The wave equation with one point interaction and the (linearized) classical electrodynamics of a point particle

We study the point limit of the linearized Maxwell-Lorentz equations describing the interaction, in the dipole approximation, of an extended charged particle with the electromagnetic field. We find that this problem perfectly fits into the framework of singular perturbations of the Laplacian; indeed we prove that the solutions of the Maxwell-Lorentz equations converge - after an infinite mass renormalization which is necessary in order to obtain a non trivial limit dynamics - to the solutions of the abstract wave equation defined by the self-adjoint operator describing the Laplacian with a singular perturbation at one point. The elements in the corresponding form domain have a natural decomposition into a regular part and a singular one, the singular subspace being three-dimensional. We obtain that this three-dimensional subspace is nothing but the velocity particle space, the particle dynamics being therefore completely determined - in an explicit way - by the behaviour of the singular component of the field. Moreover we show that the vector coefficient giving the singular part of the field evolves according to the Abraham-Lorentz-Dirac equation