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We study an interacting spinless Bose-Einstein condensate to clarifytheoretically whether the spectra of its quasiparticles (one-particleexcitations) and collective modes (two-particle excitations) are identical, asconcluded by Gavoret and Nozi\`eres [Ann. Phys. 28, 349 (1964)]. We deriveanalytic expressions for their first and second moments so as to extend theBijl-Feynman formula for the peak of the collective-mode spectrum to its width(inverse lifetime) and also to the one-particle channel. The obtained formulasindicate that the width of the collective-mode spectrum manifestly vanishes inthe long-wavelength limit, whereas that of the quasiparticle spectrumapparently remains finite. We also evaluate the peaks and widths of the twospectra numerically for a model interaction potential in terms of the Jastrowwave function optimized by a variational method. It is thereby found that thewidth of the quasiparticle spectrum increases towards a constant as thewavenumber decreases. This marked difference in the spectral widths impliesthat the two spectra are distinct. In particular, the lifetime of thequasiparticles remains finite even in the long-wavelength limit.

Are Quasiparticles and Phonons Identical in Bose--Einstein Condensates?