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BOGOLIUBOV SPECTRUM OF INTERACTING BOSE GASES 39
Assumption (A3’) implies that TrΓ >M
N
Therefore,
liminf
N→∞ (Fβ(N)−NeH) ≥ liminf
N→∞
→ 0 and TrΓ≤M
N
→ 1 as N → ∞.
�
−β −1 logTre −β(1−CN−1/5 )H �
= −β −1 logTre −βH .
Here in the later equality we have employed the Dominated Convergence
Theorem using Tre −β0H < ∞ for β0 < β in (A4).
Step 2. Convergence of Gibbs states. From the above proof, we have
�
lim Tr[H
N→∞
� ΓN]−β −1 S( � �
ΓN) = Tr[HΓ]−β −1 S(Γ)
where
�ΓN := fMΓNfM
Tr[fMΓNfM] with M = N3/5 .
We have proved that TrF+ [UNe −β(HN−NeH) U ∗ N ] → TrF+ e−βH as N → ∞,
and we will now show that ΓN → Γ in the Hilbert-Schmidt norm. It is
well known that a sequence of non-negative operators AN with Tr(AN) = 1,
which converges weakly-∗ to an operator A, converges in the trace norm if
and only if Tr(A) = 1 (see, e.g., [17], [48, Cor. 1] and [53, Add. H]). Using
this fact, we will get the result. Moreover, by using Tr[( � ΓN −ΓN) 2 ] → 0 due
to the condensation (A3’), it remains to show that Tr[( � ΓN −Γ) 2 ] → 0. The
latter convergence follows from the equality
Tr[H � ΓN]−β −1 S( � ΓN)− � Tr[HΓ]−β −1 S(Γ) � = β −1 Tr
�
ΓN(log � �
ΓN −logΓ)
and the following entropy estimate, which is inspired from [27, Theorem 1].
The proof of Theorem 3 is finished. �
Lemma 26 (Relative entropy inequality). If A and B are two trace class
operators on a Hilbert space and 0 ≤ A ≤ 1, 0 ≤ B ≤ 1−ε for some ε > 0,
then there exists Cε > 0 such that
Tr[A(logA−logB)] ≥ CεTr[(A−B) 2 ]+Tr(A−B).
Proof. A straightforward computation shows that for all x,y ∈ [0,1],
x(logx−logy)−(x−y) ≥ g(y)(x−y) 2 with g(y) :=
y −1−logy
(y −1) 2 .
Since the function y ↦→ g(y) is decreasing, we have g(y) ≥ g(1−ε) > 0 for
all y ∈ [0,1−ε]. Therefore, By Klein’s inequality [62, p. 330], one has
Tr[A(logA−logB)] ≥ g(1−ε)Tr[(A−B) 2 ]+Tr(A−B).
The proof of Lemma 26 is finished. �