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

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

262 LENZMANN and LEWIN

262 LENZMANN and LEWIN On the other hand, by deriving a suitable decay estimate for any minimizer of I(λ i ), we can deduce the strict inequality I(λ)

MINIMIZERS FOR THE HFB-THEORY 263 For later convenience, we introduce two smooth cutoff functions χ R and ζ R that are defined as follows. Let 0 χ 1 be a fixed smooth function on R 3 such that χ ≡ 1 for |x| < 1 and χ ≡ 0 for |x| 2.ForanyR>0, we then define the functions χ R (x) = χ(x/R) and ζ R (x) = √ 1 − χ R (x) 2 . (1.5) Throughout this article, we use the cutoff functions χ R and ζ R freely. Furthermore, we denote by τ y : L 2 (R 3 ; C q ) → L 2 (R 3 ; C q ) the unitary operator that is defined by (τ y f ) = f (·−y), (1.6) for a given y ∈ R 3 . We also use the notation U(γ,α)U ∗ := (UγU ∗ ,UαU ∗ ) for any unitary operator U acting on L 2 (R 3 ; C q ), for example, for the translation operator U = τ y . For the physically inclined reader, we remark that we work in units such that Planck’s constant and the speed of light c satisfy = c = 1. 2. Basic properties of HFB energy To prepare the statement of our main results, we first provide the adequate setting for the HFB variational problem, and we collect some basic properties needed for the rest of this article. 2.1. HFB states with finite kinetic energy To set up the variational calculus, we define a class of HFB states having finite pseudorelativistic kinetic energy. Therefore we introduce the following (real) Banach space of density matrices X = { (γ,α) ∈ S 1 × S 2 : γ ∗ = γ, α T =−α, ‖(γ,α)‖ X < ∞ } , (2.1) equipped with the norm ‖(γ,α)‖ X =‖(1 − ) 1/4 γ (1 − ) 1/4 ‖ S1 +‖(1 − ) 1/4 α‖ S2 . (2.2) We remind the reader that S 1 and S 2 denote the space of trace-class and Hilbert- Schmidt operators on L 2 (R 3 ; C q ), respectively. The universally fixed integer q 1 takes into account the internal spin degrees of freedom of the model. In what follows, we omit the dependence on q whenever it is of no importance. Furthermore, we define the following subsets of density matrices in X: ) ( ) )} γ α K = { (γ,α) ∈ X : ( 0 0 0 0 α ∗ 1 − γ ( 1 0 0 1 . (2.3)

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