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

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

304 LENZMANN and LEWIN

304 LENZMANN and LEWIN subsequence): N−1 ∑ { (I HF (N) − I HF (N − K) − I HF (K) } lim z n n→∞ K 0. (8.4) K=1 The nonnegative numbers zK n are defined by ( ) ∫ ∫ N z n K = ∏ K ∏ N ··· ζ K n (x i ) 2 χ n (x i ) 2 | n (x 1 ,...,x N )| 2 dx 1 ··· dx N , i=1 i=K+1 where n is the Slater determinant associated with γ n and χ n = χ(Rn −1 ·) as introduced before, and ζ n = √ 1 − χn 2 (see [15]). Note that zn K may be interpreted as the norm of the wave function associated with all the clusters for which K particles are sent to infinity while N − K particles stay close to zero. Note that one has N∑ K=0 z n K = 1 and lim n→∞ K=0 The strict inequality (8.3) and (8.4) imply that we must have lim n→∞ zn K = 0 for K = 1,...,N − 1. N∑ (N − K)z n K = λ1 . (8.5) This implies that we have ∫ ∫ ··· ζ n (x 1 ) 2 χ n (x 2 ) 2 | n (x 1 ,...,x N )| 2 dx 1 ··· dx N = 0, lim n→∞ lim || n − √ z0 n n→∞ 1 n − √ zN n 2 n || = 0, lim ||γ n − z n n→∞ 0 γ − n 1 zn N γ || n 2 S 1 = lim ||γ 1 n→∞ n γ 2 n || S1 = 0 where n 1 and 2 n are HF states defined as 1 n := χ n ⊗N n ||χn ⊗N n || , 2 n ⊗N ζn n := ||ζn ⊗N n || . Using that the operators γ n , γ 1 n and γ 2 n are projectors, we obtain z n 0 γ + n 1 zn N γ + o(1) = γ n 2 n = (γ n ) 2 = (z n 0 )2 γ 1 n + (z n N )2 γ 2 n + o(1).

MINIMIZERS FOR THE HFB-THEORY 305 Taking the trace of the previous equality, we infer that ( lim z n n→∞ 0 − (zn 0 )2) ( = lim z n n→∞ N − (zn N )2) = 0. By (8.5), we have lim n→∞ Nz0 n = λ1 > 0; therefore we arrive at the following conclusion: lim n→∞ zn 0 = 1 and lim n→∞ zn K = 0 for K = 1,...,N; hence λ 1 = N. Thisprovesthat{(γ n , 0)} is indeed compact in X. Step 4: Proof of the binding inequality (8.3). We have seen that minimizing sequences are all compact when (8.3) holds. It remains to prove (8.3). This is done like in [15] by induction on N 2.AsI HF (1) = 0 and I HF (2) < 0 = 2I HF (1), we deduce that I HF (2) possesses a minimizer. Next the strict inequality I HF (3)

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