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minimizers for the hartree-fock-bogoliubov theory of neutron stars ...

minimizers for the hartree-fock-bogoliubov theory of neutron stars ...

300 LENZMANN and LEWIN

300 LENZMANN and LEWIN by (2.15). Define η n = η(·/R n ) for some smooth function η such that η ≡ 1 on the annulus {6/5 |x| 9/5} and η ≡ 0 outside the annulus {1 |x| 2}. The last term of (7.17) can be bounded above by ∫ 6R n /5|x|8R n /5 ∫ 7R n /5|y|9R n /5 χ n (x) 2 ζ n (y) 2 |α n (x,y)| 2 dx dy |x − y| π 2 Tr(Kη nα n η 2 n α∗ n η n) π 2 Tr(Kη nγ n η n ). Finally, the last term tends to zero by (7.12) and thanks to the estimate ‖[ √ K,η n ]‖ CRn −1 . In summary, we have derived the estimate (7.15), as desired. Next, we notice that E(γn 1,α1 n ) I(Tr(γ n 1)) and E(γ n 2,α2 n ) I(Tr(γ n 2 )). Onthe other hand, by Lemma 4.1, I(λ) I ( Tr(γ 1 n )) + I ( Tr(γ 2 n )) + o(1), using that Tr(γ n ) → λ and Tr(γ n ) = Tr(γ 1 n )+Tr(γ 2 n ), which follows from χ 2 n +ζ 2 n ≡ 1 and the cyclicity of the trace. Because of lim n→∞ E(γ n ,α n ) = I(λ), we deduce that lim E(γ 1 n→∞ n ,α1 n ) = I(λ1 ) and lim E(γ 2 n→∞ n ,α n) = I(λ − λ 1 ), (7.19) where we use that Tr(γn 1) = ∫ χn 2ρ γ n → λ 1 , by Lemma 7.3, and the continuity of λ ↦→ I(λ). Therefore equation (7.19) shows that both {(γn 1,α1 n )} and {(γ n 2,α2 n )} are minimizing sequences for I(λ 1 ) and I(λ − λ 1 ), respectively. Since we already know that Tr(γn 1) → λ1 , we conclude that (γn 1,α1 n ) → (γ 1 ,α 1 ) in X from Corollary 4.1. By the continuity of E, wealsohavelim n→∞ E(γn 1,α1 n ) = E(γ 1 ,α 1 ) = I(λ 1 ),sothat (γ 1 ,α 1 ) is indeed a minimizer for I(λ 1 ). This completes our proof of (7.14). Step 4: Binding inequality and conclusion. At this stage, we have proven that the energy of our minimizing sequence behaves like I(λ) = I(λ 1 ) + I(λ − λ 1 ) and we have seen that there exists a minimizer (γ 1 ,α 1 ) for I(λ 1 ). Note that (γ 1 ,α 1 ) is the weak limit of (γ n ,α n ) and the strong limit of (γn 1,α1 n ) = χ n(γ n ,α n )χ n in X. On the other hand, the sequence {(γn 2,α2 n )} is a minimizing sequence for I(λ − λ1 ). Hence one of the two following situations must occur. The sequence {(γn 2,α2 n )} is relatively compact up to translations. In this case it converges (up to a subsequence) to a minimizer (γ 2 ,α 2 ) for I(λ − λ 1 ).

MINIMIZERS FOR THE HFB-THEORY 301 The sequence {(γn 2,α2 n )} is not relatively compact up to translations. We may then apply the whole process again and we deduce that we have I(λ − λ 1 ) = I(λ 2 ) + I(λ − λ 1 − λ 2 ) for some λ 2 > 0 and that there exists a minimizer (γ 2 ,α 2 ) ∈ K λ 2 for I(λ 2 ). We can summarize both cases by saying that I(λ) = I(λ 1 ) + I(λ 2 ) + I(λ − λ 1 − λ 2 ) (7.20) for some λ 1 ,λ 2 > 0 such that λ 1 +λ 2 λ and that I(λ k ) possess a minimizer (γ k ,α k ) for k = 1, 2. Next, we claim the following result. LEMMA 7.4 (Binding inequality) Assume that λ 1 and λ 2 are as above. Then one has I(λ 1 + λ 2 ) 0, where we recall the definition of the cutoff functions χ R and ζ R from (1.5). It suffices to show (7.23) for i = 1, since the proof for i = 2 is analogous. As for the kinetic energy, the localization formula in Lemma A.1 yields Tr(Tγ 1 ) Tr(Tχ R γ 1 χ R ) − C/R 2 . Next, we estimate the direct term as follows (using the symmetry in x and y): |D(ρ γ 1,ρ γ 1) − D(ρ χR γ 1 χ R ,ρ χR γ 1 χ R )| ∫∫ ∫∫ 2 R 3 ×R 3 (1 − χ 2 R (x)χ 2 R (y))ρ γ 1(x)ρ γ 1(y) |x − y| ρ γ 1(x)ρ γ 1(y) { } |x|R ×R |x − y| 3 dx dy ∫ dx dy C ρ γ 1(x) dx C |x|R R , 2

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