Weyl Law on Product Manifolds

The classical Weyl law is a phenomena which describes the distribution of eigenvalues of the Laplace-Beltrami operator on manifolds. Since its proposal by Hermann Weyl in 1912, it has played a central role in mathematical physics, spectral geometry, and related fields. Specifically, if $N_M(\lambda)$ represents the number of eigenvalues of the Laplacian on a $d$-dimensional compact boundaryless Riemannian manifold $(M,g)$ that are less than $\lambda$, then the Weyl law can be expressed as
  $$N_M(\lambda)=c\lambda^d+O(\lambda^{d-1}).$$
  This has been verified as the optimal estimate, with examples given by spheres of different dimensions. However, with certain restrictions on the manifolds, especially requirements on the chaos of geodesic flows on the manifold, more precise estimates on the remainder can often be made. A classic example is that Hans Duistermaat and Victor Guillemin proved in 1975 that when the set of periodic geodesics on the manifold is of measure zero, the big O estimate of the remainder can be improved to a little o estimate. From the perspective of semi-classical analysis, this is because in a chaotic classical mechanical system, the quantized system can approximate the classical mechanical system better, leading to improved estimates. This also provides ample motivation for the mathematical community to study the special type of manifolds known as product manifolds.

  This paper primarily introduces two new works that utilize the product structure of product manifolds to obtain better results, and a weak version of the Weyl law with exponential improvement is obtained by emulating their methods.

  In 2021, A. Iosevich and E. Wyman demonstrated that the remainder in Weyl law would have an exponential improvement on products of spheres. Moreover, the obtained improvement is related to the number of product components, $n$, as expressed by
  $$N_M(\lambda)=c\lambda^d+O(\lambda^{d-1-\frac{n-1}{n+1}}).$$
  By clear understanding of the spectrum of spheres, the eigenvalue counting function for spheres can be approximated with considerable precision by polynomial functions. Through the Poisson summation formula, we further transfer the approximating function for the summation over integers to the frequency space, thereby obtaining better estimates.

  Based on their results, this paper will prove that, on general product manifolds, if the number of eigenvalues near integers is used as the multiplicity of integers to fit the eigenvalue counting function, and then a fairly accurate estimate of this fitting function is given by the Hadamard parametrix, a weaker version of Weyl law with the same exponential improvement can be achieved through a similar method. In general, this result is not compatible with the normal Weyl law, as the fitting error is concentrated on the boundary where the eigenvalue density is higher, which in fact covers the improvement obtained. However, this method still has potential for development. If a more refined method for dividing eigenvalues can be found and the counting error is controlled below the obtained improvement, then it can be extended to the general Weyl law.

  Another part of this article is a review of the work contributed by Xiaoqi Huang, Christopher D. Sogge, and Michael E. Taylor in 2022. They proved that when the remainder of the eigenvalue counting function on a manifold has $\varepsilon(\lambda)$-improvement, the product with arbitrary manifolds will receive the same improvement. This result primarily relies on the composition of the spectrum of product manifolds. By setting special forms of eigenvalues, the square of the eigenvalues on the product manifold can be represented by the sum of the squares of the eigenvalues on each product component, forming a space similar to a Euclidean space with dimensions equal to the length of the product, where the coordinate of each point is given by the eigenvalues on the corresponding product components. With this unique structure, we can transform the counting function into a familiar mathematical object and get improvement.

经典外尔定律描述了流形上，拉普拉斯-贝尔特拉米算子的特征值的分布情况。自1912年由Hermann Weyl提出以来，已在数学物理、谱几何及相关领域中发挥了核心作用。具体来讲，如果$N_M(\lambda)$表示在$d$维紧致无边黎曼流形$(M,g)$上小于$\lambda$的拉普拉斯算子的特征值的个数，那么外尔定律可以被表示为
  $$N_M(\lambda)=c\lambda^d+O(\lambda^{d-1}).$$
  这已经被验证是最优的估计，其例子则由不同维数的球面给出。但如果对考虑的流形加以限制，尤其是对流形上测地流的混沌性做出要求，我们往往可以对余项作出更精确的估计。一个经典的例子是，Hans Duistermat和Victor Guillemin在1975年证明了，当流形上的周期测地线为零测集时，我们可以将余项的大O估计改进为小o估计。从半经典分析的角度来看，这是因为在一个混沌的经典力学系统中，我们可以更好地用量子化系统逼近这个经典力学系统，并以此得到更好的估计。这也给数学界研究乘积流形这一特殊的流形类型提供了充足的动力。
  
  本文主要回顾了两个利用乘积流形的乘积结构来得到更好结果的最新工作，并仿照其中的方法得到了一个带有指数提升的弱版本的外尔定律。

  A. Iosevich和E. Wyman在2021年证明，外尔定律中的余项在若干球面乘积所得的乘积流形上拥有指数提升，并且所得提升与乘积分量的个数$n$相关，即
  $$N_M(\lambda)=c\lambda^d+O(\lambda^{d-1-\frac{n-1}{n+1}}).$$
  由于我们对球面的谱具有非常清楚的认知，球面的特征值计数函数可以被多项式函数相当精确的拟合。通过泊松和公式，我们进而将对整点求和的拟合函数转移到频率空间并以此得到更好的估计。
  
  基于他们的结果，本文将会证明更一般的结论，在一般的乘积流形上，如果通过适当的划分将整点附近的特征值的个数作为整点的重数来拟合特征值计数函数，再由Hadamard拟基本解给出该拟合函数较为精确的估计，那么通过类似的方法，我们可以得到一个带有相同指数提升的弱版本的外尔定律。一般情况下，这一结果无法兼容于正常的外尔定律，因为其拟合的误差集中在特征值密度最大的边界部分，这导致了该误差事实上覆盖了我们所得到的提升。但此方法仍然拥有潜在的发展空间。如果能找到更精细的划分特征值的方法并将计数误差控制在所得提升以下，那么就能将其推广到一般的外尔定律上。

  在余下的部分中，我们将会回顾Xiaoqi Huang，Christopher D. Sogge和Michael E. Taylor在2022年做出的工作。他们证明了当特征值计数函数的余项在一个流形上有$\varepsilon(\lambda)$的提升时，该流形与任意流形的乘积都将获得相同的提升。这一结果主要依托于乘积流形的谱的特殊构成。通过设定特殊的特征值形式，乘积流形的特征值的平方可以用各乘积分量上的特征值的平方和表示，这形成了一个类似于维数为乘积长度的欧式空间，其中每一个点的坐标都由对应的乘积分量上的特征值给出，而特征值计数函数则可以看作一个计算半径为$\lambda$的球内的特征值个数的函数。通过这一特殊的结构，我们可以将计数函数转化为熟悉的数学对象并获得提升。

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