Energy estimates

Control of γe Recall that

γ_e(Ψ, ϕ) =−1

2N(N −1)(N−2)(N −3)

=(hhΨ, gβ(x1 −x2) [VN(x3 −x4),br] Ψii). Using symmetry, Lemma 2.0.5 (d) and Notation (3.94), γ_e is bounded by

γe(Ψ, ϕ)≤N⁴

hhΨ, gβ(x1−x2)

VN(x3−x4),mb^cp1p2p3p4+ 2mb^dp1p2p3q4

+2mb^ep₁q₂p₃p₄+ 4mb^fp₁q₂p₃q₄ Ψii

≤4N⁴kV_N(x₃−x₄)Ψkk1supp(VN)(x₃−x₄)p₃k_opkg_β(x₁−x₂)p₁k_op

×(kmb^ck_op+kmb^dk_op+kmb^ek_op+kmb^fk_op).

We get with (3.95), Lemma 3.3.6 and Lemma 2.0.5 that

|γ_e(Ψ, ϕ)| ≤ K(ϕ, A·)N^5/2+3ξ−β. Control of γ_f Recall that

γ_f(Ψ, ϕ) = 2N(N −1)N −2

N −1= hhΨ, g_β(x₁−x₂)

b_VN|ϕ|²(x₁),rb Ψii

. We obtain the estimate

|γ_f(Ψ, ϕ)| ≤ K(ϕ, A_·)N²kg_βkkbrk_op ≤ K(ϕ, A_·)N^1+ξ−β. (3.102) Summary of the estimates Collecting all estimates, we get with ξ <1/2

|γc(Ψ, ϕ)|+|γd(Ψ, ϕ)|+|γe(Ψ, ϕ)|+|γf(Ψ, ϕ)| ≤ K(ϕ, A·)

N^4−β+N^3−β2

. (3.103) Choosing β sufficiently large, we obtain the desired decay and hence Lemma 3.4.10.

Proof: We expanding EW_β(Ψ)− E_b^GP

Wβ(ϕ). This yields E_Wβ(Ψ)− E_b^GP

Wβ(ϕ) =k∇₁Ψk²+N −1

2 hhΨ, W_β(x1−x2)Ψii

− k∇ϕk²−1

2b_Wβkϕ²k²+hhΨ, A_t(x₁)Ψii − hϕ, A_tϕi

=k1_A^(d)

1

∇₁q₁Ψk²+k1_A^(d)

1

∇₁Ψk² +M(Ψ, ϕ) +Q_β(Ψ, ϕ), where we have defined

M(Ψ, ϕ) =2<

hh∇₁q₁Ψ,1_A^(d)

1

∇₁p₁Ψii

(3.104) +k1_A^(d)

1

∇₁p₁Ψk²− k∇ϕk² (3.105)

+hhΨ, A_t(x₁)Ψii − hϕ, A_tϕi, (3.106)

Q_β(Ψ, ϕ) =(1−)k1_A^(d)

1

∇₁Ψk²+ (1−)k1_A^(d)

1

∇₁q₁Ψk² (3.107)

+N −1

2 hhΨ,(1−p₁p₂)W_β(x₁−x₂)(1−p₁p₂)Ψii (3.108) +N −1

2 hhΨ, p₁p₂W_β(x₁ −x₂)p₁p₂Ψii − 1 2N

Z

R²

W_β(x)d²xkϕ²k² +(N −1)<hhΨ,(1−p₁p₂)W_β(x₁−x₂)p₁p₂Ψii.

We first consider the first two contributions (3.107) + (3.108). Note that (1−p₁p₂)∆₁(1− p₁p₂) =p₁∆₁p₁q₂+p₁∆₁q₁q₂+q₁q₂∆₁p₁+q₁∆q₁. We hence obtain

(3.107) =−(1−)hhΨ, q₁∆₁q₁Ψii+k1_A^(d)

1

∇₁p₁Ψk²+ 2<

hh∇₁p₁Ψ,1_A^(d)

1

∇₁q₁Ψii

=−(1−)hhΨ,(1−p₁p₂)∆₁(1−p₁p₂)Ψii

+(1−) (hhΨ, p₁∆₁p₁q₂Ψii+ 2<(hhΨ, q₁q₂∆₁p₁Ψii)) +k1_A^(d)

1

∇₁p₁Ψk²+ 2<

hh∇₁p₁Ψ,1_A^(d)

1

∇₁q₁Ψii Rearranging terms, we obtain

(3.107) + (3.108) =hhΨ,(1−p1p2)

