Bounds on Eigenfunctions of Quantum Cat Maps

Abstract

We study ${\ell }^{\infty }$ norms of ${\ell }^2$-normalized eigenfunctions of quantum cat maps. For maps with short quantum periods (constructed by Bonechi and de Bièvre), we show that there exists a sequence of eigenfunctions $u$ with $\Vert u{\Vert }_{\infty }\gtrsim (\log N{)}^{-1/2}$. For general eigenfunctions we show the upper bound $\Vert u{\Vert }_{\infty }\lesssim (\log N{)}^{-1/2}$. Here the semiclassical parameter is $h=(2\pi N{)}^{-1}$. Our upper bound is analogous to the one proved by Bérard for compact Riemannian manifolds without conjugate points.

15 pages, 5 figures. Revised according to referee's comments. To appear in Physica Scripta