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

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

264 LENZMANN and LEWIN

264 LENZMANN and LEWIN For λ>0 given, we set { K λ = (γ,α) ∈ X : ( ) ( ) 0 0 γ α 0 0 α ∗ 1 − γ ( ) } 1 0 , Tr(γ ) = λ . 0 1 (2.4) Here γ is defined by complex conjugation of the kernel γ (x,y). Note that both K and K λ are closed and convex sets in X and that we have 0 γ 1 for all (γ,α) ∈ K. In the following, we use the notation Tr(Tγ):= Tr ( (− + m 2 ) 1/4 γ (− + m 2 ) 1/4) − mTr(γ ), (2.5) for T = √ − + m 2 − m. Also, we sometimes allow the case of vanishing mass m = 0. But we remind the reader that strict positivity of m>0 is an essential assumption in all the main theorems stated below. As a first simple fact, we obtain the following Sobolev estimate for the square root of the density function ρ γ . LEMMA 2.1 For any (γ,α) ∈ K, the density function ρ γ satisfies 〈 √ ρ γ , √ − √ ρ γ 〉 Tr( √ −γ). Moreover, the map (γ,α) ↦→ √ ρ γ is continuous from K to L p (R 3 ) for all 1 p

MINIMIZERS FOR THE HFB-THEORY 265 2.2. Boundedness from below of HFB energy We define the (purely gravitational) HFB energy as E(γ,α):= Tr(Tγ) − κ 2 D(ρ γ ,ρ γ ) + κ 2 Ex(γ ) − κ |α(x,y)| 2 ∫∫R 2 dx dy, (2.8) 3 ×R |x − y| 3 where we now use the shorthand notations ∫∫ f (x)g(y) |γ (x,y)| D(f, g) := dx dy, Ex(γ ):= R 3 ×R |x − y| ∫∫R 2 dx dy, 3 3 ×R |x − y| 3 (2.9) which we refer to as the direct term and exchange term, respectively. The last term on the right-hand side of (2.8) is called the pairing term. For (γ,α) ∈ K, we deduce from Lemma 2.1 and the Hardy-Littlewood-Sobolev inequality that D(ρ γ ,ρ γ ) C‖ √ ρ γ ‖ 4 L 12/5 < ∞. (2.10) Thus the direct term D(ρ γ ,ρ γ ) is well defined. Next, we recall the pointwise estimate |γ (x,y)| 2 ρ γ (x)ρ γ (y) for almost every (x,y) ∈ R 3 × R 3 , (2.11) which is obtained by writing the spectral decomposition of γ and using the Cauchy- Schwarz inequality for sequences. This yields |γ (x,y)| ∫∫R 2 dx dy D(ρ γ ,ρ γ ) < ∞, (2.12) 3 ×R |x − y| 3 showing that the exchange term is also a well-defined quantity when (γ,α) ∈ K. To deal with the pairing term in E(γ,α), we recall the Hardy-Kato inequality 1 |x| π √ − (2.13) 2 (see [24], [22]). Applying (2.13) in the variable x with y fixed, we obtain |α(x,y)| ∫∫R 2 dx dy π 3 ×R |x − y| 2 Tr(√ −αα ∗ ) < ∞, (2.14) 3 for any (γ,α) ∈ K. Next, we introduce the (2 × 2)-matrix ( ) γ α Ɣ = 1 − γ α ∗ which defines an operator acting on L 2 (R 3 ; C q ) ⊕ L 2 (R 3 ; C q ). Following the convention in [2], we refer to any such Ɣ satisfying the constraint (1.3) as an admissible

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