Solution is presented for a Hartree-Fock problem emerging, e.g.,
in quantum dots. In the model, a beam of infinitely many spinless
particles (obeying Pauli principle via incorporation of an
explicit form of correction terms) is exposed to a special 
time-independent external electromagnetic field. In a formalism
using density matrix, an anisotropy of the Hartree's self-consistent 
potential U is admitted. The assumption of quasi-periodicity
makes this problem tractable by a suitable ansatz.  In the first
half of the text the system is assumed two-dimensional.  The 
energy functional is rigorously proved bounded
from below.  The wave functions ``of the beam" which minimize it
are shown to satisfy the corresponding non-linear coupled set of
Schroedinger-like equations in distributional sense.  The second
half of the text is devoted to a three-dimensional analogue of
this result, somewhat more artificial in requiring an
electrostatic confinement in the third dimension of course, and
somewhat more interesting as admitting even a weak Coulombic
attraction. The authors characterize the latter ``mathematical
curiosity" as having ``(probably) no physical implications" but
I cannot resist recalling a paper [C. M. Bender and K. Milton,
J. Phys. A: Math. Gen. 32 (1999) L87] where precisely the
possibility of the existence of Coulombic attraction of equal
(i.e., formally, purely imaginary) charges has been advocated 
as emerging within the so called PT symmetric quantum electrodynamics. 



CNO: 1879831
Dolbeault, J. ; Illner, R. ; Lange, H. On asymmetric quasiperiodic 
solutions of Hartree-Fock systems. J. Differential 
Equations 178 (2002), no. 2, 314--324. 

http://www.ams.org/cgi-bin/mresubs/mresubs.pl?readfile=1879831.pdf