Duke Mathematical Journal

Minimizers for the Hartree-Fock-Bogoliubov theory of neutron stars and white dwarfs

Enno Lenzmann and Mathieu Lewin

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We prove the existence of minimizers for Hartree-Fock-Bogoliubov (HFB) energy functionals with attractive two-body interactions given by Newtonian gravity. This class of HFB functionals serves as a model problem for self-gravitating relativistic Fermi systems, which are found in neutron stars and white dwarfs. Furthermore, we derive some fundamental properties of HFB minimizers such as a decay estimate for the minimizing density. A decisive feature of the HFB model in gravitational physics is its failure of weak lower semicontinuity. This fact essentially complicates the analysis compared to the well-studied Hartree-Fock theories in atomic physics

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Duke Math. J., Volume 152, Number 2 (2010), 257-315.

First available in Project Euclid: 31 March 2010

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Zentralblatt MATH identifier

Primary: 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10]
Secondary: 49J40: Variational methods including variational inequalities [See also 47J20]


Lenzmann, Enno; Lewin, Mathieu. Minimizers for the Hartree-Fock-Bogoliubov theory of neutron stars and white dwarfs. Duke Math. J. 152 (2010), no. 2, 257--315. doi:10.1215/00127094-2010-013. https://projecteuclid.org/euclid.dmj/1270041109

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  • 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.
  • V. Bach, E. H. Lieb, and J. P. Solovej, Generalized Hartree-Fock theory and the Hubbard model, J. Statist. Phys. 76 (1994), 3--89.
  • J.-P. Blaizot and G. Ripka, Quantum Theory of Finite Systems, MIT Press, Cambridge, Mass., 1985.
  • A. Bove, G. Da Prato, and G. Fano, On the Hartree-Fock time-dependent problem, Comm. Math. Phys. 49 (1976), 25--33.
  • T. Cazenave and P.-L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys. 85 (1982), 549--561.
  • J. M. Chadam, The time-dependent Hartree-Fock equations with Coulomb two-body interaction, Comm. Math. Phys. 46 (1976), 99--104.
  • 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.
  • S. Chandrasekhar, The maximum mass of ideal white dwarfs, Astrophys. J. 74 (1931), 81--82.
  • A. Coleman, Structure of fermion density matrices, Rev. Modern Phys. 35 (1963), 668--689.
  • A. Dall'Acqua, T. øStergaard SøRensen, and E. Stockmeyer, Hartree-Fock theory for pseudorelativistic atoms, Ann. Henri Poincaré 9 (2008), 711--742.
  • I. Daubechies, An uncertainty principle for fermions with generalized kinetic energy, Comm. Math. Phys. 90 (1983), 511--520.
  • D. J. Dean and M. Hjorth-Jensen, Pairing in nuclear systems: From neutron stars to finite nuclei, Rev. Mod. Phys. 75 (2003), 607--656.
  • V. Enss, A note on Hunziker's theorem, Comm. Math. Phys. 52 (1977), 233--238.
  • 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.
  • G. Friesecke, The multiconfiguration equations for atoms and molecules: charge quantization and existence of solutions, Arch. Ration. Mech. Anal. 169 (2003), 35--71.
  • J. FröHlich and E. Lenzmann, Blowup for nonlinear wave equations describing boson stars, Comm. Pure Appl. Math. 60 (2007), 1691--1705.
  • —, Dynamical collapse of white dwarfs in Hartree- and Hartree-Fock theory, Comm. Math. Phys. 274 (2007), 737--750.
  • 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.
  • —, Self-consistent solution for the polarized vacuum in a no-photon QED model, J. Phys. A 38 (2005), 4483--4499.
  • 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.
  • C. Hainzl and B. Schlein, Stellar collapse in the time dependent Hartree-Fock approximation, Comm. Math. Phys. 287 (2009), 705--717.
  • I. W. Herbst, Spectral theory of the operator $(p\sp2+m\sp2)\sp1/2-Ze\sp2/r$, Comm. Math. Phys. 53 (1977), 285--294.
  • W. Hunziker, On the spectra of Schrödinger multiparticle Hamiltonians, Helv. Phys. Acta 39 (1966), 451--462.
  • T. Kato, Perturbation Theory for Linear Operators, reprint of the 1980 2nd ed., Springer, Berlin, 1995.
  • E. Lenzmann, Well-posedness for semi-relativistic Hartree equations of critical type, Math. Phys. Anal. Geom. 10 (2007), 43--64.
  • M. Lewin, Solutions of the multiconfiguration equations in quantum chemistry, Arch. Ration. Mech. Anal. 171 (2004), 83--114.
  • 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.
  • E. H. Lieb, Convex trace functions and the Wigner-Yanase-Dyson conjecture, Advances in Math. 11 (1973), 267--288.
  • E. H. Lieb and M. Loss, Analysis, 2nd ed., Graduate Studies in Mathematics 14, Amer. Math. Soc., Providence, 2001.
  • E. H. Lieb and B. Simon, The Hartree-Fock theory for Coulomb systems, Comm. Math. Phys. 53 (1977), 185--194.
  • E. H. Lieb and W. E. Thirring, Gravitational collapse in quantum mechanics with relativistic kinetic energy, Ann. Physics 155 (1984), 494--512.
  • 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.
  • —, The stability and instability of relativistic matter, Comm. Math. Phys. 118 (1988), 177--213.
  • 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.
  • —, 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.
  • —, Solutions of Hartree-Fock equations for Coulomb systems, Comm. Math. Phys. 109 (1987), 33--97.
  • P. Ring and P. Schuck, The Nuclear Many-Body Problem, Texts and Monographs in Physics, Springer, New York, 1980.
  • 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.
  • I. M. Sigal, Geometric methods in the quantum many-body problem. Nonexistence of very negative ions, Comm. Math. Phys. 85 (1982), 309--324.
  • B. Simon, Geometric methods in multiparticle quantum systems, Comm. Math. Phys. 55 (1977), 259--274.
  • —, Trace Ideals and Their Applications, London Mathematical Society Lecture Note Series 35, Cambridge Univ. Press, Cambridge, 1979.
  • E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series 30, Princeton University Press, Princeton, N.J., 1970.
  • C. Van Winter, Theory of finite systems of particles. I. The Green function, Mat.-Fys. Skr. Danske Vid. Selsk. 2 (1964), no. 8.
  • G. M. Zhislin, Discussion of the spectrum of Schrödinger operators for systems of many particles. Trudy Moskovskogo Matematiceskogo Obscestva 9 (1960), 81--120.