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Journal of Spectral Theory


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Volume 8, Issue 4, 2018, pp. 1281–1293
DOI: 10.4171/JST/227

Published online: 2018-07-25

Distribution of the nodal sets of eigenfunctions on analytic manifolds

Xiaolong Han[1]

(1) California State University, Northridge, USA

The nodal set of the Laplacian eigenfunction has co-dimension one and has finite hypersurface measure on a compact Riemannianmanifold. In this paper, we investigate the distribution of the nodal sets of eigenfunctions, when the metric on themanifold is analytic. We prove that if the eigenfunctions are equidistributed at a small scale, then the weak limits of the hypersurface volume form of their nodal sets are comparable to the volume form on the manifold.

In particular, on the negatively curved manifolds with analytic metric and on the tori, we show that in any eigenbasis, there is a full density subsequence of eigenfunctions such that the weak limits of the hypersurface volume form of their nodal sets are comparable to the volume form on the manifold.

Keywords: Laplacian eigenfunctions, real zeros, equidistribution of nodal sets

Han Xiaolong: Distribution of the nodal sets of eigenfunctions on analytic manifolds. J. Spectr. Theory 8 (2018), 1281-1293. doi: 10.4171/JST/227