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

摘要: We evaluate finite-temperature equilibrium correlators\(\langle T_\tau \hat \psi ({\text{r}}_{\text{1}} )\hat ^\dag ({\text{r}}_{\text{2}} )\rangle \) for thermal time τ ordered Bose fields \(\hat ,{\text{ }}\hat to good approximations by new methods of functional integration in d=1, 2, 3 dimensions and with the trap potentials V(r)≢0. As translationally invariant cases, asymptotic behaviors fall as \(R^{ - 1} \equiv |{\text{r}}_1 {\text{r}}_2 |^{ longer-range condensate values only d = agreement experimental observations; but there are generally significant corrections also depending on \(S ({\text{r}}_1 {\text{ + r}}_2 )/2\) due presence traps. For 1, we regain exact results frequencies Ω → 0. In analyzing attractive investigate time-dependent c-number Gross–Pitaevskii (GP) equation potential a generalized nonlinearity −2cψ|ψ|2n c < n stationary form GP appears steepest-descent approximation integrals. show that collapse sense Zakharov can occur 0 nd ≥ 2 ENLS[ψ] ≤ even when The singularities typically arise δ-functions centered origin r=0.