Spectral gaps without the pressure condition

TL;DR

This paper proves the existence of an essential spectral gap for convex co-compact hyperbolic surfaces without assuming the pressure condition, using a fractal uncertainty principle that may be of independent interest.

Contribution

It establishes the first spectral gap result for quantum Hamiltonians without the pressure condition on the limit set dimension.

Findings

Existence of an essential spectral gap for all convex co-compact hyperbolic surfaces.

No assumption on the limit set dimension $oldsymbol{ extdelta}$, including $oldsymbol{ extdelta > 1/2}$.

Introduction of a fractal uncertainty principle for $oldsymbol{ extdelta}$-regular sets with $oldsymbol{ extdelta<1}$.

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 $\delta\leq {1\over 2}$. This is the first result of this kind for quantum Hamiltonians. Our proof follows the strategy developed by Dyatlov-Zahl [arXiv:1504.06589]. The main new ingredient is the fractal uncertainty principle for $\delta$-regular sets with $\delta<1$, which may be of independent interest.