−(1−)∆1+N −1

2 Wβ(x1 −x2)

(1−p1p2)Ψii (3.109) + (1−) (hhΨ, p₁∆₁p₁q₂Ψii+hhΨ, p₁∆₁q₁q₂Ψii+hhΨ, q₁q₂∆₁p₁Ψii) (3.110) +k1_A^(d)

1

∇₁p₁Ψk²+ 2<

hh∇₁p₁Ψ,1_A^(d)

1

∇₁q₁Ψii

. (3.111)

Note that the operator inequality−(1−)∆ +¹₂W ≥0 implies by rescaling that (3.109) is nonnegative. Furthermore, it follows

|(3.110)| ≤ K(ϕ, A·)hhΨ, q₁Ψii,

and, applying Lemma 3.5.5, part (a),

|(3.111)| ≤ K(ϕ, A·)(k∇₁q₁Ψk+ 1)N^1/2−d. Define

S_β(Ψ, ϕ) =(N −1)|hhΨ,(1−p₁p₂)W_β(x₁ −x₂)p₁p₂Ψii| (3.112) +

N −1

2 hhΨ, p₁p₂W_β(x₁−x₂)p₁p₂Ψii − 1

2b_Wβkϕ²k²

. (3.113)

Applying the estimates above, together with the assumptionsk∇₁Ψk ≤C,k∇ϕk ≤C, we can then conclude the bound

k1_A^(d)

1

∇₁q₁Ψk² ≤|M(Ψ, ϕ)|+|S_β(Ψ, ϕ)|+

E_Wβ(Ψ)− E_b^GP

Wβ(ϕ) +K(ϕ, A·) hhΨ, q₁Ψii+N^1/2−d

.

Next, we split up the energy difference E_VN(Ψ)− E_b^GP

VN(ϕ), E_VN(Ψ)− E_b^GP

VN(ϕ) = k∇₁Ψk²+ N −1

2 hhΨ, V_N(x₁−x₂)Ψii − k∇ϕk²

− bV_N

2 kϕ²k²+hhΨ, A·(x₁)Ψii − hϕ, A·ϕi.

In order to better estimate the terms corresponding to the two-particle interactions, we introduce, for µ > d, the potential Mµ(x), defined in Definition 3.3.5, and continue with

E_VN(Ψ)− E_b^GP

VN(ϕ) = k1_A^(d)

1

∇₁Ψk²+k1_B^(d)

1 1_A^(d)

1

∇₁Ψk²+k1_B^(d)

1 1_A^(d)

1

∇₁Ψk² + N −1

2 hhΨ,1_B^(d)

1

V_N(x₁ −x₂)Ψii + 1

2hhΨ,

N

X

j=2

1_B^(d)

1

(V_N −M_µ) (x₁ −x_j)Ψii

+ 1 2hhΨ,

N

X

j=2

1_B^(d)

1

M_µ(x₁−x_j)Ψii − k∇ϕk² −b_VN 2 kϕ²k² +hhΨ, A·(x₁)Ψii − hϕ, A·ϕi.

Using thatq1 = 1−p1 and symmetry, we obtain for 0< <1,

EV_N(Ψ)− E_b^GP

VN(ϕ)

=

k1_A^(d)

1

∇₁q₁Ψk²+k1_B^(d)

1 1_A^(d)

1

∇₁Ψk² + (1−)

k1_A^(d)

1

∇₁q₁Ψk²+k1_B^(d)

1 1_A^(d)

1

∇₁Ψk² +N −1

2 hhΨ,1_B^(d)

1

V_N(x₁−x₂)Ψii +N −1

2 hhΨ,1_B^(d)

1

(1−p1p2)Mµ(x1−x2)(1−p1p2)1_B^(d)

1

Ψii +k1_B^(d)

1 1_A^(d)

1

∇₁Ψk²+ 1 2hhΨ,

N

X

j=2

1_B^(d)

1

(V_N −M_µ) (x₁−x_j)Ψii +N −1

2 hhΨ,1_B^(d)

1

p₁p₂M_µ(x₁−x₂)p₁p₂1_B^(d)

1

Ψii − bVN

2 kϕ²k² + 2<

hh∇1q1Ψ,1_A^(d)

1

∇1p1Ψii + (N −1)<hhΨ,1_B^(d)

1

(1−p₁p₂)M_µ(x₁−x₂)p₁p₂1_B^(d)

1

Ψii +k1_A^(d)

1

∇₁p₁Ψk²− k∇ϕk² +hhΨ, A·(x₁)Ψii − hϕ, A·ϕi

=

k1_A^(d)

1

∇₁q₁Ψk²+k1_B^(d)

1 1_A^(d)

