Kramers-Kronig dispersion relations equation

One may deduce from equation (32) the well-known Kramers-Kronig dispersion relations which may be written in the following form ... 

This equation effectively connects the alternating and after-effect solutions provided the response is linear. We may now make use of the Kramers-Kronig dispersion relations [66,67] to rewrite equations (C.ll) and (C.12) as... 

Scattering theory has very general validity. It is based on the existence of a Schrodinger-type equation and is consistent with causality requirements through the Kramers-Kronig dispersion relations. It is widely used outside atomic physics for the description of all resonance phenomena. There is therefore some advantage in using it in atomic physics it ensures unification and is more transparent to reseachers from other fields. 

The equations (3.36a) and (3.37) are the Kramers-Kronig dispersion relations. They relate absorption and dispersion through the complex refractive index n =n — k =n — ia/(2ko). 

The quantities n(w) — 1 and k(w) satisfy the dispersion relations [equations (42) and (43)]. Indeed, the dispersion relations were first derived by Kramers and by Kronig for these particular variables. 

The relationship between rotation and refractive index or between ellipticity and absorption coefficient can be represented, in analogy to dispersion theory, by the model of coupled linear oscillators or by quantum mechanical methods. ORD and CD are related to one another by equations analogous to the Kramers-Kronig equations [34]. 

These equations imply a relationship between Gi(compatibility relations, which are the viscoelastic analogue of the Kramers-Kronig relations for optical dispersion and magnetic relaxation. 

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