Comment on “Particle-number-conserving Bogoliubov method which demonstrates the validity of the time-dependent Gross-Pitaevskii equation for a highly condensed Bose gas”

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Abstract
A recent paper of Gardiner [Phys. Rev. A 56, 1414 (1997)] introduces a particle-number-conserving Bogoliubov method for the excitation spectrum of a Bose-condensed gas, for use in theories of recently experimentally produced trapped atomic Bose condensates. Gardiner’s approach is compared and contrasted with the 1959 Girardeau-Arnowitt theory [Phys. Rev. 113, 755 (1959)], to which it is closely related and which is also fully number conserving. The number-conserving Bogoliubov quasiparticle operators of the Girardeau-Arnowitt theory satisfy Bose commutation relations exactly so long as states with the condensate totally depleted are neglected, whereas those of Gardiner satisfy Bose commutation relations only in an approximation that deteriorates progressively as the condensate is depleted.