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

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

312 LENZMANN and LEWIN

312 LENZMANN and LEWIN We start by discussing the kinetic energy. Using (A.2), we obtain Tr(Kα L α ∗ L ) (2π)3/2 Tr ( χ 2 L K(p)̂f (p)χ 2 L̂f (p) ) + (2π)3/2 π ∫ ∞ 0 √ ( ̂f (p) sdsTr K(p) 2 + s |∇χ ̂f L| 2 (p) ) K(p) 2 + s χ 2 L . Since the eigenfunction f satisfies (C.9), we obtain (2π) 3/2 Tr ( χ 2 L K(p)̂f (p)χL̂f 2 (p) ) = κ ∫∫ |αL (x,y)| 2 dx dy + βTr(α L α ∗ L 2 |x − y| ). Notice the above expressions are well defined even when f only belongs to H 1/2 (R 3 ). When 4/π κ>1, they can be justified by approximating the eigenfunction f in H 1/2 (R 3 ) by smooth functions and passing to the limit. On the other hand, we have ̂f (p) ∣ K(p) 2 + s |∇χ ̂f L| 2 (p) K(p) 2 + s χ 2 ∣ L ∣ S1 ̂f (p) ∣ ∣ ∣ K(p) 2 + s |∇χ L| 2 ∣∣ ∣∣S2∣∣ ∣∣ ̂f (p) K(p) 2 + s χ 2 ∣ L ∣ S2 ||f || L 2||∇ χ L|| 2 L 4 m + s × ||f || L 2||χ L|| 2 L 4 m + s CL−2 (m + s) 2 . Moreover, for L>0 sufficiently large, Tr(Kγ 2 L ) 2Tr( K(α L α ∗ L )2) 2 ||α L || 2 Tr(Kα L α ∗ L ) = O(L−3 ). (C.15) In summary, this shows that G (γ L ,α L ) βλ+O(L −2 ). Passing to the limit L →∞, we get the upper bound G(λ) βλ. Combined with the lower bound, this finally yields the desired equality G(λ) = βλ. TheestimateforI(λ) is obtained using the same state (γ L ,α L ).Wehave |γ L (x,y)| ∫∫R 2 dx dy π 3 ×R |x − y| 2 Tr(Kγ2 L ) = O(L−3 ), 3 which we proved above in (C.15). Moreover, note that ρ γL ρ αL αL ∗ by (C.8), and this is seen to imply that D(ρ γL ,ρ γL ) D(ρ αL α ∗ L ,ρ α L α ∗ L ) = 1 L D(χ 4 ,χ 4 ) + o(L −1 ),

MINIMIZERS FOR THE HFB-THEORY 313 as can be deduced from [14, page 052517]. Therefore, by choosing L>0 sufficiently large, we obtain as desired. I(λ) G(λ) − mλ − κ 2L D(χ 4 ,χ 4 ) + o(L −1 )

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