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R 2
BOGOLIUBOV SPECTRUM OF INTERACTING BOSE GASES 49
which we can also write as
�� ��
ρ − ε
�
f logf −
4
π
�
2
R 2
R2 f ε Mf
�
, where f = ρ
� .
2ρ
Now we use that εf logf ≤ f(1+f ε ) and that f ≤ Mf to get an error of
the form ��
R2 ��
ρ − 1
�
4 R2 f +f 1+ε − π
�
2 R2 M 1+ε
�
f .
Using now [58]
� �
π
2 R2 M 1+ε C
f ≤
ε
we end up with a total error of the form
� ��
1 1
− D(µ,µ)+ ρ−
2 4
C
��
ε
R 2
R2 f 1+ε
R 2
References
��
ρ
R2 �
ρ
�
R2 �1+ε .
ρ
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