| Title:
             | 
Weak interaction limit for nuclear matter and the time-dependent Hartree-Fock equation (English) | 
| Author:
             | 
Ducomet, Bernard | 
| Language:
             | 
English | 
| Journal:
             | 
Applications of Mathematics | 
| ISSN:
             | 
0862-7940 (print) | 
| ISSN:
             | 
1572-9109 (online) | 
| Volume:
             | 
55 | 
| Issue:
             | 
3 | 
| Year:
             | 
2010 | 
| Pages:
             | 
197-219 | 
| Summary lang:
             | 
English | 
| . | 
| Category:
             | 
math | 
| . | 
| Summary:
             | 
We consider an effective model of nuclear matter including spin and isospin degrees of freedom, described by an $N$-body Hamiltonian with suitably renormalized two-body and three-body interaction potentials. We show that the corresponding mean-field theory (the time-dependent Hartree-Fock approximation) is ``exact'' as $N$ tends to infinity. (English) | 
| Keyword:
             | 
time-dependent Hartree-Fock equation | 
| Keyword:
             | 
nuclear matter | 
| MSC:
             | 
81Q05 | 
| MSC:
             | 
81V05 | 
| MSC:
             | 
81V35 | 
| MSC:
             | 
81V70 | 
| idZBL:
             | 
Zbl 1224.81018 | 
| idMR:
             | 
MR2657834 | 
| DOI:
             | 
10.1007/s10492-010-0008-6 | 
| . | 
| Date available:
             | 
2010-07-20T13:42:38Z | 
| Last updated:
             | 
2020-07-02 | 
| Stable URL:
             | 
http://hdl.handle.net/10338.dmlcz/140395 | 
| . | 
| Reference:
             | 
[1] Amrein, W. O., Jauch, J. M., Sinha, K. B.: Scattering Theory in Quantum Mechanics. Lecture Notes and Supplements in Physics, 16.W. A. Benjamin London (1977). MR 0495999 | 
| Reference:
             | 
[2] Bardos, C., Golse, F., Mauser, N. J.: Weak coupling limit of the $N$-particle Schrödinger equation.Methods Appl. Anal. 7 (2000), 275-293. Zbl 1003.81027, MR 1869286 | 
| Reference:
             | 
[3] Bardos, C., Golse, F., Gottlieb, A. D., Mauser, N. J.: Mean field dynamics of fermions and the time-dependent Hartree-Fock equation.J. Math. Pures Appl., IX. Sér. 82 (2003), 665-683. Zbl 1029.82022, MR 1996777, 10.1016/S0021-7824(03)00023-0 | 
| Reference:
             | 
[4] Bardos, C., Ducomet, B., Golse, F., Gottlieb, A. D., Mauser, N. J.: The TDHF approximation for Hamiltonians with $m$-particle interaction potentials. Supplement.Commun. Math. Sci. 1 (2007), 1-9. MR 2301285, 10.4310/CMS.2007.v5.n5.a2 | 
| Reference:
             | 
[5] Bardos, C., Golse, F., Gottlieb, A. D., Mauser, N. J.: Derivation of the time dependent Hartree-Fock equation with Coulomb potential.Preprint. | 
| Reference:
             | 
[6] Bardos, C., Erdős, L., Golse, F., Mauser, N. J., Yau, H.-T.: Derivation of the Schrödinger-Poisson equation from the quantum $N$-body problem.C. R., Math. Acad. Sci. Paris 334 (2002), 515-520. MR 1890644, 10.1016/S1631-073X(02)02253-7 | 
| Reference:
             | 
[7] Beiner, M., Flocard, H., Giai, N. Van, Quentin, P.: Nuclear ground-state properties and self-consistent calculations with the skyrme interaction: I. Spherical description.Nucl. Phys. A238 (1975), 29-69. 10.1016/0375-9474(75)90338-3 | 
| Reference:
             | 
[8] Bitaud, L.: Etude théorique de la fission des transactinides dans le cadre d'une approche microscopique.PhD. Thesis Université Paris-Sud Paris (1996). | 
