25 research outputs found
Second order semiclassics with self-generated magnetic fields
We consider the semiclassical asymptotics of the sum of negative eigenvalues
of the three-dimensional Pauli operator with an external potential and a
self-generated magnetic field . We also add the field energy and we minimize over all magnetic fields. The parameter
effectively determines the strength of the field. We consider the weak field
regime with , where is the semiclassical
parameter. For smooth potentials we prove that the semiclassical asymptotics of
the total energy is given by the non-magnetic Weyl term to leading order with
an error bound that is smaller by a factor h^{1+\e}, i.e. the subleading term
vanishes. However, for potentials with a Coulomb singularity the subleading
term does not vanish due to the non-semiclassical effect of the singularity.
Combined with a multiscale technique, this refined estimate is used in the
companion paper \cite{EFS3} to prove the second order Scott correction to the
ground state energy of large atoms and molecules.Comment: Small typos corrected on Sep 24, 201
Scott correction for large atoms and molecules in a self-generated magnetic field
We consider a large neutral molecule with total nuclear charge in
non-relativistic quantum mechanics with a self-generated classical
electromagnetic field. To ensure stability, we assume that Z\al^2\le \kappa_0
for a sufficiently small , where \al denotes the fine structure
constant. We show that, in the simultaneous limit , \al\to 0 such
that \kappa =Z\al^2 is fixed, the ground state energy of the system is given
by a two term expansion . The leading
term is given by the non-magnetic Thomas-Fermi theory. Our result shows that
the magnetic field affects only the second (so-called Scott) term in the
expansion
Gradient corrections for semiclassical theories of atoms in strong magnetic fields
This paper is divided into two parts. In the first one the von Weizs\"acker
term is introduced to the Magnetic TF theory and the resulting MTFW functional
is mathematically analyzed. In particular, it is shown that the von
Weizs\"acker term produces the Scott correction up to magnetic fields of order
, in accordance with a result of V. Ivrii on the quantum mechanical
ground state energy. The second part is dedicated to gradient corrections for
semiclassical theories of atoms restricted to electrons in the lowest Landau
band. We consider modifications of the Thomas-Fermi theory for strong magnetic
fields (STF), i.e. for . The main modification consists in replacing
the integration over the variables perpendicular to the field by an expansion
in angular momentum eigenfunctions in the lowest Landau band. This leads to a
functional (DSTF) depending on a sequence of one-dimensional densities. For a
one-dimensional Fermi gas the analogue of a Weizs\"acker correction has a
negative sign and we discuss the corresponding modification of the DSTF
functional.Comment: Latex2e, 36 page
Scaling Limits for the System of Semi-Relativistic Particles Coupled to a Scalar Bose Field
In this paper the Hamiltonian for the system of semi-relativistic particles
interacting with a scalar bose field is investigated. A scaled total
Hamiltonian of the system is defined and its scaling limit is considered. Then
the semi-relativistic Schrodinger operator with an effective potential is
derived
The excitation spectrum for weakly interacting bosons in a trap
We investigate the low-energy excitation spectrum of a Bose gas confined in a
trap, with weak long-range repulsive interactions. In particular, we prove that
the spectrum can be described in terms of the eigenvalues of an effective
one-particle operator, as predicted by the Bogoliubov approximation.Comment: LaTeX, 32 page
Unique Solutions to Hartree-Fock Equations for Closed Shell Atoms
In this paper we study the problem of uniqueness of solutions to the Hartree
and Hartree-Fock equations of atoms. We show, for example, that the
Hartree-Fock ground state of a closed shell atom is unique provided the atomic
number is sufficiently large compared to the number of electrons. More
specifically, a two-electron atom with atomic number has a unique
Hartree-Fock ground state given by two orbitals with opposite spins and
identical spatial wave functions. This statement is wrong for some , which
exhibits a phase segregation.Comment: 18 page
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
The Second Order Upper Bound for the Ground Energy of a Bose Gas
Consider bosons in a finite box
interacting via a two-body smooth repulsive short range potential. We construct
a variational state which gives the following upper bound on the ground state
energy per particle where is the scattering
length of the potential. Previously, an upper bound of the form
for some constant was obtained in \cite{ESY}. Our result proves the
upper bound of the the prediction by Lee-Yang \cite{LYang} and Lee-Huang-Yang
\cite{LHY}.Comment: 62 pages, no figure
Monotonicity of quantum ground state energies: Bosonic atoms and stars
The N-dependence of the non-relativistic bosonic ground state energy is
studied for quantum N-body systems with either Coulomb or Newton interactions.
The Coulomb systems are "bosonic atoms," with their nucleus fixed, and the
Newton systems are "bosonic stars". In either case there exists some third
order polynomial in N such that the ratio of the ground state energy to the
respective polynomial grows monotonically in N. Some applications of these new
monotonicity results are discussed