Partial Densities of States, Scattering Matrices, and Green's Functions

TL;DR

This paper introduces local partial densities of states and sensitivities in scattering problems, relating them to Green's functions, scattering matrices, and characteristic times, with applications to specific quantum systems.

Contribution

It defines and analyzes local partial densities of states and sensitivities in scattering theory, establishing their relations to Green's functions, scattering matrices, and physical quantities like injectivity and emissivity.

Findings

Injectivities and emissivities relate to the square of scattering wave-functions.

Local partial densities of states connect to characteristic times.

Application to delta-barrier and Larmor clock demonstrates practical relevance.

Abstract

The response of an arbitrary scattering problem to quasi-static 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.