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International Mathematics Research Notices (2007) Vol. 2007 : article ID rnm142, 38 pages, doi:10.1093/imrn/rnm142 published on December 12, 2007
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Copyright © The Author 2007. Published by Oxford University Press.

A Lower Bound for the Remainder in Weyl's Law on Negatively Curved Surfaces

Dmitry Jakobson1,, Iosif Polterovich2 and John A. Toth3

1 Department of Mathematics and Statistics, McGill University, 805 Sherbrooke Str. West, Montréal QC H3A 2K6, Canada
2 Département de mathématiques et de statistique, Université de Montréal CP 6128 succ Centre-Ville, Montréal QC H3C 3J7, Canada
3 Department of Mathematics and Statistics, McGill University, 805 Sherbrooke Str. West, Montréal QC H3A 2K6, Canada

Correspondence: Correspondence to be sent to: Dmitry Jakobson, Department of Mathematics and Statistics, McGill University, 805 Sherbrooke Str. West, Montréal QC H3A 2K6, Canada. e-mail: jakobson{at}math.mcgill.ca

We obtain an estimate from below for the remainder in Weyl's law on negatively curved surfaces. In the constant curvature case, such a bound was proved independently by Hejhal and Randol in 1976 using the Selberg zeta function techniques. Our approach works in arbitrary negative curvature, and is based on wave trace asymptotics for long times, equidistribution of closed geodesics and small-scale microlocalization.


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