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BOGOLIUBOV SPECTRUM OF INTERACTING BOSE GASES 41
contains all pairs of operators (γ,α) on H such that γ is trace class, α is
Hilbert-Schmidt, α T = α and
� γ α
α ∗ 1+JγJ ∗
�
≥ 0 on H⊕H ∗ .
Let us denote by G0 the set of all pairs (γ,α) ∈ G which satisfy
αα ∗ = γ(1+JγJ ∗ ) and γα = αJγJ ∗ . (65)
The significance of G0 is that for any (γ,α) ∈ G0, there exists a unique
quasi-free pure state Φ ∈ F(H) such that
(γ,α) = (γΦ,αΦ).
Any element in G is also associated with a unique state, but the latter is a
mixed state.
The one-body density matrices offer a simple way to define quadratic
Hamiltonians. More precisely, let H be a self-adjoint operator on H and let
K be a Hilbert-Schmidt operator on H such that K = K T and such that
the following inequality holds true
�
A := ≥ η > 0 on H⊕H ∗ . (66)
� H K
K ∗ JHJ ∗
We shall consider the quadratic Hamiltonian H on F(H) defined by
〈H〉 Φ := q(γΦ,αΦ) = Tr[HγΦ]+ℜTr[KαΦ] (67)
for every state Φ living in the truncated Fock spaces and in the domain of
H:
� M� n�
D(H). (68)
M≥0 n=0 sym
It can be verified that the so-defined operator H is exactly the Bogoliubov
Hamiltonian given in (12).
The main properties of the quadratic Hamiltonian H are given in the
following
Theorem 27 (Bogoliubov Hamiltonian). Let H be defined by (67) and assume
that (66) holds true. Then we have the following statements.
(i) (Form domain). We have the quadratic-form inequalities
C −1 dΓ(H)−C ≤ H ≤ dΓ(H +C)+C (69)
As a consequence, the form domain of the Friedrichs extension of H (still
denoted by H) is the same as that of dΓ(H) on F+.
(ii) (Variational principle). For any (γ,α) ∈ G, we can find (γ ′ ,α ′ ) ∈ G0
such that q(γ ′ ,α ′ ) ≤ q(γ,α) and the inequality is strict expect when (γ,α) ∈
G0. As a consequence, the ground state energy of H is
infσ(H) = inf q(γ,α) = inf q(γ,α).
(γ,α)∈G (γ,α)∈G0
(iii) (Ground state and ground state energy). The Hamiltonian H has a
unique ground state in F(H), which is a pure quasi-free state. Moreover,
we always have infσ(H) < 0 except when K = 0 in which case infσ(H) = 0
with the vacuum being the corresponding ground state.