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On integrals of eigenfunctions over geodesics

Authors: Xuehua Chen and Christopher D. Sogge
Journal: Proc. Amer. Math. Soc. 143 (2015), 151-161
MSC (2010): Primary 35F99; Secondary 35L20, 42C99
Published electronically: August 15, 2014
MathSciNet review: 3272740
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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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Additional Information

Xuehua Chen
Affiliation: Department of Mathematics, Johns Hopkins University, Baltimore, Maryland 21218
Address at time of publication: Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, Illinois 60208-2730

Christopher D. Sogge
Affiliation: Department of Mathematics, Johns Hopkins University, Baltimore, Maryland 21218

Keywords: Eigenfunctions, negative curvature
Received by editor(s): February 27, 2013
Published electronically: August 15, 2014
Additional Notes: The authors were supported in part by the NSF grant DMS-1069175 and the Simons Foundation.
Communicated by: Alexander Iosevich
Article copyright: © Copyright 2014 American Mathematical Society