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

Singlets and reflection symmetric spin systems

We rigorously establish some exact properties of reflection symmetric spin systems with antiferromagnetic crossing bonds: At least one ground state has total spin zero and a positive semidefinite coefficient matrix. The crossing bonds obey an ice rule. This augments some previous results which were limited to bipartite spin systems and is of particular interest for frustrated spin systems.Comment: 11 pages, LaTeX 2

On the maximal ionization of atoms in strong magnetic fields

We give upper bounds for the number of spin 1/2 particles that can be bound to a nucleus of charge Z in the presence of a magnetic field B, including the spin-field coupling. We use Lieb's strategy, which is known to yield N_c<2Z+1 for magnetic fields that go to zero at infinity, ignoring the spin-field interaction. For particles with fermionic statistics in a homogeneous magnetic field our upper bound has an additional term of order $Z\times\min{(B/Z^3)^{2/5},1+|\ln(B/Z^3)|^2}$.Comment: LaTeX2e, 8 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

The Ground States of Large Quantum Dots in Magnetic Fields

The quantum mechanical ground state of a 2D $N$-electron system in a confining potential $V(x)=Kv(x)$ ($K$ is a coupling constant) and a homogeneous magnetic field $B$ is studied in the high density limit $N\to\infty$, $K\to \infty$ with $K/N$ 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 $B/N$ varies as $N\to\infty$: A 2D Thomas-Fermi (TF) theory applies in the case $B/N\to 0$; if $B/N\to{\rm const.}\neq 0$ the correct limit theory is a modified $B$-dependent TF model, and the case $B/N\to\infty$ is described by a ``classical'' continuum electrostatic theory. For homogeneous potentials this last model describes also the weak coupling limit $K/N\to 0$ for arbitrary $B$. 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