| Reference:
             | 
[9] Bove, A., Prato, G. Da, Fano, G.: On the Hartree-Fock time-dependent problem.Commun. Math. Phys. 49 (1976), 25-33. MR 0456066, 10.1007/BF01608633 | 
| Reference:
             | 
[10] Catto, I.: Some remarks on Hartree-type models in nuclear physics.In: Analyse mathématique de modèles de la Mécanique Quantique. PhD. Thesis Université Paris-Dauphine Paris (1992). | 
| Reference:
             | 
[11] Chabanat, E., Bonche, P., Haensel, P., Meyer, J., Schaeffer, R.: A Skyrme parametrization from subnuclear to neutron star densities.Nucl. Phys. A627 (1997), 710-746. 10.1016/S0375-9474(97)00596-4 | 
| Reference:
             | 
[12] Dechargé, J., Gogny, D.: Hartree-Fock-Bogolyubov calculations with the D1 effective interaction on spherical nuclei.Phys. Rev. C 21 (1980), 1568-1593. 10.1103/PhysRevC.21.1568 | 
| Reference:
             | 
[13] Ducomet, B.: Weak interaction limit for a model of nuclear matter.Oberwolfach Reports No 47 (2006), 2819-2822. | 
| Reference:
             | 
[14] Erdős, L., Yau, H.-T.: Derivation of the nonlinear Schrödinger equation from a many body Coulomb system.arXiv:math-ph/0111042v3 22 May 2002. MR 1926667 | 
| Reference:
             | 
[15] Fetter, A. L., Walecka, J. D.: Quantum Theory of Many-Particle Systems.McGraw Hill New York (1971). | 
| Reference:
             | 
[16] Golse, F.: The mean-field limit for the dynamics of large particle systems.Proceedings of the conference on partial differential equations, Forges-les-Eaux, France, June 2-6, 2003 Université de Nantes Nantes (2003). MR 2050595 | 
| Reference:
             | 
[17] Kato, T.: Fundamental properties of Hamiltonian operators of Schrödinger type.Trans. Am. Math. Soc. 70 (1951), 195-211. Zbl 0044.42701, MR 0041010 | 
| Reference:
             | 
[18] Knowles, A., Frölich, J.: A microscopic derivation of the time-dependent Hartree-Fock equation with Coulomb two-body interaction.arXiv:0810.4282. | 
| Reference:
             | 
[19] Lions, P.-L., Gogny, D.: Hartree-Fock theory in nuclear physics.RAIRO, Modélisation Math. Anal. Numér. 20 (1986), 571-637. Zbl 0607.35078, MR 0877058, 10.1051/m2an/1986200405711 | 
| Reference:
             | 
[20] Mornas, L.: Neutron stars in a Skyrme model with hyperons.arXiv:nucl-th/0407083 vl 23 Jul 2004. | 
| Reference:
             | 
[21] Ring, P., Schuck, P.: The Nuclear Many-Body Problem.Springer Berlin (1980). MR 0611683 | 
| Reference:
             | 
[22] Serot, B. D., Walecka, J. D.: The Relativistic Nuclear Many-body Problem. Adv. Nucl. Phys. 16.Plenum New York (1986). | 
| Reference:
             | 
[23] Spohn, H.: Kinetic equations from Hamiltonian dynamics.Rev. Mod. Phys. 53 (1980), 600-640. Zbl 0465.76069 | 
| Reference:
             | 
[24] Vautherin, D., Brink, D. M.: Hartree-Fock calculations with Skyrme's interaction. I. Spherical nuclei.Phys. Rev. C 5 (1972), 626-647. 10.1103/PhysRevC.5.626 | 
| Reference:
             | 
[25] Winter, C. Van, Brascamp, H. J.: The $N$-body problem with spin-orbit or Coulomb interactions.Comm. Math. Phys. 11 (1968), 19-55. MR 0260334 | 
| Reference:
             | 
[26] Winter, C. Van, Brascamp, H. J.: The $N$-body problem with spin-orbit or Coulomb interactions.Comm. Math. Phys. 11 (1968), 19-55. MR 0260334 | 
| . |