  We consider a complete study of the influence of the cavity size on the
spontaneous decay of an atom excited state, roughly approximated by a harmonic
oscillator. We confine the oscillator-field system in a spherical cavity of
radius $R$, perfectly reflective, and work in the formalism of dressed
coordinates and states, which allows to perform non-perturbative calculations
for the probability of the atom to decay spontaneously from the first excited
state to the ground state. In free space, $R\to\infty$, we obtain known exact
results an for sufficiently small $R$ we have developed a power expansion
calculation in this parameter. Furthermore, for arbitrary cavity size radius,
we developed numerical computations and showed complete agreement with the
exact one for $R\to\infty$ and the power expansion results for small cavities,
in this way showing the robustness of our results. We have found that in
general the spontaneous decay of an excited state of the atom increases with
the cavity size radius and vice versa. For sufficiently small cavities the atom
practically does not suffers spontaneous decay, whereas for large cavities the
spontaneous decay approaches the free-space $R\to\infty$ value. On the other
hand, for some particular values of the cavity radius, in which the cavity is
in resonance with the natural frequency of the atom, the spontaneous decay
transition probability is increased compared to the free-space case. Finally,
we showed how the probability spontaneous decay go from an oscillatory time
behaviour, for finite cavity radius, to an almost exponential decay, for free
space.
