arXiv subjects

Faizan Q. Nazar

Publications and source records attributed to Faizan Q. Nazar.

4 recordsLinked to original sources

arXiv2021

The nonlinear Schrödinger equation for orthonormal functions: I. Existence of ground states

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 $d\geq1$. 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.

math.AP↗

arXiv2018

Geometry Equilibration of Crystalline Defects in Quantum and Atomistic Descriptions

We develop a rigorous framework for modelling the geometry equilibration of crystalline defects. We formulate the equilibration of crystal defects as a variational problems on a discrete energy space and establish qualitatively sharp far-field decay estimates for the equilibrium configuration. This work extends Ehrlacher, Ortner, Shapeev (2016) by admitting infinite-range interaction which in particular includes some quantum chemistry based interatomic potentials.

math.NA↗

arXiv2016

Locality of the Thomas-Fermi-von Weizsäcker Equations

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. (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.

math-ph↗

arXiv2016

Convergence Rates from Yukawa to Coulomb Interaction in the Thomas-Fermi-von Weizsäcker Model

We establish uniform convergence, with explicit rate, of the solution to the Thomas--Fermi--von Weizsäcker (TFW) Yukawa model to the solution of the TFW Coulomb model, for general condensed nuclear configurations. As a consequence, we show the convergence of forces from the Yukawa to the Coulomb model. These results rely on an extension of Nazar \& Ortner (2015) to the Yukawa setting. Auxiliary results of independent interest shown also include new existence, uniqueness and stability results for the Yukawa ground state.

math-ph↗