论文标题
Weyl剩余：地球束的应用
Weyl remainders: an application of geodesic beams
论文作者
论文摘要
我们在对大地测量流的动态假设下获得了对Weyl定律剩余的新定量估计。在平滑的紧凑型Riemannian歧管$（m，g）$ n $的$中，令$π_λ$表示Laplacian的光谱投影仪的内核，$ \ Mathbb {1} _ {[0，λ^2]}（ - Δ_G）$。仅假设一组近期周期性的测量学上的$ w \ subset m $具有较小的度量，我们证明是$λ\ to \ infty $ $ $ $ $ $ $ \ int_ {w} π_λ（x，x）dx =（2π）^{ - n} \ text {vol} _ {\ mathbb {r}^n}^n}（b）\ text {vol} _g（w）\，λ^n+big（单位球。该结果的结果之一是，改进的剩余均符合所有产品歧管，特别是对产品设置中特征值计数函数的估计得到了改进。我们的结果还包括对偏离频谱投影仪$π_λ（x，y）$的渐近学提升，这是假设在$ x $和$ y $接近$ x $ y $的一组地理学都具有较小的措施，并且在非浮动类型假设下对Kuznecov进行了定量改进。我们对光谱投影仪研究中使用的关键技术是测量束的关键技术。
We obtain new quantitative estimates on Weyl Law remainders under dynamical assumptions on the geodesic flow. On a smooth compact Riemannian manifold $(M,g)$ of dimension $n$, let $Π_λ$ denote the kernel of the spectral projector for the Laplacian, $\mathbb{1}_{[0,λ^2]}(-Δ_g)$. Assuming only that the set of near periodic geodesics over $W\subset M$ has small measure, we prove that as $λ\to \infty$ $$ \int_{W} Π_λ(x,x)dx=(2π)^{-n}\text{vol}_{\mathbb{R}^n}(B)\text{vol}_g(W)\,λ^n+O\Big(\frac{λ^{n-1}}{\log λ}\Big),$$ where $B$ is the unit ball. One consequence of this result is that the improved remainder holds on all product manifolds, in particular giving improved estimates for the eigenvalue counting function in the product setup. Our results also include logarithmic gains on asymptotics for the off-diagonal spectral projector $Π_λ(x,y)$ under the assumption that the set of geodesics that pass near both $x$ and $y$ has small measure, and quantitative improvements for Kuznecov sums under non-looping type assumptions. The key technique used in our study of the spectral projector is that of geodesic beams.
