- Text
- Lemma,
- Minimizers,
- Theorem,
- Operator,
- Inequality,
- Sequence,
- Estimate,
- Hence,
- Lewin,
- Lenzmann,
- Neutron,
- Www.math.ku.dk

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

258 LENZMANN and LEWIN 1. Introduction The Hartree-Fock-Bogoliubov (HFB) **the**ory is a widely used tool (see, e.g., [37], [12]) **for** understanding many-body quantum systems where attractive two-body interactions are dominant. In particular, HFB energy functionals incorporate **the** physical phenomenon **of** “Cooper pairing,” which is likely to occur whenever attractive **for**ces among quantum particles become significant, such as in nuclear and gravitational physics as well as in superconducting materials. Despite **the** broad range **of** physical applications **for** HFB **the**ory, **the**re is not much known rigorously concerning **the** existence **of** **minimizers**, let alone a pro**of** **of** **the**ir fundamental properties such as Cooper pair **for**mation. As a starting point **for** rigorous analysis, this article is devoted to **the** existence **of** **minimizers** **for** HFB relativistic energy functionals with an interaction which behaves at infinity like **the** (attractive) Newtonian gravity. This specific class **of** functionals can be viewed as a model problem **for** self-gravitating relativistic Fermi systems which are found, **for** example, in **neutron** **stars** and white dwarfs. Moreover, speaking from a ma**the**matical point **of** view, **the** corresponding variational problem exhibits **the** delicate property **of** criticality, as we detail below. The most challenging main feature **of** **the** HFB variational problem in gravitational physics is its lack **of** weak lower semicontinuity (WLSC) due to **the** attractive interaction among particles. As a consequence **of** **the** absence **of** WLSC, **the** existence pro**of** **for** **minimizers** is much more involved than **for** **the** well-studied Hartree-Fock (HF) models arising in atomic physics, where WLSC plays an essential role (see [30], [36]). Ano**the**r difficulty in **the** analysis **of** HFB models stems from its translational invariance and **the** fact that **the** main variables are two operators related via a complicated constraint inequality. Finally, a fur**the**r complication (although conceptually less important) is **the** treatment **of** **the** pseudo-differential operator describing **the** kinetic energy **of** relativistic fermions. Following **the** general rigorous discussion **of** HFB energy functionals in [2], we specifically consider **the** energy functional given by E(γ,α):= Tr(Tγ) + κ ∫∫ W (x − y)ρ γ (x)ρ γ (y) dx dy 2 R 3 ×R 3 − κ ∫∫ W (x − y)|γ (x,y)| 2 dx dy 2 R 3 ×R 3 + κ ∫∫ W (x − y)|α(x,y)| 2 dx dy, (1.1) 2 R 3 ×R 3 which models a system **of** relativistic fermions subject to **the** interaction W .We assume that W (x) behaves like −1/|x| at infinity, that is, that far away **the** attractive

