×

Minimizers for the Hartree-Fock-Bogoliubov theory of neutron stars and white dwarfs. (English) Zbl 1202.49013

The main result of this paper is that the infimum \(I(\lambda)\) of the Hartree-Fock-Bogoliubov (HFB) functional, with attractive two-body interactions given by Newtonian gravity, is attained when the “expected value of the number of particles” \(\lambda\) is less than a quantity \(\lambda^{\text{HFB}}(\kappa)\), where \(\kappa\) is a coupling constant. The method relies on “concentration-compactness” arguments.

MSC:

49J40 Variational inequalities
85A15 Galactic and stellar structure
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] V. Bach, J. FröHlich, and L. Jonsson, Bogoliubov-Hartree-Fock mean field theory for neutron stars and other systems with attractive interactions , J. Math. Phys. 50 (2009), 102102. · Zbl 1248.81277 · doi:10.1063/1.3225565
[2] V. Bach, E. H. Lieb, and J. P. Solovej, Generalized Hartree-Fock theory and the Hubbard model , J. Statist. Phys. 76 (1994), 3–89. · Zbl 0839.60095 · doi:10.1007/BF02188656
[3] J.-P. Blaizot and G. Ripka, Quantum Theory of Finite Systems , MIT Press, Cambridge, Mass., 1985.
[4] A. Bove, G. Da Prato, and G. Fano, On the Hartree-Fock time-dependent problem , Comm. Math. Phys. 49 (1976), 25–33. · Zbl 0303.34046 · doi:10.1007/BF01608633
[5] T. Cazenave and P.-L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations , Comm. Math. Phys. 85 (1982), 549–561. · Zbl 0513.35007 · doi:10.1007/BF01403504
[6] J. M. Chadam, The time-dependent Hartree-Fock equations with Coulomb two-body interaction , Comm. Math. Phys. 46 (1976), 99–104. · Zbl 0322.35043 · doi:10.1007/BF01608490
[7] J. M. Chadam and R. T. Glassey, Global existence of solutions to the Cauchy problem for time-dependent Hartree equations , J. Math. Phys. 16 (1975), 1122–1130. · Zbl 0299.35084 · doi:10.1063/1.522642
[8] S. Chandrasekhar, The maximum mass of ideal white dwarfs , Astrophys. J. 74 (1931), 81–82. · Zbl 0002.23502 · doi:10.1086/143324
[9] A. Coleman, Structure of fermion density matrices , Rev. Modern Phys. 35 (1963), 668–689. · doi:10.1103/RevModPhys.35.668
[10] A. Dall’Acqua, T. øStergaard SøRensen, and E. Stockmeyer, Hartree-Fock theory for pseudorelativistic atoms , Ann. Henri Poincaré 9 (2008), 711–742. · Zbl 1192.81406 · doi:10.1007/s00023-008-0370-z
[11] I. Daubechies, An uncertainty principle for fermions with generalized kinetic energy , Comm. Math. Phys. 90 (1983), 511–520. · Zbl 0946.81521 · doi:10.1007/BF01216182
[12] D. J. Dean and M. Hjorth-Jensen, Pairing in nuclear systems: From neutron stars to finite nuclei , Rev. Mod. Phys. 75 (2003), 607–656.
[13] V. Enss, A note on Hunziker’s theorem , Comm. Math. Phys. 52 (1977), 233–238. · doi:10.1007/BF01609484
[14] R. L. Frank, E. H. Lieb, R. Seiringer, and H. Siedentop, Müller’s exchangecorrelation energy in density-matrix-functional theory , Phys. Rev. A 76 (2007), no. 052517.
[15] G. Friesecke, The multiconfiguration equations for atoms and molecules: charge quantization and existence of solutions , Arch. Ration. Mech. Anal. 169 (2003), 35–71. · Zbl 1035.81069 · doi:10.1007/s00205-003-0252-y
[16] J. FröHlich and E. Lenzmann, Blowup for nonlinear wave equations describing boson stars , Comm. Pure Appl. Math. 60 (2007), 1691–1705. · Zbl 1135.35011 · doi:10.1002/cpa.20186
[17] -, Dynamical collapse of white dwarfs in Hartree- and Hartree-Fock theory , Comm. Math. Phys. 274 (2007), 737–750. · Zbl 1130.85004 · doi:10.1007/s00220-007-0290-7
[18] C. Hainzl, M. Lewin, and é. SéRé, Existence of a stable polarized vacuum in the Bogoliubov-Dirac-Fock approximation , Comm. Math. Phys. 257 (2005), 515–562. · Zbl 1115.81061 · doi:10.1007/s00220-005-1343-4
[19] -, Self-consistent solution for the polarized vacuum in a no-photon QED model, J. Phys. A 38 (2005), 4483–4499. · Zbl 1073.81677 · doi:10.1088/0305-4470/38/20/014
[20] C. Hainzl, M. Lewin, and C. Sparber, Existence of global-in-time solutions to a generalized Dirac-Fock type evolution equation , Lett. Math. Phys. 72 (2005), 99–113. · Zbl 1115.81026 · doi:10.1007/s11005-005-4377-9
