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BOGOLIUBOV SPECTRUM OF INTERACTING BOSE GASES 7
Our last assumption is about the validity of Hartree theory in the limit
N → ∞. We assume that the system condensates in the unique Hartree
ground state u0. This assumption will be necessary for the proof of the
lower bound on the spectrum of HN.
(A3)(Complete Bose-Einstein condensation). For any constant R > 0, there
exists a function εR : N → [0,∞) with limN→∞εR(N) = 0 such that, for
any wave function ΨN ∈ H N satisfying 〈ΨN,HNΨN〉 ≤ E(N)+R, one has
〈u0,γΨN u0〉
≥ 1−εR(N) (10)
N
where u0 is the Hartree minimizer in Assumption (A2).
Here γΨ is the one-body density matrix of the wave function Ψ ∈ HN ,
which is the trace-class operator on L2 (Ω) with kernel
�
γΨ(x,y) := N Ψ(x,x2,...,xN)Ψ(y,x2,...,xN)dx2...dxN.
Ω N−1
Note that a Hartree state has the density matrix γu⊗N = Nu(x)u(y). There-
. For
fore (10) is the same as saying that γΨN is in some sense close to γu ⊗N
0
more explanation about the Bose-Einstein condensation, we refer to the
discussion in [39].
In many applications (especially for Coulomb systems, see Section 3), a
stronger condensation propertywill holdtrue. Namely, wewill have abound
from below valid for all ΨN ∈ HN , and not only for those which have a low
energy. We therefore introduce the following stronger assumption, which
obviously implies (A3):
(A3s) (Strong condensation). We have h ≥ ηH > 0 on H+, and there exists
a constant 0 < ε0 < 1 such that
N�
HN −NeH ≥ (1−ε0) hj +o(N).
j=1
Here h is the mean-field operator given in Assumption (A2).
In fact, in practice (A3s) follows from a Lieb-Oxford inequality
� ⎛
Ψ, ⎝ �
⎞ �
wij⎠Ψ
≥ 1
2 D(ρΨ,ρΨ)+error
1≤i