Relation between the <PERSON> approximation and Hartree-Fock ansatz
Sometimes I work in the mathematical surrounding of the Gross-Pitaevskii equation (GPE) and I'm asking myself how different derivations of the GPE are related.
Originally the GPE was discovered by <PERSON> and <PERSON> independently and <PERSON> and <PERSON> wrote a nice introduction into the phenomenon of Bose-Einstein condensation [1]. Among other things the <PERSON> approximation is central in their derivation of the GPE and I'm asking myself how this approach is related to different ways the GPE can be obtained. The traditional way of obtaining the GPE from the <PERSON> approximation is justified by noting that in an Bose-Einstein condensate the energy-groundstate is occupied macroscopically. Imagine a gas of $N$ particles where $N_0$ particles are in the energy-groundstate such that $N_0 \sim N $ with $N_0 \leq N$. If we are considering a weakly interacting gas the dynamic of the system is given by the grand canonical Hamiltonian \begin{equation} \hat{H} = \int \left( \frac{\hbar^2}{2m} \nabla \hat{\Psi}^{\dagger} ( \mathbf{r}) \nabla \hat{\Psi} ( \mathbf{r}) \right) d \mathbf{r} + \frac{1}{2} \int \hat{\Psi}^{\dagger} ( \mathbf{r'}) \hat{\Psi}^{\dagger} ( \mathbf{r}) V(\mathbf{r'}- \mathbf{r} ) \nabla \hat{\Psi} ( \mathbf{r'}) \nabla \hat{\Psi} ( \mathbf{r}) d \mathbf{r'} d \mathbf{r}. 474
\label{eq:becintro:Hamiltoniannondensegase} \end{equation}
By decomposing the N-body wavefunction of the condensate $\Psi$ into single-particle wavefunction $\phi_i$ the field operator can be written in terms of creation and annihilation operators \begin{equation} \hat{\Psi}(\mathbf{r}) = \sum_i \phi_i \hat{a}i. \end{equation} which satisfy obey the usual bosonic commutation relations \begin{equation} \left[ \hat{a}_i, \hat{a}_j \right] = \left[ \hat{a}^{\dagger}_i, \hat{a}^{\dagger}_j \right] = 0 , \quad \left[ \hat{a}_i, \hat{a}^{\dagger}_j \right] = \delta{ij}. \end{equation} In the case of Bose-Einstein condensation the <PERSON> approximation replaces the operator $\hat{a}0 = \sqrt{N_0}$ and by introducing an effective interaction potential $V{\mathrm{eff}}$ for short range distance and taking into account that the almost every particle is in the energy-groundstate, i.e. only considering the annihilation (creation) operator with momentum zero, the GPE can be derived \begin{equation} i \hbar \frac{\partial}{\partial t} \Psi(\mathbf{r},t) = \left( - \frac{-\hbar^2}{2 m} \Delta + V{\mathrm{ext}} + g_{3d} | \Psi(\mathbf{r},t)|^2 \right)\Psi(\mathbf{r},t), \label{eq:becintro:grosspitaevskiiequation} \end{equation} with a coupling constant $g_{3d} = \frac{4 \pi \alpha_s}{m}$ in terms of the s-wave scattering length $\alpha_s$. (This is derivation of the GPE was extremely shortened, since it seems to be common knowledge in the physics community)
On the other in the mathematics community the GPE equation is obtained from a Hartree-Fock ansatz in the linear $N$-body <PERSON> problem, the ansatz reads \begin{equation} \Psi(\mathbf{r}1,... 526