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

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

288 LENZMANN and LEWIN

288 LENZMANN and LEWIN We now assume that α ≠ 0. By Lemma 5.1, we have μ

MINIMIZERS FOR THE HFB-THEORY 289 The last term can be estimated as usual by Cλ/R 2 . Using that, in particular, μ0 some constant and for R>0 sufficiently large. We now estimate the right-hand side of (5.16) by using the two-body equation (5.13). Step 2: Estimates from (5.13). We multiply equation (5.13) by ζ R (x)ζ R (y) from the left, and we project onto α R (x,y) = ζ R (x)α(x,y)ζ R (y). (5.17) This gives, where 〈·, ·〉 is the inner product on L 2 (R 3 × R 3 ), where |α R (x,y)| 〈α R , (T x + T y )α R 〉−κ ∫∫R 2 dx dy 3 ×R |x − y| 3 − 2μ |α R (x,y)| 2 dx dy = I + II + III + IV, R 3 ×R 3 I =〈α R , [T x +T y ,ζ R (x)ζ R (y)]α〉, ∫∫ II = 2κ V (x)ζ 2 R (x)ζ 2 R (y)|α(x,y)|2 dx dy, III =−κ〈α R ,ζ R (x)ζ R (y) ( (X γ ) x +(X γ ) y ) α〉, IV= κ〈αR ,ζ R (x)ζ R (y)(γ x +γ y ))X α 〉. First, we note that II is easy to estimate by ∫ |II| δ(R) ζ 2 R (x)ρ γ (x) dx, (5.18) R 3 with δ(R) → 0 as R →∞, by using that ρ αα ∗ ρ γ and V → 0 as |x| →∞. Next, we claim that |III|+|IV| δ(R)( ∫ ) ζ 2 R (x)ρ γ (x) dx + Tr(Tζ R γζ R ) . (5.19) R 3

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