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An estimate for the average number of common zeros of Laplacian eigenfunctions


Authors: Dmitri Akhiezer and Boris Kazarnovskii
Translated by: the authors
Original publication: Trudy Moskovskogo Matematicheskogo Obshchestva, tom 78 (2017), vypusk 1.
Journal: Trans. Moscow Math. Soc. 2017, 123-130
MSC (2010): Primary 53C30, 58J05
DOI: https://doi.org/10.1090/mosc/269
Published electronically: December 1, 2017
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Abstract | References | Similar Articles | Additional Information

Abstract: On a compact Riemannian manifold $ M$ of dimension $ n$, we consider $ n$ eigenfunctions of the Laplace operator $ \Delta $ with eigenvalue $ \lambda $. If $ M$ is homogeneous under a compact Lie group preserving the metric then we prove that the average number of common zeros of $ n$ eigenfunctions does not exceed $ c(n)\lambda ^{n/2}{\rm vol}\,M$, the expression known from the celebrated Weyl's law. Moreover, if the isotropy representation is irreducible, then the estimate turns into the equality. The constant $ c(n)$ is explicitly given. The method of proof is based on the application of Crofton's formula for the sphere.


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Additional Information

Dmitri Akhiezer
Affiliation: Institute for Information Transmission Problems 19 B. Karetny per., 127994, Moscow, Russia
Email: akhiezer@iitp.ru

Boris Kazarnovskii
Affiliation: Institute for Information Transmission Problems 19 B. Karetny per., 127994, Moscow, Russia
Email: kazbori@gmail.com

DOI: https://doi.org/10.1090/mosc/269
Keywords: Homogeneous Riemannian manifold, Laplace operator, Crofton formula
Published electronically: December 1, 2017
Additional Notes: The research was carried out at the Institute for Information Transmission Problems under support by the Russian Foundation of Sciences, grant No. 14-50-00150
Dedicated: To Ernest Borisovich Vinberg on the occasion of his 80th birthday
Article copyright: © Copyright 2017 American Mathematical Society

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