On Hartree-Fock Systems

We show an existence and uniqueness result for mildly nonlinear Schr6dinger systems of (self-consistent) Hartree-Fock form. We also shortly resume the already existing results on the semiclassical limit and the asymptotic and dispersive behavior of such systems.


INTRODUCTION
We consider Hartree-Fock systems in d of the Hartree-Fock systems are considered an accurate description of the quantum-mechanical evolution of a Fermion system, since their derivation from many body physics takes into account the 357 Pauli exclusion principle [16], which is not the case for Hartree systems (obtained by setting Vj 0). An existence analysis for the three dimensional Coulomb case can be found in [2]. The corresponding long time behavior was analysed in [3].
In Section 2 we give an existence and uniqueness result for the system (1.1) for very general interaction potentials in any space dimension and for both the attractive and repulsive case. Section 3 is a summary of already existing results [4] on the classical limit and the asymptotic and dispersive behavior of the system (1.1).  [2] and the existence proof of the corresponding Hartree case [1,7]. In the following we skip the superscript c since c is a fixed parameter in this section. We make the following assumptions (A 1) (i) V e N" A > 0; (ii) C > 0" On the external potential we assume and on the interaction potential: We anticipate some lemmas we need in the proof of the above theorem.
We denote by A the operator The assumption (A2) on Vu suffices to state In the first case, considering the weak L part of the interaction potential U only and using the generalized Young inequality we have (2.14) for all ( for some 2 < p <_ (2d/d-2). The same holds for the strong L part of the interaction potential. Therefore, not only the total energy is conserved, but also the kinetic and potential energies are bounded uniformly in time and as a consequence ]l(t)l] r < oo, Vt>O.

OTHER PROPERTIES OF HARTREE-FOCK SYSTEMS
In this section we collect other interesting properties of the system (1.1). We only give a description of the results. Details of the following results can be found in [4]. The superscript e is important in this section, especially in the first part. The results are valid for d > l, for d special assumptions are needed (see [4]).

The Classical Limit
In this section we describe the results on the classical limit of the Hartree-Fock systems. The appropriate formulation to perform the classical limit is the Wigner formalism, which we shortly describe.
For a given potential V V(x) the pseudodifferential operator O[V] is defined by (3.4a) where v? denotes the inverse Fourier transform of w w(x, v) with respect to v" W(X, v)e -iv'r dv. (x,) (3.4b) fe is the (quadratically) nonlinear operator Under additional assumptions on the interaction potential (and on its gradient) and on the initial data it is possible to carry out the limit c 0. THEOREM 3.1 Let (A 1), (A2), (A3) of [4] hold.
Then, for every sequence --0 there exists a subsequence (denoted by the same symbol) such that wi w >_ 0 inL2(x av) weakly, (3.6a) The contribution of the exchange correlation part originating from the Pauli principle does not give any contribution in the classical limit in which no such principle exists. Therefore, the result is physically reasonable and expected.
In three dimensions with Coulomb interaction the limiting classical problem is the Vlasov-Poisson system, which also represents the limiting problem if we start from the Hartree system only (without exchange correlation part) [8,9,12].
Note that results along the lines of the ones presented (and in the more general in [4]) entirely based on the Schr6dinger formalism restricted to the 3d Coulomb case and finitely many coupled states can be found in [3,13]. Decay results for the Hartree case with Coulomb interaction can be found in [7] for all -oc < T1 < T2 < o. Integral identities of this type were obtained in [10] for the free transport equation, in [13] for the Vlasov-Poisson and Wigner-Poisson systems and in [6] for a large class of equations.
A lengthy calculation shows that the first term on the left hand side of (3.8) is nonnegative. For example in the case d 3 and 0 it is equal to 1=1 I(x (x, t)I ) dx (3.9) (see [10] also for the other cases). Now let's assume that VE=0 (no exterior field) and that the interaction potential is radial U U0(lxl) with Uo(r) <_ O. Then, an easy calculation using (x, z, t) 9 (z, x, t) and (2.17) shows that also the third term in (3.8) is nonnegative. Thus, the identity (3.8) gives the bound for the first term on its left hand side: Energy conservation shows that IIJ(t)ll,(g/ is uniformly bounded in e and t. Thus, we conclude for d 3 and all x0 E N3: (just as for the free Schr6dinger equation). Similar estimates can be obtained for dimensions different from 3.