Spurious Modes in Dirac Calculations and How to Avoid Them

In this paper we consider the problem of the occurrence of spurious modes when computing the eigenvalues of Dirac operators, with the motivation to describe relativistic electrons in an atom or a molecule. We present recent mathematical results which we illustrate by simple numerical experiments. We also discuss open problems.Comment: Chapter to be published in the book "Many-Electron Approaches in Physics, Chemistry and Mathematics: A Multidisciplinary View", edited by Volker Bach and Luigi Delle Sit

Spectral Pollution and How to Avoid It (With Applications to Dirac and Periodic Schr\"odinger Operators)

This paper, devoted to the study of spectral pollution, contains both abstract results and applications to some self-adjoint operators with a gap in their essential spectrum occuring in Quantum Mechanics. First we consider Galerkin basis which respect the decomposition of the ambient Hilbert space into a direct sum $H=PH\oplus(1-P)H$, given by a fixed orthogonal projector $P$, and we localize the polluted spectrum exactly. This is followed by applications to periodic Schr\"odinger operators (pollution is absent in a Wannier-type basis), and to Dirac operator (several natural decompositions are considered). In the second part, we add the constraint that within the Galerkin basis there is a certain relation between vectors in $PH$ and vectors in $(1-P)H$. Abstract results are proved and applied to several practical methods like the famous "kinetic balance" of relativistic Quantum Mechanics.Comment: Proceedings of the London Mathematical Society (2009) in pres

The Microscopic Origin of the Macroscopic Dielectric Permittivity of Crystals: A Mathematical Viewpoint

The purpose of this paper is to provide a mathematical analysis of the Adler-Wiser formula relating the macroscopic relative permittivity tensor to the microscopic structure of the crystal at the atomic level. The technical level of the presentation is kept at its minimum to emphasize the mathematical structure of the results. We also briefly review some models describing the electronic structure of finite systems, focusing on density operator based formulations, as well as the Hartree model for perfect crystals or crystals with a defect.Comment: Proceedings of the Workshop "Numerical Analysis of Multiscale Computations" at Banff International Research Station, December 200

Existence of Atoms and Molecules in the Mean-Field Approximation of No-Photon Quantum Electrodynamics

The Bogoliubov-Dirac-Fock (BDF) model is the mean-field approximation of no-photon Quantum Electrodynamics. The present paper is devoted to the study of the minimization of the BDF energy functional under a charge constraint. An associated minimizer, if it exists, will usually represent the ground state of a system of $N$ electrons interacting with the Dirac sea, in an external electrostatic field generated by one or several fixed nuclei. We prove that such a minimizer exists when a binding (HVZ-type) condition holds. We also derive, study and interpret the equation satisfied by such a minimizer. Finally, we provide two regimes in which the binding condition is fulfilled, obtaining the existence of a minimizer in these cases. The first is the weak coupling regime for which the coupling constant $\alpha$ is small whereas $\alpha Z$ and the particle number $N$ are fixed. The second is the non-relativistic regime in which the speed of light tends to infinity (or equivalently $\alpha$ tends to zero) and $Z$, $N$ are fixed. We also prove that the electronic solution converges in the non-relativistic limit towards a Hartree-Fock ground state.Comment: Final version, to appear in Arch. Rat. Mech. Ana

New Global Minima for Thomson's Problem of Charges on a Sphere

Using numerical arguments we find that for $N$ = 306 a tetrahedral configuration ($T_h$) and for N=542 a dihedral configuration ($D_5$) are likely the global energy minimum for Thomson's problem of minimizing the energy of $N$ unit charges on the surface of a unit conducting sphere. These would be the largest $N$ by far, outside of the icosadeltahedral series, for which a global minimum for Thomson's problem is known. We also note that the current theoretical understanding of Thomson's problem does not rule out a symmetric configuration as the global minima for N=306 and 542. We explicitly find that analogues of the tetrahedral and dihedral configurations for $N$ larger than 306 and 542, respectively, are not global minima, thus helping to confirm the theory of Dodgson and Moore (Phys. Rev. B 55, 3816 (1997)) that as $N$ grows dislocation defects can lower the lattice strain of symmetric configurations and concomitantly the energy. As well, making explicit previous work by ourselves and others, for $N<1000$ we give a full accounting of icosadeltahedral configuration which are not global minima and those which appear to be, and discuss how this listing and our results for the tetahedral and dihedral configurations may be used to refine theoretical understanding of Thomson's problem.Comment: 1- Manuscript revised. 2- A new global minimum found for a dihedral (D_5) configuration found for N=54

Variational methods in relativistic quantum mechanics

This review is devoted to the study of stationary solutions of linear and nonlinear equations from relativistic quantum mechanics, involving the Dirac operator. The solutions are found as critical points of an energy functional. Contrary to the Laplacian appearing in the equations of nonrelativistic quantum mechanics, the Dirac operator has a negative continuous spectrum which is not bounded from below. This has two main consequences. First, the energy functional is strongly indefinite. Second, the Euler-Lagrange equations are linear or nonlinear eigenvalue problems with eigenvalues lying in a spectral gap (between the negative and positive continuous spectra). Moreover, since we work in the space domain R^3, the Palais-Smale condition is not satisfied. For these reasons, the problems discussed in this review pose a challenge in the Calculus of Variations. The existence proofs involve sophisticated tools from nonlinear analysis and have required new variational methods which are now applied to other problems