Quantum integrable and nonintegrable nonlinear Schrodinger models for realizable Bose-Einstein condensation in d+1 dimensions (d=1,2,3)

We evaluate finite-temperature equilibrium correlators [T-T (Psi) over cap (r(1))(Psi) over cap (dagger)(r(2))] for thermal time tau ordered Bose fields (Psi) over cap,(Psi) over cap (dagger) to good approximations by new methods of functional integration in d=1,2,3 dimensions and with the trap potentials V(r) not equivalent to 0. As in the translationally invariant cases, asymptotic behaviors fall as R-1 equivalent to \r(1) - r(2)\(-1) to longer-range condensate values for and only for d = 3 in agreement with experimental observations; but there are generally significant corrections also depending on S equivalent to (r(1) + r(2))/2 due to the presence of the traps. For d = 1, we regain the exact translationally invariant results as the trap frequencies Omega-->0. In analyzing the attractive cases, we investigate the time dependant c-number Gross-Pitaevskii (GP) equation with the trap potential for a generalized nonlinearity -2cpsi\psi\(2n) and c < 0. For n = 1, the stationary form of the GP equation appears in the steepest-descent approximation of the functional integrals. We show that collapse in the sense of Zakharov can occur for c < 0 and nd greater than or equal to 2 and a functional E-NLS[psi] less than or equal to 0 even when V(r) not equivalent to 0. The singularities typically arise as delta-functions centred on the trap origin r = 0.