controlling the error terms. The inequalities (2.23) and (2.24) will be used in estimating the terms (2.20) and (2.21), respectively.

We begin with deriving a lower bound onkΦ_Nk²2. Fori6=p, LetF_p,i be the value that F_p would take if changing the order of particles as follows,

Fp,i(x1· · ·xN)≡F_n(p,i)(x1· · ·xi−1, xi+1· · ·xN, xi) (2.25) Here n(p, i) is defined as follows (i6=p)

n(p, i) =

p i > p

p−1 i < p , (2.26)

Similarly, we can define F_p,i,j(x₁· · ·x_N) as

F_p,i,j(x₁· · ·x_N)≡F_m(p,i,j)(x₁· · ·x_i−1, x_i+1· · ·x_j−1, x_j+1· · ·x_N, x_i, x_j) f or i < j (2.27) 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,· · · , p} \ {i, j}.

Note: As we mentioned Fp we defined is equal to the Fp in [6] in the case when v ≥0. Furthermore, one can see that our definitions ofF_p,i andF_p,i,j are equivalent to those definitions in [6] when v≥0.

With the definitions ofFp and Fp,i, we obtain thatFp,i is independent of xi and F_p is bounded from below as follows

Fp(x1· · ·xN)≥

F_p,i i > p

F_p,iθ¯_b(x_p−x_i) i < p , (2.28) and

F_p ≥Y

i<p

θ¯_b(x_p−x_i).

Then Φ²_N is bounded from below, for any fixedi, by

F₁·F₂· · ·F_N

2

≥

F_{1, i}· · ·F_{i−1, i}F_{i+1, i}· · ·F_{N, i}

2

×Y

j6=i

θ¯_b(x_i−x_j) (2.29)

≥

F_{1, i}· · ·F_{i−1, i}F_{i+1, i}· · ·F_{N, i}

2

×

1−X

j6=i

θ_b(x_i−x_j)

Integrating both sides with R Q_N

j=1dxj, we obtain that kΦ_Nk²2 ≥ kΦ_N−1k²2

1− 4πb³ 3 N/Λ

=kΦ_N−1k²2(1−Y) (2.30) Here we used the fact that

kΦ_N−1k²2=Z

F_{1, i}· · ·F_{i−1, i}F_{i+1, i}· · ·F_{N, i}

2Y

j

dx_j. (2.31) Similarly, one can also prove that for k≤N,

kΦ_kk²2 ≥ kΦ_k−1k²2(1−Y) (2.32)

Next we are going to prove (2.22) in the casem= 1, k=N, i.e.,

kΦNF_i⁻¹g1k²2 ≤4M²Λ⁻¹kΦN−1k²2kg1k²2 (2.33) One can check that F_p > F_p,i only when the following conditions are satisfied:

1. i < p

2. |x_i−x_p| ≤R,e

3. for any other j < p,|xj−xp|is greater thanR,e 4. T(R_p)<1, i.e. for any other j < p,|x_j−x_p|>2R,e i.e.,

F_p > F_p,i⇒G_p,i ≡θ_Re(x_p−x_i) Y

j<p,j6=i

θ¯_2Re(x_j−x_p) = 1 (2.34)

On the other hand, using the fact that f(R)e > ¹₂, one obtains that if F_p > F_p,i, F_p,i Y

j<p,j6=i

θ¯_2Re(x_j−x_p)> f(R)e Y

j<p,j6=i

θ¯_2Re(x_j −x_p)≥ 1 2

Y

j<p,j6=i

θ¯_2Re(x_j−x_p)

Hence when G_p,i= 1, we have 2M F_p,iG_p,i≥M ≥F_p, i.e.,

F_p ≤





Fp,i i > p

F_p,i

1 + (2M −1)G_p,i

i < p . (2.35) By the definition of G’s, one can see that if p, q > iand p6=q,

G_p,iG_q,i= 0. (2.36)

