Spectral gaps without the pressure condition

• 成果类型: Article
• 署名作者: Bourgain, Jean; Dyatlov, Semyon
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2018.187.3.5
• 发表日期: 2018
• 页码: 825-867
• 关键词: analytic continuation; complete spaces; resolvent; resonances; decay; SCATTERING; MAPS

摘要

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 <= 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 < 1, which may be of independent interest.