On integrals of eigenfunctions over geodesics

Chen, Xuehua; Sogge, Christopher

doi:10.1090/S0002-9939-2014-12233-7

On integrals of eigenfunctions over geodesics
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by Xuehua Chen and Christopher D. Sogge PDF

Proc. Amer. Math. Soc. 143 (2015), 151-161 Request permission

Abstract:

If $(M,g)$ is a compact Riemannian surface, then the integrals of $L^2(M)$-normalized eigenfunctions $e_j$ over geodesic segments of fixed length are uniformly bounded. Also, if $(M,g)$ has negative curvature and $\gamma (t)$ is a geodesic parameterized by arc length, the measures $e_j(\gamma (t)) dt$ on $\mathbb {R}$ tend to zero in the sense of distributions as the eigenvalue $\lambda _j\to \infty$, and so integrals of eigenfunctions over periodic geodesics tend to zero as $\lambda _j\to \infty$. The assumption of negative curvature is necessary for the latter result.

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