  The Bogoliubov method for the excitation spectrum of a Bose-condensed gas is
generalized to apply to a gas with an exact large number $ N$ of particles.
This generalization yields a description of the Schr\"odinger picture field
operators as the product of an annihilation operator $A$ for the total number
of particles and the sum of a ``condensate wavefunction'' $\xi(x)$ and a phonon
field operator $\chi(x)$ in the form $\psi(x) \approx A\{\xi(x) +
\chi(x)/\sqrt{N}\}$ when the field operator acts on the N particle subspace. It
is then possible to expand the Hamiltonian in decreasing powers of $\sqrt{N}$,
an thus obtain solutions for eigenvalues and eigenstates as an asymptotic
expansion of the same kind. It is also possible to compute all matrix elements
of field operators between states of different N.
