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BOGOLIUBOV SPECTRUM OF INTERACTING BOSE GASES 17
Thefact thatthenextorderintheexpansionofthegroundstate energyof
bosonic atoms is given by Bogoliubov’s theory was first conjectured in [44].
In the following, by applying Theorem 2, we shall establish not only this
conjecture but also many other properties of the system.
By a convexity argument, it can be shown (see [34]) that the Hartree
minimization problem (27) has a minimizer ut if and only if t ≤ tc, for some
critical number tc ∈ (1,2) (it was numerically computed in [5] that tc ≈
1.21). In the case of existence, the minimizer is unique, positive, radiallysymmetric.
Moreover it decays exponentially and it solves the mean-field
equation
⎧
⎪⎨ htut = 0,
⎪⎩ ht := −∆− 1
t|x| +|ut| 2 ∗ 1
|x| −µH(t),
with the Lagrange multiplier µH(t) ≤ 0. Moreover, if t < tc, then µH(t) < 0
and there is a constant ηH(t) > 0 such that
ht ≥ ηH(t) > 0 on H+ := {ut} ⊥ . (28)
Thecritical bindingnumbertc in Hartree’s theory alsoplays animportant
role for the original quantum problem. In fact, it was shown in [7, 54, 3] that
for every N there are two numbers b(N) ≤ b ′ (N) satisfying that Ht,Z always
has a ground state if t ≤ b(N) and Ht,Z has no ground states if t ≥ b ′ (N),
and that
lim b(N) = lim
N→∞ N→∞ b′ (N) = tc.
In the following we shall always assume that t is fixed strictly below tc.
In this case, Assumption (A2) holds true. In fact, due to Hardy’s inequality
1
4|x| 2 ≤ −∆x on L 2 (R 3 ), (29)
the function Kt(x,y) := ut(x)|x − y| −1 ut(y) belongs to L 2 ((R 3 ) 2 ). Hence,
Kt(x,y) is the integral kernel of a Hilbert-Schmidt operator, still denoted
by Kt. Note that Kt ≥ 0 because |x −y| −1 is a positive kernel. Thus the
spectral gap (28) implies the non-degeneracy of the Hessian, namely
�
ht +Kt Kt
�
≥ ηH(t) on H+ ⊕H+.
Kt ht +Kt
The condensation in Assumption (A3) is implicitly contained in the proof
of the asymptotic formula (26) by Benguria and Lieb. In fact, the upper
bound in (26) can be seen easily by using the Hartree state u ⊗N
t . The
lower bound is more involved and it follows from the Lieb-Oxford inequality
[33, 37] which says that for every wave function Ψ ∈ HN ,
� ⎛
Ψ, ⎝ �
⎞ �
1
⎠Ψ ≥
|xi −xj|
1
2 D(ρΨ,ρΨ)−1.68
�
R3 ρΨ(x) 4/3 dx. (30)
1≤i