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 a 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.

The trajectories of charged particles, emitted from within or from the close vicinity of pointed shaped surfaces, requires the knowledge of the electric field resulting from the potential bias between surface and detector, or screen. Frequently it is necessary the use of numerical methods for solving Laplace's equation as a result of difficulties in obtaining an analytical expression. Recently we have shown that, when any two coordinate surfaces of an orthogonal system are kept at two different but constant potentials, it is possible to obtain an analytical solution for the potential in a relatively simple manner. Using this general property of orthogonal coordinate systems, we present the solution for the electric potential and field in the vicinity of pointed surfaces for several cases of practical interest in field emission, field ionisation, atomprobe field ion spectroscopy and related phenomena.