  We report on some recent work of the authors showing the relations between
singular (point) perturbation of the Laplacian and the dynamical system
describing a charged point particle interacting with the self-generated
radiation field (the Maxwell-Lorentz system) in the dipole approximation. We
show that in the limit of a point particle, the dynamics of the system is
described by an abstract wave equation containing a selfadjoint operator $H_m$
of the class of point interactions; the classical Abraham-Lorentz-Dirac third
order equation, or better its integrated second order version, emerges as the
evolution equation of the singular part of the field and is related to the
boundary conditions entering in the definition of the operator domain of $H_m$.
We also give the Hamiltonian structure of the limit model and, in the case of
no external force, we study the reduced dynamics on the linear stable manifold.
