Bounds on Eigenfunctions of Quantum Cat Maps

TL;DR

This paper investigates the maximum amplitude of eigenfunctions of quantum cat maps, establishing bounds that relate to the quantum period and drawing parallels with results in Riemannian geometry.

Contribution

It provides new bounds on the supremum norms of eigenfunctions of quantum cat maps, especially for those with short quantum periods, extending understanding of quantum chaos.

Findings

Existence of eigenfunctions with large $ orm{u}_ty$ for short quantum periods

Upper bounds on $ orm{u}_ty$ for general eigenfunctions

Analogies with bounds in Riemannian geometry without conjugate points

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 Bi\`evre), we show that there exists a sequence of eigenfunctions $u$ with $\|u\|_{\infty}\gtrsim (\log N)^{-1/2}$. For general eigenfunctions we show the upper bound $\|u\|_\infty\lesssim (\log N)^{-1/2}$. Here the semiclassical parameter is $h=(2\pi N)^{-1}$. Our upper bound is analogous to the one proved by B\'{e}rard for compact Riemannian manifolds without conjugate points.