Reflection phase shifts of bouncing Bogoliubov waves

The Bogoliubov-de Gennes equations are solved for an inhomogeneous condensate in the vicinity of a turning point, addressing the full continuous spectrum. A basis change in the space of the two Bogoliubov "particle" and "hole" amplitudes is introduced that decouples them approximately. We find a spatially extended mode that governs mainly excitations in the condensate phase, while another mode is localised to regions with density gradients. An analytical and numerical discussion of the phase shift is provided that incident matter waves suffer upon reflection at the turning point, forming standing waves. As an application, we compute eigenfrequencies in a gravitational trap, without recourse to the local density approximation. The non-condensate density at finite temperature and its quantum depletion are discussed in a companion paper.