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Continuity properties of the integrated density of states on manifolds

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Japanese Journal of Mathematics Aims and scope

Abstract

We first analyze the integrated density of states (IDS) of periodic Schrödinger operators on an amenable covering manifold. A criterion for the continuity of the IDS at a prescribed energy is given along with examples of operators with both continuous and discontinuous IDS.

Subsequently, alloy-type perturbations of the periodic operator are considered. The randomness may enter both via the potential and the metric. A Wegner estimate is proven which implies the continuity of the corresponding IDS. This gives an example of a discontinuous “periodic” IDS which is regularized by a random perturbation.

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Authors and Affiliations

  1. Fakultät für Mathematik, TU Chemnitz, D-09107, Chemnitz, Germany

    Daniel Lenz

  2. Department of Mathematical Sciences, Durham University, Science Laboratories South Road, Durham, DH1 3LE, Great Britain

    Norbert Peyerimhoff

  3. Institut für Mathematik, Humboldt-Universität zu Berlin, Rudower Chaussee 25, 12489, Berlin, Germany

    Olaf Post

  4. Emmy-Noether Programme of the DFG, Fakultät für Mathematik, TU Chemnitz, D-09107, Chemnitz, Germany

    Ivan Veselić

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  1. Daniel Lenz
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  2. Norbert Peyerimhoff
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  3. Olaf Post
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  4. Ivan Veselić
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Correspondence to Daniel Lenz.

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Communicated by: Kaoru Ono

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Lenz, D., Peyerimhoff, N., Post, O. et al. Continuity properties of the integrated density of states on manifolds. Jpn. J. Math. 3, 121–161 (2008). https://doi.org/10.1007/s11537-008-0729-4

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  • DOI: https://doi.org/10.1007/s11537-008-0729-4

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