Interaction with a field: a simple integrable model with backreaction

The classical model of an oscillator linearly coupled to a string captures, for a low price in technique, many general features of more realistic models for describing a particle interacting with a field or an atom in an electromagnetic cavity. The scattering matrix and the asymptotic in and out-waves on the string can be computed exactly and the phenomenon of resonant scattering can be introduced in the simplest way. The dissipation induced by the coupling of the oscillator to the string can be studied completely. In the case of a d’Alembert string, the backreaction leads to an Abraham–Lorentz–Dirac-like equation. In the case of a Klein–Gordon string, one can see explicitly how radiation governs the (meta)stability of the (quasi)bounded mode.