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Hangzhou Lectures on Eigenfunctions of the Laplacian (AM-188)$
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Christopher D. Sogge

Print publication date: 2014

Print ISBN-13: 9780691160757

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691160757.001.0001

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Improved spectral asymptotics and periodic geodesics

Improved spectral asymptotics and periodic geodesics

(p.120) Chapter Five Improved spectral asymptotics and periodic geodesics
Hangzhou Lectures on Eigenfunctions of the Laplacian (AM-188)

Christopher D. Sogge

Princeton University Press

This chapter proves an improved Weyl formula under the assumption that the set of periodic geodesics for (M,g) has measure zero. It then shows trace estimates associated with shrinking spectral bands, details and proves a lemma, and gives a related generalization of the Weyl formula from Chapter 3 that involves pseudodifferential operators. The chapter then proves its main result by using a version of the Duistermaat-Guillemin theorem, which allows the use of the Hadamard parametrix and the arguments from Chapter 3. To conclude, the chapter shows that one can improve the sup-norm estimates from Chapter 3 if one assumes a condition on the geodesic flow that is similar to a hypothesis laid out in the Duistermaat-Guillemin theorem.

Keywords:   Weyl formula, spectral asymptotics, periodic geodesics, trace estimates, shrinking spectral bands, pseudodifferential operators, Duistermaat-Guillemin theorem, sup-norm estimates

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