COMMENT ON PARTICLE-NUMBER-CONSERVING BOGOLIUBOV METHOD WHICH DEMONSTRATES THE VALIDITY OF THE TIME-DEPENDENT GROSS-PITAEVSKII EQUATION FORA HIGHLY CONDENSED BOSE-GAS

A recent paper of Gardiner [Phys. Rev. A 56, 1414 (1997)] introduces a particle-number-conserving Bogoliubov method for the excitation spect rum of a Bose-condensed gas, for use in theories of recently experimen tally produced trapped atomic Bose condensates. Gardiner's approach is compared and contrasted with the 1959 Girardeau-Amowitt theory [Phys. Rev. 113, 755 (1959)], to which it is closely related and which is al so fully number conserving. The number-conserving Bogoliubov quasipart icle operators of the Girardeau-Amowitt theory satisfy Bose commutatio n relations exactly so long as states with the condensate totally depl eted are neglected, whereas those of Gardiner satisfy Bose commutation relations only in an approximation that deteriorates progressively as the condensate is depleted.