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BOGOLIUBOV SPECTRUM OF INTERACTING BOSE GASES 31
Proof. Let us write w = w+ − w− where w+ = max{w,0} and w− =
max{−w,0}. By taking the second quantization (see Lemma 13) of the
non-negative two-body operator
�
�
Q⊗(Q−εP)w+Q⊗(Q−εP)+(Q−εP)⊗Qw+(Q−εP)⊗Q
�
�
+ Q⊗(Q+εP)w−Q⊗(Q+εP)+(Q+εP)⊗Qw−(Q+εP)⊗Q
where Q = 1−P = 1−|u0〉〈u0| and ε > 0, we obtain the following Cauchy-
Schwarz inequality
�
m,n,p≥1
�
−1
≤ ε
〈um ⊗un,wup ⊗u0〉a ∗ ma ∗ napa0
m,n,p,q≥1
+ �
m,n,p≥1
〈u0 ⊗up,wun ⊗um〉a ∗ 0 a∗ p anam
〈um ⊗un,|w|up ⊗uq〉a ∗ m a∗ n apaq
+ε �
m,n≥1
〈um ⊗u0,|w|un ⊗u0〉a ∗ ma ∗ 0ana0.
After performing the unitary transformation UN(·)U ∗ N
A3 ≤
1
ε(N −1)
≤ M
N −1
�
m,n,p,q≥1
we find that
〈um ⊗un,|w|up ⊗uq〉a ∗ m a∗ n apaq
+ ε
N −1 dΓ(Q(|u0| 2 ∗|w|)Q)(N −N+). (50)
for every ε > 0.
On the other hand, by the same proof of Lemma 20 we have
1 �
〈um ⊗un,|w|up ⊗uq〉〈a
N −1
m,n,p,q≥1
∗ ma∗ napaq〉Φ (51)
� �
α2
. (52)
Moreover, by using Lemma 17 we get
〈H〉Φ +C〈N+〉Φ +C
1−α1
dΓ(Q(|u0| 2 ∗|w|)Q)(N −N+) ≤ (N −1)dΓ(Q(|u0| 2 ∗|w|)Q)
and , we get
�
dΓ(Q(|u0| 2 �
N −N+
∗|w|)Q)
N −1 Φ
From (50), (51) and (53), we can deduce that
〈A3〉Φ ≤
� M
N −1
� α2
≤ � dΓ(Q(|u0| 2 ∗|w|)Q) �
(53) Φ
� �
α2
≤ 〈H〉Φ +C〈N+〉Φ +C .
1−α1
�
〈H〉Φ +C〈N+〉Φ +C .
1−α1
By repeating the above proof with w replaced by −w, we obtain the same
upper bound on −〈A3〉Φ and then finish the proof of Lemma 21. �