1

∇₁Ψk²

+M(Ψ, ϕ) + ˜Q_µ(Ψ, ϕ).

with

Q˜_µ(Ψ, ϕ) = N −1

2 hhΨ,1_B^(d)

1

(1−p₁p₂)M_µ(x₁−x₂)(1−p₁p₂)1_B^(d)

1

Ψii + (1−)

k1_A^(d)

1

∇₁q₁Ψk² +k1_B^(d)

1 1_A^(d)

1

∇₁Ψk²

+N −1

2 hhΨ,1_B^(d)

1

V_N(x₁−x₂)Ψii (3.114) +k1_B^(d)

1 1_A^(d)

1

∇₁Ψk²+1 2hhΨ,

N

X

j=2

1_B^(d)

1

(V_N −M_µ) (x₁−x_j)Ψii (3.115) + (N −1)<hhΨ,1_B^(d)

1

(1−p₁p₂)M_µ(x₁−x₂)p₁p₂1_B^(d)

1

Ψii + N −1

2 hhΨ,1_B^(d)

1

p₁p₂M_µ(x₁−x₂)p₁p₂1_B^(d)

1

Ψii − b_VN

2 kϕ²k².

The first term in ˜Q_µ(Ψ, ϕ) is nonnegative. For µ > d Lemma 3.5.11 below shows that

(3.115) is also nonnegative. Furthermore, we are able to bound, for 0< <1, (3.114) =(1−)

k1_A^(d)

1 1_B^(d)

1

∇₁Ψk²+k1_B^(d)

1 1_A^(d)

1

∇₁Ψk²

+ N −1

2 hhΨ,1_B^(d)

1

V_N(x₁−x₂)Ψii

−(1−)2<

hh∇₁Ψ,1_A^(d)

1 1_B^(d)

1

∇₁p₁Ψii +(1−)

k1_A^(d)

1 1_B^(d)

1

∇₁q₁Ψk²+k1_A^(d)

1 1_B^(d)

1

∇₁p₁Ψk²

The third line is positive. In analogy to the proof of Lemma 3.5.5, we obtain k1_A^(d)

1 1_B^(d)

1

∇₁p₁Ψk ≤ k1_B^(d)

1

∇₁p₁Ψk ≤ K(ϕ, A·)N^1−d+δ for any δ >0. This implies

2<

hh∇₁Ψ,1_B^(d)

1 1_A^(d)

1

∇₁p₁Ψii

≤ K(ϕ, A·)N^1−d+δ. Focusing on the first term, we obtain with Corollary 4.3.15

(1−) k1_A^(d)

1 1_B^(d)

1

∇₁Ψk²+k1_B^(d)

1 1_A^(d)

1

∇₁Ψk²

+N −1

2 hhΨ,1_B^(d)

1

V_N(x₁−x₂)Ψii (3.116)

= 1

NhhΨ, −(1−)

N

X

k=1

∆k1_B^(d)

k

+

N

X

i6=j

1_B^(d)

j

1

2VN(xi−xj)

!

Ψii ≥0. (3.117)

Thus, for µ > d, we obtain the bound S˜_µ(Ψ, ϕ) =(N −1)

hhΨ,1_B^(d)

1

(1−p₁p₂)M_µ(x₁−x₂)p₁p₂1_B^(d)

1

Ψii

(3.118)

+

N−1

2 hhΨ,1_B^(d)

1 p1p2Mµ(x1−x2)p1p21_B^(d)

1 Ψii − b_VN 2 kϕ²k²

(3.119) +K(ϕ, A·)N^1−d+δ

≥ −Q˜_µ(Ψ, ϕ).

In total, we obtain

|1_A^(d)

1

∇₁q₁Ψk²+k1_B^(d)

1 1_A^(d)

1

∇₁Ψk²

≤ |M(Ψ, ϕ)|+|S˜_µ(Ψ, ϕ)|+

E_VN(Ψ)− E_b^GP

VN(ϕ) . Next, we will estimate M(Ψ, ϕ), S_β(Ψ, ϕ) and ˜S_µ(Ψ, ϕ).

• Estimate ofS_β(Ψ, ϕ) and ˜S_µ(Ψ, ϕ).

We first estimate (3.119), using the same estimate as in (3.57). Note that hhΨ,1_B^(d)

1

p₁p₂M_µ(x₁ −x₂)p₁p₂1_B^(d)

1

Ψii=hϕ, M_µ?|ϕ|²ϕihhΨ,1_B^(d)

1

p₁p₂1_B^(d)

1

Ψii.

