  Starting from a solution of the problem of a mechanical oscillator coupled to
a scalar field inside a reflecting sphere of radius $R$, we study the behaviour
of the system in free space as the limit of an arbitrarily large radius in the
confined solution. From a mathematical point of view we show that this way of
facing the problem is not equivalent to consider the system {\it a} {\it
priori} embedded in infinite space. In particular, the matrix elements of the
transformation turning the system to principal axis, do not tend to
distributions in the limit of an arbitrarily large sphere as it should be the
case if the two procedures were mathematically equivalent. Also, we introduce
"dressed" coordinates which allow an exact description of the oscillator
radiation process for any value of the coupling, strong or weak. In the case of
weak coupling, we recover from our exact expressions the well known decay
formulas from perturbation theory.
