146 research outputs found
One non-relativistic particle coupled to a photon field
We investigate the ground state energy of an electron coupled to a photon
field. First, we regard the self-energy of a free electron, which we describe
by the Pauli-Fierz Hamiltonian. We show that, in the case of small values of
the coupling constant , the leading order term is represented by
.
Next we put the electron in the field of an arbitrary external potential ,
such that the corresponding Schr\"odinger operator has at least one
eigenvalue, and show that by coupling to the radiation field the binding energy
increases, at least for small enough values of the coupling constant .
Moreover, we provide concrete numbers for , the ultraviolet cut-off
, and the radiative correction for which our procedure works.Comment: final version, to appear in Ann. Henri Poincar
Non-Perturbative Mass and Charge Renormalization in Relativistic No-Photon Quantum Electrodynamics
Starting from a formal Hamiltonian as found in the physics literature --
omitting photons -- we define a renormalized Hamiltonian through charge and
mass renormalization. We show that the restriction to the one-electron subspace
is well-defined. Our construction is non-perturbative and does not use a
cut-off. The Hamiltonian is relevant for the description of the Lamb shift in
muonic atoms.Comment: Reformulation of main theorem, minor changes in the proo
Comparing the full time-dependent Bogoliubov--de-Gennes equations to their linear approximation: A numerical investigation
In this paper we report on the results of a numerical study of the nonlinear
time-dependent Bardeen--Cooper--Schrieffer (BCS) equations, often also denoted
as Bogoliubov--de--Gennes (BdG) equations, for a one-dimensional system of
fermions with contact interaction. We show that, even above the critical
temperature, the full equations and their linear approximation give rise to
completely different evolutions. In contrast to its linearization, the full
nonlinear equation does not show any diffusive behavior in the order parameter.
This means that the order parameter does not follow a Ginzburg--Landau-type of
equation, in accordance with a recent theoretical results. We include a full
description on the numerical implementation of the partial differential BCS\
BdG equations.Comment: 10 pages, to appear in EP
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
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