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May 2018 Spectral gaps without the pressure condition
Jean Bourgain, Semyon Dyatlov
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Ann. of Math. (2) 187(3): 825-867 (May 2018). DOI: 10.4007/annals.2018.187.3.5

Abstract

For all convex co-compact hyperbolic surfaces, we prove the existence of an essential spectral gap, that is, a strip beyond the unitarity axis in which the Selberg zeta function has only finitely many zeroes. We make no assumption on the dimension $\delta$ of the limit set; in particular, we do not require the pressure condition $\delta \le \frac{1}{2}$. This is the first result of this kind for quantum Hamiltonians.

Our proof follows the strategy developed by Dyatlov and Zahl. The main new ingredient is the fractal uncertainty principle for $\delta$-regular sets with $\delta \lt 1$, which may be of independent interest.

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Jean Bourgain. Semyon Dyatlov. "Spectral gaps without the pressure condition." Ann. of Math. (2) 187 (3) 825 - 867, May 2018. https://doi.org/10.4007/annals.2018.187.3.5

Information

Published: May 2018
First available in Project Euclid: 21 December 2021

Digital Object Identifier: 10.4007/annals.2018.187.3.5

Subjects:
Primary: 42A99
Secondary: 30D60 , 35B34

Keywords: fractal uncertainty principle , scattering resonance , spectral gap

Rights: Copyright © 2018 Department of Mathematics, Princeton University

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Vol.187 • No. 3 • May 2018
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