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BOGOLIUBOV SPECTRUM OF INTERACTING BOSE GASES 27
A2 = �
m,n≥1
+ �
+ 1
2
+ 1
2
m,n≥1
A3 =
A4 =
〈um,(T −µH)un〉a ∗ m an
�
um, � |u0| 2 � � ∗ N −N+
∗w +K1 un aman N −1
�
〈um ⊗un,K2〉a ∗ na ∗ �
(N −N+)(N −N+ −1)
m
N −1
�
� (N −N+)(N −N+ −1)
〈K2,um ⊗un〉
aman,
N −1
m,n≥1
m,n≥1
1
N −1
+ 1
N −1
1
2(N −1)
�
m,n,p≥1
�
m,n,p≥1
�
m,n,p,q≥1
Wmnp0a ∗ ma∗n ap
�
N −N+
�
W0pnm N −N+a ∗ panam, Wmnpqa ∗ m a∗ n apaq.
Here recall that eH = T00 +(1/2)W0000 and µH = T00 +W0000.
In the next section, we shall carefully estimate all the terms of the right
side of (44) to show that
in the regime N+ ≪ N.
UNHNUN −NeH ≈ H
5. Bound on truncated Fock space
The main result in this section is the following bound.
Proposition 15 (Preliminary bound on truncated Fock space). Assume
that (A1) and (A2) hold true. For any vector Φ in the quadratic form domain
of H such that Φ ∈ F ≤M
+ for some 1 ≤ M ≤ N, we have
�
�
�〈UN(HN −NeH)U ∗ N〉 Φ −〈H〉 Φ
�
�
� M
� ≤ C
N 〈H+C〉 Φ .
Here for a self adjoint operator A and an element a in a Hilbert space,
we write 〈A〉a instead of 〈a,Aa〉 for short. Recall that the quadratic form
domain of H is the same as that of dΓ(h+1), on which the quadratic form
UN(HN−NeH)U ∗ N iswell-defined. AsaneasyconsequenceofProposition15,
we can prove the weak convergence in the first statement of Theorem 2.
Corollary 16 (Weak convergence towards H). Assume that (A1) and (A2)
hold true. Then we have for all fixed Φ,Φ ′ in the quadratic form domain of
the Bogoliubov Hamiltonian H,
lim
N→∞
where by convention U ∗ N
� � � ′ ∗
Φ ,UN HN −N eH UN Φ �
F+ = � Φ ′ ,HΦ �
is extended to 0 outside of F≤N
+ .
F+
(45)