On the approximate normality of eigenfunctions of the Laplacian

Details On the approximate normality of eigenfunctions of the Laplacian On the approximate normality of eigenfunctions of the Laplacian

2007-05-09

arxiv.org/abs/0705.1342v1

Trans. Amer. Math. Soc. 361, no. 10 (2009)

Mathematics

Spectral Theory

21 pages

Scientific paper

The main result of this paper is a bound on the distance between the distribution of an eigenfunction of the Laplacian on a compact Riemannian manifold and the Gaussian distribution. If $X$ is a random point on a manifold $M$ and $f$ is an eigenfunction of the Laplacian with $L^2$-norm one and eigenvalue $-\mu$, then $$d_{TV}(f(X),Z)\le\frac{2}{\mu}\E\big|\|\nabla f(X)\|^2-\E\|\nabla f(X) \|^2\big|.$$ This result is applied to construct specific examples of spherical harmonics of arbitrary (odd) degree which are close to Gaussian in distribution. A second application is given to random linear combinations of eigenfunctions on flat tori.

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