Failure of symmetry of zonal spherical harmonics
Gabriel Beiner
University of Toronto, CanadaWilliam Verreault
Université Laval, Québec, Canada
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Abstract
We show that the 2-sphere does not exhibit symmetry of norms of eigenfunctions of the Laplacian for , which answers a question of Jakobson and Nadirashvili. In other words, there exists a sequence of spherical eigenfunctions , with eigenvalues as , such that the ratio of the norms of the positive and negative parts of the eigenfunctions does not tend to as when . Our proof relies on fundamental properties of the Legendre polynomials and Bessel functions of the first kind.
Cite this article
Gabriel Beiner, William Verreault, Failure of symmetry of zonal spherical harmonics. J. Spectr. Theory 13 (2023), no. 1, pp. 383–394
DOI 10.4171/JST/446