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

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

308 LENZMANN and LEWIN

308 LENZMANN and LEWIN hence I 0,κ (λ) − mλ I(λ) I 0,κ (λ). This shows that I(λ) is bounded below if and only if λ λ HFB (κ). Furthermore, it is clear that λ HFB (·) is nonincreasing, since for all (γ,α) ∈ K, ∫∫ |α(x,y)| 2 −D(ρ γ ,ρ γ ) + Ex(γ ) − dx dy 0 |x − y| by (2.11). The continuity of λ HFB (κ) is easy to verify. Next, we prove that λ HFB (κ) > 0 and we derive the asymptotic behavior of λ HFB (κ) as κ → 0 stated in Proposition 2.1. To this end, we introduce another homogeneous minimization problem { I red 0,κ (λ) = inf Tr( √ −γ ) − κ } (γ,0)∈K λ 2 D(ρ γ ,ρ γ ) , (B.2) which, of course, yields a corresponding λ red (κ). We claim that λ HFB (κ) ∼ κ→0 λ red (κ). Indeed, by the nonnegativity of the exchange term and (2.16), ( I 0,κ (λ) 1 − κπ ) I red 0,κ/(1−κπ/4) 4 (λ). (B.3) Hence ( λ HFB (κ) λ red κ ) . (B.4) 1 − κπ/4 Now we recall that λ red (κ ′ ) > 0 for every κ ′ > 0. Indeed we have ∫∫ ρ γ (x)ρ γ (y) dx dy C ′ ‖ρ γ ‖ 2/3 L ‖ρ |x − y| 1 γ ‖ 4/3 L Cλ 2/3 Tr( √ −γ ) (B.5) 4/3 R 3 ×R 3 where we have used the Hardy-Littlewood-Sobolev inequality and the well-known (see [11]) semiclassical estimate Tr( √ −γ ) K‖ρ γ ‖ 4/3 L for fermionic density 4/3 matrices 0 γ 1. This shows that λ red (κ ′ ) (2/(Cκ ′ )) 3/2 > 0. Hence we deduce from (B.4) and 0 κ 0. Next we derive an upper bound for λ HFB (κ). Wehaveforany(γ,0) ∈ K λ , I 0,κ (λ) E(γ,0) = Tr( √ −γ ) − κ 2 D(ρ γ ,ρ γ ) + κ Ex(γ ). 2 (B.6) Now we have similarly to (2.16) ∫∫ |γ (x,y)| 2 dx dy π |x − y| 2 Tr(√ −γ 2 ) π 2 Tr(√ −γ ). (B.7)

MINIMIZERS FOR THE HFB-THEORY 309 Hence we obtain I 0,κ (λ) ( 1 + κπ 4 ) I red 0,κ/(1+κπ/4) (λ), (B.8) and we thus get the upper bound ( λ HFB (κ) λ red κ ) . (B.9) 1 + κπ/4 The last step consists of showing that λ red (κ) behaves as stated when κ → 0, which implies the result for λ HFB (κ) by (B.4) and (B.9). This was indeed essentially done by Lieb and Yau (see [32, Corollary 1]). C. Proof of Proposition 4.2 We want to prove that G(λ) = βλ, where β is the first eigenvalue of the operator K − κ/2|x| acting on L 2 odd (R3 ) when q = 1 and L 2 (R 3 ) when q 2. For the reader’s convenience, we recall that K = √ − + m 2 . We start by deriving a lower bound. Using (2.15), we infer that Tr(Kγ) Tr(Kαα ∗ ) and thus G(λ) { inf Tr(Kαα ∗ ) − κ |α(x,y)| α T =−α, 2 ∫∫R 2 } dx dy . (C.1) Tr(α ∗ 3 ×R |x − y| 3 α)λ The right-hand side of the previous inequality can also be written, using the antisymmetry of α, as follows: 〈( Kx + K y κ ) 〉 inf − α, α . (C.2) α(x,y) T =−α(x,y), 2 2|x − y| ||α|| 2 L 2 (R 3 ×R 3 λ ) This quantity is just λ times the lowest eigenvalue of the two-body operator K x + K y 2 − κ 2|x − y| (C.3) when it is restricted to the fermionic space L 2 (R 3 , C q ) ∧ L 2 (R 3 , C q ).Ifq = 1, its lowest eigenvalue equals the first eigenvalue of the same operator acting on L 2 (R 3 x , R) ∧ L2 (R 3 y , R) (triplet state), whereas when q 2, it coincides with the first eigenvalue of the same operator acting on L 2 (R 3 , R) ⊗ s L 2 (R 3 , R) (singlet state). Removing the center of mass as explained in [27], one sees that this lowest eigenvalue is given by ( 1 ∣∣∣ inf inf σ H − i∇u + l∈R 2√ l ∣ 2 + m 3 2 2 + 1 √ ∣∣∣ − i∇u − l ∣ 2 + m 2 2 2 − κ ) , 2|u|

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