On Local and Integral Forms of Energy Conservation Laws in the Scattering Theory

Consideration is given to various forms of energy balance relations that describe the influence of the scatterer on radiation of given currents and are connected with the classical description of the Purcell factor and with the optical theorem. It is shown that these relations hold not only integrally but also locally, i.e., for integrands. A transition is described from the ordinary integral form of the scattering problem (of the type of the Lippmann–Schwinger equation) to the system of two equations for the Hermitian and anti-Hermitian parts of the scattering T-operator, so that for the former the diagonal matrix element corresponding to an arbitrary incident wave directly gives the general form of the optical theorem. A particular case is also described, which concerns nonradiating currents in the absence of the scatterer with their radiation loss being only due to the presence of a scatterer.