Weyl Law Improvement for Products of Spheres
Weyl Law Improvement for Products of Spheres
复制标题
球体乘积的韦尔定律改进
作者：
A. Iosevich;E. Wyman
The classical Weyl Law says that if NM(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 NM(λ)=cλd+O(λd−1).documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$${N_M}(lambda ) = c{lambda ^d} + O({lambda ^{d - 1}}).$$end{document} 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.
