Integral equation for inhomogeneous condensed bosons generalizing the Gross-Pitaevskii differential equation

We give here the derivation of a Gross-Pitaevskii-type equation for inhomogeneous condensed bosons. Instead of the original Gross-Pitaevskii differential equation, we obtain an integral equation that implies less restrictive assumptions than are made in the very recent study of Pieri and Strinati [Phys. Rev. Lett. 91, 030401 (2003)]. In particular, the Thomas-Fermi approximation and the restriction to small spatial variations of the order parameter invoked in their study are avoided.

Figure 1

Solid lines show $r^{\prime 4}{Q}_0\left(r^{\prime }\right)$, where $Q\left(r\right)$ is defined by Eq. (6) as a function of $r^{\prime }=k_{F}r$, for several temperatures, given by ${\beta }^{\prime }=\beta \mu =10-30$ (bottom to top). Dashed lines are the asymptotic expansion Eq. (8). Units are such that $k_F^6∕\left(4\mu \right)=1$.

Figure 2

Shows diagonal element of the canonical density matrix $C\left(\mathbf{r},\mathbf{r},\beta \right)$, Eq. (18), and its Thomas-Fermi approximation, Eq. (20), as a function of $r$, for $\beta =0.5$. Energies are in units of $\hslash \omega$, while lengths are in units of ${\left(\hslash ∕m\omega \right)}^{1∕2}$.

Figure 3

Shows diagonal $f\left(z,\mathbf{r},\mathbf{r}\right)$ corresponding to the harmonic potential, as given by the inverse Laplace transform, Eq. (19), as well as the regular part of its Thomas-Fermi approximation, Eq. (21), as a function of $r$, for fixed $z=0.5$. Units as in Fig. 2.