4,663 research outputs found
Proof of Bose-Einstein Condensation for Dilute Trapped Gases
The ground state of bosonic atoms in a trap has been shown experimentally to
display Bose-Einstein condensation (BEC). We prove this fact theoretically for
bosons with two-body repulsive interaction potentials in the dilute limit,
starting from the basic Schroedinger equation; the condensation is 100% into
the state that minimizes the Gross-Pitaevskii energy functional. This is the
first rigorous proof of BEC in a physically realistic, continuum model.Comment: Revised version with some simplifications and clarifications. To
appear in Phys. Rev. Let
Ground State Asymptotics of a Dilute, Rotating Gas
We investigate the ground state properties of a gas of interacting particles
confined in an external potential in three dimensions and subject to rotation
around an axis of symmetry. We consider the so-called Gross-Pitaevskii (GP)
limit of a dilute gas. Analyzing both the absolute and the bosonic ground state
of the system we show, in particular, their different behavior for a certain
range of parameters. This parameter range is determined by the question whether
the rotational symmetry in the minimizer of the GP functional is broken or not.
For the absolute ground state, we prove that in the GP limit a modified GP
functional depending on density matrices correctly describes the energy and
reduced density matrices, independent of symmetry breaking. For the bosonic
ground state this holds true if and only if the symmetry is unbroken.Comment: LaTeX2e, 37 page
Stability of Matter in Magnetic Fields
In the presence of arbitrarily large magnetic fields, matter composed of
electrons and nuclei was known to be unstable if or is too large.
Here we prove that matter {\it is stable\/} if and
.Comment: 10 pages, LaTe
The TF Limit for Rapidly Rotating Bose Gases in Anharmonic Traps
Starting from the full many body Hamiltonian we derive the leading order
energy and density asymptotics for the ground state of a dilute, rotating Bose
gas in an anharmonic trap in the ` Thomas Fermi' (TF) limit when the
Gross-Pitaevskii coupling parameter and/or the rotation velocity tend to
infinity. Although the many-body wave function is expected to have a
complicated phase, the leading order contribution to the energy can be computed
by minimizing a simple functional of the density alone
The Ground States of Large Quantum Dots in Magnetic Fields
The quantum mechanical ground state of a 2D -electron system in a
confining potential ( is a coupling constant) and a homogeneous
magnetic field is studied in the high density limit , with fixed. It is proved that the ground state energy and
electronic density can be computed {\it exactly} in this limit by minimizing
simple functionals of the density. There are three such functionals depending
on the way varies as : A 2D Thomas-Fermi (TF) theory applies
in the case ; if the correct limit theory
is a modified -dependent TF model, and the case is described
by a ``classical'' continuum electrostatic theory. For homogeneous potentials
this last model describes also the weak coupling limit for arbitrary
. Important steps in the proof are the derivation of a new Lieb-Thirring
inequality for the sum of eigenvalues of single particle Hamiltonians in 2D
with magnetic fields, and an estimation of the exchange-correlation energy. For
this last estimate we study a model of classical point charges with
electrostatic interactions that provides a lower bound for the true quantum
mechanical energy.Comment: 57 pages, Plain tex, 5 figures in separate uufil
Stability and Instability of Relativistic Electrons in Classical Electro magnetic Fields
The stability of matter composed of electrons and static nuclei is
investigated for a relativistic dynamics for the electrons given by a suitably
projected Dirac operator and with Coulomb interactions. In addition there is an
arbitrary classical magnetic field of finite energy. Despite the previously
known facts that ordinary nonrelativistic matter with magnetic fields, or
relativistic matter without magnetic fields is already unstable when the fine
structure constant, is too large it is noteworthy that the combination of the
two is still stable provided the projection onto the positive energy states of
the Dirac operator, which defines the electron, is chosen properly. A good
choice is to include the magnetic field in the definition. A bad choice, which
always leads to instability, is the usual one in which the positive energy
states are defined by the free Dirac operator. Both assertions are proved here.Comment: LaTeX fil
Exponential localization of hydrogen-like atoms in relativistic quantum electrodynamics
We consider two different models of a hydrogenic atom in a quantized
electromagnetic field that treat the electron relativistically. The first one
is a no-pair model in the free picture, the second one is given by the
semi-relativistic Pauli-Fierz Hamiltonian. We prove that the no-pair operator
is semi-bounded below and that its spectral subspaces corresponding to energies
below the ionization threshold are exponentially localized. Both results hold
true, for arbitrary values of the fine-structure constant, , and the
ultra-violet cut-off, , and for all nuclear charges less than the
critical charge without radiation field, . We obtain
similar results for the semi-relativistic Pauli-Fierz operator, again for all
values of and and for nuclear charges less than .Comment: 37 page
Effect of electronic interactions on the persistent current in one-dimensional disordered rings
The persistent current is here studied in one-dimensional disordered rings
that contain interacting electrons. We used the density matrix renormalization
group algorithms in order to compute the stiffness, a measure that gives the
magnitude of the persistent currents as a function of the boundary conditions
for different sets of both interaction and disorder characteristics. In
contrast to its non-interacting value, an increase in the stiffness parameter
was observed for systems at and off half-filling for weak interactions and
non-zero disorders. Within the strong interaction limit, the decrease in
stiffness depends on the filling and an analytical approach is developed to
recover the observed behaviors. This is required in order to understand its
mechanisms. Finally, the study of the localization length confirms the
enhancement of the persistent current for moderate interactions when disorders
are present at half-filling. Our results reveal two different regimes, one for
weak and one for strong interactions at and off half-filling.Comment: 16 pages, 21 figures; minor changes (blanks missing, sentences
starting with a mathematical symbol
Justification of c-Number Substitutions in Bosonic Hamiltonians
The validity of substituting a c-number for the mode operator
is established rigorously in full generality, thereby verifying one aspect of
Bogoliubov's 1947 theory. This substitution not only yields the correct value
of thermodynamic quantities like the pressure or ground state energy, but also
the value of that maximizes the partition function equals the true
amount of condensation in the presence of a gauge-symmetry breaking term -- a
point that had previously been elusive.Comment: RevTeX4, 4pages; minor modifications in the text; final version, to
appear in Phys. Rev. Let
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
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