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BOGOLIUBOV SPECTRUM OF INTERACTING BOSE GASES 15
For this we need two new conditions. At positive temperature the stability
of the system does not follow from the simple relative boundedness assumptions
(A1) on w. So we need the following
(A4) (Stability). There exists β0 > 0 such that Fβ0 �
(N) ≥ −CN for all N
and TrF+ e−β0H � < ∞.
Our second new assumption is a modified version of the zero-temperature
condensation (A3), which we now only assume to hold for the Gibbs state
at temperature β −1 for simplicity.
(A3’) (Bose-Einstein condensation at positive temperature). For any β−1 <
, one has
β −1
0
〈u0,γβ,Nu0〉
lim = 1 (24)
N→∞ N
where γβ,N is the one-body density matrix of the Gibbs state Γβ,N := e−βHN /Tr � e−βHN �
, namely, in terms of kernels,
�
γβ,N(x,y) := N Γβ,N(x,x2,...,xN;y,x2,...,xN)dx2...dxN.
Ω N−1
Letusremarkthatifthestrongcondensationassumption(A3s)holdstrue
�
for some ε0 ∈ (0,1) and TrF+ e−(1−ε0)β0H � < ∞ for some β0, then we can
prove (A3’) and (A4) for the corresponding β0. We of course always assume
that (A1) and (A2) hold true. Moreover, if h ≥ ηH and K1 = K2 ≥ 0, then
dΓ(h+C)+C ≥ H ≥ dΓ(h−ε)−Cε
�
(see (69) in Appendix A), and hence the condition TrF+ e−(1−ε0)β0H � < ∞
is equivalent to Tr � e −(1−ε0)β0h � < ∞. The latter holds true if we have
Tr � e −(1−ε0)(1−α1)β0T � < ∞, because h ≥ (1−α1)T −C, where α1 ∈ (0,1) is
given in Assumption (A1).
Our main result is the following
Theorem 3 (Positive temperature case). Assume that (A1)-(A2)-(A3’)-
(A4) hold true. Then for every β−1 < β −1
0 , we have
lim
n→∞ TrF+
�
�
�UNe −β(HN−NeH) ∗
UN −e −βH
�
�
� = 0.
This implies the convergence of the corresponding Gibbs states and of the
free energy:
lim
N→∞ (Fβ(N)−NeH) = −β −1 �
logTrF+ e −βH�
.
Theorem 3 is proved using the same argument as that of the proof of Theorem
2, together with a well-known localization inequality for the entropy,
see Section 7.2.