= Solution
For each opposite-<momentum> pair use the <Bogoliubov transformation>
$$
b_{\mathbf k}=u_k\alpha_{\mathbf k}-v_k\alpha^\dagger_{-\mathbf k},\qquad
u_k^2-v_k^2=1,\qquad u_k^2+v_k^2=\frac{A_k}{E_k},\qquad 2u_kv_k=\frac{B_k}{E_k}.
$$
Equivalently $u_k=\cosh\theta_k$, $v_k=\sinh\theta_k$, with $\tanh(2\theta_k)=B_k/A_k$. The identity $u_k^2-v_k^2=1$ preserves the <canonical commutation relation>. Vanishing anomalous terms require $B_k(u_k^2+v_k^2)-2A_ku_kv_k=0$, and hence the <Bogoliubov quasiparticle dispersion> is
$$
\boxed{E_k=\sqrt{A_k^2-B_k^2}=\sqrt{\epsilon_k(\epsilon_k+2nV_k)}.}
$$
Substitution and <normal ordering> give
$$
\boxed{H_2=E_{\rm mf}+\frac12\sum_{\mathbf k\ne0}(E_k-A_k)+\sum_{\mathbf k\ne0}E_k\alpha^\dagger_{\mathbf k}\alpha_{\mathbf k}.}
$$
This is the requested form after putting each $E_k/2$ inside the oscillator term. By <momentum-pair counting in bosonic diagonalization>, all sums cover the full nonzero-<momentum> set; no further factor of two is needed. The <bosonic Bogoliubov zero-point shift> combines the two vacuum sums before taking a continuum limit.
If $V_k\to V_0>0$ smoothly, the low-energy excitations are gapless <phonons>:
$$
\boxed{E_k\simeq\hbar c|k|,\qquad c=\sqrt{nV_0/m}.}
$$
For contact repulsion, the <healing length> $\xi=\hbar/\sqrt{2mnV_0}$ separates this regime from the particle-like regime $E_k\simeq\epsilon_k+nV_0$. Stability requires $\epsilon_k+2nV_k>0$ for every nonzero mode. A general momentum-dependent interaction need not share the contact model's entire curve. An unregularized contact model also makes the vacuum-energy shift ultraviolet divergent; a physical finite-range interaction or scattering-length renormalization is needed for that absolute <ground-state energy>.
\Image[/past-exam-of-the-quantum-condensed-matter-field-theory-course-of-the-university-of-cambridge/2013/paper-1-qcm2013-bogoliubov-dispersion.png]
{title=Contact-repulsion Bogoliubov dispersion crosses from sound to particle-like excitations}
{source=paper-1-qcm2013-bogoliubov-dispersion.py}
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