论文标题
laplacians在球面上的特征值手段的半经典估计值
Semiclassical estimates for eigenvalue means of Laplacians on spheres
论文作者
论文摘要
我们计算dirichlet和Neumann边界条件的Laplacian特征值的计数函数和Riesz均值的三项半经典渐近扩展。特别是针对Riesz-Means，我们证明了涉及渐近锐利术语的上限和下限，并将其扩展到$ \ Mathbb s^d $的域。我们还证明了半球$ \ mathbb s^2 _+$中包含的域的berezin-li-yau不平等。此外，我们考虑了多谐操作员，我们证明了类似的结果，突出了维度对Pólya-type不平等的作用。最后，我们为球体和紧凑的两点均匀空间提供了拉普拉斯特征值的总和。
We compute three-term semiclassical asymptotic expansions of counting functions and Riesz-means of the eigenvalues of the Laplacian on spheres and hemispheres, for both Dirichlet and Neumann boundary conditions. Specifically for Riesz-means we prove upper and lower bounds involving asymptotically sharp shift terms, and we extend them to domains of $\mathbb S^d$. We also prove a Berezin-Li-Yau inequality for domains contained in the hemisphere $\mathbb S^2_+$. Moreover, we consider polyharmonic operators for which we prove analogous results that highlight the role of dimension for Pólya-type inequalities. Finally, we provide sum rules for Laplacian eigenvalues on spheres and compact two-point homogeneous spaces.
