2024/11/08 by Pär Kurlberg, Kurlberg, Pär, Alina Ostafe +5 · 1 citation
Computer Science · Mathematics · Physics and Astronomy · #Analysis of PDEs (math.AP) #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Number Theory (math.NT) #Quantum Computing Algorithms and Architecture #Quantum many-body systems #Random Matrices and Applications
paper · pdf · doi:10.48550/arxiv.2411.05997
openalex publication_date 2024/11/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study eigenfunction localization for higher dimensional cat maps, a popular model of quantum chaos. These maps are given by linear symplectic maps in Sp(2g,\mathbb Z), which we take to be ergodic. Under some natural assumptions, we show that there is a density one sequence of integers N so that as N tends to infinity along this sequence, all eigenfunctions of the quantized map at the inverse Planck constant N are uniformly distributed. For the two-dimensional case (g=1), this was proved by P. Kurlberg and Z. Rudnick (2001). The higher dimensional case offers several new features and requires a completely different set of tools, including from additive combinatorics, in particular Bourgain's bound (2005) for Mordell sums, and a study of tensor product structures for the cat map.