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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 $\delta \leq {1\over 2}$. This is the first result of this kind for quantum Hamiltonians. \par Our proof follows the strategy developed by Dyatlov and Zahl. The main new ingredient is the fractal uncertainty principle for $\delta $-regular sets with $\delta <1$, which may be of independent interest.
@article{10_4007_annals_2018_187_3_5,
author = {Jean Bourgain and Semyon Dyatlov},
title = {Spectral gaps without the pressure condition},
journal = {Annals of mathematics},
pages = {825--867},
publisher = {mathdoc},
volume = {187},
number = {3},
year = {2018},
doi = {10.4007/annals.2018.187.3.5},
mrnumber = {3779959},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4007/annals.2018.187.3.5/}
}
TY - JOUR AU - Jean Bourgain AU - Semyon Dyatlov TI - Spectral gaps without the pressure condition JO - Annals of mathematics PY - 2018 SP - 825 EP - 867 VL - 187 IS - 3 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.4007/annals.2018.187.3.5/ DO - 10.4007/annals.2018.187.3.5 LA - en ID - 10_4007_annals_2018_187_3_5 ER -
%0 Journal Article %A Jean Bourgain %A Semyon Dyatlov %T Spectral gaps without the pressure condition %J Annals of mathematics %D 2018 %P 825-867 %V 187 %N 3 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.4007/annals.2018.187.3.5/ %R 10.4007/annals.2018.187.3.5 %G en %F 10_4007_annals_2018_187_3_5
Jean Bourgain; Semyon Dyatlov. Spectral gaps without the pressure condition. Annals of mathematics, Tome 187 (2018) no. 3, pp. 825-867. doi: 10.4007/annals.2018.187.3.5
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