Failure of $L^p$ Symmetry of Zonal Spherical Harmonics
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866929362646859776 |
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| author | Beiner, Gabriel Verreault, William |
| author_facet | Beiner, Gabriel Verreault, William |
| contents | In this paper, we show that the 2-sphere does not exhibit symmetry of $L^p$ norms of eigenfunctions of the Laplacian for $p\geq 6$. In other words, there exists a sequence of spherical eigenfunctions $ψ_n$, with eigenvalues $λ_n\to\infty$ as $n\to\infty$, such that the ratio of the $L^p$ norms of the positive and negative parts of the eigenfunctions does not tend to $1$ as $n\to\infty$ when $p\geq 6$. Our proof relies on fundamental properties of the Legendre polynomials and Bessel functions of the first kind. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_00212 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Failure of $L^p$ Symmetry of Zonal Spherical Harmonics Beiner, Gabriel Verreault, William Classical Analysis and ODEs Spectral Theory 33C55 (Primary) 35J05, 58J50 (Secondary) In this paper, we show that the 2-sphere does not exhibit symmetry of $L^p$ norms of eigenfunctions of the Laplacian for $p\geq 6$. In other words, there exists a sequence of spherical eigenfunctions $ψ_n$, with eigenvalues $λ_n\to\infty$ as $n\to\infty$, such that the ratio of the $L^p$ norms of the positive and negative parts of the eigenfunctions does not tend to $1$ as $n\to\infty$ when $p\geq 6$. Our proof relies on fundamental properties of the Legendre polynomials and Bessel functions of the first kind. |
| title | Failure of $L^p$ Symmetry of Zonal Spherical Harmonics |
| topic | Classical Analysis and ODEs Spectral Theory 33C55 (Primary) 35J05, 58J50 (Secondary) |
| url | https://arxiv.org/abs/2209.00212 |