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- W4300891902 abstract "Let $(M,g)$ be a compact $n$-dimensional Riemannian manifold without boundary and $e_lambda$ be an $L^2$-normalized eigenfunction of the Laplace-Beltrami operator with respect to the metric $g$, i.e [ -Delta_g e_lambda = lambda^2 e_lambda qquad text{ and } qquad | e_lambda |_{L^2(M)} = 1. ] Let $Sigma$ be a $d$-dimensional submanifold and $dmu$ a smooth, compactly supported measure on $Sigma$. It is well-known (e.g. proved by Zelditch in far greater generality) that [ int_Sigma e_lambda , dmu = O(lambda^frac{n-d-1}{2}). ] We show this bound improves to $o(lambda^frac{n-d-1}{2})$ provided the set of looping directions, [ mathcal{L}_{Sigma} = { (x,xi) in SN^*Sigma : Phi_t(x,xi) in SN^*Sigma text{ for some } t > 0 } ] has measure zero as a subset of $SN^*Sigma$, where here $Phi_t$ is the geodesic flow on the cosphere bundle $S^*M$ and $SN^*Sigma$ is the unit conormal bundle over $Sigma$." @default.
- W4300891902 created "2022-10-04" @default.
- W4300891902 creator A5064320363 @default.
- W4300891902 date "2017-06-20" @default.
- W4300891902 modified "2023-09-26" @default.
- W4300891902 title "Looping directions and integrals of eigenfunctions over submanifolds" @default.
- W4300891902 doi "https://doi.org/10.48550/arxiv.1706.06717" @default.
- W4300891902 hasPublicationYear "2017" @default.
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