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Resonances for Open Quantum Maps and a Fractal Uncertainty Principle

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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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Authors and Affiliations

  1. Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Ave, Cambridge, MA, 02139, USA

    Semyon Dyatlov

  2. Department of Mathematics, Purdue University, 150 N University Street, West Lafayette, IN, 47907, USA

    Long Jin

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  1. Semyon Dyatlov
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  2. Long Jin
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Correspondence to Semyon Dyatlov.

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Communicated by J. Marklof

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Dyatlov, S., Jin, L. Resonances for Open Quantum Maps and a Fractal Uncertainty Principle. Commun. Math. Phys. 354, 269–316 (2017). https://doi.org/10.1007/s00220-017-2892-z

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  • DOI: https://doi.org/10.1007/s00220-017-2892-z

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