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On common zeros of eigenfunctions of the Laplace operator

On common zeros of eigenfunctions of the Laplace operator We consider the eigenfunctions of the Laplace operator $$\Delta $$ Δ on a compact Riemannian manifold M of dimension n. For M homogeneous with irreducible isotropy representation and for a fixed eigenvalue $$\lambda $$ λ of $$\Delta $$ Δ we find the average number of common zeros of n eigenfunctions. It turns out that, up to a constant depending on n, this number equals $$\lambda ^{n/2}\mathrm{vol}\,M$$ λ n / 2 vol M , the expression known from the celebrated Weyl’s law. To prove this we compute the volume of the image of M under an equivariant immersion into a sphere. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg Springer Journals

On common zeros of eigenfunctions of the Laplace operator

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References (10)

Publisher
Springer Journals
Copyright
Copyright © 2016 by Mathematisches Seminar der Universität Hamburg and Springer-Verlag Berlin Heidelberg
Subject
Mathematics; Mathematics, general; Algebra; Differential Geometry; Number Theory; Topology; Geometry
ISSN
0025-5858
eISSN
1865-8784
DOI
10.1007/s12188-016-0138-1
Publisher site
See Article on Publisher Site

Abstract

We consider the eigenfunctions of the Laplace operator $$\Delta $$ Δ on a compact Riemannian manifold M of dimension n. For M homogeneous with irreducible isotropy representation and for a fixed eigenvalue $$\lambda $$ λ of $$\Delta $$ Δ we find the average number of common zeros of n eigenfunctions. It turns out that, up to a constant depending on n, this number equals $$\lambda ^{n/2}\mathrm{vol}\,M$$ λ n / 2 vol M , the expression known from the celebrated Weyl’s law. To prove this we compute the volume of the image of M under an equivariant immersion into a sphere.

Journal

Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg – Springer Journals

Published: Oct 4, 2016

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