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

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

292 LENZMANN and LEWIN

292 LENZMANN and LEWIN Let us now define the sequence {u n } ∞ n=1 of nonnegative numbers given by ∫ u n = ζ 2 R n (x)ρ γ (x) dx + Tr(Tζ Rn γζ Rn ). (5.26) R 3 Collecting the previous estimates, we obtain the recursive inequality u n+1 δ n u n + C for n n (R n+1 ) 2 0 , (5.27) where δ n → 0 as n →∞and n 0 1 is sufficiently large. A simple induction argument then shows that u n satisfies the bound u n B for n 1, (5.28) (R n ) 2 where B>0 is some sufficiently large constant. It remains to extend the bound in Lemma 5.3 to all R>0 sufficiently large. To this end, let R 4 and let n 1 such that 4 n R0 sufficiently large. The proof of Lemma 5.3 is now complete. Remark 16 One can bootstrap the decay estimates obtained above by commuting |∇ζ R | 2 with (K 2 + s) −1 on the right-hand side of (5.15), leading to a bound C k /R k for all k 1 in (5.11). This better decay estimate is, however, unnecessary for our existence proof, and hence we do not give any details here. Having established Lemma 5.3, the proof of Theorem 2 is now complete. 6. Proof of Theorem 3 First, we recall the equality (3.8) and we note that, by assumption, we have that √ τ(1 − τ) ≠ 0 holds. To show that τ must have an infinite rank, we note the similarity between F (τ) and the Müller functional studied in [14]. Indeed, by following an

MINIMIZERS FOR THE HFB-THEORY 293 argument in [14], we can prove that τ has infinite rank as follows. For the reader’s convenience, we provide the details of the adaptation. We argue by contradiction and assume that τ is finite rank. Hence we can write τ = K∑ n i |ϕ i 〉〈ϕ i |, 0

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