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

302 LENZMANN and LEWIN **for** R>0 sufficiently large. Here we used **the** Kato-Hardy inequality to conclude that ‖|x| −1 ∗ ρ γ 1‖ L ∞ C‖ √ ρ γ 1‖ H 1/2 as well as **the** decay estimate from Lemma 5.3. Finally, we estimate **the** pairing term as follows: (1 − χ R (x) ∫∫R 2 χ R (y) 2 )|α 1 (x,y)| 2 dx dy 3 ×R |x − y| 3 |α 2 ∫∫{|x|R}×R 1 (x,y)| 2 |x − y| 3 ζ R/2 (x) dx dy 2 ∫∫R 2 |α 1 (x,y)| 2 3 ×R |x − y| 3 ( ∫ πTr(Kζ R/2 γ 1 ζ R/2 ) C ρ γ 1(x) dx + Tr(Tζ R/2 γ 1 ζ R/2 ) |x|R/2 ) . dx dy Here **the** first inequality follows from **the** symmetry |α 1 (x,y)| =|α 1 (y,x)| and **the** support properties **of** χ R . In **the** last line, we used Kato’s inequality and (α 1 ) ∗ α 1 γ 1 . Putting all estimates toge**the**r and using Lemma 5.3 again, we conclude that (7.23) holds. Next, we prove (7.21). To this end, we consider **the** trial state (γ R ,α R ):= U(χ R γ 1 χ R ,χ R α 1 χ R )U ∗ + τ 5Rv V (χ R γ 2 χ R ,χ R α 2 χ R )V ∗ τ ∗ 5Rv , (7.24) **for** some unit vector v and some rotations U,V ∈ SO(3). Asχ R and τ 5Rv χ R have disjoint supports, it is easily seen that **the** above operator belongs to K. Moreover, **the** number **of** particles **of** this trial state satisfies Tr(γ R ) = Tr(χ R γ 1 χ R ) + Tr(χ R γ 2 χ R ) λ 1 + λ 2 . (7.25) Note **the** invariance by rotation and translation E(τ v U(γ,α)U ∗ τv ∗ ) = E(γ,α) **for** all v ∈ R 3 and all U ∈ SO(3). Sinceλ ↦→ I(λ) is decreasing by Lemma 4.1, we have ∫ ∫ I(λ 1 + λ 2 ) E(γ R ,α R ) dU dV SO(3) SO(3) = E(χ R γ 1 χ R ,χ R α 1 χ R ) + E(χ R γ 2 χ R ,χ R α 2 χ R ) ∫∫ f (x)g(y − 5Rv) − κ dx dy R 3 ×R |x − y| 3 I(λ 1 ) + I(λ 2 ) + C R 2 − κλ1 λ 2 R , which follows from Newton’s **the**orem and (7.23), where we have defined ∫ ∫ f (x) = χ R (x) 2 ρ γ 1(U · x) dU, g(y) = χ R (y) 2 ρ γ 2(V · y) dV. SO(3) SO(3) This last estimate yields **the** desired result by taking R>0 sufficiently large.

MINIMIZERS FOR THE HFB-THEORY 303 The pro**of** **of** Theorem 1 is now complete. 8. Pro**of** **of** Theorem 4 The pro**of** **of** Theorem 4 is a combination **of** ideas from **the** pro**of** **of** Theorem 1 and **of** **the** usual geometrical methods **for** N-body systems as was developed in [23], [43], [44], [13], [40], [39]. To our knowledge, **the**y were applied **for** **the** first time to nonlinear models approximating **the** N-body case (including **the** HF approximation) by Friesecke in [15]. We only sketch **the** pro**of** **of** Theorem 4. In our opinion, it is much easier than **the** pro**of** **of** Theorem 1. Step 1: First properties **of** HF energy. The first step consists in showing that and that I HF (1) = 0, I HF (N) < 0 **for** all N 2 (8.1) I HF (N) I HF (N − K) + I HF (K) **for** all K = 1,...,N − 1. (8.2) The value **of** I HF (1) is obvious **for** density matrices **of** rank one; **the** exchange and direct terms cancel. The rest is proved similarly as in Lemma 4.1. The main part **of** **the** pro**of** consists **of** showing that if I HF (N) < min ( I HF (N − K) + I HF (K), K= 1,...,N − 1 ) (8.3) holds true, **the**n all minimizing sequences are compact and thus one gets **the** existence **of** a minimizer **for** I HF (N). In **the** following we fix some N 2, assume that (8.3) holds, and consider a minimizing sequence {γ n } **for** I HF (N). Similarly to Lemma 4.2, we obtain that {(γ n , 0)} is bounded in X. Step 2: No vanishing. We claim that our sequence {γ n } cannot vanish, similarly to Lemma 7.1. Indeed if {γ n } would vanish, applying Lemma 7.2 we get that D(ρ γn ,ρ γn ) → 0; hence we would obtain I HF (N) 0 which contradicts (8.1). There**for**e we may assume (translating γ n if necessary) that γ n ⇀γwith λ 1 := Tr(γ ) > 0. Also we have similarly to Lemma 7.3 that we can find a cut**of**f function on a ball **of** size R n such that √ Kχ(Rn −1·)γ nχ(Rn −1·)√ K → √ Kγ √ K strongly. Also, similarly to Corollary 4.1, one shows that {γ n } is compact when Tr(γ ) = N; hence it suffices to prove that λ 1 = N. Step 3: N-body geometrical methods. The next step is an easy adaptation **of** [15]; that is, it consists **of** using a cluster decomposition like in **the** usual HVZ **the**orem, based on **the** fact that γ n arises from an N-body wave function which is a Slater determinant. Indeed following [15], one arrives at an estimate **of** **the** **for**m (up to extraction **of** a

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