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24 M. LEWIN, P. T. NAM, S. SERFATY, AND J. P. SOLOVEJ
then Tre −β0(K+V) < ∞ for all β0 > 0, due to [18, Theorem 1] and the
operator inequality √ −∆ + |x| ≥ � −∆+|x| 2 in L 2 (R d ). The latter is a
consequence of the operator monotonicity of the square root and the fact
that √ −∆|x|+|x| √ −∆ ≥ 0 in L 2 (R d ), see [28, Theorem 1].
Second, in [50, Theorem 1] the authors provided upper and lower bounds
on the classical free energy at nonzero temperatures, which coincide when
β −1 → 0. Inthe quantumcase weare able to identify precisely thelimit (36)
even when β −1 > 0, which is given by the Bogoliubov Hamiltonian.
4. Operators on Fock spaces
In this preliminary section, we introduce some useful operators on Fock
spaces and we consider the unitary UN defined in (18) in detail.
For any vector f ∈ H, we may define the annihilation operator a(f) and
the creation operator a∗ (f) on the Fock space F = �∞ N=0HN by the following
actions
⎛
a(f) ⎝ �
σ∈SN
a ∗ ⎛
(fN) ⎝ �
f σ(1) ⊗...⊗f σ(N)
σ∈SN−1
⎞
⎠ = √ N �
f σ(1) ⊗...⊗f σ(N−1)
⎞
σ∈SN
⎠ = 1
√ N
� �
f,fσ(1) fσ(2) ⊗...⊗f σ(N),(37)
�
σ∈SN
f σ(1) ⊗...⊗f σ(N) (38)
for all f,f1,...,fN in H, and all N = 0,1,2,.... These operators satisfy the
canonical commutation relations
[a(f),a(g)] = 0, [a ∗ (f),a ∗ (g)] = 0, [a(f),a ∗ (g)] = 〈f,g〉H. (39)
Note that when f ∈ H+, then a(f) and a∗ (f) leave F+ invariant, and
hence we use the same notations for annihilation and creation operators on
F+. The operator-valued distributions a(x) and a∗ (x) we have used in (12)
can be defined so that for all f ∈ H+,
�
a(f) = f(x)a(x)dx and a ∗ �
(f) = f(x)a ∗ (x)dx.
Ω
To simplify the notation, let us denote an = a(un) and a ∗ n = a∗ (un), where
{un} ∞ n=0 is an orthonormal basis for L2 (Ω) such that u0 is the Hartree minimizer
and un ∈ D(h) for every n = 1,2,.... Then the Bogoliubov Hamiltonian
defined in (12) can be rewritten as
H = �
m,n≥1
〈um,(h+K1)un〉 L 2 (Ω) a ∗ man + 1
2 〈um ⊗un,K2〉 L 2 (Ω 2 ) a ∗ ma ∗ n
Ω
+ 1
2 〈K2,um ⊗un〉 L 2 (Ω 2 ) aman. (40)
The sums here are not convergent in the operator sense. They are well defined
as quadratic forms on the domain given in (14). Since the so-obtained
operator is bounded from below (by Theorem 1), it can then be properly
defined as a self-adjoint operator by the Friedrichs extension.