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

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

276 LENZMANN and LEWIN

276 LENZMANN and LEWIN Put differently, the continuous linear functional (γ,α) ↦→ Tr(γ ) on X is not weakly-∗ continuous. Throughout this article, we make use of the following important fact that helps conclude strong convergence a posteriori. Suppose that (γ n ,α n ) ⇀ (γ,α) weakly-∗ in X and, moreover, assume that and Tr ( (1 − ) 1/4 γ n (1 − ) 1/4) → Tr ( (1 − ) 1/4 γ (1 − ) 1/4) (4.3) Tr ( (1 − ) 1/4 α ∗ n α n(1 − ) 1/4) → Tr ( (1 − ) 1/4 α ∗ α(1 − ) 1/4) . (4.4) Then we have (γ n ,α n ) → (γ,α) strongly in X. This follows from [41, Theorem A.6]. 4.2. Some basic properties of HFB energy We have already seen that the energy E is well defined on K. Next, we collect some basic facts. LEMMA 4.1 (Basic properties of I(λ)) Let m>0, and let 0

MINIMIZERS FOR THE HFB-THEORY 277 when n is sufficiently large. By taking the limit n →∞, it is straightforward to conclude the subadditivity inequality. If, moreover, equality holds for some λ ′ ,then (˜γ n ,˜α n ) furnishes a minimizing sequence that fails to be relatively compact in X up to translations. To see that I(λ) < 0 holds, we choose a fixed state (γ,0) ∈ K λ with γ smooth and finite rank and such that D(ρ γ ,ρ γ )−Ex(γ ) > 0 holds. (The latter condition is indeed equivalent to saying that rank(γ ) 2.) Next, we consider the energy of the rescaled state γ δ := U δ γUδ ∗ where (U δf )(x) = δ −3/2 f (x/δ). Note that (γ δ , 0) ∈ K λ for all δ>0, as one easily verifies. Using now the operator inequality T −(1/2m),we deduce that E(γ δ , 0) 1 ( δ Tr − 1 ) 2 2m γ − κ ( D(ργ ,ρ γ ) − Ex(γ ) ) < 0, 2δ provided that δ>0 is chosen sufficiently large. This establishes Lemma 4.1(i). To show Lemma 4.1(ii), we note that the strict monotonicity of I(·) is an obvious consequence of the subadditivity inequality and the strict negativity of I(·). The continuity of I(·) follows readily from using trial states. This proves Lemma 4.1(ii). To prove Lemma 4.1(iii), we fix some 0

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