Usingk1_B^(d)

1

Ψk ≤CN^1−d+, for any >0, (see Lemma 3.3.6 (j)) we obtain, together with kp₁p₂Ψk² = 1−2kp₁q₂Ψk²− kq₁q₂Ψk²

|(3.119)| ≤3kq₁Ψk²+C N^1−d++N^2−2d+2 +1

2|Nhϕ, M_µ?|ϕ|²ϕi −NkM_µk₁kϕ²k²| +1

2|b_VN −NkM_µk₁|kϕ²k²+1

2hϕ, M_µ?|ϕ|²ϕi.

Note that, using Young’s inequality and (3.57),

|hϕ, N M_µ?|ϕ|²ϕi −NkM_µk₁kϕ²k²|

= Z

R²

d²x|ϕ(x)|² N(M_µ?|ϕ|²)(x)−NkM_µk₁|ϕ(x)|²

≤ kϕk²_∞kN(M_µ?|ϕ|²)− kN M_µk₁|ϕ|²k₁ ≤Ckϕk²_∞k∆|ϕ|²k₁N^−2µln(N)

≤ K(ϕ, A·)N^−2µln(N).

Since |NkM_µk₁−b_VN| ≤C^ln(N)_N (see Lemma 3.5.5) and

hϕ, M_µ?|ϕ|²ϕi ≤ kϕk⁴_∞kM_µk₁ ≤Ckϕk⁴_∞N⁻¹, it follows that

|(3.119)| ≤K(ϕ, A·) hhΨ,bn^ϕΨii+C(N^1−d++N^2−2d+2) +N^−2µln(N) +N⁻¹ln(N)

≤K(ϕ, A·) hhΨ,bn^ϕΨii+N⁻¹ln(N)

, (3.120)

where the last inequality holds ford large enough (recall that we chose µ > d).

Using the same estimates, we obtain

(3.113)≤ K(ϕ, A_·) hhΨ,nb^ϕΨii+N^−2βln(N) +N⁻¹ln(N) .

Line (3.118) and line (3.112) are controlled by Lemma 3.5.12, which is stated below.

(3.112),(3.118)≤ K(ϕ, A_·)(hhΨ,bnΨii+N^−1/6ln(N)).

In total, we obtain, for any µ > d≥1, the bound

Sβ(Ψ, ϕ)≤K(ϕ, A·) hhΨ,bnΨii+N^−2βln(N) +N^−1/6ln(N) S˜_µ(Ψ, ϕ)≤K(ϕ, A_·) hhΨ,bnΨii+N^−1/6ln(N)

.

• Estimate ofM(Ψ, ϕ).

First, we estimate (3.104).

|(3.104)| ≤2|hh∇₁q₁Ψ,1_A^(d)

1

∇₁p₁Ψii|+ 2|hh∇₁q₁Ψ,∇₁p₁Ψii|

≤2k∇₁q₁Ψk k1_A^(d)

1

∇₁p₁k_op+ 2|hhbn^−1/2q₁Ψ,∆₁p₁bn^1/2₁ Ψii|.

By Lemma 3.5.4, we obtain k1_A^(d)

1

∇1p1kop ≤Ck∇ϕk∞N^1/2−d. Furthermore, we use k∇₁q₁Ψk ≤ k∇₁Ψk+k∇₁p₁Ψk ≤ K(ϕ, A·) (see also Lemma 3.5.1) and

|hhbn^−1/2q₁Ψ,∆₁p₁nb^1/2₁ Ψii| ≤ K(ϕ, A·)kbn^1/2₁ Ψkkbn^1/2Ψk ≤ K(ϕ, A·)(hhΨ,bnΨii+N⁻¹).

Hence, for d large enough,

|(3.104)| ≤ K(ϕ, A·)(hhΨ,bnΨii+N^12−d+N⁻¹)≤ K(ϕ, A·)(hhΨ,bnΨii+N⁻¹).

Line (3.105) is estimated ford large enough, noting thatk∇₁p₁Ψk² =k∇ϕk²kp₁Ψk², by

(3.105) =k1_A^(d)

1

∇₁p₁Ψk²− k∇ϕk²

≤|k∇1p1Ψk²− k∇ϕk²|+k1_A^(d)

1

∇1p1Ψk²

≤C k∇ϕk²hhΨ, q₁Ψii+k∇ϕk²_∞N^1−2d

≤K(ϕ, A·) hhΨ,nΨiib +N^1−2d . For line (3.106), we use Lemma 3.5.6 to obtain

(3.106)≤CkA·k∞ hhΨ,bnΨii+N^−1/2 . In total, we obtain

M(Ψ, ϕ)≤ K(ϕ, A·) hhΨ,bnΨii+N^−1/2 .

