Theory of elementary excitations in unstable Bose-Einstein condensates and the instability of sonic horizons

Like classical fluids, quantum gases may suffer from hydrodynamic instabilities. Our paper develops a quantum version of the classical stability analysis in fluids, the Bogoliubov theory of elementary excitations in unstable Bose-Einstein condensates. In unstable condensates the excitation modes have complex frequencies. We derive the normalization conditions for unstable modes such that they can serve in a mode decomposition of the noncondensed component. Furthermore, we develop approximative techniques to determine the spectrum and the mode functions. Finally, we apply our theory to sonic horizons-sonic black and white holes. For sonic white holes the spectrum of unstable modes turns out to be intrinsically discrete, whereas black holes may be stable.