= Number-conserving Bogoliubov quadratic Hamiltonian
{title2=$A_k=\epsilon_k+nV_k,\quad B_k=nV_k$}
At fixed total particle number, $N_0=N-\sum_{k\ne0}b_k^\dagger b_k$. Expanding the condensate interaction energy subtracts $nV_0$ from the normal excited-particle coefficient, cancelling its direct Hartree term. The resulting quadratic coefficients are $A_k=\epsilon_k+nV_k$ and $B_k=nV_k$, with an opposite-momentum anomalous pair term. Replacing $N_0$ by $N$ before making this correction loses the cancellation.
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