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 $\alpha$ or $Z$ is too large. Here we prove that matter {\it is stable\/} if $\alpha<0.06$ and $Z\alpha^2<0.04$.Comment: 10 pages, LaTe

There are No Unfilled Shells in Hartree-Fock Theory

Hartree-Fock theory is supposed to yield a picture of atomic shells which may or may not be filled according to the atom's position in the periodic table. We prove that shells are always completely filled in an exact Hartree-Fock calculation. Our theorem generalizes to any system having a two-body interaction that, like the Coulomb potential, is repulsive.Comment: 5 pages, VBEHLMLJPS--16/July/9

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

Bounds for Entanglement via an Extension of Strong Subadditivity of Entropy

We prove lower bounds for the entanglement of formation and the squashed entanglement for any a bipartite density matrix in terms of the conditional entropy of the bipartite state with respect to either of its partial traces, and prove that these bounds are sharp by constructing a new class of states whose entanglements can be computed, and for which the bounds are saturated.Comment: 8 pages. Version 2 includes some additional references and comment