We study the nonlinear Schrödinger equation for systems of N orthonormal functions. We prove the existence of ground states for all N when the exponent p of the non linearity is not too large, and for an infinite sequence N j tending to infinity in the whole range of possible p's, in dimensions d ≥ 1. This allows us to prove that translational symmetry is broken for a quantum crystal in the Kohn-Sham model with a large Dirac exchange constant.

We develop a rigorous framework for modelling the geometry equilibration of crystalline defects. We formulate the equilibration of crystal defects as a variational problem on a discrete energy space and establish qualitatively sharp far-field decay estimates for the equilibrium configuration.This work extends [12] by admitting infinite-range interaction which in particular includes some quantum chemistry based interatomic interactions.2000 Mathematics Subject Classification. 65L20, 65L70, 70C20, 74G40, 74G65.

We establish a pointwise stability estimate for the Thomas-Fermi-von Weizsäcker (TFW) model, which demonstrates that a local perturbation of a nuclear arrangement results also in a local response in the electron density and electrostatic potential. The proof adapts the arguments for existence and uniqueness of solutions to the TFW equations in the thermodynamic limit by Catto et al. (The mathematical theory of thermodynamic limits: Thomas-Fermi type models. Oxford mathematical monographs. The Clarendon Press, Oxford University Press, New York, 1998). To demonstrate the utility of this combined locality and stability result we derive several consequences, including an exponential convergence rate for the thermodynamic limit, partition of total energy into exponentially localised site energies (and consequently, exponential locality of forces), and generalised and strengthened results on the charge neutrality of local defects.