Failure of $L^p$ symmetry of zonal spherical harmonics

IF 1 3区数学 Q1 MATHEMATICS

Journal of Spectral Theory Pub Date : 2022-09-01 DOI:10.4171/jst/446

G. Beiner, William Verreault

引用次数: 0

Abstract

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 $\psi_n$, with eigenvalues $\lambda_n\to\infty$ 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.

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纬向球面谐波$L^p$对称性的失效

本文证明了2球不具有$p\geq 6$的拉普拉斯本征函数的$L^p$范数的对称性。换句话说，存在一个球形特征函数$\psi_n$序列，特征值$\lambda_n\to\infty$为$n\to\infty$，使得特征函数的正负部分的$L^p$范数之比在$p\geq 6$时不趋向于$1$为$n\to\infty$。我们的证明依赖于勒让德多项式和第一类贝塞尔函数的基本性质。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Journal of Spectral Theory MATHEMATICS, APPLIED-MATHEMATICS

CiteScore

2.00

自引率

0.00%

发文量

期刊介绍： The Journal of Spectral Theory is devoted to the publication of research articles that focus on spectral theory and its many areas of application. Articles of all lengths including surveys of parts of the subject are very welcome. The following list includes several aspects of spectral theory and also fields which feature substantial applications of (or to) spectral theory. Schrödinger operators, scattering theory and resonances; eigenvalues: perturbation theory, asymptotics and inequalities; quantum graphs, graph Laplacians; pseudo-differential operators and semi-classical analysis; random matrix theory; the Anderson model and other random media; non-self-adjoint matrices and operators, including Toeplitz operators; spectral geometry, including manifolds and automorphic forms; linear and nonlinear differential operators, especially those arising in geometry and physics; orthogonal polynomials; inverse problems.