On integrals of eigenfunctions over geodesics
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- by Xuehua Chen and Christopher D. Sogge
- Proc. Amer. Math. Soc. 143 (2015), 151-161
- DOI: https://doi.org/10.1090/S0002-9939-2014-12233-7
- Published electronically: August 15, 2014
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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.References
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Bibliographic 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
- MR Author ID: 164510
- 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
- © Copyright 2014 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 143 (2015), 151-161
- MSC (2010): Primary 35F99; Secondary 35L20, 42C99
- DOI: https://doi.org/10.1090/S0002-9939-2014-12233-7
- MathSciNet review: 3272740