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

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

306 LENZMANN and LEWIN

306 LENZMANN and LEWIN LEMMA A.2 Suppose that χ ∈ C ∞ (R d ) such that ∇χ ∈ L p (R d ) with p ∈ (2d,∞], and let K = √ − + m 2 with m>0. Then the following formula holds: 1 2 (Kχ2 + χ 2 K) = χKχ − 1 ∫ ∞ 1 1 √ π 0 s + K 2 |∇χ|2 sds+ Lχ , (A.2) s + K 2 where L χ is a nonnegative bounded operator. Moreover, we have ∫ ∞ 1 1 √ ∥ ∥∥Sp/2 ∥ s + K 2 |∇χ|2 sds C‖∇χ‖ 2 s + K 2 (L 2 (R d L )) p (R d ) , (A.3) 0 ‖L χ ‖ Sp/2 (L 2 (R d )) C‖∇χ‖ 2 L p (R d ) , (A.4) for some constant C independent of χ (but depending on m, p, and d). Proof By functional calculus, we have the representation formula ∫ ∞ K = 1 K 2 ds √ , π 0 s + K 2 s which leads to [K,B] = 1 ∫ ∞ 1 1 √ π 0 s + K 2 [K2 ,B] sds. s + K 2 Using (A.6), a calculation shows 1 2 (Kχ2 + χ 2 K) = χKχ + 1 [ ] [K,χ],χ 2 = χKχ + 1 ∫ ∞ [ ] 1 1 √sds 2π s + K 2 [K2 ,χ] s + K ,χ 2 with L χ = 1 π = χKχ + 1 2π ∫ ∞ 0 0 ∫ ∞ 0 (A.5) (A.6) 1 [ [K 2 ,χ],χ ] 1 √ sds+ Lχ , s + K 2 s + K 2 1 1 1 √ s + K 2 [K2 ,χ] s + K 2 [χ,K2 ] sds. s + K 2 Since [[K 2 ,χ],χ] =−2|∇χ| 2 , this yields (A.2). Furthermore, we note that (A.7) L χ = 1 π ∫ ∞ 0 A(s) ∗ A(s) √ sds, A(s) = which shows that L χ is nonnegative. 1 1 (s + K 2 ) 1/2 [χ,K2 ] s + K , 2 (A.8)

MINIMIZERS FOR THE HFB-THEORY 307 We now show that all the above commutators are bounded (which also validates our somewhat formal calculation). Recall the Kato-Seiler-Simon inequality (see [38] and [41, Theorem 4.1]): ||f (−i∇ x )g(x)|| Sp (L 2 (R d )) (2π) −d/p ||f || L p (R d ) ||g|| L p (R d ) for p ∈ [2, ∞]. (A.9) This yields, choosing some d/(2p)

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