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16 M. LEWIN, P. T. NAM, S. SERFATY, AND J. P. SOLOVEJ 3. Applications It is easy to verify that the model considered by Seiringer in [52] and by Grech-Seiringer in [24] satisfy our assumptions (A1), (A2) and (A3s). So our approach applies and we are able to recover their results. In the rest of this section, we consider two Coulomb systems for which we are able to apply our main results. 3.1. Bosonic atoms. For a bosonic atom we mean a system including a classical nucleusattheorigininR 3 ofchargeZ > 0andN “bosonicquantum electrons” of charge −1. The system is described by the Hamiltonian N� � −∆i − Z � + |xi| � 1 |xi −xj| i=1 i=1 1≤i
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
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