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BOGOLIUBOV SPECTRUM OF INTERACTING BOSE GASES 35
From Assumption (A3) and the upper bound λL( � HN)−λ1( � HN) ≤ λL(H)−
λ1(H)+o(1), we have
||gMΦ|| 2 ≤ 2〈N+〉 Φ
M
≤ NεR(N)
M
for every normalized vector Φ ∈ Y, where εR is as in (A3). We can choose
R = λL(H)−λ1(H)+1. In particular, if M ≫ NεR(N), then ||gMΦ|| 2 → 0
as N → ∞ independently of ε and the choice of Φ in Y. Consequently, when
N is large enough we have dim(fMY) = L by Proposition 22 (ii), and by
the min-max principle, we get
〈fMΦ,HfMΦ〉
max
Φ∈Y,||Φ||=1 ||fMΦ|| 2 ≥ λL(H).
Thus from (57) and the simple bound dΓ(h) |F ≤N
+
−λ1( � HN) ≤ C( � HN +CN)
(see Remark 9) we get
λL( � � � � �
M M
HN) ≥ 1−C λL(H)−C
N N −CL
NεR(N)
M −CL
N
M2. (58)
By choosing M such that max{NεR(N), √ N} ≪ M ≪ N, we obtain
λL( � HN) ≥ λL(H)+o(1).
Remark on the convergence rate. The error obtained in the lower bound(58)
is not better than 3� εR(N) +N −1/5 . However, it can be improved by the
following bootstrap argument. First, from (57) with M = rN for some
small fixed number r > 0, and the simple bound dΓ(h) |F ≤N
+
C( � HN +CN) (see Remark 9), we obtain
− λ1( � HN) ≤
frNHfrN ≤ C( � HN +C). (59)
Next, by projecting the inequality (57), with 1 ≪ M ≪ N, onto the sub-
−λ1( � HN) ≤ C(H+C)
space frNF+ and using the refined bound dΓ(h) |F ≤N
+
(see Remark 9), we get
frN �HNfrN ≥ fM
��
1−C
� � � �
M M
H−C fM
N N
+λ1( � HN)g 2 Mf2 C
rN −
M2frN(H+C)frN (60)
Here we have used the fact that fMfrN = fM when M ≤ rN/2. Finally, we
can use Lemma 24 with M = rN, then estimate frN � HNfrN by (60), and
employ the inequality g2 M ≤ 2N+/M ≤ C(H+C)/M and (59). We have
�� � � � �
M M
�HN ≥ fM 1−C H−C fM −C
N N
N+ C
−
N M (� HN +C)
when 1 ≪ M ≪ N. Consequently,
λL( � HN) ≥ λL(H)−CL
�� M
N
1
+
M +εR(N)
