quantum gases power law potential grand potential condensation hierarchy

The changes in characteristics of Bose condensation of ideal Bose gas due to an external generic power law potential U =∑~d_（i=1）ci｜xi/ai｜~（ni） are studied carefully. Detailed calculation of Kim et al.（J. Phys. Condens. Matter 11（1999） 10269） yielded the hierarchy of condensation transitions with changing fractional dimensionality. In this manuscript, some theorems regarding specific heat at constant volume CV are presented. Careful examination of these theorems reveal the existence of hidden hierarchy of the condensation transition in trapped systems as well.