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Annales Henri Poincaré

, Volume 19, Issue 12, pp 3783–3814 | Cite as

Log-Scale Equidistribution of Zeros of Quantum Ergodic Eigensections

  • Robert Chang
  • Steve ZelditchEmail author
Article

Abstract

Under suitable hypotheses, a symplectic map can be quantized as a sequence of unitary operators acting on the Nth powers of a positive line bundle over a Kähler manifold. We show that if the symplectic map has sufficiently fast polynomial decay of correlations, then there exists a density one subsequence of eigensections whose masses and zeros become equidistributed in balls of logarithmically shrinking radii of lengths \(|\log N |^{-\gamma }\) for some constant \(\gamma > 0\) independent of N.

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Notes

Acknowledgements

We thank H. Hezari for pointing some errors and gaps in the earlier version and for suggesting corrections. We also thank F. Faure, G. Riviere, and A. Wilkinson for useful comments and references on the dynamical aspects. Finally, we thank the referees for their detailed and helpful comments that led to significantly improvements of the paper.

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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of MathematicsNorthwestern UniversityEvanstonUSA

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