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      Scaling Limit for the Kernel of the Spectral Projector and Remainder Estimates in the Pointwise Weyl Law

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          Abstract

          Let (M, g) be a compact smooth Riemannian manifold. We obtain new off-diagonal estimates as {\lambda} tend to infinity for the remainder in the pointwise Weyl Law for the kernel of the spectral projector of the Laplacian onto functions with frequency at most {\lambda}. A corollary is that, when rescaled around a non self-focal point, the kernel of the spectral projector onto the frequency interval (\lambda, \lambda + 1] has a universal scaling limit as {\lambda} goes to infinity (depending only on the dimension of M). Our results also imply that if M has no conjugate points, then immersions of M into Euclidean space by an orthonormal basis of eigenfunctions with frequencies in (\lambda, \lambda + 1] are embeddings for all {\lambda} sufficiently large.

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          The spectrum of positive elliptic operators and periodic bicharacteristics

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            Riemannian manifolds with maximal eigenfunction growth

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              About the Blowup of Quasimodes on Riemannian Manifolds

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                Author and article information

                Journal
                2014-11-03
                2015-12-27
                Article
                10.2140/apde.2015.8.1707
                1411.0658

                http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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                Analysis and PDE Vol 8 (2015) No 7 1707-1731
                Published version. Modified parametrix construction in Section 3. References added and typos corrected
                math.SP math.AP math.DG

                Analysis, Functional analysis, Geometry & Topology

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