Improved fractal Weyl bounds for hyperbolic manifolds

Abstract

We give a new fractal Weyl upper bound for resonances of convex co-compact hyperbolic manifolds in terms of the dimension $n$ of the manifold and the dimension $\delta$ of its limit set. More precisely, we show that as $R\to \infty$, the number of resonances in the box $[R,R+1]+i[-\beta ,0]$ is $O(R^{m(\beta ,\delta )+})$, where the exponent $m(\beta ,\delta )=\min (2\delta +2\beta +1-n,\delta )$ changes its behavior at $\beta =(n-1-\delta )/2$. In the case $\delta <(n-1)/2$, we also give an improved resolvent upper bound in the standard resonance free strip $\{\mathrm{Im}\lambda >\delta -(n-1)/2\}$. Both results use the fractal uncertainty principle point of view recently introduced in [arXiv:1504.06589]. The appendix presents numerical evidence for the Weyl upper bound.

42 pages, 10 figures; with an appendix by David Borthwick and Tobias Weich. Revised following suggestions of the referee. To appear in JEMS

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• Fractal Weyl laws for chaotic open systems
• Fractal Weyl law for quantum fractal eigenstates
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Cited by in corpus (3)

• Spectral gaps without the pressure condition
• Resonances for open quantum maps and a fractal uncertainty principle
• Fractal Weyl laws and wave decay for general trapping