3,381 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
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
Structure and enumeration of (3+1)-free posets
A poset is (3+1)-free if it does not contain the disjoint union of chains of
length 3 and 1 as an induced subposet. These posets play a central role in the
(3+1)-free conjecture of Stanley and Stembridge. Lewis and Zhang have
enumerated (3+1)-free posets in the graded case by decomposing them into
bipartite graphs, but until now the general enumeration problem has remained
open. We give a finer decomposition into bipartite graphs which applies to all
(3+1)-free posets and obtain generating functions which count (3+1)-free posets
with labelled or unlabelled vertices. Using this decomposition, we obtain a
decomposition of the automorphism group and asymptotics for the number of
(3+1)-free posets.Comment: 28 pages, 5 figures. New version includes substantial changes to
clarify the construction of skeleta and the enumeration. An extended abstract
of this paper appears as arXiv:1212.535
Goodwillie's Calculus of Functors and Higher Topos Theory
We develop an approach to Goodwillie's calculus of functors using the
techniques of higher topos theory. Central to our method is the introduction of
the notion of fiberwise orthogonality, a strengthening of ordinary
orthogonality which allows us to give a number of useful characterizations of
the class of -excisive maps. We use these results to show that the pushout
product of a -equivalence with a -equivalence is a
-equivalence. Then, building on our previous work, we prove a
Blakers-Massey type theorem for the Goodwillie tower. We show how to use the
resulting techniques to rederive some foundational theorems in the subject,
such as delooping of homogeneous functors.Comment: 40 pages, (a slightly modified version of) this paper is accepted for
publication by the Journal of Topolog
- …