We examine the asymptotics of the spectral counting function of a compact Riemannian manifold by V.G.~Avakumovic \cite{Avakumovic} and L.~Hörmander \cite{Hormander-eigen} and show that for the scale of orthogonal and unitary groups ${\bf SO}(N)$, ${\bf SU}(N)$, ${\bf U}(N)$ and ${\bf Spin}(N)$ it is not sharp. While for negative sectional curvature improvements are possible and known, {\it cf.} e.g., J.J.~Duistermaat $\&$ V.~Guillemin \cite{Duist-Guill}, here, we give sharp and contrasting examples in the positive Ricci curvature case [non-negative for ${\bf U}(N)$]. Furthermore here the improvements are sharp and quantitative relating to the dimension and {\it rank} of the group. We discuss the implications of these results on the closely related problem of closed geodesics and the length spectrum.
We prove a general estimate for the Weyl remainder of an elliptic, semiclassical pseudodifferential operator in terms of volumes of recurrence sets for the Hamilton flow of its principal symbol. This quantifies earlier results of Volovoy. Our result particularly improves Weyl remainder exponents for compact Lie groups and surfaces of revolution. And gives a quantitative estimate for Bérard's Weyl remainder in terms of the maximal expansion rate and topological entropy of the geodesic flow.
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 r…
We show that if $Y$ is a compact Riemannian manifold with improved $L^q$ eigenfunction estimates then, at least for large enough exponents, one always obtains improved $L^q$ bounds on the product manifold $X\times Y$ if $X$ is another compact manifold. Similarly, improved Weyl remainder term bounds on the spectral counting function of $Y$ lead to corresponding improvements on $X\times Y$. The latter results partly generalize recent ones of Iosevich and Wyman [14] involving products of spheres. Also, if $Y$ is a product of five or more spheres, we are able to obtain optimal $L^q(Y)$ and $L^q(X\…
Given a compact Riemannian manifold $(M,g)$, Chazarain, Hörmander, Duistermaat, and Guillemin study the half-wave trace $\operatorname{HWT}_{M,g}(τ) \in \mathscr{S}'(\mathbb{R}_τ)$. From the asymptotics of the half-wave trace as $τ\to 0$, Hörmander deduces the now standard remainder $\smash{O(σ^{d-1}) = O(λ^{d/2-1/2})}$ in Weyl's law, where $d=\dim M$. Given a dynamical assumption implying additional local regularity, Duistermaat and Guillemin improve this to $o(σ^{d-1})$. By examining the Tauberian step in the argument, we show how a quantitative version \[N(σ) = Z(σ) + O(σ^{d-1}\mathcal{R}(σ…
In this paper, we study the two-point Weyl Law for the Laplace-Beltrami operator on a smooth, compact Riemannian manifold $M$ with no conjugate points. That is, we find the asymptotic behavior of the Schwartz kernel, $E_λ(x,y)$, of the projection operator from $L^2(M)$ onto the direct sum of eigenspaces with eigenvalue smaller than $λ^2$ as $λ\to\infty$. In the regime where $x,y$ are restricted to a compact neighborhood of the diagonal in $M\times M$, we obtain a uniform logarithmic improvement in the remainder of the asymptotic expansion for $E_λ$ and its derivatives of all orders, which gene…
