Partial densities of states, scattering matrices, and Green's functions

The response of an arbitrary scattering problem to quasistatic perturbations in the scattering potential is naturally expressed in terms of a set of local partial densities of states and a set of sensitivities each associated with one element of the scattering matrix. We define the local partial densities of states and the sensitivities in terms of functional derivatives of the scattering matrix and discuss their relation to the Green's function. Certain combinations of the local partial densities of states represent the injectivity of a scattering channel into the system and the emissivity into a scattering channel. It is shown that the injectivities and emissivities are simply related to the absolute square of the scattering wave function. We discuss also the connection of the partial densities of states and the sensitivities to characteristic times. We apply these concepts to a delta barrier and to the local Larmor clock.