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

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

280 LENZMANN and LEWIN

280 LENZMANN and LEWIN Therefore we may use Fatou’s lemma to infer that lim inf n→∞ Tr( K(χ R γ n χ R − χ R α n χ 2 3R α∗ n χ R) ) Tr ( K(χ R γχ R − χ R αχ 2 3R α∗ χ R ) ) . Next, we introduce the following sequence of functions on R 3 × R 3 given by αn R(x,y) := χ R(x)χ 3R (y)α n (x,y), which converges weakly in H 1/2 (R 3 × R 3 ) to α R (x,y):= χ R (x)χ 3R (y)α(x,y). We can write Tr(Kχ R α n χ 2 3R α∗ n χ R) − κ χ R (x) 2 ∫∫R 2 χ 3R (y) 2 |α n (x,y)| 2 dx dy 3 ×R |x − y| 3 〈( = K x − Using that κ0. Thus we may use Fatou’s lemma one more time to obtain 〈( κ ) 〉 lim inf K x − α R n→∞ n 2|x − y| ,αR n L 2 (R 3 ×R 3 ) ) 〉 α R n ,αR n . L 2 (R 3 ×R 3 ) 〈( κ 〉 K x − )α R ,α R . (4.13) 2|x − y| L 2 (R 3 ×R 3 ) In summary, we have derived the following inequality lim inf G (γ n,α n ) Tr(Kχ R γχ R )− κ χ R (x) n→∞ 2 ∫∫R 2 χ 3R (y) 2 |α(x,y)| 2 dx dy− C 3 ×R |x − y| R . 3 By taking the limit R →∞, we obtain the desired result lim inf G (γ n,α n ) G (γ,α). n→∞ It remains to show the claim about strong convergence and equality. First, we note that if equality holds in (4.10), then by (4.12), (2.15), and κ

MINIMIZERS FOR THE HFB-THEORY 281 which itself eventually implies that (γ n ,α n ) → (γ,α) strongly in X. The proof of Proposition 4.1 is now complete. A simple but important consequence of Proposition 4.1 is the following fact. COROLLARY 4.1 (Conservation of mass implies compactness) Let m>0, and let 0

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