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BOGOLIUBOV SPECTRUM OF INTERACTING BOSE GASES 3
these situations are covered. The function w could also be replaced by an
abstract two-body operator but we do not consider this here for simplicity.
We are interested in the limit of a large number N of particles. Here we
are considering the mean-field regime, in which the interaction has a fixed
range (the function w is fixed) but its intensity is assumed to tend to zero
in the limit N → ∞, hence the factor 1/(N −1) in front of the interaction
term in the Hamiltonian HN. This factor makes the two sums of order N
in HN and, in this case, an important insight is given by Hartree theory.
Let us recall that a Hartree state is an uncorrelated many-body wave
function in which all of the particles live in the same state u ∈ L 2 (Ω) such
that �
Ω |u|2 = 1, and which takes the form
Ψ(x1,...,xN) = u(x1)···u(xN).
The energy of such a state is
�
〈Ψ,HNΨ〉 = N 〈u,Tu〉+ 1
2 D(|u|2 ,|u| 2 �
) := N EH(u)
where
�
D(f,g) :=
Ω
�
Ω
f(x)g(y)w(x−y)dxdy
is the classical interaction. Henceforth, all the Hilbert spaces we consider
haveinnerproductswhichareconjugate linear inthefirstvariableandlinear
in the second.
Provided that there is Bose-Einstein condensation, the leading term of
the ground state energy
is given by Hartree’s theory:
E(N) := infspecHN
E(N) = NeH +o(N),
where eH is the corresponding Hartree ground state energy:
eH := inf
u∈L2 EH(u) = inf
(Ω) u∈L
||u||=1
2 (Ω)
||u||=1
�
〈u,Tu〉+ 1
2 D(|u|2 ,|u| 2 )
�
. (1)
In this paper, we shall assume that there exists a unique Hartree minimizer
u0 for eH. It is then a solution of the nonlinear Hartree equation
0 = � T +|u0| 2 �
∗w −µH u0 := hu0, (2)
where µH ∈ R is a Lagrange multiplier.
Bogoliubov’s theory predicts the next order term (of order O(1)) in the
expansion of the ground state energy E(N). It also predicts the leading
term and the second term for the lower eigenvalues of HN. The Bogoliubov
method consists in describing variations of the wavefunctions around the
Hartree state u0 ⊗ ··· ⊗ u0 in a suitable manner. We will explain this in
detail in Section 2.2 below. The final result is an effective Hamiltonian H,
called the Bogoliubov Hamiltonian which is such that the lower spectrum of
HN in H N is given, in the limit N → ∞, by the spectrum of the effective