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

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

310 LENZMANN and LEWIN

310 LENZMANN and LEWIN where u = x − y and l is the Fourier variable associated with the space variable v = (x + y)/2. WehavetotakeH = L 2 odd (R3 ) when q = 1 and H = L 2 (R 3 ) when q 2. The infimum above is attained for l = 0, and hence we obtain the desired lower bound: (√ G(λ) λ inf σ H −u + m 2 − κ ) := βλ. (C.4) 2|u| Note that we can in fact deduce that the infimum on the right-hand side is attained. This follows from the min-max principle and the operator inequality √ − + m 2 − κ/2|x| −1/2m + m − κ/2|x|, where the right-hand side is the Schrödinger operator for the nonrelativistic hydrogen atom. Next, we prove the upper bound in a similar way as done for the Müller functional in [14]. Since K −κ/2|x| is a real operator, there is a real-valued eigenfunction f ∈ H such that (√ − + m2 − κ ) f = βf. 2|x| (C.5) Using the method in [10], one can show that f ∈ C ∞ (R 3 \{0}) and that it decays exponentially at infinity when 0 κ < 4/π. Note that f ∈ D( √ − + m 2 − κ/(2|x|)) ⊂ H 1/2 (R 3 ) but this domain is only known to be H 1 (R 3 ) when κ 1.(We address this regularity issue again below.) Next, we fix a nonnegative function χ ∈ C ∞ 0 (R3 ) such that ∫ χ 4 = λ, andwe denote χ L (x) = L −3/4 χ(x/L) for L>0 given. We define α L (x,y) = χ L (x)f (x − y)χ L (y), (C.6) where = 1 if q = 1 and, when q 2, ⎛ 0 1 = √ 1 −1 0 2 ⎜ ⎝ 0 ⎞ . . ⎟ .. ⎠ 0 As α L ∈ L 2 (R 3 × R 3 , C q × C q ), we may denote by α L the Hilbert-Schmidt operator whose kernel is α L (x,y).Thenα T L =−α L by the choice of f . Note that ||α L || C ||f || L 1 ||χ L || 2 L ∞ = O(L−3/2 ), (C.7)

MINIMIZERS FOR THE HFB-THEORY 311 and hence α L → 0 in the operator norm. Next, we define γ L as the unique nonnegative trace-class operator solving the following equation for L>0 large enough: γ L (1 − γ L ) = α L α ∗ L . (C.8) Note that γ L 2α L α ∗ L if L>0 is sufficiently large. In particular, we have ||γ L|| = O(L −3 ). In analogy to [14], it can be checked using ∫ χ 4 = λ that Tr(α L α ∗ L ) = λ + O(L −2 ). Therefore, we have Tr(γ L ) = λ + O(L −2 ), by (C.8), and the fact that Tr(γ 2 L ) = O(L−3 ). Next, we show that (γ L ,α L ) ∈ K. To see this, we recall from [9] and [26, Proposition 2] that there exists an orthonormal basis (ϕ i ) of L 2 (R 3 , C q ) such that α L = ∑ i1 c i ϕ 2i−1 ∧ ϕ 2i . (C.9) On the other hand, the operator α L αL ∗ is just 1/2 times the one-body density matrix of the two-body wave function α L ,sothat α L α ∗ L = 1 ∑ (c i ) 2 (|ϕ 2i−1 〉〈ϕ 2i−1 |+|ϕ 2i 〉〈ϕ 2i |) , (C.10) 2 i1 γ L = ∑ i1 γ i (|ϕ 2i−1 〉〈ϕ 2i−1 |+|ϕ 2i 〉〈ϕ 2i |) , (C.11) where for all i 1, γ i (1 − γ i ) = (c i ) 2 /2. Next, we have to verify that ( ) γL α 0 Ɣ L = L αL ∗ 1. 1 − γ L (C.12) As α L ϕ 2i−1 = (−c i / √ 2)ϕ 2i and α L ϕ 2i = (c i / √ 2)ϕ 2i−1 with respect to the basis (ϕ i ), this amounts to studying the matrix ⎛ γ i 0 0 c i / √ ⎞ 2 0 γ M = i −c i / √ 2 0 ⎜ ⎝ 0 −c i / √ ⎟ 2 1− γ i 0 c i / √ ⎠ , (C.13) 2 0 0 1− γ i which is easily shown to satisfy 0 M 1. (It is indeed a projection.) Thus we have proven that (γ L ,α L ) ∈ K for L>0 sufficiently large. Next, we estimate G (γ L ,α L ) = Tr(Kα L α ∗ L ) − κ ∫∫ |αL (x,y)| 2 dx dy + Tr(Kγ 2 L 2 |x − y| ). (C.14)

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