On matrix elements for the quantized cat map modulo prime powers

Abstract

The quantum cat map is a model for a quantum system with underlying chaotic dynamics. In this paper we study the matrix elements of smooth observables in this model, when taking arithmetic symmetries into account. We give explicit formulas for the matrix elements as certain exponential sums. With these formulas we can show that there are sequences of eigenfunctions for which the matrix elements decay significantly slower then was previously conjectured. We also prove a limiting distribution for the fluctuation of the normalized matrix elements around their average.

26 pages, final version, to appear in AHP

References in corpus (5)

• Scarred eigenstates for quantum cat maps of minimal periods
• Approach to ergodicity in quantum wave functions
• Bounds on supremum norms for Hecke eigenfunctions of quantized cat maps
• Heisenberg Realizations, Eigenfunctions and Proof of the Kurlberg-Rudnick Supremum Conjecture
• Distribution of twisted Kloosterman sums modulo prime powers