Extension of the Thomas-Fermi approximation for trapped Bose-Einstein condensates with an arbitrary number of atoms

By incorporating the zero-point energy contribution we derive simple and accurate extensions of the usual Thomas-Fermi (TF) expressions for the ground-state properties of trapped Bose-Einstein condensates that remain valid for an arbitrary number of atoms in the mean-field regime. Specifically, we obtain approximate analytical expressions for the ground-state properties of spherical, cigar-shaped, and disk-shaped condensates that reduce to the correct analytical formulas in both the TF and the perturbative regimes, and remain valid and accurate in between these two limiting cases. Mean-field quasi-1D and -2D condensates appear as simple particular cases of our formulation. The validity of our results is corroborated by an independent numerical computation based on the 3D Gross-Pitaevskii equation.Comment: 5 pages, 3 figures. Final version published in Phys. Rev.

Almost sharp nonlinear scattering in one-dimensional Born-Infeld equations arising in nonlinear Electrodynamics

We study decay of small solutions of the Born-Infeld equation in 1+1 dimensions, a quasilinear scalar field equation modeling nonlinear electromagnetism, as well as branes in String theory and minimal surfaces in Minkowski space-times. From the work of Whitham, it is well-known that there is no decay because of arbitrary solutions traveling to the speed of light just as linear wave equation. However, even if there is no global decay in 1+1 dimensions, we are able to show that all globally small $H^{s+1}\times H^s$, $s>\frac12$ solutions do decay to the zero background state in space, inside a strictly proper subset of the light cone. We prove this result by constructing a Virial identity related to a momentum law, in the spirit of works \cite{KMM,KMM1}, as well as a Lyapunov functional that controls the $\dot H^1 \times L^2$ energy.Comment: 12 pages; This is version 2. Some typos corrected and sections organized differently for ease readin

Constraining the Minimum Mass of High-Redshift Galaxies and Their Contribution to the Ionization State of the IGM

We model the latest HST WFPC3/IR observations of > 100 galaxies at redshifts z=7-8 in terms of a hierarchical galaxy formation model with starburst activity. Our model provides a distribution of UV luminosities per dark matter halo of a given mass and a natural explanation for the fraction of halos hosting galaxies. The observed luminosity function is best fit with a minimum halo mass per galaxy of 10^{9.4+0.3-0.9} Msun, corresponding to a virial temperature of 10^{4.9+0.2-0.7} K. Extrapolating to faint, undetected galaxies, the total production rate of ionizing radiation depends critically on this minimum mass. Future measurements with JWST should determine whether the entire galaxy population can comfortably account for the UV background required to keep the intergalactic medium ionized.Comment: 9 pages, 6 figures, submitted to ApJ, comments welcom