Communications in Mathematical Physics

, Volume 354, Issue 1, pp 269–316 | Cite as

Resonances for Open Quantum Maps and a Fractal Uncertainty Principle

Article

Abstract

We study eigenvalues of quantum open baker’s maps with trapped sets given by linear arithmetic Cantor sets of dimensions \({\delta\in (0,1)}\). We show that the size of the spectral gap is strictly greater than the standard bound \({\max(0,{1\over 2}-\delta)}\) for all values of \({\delta}\), which is the first result of this kind. The size of the improvement is determined from a fractal uncertainty principle and can be computed for any given Cantor set. We next show a fractal Weyl upper bound for the number of eigenvalues in annuli, with exponent which depends on the inner radius of the annulus.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA
  2. 2.Department of MathematicsPurdue UniversityWest LafayetteUSA

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