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

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

278 LENZMANN and LEWIN

278 LENZMANN and LEWIN where I κ/(1−ɛ) (λ) is the ground state energy of E on K λ with κ replaced by κ/(1 − ɛ). By continuity of λ HFB (κ) and since 0 λ0 small enough such that λ λ HFB (κ/(1 − ɛ)) and hence I κ/(1−ɛ) (λ) > −∞. Using that α ∗ α γ (see (2.15)) and the strict positivity of m>0, we see that ‖(γ,α)‖ X →∞ implies that Tr(Tγ) →∞. This fact, by (4.6), implies that E(γ,α) →∞as well. 4.3. Study of an auxillary functional The HFB energy contains two nonconvex terms: the direct and pairing terms. It turns out to be expedient to introduce the following energy functional for (γ,α) ∈ K, where we set G (γ,α):= Tr(Kγ) − κ |α(x,y)| 2 ∫∫R 2 dx dy, (4.7) 3 ×R |x − y| 3 K := √ − + m 2 . (4.8) Recall estimate (2.16), which tells us that the pairing term is controlled by the kinetic energy. Hence we have the lower bound G (γ,α) 0, (4.9) when 0 κ 4/π. We start by stating a result similar to [19, Theorem 1]. PROPOSITION 4.1 Let m>0, and let 0 κ

MINIMIZERS FOR THE HFB-THEORY 279 Thus α n (x,y) is a bounded sequence in H 1/2 (R 3 × R 3 ), and we may assume that α n (x,y) converges strongly in L 2 loc (R3 × R 3 ) by a Rellich-type theorem again. Next, we recall the definition of the cutoff functions χ R and ζ R in (1.5). We then find the estimate |α n (x,y)| ∫∫R 2 dx dy 3 ×R |x − y| 3 ζ R (x) = ∫∫R 2 |α n (x,y)| 2 χ R (x) dx dy + 3 ×R |x − y| ∫∫R 2 |α n (x,y)| 2 dx dy 3 3 ×R |x − y| 3 π 2 Tr(Kζ χ R (x) Rγ n ζ R ) + ∫∫R 2 χ 3R (y) 2 |α n (x,y)| 2 dx dy + ||α n|| 2 S 2 3 ×R |x − y| R . 3 In the last line we have used the Hardy-Kato inequality (2.13) and the inequality ζ R α n αn ∗ζ R ζ R γ n ζ R thanks to (2.15). Because (γ n ,α n ) is bounded in X, the last term in the above estimate is O(R −1 ). Now we localize the kinetic energy. Lemma A.1 tells us that the nonlocal operator K satisfies χ R Kχ R + ζ R Kζ R K + 1 ∫ ∞ 1 π s + K (|∇χ R| 2 +|∇ζ 2 R | 2 1 √ ) sds s + K 2 0 K + C/R 2 , (4.11) using ||∇χ R || 2 L + ||∇ζ R|| 2 ∞ L ∞ C/R2 .SinceTr(γ n ) is uniformly bounded, this gives us Tr(Kγ n ) Tr(Kχ R γ n χ R ) + Tr(Kζ R γ n ζ R ) − C/R 2 . Thus we obtain G (γ n ,α n ) Tr ( K(χ R γ n χ R − χ R α n χ 2 3R α∗ n χ R) ) + 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 +(1 − κπ/4) Tr(Kζ R γ n ζ R ) − C R . (4.12) Since α n converges strongly to α in L 2 loc (R3 × R 3 ),wehavethatχ R α n χ 2 3R α∗ n χ R → χ R αχ 2 3R α∗ χ R strongly in S 1 . Hence we deduce that χ R γ n χ R − χ R α n χ 2 3R α∗ n χ R ⇀χ R γχ R − χ R αχ 2 3R α∗ χ R weakly in S 1 . Notice also that, by (2.15), χ R α n χ 2 3R α∗ n χ R χ R α n α ∗ n χ R χ R γ n χ R .

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