Reflection phase shifts of bouncing Bogoliubov waves

The Bogoliubov--de Gennes equations for the elementary excitations of a Bose gas are solved for an inhomogeneous, one-dimensional system 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 spatially localised to the boundary zone where the condensate density gradient is significant. 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. We find phase shifts with relatively strong energy dispersion that show significant differences among four model potentials. As applications, we discuss the boundary-induced change in the density of states and 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.