  We discuss the quantization of an unstable field through the construction of
a "one-particle Hilbert space." The system considered here is a neutral scalar
field evolving over a globally hyperbolic static spacetime and subject to a
stationary external scalar potential. In order to prove our results we assume
spacetimes without horizons and that the theory possess a "mass gap." Our
strategy consists in building a complex structure, which arises from a suitable
positive bilinear form defined over the space of classical solutions of the
field equation. Once the space of states of the quantum field has been set, it
is possible to study the effect of the time translation symmetry on it. From
the time translation operator we obtain an expression for the Hamiltonian
operator associated with the unstable sector of the field. This last result
coincides with findings from long ago showing that the unstable degrees of
freedom of the field behave as non-relativistic particles in a parabolic
potential barrier.
