The classical Weyl Law says that if N_M(λ) denotes the number of eigenvalues of the Laplace operator on a d-dimensional compact manifold M without a boundary that are less than or equal to λ, then

$${N_M}(\lambda ) = c{\lambda ^d} + O({\lambda ^{d - 1}}).$$

This paper explores the prospects of improvements of Weyl remainders on products of manifolds. In particular we obtain a polynomial improvement to the Weyl remainder for products of spheres, demonstrate how Duistermaat and Giullemin’s result implies a little-o improvement to the remainder for products of compact Riemannian manifolds without boundary, and conjecture that polynomial improvements hold for these more general products.

中文翻译：

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球体积的外尔定律改进

经典的外尔定律说，如果N _M ( λ ) 表示拉普拉斯算子在没有边界的d维紧流形M上小于或等于λ的特征值的数量，则

$${N_M}(\lambda ) = c{\lambda ^d} + O({\lambda ^{d - 1}}).$$

本文探讨了外尔余数对流形乘积的改进前景。特别是，我们得到一个多项式改进外尔剩余的球品，展示Duistermaat和Giullemin的结果如何意味着little- Ø改进，其余的则用于紧凑黎曼流形的产品无边界，并猜想多项式改进持有这些更普遍产品。