A counterexample to symmetry of $L^p$ norms of eigenfunctions

Abstract

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 ${\psi }_n$ on the flat $d$-torus for $d\ge 3$, with eigenvalues ${\lambda }_n\to \infty$ as $n\to \infty$, such that the ratio $\Vert {\psi }_n{\chi }_{\{{\psi }_n>0\}}{\Vert }_p/\Vert {\psi }_n{\chi }_{\{{\psi }_n<0\}}{\Vert }_p$ does not tend to $1$ as $n\to \infty$ for $1<p\le \infty$. Our argument is elementary and computer-assisted.

6 pages, 1 table