  The Bogoliubov-de Gennes equations are used for a number of theoretical works
on the trapped Bose-Einstein condensates. These equations are known to give the
energies of the quasi-particles when all the eigenvalues are real. We consider
the case in which these equations have complex eigenvalues. We give the
complete set including those modes whose eigenvalues are complex. The quantum
fields which represent neutral atoms are expanded in terms of the complete set.
It is shown that the state space is an indefinite metric one and that the free
Hamiltonian is not diagonalizable in the conventional bosonic representation.
We introduce a criterion to select quantum states describing the metastablity
of the condensate, called the physical state conditions. In order to study the
instability, we formulate the linear response of the density against the
time-dependent external perturbation within the regime of Kubo's linear
response theory. Some states, satisfying all the physical state conditions,
give the blow-up and damping behavior of the density distributions
corresponding to the complex eigenmodes. It is qualitatively consistent with
the result of the recent analyses using the time-dependent Gross-Pitaevskii
equation.