Hence, we have that

Y

p>i

1 + (2M−1)Gp,i

≤2M (2.37)

Combining (2.35) and (2.37), we have the upper bound on |ΦNF_i⁻¹|as follows,

F₁·F₂· · ·F_NF_i⁻¹

2

≤4M²

F_{1, i}· · ·F_{i−1, i}F_{i+1, i}· · ·F_{N, i}

2

, (2.38)

which implies the desired result (2.33) with (2.31). Furthermore, for any m-variable function g_m(x_i1· · ·x_im)

kΦ_NF_i⁻₁¹· · ·F_i⁻_m¹g_mk²2≤(2M)^2mΛ^−mkΦ_N−mk²2kg_mk²2 (2.39)

With the inequality (2.30) and the fact F_i ≤M for any i≤N, we get

kΦ_Ng_mk²2 ≤ (2M²)^2mkΦ_N−mk²2kg_mk²2 (2.40)

≤ (1−Y)^−m(2M²)^2mΛ^−mkΦ_Nk²2kg_mk²2

Similarly, we can generalize this result to m < k≤N

kΦ_kg_mk²2 ≤ (2M²)^2mkΦ_k−mk²2kg_mk²2 (2.41)

≤ (1−Y)^−m(2M²)^2mΛ^−mkΦ_kk²2kg_mk²2

Now we shall prove the upper bound on kΦ_Nk²2 with (2.41). Choosing p=N, with the bounds of F_p in (2.18), we get that

Φ²_N ≤ F₁²·F₂²· · ·F_N²₋₁



1 + X

j<N

M²θ_Re(x_j −x_N)



 (2.42)

= Φ²_N−1+ Φ²_N−1 X

j<N

M²θ_Re(x_j−x_N)

Hence, using the inequalities (2.41)(m= 1) and (2.32), we obtain that

kΦ_Nk²2 ≤ kΦ_N−1k²2+ (1−Y)⁻¹(2M²)²YkΦ_N−2k²2 (2.43)

≤ kΦN−1k²2(1 + const. Y)

Putting (2.43) and (2.30) together, we obtain the relation betweenkΦ_NkandkΦ_N−1k kΦ_Nk²2 =kΦ_N−1k²2(1 +O(Y)) (2.44) Similarly, for k≤N

kΦ_kk²2 =kΦ_k−1k²2(1 +O(Y)) (2.45) Next, we shall prove (2.23), i.e.,

kΦ_NF_i⁻′¹g₂k²2≤ kg₂k²2Λ⁻²kΦ_Nk²2(1 + const. Y). (2.46) Using the inequalities Fi ≤ 1 +P

l<i(M−1)θ_Re(x_l−xi)

and (2.35), with the prop- erty of the G’s in (2.36), we get

Y

k6=i^′

F_k ≤ 1 +X

l<i

(M−1)θ_Re(x_l−xi)

! Y

k6=i^′,i

F_{k ,i}·

1 +X

j>i

2M Gj,i

(2.47) Similarly, replacingF_p,i’s withF_p,i,i^′’s and using the fact thatG_k,l ≤θ_Re(x_k−x_l), we get

Φ_NF_i⁻′¹ ≤ Y

k6=i^′,i

F_k,i,i′ 1 +X

l<i

M θ_Re(x_l−x_i)

!