[21] C. Hainzl and B. Schlein, Stellar collapse in the time dependent Hartree-Fock approximation, Comm. Math. Phys. 287 (2009), 705–717. · Zbl 1175.85002 · doi:10.1007/s00220-008-0668-1
[22] I. W. Herbst, Spectral theory of the operator \((p^2+m^2)^1/2-Ze^2/r\) , Comm. Math. Phys. 53 (1977), 285–294. · Zbl 0375.35047 · doi:10.1007/BF01609852
[23] W. Hunziker, On the spectra of Schrödinger multiparticle Hamiltonians , Helv. Phys. Acta 39 (1966), 451–462. · Zbl 0141.44701
[24] T. Kato, Perturbation Theory for Linear Operators , reprint of the 1980 2nd ed., Springer, Berlin, 1995. · Zbl 0435.47001
[25] E. Lenzmann, Well-posedness for semi-relativistic Hartree equations of critical type , Math. Phys. Anal. Geom. 10 (2007), 43–64. · Zbl 1171.35474 · doi:10.1007/s11040-007-9020-9
[26] M. Lewin, Solutions of the multiconfiguration equations in quantum chemistry , Arch. Ration. Mech. Anal. 171 (2004), 83–114. · Zbl 1063.81102 · doi:10.1007/s00205-003-0281-6
[27] R. T. Lewis, H. Siedentop, and S. Vugalter, The essential spectrum of relativistic multi-particle operators , Ann. Inst. H. Poincaré Phys. Théor. 67 (1997), 1–28. · Zbl 0886.35126
[28] E. H. Lieb, Convex trace functions and the Wigner-Yanase-Dyson conjecture , Advances in Math. 11 (1973), 267–288. · Zbl 0267.46055 · doi:10.1016/0001-8708(73)90011-X
[29] E. H. Lieb and M. Loss, Analysis , 2nd ed., Graduate Studies in Mathematics 14 , Amer. Math. Soc., Providence, 2001. · Zbl 0966.26002
[30] E. H. Lieb and B. Simon, The Hartree-Fock theory for Coulomb systems , Comm. Math. Phys. 53 (1977), 185–194. · doi:10.1007/BF01609845
[31] E. H. Lieb and W. E. Thirring, Gravitational collapse in quantum mechanics with relativistic kinetic energy , Ann. Physics 155 (1984), 494–512. · doi:10.1016/0003-4916(84)90010-1
[32] E. H. Lieb and H.-T. Yau, The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics , Comm. Math. Phys. 112 (1987), 147–174. · Zbl 0641.35065 · doi:10.1007/BF01217684
[33] -, The stability and instability of relativistic matter , Comm. Math. Phys. 118 (1988), 177–213. · Zbl 0686.35099 · doi:10.1007/BF01218577
[34] P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. I , Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), 109–145. · Zbl 0541.49009
[35] -, The concentration-compactness principle in the calculus of variations. The locally compact case. II , Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), 223–283. · Zbl 0704.49004
[36] -, Solutions of Hartree-Fock equations for Coulomb systems , Comm. Math. Phys. 109 (1987), 33–97. · Zbl 0618.35111 · doi:10.1007/BF01205672
[37] P. Ring and P. Schuck, The Nuclear Many-Body Problem , Texts and Monographs in Physics, Springer, New York, 1980.
[38] E. Seiler and B. Simon, Bounds in the Yukawa2 quantum field theory: upper bound on the pressure, Hamiltonian bound and linear lower bound , Comm. Math. Phys. 45 (1975), 99–114. · doi:10.1007/BF01629241
[39] I. M. Sigal, Geometric methods in the quantum many-body problem. Nonexistence of very negative ions , Comm. Math. Phys. 85 (1982), 309–324. · Zbl 0503.47041 · doi:10.1007/BF01254462
[40] B. Simon, Geometric methods in multiparticle quantum systems , Comm. Math. Phys. 55 (1977), 259–274. · Zbl 0413.47008 · doi:10.1007/BF01614550
[41] -, Trace Ideals and Their Applications , London Mathematical Society Lecture Note Series 35 , Cambridge Univ. Press, Cambridge, 1979. · Zbl 0423.47001
[42] E. M. Stein, Singular integrals and differentiability properties of functions , Princeton Mathematical Series 30 , Princeton University Press, Princeton, N.J., 1970. · Zbl 0207.13501
[43] C. Van Winter, Theory of finite systems of particles. I. The Green function , Mat.-Fys. Skr. Danske Vid. Selsk. 2 (1964), no. 8. · Zbl 0122.22403
[44] G. M. Zhislin, Discussion of the spectrum of Schrödinger operators for systems of many particles . Trudy Moskovskogo Matematiceskogo Obscestva 9 (1960), 81–120.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.