Localisation and localisation edges-a precise characterisation

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Abstract
The problem of Anderson localisation of an electron for q0 is used to calculate the frequency dependent dielectric in terms of results of quantum scattering theory. Thus, localisation is characterised as follows: (i) the localised state belongs to the pure point spectrum of the Hamiltonian operator; (ii) the inverse participation ratio E(EG)=Eg Sigma i=13(h(cross)2ki2/2mi*). At room temperatures states belong to the continuous spectrum of the Hamiltonian operator; and (iv) when there is no pure point spectrum, the inverse participation ratio is zero. A weighted inverse participation ratio, which is constructed only from generalised eigenfunctions associated with a subset of the spectrum, is formulated. A rigorous localisation criterion can then be defined, and the concepts of the localisation edge and the Anderson transition are made precise.

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