Quantum Depletion of a Soliton Condensate

We present rigorous results on the diagonalization of Bogoliubov Hamilto- nian for a soliton condensate. Using the complete and orthonomalized set of eigenfunction for the Bogoliubov de Gennes equations, we calculate ex- actly the quantum depletion of the condensate and show that two degenerate zero-modes, which originate from a U(1) guage- and a translational-symmetry breaking of the system, induce the quantum diffusion and transverse instabil- ity of the soliton condensate.