Spectral gaps without the pressure condition
Author(s)Bourgain, Jean; Dyatlov, Semyon
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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 δ of the limit set; in particular, we do not require the pressure condition δ ≤ 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 δ-regular sets with δ < 1, which may be of independent interest.
Annals of Mathematics
Mathematics Department, Princeton University
Bourgain, Jean and Semyon Dyatlov. "Spectral gaps without the pressure condition." Annals of Mathematics 187, 3 (May 2018): 825-867 © 2018 Department of Mathematics, Princeton University