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 beha
viour of the system in free space as the limit of an arbitrarily large radi
us in the confined solution. From a mathematical point of view we show that
this way of addressing the problem is not equivalent to considering the sy
stem a priori embedded in infinite space. In particular, the matrix element
s of the transformation turning the system to the principal axis do not ten
d to distributions in the limit of an arbitrarily large sphere as should be
the case if the two procedures were mathematically equivalent. Also, we in
troduce 'dressed' coordinates which allow an exact description of the oscil
lator radiation process. Expanding in powers of the coupling constant, we r
ecover from our exact expressions the well known decay formulae from pertur
bation theory.