论文标题
算术中缩小球的量子牙齿量表
Quantum ergodicity for shrinking balls in arithmetic hyperbolic manifolds
论文作者
论文摘要
我们研究了量子独特的千古猜想的完善，以在算术双曲歧管上缩小球，重点是尺寸$ 2 $和$ 3 $。对于模块化表面$ \ mathrm {psl} _2（{\ Mathbb z}）\ BackSlash \ Mathbb {h}^2 $我们证明量子唯一独特的eRgodicition的失败，与Planck-Scale和量子差异的改进结合了。 对于算术$ 3 $ -Manifolds，我们表明，Hecke-Maaß形式的量子独特的巨像失败了，以算术点和半径$ r \ asymp t_j^{ - δ} $以$ ugunignmentimentpoint点和半径为中心。对于$ \ MATHRM {PSL} _2（\ Mathcal {o} _K）\ setMinus \ setMinus \ Mathbb {h}^3 $，带有$ \ Mathcal {o} _K} _K $是一个想象中的数字一级数字的整数，我们是普通的lindef lindef lindef lindef lindef，for lightsibib如果$Δ<2/5 $，则hecke-maa {ss}形式。此外，我们证明，如果$Δ<（1-2θ）/（34+4θ）$，$θ$是$ ramanujan-petersson cointoxenture的指数，则表明等均分配无条件地适用于艾森斯坦系列。对于$ \ mathrm {psl} _2（\ mathbb {z} [i]）$，我们将最后一个指数提高到$Δ<（1-2θ）/（27+2θ）$。研究Hecke-Maaß$ L $ functions的平均林德尔估计，我们平均将最后一个指数提高到$Δ<2/5 $。 最后，我们研究了$ n $二维紧凑型算术双曲歧管的laplace本征函数的巨大不规则性。我们观察到，量子独特的千古性在缩小半径$ r \ asymp t^{ - δ_n+ε} $远离Planck-Scale的球上失败，$ n = 5/（n+1）$ for $ n \ geq 5 $。
We study a refinement of the quantum unique ergodicity conjecture for shrinking balls on arithmetic hyperbolic manifolds, with a focus on dimensions $ 2 $ and $ 3 $. For the Eisenstein series for the modular surface $\mathrm{PSL}_2( {\mathbb Z}) \backslash \mathbb{H}^2$ we prove failure of quantum unique ergodicity close to the Planck-scale and an improved bound for its quantum variance. For arithmetic $ 3 $-manifolds we show that quantum unique ergodicity of Hecke-Maaß forms fails on shrinking balls centered on an arithmetic point and radius $ R \asymp t_j^{-δ} $ with $ δ> 3/4 $. For $ \mathrm{PSL}_2(\mathcal{O}_K) \setminus \mathbb{H}^3 $ with $ \mathcal{O}_K $ being the ring of integers of an imaginary quadratic number field of class number one, we prove, conditionally on the generalized Lindelöf hypothesis, that equidistribution holds for Hecke-Maa{ss} forms if $ δ< 2/5 $. Furthermore, we prove that equidistribution holds unconditionally for the Eisenstein series if $ δ< (1-2θ)/(34+4θ) $ where $ θ$ is the exponent towards the Ramanujan-Petersson conjecture. For $ \mathrm{PSL}_2(\mathbb{Z}[i]) $ we improve the last exponent to $ δ< (1-2θ)/(27+2θ) $. Studying mean Lindelöf estimates for $ L $-functions of Hecke-Maaß forms we improve the last exponent on average to $ δ< 2/5$. Finally, we study massive irregularities for Laplace eigenfunctions on $ n $-dimensional compact arithmetic hyperbolic manifolds for $ n \geq 4 $. We observe that quantum unique ergodicity fails on shrinking balls of radii $ R \asymp t^{-δ_n+ε} $ away from the Planck-scale, with $ δ_n = 5/(n+1) $ for $ n \geq 5 $.
