  In recent studies it has become increasingly clear that presence of
infinitely many instability bands of the parametric resonance plays crucial
roles in the phenomenon of particle production under periodic classical field
oscillation. We extend previous works to a general class of models including
both the Yukawa and the quartic type of couplings of the classical field to
quantum bose fields. Decay rate from the $n-$th band is derived in the small
amplitude limit using the functional Schr$\stackrel{..}{{\rm o}}$dinger
picture. It is then shown that this analytic result of the decay rate can also
be derived as the zero momentum limit of a physical process, $n$ particles that
comprise the classical homogeneous field decaying simultaneously into 2 bose
particles. The latter approach uses ordinary perturbation theory, hence the
former result is a novel resummation of many perturbative amplitudes, which
usually becomes complicated for a large $n$ order.

High Energy Physics - Phenomenology;High Energy Physics - Theory