  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.
