Improved fractal Weyl bounds for convex cocompact hyperbolic surfaces and large resonance-free regions
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| Format: | Preprint |
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2023
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| _version_ | 1866908488418983936 |
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| author | Soares, Louis |
| author_facet | Soares, Louis |
| contents | Let $X$ be a convex cocompact hyperbolic surface, and let $δ$ denote the Hausdorff dimension of its limit set. Let $N_X(σ,T)$ denote the number of resonances of $X$ inside the box $[σ, δ] + i[0,T]$. We prove that for all $σ> δ/2$, we have \[ N_X(σ,T) \ll_εT^{1 + δ- 2(2σ- δ) + ε}. \] This strengthens the previously established "improved" fractal Weyl bounds due to Naud \cite{Naud14} and Dyatlov \cite{Dya19}. Moreover, this result implies that for every $ε> 0$, there exist resonance-free rectangular boxes of arbitrary height within the strip \[ \left\{\, s \in \mathbb{C} : \tfrac{3}{4}δ+ ε< \mathrm{Re}(s) < δ\, \right\}. \] Our proof combines Naud's approach \cite{Naud14} with the refined transfer operator machinery developed by Dyatlov-Zworski \cite{DyZw18}, as well as a new estimate for oscillatory integrals that arise naturally in our analysis. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2301_03023 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Improved fractal Weyl bounds for convex cocompact hyperbolic surfaces and large resonance-free regions Soares, Louis Spectral Theory Analysis of PDEs 58J50, 81U24 (Primary) 11M36, 37C30 (Secondary) Let $X$ be a convex cocompact hyperbolic surface, and let $δ$ denote the Hausdorff dimension of its limit set. Let $N_X(σ,T)$ denote the number of resonances of $X$ inside the box $[σ, δ] + i[0,T]$. We prove that for all $σ> δ/2$, we have \[ N_X(σ,T) \ll_εT^{1 + δ- 2(2σ- δ) + ε}. \] This strengthens the previously established "improved" fractal Weyl bounds due to Naud \cite{Naud14} and Dyatlov \cite{Dya19}. Moreover, this result implies that for every $ε> 0$, there exist resonance-free rectangular boxes of arbitrary height within the strip \[ \left\{\, s \in \mathbb{C} : \tfrac{3}{4}δ+ ε< \mathrm{Re}(s) < δ\, \right\}. \] Our proof combines Naud's approach \cite{Naud14} with the refined transfer operator machinery developed by Dyatlov-Zworski \cite{DyZw18}, as well as a new estimate for oscillatory integrals that arise naturally in our analysis. |
| title | Improved fractal Weyl bounds for convex cocompact hyperbolic surfaces and large resonance-free regions |
| topic | Spectral Theory Analysis of PDEs 58J50, 81U24 (Primary) 11M36, 37C30 (Secondary) |
| url | https://arxiv.org/abs/2301.03023 |