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 | References | Similar Articles | Additional Information

Abstract: If is a compact Riemannian surface, then the integrals of -normalized eigenfunctions over geodesic segments of fixed length are uniformly bounded. Also, if has negative curvature and is a geodesic parameterized by arc length, the measures on tend to zero in the sense of distributions as the eigenvalue , and so integrals of eigenfunctions over periodic geodesics tend to zero as . 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

DOI:
https://doi.org/10.1090/S0002-9939-2014-12233-7

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