We study the time evolution of a system of N spinless fermions in R 3 which interact through a pair potential, e.g., the Coulomb potential. We compare the dynamics given by the solution to Schrödinger's equation with the time-dependent Hartree-Fock approximation, and we give an estimate for the accuracy of this approximation in terms of the kinetic energy of the system. This leads, in turn, to bounds in terms of the initial total energy of the system. MSC class: 35Q40, 35Q55, 81Q05, 82C10

Building on the earlier work of Odeh, Barany, Golubitsky, Turski and Lasher we give a proof of existence of Abrikosov vortex lattices in the Ginzburg-Landau model of superconductivity.

We review recent works on statics and dynamics of magnetic vortices in the Ginzburg–Landau model of superconductivity and of Nielsen–Olesen (Nambu) strings in the Abelian–Higgs model of particle physics.

We consider Abrikosov-type vortex lattice solutions of the Ginzburg-Landau equations of superconductivity, consisting of single vortices, for magnetic fields below but close to the second critical magnetic field Hc2 = κ 2 and for superconductors filling the entire R 2 . Here κ is the Ginzburg-Landau parameter. The lattice shape, parameterized by τ , is allowed to be arbitrary (not just triangular or rectangular). Within the context of the time-dependent Ginzburg-Landau equations, called the Gorkov-Eliashberg-Schmidt equations, we prove that such lattices are asymptotically stable under gauge periodic perturbations for κ 2 > 1 2 (1 − 1 β(τ ) ) and unstable for κ 2 < 1 2 (1 − 1 β(τ ) ), where β(τ ) is the Abrikosov constant depending on the lattice shape τ . This result goes against the common belief among physicists and mathematicians that Abrikosov-type vortex lattice solutions are stable only for triangular lattices and κ 2 > 1 2 . (There is no real contradiction though as we consider very special perturbations.) *

Building on earlier work, we have given in [29] a proof of existence of Abrikosov vortex lattices in the Ginzburg-Landau model of superconductivity and shown that the triangular lattice gives the lowest energy per lattice cell. After [29] was published, we realized that it proves a stronger result than was stated there. This result is recorded in the present paper. The proofs remain the same as in [29], apart from some streamlining.