An improved Anderson model

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Abstract
The Anderson model has been very successful in the study of (magnetic) impurities in metals and has also proved useful for atoms adsorbed on metal surfaces. The model as originally formulated is phenomenological in that the position and width of the resonant impurity state cannot be calculated within the context of the model even in the absence of correlation effects. Anderson and McMillan and also Kanamori have proposed theories which are more quantitative. We show that the results of both theories follow from very simple assumptions. Moreover, we show that for the case of a free-electron-like metal and a spherical impurity potential both theories will give a correct density of states for the metal plus impurity if the wave function associated with the impurity is chosen properly. This is particularly important for the theory of Kanamori where the use of a non-Hermitian Hamiltonian raises questions about its validity. The best choice for the impurity wave function requires it to be energy dependent, unlike that one which appears in the usual Anderson model. However, it is shown that an energy-independent wave function can be chosen such that the Anderson-McMillan and Kanamori theories will yield a good density of states.