Zagatti, Sandro The Cauchy problem for Hartree-Fock time-dependent equations. (English) Zbl 0763.35089 Ann. Inst. Henri Poincaré, Phys. Théor. 56, No. 4, 357-374 (1992). The Cauchy problem for Hartree-Fock equations describing an infinite Fermion system with a single particle potential and a two-body interaction is investigated for \(L^ 2\)- and \(H^ 1\)-solutions. The existence, uniqueness and globality of the solutions are proved under appropriate conditions on the potentials. Reviewer: A.D.Kiselev (Chernigov) Cited in 17 Documents MSC: 35Q55 NLS equations (nonlinear Schrödinger equations) 81V70 Many-body theory; quantum Hall effect 35A07 Local existence and uniqueness theorems (PDE) (MSC2000) Keywords:existence; uniqueness; globality × Cite Format Result Cite Review PDF Full Text: Numdam EuDML References: [1] J.P. Blaizot and G. Ripka , Quantum Theory of Finite Systems , ch. 9, M.I.T. Press . MR 822225 [2] J. Ginibre and G. Velo , On a Class of Nonlinear Schrödinger Equation I. The Cauchy Problem General Case , J. Funct. Anal. , Vol. 32 , 1979 , pp. 1 - 32 . MR 533218 | Zbl 0396.35028 · Zbl 0396.35028 · doi:10.1016/0022-1236(79)90076-4 [3] J. Ginibre and G. Velo , On a Class of Nonlinear Schrödinger Equations with Nonlocal Interaction , Math. Z. , Vol. 170 , 1980 , pp. 109 - 136 . Article | MR 562582 | Zbl 0407.35063 · Zbl 0407.35063 · doi:10.1007/BF01214768 [4] T. Kato Nonlinear Schrödinger Equations , Ann. Inst. H. Poincaré , Phys. Théor. , Vol. 46 , 1987 , pp. 113 - 129 . Numdam | MR 877998 | Zbl 0632.35038 · Zbl 0632.35038 [5] T. Cazenave and F. Weissler , The Cauchy Problem for the Nonlinear Schrödinger Equation in H1 , Man. math. , Vol. 61 , 1988 , pp. 477 - 494 . Article | MR 952091 | Zbl 0696.35153 · Zbl 0696.35153 · doi:10.1007/BF01258601 [6] A. Bove , G. Da Prato and G. Fano , An existence Proof for the Hartree-Fock Time Dependent Problem with Bounded Two-Body Interaction , Comm. Math. Phys. , Vol. 37 , 1974 , pp. 183 - 191 . Article | MR 424069 | Zbl 0303.34046 · Zbl 0303.34046 · doi:10.1007/BF01646344 [7] A. Bove , G. Da Prato and G. Fano , On Hartree-Fock Time Dependent Problem , Comm. Math. Phys. , Vol. 49 , 1976 , pp. 25 - 33 . Article | MR 456066 · Zbl 0303.34046 [8] J.M. Chadam and R.T. Glassey , Global Existence of Solution to the Cauchy Problem for Time Dependent Hartree Equation , J. Math Phys. , Vol. 16 , 1975 , pp. 1122 - 1130 . MR 413843 | Zbl 0299.35084 · Zbl 0299.35084 · doi:10.1063/1.522642 [9] J.M. Chadam , The Time Dependent Hartree-Fock Equations with Coulomb Two-Body Interaction , Commun. Math. Phys. , Vol. 46 , 1976 , pp. 99 - 104 . Article | MR 411439 | Zbl 0322.35043 · Zbl 0322.35043 · doi:10.1007/BF01608490 [10] T. Cazenave and A. Haraux , Introduction aux problèmes d’évolution semilinéaires , Math. Appl. , 1 , Ellipses , Paris , 1990 . MR 1299976 | Zbl 0786.35070 · Zbl 0786.35070 [11] T. Cazenave and F.B. Weissler , Some remarks on the Nonlinear Schrödinger equation in the critical case , in Nonlinear Semigroups , Partial Differential equation and Attractors , T. L. GILL and W. W. ZACHARY Eds., Lect. Notes Math. , Vol. 1394 , Springer , 1989 , pp. 18 - 29 . MR 1021011 | Zbl 0694.35170 · Zbl 0694.35170 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.