Lemma 3.5.11

(a) LetR_β andM_β be defined as in Lemma 3.3.5. LetV satisfy Assumption 3.2.3. Then, for any Ψ∈H¹(R^2N,C)

k1^|x1−x₂|≤R_β∇₁Ψk²+ 1

2hhΨ,(V_N −M_β)(x₁−x₂)Ψii ≥0.

(b) Let M_β be defined as in Lemma 3.3.5. Let Ψ∈L²_s(R^2N,C)∩H¹(R^2N,C). Then, for sufficiently large N and for β > d,

k1_B^(d)

1 1_A^(d)

1

∇₁Ψk²+ 1 2hhΨ,

N

X

j=2

1_B^(d)

1

(V_N −M_β) (x₁−x_j)Ψii ≥0.

Proof:

(a) We first show nonnegativity of the one-particle operatorH^Zn :H²(R²,C)→L²(R²,C) given by

H^Zn =−∆ + 1 2

X

zk∈Zn

(V_N(· −z_k)−M_β(· −z_k))

for any n ∈ N and any n-elemental subset Z_n ⊂R² which is such that the supports of the potentials M_β(· −z_k) are pairwise disjoint for any two z_k ∈Z_n.

Sincefβ(· −z_k) is the zero energy scattering state of the potentialVN(· −z_k)−Wβ(· − z_k), it follows that

F_β^Zn = Y

zk∈Zn

f_β(· −z_k)

fulfills H^ZnF_β^Zn = 0 for any such Z_n. By construction, f_β is a nonnegative function, so is F_β^Zn. Since ¹₂P

zk∈Z_n(VN(· −zk)−Mβ(· −zk))∈L^∞(R²,C), this potential is a infinitesimal perturbation of −∆, thusσ_ess(H^Zn) = [0,∞). Assume now thatH^Zn is not nonnegative. Then, there exists a ground state Ψ_G ∈H²(R²,C) ofH^Zn of nega-tive energyE <0. The phase of the ground state can be chosen such that the ground state is real and positive (see e.g. Theorem 10.12. in [70]). Since such a ground state of negative energy decays exponentially, that is Ψ_G(x) ≤C₁e^−C2|x|, C₁, C₂ > 0 , the following scalar product is well defined (althoughF_β^Zn ∈/ L²(R²,C)).

hF_β^Zn, H^ZnΨGi=hF_β^Zn, EΨGi<0. (3.121) On the other hand we have since F_β^Xn

1,β is the zero energy scattering state hF_β^Zn, H^ZnΨGi=hH^ZnF_β^Zn,ΨGi= 0.

This contradicts (3.121) and the nonnegativity of H^Zn follows.

Now, assume that there exists aψ ∈H²(R²,C) such that the quadratic form Q(ψ) =k1^|·|≤Rβ∇ψk²+ 1

2hψ,(V_N(·)−M_β(·))ψi<0.

Since VN and Mβ are spherically symmetric, we can assume that ψ is spherically symmetric. Subsituting ψ → aψ, a ∈ R , we can furthermore assume that, for all

|x|=R_β, ψ(x) = 1− for >0.

Define ˜ψ such that ˜ψ(x) = ψ(x) for|x| ≤ Rβ and ˜ψ(x) = 1 for |x| > Rβ + and >0. Furthermore, ˜ψ can be constructed such thatk1^|x|≥Rβ∇ψk˜ ² ≤C(+²).

Then Q( ˜ψ) = Q(ψ) < 0 holds, because the operator associated with the quadratic form is supported inside the ball B₀(R_β).

Using ˜ψ, we can construct a set of points Z_n and a χ ∈ H²(R²,C) such that hχ, H^Znχi<0, contradicting to nonnegativity of H^Zn.

ForR >1 let

ξ_R(x) =

R²/x², for x > R;

1, else.

Let now Z_n be a subset Z_n ⊂ R² with |Z_n| =n which is such that the supports of the potentials W_β(· −z_k) lie within the sphere around zero with radius R and are pairwise disjoint for any two z_k ∈Z_n. Since we are in two dimensions we can choose a n which is of order R².

Let now χ_R(x) =ξ_R(x)Q

z_k∈Zn

ψ(x˜ −z_k). By construction, there exists a D=O(1) such that χR(x) = ˜ψ(x−zk) for|x−zk| ≤D. From this, we obtain

hχ_R, H^Znχ_Ri=k∇χ_Rk²+n1

2hψ,(V_N(·)−M_β(·))ψi

=nQ(ψ) + X

z_k∈Zn

k1|x−zk|≥Rβ∇χ_Rk²

≤nQ(ψ) +Cn(+²) +k∇ξ_Rk²

=nQ(ψ) +Cn(+²) +C.

