190 research outputs found
Renormalization and asymptotic expansion of Dirac's polarized vacuum
We perform rigorously the charge renormalization of the so-called reduced
Bogoliubov-Dirac-Fock (rBDF) model. This nonlinear theory, based on the Dirac
operator, describes atoms and molecules while taking into account vacuum
polarization effects. We consider the total physical density including both the
external density of a nucleus and the self-consistent polarization of the Dirac
sea, but no `real' electron. We show that it admits an asymptotic expansion to
any order in powers of the physical coupling constant \alphaph, provided that
the ultraviolet cut-off behaves as \Lambda\sim e^{3\pi(1-Z_3)/2\alphaph}\gg1.
The renormalization parameter $
A new approach to the modelling of local defects in crystals: the reduced Hartree-Fock case
This article is concerned with the derivation and the mathematical study of a
new mean-field model for the description of interacting electrons in crystals
with local defects. We work with a reduced Hartree-Fock model, obtained from
the usual Hartree-Fock model by neglecting the exchange term. First, we recall
the definition of the self-consistent Fermi sea of the perfect crystal, which
is obtained as a minimizer of some periodic problem, as was shown by Catto, Le
Bris and Lions. We also prove some of its properties which were not mentioned
before. Then, we define and study in details a nonlinear model for the
electrons of the crystal in the presence of a defect. We use formal analogies
between the Fermi sea of a perturbed crystal and the Dirac sea in Quantum
Electrodynamics in the presence of an external electrostatic field. The latter
was recently studied by Hainzl, Lewin, S\'er\'e and Solovej, based on ideas
from Chaix and Iracane. This enables us to define the ground state of the
self-consistent Fermi sea in the presence of a defect. We end the paper by
proving that our model is in fact the thermodynamic limit of the so-called
supercell model, widely used in numerical simulations.Comment: Final version, to appear in Comm. Math. Phy
Binding threshold for the Pauli-Fierz operator
For the Pauli-Fierz operator with a short range potential we study the
binding threshold as a function of the fine structure constant and
show that it converges to the binding threshold for the Schr\"odinger operator
in the small limit
Existence of global-in-time solutions to a generalized Dirac-Fock type evolution equation
We consider a generalized Dirac-Fock type evolution equation deduced from
no-photon Quantum Electrodynamics, which describes the self-consistent
time-evolution of relativistic electrons, the observable ones as well as those
filling up the Dirac sea. This equation has been originally introduced by Dirac
in 1934 in a simplified form. Since we work in a Hartree-Fock type
approximation, the elements describing the physical state of the electrons are
infinite rank projectors. Using the Bogoliubov-Dirac-Fock formalism, introduced
by Chaix-Iracane ({\it J. Phys. B.}, 22, 3791--3814, 1989), and recently
established by Hainzl-Lewin-Sere, we prove the existence of global-in-time
solutions of the considered evolution equation.Comment: 12 pages; more explanations added, some final (minor) corrections
include
Non-perturbative embedding of local defects in crystalline materials
We present a new variational model for computing the electronic first-order
density matrix of a crystalline material in presence of a local defect. A
natural way to obtain variational discretizations of this model is to expand
the difference Q between the density matrix of the defective crystal and the
density matrix of the perfect crystal, in a basis of precomputed maximally
localized Wannier functions of the reference perfect crystal. This approach can
be used within any semi-empirical or Density Functional Theory framework.Comment: 13 pages, 4 figure
Self-consistent solution for the polarized vacuum in a no-photon QED model
We study the Bogoliubov-Dirac-Fock model introduced by Chaix and Iracane
({\it J. Phys. B.}, 22, 3791--3814, 1989) which is a mean-field theory deduced
from no-photon QED. The associated functional is bounded from below. In the
presence of an external field, a minimizer, if it exists, is interpreted as the
polarized vacuum and it solves a self-consistent equation.
In a recent paper math-ph/0403005, we proved the convergence of the iterative
fixed-point scheme naturally associated with this equation to a global
minimizer of the BDF functional, under some restrictive conditions on the
external potential, the ultraviolet cut-off and the bare fine
structure constant . In the present work, we improve this result by
showing the existence of the minimizer by a variational method, for any cut-off
and without any constraint on the external field.
We also study the behaviour of the minimizer as goes to infinity
and show that the theory is "nullified" in that limit, as predicted first by
Landau: the vacuum totally kills the external potential. Therefore the limit
case of an infinite cut-off makes no sense both from a physical and
mathematical point of view.
Finally, we perform a charge and density renormalization scheme applying
simultaneously to all orders of the fine structure constant , on a
simplified model where the exchange term is neglected.Comment: Final version, to appear in J. Phys. A: Math. Ge
Free Energy of a Dilute Bose Gas: Lower Bound
A lower bound is derived on the free energy (per unit volume) of a
homogeneous Bose gas at density and temperature . In the dilute
regime, i.e., when , where denotes the scattering length of
the pair-interaction potential, our bound differs to leading order from the
expression for non-interacting particles by the term . Here, denotes the critical density for
Bose-Einstein condensation (for the non-interacting gas), and denotes
the positive part. Our bound is uniform in the temperature up to temperatures
of the order of the critical temperature, i.e., or smaller.
One of the key ingredients in the proof is the use of coherent states to extend
the method introduced in [arXiv:math-ph/0601051] for estimating correlations to
temperatures below the critical one.Comment: LaTeX2e, 53 page
A simple method for finite range decomposition of quadratic forms and Gaussian fields
We present a simple method to decompose the Green forms corresponding to a
large class of interesting symmetric Dirichlet forms into integrals over
symmetric positive semi-definite and finite range (properly supported) forms
that are smoother than the original Green form. This result gives rise to
multiscale decompositions of the associated Gaussian free fields into sums of
independent smoother Gaussian fields with spatially localized correlations. Our
method makes use of the finite propagation speed of the wave equation and
Chebyshev polynomials. It improves several existing results and also gives
simpler proofs.Comment: minor correction for t<
A Minimization Method for Relativistic Electrons in a Mean-Field Approximation of Quantum Electrodynamics
We study a mean-field relativistic model which is able to describe both the
behavior of finitely many spin-1/2 particles like electrons and of the Dirac
sea which is self-consistently polarized in the presence of the real particles.
The model is derived from the QED Hamiltonian in Coulomb gauge neglecting the
photon field. All our results are non-perturbative and mathematically rigorous.Comment: 18 pages, 3 figure
Ground State and Resonances in the Standard Model of Non-relativistic QED
We prove existence of a ground state and resonances in the standard model of
the non-relativistic quantum electro-dynamics (QED). To this end we introduce a
new canonical transformation of QED Hamiltonians and use the spectral
renormalization group technique with a new choice of Banach spaces.Comment: 50 pages change
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