论文标题
$ l^p $特征函数规范的对称性的反例
A counterexample to symmetry of $L^p$ norms of eigenfunctions
论文作者
论文摘要
我们回答了Jakobson和Nadirashvili关于Laplacian征收征收征收正常和负面部分的$ l^p $规范的渐近行为的问题。更准确地说，我们表明存在一系列eigenfunctions $ψ_n$在flat $ d $ -torus上，$ d \ geq 3 $，带有eigenvalues $λ_n\ to \ infty $ as $ n \ as $ n \ to \ infty $ $\|ψ_nχ_{\{ψ_n>0\}}\|_p / \|ψ_nχ_{\{ψ_n<0\}}\|_p $ does not tend to $1$ as $n\to\infty$ for $1<p\leq \infty$.我们的论点是基本的和计算机辅助的。
We answer a question of Jakobson and Nadirashvili on the asymptotic behavior of the $L^p$ norms of positive and negative parts of eigenfunctions of the Laplacian. More precisely, we show that there exists a sequence of eigenfunctions $ψ_n$ on the flat $d$-torus for $d\geq 3$, with eigenvalues $λ_n\to\infty$ as $n\to\infty$, such that the ratio $\|ψ_nχ_{\{ψ_n>0\}}\|_p / \|ψ_nχ_{\{ψ_n<0\}}\|_p $ does not tend to $1$ as $n\to\infty$ for $1<p\leq \infty$. Our argument is elementary and computer-assisted.
