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18 M. LEWIN, P. T. NAM, S. SERFATY, AND J. P. SOLOVEJ Lemma 4 (Strong condensation of bosonic atoms). If t ≤ tc, then � Ht,N −NeH ≥ 1−N −2/3� �N (ht)i −CN 1/3 . (31) Remark 8. In particular, from (31) it follows that if 〈Ht,N〉Ψ ≤ NeH +R, then 〈u0,γΨu0〉 ≥ N − CN 1/3 which is (A3). When t < tc, by Theorem 2 we can improve the estimate to 〈u0,γΨu0〉 ≥ N −C. The latter was shown by Bach, Lewis, Lieb and Siedentop in [4] using a different method. Proof. We start by estimating the terms on the right side of the Lieb-Oxford inequality (30). First, from the positivity D(f,f) ≥ 0, we have 1 2 D(ρΨ,ρΨ) ≥ D(ρΨ,N|ut| 2 )− 1 2 D(N|ut| 2 ,N|ut| 2 ) = ND(ρΨ,|ut| 2 )+N 2 (eH(t)−µH(t)). (32) Ontheotherhand,usingtheHoffmann-Ostenhofinequality[29]andSobolev’s inequality [36, Theorem 8.3] we can estimate � � N� � � Ψ, −∆i Ψ = Tr[−∆γΨ] ≥ 〈 √ ρΨ,−∆ √ �� 1/3 ρΨ〉 ≥ C . i=1 i=1 R3 ρ 3 Ψ Therefore, by Hölder inequality, we find that � ρ 4/3 Ψ ≤ �� ρ 3 �1/6�� 5/6 Ψ ρΨ ≤ εTr[−∆γΨ]+C N5/3 . ε Thus using the Lieb-Oxford inequality (30), the estimate (32) and ht ≥ −∆/2−C we get � 〈Ht,N〉Ψ ≥ NeH(t)+ 1− 2ε � Tr[htγΨ]−2Cε−Cε N −1 N 2/3 for all ε > 0. Replacing ε by N 1/3 /2, we obtain (31). � All this shows that if t < tc, then Assumptions (A1)-(A2)-(A3s) hold true, and we may apply Theorem 2 to show that the lower spectrum of Ht,Z converges to the lower spectrum of the Bogoliubov Hamiltonian � Ht := a ∗ (x) � (ht +Kt)a � (x)dx Ω + 1 2 � Ω � Ω � Kt(x,y) which acts on the Fock space F+ = � ∞ n=0 a ∗ (x)a ∗ (y)+a(x)a(y) � dxdy, � n sym H+. Beside some basic propertiesofHt alreadygiven inTheorem1, wehavethefollowingadditional information. Proposition 5 (Bogoliubov Hamiltonian of bosonic atoms). For every t ∈ (0,tc) one has σess(Ht) = [infσ(Ht)−µH(t),∞). Moreover, if t < 1, then Ht has infinitely many eigenvalues below its essential spectrum. On the other hand, if t ≥ 1, then Ht only has finitely many eigenvalues below its essential spectrum.
BOGOLIUBOV SPECTRUM OF INTERACTING BOSE GASES 19 Proof. First, sinceht = −∆+Vt(x)−µH(t), whereVt(x) := (|u0| 2 ∗|.| −1 )(x)− t −1 |x| −1 is relatively compact with respect to −∆, we obtain σess(ht) = [−µH(t),∞). Thus σess(Ht) = [infσ(Ht)−µH(t),∞) by (ii) in Theorem 1. The other statements follow from (iii) in Theorem 1 and the fact that ht has infinitely many eigenvalues below its essential spectrum if and only if t < 1. In fact, if t < 1, then Vt(x) = mt ∗|.| −1 where the measure R 3 mt := |u0| 2 −t −1 δ0 has negative mass � m < 0, and we can follow [43, Lemma II.1]. On the other hand, if t ≥ 1, then by applying Newton’s Theorem for the radially symmetric function |u0(x)| 2 , we can write � |ut(y)| Vt(x) ≥ 2 1 dy − max{|x|,|y|} |x| = � |ut(y)| 2 � � 1 1 − dy. |y| |x| |y|≥|x| Because ut(x) decays exponentially, we obtain that [V(x)]− also decays exponentially and hence belongs to L 3/2 (R 3 ). Therefore, by the CLR bound (see e.g. [38, Theorem 4.1]), we conclude that ht has finitely many eigenvalues below −µH(t). � By the celebrated HVZ Theorem [47], one has σess(Ht,N) = [Σt,N,∞) where Σt,N is the ground state energy of the (N −1)-body Hamiltonian N−1 � � −∆i − 1 � + t|xi| 1 N −1 � 1 |xi −xj| . i=1 1≤i
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