= Solution
Write $\mathcal V=L^3$ and $N_0=N-N_{\rm ex}$, where $N_{\rm ex}=\sum_{\mathbf k\ne0}b_{\mathbf k}^\dagger b_{\mathbf k}$. Macroscopic occupation permits $b_0,b_0^\dagger\simeq\sqrt{N_0}$, with relative commutator corrections of order $1/N_0$. Keep the fixed-total-number correction before replacing $n_0$ by $n$.
The four-condensate term is $V_0N_0(N_0-1)/(2\mathcal V)$. Its leading expansion gives $V_0N^2/(2\mathcal V)-nV_0N_{\rm ex}$. Terms with two condensate operators give normal scattering $n(V_0+V_{\mathbf k})b_{\mathbf k}^\dagger b_{\mathbf k}$ and conversion of two condensate particles into opposite-momentum excitations, with coefficient $nV_{\mathbf k}/2$. The $nV_0$ normal contribution cancels against the change of condensate energy at fixed $N$. Hence the <number-conserving Bogoliubov quadratic Hamiltonian> is
$$
\boxed{H\simeq\frac{\mathcal V}{2}V_0n^2+
\sum_{\mathbf k\ne0}\left[(\epsilon_{\mathbf k}+nV_{\mathbf k})b_{\mathbf k}^\dagger b_{\mathbf k}
+\frac{nV_{\mathbf k}}2(b_{\mathbf k}^\dagger b_{-\mathbf k}^\dagger+b_{-\mathbf k}b_{\mathbf k})\right].}
$$
Use a real even interaction transform, $V_{\mathbf k}=V_{-\mathbf k}$, as for a real inversion-symmetric pair potential. The constant is the condensate interaction energy; the diagonal part combines particle kinetic energy with condensate scattering; the anomalous pair terms create and annihilate correlated opposite momenta while transferring particles to or from the condensate.
At fixed volume, counting replaced condensate operators gives quadratic vertices proportional to $N_0$, omitted cubic vertices proportional to $\sqrt{N_0}$ and quartic vertices with no condensate factor. There are also the finite-$N_0$ correction in $N_0(N_0-1)$ and corrections from the condensate-number expansion. Their physical control parameter is small depletion, $N_{\rm ex}/N\ll1$, together with weak interaction; simply dropping them is not justified by low temperature in a strongly interacting gas. This is the <weakly interacting Bose gas> approximation.
Back to article page
