Views
3 years ago

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

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

## 296 LENZMANN and LEWIN

296 LENZMANN and LEWIN Proof We consider a smooth partition of unity {χ k } k∈Z 3 with ∑ k∈Z 3 χ 2 k ≡ 1 such that χ k ≡ 1 on C k and supp χ k ⊂ C ′ k , with the half-open cubes C k = [k, k + 1) 3 and C ′ k = [k − 1,k+ 2) 3 where k ∈ Z 3 . Furthermore, we assume that sup k∈Z 3 |∇χ k (x)| C. Next, we estimate as follows: ∫R 3 |f n (x)| 8/3 dx = ∑ k∈Z 3 ∫ R 3 χ k (x) 2 |f n (x)| 8/3 dx = ∑ k∈Z 3 ∫R 3 |1 C ′ k (x)f n (x)| 2/3 |χ k (x)f n (x)| 2 dx ∑ k∈Z 3 ‖1 C ′ k |f n | 2 ‖ 1/3 L 1 (R 3 ) ‖χ kf n ‖ 2 L 3 (R 3 ) C sup k∈Z 3 ‖1 C ′ k |f n | 2 ‖ 1/3 L 1 (R 3 ) ∑ k∈Z 3 〈f n ,χ k Kχ k f n 〉. (7.4) We have used the Sobolev-type inequality ‖f ‖ 2 L 3 (R 3 ) C〈f, Kf 〉. By Lemma A.1, the nonlocal operator K satisfies ∑ χ k Kχ k K + 1 ∫ ∞ 1 ( ∑ ) |∇χ π k∈Z 3 0 s + K 2 k | 2 1 √ sds K + C, (7.5) s + K 2 k∈Z 3 since the integral expression on the right-hand side in the first inequality is a bounded self-adjoint operator due to the fact that ∑ k∈Z 3 |∇χ k(x)| 2 ∈ L ∞ (R 3 ) by our choice of the partition {χ k } k∈Z 3. Therefore, we conclude that ∑ k∈Z 3 〈f n ,χ k Kχ k f n 〉 C‖f n ‖ 2 H 1/2 (R 3 ) . (7.6) Since, by assumption, we have ‖f n ‖ H 1/2 (R 3 ) C independent of n, estimate (7.4) leads to ∫ |f n (x)| 8/3 dx C sup‖1 C ′ k |f n | 2 ‖ 1/3 L 1 (R 3 ) → 0, (7.7) R 3 k∈Z 3 using that {|f n | 2 } n∈N vanishes. This shows that f n → 0 in L 8/3 (R 3 ), whence it converges to 0 strongly in L p (R 3 ) for all 2

MINIMIZERS FOR THE HFB-THEORY 297 Proof of Lemma 7.1 By the coercivity of E stated in Lemma 4.2, we know that our minimizing sequence (γ n ,α n ) is bounded in X. Hence we deduce from Lemma 2.1 that √ ρ γn is bounded in H 1/2 (R 3 ). Applying Lemma 7.2 to the sequence f n = √ ρ γn , we obtain that ρ γn → 0 strongly in L p (R 3 ) for 1

Symmetry Restoration in Hartree-Fock-Bogoliubov Based Theories
Application of the gradient method to Hartree-Fock-Bogoliubov theory
Hartree-Fock-Bogoliubov theory of polarized Fermi systems
Hartree-Fock-Bogoliubov (HFB) Theory This Lecture closely follows ...
Hartree-Fock Theory - Chemistry
I ) Direct minimization in Hartree Fock and DFT - II) Converence of ...
Experimental, Hartree-Fock, and Density Functional Theory ...
Hartree-Fock theory of nuclear deformations and high spin states
Theory of radiation transfer in neutron star atmospheres - MPE
DENSITY LOWER BOUND ESTIMATES FOR LOCAL MINIMIZERS ...
Theory of Nuclear Matter for Neutron Stars and ... - Graduate Physics
ON MINIMIZERS OF INTERACTION FUNCTIONALS WITH ...
ON MINIMIZERS OF INTERACTION FUNCTIONALS WITH ...
Stability of critical shapes for the drag minimization problem in ...
Disintegration theory for von Neumann algebras
Partial Regularity for Minimizers of Degenerate Polyconvex Energies
A dispersive estimate for the Schrödinger operator in star-shaped ...
I Integral Equations and Operator Theory
Microscopic Derivation of Ginzburg-Landau Theory
Spectral Theory in Hilbert Space
Estimating the number of negative eigenvalues of Schrodinger ...
theory of linear elasticity in the half-space - Université de Pau et des ...