Algebraic approach to the Hulthen potential

In this paper the energy eigenvalues and the corresponding eigenfunctions are calculated for Hulthen potential. Then we obtain the ladder operators and show that these operators satisfy SU(2) commutation relation.Comment: 8 Pages, 1 Tabl

Size growth of red-sequence early-type galaxies in clusters in the last 10 Gyr

We carried out a photometric and structural analysis in the rest-frame $V$ band of a mass-selected ($\log M/M_\odot >10.7$) sample of red-sequence galaxies in 14 galaxy clusters, 6 of which are at $z>1.45$. To this end, we reduced/analyzed about 300 orbits of multicolor images taken with the Advanced Camera for Survey and the Wide Field Camera 3 on the Hubble Space Telescope. We uniformly morphologically classified galaxies from $z=0.023$ to $z=1.803$, and we homogeneously derived sizes (effective radii) for the entire sample. Furthermore, our size derivation allows, and therefore is not biased by, the presence of the usual variety of morphological structures seen in early-type galaxies, such as bulges, bars, disks, isophote twists, and ellipiticy gradients. By using such a mass-selected sample, composed of 244 red-sequence early-type galaxies, we find that the $\log$ of the galaxy size at a fixed stellar mass, $\log M/M_\odot= 11$ has increased with time at a rate of $0.023\pm0.002$ dex per Gyr over the last 10 Gyr, in marked contrast with the threefold increase found in the literature for galaxies in the general field over the same period. This suggests, at face value, that secular processes should be excluded as the primary drivers of size evolution because we observed an environmental environmental dependent size growth. Using spectroscopic ages of Coma early-type galaxies we also find that recently quenched early-type galaxies are a numerically minor population not different enough in size to alter the mean size at a given mass, which implies that the progenitor bias is minor, i.e., that the size evolution measured by selecting galaxies at the redshift of observation is indistinguishable from the one that compares ancestors and descendents.Comment: A&A 593, A2 (2016) after revision of the z=1.63 cluster name, mis-typed in previous version. No result of our paper is affected by having mis-typed the cluster nam