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We derive and discuss the equations of motion for the condensate and itsfluctuations for a dilute, weakly interacting Bose gas in an external potentialwithin the self--consistent Hartree--Fock--Bogoliubov (HFB) approximation.Account is taken of the depletion of the condensate and the anomalous Bosecorrelations, which are important at finite temperatures. We give a criticalanalysis of the self-consistent HFB approximation in terms of theHohenberg--Martin classification of approximations (conserving vs gapless) andpoint out that the Popov approximation to the full HFB gives a gaplesssingle-particle spectrum at all temperatures. The Beliaev second-orderapproximation is discussed as the spectrum generated by functionaldifferentiation of the HFB single--particle Green's function. We emphasize thatthe problem of determining the excitation spectrum of a Bose-condensed gas(homogeneous or inhomogeneous) is difficult because of the need to satisfyseveral different constraints.Comment: plain tex, 19 page

Conserving and gapless approximations for an inhomogeneous Bose gas at finite temperatures