pdf file

More documents

Recommendations

Info

36 M. LEWIN, P. T. NAM, S. SERFATY, AND J. P. SOLOVEJ In the latter bound, by choosing M = N 1/3 , we get λL( � HN) ≥ λL(H)−C(εR(N)+N −1/3 ). Spectral gap. By (i) one has for every L = 1,2,..., liminf N→∞ (λL(HN)−λ1(HN)) = λL(H)−λ1(H) Taking the limit as L → ∞ we obtain the spectral gap liminf N→∞ (infσess(HN)−λ1(HN)) ≥ infσess(H)−λ1(H). Step 2. Convergence of lower eigenvectors. First we consider the convergence of ground states. If ΨN is a ground state of HN, then ΦN := UNΨN on F≤N + . From the proof in is a ground state of � HN := UN(HN −NeH)U ∗ N Step 1, if max{NεR(N), √ N} ≪ M ≪ N, then we have ||gMΦN|| 2 → 0 and lim N→∞ 〈fMΦN,HfMΦN〉 � = λ1(H) = Φ (1) ,HΦ (1)� where Φ (1) is the unique ground state of H on F+. To prove ΦN → Φ (1) , it suffices to show that fMΦN → Φ (1) . Let us write fMΦN = aN +bN where aN ∈ Span({Φ (1) }) and bN⊥Φ (1) . Then 〈fMΦN,HfMΦN〉 = 〈aN,HaN〉+〈bN,HbN〉 ≥ λ1(H)||aN|| 2 +λ2(H)||bN|| 2 = ||fMΦN|| 2 λ1(H)+(λ2(H)−λ1(H))||bN|| 2 . Since 〈fMΦN,HfMΦN〉� ΦN − ||fMΦN|| 2 λ1(H) → 0 and λ2(H) > λ1(H), we conclude that bN → 0 as N → ∞. Therefore, fMΦN → Φ (1) , and hence ΦN → Φ (1) , as N → ∞. Remark on the convergence in quadratic domain. We can show that if (A3s) holds true, then we have the strong convergence ΦN → Φ (1) in the norm inducedbythequadraticformofHonF+, namely〈ΦN,HΦN〉 → � Φ (1) ,HΦ (1)� . In fact, from the above proof we already had 〈fMΦN,HfMΦN〉 → � Φ (1) ,HΦ (1)� and � gMΦN, � � HNgMΦN → 0. On the other hand, from (A3s) and (16), we get � HN ≥ c0(H+N+)+g(N) on F ≤N + , where c0 > 0 and g(N) → 0 as N → ∞. By choosing M such that max{g(N),NεR(N), √ N} ≪ M ≪ N, we obtain gM � HNgM ≥ c0(H+M)+g(N) ≥ c0H. It implies that 〈gMΦN,HgMΦN〉 → 0. By using the localization for H in Proposition 23 we can conclude that 〈ΦN,HΦN〉 → � Φ (1) ,HΦ (1)� . Theconvergence of excited states of HN can beproved by usingthe above argument and the following abstract result. Lemma 25 (Convergence of approximate eigenvectors). Assume that A is a self-adjoint operator, which is bounded from below, on a (separable) Hilbert space, with the min-max values λ1(A) ≤ ... ≤ λL(A) < infσess(A). If the normalized vectors {x (j) n } L≥j≥1 n≥1 satisfy, for all i,j ∈ {1,2,...,L}, lim n→∞ 〈x(i) n ,x(j) n 〉 = δij and lim n→∞ 〈x(j) n ,Ax(j) n 〉 = λj(A),
BOGOLIUBOV SPECTRUM OF INTERACTING BOSE GASES 37 then there exists a subsequence {x (L) nk }k≥1 which converges, in the norm induced by the quadratic form of A, to an eigenvector of A with the eigenvalue λL(A). Theproofof Lemma25is elementary, usingthesameargument ofproving the convergence of ground states, and an induction process. The proof of Theorem 2 is finished. � 7.2. Proof of Theorem 3. Step 1. Convergence of the free energy. We need to show that We can rewrite lim N→∞ (Fβ(N)−NeH) = −β −1 logTrF+ [e−βH ]. Fβ(N)−NeH = inf Γ≥0,Tr F ≤N + where � HN := UN(HN −NeH)U ∗ N , and (Γ)=1 {Tr[� HNΓ]−β −1 S(Γ)} −β −1 logTrF+ [e−βH ] = Tr[HΓ]−β −1 S(Γ) where Γ := Z −1 e −βH with Z = Tr � e −βH� . Upper bound. Let us write Γ = �∞ i=1ti � �Φ (i) 〉〈Φ (i)�� where {Φ (i) } ∞ i=1 is an orthonormal family in F+ and t1 ≥ t2 ≥ ... ≥ 0, � ti = 1. Then TrF+ � HΓ � −β −1 S(Γ) = ∞� i=1 � ti〈H〉 Φ (i) +β −1 � tilogti . (61) Fix L ∈ N. By using Lemma 23 and the fact that H is bounded from below, we can findfor every M ≥ 1afamily of normalized states {Φ (i) M }L i=1 ⊂ F ≤M + such that limM→∞〈Φ (i) M ,Φ(j) M 〉 = δij and limsup〈H〉 (i) ≤ 〈H〉 Φ Φ (i) M→∞ M for all i,j ∈ {1,2,...,L}. Denote θL = � L i=1 ti and L� � � ti ΓL,M := �Φ θL i=1 (i) M �� Φ (i) � � �. M (62) Then it is easy to see that ΓL,M ≥ 0 and Tr[ΓL,M] = 1. Moreover, because limM→∞〈Φ (i) M ,Φ(j) M 〉 = δij we get lim M→∞ S(ΓL,M) = − L� ti θL i=1 � � ti log . (63) θL Choosing M = N 1/3 and applying Proposition 15 we obtain Tr ≤N[ F+ � HNΓL,M]−TrF+ [HΓL,M] → 0 (64)
Page 1 and 2: BOGOLIUBOV SPECTRUM OF INTERACTING
Page 21 and 22: Lemma 8.2] BOGOLIUBOV SPECTRUM OF I
Page 35: BOGOLIUBOV SPECTRUM OF INTERACTING
Page 49 and 50: R 2 BOGOLIUBOV SPECTRUM OF INTERACT
Page 51: BOGOLIUBOV SPECTRUM OF INTERACTING

pdf file

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?