(2.48)

×

1 +X

j>i

2M θ_Re(xj −xi)

×

1 +X

j^′>i^′

2M θ_Re(x^′_j −x^′_i)

Expanding (2.48), multiplying g₂(x_i, x_i^′) to each side and integrating them with Q_N

k=1dx_k, with the result of (2.41, 2.45), we obtain that

kΦ_NF_i⁻′¹g₂k²2 ≤ kg₂k²2Λ⁻²kΦ_Nk²2(1 + const. Y) (2.49) So far we proved some upper bounds of the expectation value of Φ_N. Next we are going to prove the following lower bound on kΦ_Nf_i,i⁻¹_′g₂k²2:

kΦ_Nf_i,i⁻¹′g₂k²2 ≥ kg₂k²2Λ⁻²kΦ_Nk²2(1−const. Y) (2.50) Here we denotef_i,i^′ =f(xi−x_i^′) (i < i^′). First, by the definition ofF_i^′, one can see that F_i^′ =f(x_i−x_i^′) when Q

k<i^′,k6=iθ¯_b(x_k−x^′_i) = 1, i.e., F_i²′ ≥f(x_i−x_i^′)²



1− X

k<i^′,k6=i

θ_b(x_k−x_i^′)



 (2.51)

Using this inequality and (2.39) with m = 3, i₁ =i, i₂ = i^′ and i₃ = k, we obtain that

Φ_Nf_i,i⁻¹′g₂ ²

2 ≥Φ_NF_i⁻′¹g₂²

2−const. Y kg₂k²2Λ⁻²kΦ_Nk²2 (2.52) Then with the lower bound onF_i²in (2.18), i.e.,F_i² ≥1−P

k<iθ_b(x_k−x_i), we obtain that

kΦ_Nf_i,i⁻¹_′g₂k²2 ≥ kΦ_NF_i⁻¹F_i⁻_′¹g₂k²2−const. Ykg₂k²2Λ⁻²kΦ_Nk²2 (2.53) Again, using the bound on F_p in (2.28), we see that

Φ_NF_i⁻¹F_i⁻′¹ ≥ Y

k6=i,i^′

F_k,i,i^′ 1−X

l<i

θ_b(x_l−x_i)

!

×



1− X

l^′<i^′, l^′6=i

θ_b(x_l^′−x_i^′)





Then using (2.41) and (2.45), we arrive at the desired result (2.50).

So far, we have proved the inequalities we need for calculating the value of hΦ_N|P

i,jv^a(x_i−x_j)|Φ_Ni. Then we need to calculate ∇iΦ_N. We denote i_p as the particle satisfyingi_p < pand|x_ip−x_p|=r_p andn^r_pas the unit vector in the direction of xp−xip. Similarly, denotejp as the particle satisfyingjp < pand |xjp−xp|=Rp

and n^R_p as the unit vector in the direction of x_p−x_jp. We remark that such i_p or j_p may not exist in some cases, but we do define them as 0. We denote ∇0F_p = 0.

Recall the definition of Fp in (2.13). We have

− ∇^pFp =∇ⁱpFp+∇^jpFp (2.54)

∇ipF_p =−n^r_pf^′(r_p)

Θⁱⁿ_p + Θ⁻_p(1−T(R_p)) + Θ⁺_p

∇jpF_p =−n^R_p

Θ^out_p f^′(R_p) + Θ⁻_pT(R_p)f^′(R_p) + Θ⁻_pT^′(R_p) f(R_p)−f(r_p)

Here Θ⁺_p is the function of (x₁· · ·x_N) which is defined as Θ⁺_p ≡

1−Θⁱⁿ−Θ^out

·h

f(R_p)−f(r_p)

(2.55) and Θ⁻_p is defined as

Θ⁻_p ≡

1−Θⁱⁿ−Θ^out

·h

f(rp)−f(Rp)

(2.56) Here h is the Heaviside step function. By the definition of Φ_N, we obtain that

|∇pΦ_N|²

|Φ_N|² =

−F_p⁻¹∇ipF_p−F_p⁻¹∇jpF_p+ X

q,iq=p

F_q⁻¹∇pF_q+ X

q,jq=p

F_q⁻¹∇pF_q

2

Then with (2.54), we have that X

p

|∇pΦ_N|²≤ 2|Φ_N|²X

p

F_p⁻²

|f^′(r_p)|² Θⁱⁿ_p + Θ⁻_p|1−T(R_p)|²+ Θ⁺_p +|T(R_p)f^′(R_p)|²Θ⁻_p +|T^′(R_p)|²|f(R_p)−f(r_p)|²Θ⁻_p

