Quantum Integrable and Nonintegrable Nonlinear Schrödinger Models for Realizable Bose–Einstein Condensation in $d+1$ Dimensions $(d=1,2,3)$

We evaluate finite-temperature equilibrium correlators корреляторы $\langle T_\tau \hat{\psi}({\bold r}_1) \hat{\psi}^\dagger({\bold r}_2)\rangle$ for thermal time $\tau$ ordered Bose fields полей $\hat{\psi}$, $\hat{\psi}^\dagger$ to good approximations by new methods of functional integration in $d=1,2,3$ dimensions and with the trap potentials $V({\bold r})\not\equiv0$. As in the translationally invariant cases, asymptotic behaviors fall as $R^{-1}\equiv|{\bold r}_1-{\bold r}_2|^{-1}$ 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 ${\bold S}\equiv({\bold r}_1+{\bold r}_2)/2$ due to the presence of the traps. For $d=1$, we regain the exact translationally invariant results as the trap frequencies $\Omega\rightarrow0$. In analyzing the attractive cases, we investigate the time-dependent $c$-number Gross–Pitaevskii (GP) equation with the trap potential for a generalized nonlinearity $-2c\psi|\psi|^{2n}$ 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 $nd\geq2$ and a functional $E_{\textup{NLS}}[\psi]\leq0$ even when $V({\bold r})\not\equiv0$. The singularities typically arise as $\delta$-functions centered on the trap origin ${\bold r}={\bold 0}$.