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A Dixmier Trace Formula for the Density of States

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Abstract

A version of Connes trace formula allows to associate a measure on the essential spectrum of a Schrödinger operator with bounded potential. In solid state physics there is another celebrated measure associated with such operators—the density of states. In this paper we demonstrate that these two measures coincide. We show how this equality can be used to give explicit formulae for the density of states in some circumstances.

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Notes

  1. Moreover it can be proved that \(\mathrm{Tr}_\omega \) is a Dixmier trace for every extended limit \(\omega \). This will appear in the upcoming second edition of [27].

  2. This follows from the identity \(\Gamma (z) = \frac{1}{z}\Gamma (z+1)\).

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Acknowledgements

The authors wish to thank the anonymous referee for helpful comments and suggestions. F. S. is partially supported by the Australian Research Council Grant FL170100052.

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Authors and Affiliations

  1. University of New South Wales, Kensington, NSW, 2052, Australia

    N. Azamov, E. McDonald, F. Sukochev & D. Zanin

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  1. N. Azamov
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  2. E. McDonald
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  3. F. Sukochev
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  4. D. Zanin
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Correspondence to E. McDonald.

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Communicated by M. Salmhofer.

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Azamov, N., McDonald, E., Sukochev, F. et al. A Dixmier Trace Formula for the Density of States. Commun. Math. Phys. 377, 2597–2628 (2020). https://doi.org/10.1007/s00220-020-03756-7

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