Japanese Journal of Mathematics

, Volume 3, Issue 1, pp 121–161

Continuity properties of the integrated density of states on manifolds

  • Daniel Lenz
  • Norbert Peyerimhoff
  • Olaf Post
  • Ivan Veselić
Original Article

DOI: 10.1007/s11537-008-0729-4

Cite this article as:
Lenz, D., Peyerimhoff, N., Post, O. et al. Jpn. J. Math. (2008) 3: 121. doi:10.1007/s11537-008-0729-4


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.

Keyword and phrases

integrated density of states periodic and random operators Schrödinger operators on manifolds continuity properties 

Mathematics Subject Classification (2000)

35J10 82B44 

Copyright information

© The Mathematical Society of Japan and Springer Japan 2008

Authors and Affiliations

  • Daniel Lenz
    • 1
  • Norbert Peyerimhoff
    • 2
  • Olaf Post
    • 3
  • Ivan Veselić
    • 4
  1. 1.Fakultät für Mathematik, TU ChemnitzChemnitzGermany
  2. 2.Department of Mathematical SciencesDurham UniversityDurhamGreat Britain
  3. 3.Institut für MathematikHumboldt-Universität zu BerlinBerlinGermany
  4. 4.Emmy-Noether Programme of the DFGFakultät für Mathematik, TU ChemnitzChemnitzGermany

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