507 research outputs found
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 , given by a fixed orthogonal projector
, 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 and vectors in
. 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 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 is small whereas
and the particle number are fixed. The second is the
non-relativistic regime in which the speed of light tends to infinity (or
equivalently tends to zero) and , 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 = 306 a tetrahedral
configuration () and for N=542 a dihedral configuration () are likely
the global energy minimum for Thomson's problem of minimizing the energy of
unit charges on the surface of a unit conducting sphere. These would be the
largest 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 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 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 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
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