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

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

274 LENZMANN and LEWIN

274 LENZMANN and LEWIN corresponding solution Ɣ(t) = Ɣ(γ (t),α(t)) of (3.12) is given by or equivalently in a (2 × 2)-matrix form γ (t) = γ 0 and α(t) = e −2μit α 0 , (3.13) Ɣ(t) = e iμNt Ɣ 0 e −iμNt (3.14) with μ 0, there is δ>0 such that if the initial condition satisfies dist Mλ (γ 0 ,α 0 )

MINIMIZERS FOR THE HFB-THEORY 275 every R>0, ∫ lim inf ρ γ (t) dx N HF (κ), t↗T |x| 0 is the same constant as in Theorem 4 above. Remark 13 In order for E(γ 0 , 0) < −mN to hold, we must have large initial data in the sense that Tr(γ 0 ) = N>N HF (κ) holds (which implies that I HF (N) =−∞for such large N; see Section 3.3 for the definition of N HF ). Remark 14 It is of interest to extend the above blowup result to HFB states with nonvanishing pairing α ≠ 0 or, even more ambitious, to relax the radiality assumptions on the initial data. 4. Preliminaries In this section we collect some preliminary results that are needed to set up the variational calculus for the proof of the main theorems. 4.1. Weak-∗ topology on X Recall that we assume that q 1 (describing the spin degrees of freedom) is a fixed integer and that S p with 1 p

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