Choosing R and hence n large enough and small, we can find a Z_n such that hχ_R, H^Znχ_Ri is negative, contradicting nonnegativity of H^Zn.

Now, we can prove that k1^|x1−x₂|≤R_β

1∇₁Ψk²+1

2hhΨ,(V_N −M_β)(x₁−x₂)Ψii ≥0. (3.122) holds for any Ψ ∈ H²(R^2N,C). Using the coordinate transformation ˜x₁ = x₁ − x₂ ,x˜_i =x_i ∀i≥2, we have∇_x1 =∇_x˜1. Thus (3.122) is equivalent tok1^|x1|≤R_β

1∇₁Ψk²+

1

2hhΨ,(VN −Mβ)(x1)Ψii ≥0 ∀Ψ∈H²(R^2N,C) which follows directly from Q(ψ)≥0 for all ψ ∈ H²(R²,C). By a standard density argument, we can conclude that Q(Ψ)≥0 ∀Ψ∈H¹(R^2N,C).

(b) Definec_k ={(x₁, . . . , x_N)∈R^2N||x₁−x_k| ≤R_β}andC₁ =∪^N_k=2c_k. For (x₁, . . . , x_N)∈ B^(d)₁ it holds that |x_i −x_j| ≥ N^−d for 2 ≤ i, j ≤ N. Let β > d. Assume that N^−d>2R_β, which hold for N sufficiently large, since R_β ≤CN^−β. Then, it follows that, for 2 ≤ i, j ≤ N and i 6= j,

c_i∩ B₁^(d)

∩

c_j ∩ B₁^(d)

= ∅. Under the same conditions, we also have1_A^(d)

1

≥1^C1. Therefore, 1_A^(d)

1 1_B^(d)

1

≥1C11_B^(d)

1

=1_C1∩B^(d)

1

=1_∪N k=2

ck∩B₁^(d)=

N

X

k=2

1_c

k∩B^(d)₁ =1_B^(d)

1

N

X

k=2

1ck. Note that1_B^(d)

1

depends only on x₂, . . . , x_N. By this k1_A^(d)

1 1_B^(d)

1

∇₁Ψk² ≥

N

X

k=2

k1ck∇₁1_B^(d)

1

Ψk² = (N −1)k1^|x1−x2|≤R_β∇₁1_B^(d)

1

Ψk².

This yields

(3.115)≥(N −1)

k1^|x1−x₂|≤R_β∇₁1_B^(d)

1

Ψk²+1 2hh1_B^(d)

1

Ψ,(V_N −M_β)(x₁−x₂)1_B^(d)

1

Ψii

≥0.

where the last inequality follows from (a)

Lemma 3.5.12 Let Wβ ∈ Vβ as in Definition 3.3.4. Let Ψ ∈ L²_s(R^2N,C)∩H¹(R^2N,C) andk∇₁Ψkbe bounded uniformly in N. Let d in Definition 3.5.4 of1_B^(d)

1

sufficiently large.

Let Γ∈ {Ψ,1_B^(d)

1

Ψ}. Then, for all β >0, (a)

N|hhΓ, q₁p₂W_β(x₁ −x₂)p₁p₂Γii| ≤ Ckϕk²_∞hhΨ,nΨii.ˆ (b)

N|hhΓ, p1p2Wβ(x1−x2)q1q2Γii| ≤ K(ϕ, A·) hhΨ,bnΨii+N^−1/6ln(N) . (c)

N|hhΓ,(1−p₁p₂)W_β(x₁−x₂)p₁p₂Γii| ≤ K(ϕ, A·) hhΨ,bnΨii+N^−1/6ln(N) . Proof:

(a) Let first Γ =1_B^(d)

1

Ψ. Then, N

hh1_B^(d)

1

Ψ, q₁p₂W_β(x₁−x₂)p₁p₂1_B^(d)

1

Ψii

≤N hh1_B^(d)

1

Ψ, q₁p₂W_β(x₁−x₂)p₁p₂1_B^(d)

1

Ψii

(3.123)

+N

hhΨ, q₁p₂W_β(x₁−x₂)p₁p₂1_B^(d)

1

Ψii

. (3.124)

Using Lemma 3.5.5 together with kp₂W_β(x₁ −x₂)p₂k_op ≤ kϕk²_∞kW_βk₁, the first line can be bounded, for any >0, by

(3.123)≤ K(ϕ, A_·)Nk1_B^(d)

