arXiv subjects

Virginie Ehrlacher

Publications and source records attributed to Virginie Ehrlacher.

57 records · Page 4Linked to original sources

arXiv2011

Periodic Schrödinger operators with local defects and spectral pollution

This article deals with the numerical calculation of eigenvalues of perturbed periodic Schrödinger operators located in spectral gaps. Such operators are encountered in the modeling of the electronic structure of crystals with local defects, and of photonic crystals. The usual finite element Galerkin approximation is known to give rise to spectral pollution. In this article, we give a precise description of the corresponding spurious states. We then prove that the supercell model does not produce spectral pollution. Lastly, we extend results by Lewin and Séré on some no-pollution criteria. In particular, we prove that using approximate spectral projectors enables one to eliminate spectral pollution in a given spectral gap of the reference periodic Schödinger operator.

math-ph↗

arXiv2011

Convergence of a greedy algorithm for high-dimensional convex nonlinear problems

In this article, we present a greedy algorithm based on a tensor product decomposition, whose aim is to compute the global minimum of a strongly convex energy functional. We prove the convergence of our method provided that the gradient of the energy is Lipschitz on bounded sets. The main interest of this method is that it can be used for high-dimensional nonlinear convex problems. We illustrate this method on a prototypical example for uncertainty propagation on the obstacle problem.

math.FA↗

arXiv2010

Local defects are always neutral in the Thomas-Fermi-von Weiszäcker theory of crystals

The aim of this article is to propose a mathematical model describing the electronic structure of crystals with local defects in the framework of the Thomas-Fermi-von Weizsäcker (TFW) theory. The approach follows the same lines as that used in {\it E. Cancès, A. Deleurence and M. Lewin, Commun. Math. Phys., 281 (2008), pp. 129--177} for the reduced Hartree-Fock model, and is based on thermodynamic limit arguments. We prove in particular that it is not possible to model charged defects within the TFW theory of crystals. We finally derive some additional properties of the TFW ground state electronic density of a crystal with a local defect, in the special case when the host crystal is modelled by a homogeneous medium.

math-ph↗