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

Akhiezer, Dmitri; Kazarnovskii, Boris

doi:10.1090/mosc/269

An estimate for the average number of common zeros of Laplacian eigenfunctions
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by Dmitri Akhiezer and Boris Kazarnovskii
Translated by: the authors

Trans. Moscow Math. Soc. 2017, 123-130

DOI: https://doi.org/10.1090/mosc/269

Published electronically: December 1, 2017

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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 $c(n)\lambda ^{n/2}\textrm {vol} M$, 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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