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On a smooth, compact, Riemannian manifold without boundary(M,g), let\Delta_{g}be the Laplace–Beltrami operator. We define the orthogonal projection operator \Pi_{I_\lambda}\colon L^{2}(M)\to \bigoplus_{\mathclap{\lambda_j\in I_\lambda}}\ker(\Delta_{g}+\lambda_{j}^{2}) for an intervalI_{\lambda}centered around\lambda\in\Rof a small, fixed length. The Schwartz kernel,\Pi_{I_\lambda}(x,y), of this operator plays a key role in the analysis of monochromatic random waves, a model for high energy eigenfunctions. It is expected that\Pi_{I_\lambda}(x,y)has universal asymptotics as\lambda \to \inftyin a shrinking neighborhood of the diagonal inM\times M(providedI_{\lambda}is chosen appropriately) and hence that certain statistics for monochromatic random waves have universal behavior. These asymptotics are well known for the torus and the round sphere, and were recently proved to hold near points inMwith few geodesic loops by Canzani–Hanin. In this article, we prove that the same universal asymptotics hold in the opposite case of Zoll manifolds (manifolds all of whose geodesics are closed with a common period) under an assumption on the volume of loops with length incommensurable with the minimal common period. 

Abstract A spectral minimal partition of a manifold is its decomposition into disjoint open sets that minimizes a spectral energy functional. It is known that bipartite spectral minimal partitions coincide with nodal partitions of Courant‐sharp Laplacian eigenfunctions. However, almost all minimal partitions are non‐bipartite. To study those, we define a modified Laplacian operator and prove that the nodal partitions of its Courant‐sharp eigenfunctions are minimal within a certain topological class of partitions. This yields new results in the non‐bipartite case and recovers the above known result in the bipartite case. Our approach is based on tools from algebraic topology, which we illustrate by a number of examples where the topological types of partitions are characterized by relative homology. 

Abstract 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 $$\Pi _\lambda $$ Π λ denote the kernel of the spectral projector for the Laplacian, $$\mathbb {1}_{[0,\lambda ^2]}(-\Delta _g)$$ 1 [ 0 , λ 2 ] ( - Δ g ) . Assuming only that the set of near periodic geodesics over $${W}\subset M$$ W ⊂ M has small measure, we prove that as $$\lambda \rightarrow \infty $$ λ → ∞ $$\begin{aligned} \int _{{W}} \Pi _\lambda (x,x)dx=(2\pi )^{-n}{{\,\textrm{vol}\,}}_{_{{\mathbb {R}}^n}}\!(B){{\,\textrm{vol}\,}}_g({W})\,\lambda ^n+O\Big (\frac{\lambda ^{n-1}}{\log \lambda }\Big ), \end{aligned}$$ ∫ W Π λ ( x , x ) d x = ( 2 π ) - n vol R n ( B ) vol g ( W ) λ n + O ( λ n - 1 log λ ) , where B is the unit ball. One consequence of this result is that the improved remainder holds on all product manifolds, in particular giving improved estimates for the eigenvalue counting function in the product setup. Our results also include logarithmic gains on asymptotics for the off-diagonal spectral projector $$\Pi _\lambda (x,y)$$ Π λ ( x , y ) under the assumption that the set of geodesics that pass near both x and y has small measure, and quantitative improvements for Kuznecov sums under non-looping type assumptions. The key technique used in our study of the spectral projector is that of geodesic beams.