3,218 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
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
Sharp constants in several inequalities on the Heisenberg group
We derive the sharp constants for the inequalities on the Heisenberg group
H^n whose analogues on Euclidean space R^n are the well known
Hardy-Littlewood-Sobolev inequalities. Only one special case had been known
previously, due to Jerison-Lee more than twenty years ago. From these
inequalities we obtain the sharp constants for their duals, which are the
Sobolev inequalities for the Laplacian and conformally invariant fractional
Laplacians. By considering limiting cases of these inequalities sharp constants
for the analogues of the Onofri and log-Sobolev inequalities on H^n are
obtained. The methodology is completely different from that used to obtain the
R^n inequalities and can be (and has been) used to give a new, rearrangement
free, proof of the HLS inequalities.Comment: 30 pages; addition of Corollary 2.3 and some minor changes; to appear
in Annals of Mathematic
Norms of quantum Gaussian multi-mode channels
We compute the norm of a general Gaussian
gauge-covariant multi-mode channel for any , where is a Schatten space. As a consequence, we verify the Gaussian optimizer
conjecture and the multiplicativity conjecture in these cases.Comment: 9 pages; minor changes; to appear in J. Math. Phy
A compactness lemma and its application to the existence of minimizers for the liquid drop model
The ancient Gamow liquid drop model of nuclear energies has had a renewed
life as an interesting problem in the calculus of variations: Find a set
with given volume A that minimizes the sum of its
surface area and its Coulomb self energy. A ball minimizes the former and
maximizes the latter, but the conjecture is that a ball is always a minimizer
-- when there is a minimizer. Even the existence of minimizers for this
interesting geometric problem has not been shown in general. We prove the
existence of the absolute minimizer (over all ) of the energy divided by
(the binding energy per particle). A second result of our work is a general
method for showing the existence of optimal sets in geometric minimization
problems, which we call the `method of the missing mass'. A third point is the
extension of the pulling back compactness lemma from to .Comment: 16 page
Stability of Relativistic Matter With Magnetic Fields
Stability of matter with Coulomb forces has been proved for non-relativistic
dynamics, including arbitrarily large magnetic fields, and for relativistic
dynamics without magnetic fields. In both cases stability requires that the
fine structure constant alpha be not too large. It was unclear what would
happen for both relativistic dynamics and magnetic fields, or even how to
formulate the problem clearly. We show that the use of the Dirac operator
allows both effects, provided the filled negative energy `sea' is defined
properly. The use of the free Dirac operator to define the negative levels
leads to catastrophe for any alpha, but the use of the Dirac operator with
magnetic field leads to stability.Comment: This is an announcement of the work in cond-mat/9610195 (LaTeX
Inversion positivity and the sharp Hardy-Littlewood-Sobolev inequality
We give a new proof of certain cases of the sharp HLS inequality. Instead of
symmetric decreasing rearrangement it uses the reflection positivity of
inversions in spheres. In doing this we extend a characterization of the
minimizing functions due to Li and Zhu.Comment: 15 pages; references added and minor change
A Rigorous Derivation of the Gross-Pitaevskii Energy Functional for a Two-Dimensional Bose Gas
We consider the ground state properties of an inhomogeneous two-dimensional
Bose gas with a repulsive, short range pair interaction and an external
confining potential. In the limit when the particle number is large but
is small, where is the average particle density and
the scattering length, the ground state energy and density are rigorously
shown to be given to leading order by a Gross-Pitaevskii (GP) energy functional
with a coupling constant . In contrast to the 3D
case the coupling constant depends on through the mean density. The GP
energy per particle depends only on . In 2D this parameter is typically so
large that the gradient term in the GP energy functional is negligible and the
simpler description by a Thomas-Fermi type functional is adequate.Comment: 14 pages, no figures, latex 2e. References, some clarifications and
an appendix added. To appear in Commun. Math. Phy
Phase space localization of antisymmetric functions
Upper and lower bounds are written down for the minimum information entropy
in phase space of an antisymmetric wave function in any number of dimensions.
Similar bounds are given when the wave function is restricted to belong to any
of the proper subspaces of the Fourier transform operator.Comment: 5 pages, REVTEX, no figure
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