UNIFORM WEYL ASYMPTOTICS FOR OFF-DIAGONAL SPECTRAL PROJECTIONS

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Date
2020-07-06
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Johns Hopkins University
Abstract
Let $(M,g)$ denote a smooth, compact Riemannian manifold, without boundary of dimension $n \geq 2$. Furthermore, we assume that all points on the manifold are non self-focal. In our paper we provide a new proof for the off-diagonal behavior of the Schwartz kernel for the spectral projection operator onto the unit frequency interval $(\lambda, \lambda+1]$ as $\lambda \to \infty$; we denote this kernel by $K_{\lambda}(x,y) = K(x, y; \lambda)$. By using standard techniques from harmonic and microlocal analysis we manage to obtain a result similar to \cite{CH}. However, whereas their result requires that $x$ and $y$ be taken in an ever-shrinking neighborhood of the diagonal, we prove that the same asymptotics can be found over a uniformly-small neighborhood, dependent only on the manifold $(M,g)$.
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Keywords
harmonic analysis, microlocal analysis, eigenfunctions of the laplacian
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