  We study the energies and decay of elementary excitations in weakly
interacting Bose-Einstein condensates within a finite-temperature gapless
second-order theory. The energy shifts for the high-lying collective modes turn
out to be systematically negative compared with the
Hartree-Fock-Bogoliubov-Popov approximation and the decay of the low-lying
modes is found to exhibit collapse and revival effects. In addition,
perturbation theory is used to qualitatively explain the experimentally
observed Beliaev decay process of the scissors mode.