MINIMIZERS FOR THE HFB-THEORY 259 Newtonian **for**ce dominates. † Indeed, in most **of** **the** article we take **for** simplicity **the** purely gravitational interaction W (x) =− 1 |x| . In (1.1) Tr(·) denotes **the** trace, and **the** pseudo-differential operator T = √ − + m 2 − m describes **the** kinetic energy **of** a fermion with mass m>0. The coupling constant κ>0 parameterizes **the** strength **of** **the** interaction among **the** particles. Fur**the**rmore, **the** variables γ and α are two operators acting on L 2 (R 3 ; C 2 ). †† The operator γ is referred to as **the** one-body density matrix. It is self-adjoint and nonnegative and has a finite trace fixed to be Tr(γ ) = λ (1.2) **for** some given λ>0, which is a real parameter and corresponds to **the** expected value **of** **the** number **of** particles in **the** star. The density ρ γ is **the** unique L 1 (R 3 ) nonnegative function that satisfies Tr(γV) = ∫ ρ R 3 γ (x)V (x) dx **for** any bounded function V .As γ is trace class, it has a kernel γ (x,y) appearing in **the** second line **of** (1.1), which is a 2 × 2 Hermitian matrix **for** almost every x,y ∈ R 3 . Finally, **the** operator α is called **the** pairing density matrix. It is only assumed to be Hilbert-Schmidt; that is, we have Tr(α ∗ α) < ∞. Its kernel is a (2 × 2)-matrix which is supposed to be antisymmetric in **the** following sense: α(x,y) T =−α(y,x), where T is **the** usual transposition **of** matrices. To **for**mulate **the** HFB variational problem **for** **the** energy E(γ,α), wehaveto supplement **the** side condition (1.2) by **the** following operator inequality relating γ and α: ( ) ( ) 0 0 γ α 0 0 α ∗ 1 − γ ( ) 1 0 0 1 on L 2 (R 3 ; C 2 ) ⊕ L 2 (R 3 ; C 2 ). (1.3) This inequality guarantees that **the** pair (γ,α) is associated to a unique quasi-free state in Fock space [2]. The corresponding HFB minimization problem **the**n reads I(λ) := inf { E(γ,α):γ ∗ = γ, α T =−α such that (1.2) and (1.3) } . † See Remark 6 **for** precise assumptions on W. †† More generally, we consider L 2 (R 3 ; C q )below,whereq 1 denotes **the** internal spin degree **of** freedom. The case q = 2 (corresponding to spin 1/2) is physically **the** most relevant one.

- Page 1: MINIMIZERS FOR THE HARTREE-FOCK-BOG
- Page 5 and 6: MINIMIZERS FOR THE HFB-THEORY 261 a
- Page 7 and 8: MINIMIZERS FOR THE HFB-THEORY 263 F
- Page 9 and 10: MINIMIZERS FOR THE HFB-THEORY 265 2
- Page 11 and 12: MINIMIZERS FOR THE HFB-THEORY 267 F
- Page 13 and 14: MINIMIZERS FOR THE HFB-THEORY 269 R
- Page 15 and 16: MINIMIZERS FOR THE HFB-THEORY 271 3
- Page 17 and 18: MINIMIZERS FOR THE HFB-THEORY 273 3
- Page 19 and 20: MINIMIZERS FOR THE HFB-THEORY 275 e
- Page 21 and 22: MINIMIZERS FOR THE HFB-THEORY 277 w
- Page 23 and 24: MINIMIZERS FOR THE HFB-THEORY 279 T
- Page 25 and 26: MINIMIZERS FOR THE HFB-THEORY 281 w
- Page 27 and 28: MINIMIZERS FOR THE HFB-THEORY 283
- Page 29 and 30: MINIMIZERS FOR THE HFB-THEORY 285 H
- Page 31 and 32: MINIMIZERS FOR THE HFB-THEORY 287
- Page 33 and 34: MINIMIZERS FOR THE HFB-THEORY 289 T
- Page 35 and 36: MINIMIZERS FOR THE HFB-THEORY 291 w
- Page 37 and 38: MINIMIZERS FOR THE HFB-THEORY 293 a
- Page 39 and 40: MINIMIZERS FOR THE HFB-THEORY 295 7
- Page 41 and 42: MINIMIZERS FOR THE HFB-THEORY 297 P
- Page 43 and 44: MINIMIZERS FOR THE HFB-THEORY 299 T
- Page 45 and 46: MINIMIZERS FOR THE HFB-THEORY 301
- Page 47 and 48: MINIMIZERS FOR THE HFB-THEORY 303 T
- Page 49 and 50: MINIMIZERS FOR THE HFB-THEORY 305 T
- Page 51 and 52: MINIMIZERS FOR THE HFB-THEORY 307 W
- Page 53 and 54:
MINIMIZERS FOR THE HFB-THEORY 309 H

- Page 55 and 56:
MINIMIZERS FOR THE HFB-THEORY 311 a

- Page 57 and 58:
MINIMIZERS FOR THE HFB-THEORY 313 a

- Page 59:
MINIMIZERS FOR THE HFB-THEORY 315 [