1

ΨkkW_βk₁ ≤ K(ϕ, A_·)N^1−d+. (3.125) The second term is bounded by

(3.124) =N

hhq

|W_β(x₁−x₂)|q₁p₂(ˆn)⁻¹²Ψ, q

|W_β(x₁−x₂)|p₁p₂nˆ

1 2

11B^(d)1Ψii

≤CNk q

|W_β(x₁−x₂)|p₂k²_op

kq₁(ˆn)⁻¹²Ψk² +kˆn

1 2

11B^(d)1Ψk²

≤CNkq

|W_β(x₁−x₂)|p₂k²_op

hhΨ,nΨiiˆ +kˆn

1 2

1Ψk²+kˆn

1 2

11_B^(d)

1

Ψk²

≤CNkWβk1kϕk²_∞

hhΨ,ˆnΨii+k1_B^(d)

1

Ψk²

≤Ckϕk²_∞ hhΨ,nΨiiˆ +N^1−d+

.

Choosing d large enough, N^1−d+ is smaller than hhΨ,nΨii. This yields (a) in theˆ case Γ =1_B^(d)

1

Ψ. The inequality (a) can be proven analogously for Γ = Ψ.

(b) Let Γ = 1_B^(d)

1

Ψ. We first consider (b) for potentials with β < 1/4. We have to estimate

N|hh1_B^(d)

1

Ψ, p₁p₂W_β(x₁−x₂)q₁q₂1_B^(d)

1

Ψii| ≤ N|hhΨ, p₁p₂W_β(x₁−x₂)q₁q₂Ψii|

+N|hh1_B^(d)

1

Ψ, p₁p₂W_β(x₁ −x₂)q₁q₂Ψii|+N|hhΨ, p₁p₂W_β(x₁−x₂)q₁q₂1_B^(d)

1

Ψii|

+N|hh1_B^(d)

1

Ψ, p₁p₂W_β(x₁ −x₂)q₁q₂1_B(d) 1

Ψii|

≤N|hhΨ, p₁p₂W_β(x₁−x₂)q₁q₂Ψii| (3.126) +CNk1_B^(d)

1

ΨkkW_βk∞. (3.127)

The last term is bounded, for any >0, by

(3.127)≤CN N^1−d+N^−1+2β ≤N⁻² , where the last inequality holds choosing d large enough.

Using Lemma 2.0.5 (c) and Lemma 2.0.10 withO_1,2 =q₂W_β(x₁−x₂)p₂, Ω =N^−1/2q₁Ψ and χ=N^1/2p₁Ψ we get

(3.126)≤ kq₁Ψk²+N²

hhq₂Ψ, p₁k q

|W_β(x₁−x₂)|p₃k q

|W_β(x₁−x₃)|

× k q

|W_β(x₁ −x₂)|p₂k q

|W_β(x₁−x₃)|p₁q₃Ψii +N²(N −1)⁻¹kq₂W_β(x₁ −x₂)p₂p₁Ψk²

≤ kq₁Ψk²+N²kkq

|W_β(x₁−x₂)|p₁k⁴_op kq₂Ψk² +CNkW_β(x₁−x₂)p₂k²_op.

With Lemma 2.0.5 (e) we get the bound

(3.126)≤kq₁Ψk² +N²kϕk⁴_∞kW_βk²₁ kq₁Ψk²+CNkW_βk²kϕk²_∞.

Note, that kWβk1 ≤CN⁻¹, kWβk² ≤CN^−2+2β Hence

(3.126)≤C hhΨ, q₁Ψii+K(ϕ)N^−1+2β .

Note that, for β < 1/4, N^−1+2β ≤N^−1/6ln(N). Using the same bounds for Γ = Ψ, we obtain (b) for the case β <1/4.

b) for 1/4≤β:

We useU_β1,β from Definition 3.5.2 for some 0< β₁ <1/4.

Z_β^ϕ(x₁, x₂)−W_β+U_β1,β has the form ofZ_β^ϕ

1(x₁, x₂) which has been controlled above.

It is left to control N

hh1_B^(d)

1

Ψ, p₁p₂(W_β(x₁−x₂)−U_β1,β(x₁−x₂))q₁q₂1_B^(d)

1

Ψii . Let ∆h_β1,β =W_β−U_β1,β. Integrating by parts and using that

∇₁h_β1,β(x₁−x₂) = −∇₂h_β1,β(x₁−x₂) gives N

hh1_B^(d)

1

Ψ, p₁p₂(W_β(x₁−x₂)−U_β1,β(x₁−x₂))q₁q₂1_B^(d)

1

Ψii

=N

hh∇₁p₁1_B^(d)

1

Ψ, p₂∇₂h_β1,β(x₁−x₂)q₁q₂1_B^(d)

1

Ψii

(3.128)

+N hh1_B^(d)

1

Ψ, p₁p₂∇₂h_β1,β(x₁−x₂)∇₁q₁q₂1_B^(d)

1

Ψii

. (3.129)

Let (a1, b1) = (q1,∇p1) or (a1, b1) = (∇q1, p1). Then, both terms can be estimated as follows:

We use Lemma 2.0.10 with Ω = N^−η/2a₁1_B^(d)

1

Ψ, O_1,2 =N^1+η/2q₂∇₂h_β1,β(x₁−x₂)p₂ and χ=b11_B^(d)

1 Ψ. We choose η <2β1.

