L^p norms of eigenfunctions in the completely integrable case

Abstract

The eigenfunctions e^{i λx} of the Laplacian on a flat torus have uniformly bounded L^p norms. In this article, we prove that for every other quantum integrable Laplacian, the L^p norms of the joint eigenfunctions must blow up at a rate \gg λ^{p-2/4p - ε} for every ε>0 as λ\to \infty.