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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.
BOGOLIUBOV SPECTRUM OF INTERACTING BOSE GASES 25 Itis also useful to lift operators on H N to the Fock space F. Thefollowing identities are well-known; their proofs are elementary and can befound, e.g., in [8] and [56, Lemmas 7.8 and 7.12]. Lemma 13 (Second quantizations of one- and two-body operators). Let A be a symmetric operator on H such that un ∈ D(A) for all n ≥ 0, and let ω be a symmetric operator on H ⊗ H such that um ⊗ un ∈ D(ω) and 〈um ⊗un,ωup ⊗uq〉 = 〈un ⊗um,ωup ⊗uq〉 for all m,n,p,q ≥ 0. Then and 0⊕0⊕ dΓ(A) := 0⊕ ∞� � N=2 1≤i
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