N hh1_B^(d)

1 Ψ, a1p2∇2hβ1,β(x1−x2)b1q21_B^(d)

1 Ψii

≤N^−ηka₁1_B^(d)

1

Ψk² (3.130)

+ N^2+η

N −1kq₂∇₂h_β1,β(x₁−x₂)b₁p₂1_B^(d)

1

Ψk² (3.131)

+N^2+η hh1_B^(d)

1

Ψ, b₁p₂q₃∇₂h_β1,β(x₁−x₂)∇₃h_β1,β(x₁−x₃)b₁q₂p₃1_B^(d)

1

Ψii

1/2

. (3.132) We obtain (note that 1_B^(d)

1

does not depend on x₁) (3.130)≤N^−ηka₁1_B^(d)

1

Ψk² =N^−ηk1_B^(d)

1

a₁Ψk² ≤ K(ϕ, A·)N^−η.

since both k∇q1Ψk and kq1Ψk are bounded uniformly in N. Since q2 is a projector it follows that

(3.131) ≤N^2+η

N −1k∇₂h_β1,β(x₁−x₂)p₂k²_opkb₁1_B^(d)

1

Ψk²

≤C N^2+η

N −1kϕk²_∞k∇h_β1,βk²kb₁1_B^(d)

1

Ψk²

≤K(ϕ, A·)N^η−1ln(N)kϕk²_∞, where we used Lemma 3.5.3 in the last step.

Next, we estimate

(3.132)≤N^2+ηkp₂∇₂h_β1,β(x₁−x₂)b₁q₂1_B^(d)

1

Ψk²

≤2N^2+ηkp₂∇₂h_β1,β(x₁−x₂)b₁q₂1_B^(d)

1

Ψk² (3.133)

+2N^2+ηkp₂∇₂h_β1,β(x₁−x₂)b₁q₂Ψk². (3.134) The first term can be estimated as

(3.133)≤CN^2+ηk∇₂h_β1,β(x₁−x₂)b₁k²_opk1_B^(d)

1

Ψk²

≤CN^2+ηk∇₂h_β1,βk²(kϕk²_∞+k∇ϕk²_∞)k1_B^(d)

1

Ψk²

≤K(ϕ, A·)N^2+ηN⁻²ln(N)N^2−2d+2 =K(ϕ, A·)N^2−2d+2+ηln(N), for any > 0. For d large enough, this term is subleading. The last term can be estimated as

(3.134)≤2N^2+ηkp₂h_β1,β(x₁−x₂)b₁∇₂q₂Ψk²

+2N^2+ηk|ϕ(x₂)ih∇ϕ(x₂)|h_β1,β(x₁ −x₂)b₁q₂Ψk²

≤CN^2+ηkp₂h_β1,β(x₁−x₂)k²_opkb₁∇₂q₂Ψk²

+CN^2+ηk|ϕ(x₂)ih∇ϕ(x₂)|h_β1,β(x₁−x₂)k²_opkb₁q₂Ψk²

≤CN^2+η k∇ϕk²_∞+kϕk²_∞

kh_β1,βk²(1 +k∇ϕk²)

≤K(ϕ, A·)N^η−2β1ln(N)².

Combining both estimates we obtain, for any β >1, N

hh1_B^(d)

1

Ψ, p1p2Wβ(x1−x2)q1q21B^(d)₁Ψii

≤ inf

η>0 inf

0<µ<1/4 K(ϕ, A·) hhΨ,nΨiib +N^−1+2µ+N^−η+N^η−1ln(N) +N^η−2µln(N)

≤ K(ϕ, A·) hhΨ,nΨiib +N^−1/6ln(N) .

where the last inequality comes from choosing η= 1/3 and µ= 1/4. For Γ = Ψ, (b) can be estimated the same way, yielding the same bound.

(c) This follows from (a) and (b), using that 1−p₁p₂ =q₁q₂ +p₁q₂+q₁p₂.

Im Dokument Derivation of mean field dynamics for attractive quantum systems (Seite 74-87)