As is well-known, in Bogoliubov's theory of an interacting Bose gas the ground state of the Hamiltonian $\hat{ H}=\sum_{ \bf k\neq 0}\hat{ H}_{ \bf k}$ is found by diagonalizing each of the Hamiltonians $\hat{ H}_{ \bf k}$ corresponding to a given momentum mode ${ \bf k}$ independently of the Hamiltonians $\hat{ H}_{ \bf k'(\neq k)}$ of the remaining modes. We argue that this way of diagonalizing $\hat{ H}$ may not be adequate, since the Hilbert spaces where the single-mode Hamiltonians $\hat{ H}_{ \bf k}$ are diagonalized are not disjoint, but have the ${ \bf k}=0$ in common. A number-conserving generalization of Bogoliubov's method is presented where the total Hamiltonian $\hat{ H}$ is diagonalized directly. When this is done, the spectrum of excitations changes from a gapless one, as predicted by Bogoliubov's method, to one which has a finite gap in the $k\to 0$ limit.
Categories: cond-mat.stat-mech cond-mat.other cond-mat.quant-gas