+|T(Rp)| · |1−T(Rp)| · |f^′(rp)| · |f^′(Rp)|Θ⁻_p +|f^′(Rp)|²Θ^out_p

+2|Φ_N|² X

k<p<q

F_p⁻¹F_q⁻¹

|∇kF_p| · |∇pF_q|+|∇kF_p| · |∇kF_q|

Because 0≤T ≤1 andi_p6=j_p, one can easily prove that for any fixedp,

|f^′(r_p)|² Θⁱⁿ_p + Θ⁻_p|1−T(R_p)|²+ Θ⁺_p

+|f^′(R_p)|²Θ^out_p

+ |T(R_p)f^′(R_p)|²Θ⁻_p +|T(R_p)| · |1−T(R_p)| · |f^′(r_p)| · |f^′(R_p)|Θ⁻_p

≤ X

k:k<p

f^′(|xp−x_k|)², and

|T^′(R_p)|²|f(R_p)−f(r_p)|²Θ⁻_p ≤M² X

k:k<p



T^′(|x_p−x_k|)² X

j:j6=k, p

θ_Re(x_j −x_p)





(2.57)

Hence, we obtain that hΦN|HN|ΦNi ≤2X

i<j

Z

|ΦN|²F_j⁻²

1

2f(xi−xj)²[v(xi−xj)]++f^′(xi−xj)²

−2X

i<j

Z

|Φ_N|²f⁻²(x_i−x_j)

1

2f²(x_i−x_j)

[v(x_i−x_j)]₋

+2X

i<j

Z

|Φ_N|²M²X

k<p



T^′(|x_p−x_k|)² X

j:j6=k, p

θ_Re(x_j−x_p)



 (2.58)

+2 X

k<p<q

Z

|Φ_N|²F_p⁻¹F_q⁻¹

|∇kF_p| · |∇pF_q|+|∇kF_p| · |∇kF_q|

Here [·]₊ and [·]₋ denote the positive and negative part, respectively and we used the fact that Fj ≤f(xi−xj) wheni < j and |xi−xj| ≤R, which implies thate

[v(xi−xj)]+≤F_j⁻²f(xi−xj)²[v(xi−xj)]+ (2.59) With the results in (2.49) and (2.50), we can obtain the upper bound on the main part of hΦN|HN|ΦNi, i.e.,

2X

i<j

Z

|Φ_N|²× 1

2

F_j⁻²

f²(x_i−x_j)[v(x_i−x_j)]₊+f^′(x_i−x_j)²

−f²(x_i−x_j) f²(x_i−x_j)

[v(x_i−x_j)]₋

≤4πaN²/Λ(1 + const. Y)kΦ_Nk²2 (2.60) With the definition of T in (2.12) and (2.40), we obtain that the third line of (2.58) is bounded as const. aN²YkΦ_Nk²2/Λ. For the other terms, we have

|∇ⁱpFp| ≤ |f^′(|xip−xp|)|, |∇^jpFp| ≤ |f^′(|xjp−xp|)|+M T^′(|xjp−xp|) (2.61) Hence, with the inequality (2.39), we can prove that the last line in (2.58) are bounded as

const. N³Λ⁻²(K+L)²kΦ_Nk²2 (2.62) Here K and Lare defined as follows

K ≡ Z

R³|f^′(|x−y|)|dy L≡ Z

R³

T^′(|x−y|)dy. (2.63) 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(Rb) =e O(ab) (2.64) Hence we obtain that the last line in (2.58) are bounded by const. aN²Y². Combining this result with (2.60), we get the following result,

hΦN|HN|ΦNi

kΦ_Nk²2 ≤4πaN²/Λ(1 + const. Y) (2.65)